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# Divergence form nonlinear nonsmooth parabolic equations with locally
arbitrary growth conditions and nonlinear maximal regularity
Qiao-fu Zhang
(Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, P. R. China)
###### Abstract
This is a generalization of our prior work on the compact fixed point theory
for the elliptic Rosseland-type equations. Inspired by the Rosseland equation
in the conduction-radiation coupled heat transfer, we use the locally
arbitrary growth conditions instead of the common global restricted growth
conditions. Its physical meaning is: the absolute temperature should be
positive and bounded.
There exists a fixed point for the linearized map (compact and continuous in
$L^{2}$) in a closed convex set. We also consider the nonlinear maximal
regularity in Sobolev space.
Key words: arbitrary growth conditions; fixed point;
Rosseland equation; nonlinear maximal regularity;
nonlinear parabolic equations; nonsmooth data.
## 1 Introduction
Suppose $S=(0,T)$ where $T$ is a positive constant. Consider the following
parabolic problem:
$\partial_{t}u-\mbox{div\,}[A(u(x),x,t)\nabla
u]=0,\quad\mbox{in\,}\,Q_{T}=\Omega\times S.$ (1.1)
The weak solution can be defined as the following: find $u$,
$(u-g)\in L^{2}(S;\,H^{1}_{0}(\Omega)),\quad(u-g)(x,0)=0,$ (1.2)
(so we know the boundary and initial conditions)
$\partial_{t}u\in L^{2}(S;H^{-1}(\Omega)),$ (1.3)
where $L^{2}(S;H^{-1}(\Omega))$ is the dual space of
$L^{2}(S;\,H^{1}_{0}(\Omega))$, such that $\forall\,\varphi\in
L^{2}(S;\,H^{1}_{0}(\Omega))$,
$\langle\partial_{t}u,\,\varphi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}+\iint_{Q_{T}}A(u(x),x)\nabla
u\cdot\nabla\varphi=0.$ (1.4)
For the definitions of these spaces, see [1, 4].
For the Rosseland equation: $A(z,x,t)=K(x,t)+z^{3}B(x,t)$, where $K(x,t)$ and
$B(x,t)$ are symmetric and positive definite.
(1) $K(x,t)+z^{3}B(x,t)$ is positive definite only in an interval for $z$.
(2) it doesn’t satisfy the common growth and smooth conditions and there may
be no $C^{2,\gamma}$ estimate (Theorem 15.11 [6]).
The problem of the existence theory for the Rosseland equation (also named
diffusion approximation) was proposed by Laitinen [11] in 2002. It may be
useful to keep this equation in mind while reading this paper.
It’s a little technical to prove the existence by the fixed point method in
$L^{\infty}(Q_{T})$ (or $C^{0}(\overline{Q}_{T})$ ) [12, 13]. We will use
$L^{2}(Q_{T})$ in this paper.
Firstly, we make the following assumptions.
(A1) $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain. $S=(0,T)$
where $T$ is a positive constant. $Q_{T}=\Omega\times S$.
(A2) $A=(a_{ij})$. $a_{ij}=a_{ji}$. $T_{min}\leq T_{max}$ are two constants.
$\lambda|\xi|^{2}\leq a_{ij}(z,x,t)\xi_{i}\xi_{j}\leq\Lambda|\xi|^{2},\quad
0<\lambda\leq\Lambda,$ (1.5) $\forall\,\,(z,x,t,\xi)\in[T_{min},T_{max}]\times
Q_{T}\times\mathbb{R}^{n}.$ (1.6)
Here we use the Einstein summation convention.
(A3) $\partial_{p}Q_{T}=\\{\partial\Omega\times
S\\}\cup\\{(x,0);x\in\Omega\\}$,
$g\in H^{1}(Q_{T}).\quad T_{min}\leq g(x,t)\leq
T_{max},\quad\mbox{a.\,e.\,\,in}\,\,\,\partial_{p}Q_{T}.$ (1.7)
(A4) $A(z,x,t)$ is uniformly continuous with respect to $z$ in $\mathfrak{C}$,
where
$\mathfrak{C}=\\{\varphi\in L^{2}(Q_{T});\,\,T_{min}\leq\varphi(x,t)\leq
T_{max},\,\,\mbox{a.\,e.\,\, in}\,\,Q_{T}\\}.$ (1.8)
In other words, if $z_{i},\,z\in\mathfrak{C}$, $\|z_{i}-z\|_{2}\to 0$,
$\sup_{1\leq p,q\leq n}\|a_{pq}(z_{i}(x,t),x,t)-a_{pq}(z(x,t),x,t)\|_{2}\to
0.$ (1.9)
###### Remark 1.1
In fact, we had considered a general case: parabolic equations with
$(c(x)\rho(x)u)^{\prime}$, nonnegative bounded mixed boundary conditions and
right-hand term $f(z,x,t)$ [13].
If $a_{pq}$ is uniformly Hölder continuous with respect to $z$, (A4) is
natural since
$\displaystyle\|a_{pq}(z_{i}(x,t),x,t)-a_{pq}(z(x,t),x,t)\|_{2}^{2}$
$\displaystyle\leq$ $\displaystyle\iint_{Q_{T}}C|z_{i}(x,t)-z(x,t)|^{2\alpha}$
(1.10) $\displaystyle\leq$ $\displaystyle C\|z_{i}-z\|_{2}^{2}\to 0.$
###### Theorem 1.1 (Parabolic spaces)
Let
$W\equiv\\{w\in L^{2}(S;\,H^{1}_{0}(\Omega));\,\partial_{t}w\in
L^{2}(S;H^{-1}(\Omega))\\},$ (1.11)
$\|w\|_{W}^{2}=\|w\|_{L^{2}(S;\,H^{1}_{0}(\Omega))}^{2}+\|\partial_{t}w\|_{L^{2}(S;H^{-1}(\Omega))}^{2},$
(1.12)
then $($1$)$ $($page 173 [1], page 61 [4]$)$
$W\hookrightarrow C([0,T];L^{2}(\Omega)),\quad W\hookrightarrow L^{2}(Q_{T}).$
(1.13)
The last imbedding is compact.
$($2$)$ $($page 173 [1]$)$ $C^{\infty}([0,T];H^{1}_{0}(\Omega))$ is dense in
$W$.
$($3$)$ $($Theorem 1.6 [8]$)$ Let
$W_{c\rho}\equiv\\{w\in L^{2}(S;\,X);\,\partial_{t}[c\rho(x)w]\in
L^{2}(S;X^{\prime})\\},$ (1.14)
then $C^{\infty}([0,T];X)$ is dense in $W_{c\rho}$. For the mixed boundary
conditions, we can let $X=H^{1}_{D}(\Omega)$.
###### Theorem 1.2 ($V^{1,0}_{2}(Q_{T})$)
$($page 42-44 [2]$)$ Let
$V^{1,0}_{2}(Q_{T})\equiv L^{2}(S;\,H^{1}(\Omega))\cap
C([0,T];L^{2}(\Omega)),$ (1.15) $\|w\|_{V^{1,0}_{2}(Q_{T})}^{2}=\|\nabla
w\|_{L^{2}(Q_{T};\mathbb{R}^{n})}^{2}+\sup_{t\in[0,T]}\|w(x,t)\|_{L^{2}(\Omega)}^{2},$
(1.16)
then
$($1$)$ $H^{1}(Q_{T})\subset V^{1,0}_{2}(Q_{T})$.
$($2$)$ If $u(x,t)\in V^{1,0}_{2}(Q_{T})$, $\forall\,k\in\mathbb{R}$,
$(u-k)_{+}(x,t)=\max\\{(u-k)(x,t),0\\}\in V^{1,0}_{2}(Q_{T}).$ (1.17)
$($3$)$ If $\|u_{i}-u\|_{V^{1,0}_{2}(Q_{T})}\to 0$, then
$\forall\,k\in\mathbb{R}$,
$\|(u_{i}-k)_{+}-(u-k)_{+}\|_{V^{1,0}_{2}(Q_{T})}\to 0.$ (1.18)
## 2 Linearized map and fixed point
###### Theorem 2.1
$($Corollary 11.2 [6]$)$ Let $\mathfrak{C}$ be a closed convex set in a Banach
space $\mathfrak{B}$ and let $\mathcal{L}$ be a continuous mapping of
$\mathfrak{C}$ into itself such that the image $\mathcal{L}\mathfrak{C}$ is
precompact. Then $\mathcal{L}$ has a fixed point.
###### Lemma 2.1
The following set
$\mathfrak{C}=\\{\varphi\in L^{2}(Q_{T});\,\,T_{min}\leq\varphi(x,t)\leq
T_{max},\,\,\mbox{a.\,e.\,\, in}\,\,Q_{T}\\}.$ (2.19)
is a closed convex set in the Banach space $L^{2}(Q_{T})$.
Proof Suppose $v_{i}\in\mathfrak{C}$, $v\in L^{2}(Q_{T})$, $\|v_{i}-v\|_{2}\to
0$. If $v\notin\mathfrak{C}$, there exist two constants $\delta_{0}>0$,
$\delta_{1}>0$, such that the Lebesgue measure of the set
$Q_{0}\equiv\\{(x,t)\in Q_{T};\,v(x,t)\geq T_{max}+\delta_{0}\\}$ is bigger
than $\delta_{1}>0$. Then
$\|v_{i}-v\|_{2}^{2}=\iint_{Q_{T}}|v_{i}-v|^{2}\geq\iint_{Q_{0}}|v_{i}-v|^{2}\geq\delta_{0}^{2}\delta_{1}.$
(2.20)
It’s impossible since $\|v_{i}-v\|_{2}\to 0$. Similarly, $v\geq T_{min}$ and
$\mathfrak{C}$ is closed.
$\forall\,\theta\in[0,1],\quad\theta v_{1}+(1-\theta)v_{2}\leq\theta
T_{max}+(1-\theta)T_{max}=T_{max}.$ (2.21)
So $\mathfrak{C}$ is convex. $\square$
###### Theorem 2.2
If $(A1)-(A4)$ are satisfied, then
$(1)$ $\forall\,z\in\mathfrak{C}$, there exists a unique $w$,
$(w-g)\in L^{2}(S;\,H^{1}_{0}(\Omega)),\quad(w-g)(x,0)=0,$ (2.22)
$w\in\mathfrak{C},\quad\partial_{t}w\in L^{2}(S;H^{-1}(\Omega)),$ (2.23)
such that $\forall\,\varphi\in L^{2}(S;\,H^{1}_{0}(\Omega))$,
$\langle\partial_{t}w,\,\varphi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}+\iint_{Q_{T}}A(z(x,t),x,t)\nabla
w\cdot\nabla\varphi=0.$ (2.24)
$(2)$ Define a map $\mathcal{L}:\,\mathfrak{C}\to\mathfrak{C}$,
$\mathcal{L}z=w$, then $\mathcal{L}\mathfrak{C}$ is precompact in
$L^{2}(Q_{T})$.
$(3)$ $\mathcal{L}$ is continuous in $L^{2}(Q_{T})$. So $\mathcal{L}$ has a
fixed point in $\mathfrak{C}$.
Proof (1) For the a priori estimate, since $H^{-1}(\Omega)\hookrightarrow
L^{2}(\Omega)$ (page 55, 60 [4]),
$\partial_{t}g\in L^{2}(Q_{T})=L^{2}(S;L^{2}(\Omega))\hookrightarrow
L^{2}(S;H^{-1}(\Omega)),$ (2.25) $v\equiv(w-g)\in W\hookrightarrow
C([0,T];L^{2}(\Omega)),$ (2.26) $g\in H^{1}(Q_{T})\hookrightarrow
C([0,T];L^{2}(\Omega)),\quad w\in C([0,T];L^{2}(\Omega)).$ (2.27) $w\in
L^{2}(S;H^{1}(\Omega)),\quad w\in V^{1,0}_{2}(Q_{T}).$ (2.28)
Let
$\varphi=(w-T_{max})_{+}\in V^{1,0}_{2}(Q_{T}),$ (2.29)
then
$\varphi(x,t)|_{\partial_{p}Q_{T}}=0,\quad\varphi\in
L^{2}(S;\,H^{1}_{0}(\Omega)).$ (2.30)
For any $v_{i}\in C^{\infty}([0,T];\,H^{1}_{0}(\Omega))\subset H^{1}(Q_{T})$,
we have
$\displaystyle\langle\partial_{t}(v_{i}+g),\,(v_{i}+g-T_{max})_{+}\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
(2.31) $\displaystyle=$
$\displaystyle\iint_{Q_{T}}\partial_{t}(v_{i}+g)\cdot(v_{i}+g-T_{max})_{+}$
$\displaystyle=$
$\displaystyle\iint_{Q_{T}}\partial_{t}\frac{(v_{i}+g-T_{max})_{+}(x,t)^{2}}{2}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\frac{(v_{i}+g-T_{max})_{+}(x,T)^{2}}{2}$
$\displaystyle-\int_{\Omega}\frac{(v_{i}+g-T_{max})_{+}(x,0)^{2}}{2}.$
By the density of $C^{\infty}([0,T];\,H^{1}_{0}(\Omega))$ in $W$, for
$v\equiv(w-g)\in W$, we can find $\\{v_{i}\\}\subset
C^{\infty}([0,T];\,H^{1}_{0}(\Omega))$ such that
$\|v_{i}-v\|_{C([0,T];L^{2}(\Omega))}\leq C\|v_{i}-v\|_{W}\to 0,$ (2.32)
$\|(v_{i}+g)-(v+g)\|_{V^{1,0}_{2}(Q_{T})}=\|v_{i}-v\|_{V^{1,0}_{2}(Q_{T})}\to
0,$ (2.33)
$\|(v_{i}+g-T_{max})_{+}-(v+g-T_{max})_{+}\|_{V^{1,0}_{2}(Q_{T})}\to 0.$
(2.34)
$\|(v_{i}+g-T_{max})_{+}-(v+g-T_{max})_{+}\|_{L^{2}(S;\,H^{1}(\Omega))}\to 0.$
(2.35) $v_{i},\,v|_{\partial\Omega\times S}=0,\quad g|_{\partial\Omega\times
S}\leq T_{max}.$ (2.36)
$\|(v_{i}+g-T_{max})_{+}-(v+g-T_{max})_{+}\|_{L^{2}(S;\,H^{1}_{0}(\Omega))}\to
0.$ (2.37)
$\|\partial_{t}(v_{i}+g)-\partial_{t}(v+g)\|_{L^{2}(S;H^{-1}(\Omega))}\to 0,$
(2.38)
$\displaystyle\int_{\Omega}[(v_{i}+g-T_{max})_{+}(x,t)^{2}-(v+g-T_{max})_{+}(x,t)^{2}]$
(2.39) $\displaystyle=$
$\displaystyle\int_{\Omega}[(v_{i}+g-T_{max})_{+}(x,t)+(v+g-T_{max})_{+}(x,t)]$
$\displaystyle\quad[(v_{i}+g-T_{max})_{+}(x,t)-(v+g-T_{max})_{+}(x,t)]$
$\displaystyle\leq$
$\displaystyle\|(v_{i}+g-T_{max})_{+}(x,t)+(v+g-T_{max})_{+}(x,t)\|_{L^{2}(\Omega)}$
$\displaystyle\quad\|(v_{i}+g-T_{max})_{+}(x,t)-(v+g-T_{max})_{+}(x,t)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle(\|(v_{i}+g-T_{max})(x,t)\|_{L^{2}(\Omega)}+\|(v+g-T_{max})(x,t)\|_{L^{2}(\Omega)})$
$\displaystyle\quad\|(v_{i}+g-T_{max})(x,t)-(v+g-T_{max})(x,t)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle(\|(v_{i}+g-T_{max})(x,s)\|_{C([0,T];L^{2}(\Omega))}$
$\displaystyle\quad\quad+\|(v+g-T_{max})(x,s)\|_{C([0,T];L^{2}(\Omega))})\|v_{i}-v\|_{C([0,T];L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle C\|v_{i}-v\|_{W}\to 0.$
$\displaystyle\langle\partial_{t}w,\,(w-T_{max})_{+}\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
(2.40) $\displaystyle=$
$\displaystyle\langle\partial_{t}(v+g),\,(v+g-T_{max})_{+}\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle=$
$\displaystyle\lim_{i\to\infty}\langle\partial_{t}(v_{i}+g),\,(v_{i}+g-T_{max})_{+}\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle=$
$\displaystyle\lim_{i\to\infty}\int_{\Omega}\left[\frac{(v_{i}+g-T_{max})_{+}(x,T)^{2}}{2}-\frac{(v_{i}+g-T_{max})_{+}(x,0)^{2}}{2}\right]$
$\displaystyle=$
$\displaystyle\int_{\Omega}\frac{(v+g-T_{max})_{+}(x,T)^{2}}{2}-\int_{\Omega}\frac{(v+g-T_{max})_{+}(x,0)^{2}}{2}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\frac{(v+g-T_{max})_{+}(x,T)^{2}}{2}\geq 0.$
$\displaystyle\iint_{Q_{T}}A(z(x,t),x,t)\nabla w\cdot\nabla(w-T_{max})_{+}$
(2.41) $\displaystyle=$
$\displaystyle\iint_{Q_{T}}A(z(x,t),x,t)\nabla(w-T_{max})_{+}\cdot\nabla(w-T_{max})_{+}$
$\displaystyle\geq$
$\displaystyle\lambda\int_{S}\int_{\Omega}|\nabla(w-T_{max})_{+}|^{2}$
$\displaystyle\geq$ $\displaystyle
C(\Omega)\lambda\int_{S}\int_{\Omega}(w-T_{max})_{+}^{2}.$ $\displaystyle
C(\Omega)\lambda\int_{S}\int_{\Omega}(w-T_{max})_{+}^{2}$ (2.42)
$\displaystyle\leq$
$\displaystyle\langle\partial_{t}w,\,(w-T_{max})_{+}\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle\quad\quad+\iint_{Q_{T}}A(z(x,t),x,t)\nabla
w\cdot\nabla(w-T_{max})_{+}$ $\displaystyle=$ $\displaystyle 0.$
So $w\leq T_{max}$, a. e. in $Q_{T}$. Similarly, $w\in\mathfrak{C}$.
For the well-posedness (the existence, uniqueness and the estimate in $W$) of
$w,\,\,(w-g)\in W$, we refer to Galerkin method (page 171 [1], page 77 [3],
page 205-211 [4]; for mixed problems, see Theorem 2.2 [8]).
(2) $\|(w-g)\|_{W}\leq C$. $W$ can be compactly imbedded in $L^{2}(Q_{T})$, so
$\mathcal{L}\mathfrak{C}$ is precompact in $L^{2}(Q_{T})$.
(3) Suppose
$z_{i},\,z\in\mathfrak{C},\quad\|z_{i}-z\|_{2}\to
0,\quad\mathcal{L}z_{i}=w_{i},\quad\mathcal{L}z=w.$ (2.43)
$W$ is a Hilbert and thus a reflexive space, so there exists a subsequence
$\\{i_{k}\\}$ and $v_{0}=(w_{0}-g)\in W$ such that
$(w_{i_{k}}-g)\to(w_{0}-g),\quad\mbox{weakly\,\,in\,\,}W.$ (2.44) $W\subset
L^{2}(S;\,H^{1}_{0}(\Omega)),\quad(L^{2}(S;\,H^{1}_{0}(\Omega)))^{\prime}\subset
W^{\prime}.$ (2.45)
$(w_{i_{k}}-g)\to(w_{0}-g),\quad\mbox{weakly\,\,in\,\,}L^{2}(S;\,H^{1}_{0}(\Omega)).$
(2.46)
$\nabla(w_{i_{k}}-g)\to\nabla(w_{0}-g),\quad\mbox{weakly\,\,in\,\,}L^{2}(Q_{T};\mathbb{R}^{n}).$
(2.47) $\nabla w_{i_{k}}\to\nabla
w_{0},\quad\mbox{weakly\,\,in\,\,}L^{2}(Q_{T};\mathbb{R}^{n}).$ (2.48)
$\|w_{i_{k}}-g-w_{0}+g\|_{2}\to 0,\quad\|w_{i_{k}}-w_{0}\|_{2}\to 0.$ (2.49)
$\partial_{t}(w_{i_{k}}-g)\to\partial_{t}(w_{0}-g),\quad\mbox{weakly\,\,in\,\,}L^{2}(S;\,H^{-1}(\Omega)).$
(2.50) $\partial_{t}g\in L^{2}(Q_{T})\subset L^{2}(S;\,H^{-1}(\Omega)).$
(2.51)
$\partial_{t}w_{i_{k}}\to\partial_{t}w_{0},\quad\mbox{weakly\,\,in\,\,}L^{2}(S;\,H^{-1}(\Omega)).$
(2.52)
$\forall\,\phi\in C^{\infty}([0,T];\,C^{\infty}_{0}(\Omega))$, using the
natural map into its second dual (page 89 [5]),
$\displaystyle\langle
F(\phi),\partial_{t}w_{i_{k}}-\partial_{t}w_{0}\rangle_{L^{2}(S;\,H^{-1}(\Omega))}$
(2.53) $\displaystyle\equiv$
$\displaystyle\langle\partial_{t}w_{i_{k}}-\partial_{t}w_{0},\phi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}.$
$\displaystyle\langle
F(\phi),\partial_{t}w_{i_{k}}-\partial_{t}w_{0}\rangle_{L^{2}(S;\,H^{-1}(\Omega))}\to
0,$ (2.54) $\displaystyle\Rightarrow$
$\displaystyle\langle\partial_{t}w_{i_{k}}-\partial_{t}w_{0},\phi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}\to
0.$
$\displaystyle|\iint_{Q_{T}}[A(z_{i_{k}}(x,t),x,t)\nabla
w_{i_{k}}-A(z(x,t),x,t)\nabla w_{0}]\cdot\nabla\phi|$ (2.55)
$\displaystyle\leq$ $\displaystyle|\iint_{Q_{T}}[A(z_{i_{k}}(x,t),x,t)\nabla
w_{i_{k}}-A(z(x,t),x,t)\nabla w_{i_{k}}]\cdot\nabla\phi|$
$\displaystyle\,+\,|\iint_{Q_{T}}[A(z(x,t),x,t)\nabla
w_{i_{k}}-A(z(x,t),x,t)\nabla w_{0}]\cdot\nabla\phi|$ $\displaystyle=$
$\displaystyle|\iint_{Q_{T}}[A(z_{i_{k}}(x,t),x,t)-A(z(x,t),x,t)]\nabla
w_{i_{k}}\cdot\nabla\phi|$ $\displaystyle\,+\,|\iint_{Q_{T}}[\nabla
w_{i_{k}}-\nabla w_{0}]\cdot A(z(x,t),x,t)^{\top}\nabla\phi|$
$\displaystyle\leq$ $\displaystyle C\sup_{1\leq p,q\leq
n}\|a_{pq}(z_{i_{k}}(x,t),x,t)-a_{pq}(z(x,t),x,t)\|_{2}+\epsilon(i_{k})$
$\displaystyle\to$ $\displaystyle 0.$
$\displaystyle\langle\partial_{t}w_{i_{k}},\phi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle\,+\,\iint_{Q_{T}}A(z_{i_{k}}(x,t),x,t)\nabla
w_{i_{k}}\cdot\nabla\phi=0.$ (2.56)
$\displaystyle\langle\partial_{t}w_{0},\phi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle\,+\,\iint_{Q_{T}}A(z(x,t),x,t)\nabla w_{0}\cdot\nabla\phi=0.$
(2.57)
From the density of $C^{\infty}([0,T];\,C^{\infty}_{0}(\Omega))$ in
$L^{2}(S;\,H^{1}_{0}(\Omega))$, $\forall\,\,\varphi\in
L^{2}(S;\,H^{1}_{0}(\Omega))$,
$\displaystyle\langle\partial_{t}w_{0},\varphi\rangle_{L^{2}(S;\,H^{1}_{0}(\Omega))}$
$\displaystyle\,+\,\iint_{Q_{T}}A(z(x,t),x,t)\nabla
w_{0}\cdot\nabla\varphi=0.$ (2.58)
For the boundary condition,
$(w_{0}-g)\in
L^{2}(S;\,H^{1}_{0}(\Omega)),\quad(w_{0}-g)|_{\partial\Omega\times S}=0.$
(2.59)
For the initial condition, $\forall\,\,\psi(x)\in L^{2}(\Omega)$, we can
define a linear functional on $W$,
$\langle\Psi,h\rangle_{W}\equiv\int_{\Omega}h(x,0)\psi,\quad\forall\,\,h(x,t)\in
W.$ (2.60)
This functional is bounded since
$\displaystyle|\langle\Psi,h\rangle_{W}|$ $\displaystyle=$
$\displaystyle|\int_{\Omega}h(x,0)\psi|$ (2.61) $\displaystyle\leq$
$\displaystyle\|h(x,0)\|_{2}\|\psi\|_{2}\leq\|\psi\|_{2}\sup_{s\in[0,T]}\|h(x,s)\|_{2}$
$\displaystyle=$
$\displaystyle\|\psi\|_{2}\|h(x,t)\|_{C([0,T];\,L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle C\|h(x,t)\|_{W}.$
Since
$(w_{i_{k}}-g)\to(w_{0}-g),\quad\mbox{weakly\,\,in\,\,}W,$ (2.62)
$\forall\,\,\psi(x)\in L^{2}(\Omega)$,
$\langle\Psi,(w_{i_{k}}-g)-(w_{0}-g)\rangle_{W}\equiv\int_{\Omega}[(w_{i_{k}}-g)-(w_{0}-g)](x,0)\psi\to
0.$ (2.63)
From the Riesz Representation Theorem in $L^{2}(\Omega)$,
$(w_{i_{k}}-g)(x,0)\to(w_{0}-g)(x,0),\quad\mbox{weakly\,\,in\,\,}L^{2}(\Omega).$
(2.64)
Note that from the initial condition,
$(w_{i_{k}}-g)(x,0)=0,\quad\mbox{in}\,\,L^{2}(\Omega).$ (2.65)
$(w_{i_{k}}-g)(x,0)\to 0,\quad\mbox{strongly\,\,in\,\,}L^{2}(\Omega).$ (2.66)
$(w_{0}-g)(x,0)=0,\quad\mbox{in}\,\,L^{2}(\Omega).$ (2.67)
To sum up, $w_{0}=\mathcal{L}z$: $w_{0}$ satifies the linearized equation and
the initial-boundary conditions.
Since the solution is unique from the step (1), $w_{0}=\mathcal{L}z=w$. So
$\|w_{i_{k}}-w\|_{2}\to 0$. Each subsequence of $\\{\|w_{i}-w\|_{2}\\}$ has a
sub-subsequence which converges to $0$, so $\|w_{i}-w\|_{2}\to 0$. We have
obtain the continuity of $\mathcal{L}$.
From Theorem 2.1, there exists a fixed point. $\square$
###### Remark 2.1
For the continuity of $\mathcal{L}$ in $C^{0}(\overline{Q}_{T})$, we can use
the well-known De Giorgi-Nash estimate: $\\{w_{i}\\}$ is bounded in
$C^{2\alpha,\alpha}(\overline{Q}_{T})$ if $g\in
C^{2\alpha_{0},\alpha_{0}}(\partial_{p}Q_{T})$ and $\Omega$ is an (A) domain
(page 145 [2]).
Then from the Arzel$\grave{\rm{a}}$-Ascoli Lemma,
$\|w_{i_{k}}-w_{0}\|_{C^{0}(\overline{Q}_{T})}\to 0$. By the same method,
$w_{0}=w$ and $\|w_{i}-w\|_{C^{0}(\overline{Q}_{T})}\to 0$.
From the linear maximal regularity [7, 8], a natural conjecture is:
$\mathcal{L}$ is continuous in $C^{2\alpha,\alpha}(\overline{Q}_{T})$ and $W$.
## 3 Nonlinear maximal regularity
For the linear parabolic/ellptic equations with nonsmooth data, the theory of
maximal regularity has been established [7, 8, 9, 10]. In brief, maximal
regularity is about the smoothness of the data-to-solution-map [10]. This
smooth dependence has its physical meaning: many physical processes are stable
with respect to the parameters (except the chaos and critical theory). For the
mathematicians, ”the door is open to apply the powerful theorems of
differential calculus”([10], e.g. the Implicit Function Theorem).
In the following, we will discuss the continuous dependence (between the
solutions and the data) for the parabolic equations with locally arbitrary
growth conditions (e.g. Rosseland-type).
## 4 Acknowledge
This work is supported by the National Nature Science Foundation of China (No.
90916027). This is a part of my PhD thesis [13] in AMSS, Chinese Academy of
Sciences, and a simplification of our prior paper [12]. So I will thank my
advisor Professor Jun-zhi Cui (he is also a member of the Chinese Academy of
Engineering) and the referees for their careful reading and helpful comments.
My E-mail is: zhangqf@lsec.cc.ac.cn.
## References
* [1] Shu-xing Chen. An introduction to mordern PDE (in Chinese). China Science Press, 2005.
* [2] Ya-zhe Chen. Parabolic partial differential equations of second order (in Chinese). Peking University Press, 2003.
* [3] Zhi-ming. Chen and Hai-jun Wu. Selected topics in Finite Element Methods. China Science Press, 2010.
* [4] D. Cioranescu and P. Donato. An introduction to homogenization. Oxford University Press, 1999.
* [5] J.B. Conway. A course in functional analysis, volume 96. Springer, 1990.
* [6] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order, volume 224. Springer Verlag, 2001.
* [7] J.A. Griepentrog. Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Advances in Differential Equations, 12(9):1031–1078, 2007\.
* [8] J.A. Griepentrog. Sobolev-Morrey spaces associated with evolution equations. Advances in Differential Equations, 12(7):781-840, 2007.
* [9] J.A. Griepentrog and L. Recke. Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Mathematische Nachrichten, 225:39–74, 2001.
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|
arxiv-papers
| 2012-05-15T02:10:51 |
2024-09-04T02:49:30.912724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qiao-fu Zhang",
"submitter": "Qiaofu Zhang",
"url": "https://arxiv.org/abs/1205.3237"
}
|
1205.3369
|
# The best simultaneous approximation in linear 2-normed spaces
Mehmet Acikgoz University of Gaziantep Faculty of Science and Arts, Department
of Mathematics 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
(Date: May 07, 2012)
###### Abstract.
In this paper, we shall investigate and analyse a new study on the best
simultaneous approximation in the context of linear 2-normed spaces inspired
by Elumalai and his coworkers in [10]. The basis of this investigation is to
extend and refinement the definition of the classical aproximation, best
approximation and some related concepts to linear 2-normed spaces.
###### Key words and phrases:
2-normed space, best approximation, simultaneous best approximation, 2-Banach
space.
###### 2000 Mathematics Subject Classification:
Primary 46A15, Secondary 41A65.
## 1\. Introduction
The problem of best and simultaneous best approximation has been studied by
several mathematicians (for more informations, see [5], [6], [7], [8], [9],
[18], [19], [20], [26]). Most of these works have dealt with the existence,
uniqueness and characterization of best approximations in spaces of continuous
functions with values in Banach spaces. Recently, many works on approximation
has been done on 2-structures such as 2-normed spaces, generalized 2-normed
spaces (for details, see [4], [23]) and 2-Banach spaces (see [2], [3], [10],
[11], [12], [13], [14], [15]). Diaz and McLaughlin [7] and Dunham [9] have
considered simultaneously approximating of two real-valued continuously
functions $f_{1}$, $f_{2}$ defined on $[a,b]$, by elements of set $C[a,b].$
Several results of best simultaneous approximation in the context of linear
space were obtained by Goel et al. (for details, see [19], [20]). The subject
of approximation theory has attracted the attention of several mathematicians
during the last 130 years or so. This theory is now an extremely extensive
branch of mathematical analysis. It has many applications in many areas,
especially in engineering.
The concept of linear 2-normed spaces has been investigated by Gahler in 1965
[17] and given many important properties and examples for these spaces. After,
these spaces have been developed extensively in different subjects by other
researchers from many points of view and then the field has considerably
grown. Z. Lewandowska published some of papers on 2-normed sets and
generalized 2-normed spaces in 1999-2003 (see [23]). In [10], Elumalai and his
coworkers published a series of papers related this subject. They have
developed best approximation theory in the context of linear 2-normed spaces.
These are some works on characterization of 2-normed spaces, extension of
2-functionals and approximation in 2-normed spaces (see [1], [2]). Also, the
author has some works in $\varepsilon$-approximation theory [3] and Rezapour
has also such studies in [25]. The essential aim of this paper is to derive
new different definitions of approximation and obtain some results related to
these definitions. The essential results of the set of best simultaneous
approximation are given in the fourth section of this paper.
Throughout this paper, we first fix some notations. Let $X$ be a linear space
and $L\left\\{y\right\\}$ be the subspace of $Y$ generated by $y$. It is also
let $\left(X,\left\|.\right\|\right)$ and $(X,\|.,.\|)$ denote a normed space
and $2$-normed space with the corresponding norms, respectively. $\mathbb{R}$
denotes the set of real numbers, $\mathbb{N}$ denotes the set of natural
numbers and $\mathbb{C}$ denotes the set of complex numbers. Throughout this
work, $K$ is variously considered as an indeterminate, as a real number
$K\in\mathbb{R}$, or as a complex number $K\in\mathbb{C}$.
We now summarize our work in four section as follows:
In first section, we gave history of normed and $2$-normed spaces and
motivation of our work. In section 2 and 3, we specify definitions and
properties of normed and $2$-normed spaces, respectively. In section 4, we
gave suitable a definition for studying in linear $2$-normed spaces and so we
derived three lemma and two proposition by using our definition.
Thus, we are now ready in order to begin with the second section as follows.
## 2\. Some Definitions of Normed Spaces
###### Definition 1.
Let $\left(X,\left\|.\right\|\right)$ be a normed space and $K\subset X$. For
$u\in X$,
$\underset{v\in K}{\inf}\left\\{\left\|u-v\right\|\right\\}$
is a general best approximation.
Mohebi and Rubinov ([24]) and Rezapour in [25] gave the main preliminaries on
the approximation theory in the usual sense as follows:
###### Definition 2.
Let $\left(X,\left\|.\right\|\right)$ be a linear normed space. For a nonempty
subset $A$ of $X$ and $x\in X$,
$d\left(x,A\right)=\underset{a\in A}{\inf}\left\\{\left\|u-a\right\|\right\\}$
denotes the distance from $x$ to the set $A$. If
$\left\|x-a_{0}\right\|=d\left(x,A\right)\text{.}$
Then, we say that a point $a_{0}\in A$ is called a best approximation for
$x\in X$. If each $x\in X$ has at least one best approximation $a_{0}\in A,$
then $A$ is called a proximinal subset of $X$. If each $x\in X$ has a unique
best approximation $a_{0}\in A$, then $A$ is called a Chebyshev subset of $X$.
###### Definition 3.
Let $A\subset X$. For $x\in X$,
$P_{A}\left(x\right)=\left\\{a\in
A:\left\|x-a\right\|=d\left(x,A\right)\right\\}$
where $P_{A}\left(x\right)$, the set of all best approximations of $x$ in $A$.
We know that $P_{A}\left(x\right)$ is a closed and bounded subset of $X$. For
$x\notin A$, $P_{A}\left(x\right)$ is located in the boundary of $A$.
###### Definition 4.
Let $\left(X,\left\|.\right\|\right)$ be a linear normed space. For a nonempty
subset $A$ of $X$ and a nonempty set $W$ of $X$,
$d\left(A,W\right)=\underset{w\in W}{\inf}\underset{a\in
A}{\sup}\left\\{\left\|a-w\right\|\right\\}$
denotes the distance from the set $A$ to the set $W$. If
$\underset{w\in W}{\inf}\underset{a\in
A}{\sup}\left\\{\left\|a-w\right\|\right\\}=\underset{a\in
A}{\sup}\left\\{\left\|a-w_{0}\right\|\right\\}\text{.}$
Then, we say that a point $w_{0}\in W$ is called a best approximation from $A$
to $W$.
## 3\. Properties of $2$-Normed Spaces
In [16], Cho et al. defined linear 2-normed spaces and gave interesting
properties of them. After, Lewandowska defined generalized $2$-normed spaces
and derived properties of these spaces in [23]. Now, let us give the
definition of 2-normed space.
###### Definition 5.
Let $X$ be a linear space over $F$, where $F$ is the real or complex numbers
field, $\dim X>1$, and let
$\left\|.,.\right\|:X^{2}\rightarrow\mathbb{R}^{+}\cup\left\\{0\right\\}$
be a non-negative real-valued function on $X\times X$ with the following
properties:
N1) $\left\|x,y\right\|=0$ if and only if $x$ and $y$ are linearly dependent
vectors,
N2) $\left\|x,y\right\|=\left\|y,x\right\|$ for all $x,y\in X$,
N3) $\left\|\alpha x,y\right\|=\left|\alpha\right|\left\|x,y\right\|$ for all
$\alpha\in K$ and all $x,y\in X$,
N4) $\left\|x+y,z\right\|\leq\left\|x,z\right\|+\left\|y,z\right\|$ for all
$x,y,z\in X$.
Then, $\left\|.,.\right\|$ is called a 2-norm on $X$ and
$\left(X,\left\|.,.\right\|\right)$ is called a linear 2-normed space.
Every 2-normed space is a locally convex topological linear space. In fact,
for a fixed $b\in X$. For all $x\in X$,
$p_{b}\left(x\right)=\left\|x,b\right\|$
which is a seminorm and the family of $P$, that is
$P=\left\\{p_{b}:b\in X\right\\}$
generates a locally convex topology on $X.$ This space will be denoted by
$\left(X,p_{b}\right)$. In each 2-normed space
$\left(X,\left\|.,.\right\|\right)$. For all $x,y\in X$ and for every real
$\alpha$, we have non-negative norm,
$\left\|x,y\right\|\geq 0\text{ and }\left\|x,y+\alpha
x\right\|=\left\|x,y\right\|\text{.}$
Also, if $x$, $y$ and $z$ are linearly dependent, this occurs for $\dim X=2$.
Then,
$\left\|x,y+z\right\|=\left\|x,y\right\|+\left\|x,z\right\|or\left\|x,y-z\right\|=\left\|x,y\right\|+\left\|x,z\right\|\text{.}$
###### Example 1.
([27])Let $P_{n}$ denotes the set of real polynomials of degree less than or
equal to $n$, on the interval $\left[0,1\right]$. By considering usual
addition and scalar multiplication, $P_{n}$ is a linear vector space over the
reals. Let $\left\\{x_{1},x_{2},\cdots,x_{2n}\right\\}$ be distinct fixed
points in $\left[0,1\right]$ and define the 2-norm on $P_{n}$ as
$\left\|f,g\right\|=\mathop{\displaystyle\sum}\limits_{k=1}^{2n}\left|f\left(x_{k}\right)g^{\prime}\left(x_{k}\right)-f^{\prime}\left(x_{k}\right)g\left(x_{k}\right)\right|\text{.}$
Then, $\left(P_{n},\left\|.,.\right\|\right)$ is a 2-normed space.
Let $\left(X,\left\|.,.\right\|\right)$ be a 2-normed space. Under this
assumption, we can give the following defitinions:
###### Definition 6.
([27])A sequence $\left\\{x_{n}\right\\}_{n\geq 1}$ in a linear 2-normed space
$X$ is called Cauchy sequence if there exist independent elements $y,z\in X$
such that
$\lim_{n,m\rightarrow\infty}\left\|x_{n}-x_{m},y\right\|=0\text{ and
}\lim_{n,m\rightarrow\infty}\left\|x_{n}-x_{m},z\right\|=0\text{.}$
###### Definition 7.
([27])A sequence $\left\\{x_{n}\right\\}_{n\geq 1}$ in a linear 2-normed space
$X$ is called convergent if there exists an element $x\in X$ such that
$\lim_{n\rightarrow\infty}\left\|x_{n}-x,z\right\|=0$
for all $z\in X$.
###### Proposition 1.
([6])Let $\left(X,\left\|.,.\right\|\right)$ be 2-normed space and $W$ be a
subspace of $X$, $b\in X$ and $L\left\\{b\right\\}$ be the subspace of $X$
generated by $b$. If $x_{0}\in X$ is such that
$\delta=\underset{w\in
W}{\inf}\left\\{\left\|x_{0}-w,b\right\|\right\\}>0\text{.}$
Then, there exists a bounded bilinear functional as follows
$f:X\times L\left\\{b\right\\}\rightarrow K$
such that
$F|_{w\times L\left\\{b\right\\}}=0\text{, }F\left(x_{0},b\right)=1\text{ and
}\left\|F\right\|=\frac{1}{\delta}\text{.}$
###### Definition 8.
A 2-normed space $\left(X,\left\|.,.\right\|\right)$ is which every Cauchy
sequence $\left(x_{n}\right)$ converges to some $x\in X$ then $X$ is said to
be complete with respect to the 2-norm.
###### Definition 9.
A complete 2-normed space $\left(X,\left\|.,.\right\|\right)$ is called a
2-Banach space.
The examples 1 and 2 are 2-Banach spaces while the example 3 does not (For
details, see [27]).
###### Lemma 1.
([27]) $\left(i\right)$ Every 2-normed space of dimension 2 is a 2-Banach
space, when the underlying field is complete.
$\left(ii\right)$ If $\left\\{x_{n}\right\\}$ is a sequence in 2-normed space
$\left(X,\left\|.,.\right\|\right)$ and if
$\lim_{n\rightarrow\infty}\left\|x_{n}-x,y\right\|=0\text{{.}}$
then, we have
$\lim_{n\rightarrow\infty}\left\|x_{n},y\right\|=\left\|x,y\right\|\text{.}$
## 4\. Fundamental Results
In this section, let us also consider a definition, and however, we give Lemma
and Proposition for the best simultaneous approximation in linear $2$-normed
spaces.
###### Definition 10.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space and $W$
be any bounded subset of $X$. An element $g^{\ast}\in G$ is said to be a best
approximation to the set $W$, if
$\underset{f\in W}{\sup}\left\|f-g^{\ast},b\right\|=\underset{g\in
G}{\inf}\left\\{\underset{f\in W}{\sup}\left\|f-g,b\right\|\right\\}$
where $b\in X\backslash L\left\\{f,g^{\ast}\right\\}$ is the subspace of $X$
generated by $f$ and $g^{\ast}$.
###### Lemma 2.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. Then,
$\Phi\left(g,b\right)=\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$
is a continuous functional on $X$, where $b\in X\backslash
L\left\\{f,g^{\ast}\right\\}$.
###### Proof.
Since the norms $\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|$are
continuous functionals of $g$ on $X$, $\phi\left(g,b\right)$ is clearly a
continuous functional. To show this, for any $f_{1},f_{2}\in W$ and
$g,g^{{}^{\prime}}\in X$ , we have
$\displaystyle\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$
$\displaystyle\leq$
$\displaystyle\left\\{\left\|f_{1}-g^{{}^{\prime}},b\right\|+\left\|g-g^{{}^{\prime}},b\right\|,\left\|f_{2}-g^{{}^{\prime}},b\right\|,\left\|g-g^{{}^{\prime}},b\right\|\right\\}\text{.}$
Then
$\displaystyle\underset{f_{1},f_{2}\in
w}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$
$\displaystyle\leq$ $\displaystyle\underset{f_{1},f_{2}\in
w}{\sup}\left\\{\left\|f_{1}-g^{{}^{\prime}},b\right\|+\left\|g-g^{{}^{\prime}},b\right\|,\left\|f_{2}-g^{{}^{\prime}},b\right\|,\left\|g-g^{{}^{\prime}},b\right\|\right\\}\text{.}$
Now, if
$\left\|g-g^{{}^{\prime}},b\right\|<\frac{\varepsilon}{2},\text{ then
}\phi\left(g,b\right)\leq\phi\left(g^{{}^{\prime}},b\right)+\varepsilon\text{.}$
By interchanging $g$ and $g^{{\acute{}}}$, proof of Theorem will be completed.
###### Lemma 3.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. Then there exists a best
simultaneous approximation $g^{\ast}\in G$ to any given compact subset
$W\subset X$.
###### Proof.
By using the proof of Elumalai and his coworkers in same manner, we can make
the proof, using the definition of the continuous functional
$\phi\left(g,b\right)=\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}\text{.}$
###### Lemma 4.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. If $g_{1},g_{2}\in G$ are best
simultaneous approximations to $W$ by elements of $G$. Then
$g=\lambda_{1}g_{1}+\lambda_{2}g_{2}$ is also a best simultaneous
approximation to $f_{1}$ and $f_{2}$, where $0\leq\lambda\leq 1$ and
$\lambda_{1}+\lambda_{2}=1$.
###### Proof.
By using expression of $\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-\overset{\\_}{g},b\right\|,\left\|f_{2}-\overset{\\_}{g},b\right\|\right\\}$,
we discover the followings
$\displaystyle=$ $\displaystyle\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-\lambda_{1}g_{1}-\lambda_{2}g_{2},b\right\|,\left\|f_{2}-\lambda_{1}g_{1}-\lambda_{2}g_{2},b\right\|\right\\}$
$\displaystyle=$ $\displaystyle\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\text{.}$
From last equality, we easily derive as
$\leq\left[\begin{array}[]{c}\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\\\
+\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\end{array}\right]$
By using definition of 2-norm and definition 10, we deduce as follows
$\underset{g\in G}{\inf}\left\\{\underset{f\in
W}{\sup}\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}\text{.}$
Subsequently, we complete the proof of Lemma.
###### Proposition 2.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space, $G$ is a
non-empty strictly convex subset of $X$ and $Y$ be a compact subset of $X$.
Then there is only one $y_{0}\in Y$ such that
$\left\|x_{0}-y_{0},z\right\|=\underset{y\in
Y}{\inf}\left\\{\left\|x_{0}-y,z\right\|\right\\}$
for $x_{0}\in X\backslash Y$ and for every $z\in X\backslash L\left\\{x\in
G\right\\}$ and $y_{0}\in Y$.
###### Proof.
If $x_{0}\in Y$. Then, we have $\left\|x_{0}-y_{0},z\right\|=0$. Hence, assume
that
$x_{0}\in Y\ \text{or }x\in X\backslash Y\text{.}$
If we say
$d_{0}=\underset{y\in Y}{\inf}\left\\{\left\|x_{0}-y,z\right\|\right\\}$
and
$d_{0}=\underset{y\in
Y}{\inf}\left\\{\left\|x_{0}-y,y^{{}^{\prime}}\right\|\right\\}\text{.}$
Then, there are linearly independent elements $y^{{}^{\prime}}$ and $z$ in
$X$. So, there is a Cauchy sequence $\left\\{y_{n}\right\\}$ such that
$\underset{n\rightarrow\infty}{\lim}\left\|x_{0}-y_{n},z\right\|=d_{0},\text{
\ \ }\underset{m\rightarrow\infty}{\lim}\left\|x_{0}-y_{m},z\right\|=d_{0}$
and
$\underset{n\rightarrow\infty}{\lim}\left\|x_{0}-y_{n},y^{{}^{\prime}}\right\|=d_{0},\text{
\ \
}\underset{m\rightarrow\infty}{\lim}\left\|x_{0}-y_{m},y^{{}^{\prime}}\right\|=d_{0}\text{.}$
Thus, we procure the following
$\left\|x_{0}-y_{0},y\right\|=d_{0}\text{ \ and \
}\left\|x_{0}-y_{0},z\right\|=d_{0}\text{.}$
By using the following inequalities
$d_{0}\leq\left\|x_{0}-y_{0},z\right\|\leq\left\|x_{0}-y_{n},z\right\|+\left\|y_{n}-y_{0},z\right\|$
and
$d_{0}\leq\left\|x_{0}-y_{0},y\right\|\leq\left\|x_{0}-y_{n},y\right\|+\left\|y_{n}-y_{0},y\right\|\text{.}$
From this, we see that
$\displaystyle\left\|y_{0}-y_{0}^{{}^{\prime}},z\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}+\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}$
$\displaystyle-\left\|y_{0}+y_{0}^{{}^{\prime}}-2x_{0},z\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(\left\|x_{0}-y_{0},z\right\|^{2}+\left\|y_{0}^{{}^{\prime}}-x_{0},z\right\|^{2}\right)-4\left\|\frac{y_{0}+y_{0}^{{}^{\prime}}}{2}-x_{0},z\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(2d_{0}^{2}\right)-4d_{0}^{2}=0\text{.}$
We find $y_{0}=y_{0}^{{}^{\prime}}$. In similar way, we again obtain
$y_{0}=y_{0}^{{}^{\prime}}$ with respect to $y$
$\displaystyle\left\|y_{0}-y_{0}^{{}^{\prime}},y\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}+\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}$
$\displaystyle-\left\|y_{0}+y_{0}^{{}^{\prime}}-2x_{0},y\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(\left\|x_{0}-y_{0},y\right\|^{2}+\left\|y_{0}^{{}^{\prime}}-x_{0},y\right\|^{2}\right)-4\left\|\frac{y_{0}+y_{0}^{{}^{\prime}}}{2}-x_{0},y\right\|^{2}$
$\displaystyle\leq$ $\displaystyle 2\left(2d_{0}^{2}\right)-4d_{0}^{2}=0$
Then, we complete the proof of theorem.
Let $X$ be a linear 2-normed space and $W_{1}$ and $W_{2}$ are linear
subspaces in $X$, and $f$ be a 2-functional with domain $W_{1}\times W_{2}$.
If $\left\|.,.\right\|$ denotes 2-norm, then the problem is to find an element
$g^{\ast}\in G$, if it exists for which
$\underset{f_{1},f_{2}\in
W}{\sup}\left\\{\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right\\}=\underset{g\in
G}{\inf}\left\\{\underset{f_{1},f_{2}\in
W}{\sup}\left(\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right)\right\\}\text{.}$
Thus, we reach the following proposition which is interesting and worthwhile
for studying in linear $2$-normed spaces.
###### Proposition 3.
Let $\left(X,\left\|.,.\right\|\right)$ be a linear 2-normed space over $R$
and $G$ be a linear subspace of $X$. Let $f_{1},f_{2}\in X\backslash G$ such
that $f_{1},f_{2}$ and $b\in X$ are linearly independent. Then there exists a
best simultaneous approximation by elements of $G$ to $f_{1},f_{2}\in W$ such
that
$\underset{g\in G}{\inf}\left\\{\underset{f_{1},f_{2}\in
W}{\sup}\left(\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right)\right\\}=\underset{f_{1},f_{2}\in
W}{\sup}\left\\{\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right\\}$
where $W=\left\\{f_{1},f_{2}\right\\}$.
###### Acknowledgement 1.
Author would like to thank to Serkan Araci for his help in this paper.
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* [8] Diaz, J. B. and H. W. McLaughlin, On simultaneous chebyshev approximation and chebyshev approximation with additive weight function, J. App. Theory, (6) (1972), 68-71.
* [9] Dunham, C. B., Simultaneous chebyshev approximations of functions on an interval, Proc. Aner. Math. Soc., (18) (1967), 472-477.
* [10] Elumalai, S., Asha, A., Patricia J., Some results on linear 2-normed spaces, Bull. Cal. Math. Soc., 92 (6) (2000), 441-454.
* [11] Elumalai, S., Souruparani, M., A characterization of best approximation and the operators in linear 2-normed spaces, Bull. Cal. Math. Soc., 92 (4) (2000), 441-454.
* [12] Elumalai, S., and Vijayaragavan, R., Best approximation in linear 2- normed spaces, General Mathematics 16 (1) (2008), 73-81.
* [13] Elumalai, S., Best approximation sets in linear 2- normed spaces, Commu. Korean. Math. Soc., (12) (1997), 619-629.
* [14] Elumalai, S., Souruparani, M., A characterization of best approximation and operators in linear 2-normed spaces, Cal. Math. Soc., 92 (4) (2000), 235-248.
* [15] Ehret, R., Linear 2-normed spaces, Dissertation, St. Louis University, 1969.
* [16] Freese, R., Cho, Y., Geometry of linear 2-normed spaces, Nova Science Publishers, 2001.
* [17] Gahler, S., Lineare 2-normierte raume. Math. Nachr. (28) (1964), 1-43.
* [18] Lorentz, G. G., Approximation of functions, Holt, Rinehart and Winstoni New York, 1966.
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* [20] Goel, D. S., Holland, A. S. B., Nasim, C., Sahney, B.N., Characterization of an element of best $l_{p}$ simultaneous approximation, S. Ramanujan Memorial Volume Madras, 1974, 10-14.
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* [25] Rezapour, Sh., Proximinal Subspaces of 2-normed Spaces, Anal. Theory Appl. 22, No. 2 (2006), 114-119.
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|
arxiv-papers
| 2012-05-15T13:42:29 |
2024-09-04T02:49:30.921702
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehmet Acikgoz",
"submitter": "Mehmet Acikgoz",
"url": "https://arxiv.org/abs/1205.3369"
}
|
1205.3398
|
040003 2012 V. Lakshminarayanan C. Negreira, Laboratorio de Acústica
Ultrasonora,
Universidad de la República, Uruguay 040003
Water in plant xylem is often superheated, and therefore in a meta-stable
state. Under certain conditions, it may suddenly turn from the liquid to the
vapor state. This cavitation process produces acoustic emissions. We report
the measurement of ultrasonic acoustic emissions (UAE) produced by natural and
induced cavitation in corn stems. We induced cavitation and UAE in vivo, in
well controlled and reproducible experiments, by irradiating the bare stem of
the plants with a continuous-wave laser beam. By tracing the source of UAE, we
were able to detect absorption and frequency filtering of the UAE propagating
through the stem. This technique allows the unique possibility of studying
localized embolism of plant conduits, and thus to test hypotheses on the
hydraulic architecture of plants. Based on our results, we postulate that the
source of UAE is a transient “cavity oscillation” triggered by the disruptive
effect of cavitation inception.
# Natural and laser-induced cavitation in corn stems: On the mechanisms of
acoustic emissions
E. Fernández [inst1] R. J. Fernández [inst1] G. M. Bilmes[inst2] E-mail:
gabrielb@ciop.unlp.edu.ar
(14 March 2011; 23 April 2012)
††volume: 4
99 inst1 IFEVA, Facultad de Agronomía, Universidad de Buenos Aires y CONICET,
Av. San Martín 4453, C1417DSE Buenos Aires, Argentina. inst2 Centro de
Investigaciones Opticas (CONICET-CIC) and Facultad de Ingeniería, Universidad
Nacional de La Plata, Casilla de Correo 124, 1900 La Plata, Argentina.
## 1 Introduction
The cohesion-tension theory suggests that water in the xylem of transpiring
plants is under tension with a hydrostatic pressure below atmospheric and,
thus, most of the time at “negative” values [1]. Negative pressures means that
water in the xylem has a reduced density compared to equilibrium [2].
According to its phase diagram, water under these conditions is overheated
(i.e., in a meta-stable state). Therefore, it should not be in the liquid but
in the vapor phase [3]. The molecules in the liquid phase are further away
from each other, but their mutual attraction allows the system to remain
unchanged.
Under sufficiently high tension (i.e., low pressures caused by water deficit),
xylem may fail to maintain this state, causing liquid water to turn into vapor
in a violent way. This phenomenon, usually known as cavitation, causes the
embolism of the conduits, reducing tissue hydraulic conductivity and
exacerbating plant physiological stress [4, 5]. Some herbaceous species are
known to sustain cavitation almost every day, repairing embolism during the
night, while most woody species preclude cavitation occurrence by a
combination of stomatal behavior and anatomical and morphological adjustment
[6].
Cavitation events in xylem produce sound [7, 8]. In 1966, Milburn and Johnson
developed a technique to detect sound by registering ‘clicks’ in a record
player pick-up head attached to stressed plants and connected to an amplifier
[9]. They associated this sound emission with the rupture of the water column
in xylem vessels. Since then, several authors have used this audible acoustic
emission technique to measure xylem cavitation [10, 11]. Later on, some
authors have improved the technique by detecting ultrasonic acoustic emissions
(UAE) [12, 13, 14, 15, 16, 17]. These authors have demonstrated a good
correlation between UAE and cavitation. However, the connection between
audible or ultrasonic acoustic emissions and cavitation phenomena on xylem
vessels remains unexplained. Tyree and Dixon [12] proposed four possible
sources of acoustic emissions that we will consider in the Discussion section.
Others authors have developed explanations based on alternatives to the
cohesion-tension theory [3, 18]
One of the problems of studying cavitation in plants is the spontaneous
character of the phenomenon, so far precluding our ability to produce it in a
controlled way. Most cavitation experiments use transpiration to raise xylem
water tension, the trigger for cavitation events. In some cases, xylem tension
was increased by centrifugation [19, 20], but even there, cavitation events
took place rather randomly along the water column. On the other hand, in order
to study bubble behavior in isolated physical systems, several authors
explored the generation of cavitation phenomena using lasers [21, 22, 23, 24].
In these experiments, cavitation bubbles are generated in a very well defined
location taking advantage of the accuracy of the laser beam. Even though this
technique was developed to generate cavitation in transparent environments, we
wondered whether it could be used in biological systems to generate cavitation
at specific locations along the stem.
In this article, we report spontaneous UAE produced by natural cavitation in
xylem vessels of corn (Zea mays L.) stems and we characterized and classified
the signals. We also developed a method to produce laser-induced cavitation
and UAE events in a controlled way by irradiating plants with a continuous-
wave laser. We performed experiments with this method to study the generation
and propagation of UAE. Our results allowed us to explain the connection
between cavitation and UAE, as well as the relationship between signal
frequency and the localization of the source in the stem.
## 2 Materials and methods
Figure 1: Set-up for the different experiments. (A) Experiment 1. (B)
Experiments 2 and 4. (C) Experiment 3. (D) Experiment 5. CS: corn stem; T1 and
T2: PZT transducers for UAE detection.
Corn plants were grown in a greenhouse in 3l pots containing sand. They were
watered at field capacity every 1-2 days with nutritive solution (3 g l-1 of
KSC II – Roulier). After four months, tasseling plants (around 1 m high and 12
mm stem width) were used to perform experiments under different conditions:
total darkness (D); room diffuse light (RL; PAR ca. 100 mE m-2 s-1); leaf
illumination with a 150 W incandescent lamp (IL; placed ca. 0.5 m away), and
laser irradiation (L). In the latter case, experiments were carried out by
directing the beam of a 50 mW He–Ne red laser (630 nm), or a continuous-wave
(CW) Ar-ion laser (Spectra Physics Model 165/09) directly on to the stems.
Most experiments were conducted sequentially in 3-5 plants, and we report the
full range of observed results.
Ultrasonic acoustic emissions generated in the stems of plants were recorded
by home-made PZT-based piezoelectric transducers (4 $\times$ 4 mm, 230 kHz)
[25] coated with glycerin and clamped to the bare stem by a three-prong
thumbtack. Signals, of the order of 1 mV, were amplified (gain $10^{3}$) and
recorded in a storage digital oscilloscope. Different transducer positions in
the stems were explored, as well as simultaneous measurement of UAE with two
detectors attached to different points of the stems, providing a method to
trace the origin of the signals (Fig. 1).
Figure 2: Examples of the detected UAE related to cavitation events in corn
stem. (a) Type 1 broadband frequency emission signals detected at room light.
(b) Type 2 low frequency signals detected at room light. (c) Type 1 signals
detected with the laser impinging near the detector. (d) Type 2 signals
detected with the laser impinging far from the detector. (e) and (f) Frequency
spectra of type 1 and type 2 signals detected with laser. See the similarity
of the signals produced with laser and those detected at room light.
## 3 Results
### 3.1 Measurements of spontaneous UAE
In the first experiment (Experiment 1), a transducer was attached to the bare
stem on an internode with 5–7 developed leaves above it. UAE were monitored in
the dark (D), under room light (RL) and under incandescent lamp (IL)
illumination. The experiment was performed with several plants changing the
sequence conditions of light (D–RL–IL; RL–D–IL, etc.). We registered no
emissions in the dark. In experiments 2–3 h long, a rate of 1.15 $\pm$ 0.09
emissions min-1 were detected when the plant was transpiring under ambient
light. This rate was increased to 1.45 $\pm$ 0.15 emissions min-1 when
transpiration was stimulated with an incandescent lamp. The change in the rate
of emissions between RL and IL took place less than a minute after turning the
lamp on or off. When two transducers were attached to the bare stem at the
same height but in different radial positions [Fig. 1(B)], the rate of
emission and the type of signals (see below) were the same for both detectors.
In a second set of experiments performed under room light, and each lasting
ca. 2 h (Experiment 2), two transducers were attached to the bare stem at
different heights. Both transducers registered UAE, but the rate of emissions
was dependent on the transducer position: near the leaves was higher than
closer to the plant base. For instance, when T1 was located at 14 cm from the
plant base and T2 at 35–40 cm, no signals were detected by T1, while 0.3–3.5
emissions min-1 were detected by T2. The amplitude of the signals detected by
each transducer was registered as a function of time, and the UAE were
classified by their form and main frequency. Two types of signals were
identified: those who have a broad band of frequencies up to 0.2 MHz, named
type 1, [Fig. 2(a)], and low frequency signals, with values below 0.075 MHz,
named type 2, [Fig. 2(b)].
### 3.2 Laser induced UAE
With the aim of developing a method to induce UAE in a controlled way, in the
next series of experiments (Experiment 3) we impinged a laser beam at a point
on a corn stem with a transducer attached on the opposite side [Fig. 1(C)]. We
started measuring UAE in the dark, and without laser irradiation. Under these
conditions, no UAE were detected. Then, again in the dark, we irradiated the
stem with the He–Ne red laser, but even at its maximum power, no UAE were
detected. After that, the CW Ar-ion laser was tested at different wavelengths
and powers. We found that with powers up to 600 mW, only the blue line at 488
nm produced results. Under these conditions, when the laser was turned on,
acoustic signals were registered and when it was turned off, the rate of
emission decayed and disappeared after a few seconds (Fig. 3).
Figure 3: Laser induced UAE in corn stem. The beam of a CW Ar ion laser (600
mW) at 488 nm impinges on the stem opposite to the transducer. Grey line: the
laser is on. Black line: the laser is off.
This sequence (switching the laser on and off, always impinging on the same
point of the stem) was repeated with the same qualitative results, although
the rate of UAE decreased with every cycle (in Fig. 3 compare the slope of the
sequence starting at minute 40 with the one starting at minute 55). Even when
the rate of emissions in different plants encompassed a wide range (ca. 2–17
emissions min-1 with the laser on), the same pattern always held (i.e.,
emissions when laser is on, and no emissions a few seconds after the laser is
off). The same behavior was observed when the laser impinged at a right angle
from the transducer axis.
Figure 4: (A) Spontaneous and (B) laser induced UAE in corn stems measured
simultaneously with two transducers attached at the same height of the stem.
In (B) the laser impinged between both transducers. Open triangles: transducer
T1. Open circles: transducer T2.
The signals were classified according to their form and frequencies. Figure
2(c) shows a typical signal generated by the laser in this experiment. As it
can be seen, these signals are similar to the broadband frequency signals [the
type 1 shown in Fig. 2(a)] measured in experiment 2 with room light.
With the aim of comparing spontaneous and laser-induced UAE, we attached two
transducers to the bare stem on opposite sides, at the same height [Experiment
4, Fig. 1(B)]. We first registered acoustic emissions detected simultaneously
by both transducers at room light, without laser irradiation [Fig. 4(A)].
After that, in the dark, we measured the UAE generated after impinging the CW
laser between both detectors, in a direction perpendicular to their axes [Fig.
4(B)]. The characteristic signals observed in both cases were type 1 signals
(broadband frequency signals).
Then, we proceeded to study how the distance between detector and source
modified the rate and shape of the UAE (Experiment 5). Two transducers were
attached to the bare stem: one at 8.5 cm (T1) and the other at 12.5 cm (T2)
from the base. The CW laser beam impinged on different points of the stem.
Points a and b were at the same height of T1 and T2, respectively, but at the
opposite side; point c was between T1 and T2 [Fig. 1(D)].
Figure 5: Laser induced UAE as a function of the transducer position. Two PZT
transducers were attached to the stem at two heights as shown in Fig. 1(D).
Open triangles: transducer T1. Open circles: transducer T2. (A) The Laser beam
impinged near T1. (B) The Laser beam impinged near T2. (C) The Laser beam
impinged between T1 and T2. The arrow in (B) indicates the laser was off.
When the laser beam impinged on a, both transducers detected UAE. Broadband
frequency signals (type 1) were observed with T1 and low frequency signals
(type 2) were observed with T2. Figure 2(d) shows an example of type 2 signals
generated with laser. As it can be seen, these signals are similar to those
detected at room light [Fig. 2(b)].
Besides, the number of emissions detected by T1 was higher than that detected
by T2 [Fig. 5(A)]. When the laser beam impinged on b, once again, both
transducers detected UAE. In this case, T2 detected type 1 signals while T1
detected type 2 signals, and the number of emissions detected by T2 was higher
than that detected by T1 [Fig. 5(B)]. When the laser beam impinged on c, both
transducers simultaneously detected UAE of low frequency similar to those
described as type 2 [Fig. 5(C)].
## 4 Discussion and conclusions
Experiments 1 and 2 show that the spontaneous UAE can be attributed to natural
cavitation events occurring in the xylem vessels of the corn stem: no
emissions were observed when the plant was in the dark. Around 1 emission
min-1 was detected under room light, and a rate ca. 25% higher under the lamp.
This behavior is in agreement with the cohesion-tension theory and current
plant cavitation models. As transpiration rate increases, xylem tension rises
and cavitation events are expected to increase, as it happens in our
experiments. Besides, the UAE signals registered (Fig. 2) were very similar to
those described [12]. These authors demonstrated that these kinds of emissions
are strongly related to cavitation events [12, 13, 26].
In the transpiring plant, the tension developed in the water stream generates
a metastable equilibrium. When liquid water is subjected to a sufficiently low
pressure, this equilibrium can be broken, and form a cavity. This initial
stage of the cavitation phenomenon is termed cavitation inception. When the
plant is in the dark, water in the xylem is slightly under tension at a
pressure value close to atmospheric. Under these conditions, the local
pressure does not fall enough, compared to the saturated vapor pressure, to
produce cavitation inception.
As the CW laser impinges on the stem, this absorbs light and release energy to
the xylem, heating it. This extra energy allows the phase change to gas in the
water column, triggering cavitation inception. In this sense, the physical
process of cavitation inception is similar to boiling, the major difference
being the thermodynamic path which precedes the formation of the vapor. We
found that UAE generated using a CW laser are of the same kind of those
registered on transpiring plants. We can conclude that this method allows, for
the first time, to induce cavitation events in xylem in a controlled and
reproducible way.
Regarding the mechanisms of UAE generation, either natural or laser-induced,
previous work has clearly shown that once cavitation inception is produced,
embolism of the xylem immediately takes place. This means that the formed
cavity remains, and there is no collapse of the void in the water column (as
would occur in the so called inertial cavitation). Then, UAE generation can be
produced by an oscillating source activated by the rupture of the water
column. As mentioned in the introduction, Tyree et al. [12] proposed four
possible UAE sources. The first one, oscillation of hydrogen bonds in water
after tension release, seems unlikely because of its very low magnitude,
undetectable by the kind of transducers we used. The second one, oscillations
caused by a “snap back” of vessel walls, is also unlikely because of the
rigidity of the xylem, and especially hard to explain under laser-induced
cavitation inception in the dark, when xylem tension was nil or very small.
The third one, torus aspiration, is impossible in our case because of the
absence of these structures in corn. Finally, the fourth one, structural
failure in the sapwood, was elegantly rejected by Tyree himself [13], who
exposed xylem to pressure and detected a different kind of emission.
We postulate that another possible source of the UAE must be taken into
account. It is the local oscillation of the liquid–gas interface of the water
column produced by the expansion and compression of the formed cavity, i.e.,
the stress wave generated by rapid bonding energy release. During cavitation
inception, after the cavity expands, it is expected to be compressed almost
immediately by the water column. This “cavity oscillation” starts as a high
frequency burst produced by the disruptive effect of the cavitation inception.
As a consequence, ultrasonic acoustic signals are produced.
In order for cavitation inception to occur, the cavitation “bubbles” generally
need a surface on which they can nucleate. This could be provided by
impurities in the liquid or the xylem walls, or by small undissolved micro-
bubbles within the water, but most likely by air seeding through pit membranes
[4]. These act as capillary valves that allow or prevent air seeding by
adjusting local curvatures and interface positions [27]. Air seeding induced
by the heating at pit membranes under CW laser irradiation should also be
taken into account as an initial stage in laser induced cavitation inception.
The CW laser-induced cavitation opens the opportunity to study embolism in
plants in a controlled manner. It also has the advantage of tracing the
source, allowing the characterization of the signals and studying their
propagation. By directing the laser beam to one point in the stem and
recording acoustic emissions at different distances, we found that when
cavitation was produced near the transducer, broadband frequency emissions
were registered. But, if the transducer was installed further away, the rate
and frequency of the emissions decreased with the distance to the cavitation
source. This means that during signal propagation, absorption by the tissue
takes place (rate decay) as well as frequency filtering. Figures 2(e) and 2(f)
show the frequency spectra of type 1 and type 2 signals. When comparing these
figures, the frequency filtering effect is evident.
Our results confirm the hypothesis by Ritman and Milburn [28], who proposed
that cavitation of xylem sap generally results in the production of a
broadband acoustic emission with lower cut-off frequency determined by the
dimensions of the resonating element. The larger a conduit dimension, the
lower the frequency of its major resonance. Thus, small cavitating elements,
such as corn stem xylem, are expected to produce acoustic signals with a
broadband frequency spectrum. Our results can also explain the observations by
Tyree and Dixon [12] who found and classified UAE of different frequencies
(between 0.1 and 1 MHz). According to our experiments, the different signals
would be generated by cavitation events produced in different regions of the
stem. Broad band frequency signals would come from near the transducer while
low frequency signals would come from regions far from to the transducer.
According to these results, one might use the waveform of the emissions to
determine the location of each cavitation event. In that case, a whole new
field would be opened in the study of hydraulic architecture of plants.
###### Acknowledgements.
The authors are indebted to Dr. H. F. Ranea Sandoval of FCE-UNCBA-Tandil-
Argentina, Professor Silvia E. Braslavsky from Max-Planck-Institut für
Bioanorganische Chemie Mülheim an der Ruhr, Germany and Dr. J. Alvarado-Gil
from CINVESTAV-Unidad, Merida, Merida, Mexico for fruitful comments and
suggestions. This work was partially supported by ANPCyT, UNCPBA, UBA and
UNLP. G.M.B. is member of the Carrera del Investigador Científico CIC-BA, and
R.J.F. of CONICET.
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|
arxiv-papers
| 2012-05-15T14:46:42 |
2024-09-04T02:49:30.928575
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Fern\\'andez, R. J. Fern\\'andez, G. M. Bilmes",
"submitter": "Gabriel Bilmes",
"url": "https://arxiv.org/abs/1205.3398"
}
|
1205.3411
|
# Studies of the decay $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$
B. Storaci∗ on behalf of the LHCb collaboration
The decay mode $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ allows for one of the
theoretically cleanest time dependent measurements of the CKM angle $\gamma$.
This contribution reports the world best branching fraction of this decay
relative to the Cabibbo–favoured mode $B^{0}_{s}\to D_{s}^{-}\pi^{+}$ based on
data sample of 0.37 $\rm fb^{-1}$ proton–proton collisions at $\sqrt{s}=7$ TeV
collected with the LHCb detector in 2011, resulting in $BR(B^{0}_{s}\to
D^{\mp}_{s}K^{\pm})=(1.90\pm
0.12^{stat}\pm{{0.13^{syst}}^{+0.12}_{-0.14}}^{f_{s}/f_{d}})\times 10^{-4}$.
## 1 Motivation
The least precise direct measured parameter of the unitary triangle is the
angle $\gamma$. The high abundance of $b\overline{b}$ pairs, together with an
excellent proper time resolution, an excellent particle identification and
trigger capability to select hadronic final states, allows the LHCb experiment
to determine this parameter through a time dependent analysis using the
$B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ decay. Unlike the flavour-specific decay
$B^{0}_{s}\to D_{s}^{-}\pi^{+}$, the Cabibbo-suppressed decay $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$ proceeds through two different tree-level amplitudes of
similar strength.
These two decay amplitudes can have a large $C\\!P$-violating interference via
$B^{0}_{s}-\bar{B}^{0}_{s}$ mixing, allowing the determination of the CKM
angle $\gamma$ with small theoretical uncertainties through the measurement of
tagged and untagged time-dependent decay rates to both the $D^{-}_{s}K^{+}$
and $D^{+}_{s}K^{-}$ final states $\\!{}^{{\bf?}}$. Although the $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$ decay mode has been observed by the CDF $\\!{}^{{\bf?}}$
and BELLE $\\!{}^{{\bf?}}$ collaborations, at present its branching fraction
is known with an uncertainty around 23% $\\!{}^{{\bf?}}$. Moreover, only the
LHCb experiment has both the necessary decay time resolution and access to
large enough signal yields to perform the time-dependent $C\\!P$ measurement.
## 2 The LHCb experiment
The LHCb detector $\\!{}^{{\bf?}}$ is a single-arm forward spectrometer
covering the pseudo-rapidity range $2<\eta<5$, designed for studying particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system (silicon and straw tube technologies) and a dipole magnet with a
bending power of about $4{\rm\,Tm}$. The tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at 5 GeV to 0.6% at 100 GeV, an
impact parameter resolution of 20$\mathrm{\mu m}$ for tracks with high
transverse momentum, and a decay time resolution of 50 fs. Charged hadrons are
identified using two ring-imaging Cherenkov detectors. Calorimeter and muon
systems provide the identification of photon, electron, hadron and muon
candidates.
The analysis is based on a sample of $pp$ collisions corresponding to an
integrated luminosity of 0.37 fb-1, collected at the LHC in 2011 at a centre-
of-mass energy $\sqrt{s}=7$ TeV.
## 3 Selection
The channels considered as signal in this document are the decays $B^{0}\to
D^{-}\pi^{+}$, $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ and the $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$. These decays are all characterized by a similar topology
and therefore the same trigger, stripping and offline selection are used to
select them, minimizing the efficiency corrections.
The LHCb trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies full
event reconstruction.
The decays of $B$ mesons can be distinguished from the background by using
variables such as the $p_{T}$ and impact parameter $\chi^{2}$ of the $B$, $D$,
and the final state particles with respect to the primary interaction. In
addition, the vertex quality of the $B$ and $D$ candidates, the $B$ lifetime,
and the angle between the $B$ momentum vector and the vector joining the $B$
production and decay vertices are used in the selection. In order to remove
charmless background a requirement in the flight distance $\chi^{2}$ of the
$D^{-}_{s}$ from the $B^{0}_{s}$ is applied $\\!{}^{{\bf?}}$.
Further suppression of combinatorial backgrounds is achieved using a gradient
boosted decision tree technique $\\!{}^{{\bf?}}$. The optimal working point is
evaluated directly from a sub-sample of $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ events
in data selected using particle identification and trigger requirements. The
chosen figure of merit is the significance of the $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$ signal, scaled according to the Cabibbo suppression
relative to the $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ signal, with respect to the
combinatorial background. Multiple candidates occur in about $2\%$ of the
events and in such cases a single candidate is selected at random.
Particle identification (PID) criteria serve two purposes: separate the
Cabibbo-favoured from the Cabibbo-suppressed modes (when applied to the
bachelor particle) and suppress the misidentified backgrounds which have the
same bachelor particle (when applied to the decay products of the $D^{-}_{s}$
or $D^{-}$). All PID criteria are based on the differences in log-likelihood
(DLL) between the kaon, proton, or pion hypotheses. Their efficiencies are
obtained from calibration samples of $D^{*+}\to(D^{0}\to K^{-}\pi^{+})\pi^{+}$
and $\Lambda\to p\pi^{-}$ signals, which are themselves selected without any
PID requirements. These samples are split according to the magnet polarity,
binned in momentum and $p_{\rm T}$, and then reweighted to have the same
momentum and $p_{\rm T}$ distributions as the signal decays under study.
## 4 Mass fit
The three signal decays are distinguished with particle identification
requirements applied at the final stage of the analysis. The signal yields are
obtained from extended maximum likelihood unbinned fits to the data. In order
to achieve the highest sensitivity, the sample is fitted separately for the
magnet up and down data. The signal line shapes are taken from simulated
signal events. A mass constraint on the $D_{(s)}$ meson mass is used in order
to improve the $B$ mass resolution.
The shape of the signal mass distribution is obtained fitting a double Crystal
Ball function which consist of a common Gaussian with two exponential tails,
one to describe the radiative tail present in the lower mass region and the
other one describing the higher mass region where only the detector resolution
is involved.
A common signal shape describes properly both polarities so a simultaneous fit
with a common mean and width of the double crystal ball function is used. The
mean is free to float in all the fits, while the width is fixed in the
$B^{0}_{s}$-modes from the result obtained in data in the $B^{0}\to
D^{-}\pi^{+}$ fit corrected for the $B^{0}-B^{0}_{s}$ differences observed in
the simulation samples. The other parameters are fixed from simulation.
Four sources of backgrounds are present: the combinatorial background, the
charmless background, the fully reconstructed (misidenfied) background and the
partially reconstructed background. The offline selection is optimized to
reduce the combinatorial background contribution, and the remaining
contamination is fitted with an exponential shape for the modes with a
bachelor pion, while it is taken flat for the $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$ mode. The validity of this assumption is checked with
wrong-side samples and accounted for in the systematic uncertainty associated
to the fit model. The other two background categories have different
components in the three fits and therefore are explained separately. In all
the fits the partially reconstructed background shapes are obtained fitting a
non-parametric function on samples of simulated events generated in the
specific exclusive modes, corrected for the observed mass shifts, momentum
spectra, and particle identification efficiencies observed in data when it is
needed. The yields are left free when possible or a gaussian constraint is
applied if an expected amount is computable.
In the $B^{0}\to D^{-}\pi^{+}$ mass fit the two relevant sources of partially
reconstructed background are the $B^{0}\to D^{*-}\pi^{+}$ and $B^{0}\to
D^{-}\rho^{+}$ decays and their yields are left free to float in the fit.
In the $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ mass fit the misidentified $B^{0}\to
D^{-}\pi^{+}$ shape is fixed from data using a reweighting procedure to
account for misidentification momentum dependency $\\!{}^{{\bf?}}$ . The
number of expected events is computed from the yield obtained in the $B^{0}\to
D^{-}\pi^{+}$ fit and the PID efficiency obtained from calibration sample. Its
yield is therefore constrained to this expected value with a 10% uncertainty.
The $B^{0}\to D^{-}_{s}\pi^{+}$ yield is calculated based on the $B^{0}\to
D^{-}_{s}\pi^{+}$ branching fraction $\\!{}^{{\bf?}}$, the measured LHCb value
of $f_{s}/f_{d}$ $\\!{}^{{\bf?}}$, and the value of the $B^{0}_{s}\to
D^{-}_{s}\pi^{+}$ branching fraction $\\!{}^{{\bf?}}$. The shape used is the
same of the signal $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ with the mean position
fixed from the $B^{0}\to D^{-}\pi^{+}$ fit. The partially reconstructed
backgrounds relevant for this fit are the decays $B^{0}_{s}\to
D^{*-}_{s}\pi^{+}$ and the $B^{0}_{s}\to D^{-}_{s}\rho^{+}$. Due to the large
correlation between these two components, a gaussian constraint is used for
the fraction of these two backgrounds. The fraction is assumed to be the same
as in the $B^{0}$ case, while the variation is assumed to correspond to 20%
$SU(3)$ breaking.
In the $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ mass fit there are numerous
reflections which contribute to the mass distribution. The most important
reflection is the misidentified $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ decay. Its
shape is fixed from data using the reweighting procedure while the yield is
left free to float. The same procedure is also applied on simulation sample to
extract the shape of the $B^{0}\to D^{-}K^{+}$ misidentified background. The
yield is constraint according to the expected $B^{0}\to D^{-}K^{+}$ yield
corrected for the PID efficiency. In addition, there is potential cross-feed
from partially reconstructed modes with a misidentified pion such as
$B^{0}_{s}\to D_{s}^{-}\rho^{+}$, as well as several small contributions from
partially reconstructed backgrounds with similar mass shapes. The yields of
these modes, whose branching fractions are known or can be estimated are
constrained to values obtained based on criteria such as relative branching
fractions and reconstruction efficiencies and PID probabilities
$\\!{}^{{\bf?}}$. The fit results are shown in Fig. 1
Figure 1: Mass distribution of the $B^{0}\to D^{-}\pi^{+}$ candidates (top-
left), $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ candidates (top-right) and
$B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ (bottom). The stacked background shapes
follow the same top-to-bottom order in the legend and in the plot. For
illustration purposes the plot includes events from both magnet polarities.
## 5 Systematic uncertainty
Systematic uncertainties related to the fit are evaluated by generating large
sets of simulated experiments using the nominal fit, and then fitting them
with a model where certain parameters are varied. The sources of systematic
uncertainty considered for the fit are signal widths, the slope of the
combinatorial backgrounds, and constraints placed on specific backgrounds. The
largest deviations are due to the signal widths and the fixed slope of the
combinatorial background in the $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ fit. The
systematic uncertainty related to PID is evaluated using simulated signal and
calibration samples. The observed signal yields are corrected by the
difference observed in the (non-PID) selection efficiencies of different modes
as measured from simulated events. A systematic uncertainty is assigned on the
ratio to account for percent level differences between the data and the
simulation. These are dominated by the simulation of the hardware trigger. A
total systematic uncertainty of $3.9\%$ for the ratio $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}/B^{0}_{s}\to D^{-}_{s}\pi^{+}$, of $3.4\%$ for the ratio
$B^{0}_{s}\to D^{-}_{s}\pi^{+}\ B^{0}\to D^{-}\pi^{+}$ and of $4.6\%$ for the
ratio $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}\ B^{0}\to D^{-}\pi^{+}$ is found.
## 6 Results
The sum of the $B^{0}_{s}\to D_{s}^{-}K^{+}$ and $B^{0}_{s}\to D_{s}^{+}K^{-}$
branching fractions relative to $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ is obtained by
correcting the raw signal yields for PID and selection efficiency differences
and it leads to
$\frac{BR(B^{0}_{s}\to D^{\mp}_{s}K^{\pm})}{BR(B^{0}_{s}\to
D^{-}_{s}\pi^{+})}=0.0646\pm 0.0043\pm 0.0025\;,$ (1)
where the first uncertainty is statistical and the second is the total
systematic uncertainty.
The relative yields of the three decays $B^{0}\to D^{-}\pi^{+}$, $B^{0}_{s}\to
D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ are used to extract
the branching fraction of $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\to
D^{\mp}_{s}K^{\pm}$ together with the recent $f_{s}/f_{d}$ measurement from
semileptonic decays $\\!{}^{{\bf?}}$, leading to
$\displaystyle BR(B^{0}_{s}\to D^{-}_{s}\pi^{+})$ $\displaystyle=$
$\displaystyle(2.95\pm 0.05\pm 0.17^{+0.18}_{-0.22})\times 10^{-3}\;,$ (2)
$\displaystyle BR(B^{0}_{s}\to D^{\mp}_{s}K^{\pm})$ $\displaystyle=$
$\displaystyle(1.90\pm 0.12\pm 0.13^{+0.12}_{-0.14})\times 10^{-4}\;,$ (3)
where the first uncertainty is statistical, the second is the experimental
systematic plus the uncertainty arising from the $B^{0}\to D^{-}\pi^{+}$
branching fraction, and the third is the uncertainty (statistical and
systematic) from the semileptonic $f_{s}/f_{d}$ measurement. Both measurements
are significantly more precise than the existing world averages
$\\!{}^{{\bf?}}$.
## References
## References
* [1] R. Fleischer, Nucl. Phys. B 671, 0 (2003).
* [2] T. Aaltonen et al, PRL 103, 19 (2009)
* [3] R. Louvot et al, Phys. Rev. Lett. 102, 2 (2009)
* [4] K. Nakamura et al, J.Phys. G 37, 7 (2010)
* [5] A. A. Alves Jr. et al, JINST 3, 8 (2008)
* [6] R. Aaij et al, arXiv:1204.1237v1
* [7] A. Hoecker et al, PoS ACAT, 040 (2007)
* [8] R. Aaij et al, Phys. Rev. Lett. 107, 21 (2011)
* [9] R. Aaij et al, Phys. Rev. D 85, 3 (2012)
|
arxiv-papers
| 2012-05-15T15:21:28 |
2024-09-04T02:49:30.935065
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Barbara Storaci",
"submitter": "Barbara Storaci",
"url": "https://arxiv.org/abs/1205.3411"
}
|
1205.3422
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-129 LHCb-PAPER-2012-011 May 15, 2012
Measurement of the isospin asymmetry in $B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The isospin asymmetries of $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays and the
partial branching fractions of $B^{0}\rightarrow K^{0}\mu^{+}\mu^{-}$ and
$B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$ are measured as a function of the di-
muon mass squared $q^{2}$ using an integrated luminosity of 1.0 fb-1 collected
with the LHCb detector. The $B\rightarrow K\mu^{+}\mu^{-}$ isospin asymmetry
integrated over $q^{2}$ is negative, deviating from zero with over 4 $\sigma$
significance. The $B\rightarrow K^{*}\mu^{+}\mu^{-}$ decay measurements are
consistent with the Standard Model prediction of negligible isospin asymmetry.
The observation of the decay $B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$ is reported with 5.7 $\sigma$ significance. Assuming that
the branching fraction of $B^{0}\rightarrow K^{0}\mu^{+}\mu^{-}$ is twice that
of $B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$, the
branching fractions of $B^{0}\rightarrow K^{0}\mu^{+}\mu^{-}$ and
$B\rightarrow K^{*+}\mu^{+}\mu^{-}$ are found to be
($0.31^{+0.07}_{-0.06})\times 10^{-6}$ and ($1.16\pm 0.19)\times 10^{-6}$,
respectively.
Submitted to Journal of High Energy Physics
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,
F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10, F.
Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T.
Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,
D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42,
R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J.
Lefrançois7, O. Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9,
M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35,
J. Magnin1, S. Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev27, E.
Maurice6, B. Maynard53, A. Mazurov16,30,35, J. McCarthy42, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A. Milanes13, M.-N.
Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski23, R.
Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B.
Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I.
Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M.
Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A. Nomerotski52,35, A.
Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48,
O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P.
Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35,
A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M. Plo
Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D.
Popov10, B. Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V.
Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43, B.
Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M.
Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa
Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T.
Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T.
Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, 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
## 1 Introduction
The flavour-changing neutral current decays $B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-}$ are forbidden at tree level in the Standard Model (SM).
Such transitions must proceed via loop or box diagrams and are powerful probes
of physics beyond the SM. Predictions for the branching fractions of these
decays suffer from relatively large uncertainties due to form factor
estimates. Theoretically clean observables can be constructed from ratios or
asymmetries where the leading form factor uncertainties cancel. The $C\\!P$
averaged isospin asymmetry ($A_{\rm I}$) is such an observable. It is defined
as
$\begin{split}A_{\rm I}=\frac{\Gamma(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})-\Gamma(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}{\Gamma(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})+\Gamma(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}\phantom{A_{\rm I}}\phantom{,}\\\
=\frac{\mathcal{B}(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})-\frac{\tau_{0}}{\tau_{+}}\mathcal{B}(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}{\mathcal{B}(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})+\frac{\tau_{0}}{\tau_{+}}\mathcal{B}(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})},\end{split}$ (1)
where $\Gamma(B\rightarrow f)$ and $\mathcal{B}(B\rightarrow f)$ are the
partial width and branching fraction of the $B\rightarrow f$ decay and
$\tau_{0}/\tau_{+}$ is the ratio of the lifetimes of the $B^{0}$ and $B^{+}$
mesons.111Charge conjugation is implied throughout this paper. For
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$, the SM prediction for $A_{\rm I}$ is
around $-1\%$ in the di-muon mass squared ($q^{2}$) region below the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance, apart from the very
low $q^{2}$ region where it rises to $\mathcal{O}(10\%)$ as $q^{2}$ approaches
zero [1]. There is no precise prediction for $A_{\rm I}$ in the
$B\\!\rightarrow K\mu^{+}\mu^{-}$ case, but it is also expected to be close to
zero. The small isospin asymmetry predicted in the SM is due to initial state
radiation of the spectator quark, which is different between the neutral and
charged decays. Previously, $A_{\rm I}$ has been measured to be significantly
below zero in the $q^{2}$ region below the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance [2, 3]. In
particular, the combined $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ isospin asymmetries measured by the
BaBar experiment were 3.9 $\sigma$ below zero. For $B\\!\rightarrow
K^{*}\mu^{+}\mu^{-}$, $A_{\rm I}$ is expected to be consistent with the
$B\rightarrow K^{*0}\gamma$ measurement of $5\pm 3\%$ [4] as $q^{2}$
approaches zero. No such constraint is present for $B\\!\rightarrow
K\mu^{+}\mu^{-}$.
The isospin asymmetries are determined by measuring the differential branching
fractions of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$,
$B^{0}\\!\rightarrow(K^{*0}\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow(K^{*+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})\mu^{+}\mu^{-}$; the decays involving a
$K^{0}_{\rm\scriptscriptstyle L}$ or $\pi^{0}$ are not considered. The
$K^{0}_{\rm\scriptscriptstyle S}$ meson is reconstructed via the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay mode. The
signal selections (Section 3) are optimised to provide the lowest overall
uncertainty on the isospin asymmetries; this leads to a very tight selection
for the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow(K^{*0}\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ channels
where signal yield is sacrificed to achieve overall uniformity with the
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow(K^{*+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})\mu^{+}\mu^{-}$ channels, respectively. In order to convert a signal
yield into a branching fraction, the four signal channels are normalised to
the corresponding $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ channels (Section 5). The relative normalisation in each
$q^{2}$ bin is performed by calculating the relative efficiency between the
signal and normalisation channels using simulated events. The normalisation of
$B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ assumes that
$\mathcal{B}(B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-})=2\mathcal{B}(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-})$. Finally, $A_{\rm I}$ is
determined by simultaneously fitting the $K^{(*)}\mu^{+}\mu^{-}$ mass
distributions for all signal channels. Confidence intervals are estimated for
$A_{\rm I}$ using a profile likelihood method (Section 7). Systematic
uncertainties are included in the fit using Gaussian constraints.
## 2 Experimental setup
The measurements described in this paper are performed with
1.0$\mbox{\,fb}^{-1}$ of $pp$ collision data collected with the LHCb detector
at the CERN LHC during 2011. The LHCb detector [5] is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector (VELO)
surrounding the $pp$ interaction region, a large-area silicon-strip detector
(TT) located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. For this analysis, candidate events are first
required to pass a hardware trigger which selects muons with a transverse
momentum, $\mbox{$p_{\rm T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for
one muon, and 0.56 and 0.48${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for two
muons. In the subsequent software trigger [6], at least one of the final state
particles is required to have both $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP $>100\,\upmu\rm m$ with
respect to all of the primary proton-proton interaction vertices in the event.
Finally, the tracks of two or more of the final state particles are required
to form a vertex which is significantly displaced from the primary vertices in
the event.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [7] with a
specific LHCb configuration [8]. Decays of hadronic particles are described by
EvtGen [9] in which final state radiation is generated using Photos [10]. The
EvtGen physics model used is based on Ref. [11]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [12, *Agostinelli:2002hh] as described in Ref. [14].
## 3 Event selection
Candidates are reconstructed with an initial cut-based selection, which is
designed to reduce the dataset to a manageable level. Channels involving a
$K^{0}_{\rm\scriptscriptstyle S}$ meson are referred to as
$K^{0}_{\rm\scriptscriptstyle S}$ channels whereas those with a $K^{+}$ meson
are referred to as $K^{+}$ channels. Only events which are triggered
independently of the $K^{+}$ candidate are accepted. Therefore, apart from a
small contribution from candidates which are triggered by the
$K^{0}_{\rm\scriptscriptstyle S}$ meson, the $K^{0}_{\rm\scriptscriptstyle S}$
and the $K^{+}$ channels are triggered in a similar way. The initial selection
places requirements on the geometry, kinematics and particle identification
(PID) information of the signal candidates. Kaons are identified using
information from the RICH detectors, such as the difference in log-likelihood
(DLL) between the kaon and pion hypothesis, $\mathrm{DLL}_{K\pi}$. Kaon
candidates are required to have $\mathrm{DLL}_{K\pi}$ $>1$, which has a kaon
efficiency of $\sim 85\%$ and a pion efficiency of $\sim 10\%$. Muons are
identified using the amount of hits in the muon stations combined with
information from the calorimeter and RICH systems. The muon PID efficiency is
around 90%. Candidate $K^{0}_{\rm\scriptscriptstyle S}$ are required to have a
di-pion mass within 30 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
nominal $K^{0}_{\rm\scriptscriptstyle S}$ mass and $K^{*}$ candidates are
required to have an mass within 100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the nominal $K^{*}$ mass. At this stage, the $K^{0}_{\rm\scriptscriptstyle
S}$ channels are split into two categories depending on how the pions from the
$K^{0}_{\rm\scriptscriptstyle S}$ decay are reconstructed. For decays where
both pions have hits inside the VELO and the downstream tracking detectors the
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are classified as long (L). If
the daughter pions are reconstructed without VELO hits (but still with TT hits
upstream of the magnet) they are classified as downstream (D)
$K^{0}_{\rm\scriptscriptstyle S}$ candidates. Separate selections are applied
to the L and D categories in order to maximise the sensitivity. The selection
criteria described in the next paragraph refer to the
$K^{0}_{\rm\scriptscriptstyle S}$ channels.
After the initial selection, the L category has a much lower level of
background than the D category. For this reason simple cut-based selections
are applied to the former, whereas multivariate selections are employed for
the latter. Both $B^{0}$ and $B^{+}$ L selections require the
$K^{0}_{\rm\scriptscriptstyle S}$ decay time to be greater than 3${\rm\,ps}$,
and for the IP $\chi^{2}$ to be greater than 10 when the IP of the
$K^{0}_{\rm\scriptscriptstyle S}$, with respect to the PV, is forced to be
zero. The $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ L selection requires that
$K^{0}_{\rm\scriptscriptstyle S}$ $p_{\rm T}$ $>$
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $B$ $p_{\rm T}$ $>$
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $K^{0}_{\rm\scriptscriptstyle S}$
mass window is also tightened to
$\pm$20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $B^{+}\\!\rightarrow
K^{*+}\mu^{+}\mu^{-}$ L selection requires that the pion from the $K^{*+}$ has
an IP $\chi^{2}>30$. Multi-variate selections are applied to the D categories
using a boosted decision tree (BDT) [15] which uses geometrical and kinematic
information of the $B$ candidate and of its daughters. The most discriminating
variables according to the $B^{0}$ and $B^{+}$ BDTs are the
$K^{0}_{\rm\scriptscriptstyle S}$ $p_{\rm T}$ and the angle between the $B$
momentum and its line of flight (from the primary vertex to the decay vertex).
The BDTs are trained and tested on simulated events for the signal and data
for the background. The simulated events have been corrected to match the data
as described in Sect. 5. All the variables used in the BDTs are well described
in the simulation after correction. The background sample used is 25% of $B$
candidates which have $|m_{K^{(*)}\mu^{+}\mu^{-}}-m_{B}|$ $>$
60${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where $m_{B}$ is obtained from
fits to the appropriate $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ normalisation channel. These data are excluded from the
analysis. The selection based on the BDT output maximises the metric
$S/\sqrt{S+B}$, where $S$ and $B$ are the expected signal and background
yields, respectively.
The $K^{+}$ channels have, as far as possible, the same selection criteria as
used to select the $K^{0}_{\rm\scriptscriptstyle S}$ channels. The cut-based
selections applied to the L categories have the $K^{0}_{\rm\scriptscriptstyle
S}$ specific variables (e.g. $K^{0}_{\rm\scriptscriptstyle S}$ decay time)
removed and the remaining requirements are applied to the $K^{+}$ channels.
The BDTs trained on the D categories contain variables which can be applied to
both $K^{0}_{\rm\scriptscriptstyle S}$ and $K^{+}$ candidates and the BDTs
trained on the $K^{0}_{\rm\scriptscriptstyle S}$ channels are simply applied
to the corresponding $K^{+}$ channels. The $K^{+}$ channels are therefore also
split into two different categories, one of which has the L selection applied,
while the other one has the D selection applied. The overlap of events between
these categories induces a correlation between the L and D categories for the
$K^{+}$ channels. This correlation is accounted for in the fit to $A_{\rm I}$.
The final selection reduces the combinatorial background remaining after the
initial selection by a factor of 5–20, while retaining 60–90% of the signal,
depending on the category and decay mode. It is ineffective at reducing
background from fully reconstructed $B$ decays, where one or more final state
particles have been misidentified. Additional selection criteria are therefore
applied. For the $K^{0}_{\rm\scriptscriptstyle S}$ channels, the $\mathchar
28931\relax\rightarrow p\pi^{-}$ decay can be mistaken for a
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay if the proton
is misidentified as a pion. If one of the pion daughters from the
$K^{0}_{\rm\scriptscriptstyle S}$ candidate has a $\mathrm{DLL}_{p\pi}$ $>$
10, the proton mass hypothesis is assigned to it. For the L(D) categories, if
the $p\pi^{-}$ mass is within 10(15)${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the nominal $\mathchar 28931\relax$ mass the candidate is rejected. This
selection eliminates background from $\mathchar
28931\relax^{0}_{b}\\!\rightarrow(\mathchar 28931\relax\rightarrow
p\pi^{-})\mu^{+}\mu^{-}$ which peaks above the $B$ mass. For the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay, the same peaking background
vetoes are used as in Ref. [16], which remove contaminations from
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays where the kaon and pion are
swapped. Finally, for the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay,
backgrounds from $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B\rightarrow\psi{(2S)}K^{+}$ are present, where the $K^{+}$
and $\mu^{+}$ candidates are swapped. If a candidate has a $K^{+}\mu^{-}$
track combination consistent with originating from a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$ resonance, the
kaon is required to be inside the acceptance of the muon system but to have
insufficient hits in the muon stations to be classified as a muon. These
vetoes remove less than 1% of the signal and reduce peaking backgrounds to a
negligible level.
Figure 1: Mass of the di-muon versus the mass of the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates. Only the di-muon mass region close to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ masses is
shown. The lines show the boundaries of the regions which are removed. Regions
(a)–(c) are explained in the text.
The mass distribution of $B$ candidates is shown versus the di-muon mass for
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ data in Fig. 1. The other signal
channels have similar distributions, but with a smaller number of events. The
excess of candidates seen as horizontal bands around
3090${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
3690${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are due to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ decays,
respectively. These events are removed from the signal channels by excluding
the di-muon regions in the ranges
2946$-$3181${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
3586$-$3766${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. If a $B$ candidate has
an mass below 5220${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ the veto is
extended to 2800$-$3181${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
3450$-$3766${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to eliminate candidates
for which the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or the
$\psi{(2S)}$ decay undergoes final state radiation. Such events are shown in
Fig. 1 as regions (a). In a small fraction of events, the di-muon mass is
poorly reconstructed. This causes the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\psi{(2S)}$ decay to leak into the region just above the $B$
mass. These events are shown in Fig. 1 as regions (b). The veto is extended to
2946$-$3250${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
3586$-$3816${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in the candidate $B$
mass region from 5330$-$5460${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
eliminate these events. These vetoes largely remove the charmonium resonances
and reduce the combinatorial background. Regions (c) in Fig. 1 are composed of
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}X$ and
$B\rightarrow\psi{(2S)}K^{+}X$ decays where $X$ is not reconstructed. In the
subsequent analysis only candidates with masses above
5170${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are included to avoid
dependence on the shape of this background.
## 4 Signal yield determination
Table 1: Signal yields of the $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays. The upper bound of the highest $q^{2}$ bin, $q^{2}_{\mathrm{max}}$, is 19.3${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and 23.0${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ and $B\\!\rightarrow K\mu^{+}\mu^{-}$, respectively. $q^{2}$ range | $K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ | $K^{+}\mu^{+}\mu^{-}$ | $K^{*+}\mu^{+}\mu^{-}$ | $K^{*0}\mu^{+}\mu^{-}$
---|---|---|---|---
$[{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}]$ | L | D | L + D | L | D | L + D
$\phantom{0}0.05-\phantom{0}2.00$ | $\phantom{1}1\pm 2$ | $\phantom{1}2\pm 3\phantom{1}$ | $\phantom{1}135\pm 13$ | $4\pm 3$ | $\phantom{1}5\pm 4\phantom{1}$ | $108\pm 11$
$\phantom{0}2.00-\phantom{0}4.30$ | $\phantom{1}2\pm 3$ | $-1\pm 3\phantom{1}$ | $\phantom{1}175\pm 16$ | $3\pm 2$ | $\phantom{1}5\pm 3\phantom{1}$ | 0$53\pm\phantom{1}9$
$\phantom{0}4.30-\phantom{0}8.68$ | $\phantom{1}9\pm 4$ | $16\pm 6\phantom{1}$ | $\phantom{1}303\pm 22$ | $4\pm 3$ | $17\pm 6\phantom{1}$ | $203\pm 17$
$10.09-12.86$ | $\phantom{1}4\pm 3$ | $10\pm 4\phantom{1}$ | $\phantom{1}214\pm 18$ | $4\pm 3$ | $15\pm 5\phantom{1}$ | $128\pm 14$
$14.18-16.00$ | $\phantom{1}3\pm 2$ | $\phantom{1}3\pm 3\phantom{1}$ | $\phantom{1}166\pm 15$ | $5\pm 3$ | $\phantom{1}4\pm 3\phantom{1}$ | 0$90\pm 10$
$16.00-\phantom{1}q^{2}_{\mathrm{max}}$ | $\phantom{1}5\pm 3$ | $\phantom{1}4\pm 3\phantom{1}$ | $\phantom{1}257\pm 19$ | $2\pm 1$ | $\phantom{1}4\pm 3\phantom{1}$ | 0$80\pm 11$
$\phantom{0}1.00-\phantom{0}6.00$ | $\phantom{1}8\pm 4$ | $\phantom{1}3\pm 6\phantom{1}$ | $\phantom{1}356\pm 23$ | $5\pm 3$ | $15\pm 5\phantom{1}$ | $155\pm 15$
$\phantom{0}0.05-\phantom{1}q^{2}_{\mathrm{max}}$ | $25\pm 8$ | $35\pm 11$ | $1250\pm 42$ | $23\pm 6$ | $53\pm 10$ | $673\pm 30$
Figure 2: Mass distributions and fits of the signal channels integrated over
the full $q^{2}$ region. For the $K^{0}_{\rm\scriptscriptstyle S}$ channels,
the plots are shown separately for the L and D $K^{0}_{\rm\scriptscriptstyle
S}$ reconstruction categories, (a,b) and (c,d) respectively. The signal
component is shown by the dashed line, the partially reconstructed component
in 2 and 2 is shown by the dotted line while the solid line shows the entire
fit model.
The yields for the signal channels are determined using extended unbinned
maximum likelihood fits to the $K^{(*)}\mu^{+}\mu^{-}$ mass in the range
5170–5700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. These fits are performed
in seven $q^{2}$ bins and over the full range as shown in Table 1. The results
of the fits integrated over the full $q^{2}$ range are shown in Fig. 2. After
selection, the mass of $K^{0}_{\rm\scriptscriptstyle S}$ candidates is
constrained to the nominal $K^{0}_{\rm\scriptscriptstyle S}$ mass. The signal
component is described by the sum of two Crystal Ball functions [17] with
common peak and tail parameters, but different widths. The shape is taken to
be the same as the $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ normalisation channels. The combinatorial background is fitted
with a single exponential function. As stated in Sect. 3, part of the
combinatorial background is removed by the charmonium vetoes. This is
accounted for by scaling the remaining background. For the $B\\!\rightarrow
K\mu^{+}\mu^{-}$ decays, a component arising mainly from partially
reconstructed $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decays is present at
masses below the $B$ mass. This partially reconstructed background is
characterised using a threshold model detailed in Ref. [18]. The shape of the
partial reconstruction component is again assumed to be the same as for the
normalisation channels. For the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
channel, the impact of this component is negligible due to the relatively high
signal and low background yields. For the $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ channel, the amount of
partially reconstructed decays is found to be less than $25\%$ of the total
combinatorial background in the fit range.
The signal-shape parameters are allowed to vary in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ mass fits and are subsequently fixed
for the $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$
mass fits when calculating the significance. The significance $\sigma$ of a
signal $S$ for $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$ is defined as
$\sigma^{2}=2\mathrm{ln}\mathcal{L}^{\textrm{L}}(S)+2\mathrm{ln}\mathcal{L}^{\textrm{D}}(S)-2\mathrm{ln}\mathcal{L}^{\textrm{L}}(0)-2\mathrm{ln}\mathcal{L}^{\textrm{D}}(0)$
where $\mathcal{L}^{\textrm{L,D}}(S)$ and $\mathcal{L}^{\textrm{L,D}}(0)$ are
the likelihoods of the fit with and without the signal component,
respectively. The $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$ channel is observed with a significance of 5.7 $\sigma$.
## 5 Normalisation
In order to simplify the calculation of systematic uncertainties, each signal
mode is normalised to the
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ channel,
where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays into two
muons. These decays have well measured branching fractions which are
approximately two orders of magnitude higher than those of the signal decays.
Each normalisation channel has similar kinematics and the same final state
particles as the signal modes.
Figure 3: Efficiency of the $K^{0}_{\rm\scriptscriptstyle S}$ channels with
respect to the $K^{+}$ channels for (left) $B\\!\rightarrow K\mu^{+}\mu^{-}$
and (right) $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$, calculated using the
simulation. The efficiencies are shown for both L and D
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories and include the
visible branching fraction of $K^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$. The error bars are not visible as they are
smaller than the marker size.
The relative efficiency between signal and normalisation channels is estimated
using simulated events. After smearing the IP resolution of all tracks by 20%,
the IP distributions of candidates in the simulation and data agree well. The
performance of the PID is studied using the decay
$D^{*+}\\!\rightarrow(D^{0}\\!\rightarrow\pi^{+}K^{-})\pi^{+}$, which provides
a clean source of kaons to study the kaon PID efficiency, and a _tag-and-
probe_ sample of $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ to study the muon PID efficiency. The simulation is reweighted to
match the PID performance of the data.
Integrating over $q^{2}$, the relative efficiency between the signal and
normalisation channels is between 70 and 80% depending on the decay mode and
category. The relative efficiency includes differences in the geometrical
acceptance, as well as the reconstruction, selection and trigger efficiencies.
Most of these effects cancel in the efficiency ratio between
$K^{0}_{\rm\scriptscriptstyle S}$ and $K^{+}$ channels, as shown in Fig. 3.
The dominant effect remaining is due to the $K^{0}_{\rm\scriptscriptstyle S}$
reconstruction efficiency, which depends on the $K^{0}_{\rm\scriptscriptstyle
S}$ momentum. At low $q^{2}$, the efficiency for $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ (D) decreases with respect to
that for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ due to the high
$K^{0}_{\rm\scriptscriptstyle S}$ momentum in this region. This results in the
$K^{0}_{\rm\scriptscriptstyle S}$ meson more often decaying beyond the TT and
consequently it has a lower reconstruction efficiency. This effect is not seen
in the $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ D category as the
$K^{0}_{\rm\scriptscriptstyle S}$ typically has lower momentum in this decay
and so the $K^{0}_{\rm\scriptscriptstyle S}$ reconstruction efficiency is
approximately constant across $q^{2}$. This $K^{0}_{\rm\scriptscriptstyle S}$
reconstruction effect is also seen in the L category for both modes but is
partially compensated by the fact that the $K^{0}_{\rm\scriptscriptstyle S}$
daughters can cause the event to be triggered, which increases the trigger
efficiency with respect to the $K^{+}$ channels at low $q^{2}$. Summed over
both the L and D categories, the efficiency of the decays involving a $K^{0}$
meson is approximately 10% with respect to those involving a charged kaon.
This is partly due to the visible branching fraction of $K^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ ($\sim$30%) and
partly due to the lower reconstruction efficiency of the
$K^{0}_{\rm\scriptscriptstyle S}$ due to the long lifetime and the need to
reconstruct an additional track ($\sim$30%). The relative efficiency between
the L and D signal categories is cross-checked by comparing the ratio for the
$B\\!\rightarrow\psi{(2S)}K^{(*)}$ decay to the corresponding ratio for the
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ decays
seen in data. The results agree within the statistical accuracy of 5%.
## 6 Systematic uncertainties
Gaussian constraints are used to include all systematic uncertainties in the
fits for $A_{\rm I}$ and the branching fractions. In most cases the dominant
systematic uncertainty is that from the branching fraction measurements of the
normalisation channels, ranging from 3 to 6%. There is also a statistical
uncertainty on the yield of the normalisation channels, which is in the range
0.5–2.0%, depending on the channel.
The finite size of the simulation samples introduces a statistical uncertainty
on the relative efficiency and leads to a systematic uncertainty in the range
0.8–2.5% depending on $q^{2}$ and decay mode.
The relative tracking efficiency between the signal and normalisation channels
is corrected using data. The statistical precision of these corrections leads
to a systematic uncertainty of $\sim$ 0.2% per long track. The differences
between the downstream tracking efficiency between the simulation and data are
expected to mostly cancel in the normalisation procedure. A conservative
systematic uncertainty of 1% per downstream track is assigned for the
variation across $q^{2}$.
The PID efficiency is derived from data, and its corresponding systematic
uncertainty arises from the statistical error associated with the PID
efficiency measurements. The uncertainty on the relative efficiency is
determined by randomly varying PID efficiencies within their uncertainties,
and recomputing the relative efficiency. The resulting uncertainty is found to
be negligible.
The trigger efficiency is calculated using the simulation. Its uncertainty
consists of two components, one associated with the trigger efficiency of the
$K^{0}_{\rm\scriptscriptstyle S}$ meson, and one associated with the trigger
efficiency of the muons (and pion from the $K^{*}$). For the muons and pion
the uncertainty is obtained using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
in data that are triggered independently of the signal. These candidates are
used to calculate the trigger efficiency and are compared to the efficiency
calculated using the same method in simulation. The difference is found to be
$\sim 2\%$ for both $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ decays and is assigned as a systematic uncertainty. This
uncertainty is assumed to cancel for the isospin asymmetry as the presence of
muons is common between the $K^{0}_{\rm\scriptscriptstyle S}$ channels and the
$K^{+}$ channels. The uncertainty associated with the
$K^{0}_{\rm\scriptscriptstyle S}$ trigger efficiency is calculated by
comparing the fraction of candidates triggered by
$K^{0}_{\rm\scriptscriptstyle S}$ daughters in the simulation and the data.
The difference is used as an estimate of the capability of simulation to
reproduce these trigger decisions. The simulation is found to underestimate
the $K^{0}_{\rm\scriptscriptstyle S}$ trigger decisions by 10–20% depending on
the decay mode. This percentage is multiplied by the fraction of trigger
decisions where the $K^{0}_{\rm\scriptscriptstyle S}$ participates in a given
bin of $q^{2}$ leading to an uncertainty of 0.2–4.1% depending on $q^{2}$ and
decay mode.
The effect of the unknown angular distribution of $B^{+}\\!\rightarrow
K^{*+}\mu^{+}\mu^{-}$ decays on the relative efficiency is estimated by
altering the Wilson coefficients appearing in the operator product expansion
method [19, 20]. The Wilson coefficients, $\mathcal{C}_{7}$ and
$\mathcal{C}_{10}$, have their real part inverted and the relative efficiency
is recalculated. This can be seen as an extreme variation which is used to
obtain a conservative estimate of the associated uncertainty. The calculation
was performed using an EvtGen physics model which uses the transition form
factors detailed in Ref. [21]. The difference in the relative efficiency
varies from 0–6%, depending on $q^{2}$, and it is assigned as a systematic
uncertainty.
The shape parameters for the signal modes are assumed to be the same as the
normalisation channels. This assumption is validated using the simulation and
no systematic uncertainty is assigned. The statistical uncertainties of these
shape parameters are propagated through the fit using Gaussian constraints,
accounting for correlations between the parameters. The uncertainty on the
amount of partially reconstructed background is also added to the fit using
Gaussian constraints, therefore no further uncertainty is added. The
parametrisation of the fit model is cross-checked by varying the fit range and
background model. Consistent yields are observed and no systematic uncertainty
is assigned.
Overall the systematic error on the branching fraction is 4–8% depending on
$q^{2}$ and the decay mode. This is small compared to the typical statistical
error of $\sim$ 40%.
## 7 Results and conclusions
The differential branching fraction in the $i^{\mathrm{th}}$ $q^{2}$ bin can
be written as
$\frac{d\mathcal{B}^{i}}{dq^{2}}=\frac{N^{i}(B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-})}{N(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})}\times\frac{\mathcal{B}(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})\mathcal{B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})}{\epsilon^{i}_{\mathrm{rel}}\Delta^{i}},$ (2)
where ${N^{i}(B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}})$ is the number of signal
candidates in bin $i$,
${N(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}})$ is
the number of normalisation candidates, the product of
$\mathcal{B}(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})$ and $\mathcal{B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})$ is the visible branching fraction of the
normalisation channel [22], $\epsilon^{i}_{\mathrm{rel}}$ is the relative
efficiency between the signal and normalisation channels in bin $i$ and
finally $\Delta^{i}$ is the bin $i$ width. The differential branching fraction
is determined by simultaneously fitting the L and D categories of the signal
channels. The branching fraction of the signal channel is introduced as a fit
parameter by re-arranging Eq. (2) in terms of ${N(B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-}})$. Confidence intervals are evaluated by scanning the
profile likelihood. The results of these fits for $B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays are
shown in Fig. 4 and given in Tables 2 and 3. Theoretical predictions [23, 24,
25] are superimposed on Figs. 4 and 5. In the low $q^{2}$ region, these
predictions rely on the QCD factorisation approaches from Refs. [26, 27] for
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ and Ref. [28] for $B\\!\rightarrow
K\mu^{+}\mu^{-}$ which lose accuracy when approaching the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance. In the high $q^{2}$
region, an operator product expansion in the inverse $b$-quark mass,
$1/m_{b}$, and in $1/\sqrt{q^{2}}$ is used based on Ref. [29]. This expansion
is only valid above the open charm threshold. In both $q^{2}$ regions the form
factor calculations for $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ and
$B\\!\rightarrow K\mu^{+}\mu^{-}$ are taken from Refs. [30] and [31]
respectively. These form factors lead to a high correlation in the uncertainty
of the predictions across $q^{2}$. A dimensional estimate is made of the
uncertainty from expansion corrections [32]. The non-zero isospin asymmetry
arises in the low $q^{2}$ region due to spectator-quark differences in the so-
called hard-scattering part. There are also sub-leading corrections included
from Refs. [1] and [27] which only affect the charged modes and further
contribute to the isospin asymmetry.
The total branching fractions are also measured by extrapolating underneath
the charmonium resonances assuming the same $q^{2}$ distribution as in the
simulation. The branching fractions of $B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ are found
to be
$\begin{split}\phantom{+}\mathcal{B}(B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-})=(0.31^{+0.07}_{-0.06})\times 10^{-6}\phantom{,\pm
b}\rm{and}\\\ \mathcal{B}(B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-})=(1.16\pm
0.19)\times 10^{-6},\phantom{and}\end{split}$
respectively, where the errors include statistical and systematic
uncertainties. These results are in agreement with previous measurements and
with better precision [22].
Figure 4: Differential branching fractions of (left) $B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-}$ and (right) $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$.
The theoretical SM predictions are taken from Refs. [23, 24].
The isospin asymmetries as a function of $q^{2}$ for $B\\!\rightarrow
K\mu^{+}\mu^{-}$ and $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ are shown in Fig. 5
and given in Tables 2 and 3. As for the branching fractions, the fit is done
simultaneously for both the L and D categories where $A_{\rm I}$ is a common
parameter for the two cases. The confidence intervals are also determined by
scanning the profile likelihood. The significance of the deviation from the
null hypothesis is obtained by fixing $A_{\rm I}$ to be zero and computing the
difference in the negative log-likelihood from the nominal fit.
Figure 5: Isospin asymmetry of (left) $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
(right) $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$. For $B\\!\rightarrow
K^{*}\mu^{+}\mu^{-}$ the theoretical SM prediction, which is very close to
zero, is shown for $q^{2}$ below
8.68${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, from Ref. [25].
In summary, the isospin asymmetries of $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$
decays and the branching fractions of $B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ are
measured, using 1.0$\mbox{\,fb}^{-1}$ of data taken with the LHCb detector.
The two $q^{2}$ bins below 4.3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
the highest bin above 16${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ have the
most negative isospin asymmetry in the $B\\!\rightarrow K\mu^{+}\mu^{-}$
channel. These $q^{2}$ regions are furthest from the charmonium regions and
are therefore cleanly predicted theoretically. This asymmetry is dominated by
a deficit in the observed $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ signal.
Ignoring the small correlation of errors between each $q^{2}$ bin, the
significance of the deviation from zero integrated across $q^{2}$ is
calculated to be 4.4 $\sigma$. The $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ case
agrees with the SM prediction of almost zero isospin asymmetry [1]. All
results agree with previous measurements [3, 33, 34].
Table 2: Partial branching fractions of $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ and isospin asymmetries of $B\\!\rightarrow K\mu^{+}\mu^{-}$ decays. The significance of the deviation of $A_{\rm I}$ from zero is shown in the last column. The errors include the statistical and systematic uncertainties. $q^{2}$ range [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $d\mathcal{B}/dq^{2}[10^{-8}/{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}]$ | $A_{\rm I}$ | $\sigma$($A_{\rm I}$ = 0)
---|---|---|---
$\phantom{0}0.05-\phantom{0}2.00$ | $1.1^{+1.4}_{-1.2}$ | $-0.55^{+0.40}_{-0.56}$ | 1.5
$\phantom{0}2.00-\phantom{0}4.30$ | $0.3^{+1.1}_{-0.9}$ | $-0.76^{+0.45}_{-0.79}$ | 1.9
$\phantom{0}4.30-\phantom{0}8.68$ | $2.8\pm 0.7$ | $\phantom{-}0.00^{+0.14}_{-0.15}$ | 0.1
$10.09-12.86$ | $1.8^{+0.8}_{-0.7}$ | $-0.15^{+0.19}_{-0.22}$ | 0.8
$14.18-16.00$ | $1.1^{+0.7}_{-0.5}$ | $-0.40\pm 0.22$ | 1.9
$16.00-23.00$ | $0.5^{+0.3}_{-0.2}$ | $-0.52^{+0.18}_{-0.22}$ | 3.0
$\phantom{0}1.00-\phantom{0}6.00$ | $1.3^{+0.9}_{-0.7}$ | $-0.35^{+0.23}_{-0.27}$ | 1.7
Table 3: Partial branching fractions of $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ and isospin asymmetries of $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decays. The significance of the deviation of $A_{\rm I}$ from zero is shown in the last column. The errors include the statistical and systematic uncertainties. $q^{2}$ range [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $d\mathcal{B}/dq^{2}[10^{-8}/{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}]$ | $A_{\rm I}$ | $\sigma$($A_{\rm I}$ = 0)
---|---|---|---
$\phantom{0}0.05-\phantom{0}2.00$ | $7.0^{+3.1}_{-3.0}$ | $\phantom{-}0.05^{+0.27}_{-0.21}$ | 0.2
$\phantom{0}2.00-\phantom{0}4.30$ | $5.4^{+2.6}_{-2.4}$ | $-0.27^{+0.29}_{-0.18}$ | 0.9
$\phantom{0}4.30-\phantom{0}8.68$ | $5.7^{+2.0}_{-1.7}$ | $-0.06^{+0.19}_{-0.14}$ | 0.4
$10.09-12.86$ | $7.7^{+2.6}_{-2.4}$ | $-0.16^{+0.17}_{-0.16}$ | 0.9
$14.18-16.00$ | $5.5^{+2.6}_{-2.1}$ | $\phantom{-}0.02^{+0.23}_{-0.21}$ | 0.1
$16.00-19.30$ | $3.8\pm 1.4$ | $\phantom{-}0.02^{+0.21}_{-0.20}$ | 0.1
$\phantom{0}1.00-\phantom{0}6.00$ | $5.8^{+1.8}_{-1.7}$ | $-0.15\pm 0.16$ | 1.0
## Acknowledgements
We would like to thank Christoph Bobeth, Danny van Dyk and Gudrun Hiller for
providing SM predictions for the branching fractions and the isospin asymmetry
of $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decays. 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.
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|
arxiv-papers
| 2012-05-15T15:44:52 |
2024-09-04T02:49:30.941176
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, 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, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook,\n H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N.\n Chiapolini, M. Chrzaszcz, 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, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G. A. Cowan, D. Craik, R. Currie, C. D'Ambrosio, P.\n David, P. N. Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.\n M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, 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, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El\n Rifai, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, 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, O. Francisco, 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, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T. 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, M. Hoballah, 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, O. Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi, M. Lieng, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n 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, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, J. McCarthy, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, 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, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, F. Polci, 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, A.\n Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues,\n P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K. Sobczak, F. J.\n P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U.\n Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, M. Vesterinen, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace,\n S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale,\n M. Whitehead, J. Wicht, 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 S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Ulrik Egede",
"url": "https://arxiv.org/abs/1205.3422"
}
|
1205.3452
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-128 LHCb-PAPER-2012-012 28 October 2012
Observation of excited $\mathchar 28931\relax^{0}_{b}$ baryons
The LHCb collaboration †††Authors are listed on the following pages.
Using $pp$ collision data corresponding to 1.0 $\mbox{\,fb}^{-1}$ integrated
luminosity collected by the LHCb detector, two narrow states are observed in
the $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ spectrum with masses
$5911.97\pm 0.12(\mbox{stat})\pm 0.02(\mbox{syst})\pm 0.66(\mathchar
28931\relax^{0}_{b}\mbox{ mass})$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $5919.77\pm 0.08(\mbox{stat})\pm 0.02(\mbox{syst})\pm 0.66(\mathchar
28931\relax^{0}_{b}\mbox{ mass})$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The significances of the observations are $5.2$ and $10.2$ standard
deviations, respectively. These states are interpreted as the orbitally
excited $\mathchar 28931\relax^{0}_{b}$ baryons, $\mathchar
28931\relax_{b}^{*0}(5912)$ and $\mathchar 28931\relax_{b}^{*0}(5920)$.
To be submitted to Phys. Rev. Lett.
The LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,
F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10, F.
Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T.
Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,
D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42,
R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J.
Lefrançois7, O. Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9,
M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35,
J. Magnin1, S. Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev27, E.
Maurice6, B. Maynard53, A. Mazurov16,30,35, J. McCarthy42, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A. Milanes13, M.-N.
Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski23, R.
Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B.
Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I.
Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M.
Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A. Nomerotski52,35, A.
Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48,
O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P.
Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35,
A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M. Plo
Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D.
Popov10, B. Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V.
Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43, B.
Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M.
Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa
Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T.
Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T.
Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, 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
The system of baryons containing a $b$ quark (beauty baryons) remains largely
unexplored, despite recent progress made at the experiments at the Tevatron.
In addition to the ground state, $\mathchar 28931\relax^{0}_{b}$, the
$\mathchar 28932\relax_{b}^{-}$ baryon with the quark content $bsd$ has been
observed by the D0 [1] and CDF [2] collaborations, followed by the observation
of the doubly-strange $\mathchar 28938\relax_{b}^{-}$ baryon ($bss$) [3, 4].
The last ground state of beauty-strange content, $\mathchar
28932\relax_{b}^{0}$ ($bsu$), has been observed by CDF [5]. Recently, the CMS
collaboration has found the corresponding excited state, most likely
$\mathchar 28932\relax_{b}^{*0}$ with $J^{P}=3/2^{+}$ [6]. Beauty baryons with
two light quarks ($bqq$, where $q=u,d$), other than the $\mathchar
28931\relax^{0}_{b}$, have been studied so far by CDF only. Of the triplets
$\mathchar 28934\relax_{b}^{\pm,0}$ with spin $J=1/2$ and $\mathchar
28934\relax_{b}^{*\pm,0}$ with $J=3/2$ predicted by theory, only the charged
states $\mathchar 28934\relax_{b}^{(*)\pm}$ have so far been observed via
their decay to $\mathchar 28931\relax^{0}_{b}\pi^{\pm}$ final states [7, 8].
None of the quantum numbers of beauty baryons have been measured.
The quark model predicts the existence of two orbitally excited $\mathchar
28931\relax^{0}_{b}$ states, $\mathchar 28931\relax_{b}^{*0}$, with the
quantum numbers $J^{P}=1/2^{-}$ and $3/2^{-}$, respectively, that should decay
to $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ or $\mathchar
28931\relax^{0}_{b}\gamma$. These states have not previously been established
experimentally. The properties of excited $\mathchar 28931\relax^{0}_{b}$
baryons are discussed in Refs. [9, 10, 11, 12, 13, 14, 15]. Most predictions
give masses above the $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ threshold,
but below the $\mathchar 28934\relax_{b}\pi$ threshold. Observation of
$\mathchar 28931\relax_{b}^{*0}$ states and measurement of their quantum
numbers would provide a further confirmation of the validity of the quark
model, and the precise measurement of their masses would test the
applicability of various theoretical models used to describe the interaction
of heavy quarks.
This Letter reports the first observation of the $\mathchar
28931\relax_{b}^{*0}$ states decaying into $\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$, and the measurement of their masses and
upper limits on their natural widths. The data set of 1.0 $\mbox{\,fb}^{-1}$
collected in $pp$ collisions at the LHC collider at the center-of-mass energy
$\sqrt{s}=7$ $\mathrm{\,Te\kern-1.00006ptV}$ in 2011 is used for the analysis.
The LHCb detector [16] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a muon system composed of alternating layers of iron and multiwire
proportional chambers.
The online event selection (trigger) consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage which applies full event reconstruction. The software trigger used in
this analysis requires a two-, three- or four-track secondary vertex with a
high sum of the momenta transverse to the beam axis, $p_{\rm T}$, of the
tracks, and significant displacement from the primary interaction vertex (PV).
In addition, the secondary vertex should have at least one track with
$\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, IP $\chi^{2}$
with respect to any PV greater than 16 (where the IP $\chi^{2}$ is defined as
the difference of the PV fit $\chi^{2}$ with and without the track included),
and a track fit $\chi^{2}/\rm{ndf}<2$ where $\rm{ndf}$ is the number of
degrees of freedom in the fit. A multivariate algorithm is used for the
identification of the secondary vertices [17].
The $\mathchar 28931\relax^{0}_{b}$ candidates are reconstructed in the
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}\pi^{-}$, $\mathchar 28931\relax^{+}_{c}\\!\rightarrow
pK^{-}\pi^{+}$ decay chain (addition of charge-conjugate states is implied
throughout this Letter). The selection of $\mathchar 28931\relax^{0}_{b}$
candidates is performed in two stages. First, a loose preselection of events
containing beauty hadron candidates decaying to charm hadron candidates is
performed. It requires that the tracks forming the candidate, as well as the
beauty and charm vertices, have good quality and are well separated from any
PV, and the invariant masses of the beauty and charm candidates are consistent
with the masses of the corresponding particles.
The final selection requires that all the tracks forming the $\mathchar
28931\relax^{0}_{b}$ candidate have an IP $\chi^{2}$ with respect to any PV
greater than 9, and the IP $\chi^{2}$ of the $\mathchar 28931\relax^{0}_{b}$
candidate to the best PV (PV having the minimum IP $\chi^{2}$ for the
$\mathchar 28931\relax^{0}_{b}$ candidate) is less than 16. Particle
identification (PID) information from the RICH detectors is used to identify
kaons and protons in the final state in the form of differences of logarithms
of likelihoods between the proton and pion ($\mathrm{DLL}_{p\pi}$) and kaon
and pion ($\mathrm{DLL}_{K\pi}$) hypotheses. No PID requirements are applied
to the pions from $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}\pi^{-}$ decays to increase the $\mathchar
28931\relax^{0}_{b}$ yield: a significant fraction of these pions have momenta
above 100 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ where the PID performance is
reduced. Finally, a kinematic fit is used which constrains the decay products
of the $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28931\relax^{+}_{c}$
baryons to originate from common vertices, the $\mathchar 28931\relax^{0}_{b}$
to originate from the PV and the invariant mass of the $\mathchar
28931\relax^{+}_{c}$ candidate to be equal to the established $\mathchar
28931\relax^{+}_{c}$ mass [18].
A momentum scale correction is applied to all invariant mass spectra in this
analysis to improve the mass measurement using the procedure similar to [19].
The momentum scale has been calibrated using
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
decays, and its accuracy has been quantified with other two-body resonance
decays ($\mathchar 28935\relax{(1S)}\rightarrow\mu^{+}\mu^{-}$,
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$, $\phi\rightarrow
K^{+}K^{-}$).
Signal and background distributions are studied using simulation. Proton-
proton collisions are generated using Pythia 6.4 [20] with a specific LHCb
configuration [21]. Decays of hadronic particles are described by EvtGen [22]
in which final state radiation is generated using Photos [23]. The interaction
of the generated particles with the detector and its response are implemented
using the Geant4 toolkit [24, *Agostinelli:2002hh] as described in Ref. [26].
The distribution of the $\mathchar 28931\relax^{+}_{c}\pi^{-}$ invariant mass
after the kinematic fit is shown in Fig. 1, where a requirement of good
quality of the kinematic fit is applied. In addition to the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ signal
contribution, the spectrum contains backgrounds from random combinations of
tracks (random background), from partially-reconstructed decays where one or
more particles are not reconstructed, and from $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decays
with the kaon reconstructed under the pion mass hypothesis. A fit of the
spectrum yields $70\,540\pm 330$ signal events, and the signal-to-background
ratio in a $\pm 25$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ interval around
the nominal $\mathchar 28931\relax^{0}_{b}$ mass is $S/B=11$. The fit to the
$\mathchar 28931\relax^{+}_{c}\pi^{-}$ spectrum is only used to estimate the
$\mathchar 28931\relax^{0}_{b}$ yield and the $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$
contribution, and is not used in the subsequent analysis.
Figure 1: Invariant mass spectrum of $\mathchar 28931\relax^{+}_{c}\pi^{-}$
combinations. The points with error bars are the data, and the fitted
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}\pi^{-}$ signal and three background components ($\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$,
partially-reconstructed and random background) are shown with different fill
styles.
The $\mathchar 28931\relax^{0}_{b}$ candidates obtained with the above
selection are combined with two tracks under the pion mass hypothesis
(referred to as slow pions from now on) to search for excited $\mathchar
28931\relax^{0}_{b}$ states. The tracks are required to have transverse
momentum $\mbox{$p_{\rm T}$}>150{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and no
PID requirements are applied. A kinematic fit is applied that, in addition to
all constraints described above for $\mathchar 28931\relax^{0}_{b}$
candidates, constrains the two slow pion tracks to originate from the PV and
the invariant mass of the $\mathchar 28931\relax^{0}_{b}$ candidate to a fixed
value of $5619.37$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is a
combination of the world average [18] and the LHCb measurement [27]. The
uncertainty on the combined $\mathchar 28931\relax^{0}_{b}$ mass obtained in
this way, $0.69$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is treated as a
systematic effect. Combinations with a good quality of kinematic fit,
$\chi^{2}/{\rm ndf}<3.3$, are retained. From the simulation study, this
requirement is optimal for the observation of a narrow state near the
kinematic threshold with signal-to-background ratio around one.
The fit of the $\mathchar 28931\relax^{+}_{c}\pi^{-}$ mass spectrum (Fig. 1)
indicates the presence of the background from $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decays at
a rate around 12%, relative to the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ signal.
Alternatively, its rate can be estimated from the ratio of
$B^{+}\rightarrow\overline{D}{}^{0}K^{+}$ and
$B^{+}\rightarrow\overline{D}{}^{0}\pi^{+}$ decays that equals to 8% [18]. Due
to the $\mathchar 28931\relax^{0}_{b}$ mass constraint in the kinematic fit,
the $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ invariant mass distribution
for this mode is biased by less than 0.1
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ if reconstructed under the
$\mathchar 28931\relax^{+}_{c}\pi^{-}$ mass hypothesis, and has a resolution
only a factor of two worse than that with the $\mathchar
28931\relax^{+}_{c}\pi^{-}$ signal. After the kinematic fit quality
requirement, the fraction of $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$
with $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}K^{-}$ decays compared to those with the $\mathchar
28931\relax^{+}_{c}\pi^{-}$ is reduced to 8%. This mode is thus not treated
separately, and its effect is taken into account as a part of the systematic
uncertainty due to the signal shape.
Combinations of $\mathchar 28931\relax^{0}_{b}$ candidates with both opposite-
sign and same-sign slow pions are selected in data. The latter are used to
constrain the background shape coming from random combinations of $\mathchar
28931\relax^{0}_{b}$ baryon and two tracks. The assumption that the shape of
the background in $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ and $\mathchar
28931\relax^{0}_{b}\pi^{\pm}\pi^{\pm}$ modes is the same is validated with
simulation. The $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ and $\mathchar
28931\relax^{0}_{b}\pi^{\pm}\pi^{\pm}$ invariant mass spectra are shown in
Fig. 2; two narrow structures with masses around 5912 and 5920
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are evident in the $\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$ spectrum. They are interpreted as the
orbitally excited $\mathchar 28931\relax^{0}_{b}$ states, and are denoted
hereafter as $\mathchar 28931\relax_{b}^{*0}(5912)$ and $\mathchar
28931\relax_{b}^{*0}(5920)$.
A combined unbinned fit of the $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$
and $\mathchar 28931\relax^{0}_{b}\pi^{\pm}\pi^{\pm}$ samples is performed to
extract the masses and event yields of the two states. The background is
described with a quadratic polynomial function with common parameters for both
samples except for an overall normalization. The probability density function
(PDF) for each of the $\mathchar 28931\relax_{b}^{*0}(5912)$ and $\mathchar
28931\relax_{b}^{*0}(5920)$ signals is a sum of two Gaussian PDFs with the
same mean. The relative normalization of the two Gaussian PDFs are fixed to
the values obtained from the simulation of states with masses 5912 and 5920
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and zero natural widths, while the
mean value and overall normalization for each signal are left free in the fit.
The core resolution (width of the narrower Gaussian PDF) obtained from
simulation is 0.19 and 0.27 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\mathchar 28931\relax_{b}^{*0}(5912)$ and $\mathchar
28931\relax_{b}^{*0}(5920)$, respectively. Study of several high-statistics
samples ($\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}\pi^{-}$,
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$, $D^{*+}\rightarrow D^{0}\pi^{+}$) shows that the
invariant mass resolution in data is typically worse by $20\%$ than in the
simulation. Thus the nominal data fit uses the widths of Gaussian PDFs from
the simulation multiplied by 1.2. The data fit yields $17.6\pm 4.8$ events
with mass $M_{\mathchar 28931\relax_{b}^{*0}(5912)}=5911.97\pm 0.12$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $52.5\pm 8.1$ events with mass
$M_{\mathchar 28931\relax_{b}^{*0}(5920)}=5919.77\pm 0.08$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
(a)
(b)
Figure 2: Invariant mass spectrum of (a) $\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$ and (b) $\mathchar
28931\relax^{0}_{b}\pi^{\pm}\pi^{\pm}$ combinations. The points with error
bars are the data, the solid line is the fit result, the dashed line is the
background contribution.
Limits on natural widths $\Gamma$ of the two states are obtained by performing
an alternative fit where the signal PDFs are convolved with relativistic
Breit-Wigner distributions. The dependence of Breit-Wigner width $\Gamma$ on
the $\Lambda_{b}^{0}\pi^{+}\pi^{-}$ invariant mass $M$ is taken into account
as $\Gamma_{\mathchar 28931\relax_{b}^{*0}}(M)=\Gamma_{\mathchar
28931\relax_{b}^{*0}}\times(q/q_{0})^{2}\times(M_{\mathchar
28931\relax_{b}^{*0}}/M)$. Here $M_{\mathchar 28931\relax_{b}^{*0}}$ is the
mass of the $\mathchar 28931\relax_{b}^{*0}$ state, and $q_{(0)}$ is the
kinematic energy for the decay of the state with mass $M_{(\mathchar
28931\relax_{b}^{*0})}$: $q_{(0)}=M_{(\mathchar
28931\relax_{b}^{*0})}-M_{\mathchar 28931\relax^{0}_{b}}-2M_{\pi}$, where
$M_{\mathchar 28931\relax^{0}_{b}}$ and $M_{\pi}$ are the masses of $\mathchar
28931\relax^{0}_{b}$ and $\pi^{+}$, respectively. Scans of Breit-Wigner widths
$\Gamma_{\mathchar 28931\relax_{b}^{*0}(5912)}$ and $\Gamma_{\mathchar
28931\relax_{b}^{*0}(5920)}$ are performed with all the other parameters free
to vary in the fit. The upper limits are obtained without applying the mass
resolution scaling factor of 1.2 as in the nominal fit to account for the
uncertainty of this quantity: this gives a more conservative value for the
upper limit. The 90% (95%) confidence level (CL) upper limit on $\Gamma$,
which corresponds to 1.28 (1.64) standard deviations, is obtained as the value
of $\Gamma$ where the negative logarithm of the likelihood is
$1.28^{2}/2=0.82$ ($1.64^{2}/2=1.34$) greater than at its minimum. The 90%
(95%) CL upper limit is $\Gamma_{\mathchar 28931\relax_{b}^{*0}(5912)}<0.66$
$\mathrm{\,Me\kern-1.00006ptV}$ ($0.83$ $\mathrm{\,Me\kern-1.00006ptV}$) for
the $\mathchar 28931\relax_{b}^{*0}(5912)$ state, and $\Gamma_{\mathchar
28931\relax_{b}^{*0}(5920)}<0.63$ $\mathrm{\,Me\kern-1.00006ptV}$ ($0.75$
$\mathrm{\,Me\kern-1.00006ptV}$) for the $\mathchar
28931\relax_{b}^{*0}(5920)$ state.
The invariant mass of the two pions, $M(\pi^{+}\pi^{-})$, in the $\mathchar
28931\relax_{b}^{*0}(5920)\rightarrow\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$ decay is shown in Fig. 3. The background is
subtracted using the sWeights procedure [28]. The weights are calculated from
the fit to $\mathchar 28931\relax^{0}_{b}\pi^{+}\pi^{-}$ invariant mass
distribution, which is practically uncorrelated with $M(\pi^{+}\pi^{-})$. The
$M(\pi^{+}\pi^{-})$ distribution is consistent with the result of phase-space
decay simulation, with $\chi^{2}/{\rm ndf}=1.6$ for ${\rm ndf}=9$. No peaking
structures are evident.
Figure 3: Invariant mass of the two pions from $\mathchar
28931\relax_{b}^{*0}(5920)\rightarrow\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$ decay. The points with the error bars are
background-subtracted data, the solid histogram is the result of phase-space
decay simulation.
Systematic uncertainties on the mass measurement are shown in Table 1. The
dominant uncertainty in the absolute $\mathchar 28931\relax_{b}^{*0}$ mass
measurement comes from the uncertainty on the $\mathchar 28931\relax^{0}_{b}$
mass $\delta M_{\mathchar 28931\relax^{0}_{b}}=0.69$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$; it is propagated to the $\mathchar
28931\relax_{b}^{*0}$ mass uncertainty as $\delta M_{\mathchar
28931\relax_{b}^{*0}}=\delta M_{\mathchar
28931\relax^{0}_{b}}\times(M_{\mathchar 28931\relax^{0}_{b}}/M_{\mathchar
28931\relax_{b}^{*0}})\simeq 0.66$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
This uncertainty mostly cancels in the mass difference $\Delta M_{\mathchar
28931\relax_{b}^{*0}}=M_{\mathchar 28931\relax_{b}^{*0}}-M_{\mathchar
28931\relax^{0}_{b}}$, where the residual uncertainty is $\delta\Delta
M_{\mathchar 28931\relax_{b}^{*0}}=\delta M_{\mathchar
28931\relax^{0}_{b}}\times(\Delta M_{\mathchar
28931\relax^{0}_{b}}/M_{\mathchar 28931\relax_{b}^{*0}})$. The uncertainty of
the signal parameterization is estimated by using the simulated signal
parametrization without applying the resolution scaling factor, by using the
natural width for both states when left free in the fit, and by conservatively
including the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{+}_{c}K^{-}$ contribution with the rate 12% parameterized from
simulation. The uncertainty due to the background parameterization is
estimated by:
* •
using an alternative fit model for background description,
* •
using the fit without the $\mathchar 28931\relax^{0}_{b}\pi^{\pm}\pi^{\pm}$
constraint,
* •
using the fit with the background obtained from the simulation,
* •
fitting in the reduced invariant mass range 5910–5930
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
and taking the largest difference from the nominal fit result as systematic
uncertainty. The effect of the momentum scale correction is evaluated by
varying the scale coefficient by its relative uncertainty $5\times 10^{-4}$ in
simulated signal samples.
Table 1: Systematic uncertainties on the mass difference $\Delta M_{\mathchar 28931\relax_{b}^{*0}}$ between $\mathchar 28931\relax_{b}^{*0}$ and $\mathchar 28931\relax^{0}_{b}$. Source of | Systematic bias, MeV/$c^{2}$
---|---
uncertainty | $\Delta M_{\mathchar 28931\relax_{b}^{*0}(5912)}$ | $\Delta M_{\mathchar 28931\relax_{b}^{*0}(5920)}$
$\mathchar 28931\relax^{0}_{b}$ mass | 0.034 | 0.035
Signal PDF | 0.021 | 0.011
Background PDF | 0.002 | 0.002
Momentum scale | 0.008 | 0.013
Total | 0.041 | 0.039
The significance of the observation of the two states is evaluated with
simulated pseudo-experiments. A large number of background-only invariant mass
distributions are simulated with parameters equal to the fit result, and each
distribution is fitted with models that include background only, as well as
background and signal. The mean mass value of the signal PDF is not
constrained in the fit to account for a trial factor in the range 5900–5950
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The significance is calculated as
the fraction of samples where the difference of the logarithms of fit
likelihoods $\Delta\log\mathcal{L}$ with and without the signal is larger than
in data. The fraction is obtained by an exponential extrapolation of the
$\Delta\log\mathcal{L}$ distribution [29] that allows a limited number of
pseudo-experiments to be used for a signal with high significance. The
significance is then expressed in terms of the number of standard deviations
($\sigma$). The significance of the $\mathchar 28931\relax_{b}^{*0}(5912)$
state obtained in this way is $5.4\sigma$ for the $\Delta\log\mathcal{L}$
obtained from the nominal fit. To account for systematic effects, the minimum
$\Delta\log\mathcal{L}$ among all systematic variations is taken; in that case
the significance reduces to $5.2\sigma$. Similarly, the statistical
significance of the $\mathchar 28931\relax_{b}^{*0}(5920)$ state is
11.7$\sigma$, and the significance including systematic uncertainties is
10.2$\sigma$.
The fit biases and the validity of the statistical uncertainties are checked
with pseudo-experiments where the PDF contains both signal and background
components. The fit does not introduce any noticeable bias on the measurement
of the masses. The mass uncertainty for $\mathchar 28931\relax_{b}^{*0}(5920)$
state is estimated correctly within 1% precision; however, the mass
uncertainty for the $\mathchar 28931\relax_{b}^{*0}(5912)$ is underestimated
by 4%. This factor is taken into account in the final result.
In summary, we report the observation of two narrow states in the $\mathchar
28931\relax^{0}_{b}\pi^{+}\pi^{-}$ mass spectrum, $\mathchar
28931\relax_{b}^{*0}(5912)$ and $\mathchar 28931\relax_{b}^{*0}(5920)$, with
masses
$\begin{split}M_{\mathchar 28931\relax_{b}^{*0}(5912)}&=5911.97\pm 0.12\pm
0.02\pm 0.66{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\\\ M_{\mathchar
28931\relax_{b}^{*0}(5920)}&=5919.77\pm 0.08\pm 0.02\pm
0.66{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\\\ \end{split}$
where the first uncertainty is statistical, the second is systematic, and the
third is the uncertainty due to knowledge of the $\mathchar
28931\relax^{0}_{b}$ mass. The values of the mass differences with respect to
the $\mathchar 28931\relax^{0}_{b}$ mass, where most of the last uncertainty
cancels, and the remaining part is included in the systematic uncertainty, are
$\begin{split}\Delta M_{\mathchar 28931\relax_{b}^{*0}(5912)}&=292.60\pm
0.12(\mbox{stat})\pm
0.04(\mbox{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\\\ \Delta
M_{\mathchar 28931\relax_{b}^{*0}(5920)}&=300.40\pm 0.08(\mbox{stat})\pm
0.04(\mbox{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.\\\ \end{split}$
The signal yield for the $\mathchar 28931\relax_{b}^{*0}(5912)$ state is
$17.6\pm 4.8$ events, and the significance of the signal (including systematic
uncertainty and trial factor in the mass range 5900–5950
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) is 5.2 standard deviations. For
the $\mathchar 28931\relax_{b}^{*0}(5920)$ state, the yield is $52.5\pm 8.1$
events and the significance is 10.2 standard deviations. The limits on the
natural widths of these states are $\Gamma_{\mathchar
28931\relax_{b}^{*0}(5912)}<0.66$ $\mathrm{\,Me\kern-1.00006ptV}$ ($<0.83$
$\mathrm{\,Me\kern-1.00006ptV}$) and $\Gamma_{\mathchar
28931\relax_{b}^{*0}(5920)}<0.63$ $\mathrm{\,Me\kern-1.00006ptV}$ ($<0.75$) at
the 90% (95%) CL.
The masses of $\mathchar 28931\relax_{b}^{*0}$ states obtained in our analysis
are 30–40 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ higher than in the
prediction using the constituent quark model [12], and 20–30
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ lower than the predictions based on
the relativistic quark model [11], modeling the color hyperfine interaction
[14] and an approach based on the heavy quark effective theory [15].
Calculation involving a combined heavy quark and large number of colors
expansion [9, 10] gives a value roughly in agreement, although only the spin-
averaged prediction is available. The earlier prediction based on the
relativized quark potential model [13] matches well the absolute mass values
for both states, but the $\mathchar 28931\relax^{0}_{b}$ mass prediction using
this model is 35 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ lower than the
measured value.
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.
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|
arxiv-papers
| 2012-05-15T17:24:43 |
2024-09-04T02:49:30.951921
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann,\n J.J. Back, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, 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, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N.\n Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke,\n M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G.A. Cowan, D. Craik, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M.\n De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, 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, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El\n Rifai, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, 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, O. Francisco, 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, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson,\n S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.\n Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J.\n Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U.\n Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, J.\n McCarthy, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D.A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva,\n M. Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M.\n Otalora Goicochea, P. Owen, B.K. Pal, J. Palacios, A. Palano, M. Palutan, J.\n Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues,\n F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, K. Sobczak, F.J.P.\n Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer,\n V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi,\n J.J. Velthuis, M. Veltri, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X.\n Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Anton Poluektov",
"url": "https://arxiv.org/abs/1205.3452"
}
|
1205.3478
|
On the logarithmic oscillator as a thermostat
Marc Meléndez
_Dpto. Física Fundamental, Universidad Nacional de Educación a Distancia,_
_Madrid, Spain_.
###### Abstract
Campisi, Zhan, Talkner and Hänggi have recently proposed [1] the use of the
logarithmic oscillator as an ideal Hamiltonian thermostat, both in simulations
and actual experiments. However, the system exhibits several theoretical
drawbacks which must be addressed if this thermostat is to be implemented
effectively.
## 1 The logarithmic oscillator
A logarithmic oscillator is a point mass $m$ in a central logarithmic
potential. The Hamiltonian for such a particle is
$H_{osc.}\left(\boldsymbol{q},\,\boldsymbol{p}\right)=\frac{\boldsymbol{p}^{2}}{2m}+k_{B}T\,\ln\left(\frac{\left\|\boldsymbol{q}\right\|}{b}\right)=E,$
(1.1)
where $k_{B}T$ and $b$ can be considered arbitrary parameters for the time
being. The Hamiltonian equations of motion are therefore
$\left\\{\begin{array}[]{ccc}\dot{q}_{i}=\frac{\partial H}{\partial
p_{i}}&=&\frac{p_{i}}{m},\\\ \dot{p}_{i}=-\frac{\partial H}{\partial
q_{i}}&=&-k_{B}T\frac{q_{i}}{\boldsymbol{q}^{2}}.\end{array}\right.$ (1.2)
This mechanical system has several interesting properties.
In the one-dimensional version of the oscillator, it is particularly easy to
find the equations of motion by direct integration (we will disregard the
singularity in the potential for the moment). From (1.1), we get the value of
the momentum,
$p=\sqrt{2m\left(E-k_{B}T\,\ln\left(\frac{q}{b}\right)\right)},$
and using the first of Hamilton’s equations of motion (1.2),
$\dot{q}=\sqrt{\frac{2}{m}\left(E-k_{B}T\,\ln\left(\frac{q}{b}\right)\right)},$
we get a differential equation which can be solved by separation of variables
$t=\sqrt{\frac{m}{2}}\int\frac{dq}{\sqrt{E-k_{B}T\,\ln\left(\frac{q}{b}\right)}}.$
(1.3)
Now, the amplitude of the oscillation is determined by the points $q_{\alpha}$
that satisfy the following equation:
$k_{B}T\,\ln\left(\frac{q_{\alpha}}{b}\right)=E,$
that is,
$\displaystyle q_{A}$ $\displaystyle=$ $\displaystyle-be^{\beta E},$
$\displaystyle q_{B}$ $\displaystyle=$ $\displaystyle be^{\beta E},$
where $\beta$ represents $\left(k_{B}T\right)^{-1}$. The period of oscillation
is just twice the time taken by the particle to go from $q_{A}$ to $q_{B}$,
$2t_{AB}=\sqrt{2m}\int_{q_{A}}^{q_{B}}\frac{dx}{\sqrt{E-k_{B}T\,\ln\left(\frac{\left|q\right|}{b}\right)}}.$
(1.4)
The function in the integral is even, so
$\displaystyle 2t_{AB}$ $\displaystyle=$
$\displaystyle\sqrt{8m}\int_{0}^{q_{B}}\frac{dx}{\sqrt{E-k_{B}T\,\ln\left(\frac{q}{b}\right)}}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{8\pi m}{k_{B}T}}be^{\beta E}.$
In the more general case, the motion of the particle lies on a plane. If it
moves in circular orbits around the singularity with a radius $r$, then its
velocity can be deduced from the fact that the central and centrifugal forces
must balance,
$F=\frac{k_{B}T}{r}=m\frac{v^{2}}{r}.$
Therefore, the speed
$v=\sqrt{\frac{k_{B}T}{m}}$ (1.5)
does not depend on the radius of the orbit. The radius of the orbit is a
function of the total energy $E$, because inserting (1.5) into (1.1), setting
$q$ equal to $r$ and then solving for $r$ gets us
$r=\frac{be^{\beta E}}{\sqrt{e}}.$
Therefore, the time it takes the particle to complete an orbit is
$t_{orb.}=\frac{2\pi r}{v}=2\pi\sqrt{\frac{m}{ek_{B}T}}be^{\beta E}.$
For arbitrary initial conditions, the trajectory followed by the oscillator
will not usually be a closed path, but the particle will never move further
out than
$r_{max.}=be^{\beta E},$
for a given energy $E$, and the time between two consecutive maximum distances
will be somewhere between $2t_{AB}$ and $t_{orb.}$ (note that both times are
of the same order of magnitude),
$\frac{2t_{AB}}{t_{orb.}}=\sqrt{\frac{2e}{\pi}}.$ (1.6)
## 2 Statistical properties
The fact that the speed on a circular orbit does not depend on the radius is
quite surprising. It implies that, if an external perturbation were to
relocate the oscillator on a new circular orbit, the kinetic energy would
remain the same and all the energy absorbed would be completely converted into
potential energy.
In a sense, this result can be generalised to the oscillator’s other
trajectories. If we define the virial $G$ as
$G=pr,$ (2.1)
and calculate its time derivative using (1.2),
$\frac{dG}{dt}=p\dot{r}+\dot{p}r=2\left(\frac{p^{2}}{2m}\right)-k_{B}T.$
The time average of the previous formula is
$\left\langle\frac{dG}{dt}\right\rangle_{t}=2\left\langle\frac{p^{2}}{2m}\right\rangle_{t}-k_{B}T,$
and if $\left\langle dG/dt\right\rangle_{t}=0$, then the average kinetic
energy must be
$\left\langle\frac{p^{2}}{2m}\right\rangle_{t}=\frac{1}{2}k_{B}T,$ (2.2)
_whatever the value of_ $E$! This means that the logarithmic oscillator can
absorb an arbitrary amount of energy without changing its temperature at all,
behaving (in a way) like an ideal thermostat.
Is it true, then, that $\left\langle dG/dt\right\rangle_{t}=0$? It certainly
is, as
$\displaystyle\left\langle\frac{dG}{dt}\right\rangle_{t}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}\frac{dG}{d\tau}d\tau$
$\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\frac{G\left(t\right)-G\left(0\right)}{t}=0,$
because $G$ has upper and lower bounds, as one can see by noting that $G$ is a
continuous function, except at the origin. Given that
$\displaystyle\lim_{r\rightarrow 0}G\left(r\right)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle G\left(r_{max.}\right)$ $\displaystyle=$
$\displaystyle 0,$
we can infer that $G\left(r\right)$ has upper and lower bounds in the interval
$\left(0,\,r_{max}\right]$, and (2.2) is correct. However, we must not forget
that there is a limiting process involved in (2), and hence it might take a
very long time for the average kinetic energy to converge to $k_{B}T/2$. In
fact, we will argue that this is generally the case, and that the logarithmic
oscillator is therefore a somewhat less-than-ideal thermostat.
A recent article in the ar$\chi$iv [1] argued that weak coupling between a
system of interest and a logarithmic oscillator will result in canonical
sampling of the former’s phase space. The dynamics of the compound system
would then be determined by a total Hamiltonian
$\displaystyle H\left(\boldsymbol{q},\,\boldsymbol{p},\,r,\,p_{r}\right)$
$\displaystyle=$ $\displaystyle
H_{S}\left(\boldsymbol{q},\,\boldsymbol{p}\right)+H_{osc.}\left(r,\,p_{r}\right)$
$\displaystyle+H_{int.}\left(\boldsymbol{q},\,\boldsymbol{p},\,r,\,p_{r}\right)=E,$
where $H_{S}\left(\boldsymbol{q},\,\boldsymbol{p}\right)$ is the Hamiltonian
for the system of interest, $H_{osc.}\left(r,\,p_{r}\right)$ is the one-
dimensional version of (1.1), and $H_{int.}$ is the potential energy of the
weak interaction between the system and the oscillator, which we will assume
is negligible compared to $H_{S}$ and $H_{osc.}$. The density of states for
the logarithmic oscillator is
$\displaystyle\Omega_{osc.}\left(E_{osc.}\right)$ $\displaystyle=$
$\displaystyle\int\delta\left(H_{osc.}\left(r,\,p_{r}\right)-E_{osc.}\right)\,dp_{r}\,dr,$
with $\delta$ representing the Dirac delta function. The integral turns out to
be exactly the same as (1.4), so
$\Omega_{osc.}\left(E_{osc.}\right)=\sqrt{\frac{8\pi m}{k_{B}T}}be^{\beta
E_{osc.}}.$ (2.4)
Furthermore, the probability density $\rho$ for a point in the phase space of
the system corresponding to $H_{S}$ is
$\rho\left(\boldsymbol{q},\,\boldsymbol{p}\right)=\frac{\Omega_{osc.}\left(E-H_{S}\left(\boldsymbol{q},\,\boldsymbol{p}\right)\right)}{\Omega\left(E\right)}.$
(2.5)
The function $\Omega\left(E\right)$ represents the density of states of the
compound system,
$\Omega\left(E\right)=\int\delta\left(E-H\left(\boldsymbol{q},\,\boldsymbol{p},\,r,\,p_{r}\right)\right)\,dr\,dp_{r}\,d\boldsymbol{q}\,d\boldsymbol{p}.$
(2.6)
Expressions (2.4) and (2.6) can be used to convert (2.5) into
$\rho\left(\boldsymbol{q},\,\boldsymbol{p}\right)=\frac{e^{-\beta
H_{S}\left(\boldsymbol{q},\,\boldsymbol{p}\right)}}{\int e^{-\beta
H_{S}\left(\boldsymbol{q},\,\boldsymbol{p}\right)}d\boldsymbol{q}\,d\boldsymbol{p}},$
which is precisely the canonical distribution for $H_{S}$.
According to the authors of [1], the logarithmic oscillator thermostat has two
obvious advantages. Firstly, contrary to the popular Nosé-Hoover thermostat,
the dynamical equations of motion are Hamiltonian. Secondly, it is possible to
design experimental setups in which the thermostat is an actual _physical_
system. Hoover wrote a reply [2] to the first claim arguing that Nosé-Hoover
mechanics _are_ in fact Hamiltonian, and included an example of an alternative
Hamiltonian thermostat of the Nosé-Hoover type. Campisi _et alii_ answered
explaining their claim further in [3]. Here we will be considering the second
claim instead, that is, we will concentrate on the implementation of the
logarithmic oscillator as a thermostat, both in experiments and simulations.
## 3 Experiments
An experimental thermostat that relies on the dynamics of only a few degrees
of freedom is no doubt a very interesting system. However, the nature of the
logarithmic oscillator imposes some serious limitations which must be taken
into account before one attempts to design such an experiment.
The first problem is a consequence of the length-scales involved. Assume that
we wish to bring a system with $N$ degrees of freedom to the equilibrium
temperature $T$. If the kinetic energy per degree of freedom is initially off
by a fraction $\alpha$ of the energy,
$\left\langle\frac{p_{i}^{2}}{2m}\right\rangle_{t}=\left(1+\alpha\right)\frac{1}{2}k_{B}T,$
then the logarithmic oscillator will have to absorb at least an amount of
energy equal to $\Delta E=N\alpha k_{B}T/2$. We have seen that the oscillator
typically covers distances of the order of $b\,\exp\left\\{\beta
E_{osc.}\right\\}$. The change in energy implies that the distances covered
will change by
$\Delta r_{max.}=r_{max.}\left(e^{\beta\Delta E}-1\right).$ (3.1)
This can be problematic if $r_{max.}$ is initially comparable to the size of
the experimental apparatus and the oscillator is cooling the system.
The enormous changes in lengths imply similar changes in time scales. Having
assumed a weak interaction between the system of interest and the oscillator,
the effect of the interaction on the latter during one period of oscillation
should not be significant. The period is
$t_{per.}=\lambda\sqrt{\frac{m}{k_{B}T}}be^{\beta E_{osc.}},$
where $\lambda$ is a factor that depends on the trajectory, but which is of
the order of magnitude of $\sqrt{8\pi}$, in agreement with (1.6). The change
in distances carries with it a corresponding change in periods of oscillation,
$\Delta t_{per.}=t_{per.}\left(e^{\beta\Delta E}-1\right).$ (3.2)
Therefore, when the oscillator is cooling down the system of interest, it will
usually move very far out and oscillate very slowly. On the other hand, when
it is “hotter” than the system, it will squeeze into a small neighborhood of
the singularity and vibrate very quickly.
Let us illustrate the problem with some numbers. The authors of [1] propose an
experiment in which a small system composed of neutral atoms is contained in a
box of length $L$. The logarithmic oscillator is an ion in a two-dimensional
Coulomb field generated by a charged wire.
Assume, for example, that we have a dilute gas of $10$ atoms of argon at an
initial temperature $T_{0}=3\,\mathrm{K}$ and that we wish to bring them to
$T=1\,\mathrm{K}$. This means that the logarithmic oscillator must absorb
about
$\Delta E=\frac{3}{2}Nk_{B}T_{0}-\frac{3}{2}Nk_{B}T=30\,k_{B}T$ (3.3)
units of energy. Let us assume further that the cross section of the charged
wire has a radius equal to $10^{-3}\,L$. Then the logarithmic oscillator must
move in orbits with
$r_{max.}>10^{-3}\,L.$
However, when we insert (3.3) into (3.1) we find that
$\Delta r_{max.}=r_{max.}\left(e^{30}-1\right)>10^{10}\,L.$
If we also take equation (3.2) into account, it is easy to see that we should
expect to find the oscillator outside the box most of the time.
## 4 Simulations
The wide range of time and length scales affects the precision and time of
computation of numerical simulations as well, but the presence of a
singularity in the logarithmic potential introduces another complication in
the numerical implementation of the oscillator, as stepping over the
singularity will usually lead to the wrong energy $E_{osc.}$.
When the particle is in the vicinity of the singularity, the slope $\partial
H/\partial r$ changes very quickly. If the oscillator ends up too close to the
singularity, it will feel a great force which will push it away from the
singularity during the next time step, making it skip the area in which the
potential would slow it down again, unless a very small time step is chosen.
For the one-dimensional version of the logarithmic oscillator, the problem can
be solved by calculating the new position of the logarithmic oscillator first.
If the oscillator has stepped over the singularity, then expression (1.3) can
be used to calculate the time it would have taken to get to the new position,
and one can reset its kinetic energy to the correct value and calculate the
evolution of the system of interest during that time. This solution is far
from satisfactory, though, because it involves finding numerical values of the
error function every time the particle passes the singularity.
A different approach ([1]) replaces the logarithmic potential with the
approximate potential
$V\left(r\right)=\frac{1}{2}k_{B}T\,\ln\left(\frac{r^{2}+b^{2}}{b^{2}}\right),$
thereby eliminating the singularity and introducing only a slight correction
in the density of states for low values of $E_{osc.}$. Unfortunately, this
imposes a limit on the amount of energy available for exchange between the
oscillator and the system. If the system and oscillator are enclosed in a box
of length $L$, one only has about $k_{B}T\,\ln\left(L/b\right)$ units of
energy to play with. In order to allow for larger energy ranges, one must
choose smaller values of $b$ (of the order of $\exp\left\\{-2\alpha
3N\right\\}$ if we wish to allow the energy to fluctuate by a fraction
$\alpha$ either way), and this will tend to generate a small neighbourhood of
$r=0$ in which the forces on the oscillator are huge.
## 5 Conclusions
The logarithmic oscillator proposed by Campisi, Zhan, Talkner and Hänggi
displays very interesting properties from the point of view of theoretical
statistical mechanics. However, before it can be used as a thermostat in
actual experiments and numerical simulations, three problems must be
addressed. Firstly, the distances covered by the oscillator depend
exponentially on its energy. Given that it must not interact strongly with
container walls or other objects, one would expect that it would be very
difficult to control such a system in practice. Secondly, the vast increase in
the period of oscillation when a system is being cooled down suggests that the
desired thermostated dynamics will be achieved very slowly. Lastly, the
presence of a singularity introduces some technical complications in the
numerical implementation of the dynamical behaviour of the oscillator. It
seems, therefore, that Nosé-Hoover dynamics will remain a popular option in
molecular dynamics at least until the problems mentioned here are resolved
satisfactorily.
## Aknowledgments
The author would like to express his gratitude to Pep Español for his helpful
comments.
## References
* [1] M. Campisi, F. Zhan, P. Talkner, and P. Hänggi, _Logarithmic Oscillators: Ideal Hamiltonian Thermostats_ , ar$\chi$iv 1203.5968v3 (2012) [cond-mat.stat-mech].
* [2] Wm. G. Hoover, _Another Hamiltonian Thermostat – Comment on ar $\chi$iv 1203.5968 and 1204.0312_, ar$\chi$iv 1204.0312v3 (2012) [cond-mat.stat-mech].
* [3] M. Campisi, F. Zhan, P. Talkner, and P. Hänggi, _Reply to Hoover [arXiv:1204.0312v2]_ , ar$\chi$iv:1204.4412v1 (2012) [cond-mat.stat-mech].
|
arxiv-papers
| 2012-05-15T19:11:38 |
2024-09-04T02:49:30.960713
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marc Mel\\'endez Schofield",
"submitter": "Marc Mel\\'endez Schofield",
"url": "https://arxiv.org/abs/1205.3478"
}
|
1205.3518
|
# Lower Bounds for Adaptive Sparse Recovery
Eric Price
MIT
ecprice@mit.edu David P. Woodruff
IBM Almaden
dpwoodru@us.ibm.com
###### Abstract
We give lower bounds for the problem of stable sparse recovery from _adaptive_
linear measurements. In this problem, one would like to estimate a vector
$x\in\mathbb{R}^{n}$ from $m$ linear measurements $A_{1}x,\dotsc,A_{m}x$. One
may choose each vector $A_{i}$ based on $A_{1}x,\dotsc,A_{i-1}x$, and must
output $\hat{x}$ satisfying
$\left\lVert\hat{x}-x\right\rVert_{p}\leq(1+\epsilon)\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{p}$
with probability at least $1-\delta>2/3$, for some $p\in\\{1,2\\}$. For $p=2$,
it was recently shown that this is possible with
$m=O(\frac{1}{\epsilon}k\log\log(n/k))$, while nonadaptively it requires
$\Theta(\frac{1}{\epsilon}k\log(n/k))$. It is also known that even adaptively,
it takes $m=\Omega(k/\epsilon)$ for $p=2$. For $p=1$, there is a non-adaptive
upper bound of $\widetilde{O}(\frac{1}{\sqrt{\epsilon}}k\log n)$. We show:
* •
For $p=2$, $m=\Omega(\log\log n)$. This is tight for $k=O(1)$ and constant
$\epsilon$, and shows that the $\log\log n$ dependence is correct.
* •
If the measurement vectors are chosen in $R$ “rounds”, then
$m=\Omega(R\log^{1/R}n)$. For constant $\epsilon$, this matches the previously
known upper bound up to an $O(1)$ factor in $R$.
* •
For $p=1$, $m=\Omega(k/(\sqrt{\epsilon}\cdot\log k/\epsilon))$. This shows
that adaptivity cannot improve more than logarithmic factors, providing the
analogue of the $m=\Omega(k/\epsilon)$ bound for $p=2$.
## 1 Introduction
_Compressed sensing_ or _sparse recovery_ studies the problem of solving
underdetermined linear systems subject to a sparsity constraint. It has
applications to a wide variety of fields, including data stream algorithms
[Mut05], medical or geological imaging [CRT06, Don06], and genetics testing
[SAZ10]. The approach uses the power of a _sparsity_ constraint: a vector
$x^{\prime}$ is _$k$ -sparse_ if at most $k$ coefficients are non-zero. A
standard formulation for the problem is that of _stable sparse recovery_ : we
want a distribution $\mathcal{A}$ of matrices $A\in\mathbb{R}^{m\times n}$
such that, for any $x\in\mathbb{R}^{n}$ and with probability $1-\delta>2/3$
over $A\in\mathcal{A}$, there is an algorithm to recover $\hat{x}$ from $Ax$
with
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{p}\leq(1+\epsilon)\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{p}$ (1)
for some parameter $\epsilon>0$ and norm $p$. We refer to the elements of $Ax$
as _measurements_. We say Equation (1) denotes _$\ell_{p}/\ell_{p}$ recovery_.
The goal is to minimize the number of measurements while still allowing
efficient recovery of $x$. This problem has recently been largely closed: for
$p=2$, it is known that $m=\Theta(\frac{1}{\epsilon}k\log(n/k))$ is tight
(upper bounds in [CRT06, GLPS10], lower bounds in [PW11, CD11]), and for $p=1$
it is known that $m=\widetilde{O}(\frac{1}{\sqrt{\epsilon}}k\log n)$ and
$m=\widetilde{\Omega}(\frac{k}{\sqrt{\epsilon}})$ [PW11] (recall that
$\widetilde{O}(f)$ means $O(f\log^{c}f)$ for some constant c, and similarly
$\widetilde{\Omega}(f)$ means $\Omega(f/\log^{c}f)$).
In order to further reduce the number of measurements, a number of recent
works have considered making the measurements _adaptive_ [JXC08, CHNR08,
HCN09, HBCN09, MSW08, AWZ08, IPW11]. In this setting, one may choose each row
of the matrix after seeing the results of previous measurements. More
generally, one may split the adaptivity into $R$ “rounds”, where in each round
$r$ one chooses $A^{r}\in\mathbb{R}^{m_{r}\times n}$ based on
$A^{1}x,\dotsc,A^{r-1}x$. At the end, one must use $A^{1}x,\dotsc,A^{R}x$ to
output $\hat{x}$ satisfying Equation (1). We would still like to minimize the
total number of measurements $m=\sum m_{i}$. In the $p=2$ setting, it is known
that for arbitrarily many rounds $O(\frac{1}{\epsilon}k\log\log(n/k))$
measurements suffice, and for $O(r\log^{*}k)$ rounds
$O(\frac{1}{\epsilon}kr\log^{1/r}(n/k))$ measurements suffice [IPW11].
Given these upper bounds, two natural questions arise: first, is the
improvement in the dependence on $n$ from $\log(n/k)$ to $\log\log(n/k)$
tight, or can the improvement be strengthened? Second, can adaptivity help by
more than a logarithmic factor, by improving the dependence on $k$ or
$\epsilon$?
A recent lower bound showed that $\Omega(k/\epsilon)$ measurements are
necessary in a setting essentially equivalent to the $p=2$ case [ACD11]111Both
our result and their result apply in both settings. See Appendix A for a more
detailed discussion of the relationship between the two settings.. Thus, they
answer the second question in the negative for $p=2$. Their techniques rely on
special properties of the $2$-norm; namely, that it is a rotationally
invariant inner product space and that the Gaussian is both $2$-stable and a
maximum entropy distribution. Such techniques do not seem useful for proving
lower bounds for $p=1$.
#### Our results.
For $p=2$, we show that any adaptive sparse recovery scheme requires
$\Omega(\log\log n)$ measurements, or $\Omega(R\log^{1/R}n)$ measurements
given only $R$ rounds. For $k=O(1)$, this matches the upper bound of [IPW11]
up to an $O(1)$ factor in $R$. It thus shows that the $\log\log n$ term in the
adaptive bound is necessary.
For $p=1$, we show that any adaptive sparse recovery scheme requires
$\widetilde{\Omega}(k/\sqrt{\epsilon})$ measurements. This shows that
adaptivity can only give $\text{polylog}(n)$ improvements, even for $p=1$.
Additionally, our bound of
$\Omega(k/(\sqrt{\epsilon}\cdot\log(k/\sqrt{\epsilon})))$ improves the
previous _non-adaptive_ lower bound for $p=1$ and small $\epsilon$, which lost
an additional $\log k$ factor [PW11].
#### Related work.
Our work draws on the lower bounds for non-adaptive sparse recovery, most
directly [PW11].
The main previous lower bound for adaptive sparse recovery gets
$m=\Omega(k/\epsilon)$ for $p=2$ [ACD11]. They consider going down a similar
path to our $\Omega(\log\log n)$ lower bound, but ultimately reject it as
difficult to bound in the adaptive setting. Combining their result with ours
gives a $\Omega(\frac{1}{\epsilon}k+\log\log n)$ lower bound, compared with
the $O(\frac{1}{\epsilon}k\cdot\log\log n)$ upper bound. The techniques in
their paper do not imply any bounds for the $p=1$ setting.
For $p=2$ in the special case of adaptive Fourier measurements (where
measurement vectors are adaptively chosen from among $n$ rows of the Fourier
matrix), [HIKP12] shows $\Omega(k\log(n/k)/\log\log n)$ measurements are
necessary. In this case the main difficulty with lower bounding adaptivity is
avoided, because all measurement rows are chosen from a small set of vectors
with bounded $\ell_{\infty}$ norm; however, some of the minor issues in using
[PW11] for an adaptive bound were dealt with there.
#### Our techniques.
We use very different techniques for our two bounds.
To show $\Omega(\log\log n)$ for $p=2$, we reduce to the information capacity
of a Gaussian channel. We consider recovery of the vector $x=e_{i^{*}}+w$, for
$i^{*}\in[n]$ uniformly and $w\sim N(0,I_{n}/\Theta(n))$. Correct recovery
must find $i^{*}$, so the mutual information $I(i^{*};Ax)$ is $\Omega(\log
n)$. On the other hand, in the nonadaptive case [PW11] showed that each
measurement $A_{j}x$ is a power-limited Gaussian channel with constant signal-
to-noise ratio, and therefore has $I(i^{*};A_{j}x)=O(1)$. Linearity gives that
$I(i^{*};Ax)=O(m)$, so $m=\Omega(\log n)$ in the nonadaptive case. In the
adaptive case, later measurements may “align” the row $A_{j}$ with $i^{*}$, to
increase the signal-to-noise ratio and extract more information—this is
exactly how the upper bounds work. To deal with this, we bound how much
information we can extract as a function of how much we know about $i^{*}$. In
particular, we show that given a small number $b$ bits of information about
$i^{*}$, the posterior distribution of $i^{*}$ remains fairly well “spread
out”. We then show that any measurement row $A_{j}$ can only extract $O(b+1)$
bits from such a spread out distribution on $i^{*}$. This shows that the
information about $i^{*}$ increases at most exponentially, so $\Omega(\log\log
n)$ measurements are necessary.
To show an $\widetilde{\Omega}(k/\sqrt{\epsilon})$ bound for $p=1$, we first
establish a lower bound on the multiround distributional communication
complexity of a two-party communication problem that we call
$\mathsf{Multi}\ell_{\infty}$, for a distribution tailored to our application.
We then show how to use an adaptive $(1+\epsilon)$-approximate
$\ell_{1}/\ell_{1}$ sparse recovery scheme $\mathcal{A}$ to solve the
communication problem $\mathsf{Multi}\ell_{\infty}$, modifying the general
framework of [PW11] for connecting non-adaptive schemes to communication
complexity in order to now support adaptive schemes. By the communication
lower bound for $\mathsf{Multi}\ell_{\infty}$, we obtain a lower bound on the
number of measurements required of $\mathcal{A}$.
In the $\mathsf{Gap}\ell_{\infty}$ problem, the two players are given $x$ and
$y$ respectively, and they want to approximate $\|x-y\|_{\infty}$ given the
promise that all entries of $x-y$ are small in magnitude or there is a single
large entry. The $\mathsf{Multi}\ell_{\infty}$ problem consists of solving
multiple independent instances of $\mathsf{Gap}\ell_{\infty}$ in parallel.
Intuitively, the sparse recovery algorithm needs to determine if there are
entries of $x-y$ that are large, which corresponds to solving multiple
instances of $\mathsf{Gap}\ell_{\infty}$. We prove a multiround direct sum
theorem for a distributional version of $\mathsf{Gap}\ell_{\infty}$, thereby
giving a distributional lower bound for $\mathsf{Multi}\ell_{\infty}$. A
direct sum theorem for $\mathsf{Gap}\ell_{\infty}$ has been used before for
proving lower bounds for non-adaptive schemes [PW11], but was limited to a
bounded number of rounds due to the use of a bounded round theorem in
communication complexity [BR11]. We instead use the information complexity
framework [BJKS04] to lower bound the conditional mutual information between
the inputs to $\mathsf{Gap}\ell_{\infty}$ and the transcript of any correct
protocol for $\mathsf{Gap}\ell_{\infty}$ under a certain input distribution,
and prove a direct sum theorem for solving $k$ instances of this problem. We
need to condition on “help variables” in the mutual information which enable
the players to embed single instances of $\mathsf{Gap}\ell_{\infty}$ into
$\mathsf{Multi}\ell_{\infty}$ in a way in which the players can use a correct
protocol on our input distribution for $\mathsf{Multi}\ell_{\infty}$ as a
correct protocol on our input distribution for $\mathsf{Gap}\ell_{\infty}$;
these help variables are in addition to help variables used for proving lower
bounds for $\mathsf{Gap}\ell_{\infty}$, which is itself proved using
information complexity. We also look at the conditional mutual information
with respect to an input distribution which doesn’t immediately fit into the
information complexity framework. We relate the conditional information of the
transcript with respect to this distribution to that with respect to a more
standard distribution.
## 2 Notation
We use lower-case letters for fixed values and upper-case letters for random
variables. We use $\log x$ to denote $\log_{2}x$, and $\ln x$ to denote
$\log_{e}x$. For a discrete random variable $X$ with probability $p$, we use
$H(X)$ or $H(p)$ to denote its entropy
$H(X)=H(p)=\sum-p(x)\log p(x).$
For a continuous random variable $X$ with pdf $p$, we use $h(X)$ to denote its
differential entropy
$h(X)=\int_{x\in X}-p(x)\log p(x)dx.$
Let $y$ be drawn from a random variable $Y$. Then $(X\mid y)=(X\mid Y=y)$
denotes the random variable $X$ conditioned on $Y=y$. We define $h(X\mid
Y)=\operatorname*{\mathbb{E}}_{y\sim Y}h(X\mid y)$. The mutual information
between $X$ and $Y$ is denoted $I(X;Y)=h(X)-h(X\mid Y)$.
For $p\in\mathbb{R}^{n}$ and $S\subseteq[n]$, we define
$p_{S}\in\mathbb{R}^{n}$ to equal $p$ over indices in $S$ and zero elsewhere.
We use $f\lesssim g$ to denote $f=O(g)$.
## 3 Tight lower bound for $p=2,k=1$
We may assume that the measurements are orthonormal, since this can be
performed in post-processing of the output, by multiplying $Ax$ on the left to
orthogonalize $A$. We will give a lower bound for the following instance:
Alice chooses random $i^{*}\in[n]$ and i.i.d. Gaussian noise
$w\in\mathbb{R}^{n}$ with $\operatorname*{\mathbb{E}}[\left\lVert
w\right\rVert_{2}^{2}]=\sigma^{2}=\Theta(1)$, then sets $x=e_{i^{*}}+w$. Bob
performs $R$ rounds of adaptive measurements on $x$, getting
$y^{r}=A^{r}x=(y^{r}_{1},\dotsc,y^{r}_{m_{r}})$ in each round $r$. Let $I^{*}$
and $Y^{r}$ denote the random variables from which $i^{*}$ and $y^{r}$ are
drawn, respectively. We will bound $I(I^{*};Y^{1},Y^{2},\dotsc,Y^{r})$.
We may assume Bob is deterministic, since we are giving a lower bound for a
distribution over inputs – for any randomized Bob that succeeds with
probability $1-\delta$, there exists a choice of random seed such that the
corresponding deterministic Bob also succeeds with probability $1-\delta$.
First, we give a bound on the information received from any single
measurement, depending on Bob’s posterior distribution on $I^{*}$ at that
point:
###### Lemma 3.1.
Let $I^{*}$ be a random variable over $[n]$ with probability distribution
$p_{i}=\Pr[I^{*}=i]$, and define
$b=\sum_{i=1}^{n}p_{i}\log(np_{i}).$
Define $X=e_{I^{*}}+N(0,I_{n}\sigma^{2}/n)$. Consider any fixed vector
$v\in\mathbb{R}^{n}$ independent of $X$ with $\left\lVert
v\right\rVert_{2}=1$, and define $Y=v\cdot X$. Then
$I(v_{I^{*}};Y)\leq C(b+1)$
for some constant $C$.
###### Proof.
Let $S_{i}=\\{j\mid 2^{i}\leq np_{j}<2^{i+1}\\}$ for $i>0$ and $S_{0}=\\{i\mid
np_{i}<2\\}$. Define $t_{i}=\sum_{j\in S_{i}}p_{j}=\Pr[I^{*}\in S_{i}]$. Then
$\displaystyle\sum_{i=0}^{\infty}it_{i}$ $\displaystyle=\sum_{i>0}\sum_{j\in
S_{i}}p_{j}\cdot i$ $\displaystyle\leq\sum_{i>0}\sum_{j\in
S_{i}}p_{j}\log(np_{j})$ $\displaystyle=b-\sum_{j\in S_{0}}p_{j}\log(np_{j})$
$\displaystyle\leq b-t_{0}\log(nt_{0}/\left|S_{0}\right|)$ $\displaystyle\leq
b+\left|S_{0}\right|/(ne)$
using convexity and minimizing $x\log ax$ at $x=1/(ae)$. Hence
$\displaystyle\sum_{i=0}^{\infty}it_{i}<b+1$ (2)
Let $W=N(0,\sigma^{2}/n)$. For any measurement vector $v$, let $Y=v\cdot X\sim
v_{I^{*}}+W$. Let $Y_{i}=(Y\mid I^{*}\in S_{i})$. Because $\sum v_{j}^{2}=1$,
$\displaystyle\operatorname*{\mathbb{E}}[Y_{i}^{2}]$
$\displaystyle=\sigma^{2}/n+\sum_{j\in
S_{i}}v_{j}^{2}p_{j}/t_{i}\leq\sigma^{2}/n+\left\lVert
p_{S_{i}}\right\rVert_{\infty}/t_{i}\leq\sigma^{2}/n+2^{i+1}/(nt_{i}).$ (3)
Let $T$ be the (discrete) random variable denoting the $i$ such that $I^{*}\in
S_{i}$. Then $Y$ is drawn from $Y_{T}$, and $T$ has probability distribution
$t$. Hence
$\displaystyle h(Y)$ $\displaystyle\leq h((Y,T))$
$\displaystyle=H(T)+h(Y_{T}\mid T)$ $\displaystyle=H(t)+\sum_{i\geq
0}t_{i}h(Y_{i})$ $\displaystyle\leq H(t)+\sum_{i\geq
0}t_{i}h(N(0,\operatorname*{\mathbb{E}}[Y_{i}^{2}]))$
because the Gaussian distribution maximizes entropy subject to a power
constraint. Using the same technique as the Shannon-Hartley theorem,
$\displaystyle I(v_{I^{*}},Y)=I(v_{I^{*}};v_{I^{*}}+W)$
$\displaystyle=h(v_{I^{*}}+W)-h(v_{I^{*}}+W\mid v_{I^{*}})$
$\displaystyle=h(Y)-h(W)$ $\displaystyle\leq H(t)+\sum_{i\geq
0}t_{i}(h(N(0,\operatorname*{\mathbb{E}}[Y_{i}^{2}]))-h(W))$
$\displaystyle=H(t)+\frac{1}{2}\sum_{i\geq
0}t_{i}\ln(\frac{\operatorname*{\mathbb{E}}[Y_{i}^{2}]}{\operatorname*{\mathbb{E}}[W^{2}]})$
and hence by Equation (3),
$\displaystyle I(v_{I^{*}};Y)\leq H(t)+\frac{\ln 2}{2}\sum_{i\geq
0}t_{i}\log(1+\frac{2^{i+1}}{t_{i}\sigma^{2}}).$ (4)
All that requires is to show that this is $O(1+b)$. Since $\sigma=\Theta(1)$,
we have
$\displaystyle\sum_{i}t_{i}\log(1+\frac{2^{i}}{\sigma^{2}t_{i}})$
$\displaystyle\leq\log(1+1/\sigma^{2})+\sum_{i}t_{i}\log(1+\frac{2^{i}}{t_{i}})$
$\displaystyle\leq
O(1)+\sum_{i}t_{i}\log(1+2^{i})+\phantom{}\sum_{i}t_{i}\log(1+1/t_{i}).$ (5)
Now, $\log(1+2^{i})\lesssim i$ for $i>0$ and is $O(1)$ for $i=0$, so by
Equation (2),
$\sum_{i}t_{i}\log(1+2^{i})\lesssim 1+\sum_{i>0}it_{i}<2+b.$
Next, $\log(1+1/t_{i})\lesssim\log(1/t_{i})$ for $t_{i}\leq 1/2$, so
$\displaystyle\sum_{i}t_{i}\log(1+1/t_{i})$ $\displaystyle\lesssim\sum_{i\mid
t_{i}\leq 1/2}t_{i}\log(1/t_{i})+\sum_{i\mid t_{i}>1/2}1\leq H(t)+1.$
Plugging into Equations (5) and (4),
$\displaystyle I(v_{I^{*}},Y)\lesssim 1+b+H(t).$ (6)
To bound $H(t)$, we consider the partition $T_{+}=\\{i\mid t_{i}>1/2^{i}\\}$
and $T_{-}=\\{i\mid t_{i}\leq 1/2^{i}\\}$. Then
$\displaystyle H(t)$ $\displaystyle=\sum_{i}t_{i}\log(1/t_{i})$
$\displaystyle\leq\sum_{i\in T_{+}}it_{i}+\sum_{t\in T_{-}}t_{i}\log(1/t_{i})$
$\displaystyle\leq 1+b+\sum_{t\in T_{-}}t_{i}\log(1/t_{i})$
But $x\log(1/x)$ is increasing on $[0,1/e]$, so
$\displaystyle\sum_{t\in T_{-}}t_{i}\log(1/t_{i})$ $\displaystyle\leq
t_{0}\log(1/t_{0})+t_{1}\log(1/t_{1})+\sum_{i\geq
2}\frac{1}{2^{i}}\log(1/2^{i})\leq 2/e+3/2=O(1)$
and hence $H(t)\leq b+O(1)$. Combining with Equation (6) gives that
$I(v_{I^{*}};Y)\lesssim b+1$
as desired. ∎
###### Theorem 3.2.
Any scheme using $R$ rounds with number of measurements
$m_{1},m_{2},\dotsc,m_{R}>0$ in each round has
$I(I^{*};Y^{1},\dotsc,Y^{R})\leq C^{R}\prod_{i}m_{i}$
for some constant $C>1$.
###### Proof.
Let the signal in the absence of noise be
$Z^{r}=A^{r}e_{I^{*}}\in\mathbb{R}^{m_{r}}$, and the signal in the presence of
noise be $Y^{r}=A^{r}(e_{I^{*}}+N(0,\sigma^{2}I_{n}/n))=Z^{r}+W^{r}$ where
$W^{r}=N(0,\sigma^{2}I_{m_{r}}/n)$ independently. In round $r$, after
observations $y^{1},\dotsc,y^{r-1}$ of $Y^{1},\dotsc,Y^{r-1}$, let $p^{r}$ be
the distribution on $(I^{*}\mid y^{1},\dotsc,y^{r-1})$. That is, $p^{r}$ is
Bob’s posterior distribution on $I^{*}$ at the beginning of round $r$.
We define
$\displaystyle b_{r}$ $\displaystyle=H(I^{*})-H(I^{*}\mid
y^{1},\dotsc,y^{r-1})$ $\displaystyle=\log n-H(p^{r})$ $\displaystyle=\sum
p^{r}_{i}\log(np^{r}_{i}).$
Because the rows of $A^{r}$ are deterministic given $y^{1},\dotsc,y^{r-1}$,
Lemma 3.1 shows that any single measurement $j\in[m_{r}]$ satisfies
$I(Z^{r}_{j};Y^{r}_{j}\mid y^{1},\dotsc,y^{r-1})\leq C(b_{r}+1).$
for some constant $C$. Thus by Lemma B.1
$I(Z^{r};Y^{r}\mid y^{1},\dotsc,y^{r-1})\leq Cm_{r}(b_{r}+1).$
There is a Markov chain $(I^{*}\mid y^{1},\dotsc,y^{r-1})\to(Z^{r}\mid
y^{1},\dotsc,y^{r-1})\to(Y^{r}\mid y^{1},\dotsc,y^{r-1})$, so
$\displaystyle I(I^{*};Y^{r}\mid y^{1},\dotsc,y^{r-1})\leq I(Z^{r};Y^{r}\mid
y^{1},\dotsc,y^{r-1})\leq Cm_{r}(b_{r}+1).$
We define
$B_{r}=I(I^{*};Y^{1},\dotsc,Y^{r-1})=\operatorname*{\mathbb{E}}_{y}b_{r}$.
Therefore
$\displaystyle B_{r+1}$ $\displaystyle=I(I^{*};Y^{1},\dotsc,Y^{r})$
$\displaystyle=I(I^{*};Y^{1},\dotsc,Y^{r-1})+I(I^{*};Y^{r}\mid
Y^{1},\dotsc,Y^{r-1})$
$\displaystyle=B_{r}+\operatorname*{\mathbb{E}}_{y^{1},\dotsc,y^{r-1}}I(I^{*};Y^{r}\mid
y^{1},\dotsc,y^{r-1})$ $\displaystyle\leq
B_{r}+Cm_{r}\operatorname*{\mathbb{E}}_{y^{1},\dotsc,y^{r-1}}(b_{r}+1)$
$\displaystyle=(B_{r}+1)(Cm_{r}+1)-1$ $\displaystyle\leq
C^{\prime}m_{r}(B_{r}+1)$
for some constant $C^{\prime}$. Then for some constant $D\geq C^{\prime}$,
$I(I^{*};Y^{1},\dotsc,Y^{R})=B_{R+1}\leq D^{R}\prod_{i}m_{i}$
as desired. ∎
###### Corollary 3.3.
Any scheme using $R$ rounds with $m$ measurements has
$I(I^{*};Y^{1},\dotsc,Y^{R})\leq(Cm/R)^{R}$
for some constant $C$. Thus for sparse recovery, $m=\Omega(R\log^{1/R}n)$.
Minimizing over $R$, we find that $m=\Omega(\log\log n)$ independent of $R$.
###### Proof.
The equation follows from the AM-GM inequality. Furthermore, our setup is such
that Bob can recover $I^{*}$ from $Y$ with large probability, so
$I(I^{*};Y)=\Omega(\log n)$; this was formally shown in Lemma 6.3 of [HIKP12]
(modifying Lemma 4.3 of [PW11] to adaptive measurements and
$\epsilon=\Theta(1)$). The result follows. ∎
## 4 Lower bound for dependence on $k$ and $\epsilon$ for $\ell_{1}/\ell_{1}$
In Section 4.1 we establish a new lower bound on the communication complexity
of a two-party communication problem that we call
$\mathsf{Multi}\ell_{\infty}$. In Section 4.2 we then show how to use an
adaptive $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$ sparse recovery scheme
$\mathcal{A}$ to solve the communication problem
$\mathsf{Multi}\ell_{\infty}$. By the communication lower bound in Section
4.1, we obtain a lower bound on the number of measurements required of
$\mathcal{A}$.
### 4.1 Direct sum for distributional $\ell_{\infty}$
We assume basic familiarity with communication complexity; see the textbook of
Kushilevitz and Nisan [KN97] for further background. Our reason for using
communication complexity is to prove lower bounds, and we will do so by using
information-theoretic arguments. We refer the reader to the thesis of Bar-
Yossef [Bar02] for a comprehensive introduction to information-theoretic
arguments used in communication complexity.
We consider two-party randomized communication complexity. There are two
parties, Alice and Bob, with input vectors $x$ and $y$ respectively, and their
goal is to solve a promise problem $f(x,y)$. The parties have private
randomness. The communication cost of a protocol is its maximum transcript
length, over all possible inputs and random coin tosses. The randomized
communication complexity $R_{\delta}(f)$ is the minimum communication cost of
a randomized protocol $\Pi$ which for every input $(x,y)$ outputs $f(x,y)$
with probability at least $1-\delta$ (over the random coin tosses of the
parties). We also study the distributional complexity of $f$, in which the
parties are deterministic and the inputs $(x,y)$ are drawn from distribution
$\mu$, and a protocol is correct if it succeeds with probability at least
$1-\delta$ in outputting $f(x,y)$, where the probability is now taken over
$(x,y)\sim\mu$. We define $D_{\mu,\delta}(f)$ to be the minimum communication
cost of a correct protocol $\Pi$.
We consider the following promise problem $\mathsf{Gap}\ell_{\infty}^{B}$,
where $B$ is a parameter, which was studied in [SS02, BJKS04]. The inputs are
pairs $(x,y)$ of $m$-dimensional vectors, with
$x_{i},y_{i}\in\\{0,1,2,\ldots,B\\}$ for all $i\in[m]$, with the promise that
$(x,y)$ is one of the following types of instance:
* •
NO instance: for all $i$, $|x_{i}-y_{i}|\in\\{0,1\\}$, or
* •
YES instance: there is a unique $i$ for which $|x_{i}-y_{i}|=B$, and for all
$j\neq i$, $|x_{j}-y_{j}|\in\\{0,1\\}$.
The goal of a protocol is to decide which of the two cases (NO or YES) the
input is in.
Consider the distribution $\sigma$: for each $j\in[m]$, choose a random pair
$(Z_{j},P_{j})\in\\{0,1,2,\ldots,B\\}\times\\{0,1\\}\setminus\\{(0,1),(B,0)\\}$.
If $(Z_{j},P_{j})=(z,0)$, then $X_{j}=z$ and $Y_{j}$ is uniformly distributed
in $\\{z,z+1\\}$; if $(Z_{j},P_{j})=(z,1)$, then $Y_{j}=z$ and $X_{j}$ is
uniformly distributed on $\\{z-1,z\\}$. Let $Z=(Z_{1},\ldots,Z_{m})$ and
$P=(P_{1},\ldots,P_{m})$. Next choose a random coordinate $S\in[m]$. For
coordinate $S$, replace $(X_{S},Y_{S})$ with a uniform element of
$\\{(0,0),(0,B)\\}$. Let $X=(X_{1},\ldots,X_{m})$ and
$Y=(Y_{1},\ldots,Y_{m})$.
Using similar arguments to those in [BJKS04], we can show that there are
positive, sufficiently small constants $\delta_{0}$ and $C$ so that for any
randomized protocol $\Pi$ which succeeds with probability at least
$1-\delta_{0}$ on distribution $\sigma$,
$\displaystyle I(X,Y;\Pi|Z,P)\geq\frac{Cm}{B^{2}},$ (7)
where, with some abuse of notation, $\Pi$ is also used to denote the
transcript of the corresponding randomized protocol, and here the input
$(X,Y)$ is drawn from $\sigma$ conditioned on $(X,Y)$ being a NO instance.
Here, $\Pi$ is randomized, and succeeds with probability at least
$1-\delta_{0}$, where the probability is over the joint space of the random
coins of $\Pi$ and the input distribution.
Our starting point for proving (7) is Jayram’s lower bound for the conditional
mutual information when the inputs are drawn from a related distribution
(reference [70] on p.182 of [Bar02]), but we require several non-trivial
modifications to his argument in order to apply it to bound the conditional
mutual information for our input distribution, which is $\sigma$ conditioned
on $(X,Y)$ being a NO instance. Essentially, we are able to show that the
variation distance between our distribution and his distribution is small, and
use this to bound the difference in the conditional mutual information between
the two distributions. The proof is rather technical, and we postpone it to
Appendix C.
We make a few simple refinements to (7). Define the random variable $W$ which
is $1$ if $(X,Y)$ is a YES instance, and $0$ if $(X,Y)$ is a NO instance. Then
by definition of the mutual information, if $(X,Y)$ is drawn from $\sigma$
without conditioning on $(X,Y)$ being a NO instance, then we have
$\displaystyle I(X,Y;\Pi|W,Z,P)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}I(X,Y;\Pi|Z,P,W=0)$ $\displaystyle=$
$\displaystyle\Omega(m/B^{2}).$
Observe that
$\displaystyle I(X,Y;\Pi|S,W,Z,P)$ $\displaystyle\geq
I(X,Y;\Pi|W,Z,P)-H(S)=\Omega(m/B^{2}),$ (8)
where we assume that $\Omega(m/B^{2})-\log m=\Omega(m/B^{2})$. Define the
constant $\delta_{1}=\delta_{0}/4$. We now define a problem which involves
solving $r$ copies of $\mathsf{Gap}\ell_{\infty}^{B}$.
###### Definition 4.1 ($\mathsf{Multi}\ell_{\infty}^{r,B}$ Problem).
There are $r$ pairs of inputs
$(x^{1},y^{1}),(x^{2},y^{2}),\ldots,(x^{r},y^{r})$ such that each pair
$(x^{i},y^{i})$ is a legal instance of the $\mathsf{Gap}\ell_{\infty}^{B}$
problem. Alice is given $x^{1},\ldots,x^{r}$. Bob is given
$y^{1},\ldots,y^{r}$. The goal is to output a vector $v\in\\{NO,YES\\}^{r}$,
so that for at least a $1-\delta_{1}$ fraction of the entries $i$,
$v_{i}=\mathsf{Gap}\ell_{\infty}^{B}(x^{i},y^{i})$.
###### Remark 4.2.
Notice that Definition 4.1 is defining a promise problem. We will study the
distributional complexity of this problem under the distribution $\sigma^{r}$,
which is a product distribution on the $r$ instances
$(x^{1},y^{1}),(x^{2},y^{2}),\ldots,(x^{r},y^{r})$.
###### Theorem 4.3.
$D_{\sigma^{r},\delta_{1}}(\mathsf{Multi}\ell_{\infty}^{r,B})=\Omega(rm/B^{2}).$
###### Proof.
Let $\Pi$ be any deterministic protocol for
$\mathsf{Multi}\ell_{\infty}^{r,B}$ which succeeds with probability at least
$1-\delta_{1}$ in solving $\mathsf{Multi}\ell_{\infty}^{r,B}$ when the inputs
are drawn from $\sigma^{r}$, where the probability is taken over the input
distribution. We show that $\Pi$ has communication cost $\Omega(rm/B^{2})$.
Let $X^{1},Y^{1},S^{1},W^{1},Z^{1},P^{1}\ldots,X^{r},Y^{r},S^{r},W^{r},Z^{r},$
and $P^{r}$ be the random variables associated with $\sigma^{r}$, i.e.,
$X^{j},Y^{j},S^{j},W^{j},P^{j}$ and $Z^{j}$ correspond to the random variables
$X,Y,S,W,Z,P$ associated with the $j$-th independent instance drawn according
to $\sigma$, defined above. We let $X=(X^{1},\ldots,X^{r})$,
$X^{<j}=(X^{1},\ldots,X^{j-1})$, and $X^{-j}$ equal $X$ without $X^{j}$.
Similarly we define these vectors for $Y,S,W,Z$ and $P$.
By the chain rule for mutual information,
$I(X^{1},\ldots,X^{r},Y^{1},\ldots,Y^{r};\Pi|S,W,Z,P)$ is equal to
$\sum_{j=1}^{r}I(X^{j},Y^{j};\Pi|X^{<j},Y^{<j},S,W,Z,P).$ Let $V$ be the
output of $\Pi$, and $V_{j}$ be its $j$-th coordinate. For a value $j\in[r]$,
we say that $j$ is good if
$\Pr_{X,Y}[V_{j}=\mathsf{Gap}\ell_{\infty}^{B}(X^{j},Y^{j})]\geq
1-\frac{2\delta_{0}}{3}.$ Since $\Pi$ succeeds with probability at least
$1-\delta_{1}=1-\delta_{0}/4$ in outputting a vector with at least a
$1-\delta_{0}/4$ fraction of correct entries, the expected probability of
success over a random $j\in[r]$ is at least $1-\delta_{0}/2$, and so by a
Markov argument, there are $\Omega(r)$ good indices $j$.
Fix a value of $j\in[r]$ that is good, and consider
$I(X^{j},Y^{j};\Pi|X^{<j},Y^{<j},S,W,Z,P)$. By expanding the conditioning,
$I(X^{j},Y^{j};\Pi|X^{<j},Y^{<j},S,W,Z,P)$ is equal to
$\displaystyle{\bf
E}_{x,y,s,w,z,p}[I(X^{j},Y^{j};\Pi\mid(X^{<j},Y^{<j},S^{-j},W^{-j},Z^{-j},P^{-j})=(x,y,s,w,z,p),S^{j},W^{j},Z^{j},P^{j})].$
(9)
For each $x,y,s,w,z,p$, define a randomized protocol $\Pi_{x,y,s,w,z,p}$ for
$\mathsf{Gap}\ell_{\infty}^{B}$ under distribution $\sigma$. Suppose that
Alice is given $a$ and Bob is given $b$, where $(a,b)\sim\sigma$. Alice sets
$X^{j}=a$, while Bob sets $Y^{j}=b$. Alice and Bob use $x,y,s,w,z$ and $p$ to
set their remaining inputs as follows. Alice sets $X^{<j}=x$ and Bob sets
$Y^{<j}=y$. Alice and Bob can randomly set their remaining inputs without any
communication, since for $j^{\prime}>j$, conditioned on
$S^{j^{\prime}},W^{j^{\prime}},Z^{j^{\prime}}$, and $P^{j^{\prime}}$, Alice
and Bob’s inputs are independent. Alice and Bob run $\Pi$ on inputs $X,Y$, and
define $\Pi_{x,y,s,w,z,p}(a,b)=V_{j}.$ We say a tuple $(x,y,s,w,z,p)$ is good
if
$\displaystyle\Pr_{X,Y}[V_{j}=\mathsf{Gap}\ell_{\infty}^{B}(X^{j},Y^{j})\
\mid\ X^{<j}=x,Y^{<j}=y,S^{-j}=s,W^{-j}=w,Z^{-j}=z,P^{-j}=p]\geq
1-\delta_{0}.$
By a Markov argument, and using that $j$ is good, we have
$\Pr_{x,y,s,w,z,p}[(x,y,s,w,z,p)\textrm{ is good }]=\Omega(1).$ Plugging into
(9), $I(X^{j},Y^{j};\Pi|X^{<j},Y^{<j},S,W,Z,P)$ is at least a constant times
$\displaystyle{\bf E}_{x,y,s,w,z,p}[I(X^{j}Y^{j};\Pi|$
$\displaystyle(X^{<j},Y^{<j},S^{-j},W^{-j},Z^{-j},P^{-j})=(x,y,s,w,z,p),$
$\displaystyle S^{j},W^{j},Z^{j},P^{j},(x,y,s,w,z,p)\textrm{ is good})].$
For any $(x,y,s,w,z,p)$ that is good, $\Pi_{x,y,s,w,z,p}(a,b)=V_{j}$ with
probability at least $1-\delta_{0}$, over the joint distribution of the
randomness of $\Pi_{x,y,s,w,z,p}$ and $(a,b)\sim\sigma$. By (8),
$\displaystyle{\bf E}_{x,y,s,w,z,p}[I(X^{j},Y^{j};\Pi|$
$\displaystyle(X^{<j},Y^{<j},S^{-j},W^{-j},Z^{-j},P^{-j})=(x,y,s,w,z,p),$
$\displaystyle S^{j},W^{j},Z^{j},P^{j},(x,y,s,w,z,p)\textrm{ is
good}]=\Omega\left(\frac{m}{B^{2}}\right).$
Since there are $\Omega(r)$ good indices $j$, we have
$I(X^{1},\ldots,X^{r};\Pi|S,W,Z,P)=\Omega(mr/B^{2}).$ Since the distributional
complexity $D_{\sigma^{r},\delta_{1}}(\mathsf{Multi}\ell_{\infty}^{r,B})$ is
at least the minimum of $I(X^{1},\ldots,X^{r};\Pi|S,W,Z,P)$ over deterministic
protocols $\Pi$ which succeed with probability at least $1-\delta_{1}$ on
input distribution $\sigma^{r}$, it follows that
$D_{\sigma^{r},\delta_{1}}(\mathsf{Multi}\ell_{\infty}^{r,B})=\Omega(mr/B^{2})$.
∎
### 4.2 The overall lower bound
We use the theorem in the previous subsection with an extension of the method
of section 6.3 of [PW11].
Let $X\subset\mathbb{R}^{n}$ be a distribution with
$x_{i}\in\\{-n^{d},\dotsc,n^{d}\\}$ for all $i\in[n]$ and $x\in X$. Here
$d=\Theta(1)$ is a parameter. Given an adaptive compressed sensing scheme
$\mathcal{A}$, we define a $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$
sparse recovery multiround bit scheme on $X$ as follows.
Let $A^{i}$ be the $i$-th (adaptively chosen) measurement matrix of the
compressed sensing scheme. We may assume that the union of rows in matrices
$A^{1},\ldots,A^{r}$ generated by $\mathcal{A}$ is an orthonormal system,
since the rows can be orthogonalized in a post-processing step. We can assume
that $r\leq n$.
Choose a random $u\in\mathbb{R}^{n}$ from distribution
$\mathcal{N}(0,\frac{1}{n^{c}}\cdot I_{n\times n})$, where $c=\Theta(1)$ is a
parameter. We require that the compressed sensing scheme outputs a valid
result of $(1+\epsilon)$-approximate recovery on $x+u$ with probability at
least $1-\delta$, over the choice of $u$ and its random coins. By Yao’s
minimax principle, we can fix the randomness of the compressed sensing scheme
and assume that the scheme is deterministic.
Let $B^{1}$ be the matrix $A^{1}$ with entries rounded to $t\log n$ bits for a
parameter $t=\Theta(1)$. We compute $B^{1}x$. Then, we compute
$B^{1}x+A^{1}u$. From this, we compute $A^{2}$, using the algorithm specified
by $\mathcal{A}$ as if $B^{1}x+A^{1}u$ were equal to $A^{1}x^{\prime}$ for
some $x^{\prime}$. For this, we use the following lemma, which is Lemma 5.1 of
[DIPW10].
###### Lemma 4.4.
Consider any $m\times n$ matrix $A$ with orthonormal rows. Let $B$ be the
result of rounding $A$ to $b$ bits per entry. Then for any
$v\in\mathbb{R}^{n}$ there exists an $s\in\mathbb{R}^{n}$ with $Bv=A(v-s)$ and
$\|s\|_{1}<n^{2}2^{-b}\|v\|_{1}$.
In general for $i\geq 2$, given
$B^{1}x+A^{1}u,B^{2}x+A^{2}u,\ldots,B^{i-1}x+A^{i-1}u$ we compute $A^{i}$, and
round to $t\log n$ bits per entry to get $B^{i}$. The output of the multiround
bit scheme is the same as that of the compressed sensing scheme. If the
compressed sensing scheme uses $r$ rounds, then the multiround bit scheme uses
$r$ rounds. Let $b$ denote the total number of bits in the concatenation of
discrete vectors $B^{1}x,B^{2}x,\ldots,B^{r}x$.
We give a generalization of Lemma 5.2 of [PW11] which relates bit schemes to
sparse recovery schemes. Here we need to generalize the relation from non-
adaptive schemes to adaptive schemes, using Gaussian noise instead of uniform
noise, and arguing about multiple rounds of the algorithm.
###### Lemma 4.5.
For $t=O(1+c+d)$, a lower bound of $\Omega(b)$ bits for a multiround bit
scheme with error probability at most $\delta+1/n$ implies a lower bound of
$\Omega(b/((1+c+d)\log n))$ measurements for $(1+\epsilon)$-approximate sparse
recovery schemes with failure probability at most $\delta$.
###### Proof.
Let $\mathcal{A}$ be a $(1+\epsilon)$-approximate adaptive compressed sensing
scheme with failure probability $\delta$. We will show that the associated
multiround bit scheme has failure probability $\delta+1/n$.
By Lemma 4.4, for any vector $x\in\\{-n^{d},\ldots,n^{d}\\}$ we have
$B^{1}x=A^{1}(x+s)$ for a vector $s$ with $\left\lVert s\right\rVert_{1}\leq
n^{2}2^{-t\log n}\left\lVert x\right\rVert_{1}$, so $\left\lVert
s\right\rVert_{2}\leq n^{2.5-t}\left\lVert x\right\rVert_{2}\leq n^{3.5+d-t}$.
Notice that $u+s\sim\mathcal{N}(s,\frac{1}{n^{c}}\cdot I_{n\times n})$. We use
the following quick suboptimal upper bound on the statistical distance between
two univariate normal distributions, which suffices for our purposes.
###### Fact 4.6.
(see section 3 of [Pol05]) The variation distance between
$\mathcal{N}(\theta_{1},1)$ and $\mathcal{N}(\theta_{2},1)$ is
$\frac{4\tau}{\sqrt{2\pi}}+O(\tau^{2}),$ where
$\tau=|\theta_{1}-\theta_{2}|/2$.
It follows by Fact 4.6 and independence across coordinates, that the variation
distance between $\mathcal{N}(0,\frac{1}{n^{c}}\cdot I_{n\times n})$ and
$\mathcal{N}(s,\frac{1}{n^{c}}\cdot I_{n\times n})$ is the same as that
between $\mathcal{N}(0,I_{n\times n})$ and $\mathcal{N}(s\cdot
n^{c/2},I_{n\times n})$, which can be upper-bounded as
$\displaystyle\sum_{i=1}^{n}\cdot\frac{2n^{c/2}|s_{i}|}{\sqrt{2\pi}}+O(n^{c}s_{i}^{2})$
$\displaystyle=$ $\displaystyle O(n^{c/2}\|s\|_{1}+n^{c}\|s\|_{2}^{2})$
$\displaystyle=$ $\displaystyle
O(n^{c/2}\cdot\sqrt{n}\|s\|_{2}+n^{c}\|s\|_{2}^{2})$ $\displaystyle=$
$\displaystyle O(n^{c/2+4+d-t}+n^{c+7+2d-2t}).$
It follows that for $t=O(1+c+d)$, the variation distance is at most $1/n^{2}$.
Therefore, if $\mathcal{T}^{1}$ is the algorithm which takes $A^{1}(x+u)$ and
produces $A^{2}$, then
$\mathcal{T}^{1}(A^{1}(x+u))=\mathcal{T}^{1}(B^{1}x+A^{1}u)$ with probability
at least $1-1/n^{2}$. This follows since $B^{1}x+A^{1}u=A^{1}(x+u+s)$ and
$u+s$ and $u$ have variation distance at most $1/n^{2}$.
In the second round, $B^{2}x+A^{2}u$ is obtained, and importantly we have for
the algorithm $\mathcal{T}^{2}$ in the second round,
$\mathcal{T}^{2}(A^{2}(x+u))=\mathcal{T}^{2}(B^{2}x+A^{2}u)$ with probability
at least $1-1/n^{2}$. This follows since $A^{2}$ is a deterministic function
of $A^{1}u$, and $A^{1}u$ and $A^{2}u$ are independent since $A^{1}$ and
$A^{2}$ are orthonormal while $u$ is a vector of i.i.d. Gaussians (here we use
the rotational invariance / symmetry of Gaussian space). It follows by
induction that with probability at least $1-r/n^{2}\geq 1-1/n$, the output of
the multiround bit scheme agrees with that of $\mathcal{A}$ on input $x+u$.
Hence, if $m_{i}$ is the number of measurements in round $i$, and
$m=\sum_{i=1}^{r}m_{i}$, then we have a multiround bit scheme using a total of
$b=mt\log n=O(m(1+c+d)\log n)$ bits and with failure probability $\delta+1/n$.
∎
The rest of the proof is similar to the proof of the non-adaptive lower bound
for $\ell_{1}/\ell_{1}$ sparse recovery given in [PW11]. We sketch the proof,
referring the reader to [PW11] for some of the details. Fix parameters
$B=\Theta(1/\epsilon^{1/2})$, $r=k$, $m=1/\epsilon^{3/2}$, and
$n=k/\epsilon^{3}$. Given an instance $(x^{1},y^{1}),\ldots,(x^{r},y^{r})$ of
$\mathsf{Multi}\ell_{\infty}^{r,B}$ we define the input signal $z$ to a sparse
recovery problem. We allocate a set $S^{i}$ of $m$ disjoint coordinates in a
universe of size $n$ for each pair $(x^{i},y^{i})$, and on these coordinates
place the vector $y^{i}-x^{i}$. The locations turn out to be essential for the
proof of Lemma 4.8 below, and are placed uniformly at random among the $n$
total coordinates (subject to the constraint that the $S^{i}$ are disjoint).
Let $\rho$ be the induced distribution on $z$.
Fix a $(1+\epsilon)$-approximate $k$-sparse recovery multiround bit scheme
$Alg$ that uses $b$ bits and succeeds with probability at least
$1-\delta_{1}/2$ over $z\sim\rho$. Let $S$ be the set of top $k$ coordinates
in $z$. As shown in equation (14) of [PW11], $Alg$ has the guarantee that if
$w=Alg(z)$, then
$\displaystyle\|(w-z)_{S}\|_{1}+\|(w-z)_{[n]\setminus
S}\|_{1}\leq(1+2\epsilon)\|z_{[n]\setminus S}\|_{1}.$ (10)
(the $1+2\epsilon$ instead of the $1+\epsilon$ factor is to handle the
rounding of entries of the $A^{i}$ and the noise vector $u$). Next is our
generalization of Lemma 6.8 of [PW11].
###### Lemma 4.7.
For $B=\Theta(1/\epsilon^{1/2})$ sufficiently large, suppose that
$\Pr_{z\sim\rho}[\|(w-z)_{S}\|_{1}\leq 10\epsilon\cdot\|z_{[n]\setminus
S}\|_{1}]\geq 1-\frac{\delta_{1}}{2}.$
Then $Alg$ requires $b=\Omega(k/\epsilon^{1/2})$.
###### Proof.
We show how to use $Alg$ to solve instances of
$\mathsf{Multi}\ell_{\infty}^{r,B}$ with probability at least $1-\delta_{1}$,
where the probability is over input instances to
$\mathsf{Multi}\ell_{\infty}^{r,B}$ distributed according to $\sigma^{r}$,
inducing the distribution $\rho$ on $z$. The lower bound will follow by
Theorem 4.3. Let $w$ be the output of $Alg$.
Given $x^{1},\ldots,x^{r}$, Alice places $-x^{i}$ on the appropriate
coordinates in the set $S^{i}$ used in defining $z$, obtaining a vector
$z_{Alice}$. Given $y^{1},\ldots,y^{r}$, Bob places the $y^{i}$ on the
appropriate coordinates in $S^{i}$. He thus creates a vector $z_{Bob}$ for
which $z_{Alice}+z_{Bob}=z$. In round $i$, Alice transmits $B^{i}z_{Alice}$ to
Bob, who computes $B^{i}(z_{Alice}+z_{Bob})$ and transmits it back to Alice.
Alice can then compute $B^{i}(z)+A^{i}(u)$ for a random
$u\sim\mathcal{N}(0,\frac{1}{n^{c}}\cdot I_{n\times n})$. We can assume all
coordinates of the output vector $w$ are in the real interval $[0,B]$, since
rounding the coordinates to this interval can only decrease the error.
To continue the analysis, we use a proof technique of [PW11] (see the proof of
Lemma 6.8 of [PW11] for a comparison). For each $i$ we say that $S^{i}$ is bad
if either
* •
there is no coordinate $j$ in $S^{i}$ for which $|w_{j}|\geq\frac{B}{2}$ yet
$(x^{i},y^{i})$ is a YES instance of $\mathsf{Gap}\ell_{\infty}^{B}$, or
* •
there is a coordinate $j$ in $S^{i}$ for which $|w_{j}|\geq\frac{B}{2}$ yet
either $(x^{i},y^{i})$ is a NO instance of $\mathsf{Gap}\ell_{\infty}^{B}$ or
$j$ is not the unique $j^{*}$ for which $y_{j^{*}}^{i}-x_{j^{*}}^{i}=B$.
The proof of Lemma 6.8 of [PW11] shows that the fraction $C>0$ of bad $S^{i}$
can be made an arbitrarily small constant by appropriately choosing an
appropriate $B=\Theta(1/\epsilon^{1/2})$. Here we choose $C=\delta_{1}$. We
also condition on $\|u\|_{2}\leq n^{-c}$ for a sufficiently large constant
$c>0$, which occurs with probability at least $1-1/n$. Hence, with probability
at least $1-\delta_{1}/2-1/n>1-\delta_{1}$, we have a $1-\delta_{1}$ fraction
of indices $i$ for which the following algorithm correctly outputs
$\mathsf{Gap}\ell_{\infty}(x^{i},y^{i})$: if there is a $j\in S^{i}$ for which
$|w_{j}|\geq B/2$, output YES, otherwise output NO. It follows by Theorem 4.3
that $Alg$ requires $b=\Omega(k/\epsilon^{1/2})$, independent of the number of
rounds. ∎
The next lemma is the same as Lemma 6.9 of [PW11], replacing $\delta$ in the
lemma statement there with the constant $\delta_{1}$ and observing that the
lemma holds for compressed sensing schemes with an arbitrary number of rounds.
###### Lemma 4.8.
Suppose $\Pr_{z\sim\rho}[\|(w-z)_{[n]\setminus
S}\|_{1}\leq(1-8\epsilon)\cdot\|z_{[n]\setminus S}\|_{1}]\geq\delta_{1}.$ Then
$Alg$ requires $b=\Omega(k\log(1/\epsilon)/\epsilon^{1/2})$.
###### Proof.
As argued in Lemma 6.9 of [PW11], we have $I(w;z)=\Omega(\epsilon
mr\log(n/(mr)))$, which implies that $b=\Omega(\epsilon mr\log(n/(mr)))$,
independent of the number $r$ of rounds used by $Alg$, since the only
information about the signal is in the concatenation of
$B^{1}z,\ldots,B^{r}z$. ∎
Finally, we combine our Lemma 4.7 and Lemma 4.8 to prove the analogue of
Theorem 6.10 of [PW11], which completes this section.
###### Theorem 4.9.
Any $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$ recovery scheme with
success probability at least $1-\delta_{1}/2-1/n$ must make
$\Omega(k/(\epsilon^{1/2}\cdot\log(k/\epsilon)))$ measurements.
###### Proof.
We will lower bound the number of bits used by any $\ell_{1}/\ell_{1}$
multiround bit scheme $Alg$. If $Alg$ succeeds with probability at least
$1-\delta_{1}/2$, then in order to satisfy (10), we must either have
$\|(w-z)_{S}\|_{1}\leq 10\epsilon\cdot\|z_{[n]\setminus S}\|_{1}$ or
$\|(w-z)_{[n]\setminus S}\|_{1}\leq(1-8\epsilon)\|z_{[n]\setminus S}\|_{1}$.
Since $Alg$ succeeds with probability at least $1-\delta_{1}/2$, it must
either satisfy the hypothesis of Lemma 4.7 or Lemma 4.8. But by these two
lemmas, it follows that $b=\Omega(k/\epsilon^{1/2})$. Therefore by Lemma 4.5,
any $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$ sparse recovery algorithm
succeeding with probability at least $1-\delta_{1}/2-1/n$ requires
$\Omega(k/(\epsilon^{1/2}\cdot\log(k/\epsilon)))$ measurements. ∎
## 5 Acknowledgements
Some of this work was performed while E. Price was an intern at IBM research,
and the rest was performed while he was supported by an NSF Graduate Research
Fellowship.
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## Appendix A Relationship between Post-Measurement and Pre-Measurement noise
In the setting of [ACD11], the goal is to recover a $k$-sparse $x$ from
observations of the form $Ax+w$, where $A$ has unit norm rows and $w$ is
i.i.d. Gaussian with variance $\left\lVert
x\right\rVert_{2}^{2}/\epsilon^{2}$. By ignoring the (irrelevant) component of
$w$ orthogonal to $A$, this is equivalent to recovering $x$ from observations
of the form $A(x+w)$. By contrast, our goal is to recover $x+w$ from
observations of the form $A(x+w)$, and for general $w$ rather than only for
Gaussian $w$.
By arguments in [PW11, HIKP12], for Gaussian $w$ the difference between
recovering $x$ and recovering $x+w$ is minor, so any lower bound of $m$ in the
[ACD11] setting implies a lower bound of $\min(m,\epsilon n)$ in our setting.
The converse is only true for proofs that use Gaussian $w$, but our proof fits
this category.
## Appendix B Information Chain Rule with Linear Observations
###### Lemma B.1.
Suppose $a_{i}=b_{i}+w_{i}$ for $i\in[s]$ and the $w_{i}$ are independent of
each other and the $b_{i}$. Then
$I(a;b)\leq\sum_{i}I(a_{i};b_{i})$
###### Proof.
Note that $h(a\mid b)=h(a-b\mid b)=h(w\mid b)=h(w)$. Thus
$\displaystyle I(a;b)$ $\displaystyle=h(a)-h(a\mid b)=h(a)-h(w)$
$\displaystyle\leq\sum_{i}h(a_{i})-h(w_{i})$
$\displaystyle=\sum_{i}h(a_{i})-h(a_{i}\mid b_{i})=\sum_{i}I(a_{i};b_{i})$
∎
## Appendix C Switching Distributions from Jayram’s Distributional Bound
We first sketch a proof of Jayram’s lower bound on the distributional
complexity of $\mathsf{Gap}\ell_{\infty}^{B}$ [Jay02], then change it to a
different distribution that we need for our sparse recovery lower bounds in
Subsection C.1. Let $X,Y\in\\{0,1,\ldots,B\\}^{m}$. Define distribution
$\mu^{m,B}$ as follows: for each $j\in[m]$, choose a random pair
$(Z_{j},P_{j})\in\\{0,1,2,\ldots,B\\}\times\\{0,1\\}\setminus\\{(0,1),(B,0)\\}$.
If $(Z_{j},P_{j})=(z,0)$, then $X_{j}=z$ and $Y_{j}$ is uniformly distributed
in $\\{z,z+1\\}$; if $(Z_{j},P_{j})=(z,1)$, then $Y_{j}=z$ and $X_{j}$ is
uniformly distributed on $\\{z-1,z\\}$. Let $X=(X_{1},\ldots,X_{m})$,
$Y=(Y_{1},\ldots,Y_{m})$, $Z=(Z_{1},\ldots,Z_{m})$ and
$P=(P_{1},\ldots,P_{m})$.
The other distribution we define is $\sigma^{m,B}$, which is the same as
distribution $\sigma$ in Section 4 (we include $m$ and $B$ in the notation
here for clarity). This is defined by first drawing $X$ and $Y$ according to
distribution $\mu^{m,B}$. Then, we pick a random coordinate $S\in[m]$ and
replace $(X_{S},Y_{S})$ with a uniformly random element in the set
$\\{(0,0),(0,B)\\}$.
Let $\Pi$ be a deterministic protocol that errs with probability at most
$\delta$ on input distribution $\sigma^{m,B}$.
By the chain rule for mutual information, when $X$ and $Y$ are distributed
according to $\mu^{m,B}$,
$\displaystyle I(X,Y;\Pi|Z,P)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}I(X_{j},Y_{j};\Pi|X^{<j},Y^{<j},Z,P),$
which is equal to
$\displaystyle\sum_{j=1}^{m}{\bf E}_{x,y,z,p}[I(X_{j},Y_{j};\Pi\
|Z_{j},P_{j},X^{<j}=x,Y^{<j}=y,Z^{-j}=z,P^{-j}=p)].$
Say that an index $j\in[m]$ is good if conditioned on $S=j$, $\Pi$ succeeds on
$\sigma^{m,B}$ with probability at least $1-2\delta$. By a Markov argument, at
least $m/2$ of the indices $j$ are good. Fix a good index $j$.
We say that the tuple $(x,y,z,p)$ is good if conditioned on $S=j$, $X^{<j}=x$,
$Y^{<j}=y$, $Z^{-j}=z$, and $P^{-j}=p$, $\Pi$ succeeds on $\sigma^{m,B}$ with
probability at least $1-4\delta$. By a Markov bound, with probability at least
$1/2$, $(x,y,z,p)$ is good. Fix a good $(x,y,z,p)$.
We can define a single-coordinate protocol $\Pi_{x,y,z,p,j}$ as follows. The
parties use $x$ and $y$ to fill in their input vectors $X$ and $Y$ for
coordinates $j^{\prime}<j$. They also use $Z^{-j}=z$, $P^{-j}=p$, and private
randomness to fill in their inputs without any communication on the remaining
coordinates $j^{\prime}>j$. They place their single-coordinate input $(U,V)$
on their $j$-th coordinate. The parties then output whatever $\Pi$ outputs.
It follows that $\Pi_{x,y,z,p,j}$ is a single-coordinate protocol
$\Pi^{\prime}$ which distinguishes $(0,0)$ from $(0,B)$ under the uniform
distribution with probability at least $1-4\delta$. For the single-coordinate
problem, we need to bound $I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j})$ when
$(X_{j},Y_{j})$ is uniformly random from the set
$\\{(Z_{j},Z_{j}),(Z_{j},Z_{j}+1)\\}$ if $P_{j}=0$, and $(X_{j},Y_{j})$ is
uniformly random from the set $\\{(Z_{j},Z_{j}),(Z_{j}-1,Z_{j})\\}$ if
$P_{j}=1$. By the same argument as in the proof of Lemma 8.2 of [BJKS04], if
$\Pi^{\prime}_{u,v}$ denotes the distribution on transcripts induced by inputs
$u$ and $v$ and private coins, then we have
$\displaystyle
I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j})\geq\Omega(1/B^{2})\cdot(h^{2}(\Pi^{\prime}_{0,0},\Pi^{\prime}_{0,B})+h^{2}(\Pi^{\prime}_{B,0},\Pi^{\prime}_{B,B})),$
(11)
where
$h(\alpha,\beta)=\sqrt{\frac{1}{2}\sum_{\omega\in\Omega}(\sqrt{\alpha(\omega)}-\sqrt{\beta(\omega)})^{2}}$
is the Hellinger distance between distributions $\alpha$ and $\beta$ on
support $\Omega$. For any two distributions $\alpha$ and $\beta$, if we define
$D_{TV}(\alpha,\beta)=\frac{1}{2}\sum_{\omega\in\Omega}|\alpha(\omega)-\beta(\omega)|$
to be the variation distance between them, then $\sqrt{2}\cdot
h(\alpha,\beta)\geq D_{TV}(\alpha,\beta)$ (see Proposition 2.38 of [Bar02]).
Finally, since $\Pi^{\prime}$ succeeds with probability at least $1-4\delta$
on the uniform distribution on input pair in $\\{(0,0),(0,B)\\}$, we have
$\sqrt{2}\cdot h(\Pi^{\prime}_{0,0},\Pi^{\prime}_{0,B})\geq
D_{TV}(\Pi^{\prime}_{0,0},\Pi^{\prime}_{0,B})=\Omega(1).$
Hence,
$\displaystyle
I(X_{j},Y_{j};\Pi|Z_{j},P_{j},X^{<j}=x,Y^{<j}=y,Z^{-j}=z,P^{-j}=p)$
$\displaystyle=\Omega(1/B^{2})$
for each of the $\Omega(m)$ good $j$. Thus $I(X,Y;\Pi|Z,P)=\Omega(m/B^{2})$
when inputs $X$ and $Y$ are distributed according to $\mu^{m,B}$, and $\Pi$
succeeds with probability at least $1-\delta$ on $X$ and $Y$ distributed
according to $\sigma^{m,B}$.
### C.1 Changing the distribution
Consider the distribution
$\zeta^{m,B}=(\sigma^{m,B}\mid(X_{S},Y_{S})=(0,0)).$
We show $I(X,Y;\Pi|Z)=\Omega(m/B^{2})$ when $X$ and $Y$ are distributed
according to $\zeta^{m,B}$ rather than according to $\mu^{m,B}$.
For $X$ and $Y$ distributed according to $\zeta^{m,B}$, by the chain rule we
again have that $I(X,Y;\Pi|Z,P)$ is equal to
$\displaystyle\sum_{j=1}^{m}{\bf
E}_{x,y,z,p}[I(X_{j},Y_{j};\Pi|Z_{j},P_{j},X^{<j}=x,Y^{<j}=y,Z^{-j}=z,P^{-j}=p)].$
Again, say that an index $j\in[m]$ is good if conditioned on $S=j$, $\Pi$
succeeds on $\sigma^{m,B}$ with probability at least $1-2\delta$. By a Markov
argument, at least $m/2$ of the indices $j$ are good. Fix a good index $j$.
Again, we say that the tuple $(x,y,z,p)$ is good if conditioned on $S=j$,
$X^{<j}=x$, $Y^{<j}=y$, $Z^{-j}=z$ and $P^{-j}=p$, $\Pi$ succeeds on
$\sigma^{m,B}$ with probability at least $1-4\delta$. By a Markov bound, with
probability at least $1/2$, $(x,y,z,p)$ is good. Fix a good $(x,y,z,p)$.
As before, we can define a single-coordinate protocol $\Pi_{x,y,z,p,j}$. The
parties use $x$ and $y$ to fill in their input vectors $X$ and $Y$ for
coordinates $j^{\prime}<j$. They can also use $Z^{-j}=z$, $P^{-j}=p$, and
private randomness to fill in their inputs without any communication on the
remaining coordinates $j^{\prime}>j$. They place their single-coordinate input
$(U,V)$, uniformly drawn from $\\{(0,0),(0,B)\\}$, on their $j$-th coordinate.
The parties output whatever $\Pi$ outputs. Let $\Pi^{\prime}$ denote
$\Pi_{x,y,z,p,j}$ for notational convenience.
The first issue is that unlike before $\Pi^{\prime}$ is not guaranteed to have
success probability at least $1-4\delta$ since $\Pi$ is not being run on input
distribution $\sigma^{m,B}$ in this reduction. The second issue is in bounding
$I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j})$ since $(X_{j},Y_{j})$ is now drawn
from the marginal distribution of $\zeta^{m,B}$ on coordinate $j$.
Notice that $S\neq j$ with probability $1-1/m$, which we condition on. This
immediately resolves the second issue since now the marginal distribution on
$(X_{j},Y_{j})$ is the same under $\zeta^{m,B}$ as it was under
$\sigma^{m,B}$; namely it is the following distribution: $(X_{j},Y_{j})$ is
uniformly random from the set $\\{(Z_{j},Z_{j}),(Z_{j},Z_{j}+1)\\}$ if
$P_{j}=0$, and $(X_{j},Y_{j})$ is uniformly random from the set
$\\{(Z_{j},Z_{j}),(Z_{j}-1,Z_{j})\\}$ if $P_{j}=1$.
We now address the first issue. After conditioning on $S\neq j$, we have that
$(X^{-j},Y^{-j})$ is drawn from $\zeta^{m-1,B}$. If instead $(X^{-j},Y^{-j})$
were drawn from $\mu^{m-1,B}$, then after placing $(U,V)$ the input to $\Pi$
would be drawn from $\sigma^{m,B}$ conditioned on a good tuple. Hence in that
case, $\Pi^{\prime}$ would succeed with probability $1-4\delta$. Thus for our
actual distribution on $(X^{-j},Y^{-j})$, after conditioning on $S\neq j$, the
success probability of $\Pi^{\prime}$ is at least
$1-4\delta-D_{TV}(\mu^{m-1,B},\zeta^{m-1,B}).$
Let $C^{\mu,m-1,B}$ be the random variable which counts the number of
coordinates $i$ for which $(X_{i},Y_{i})=(0,0)$ when $X$ and $Y$ are drawn
from $\mu^{m-1,B}$. Let $C^{\zeta,m-1,B}$ be a random variable which counts
the number of coordinates $i$ for which $(X_{i},Y_{i})=(0,0)$ when $X$ and $Y$
are drawn from $\zeta^{m-1,B}$. Observe that $(X_{i},Y_{i})=(0,0)$ in $\mu$
only if $P_{i}=0$ and $Z_{i}=0$, which happens with probability $1/(2B)$.
Hence, $C^{\mu,m-1,B}$ is distributed as Binomial$(m-1,1/(2B))$, while
$C^{\zeta,m-1,B}$ is distributed as Binomial$(m-2,1/(2B))+1$. We use
$\mu^{\prime}$ to denote the distribution of $C^{\mu,m-1,B}$ and
$\zeta^{\prime}$ to denote the distribution of $C^{\zeta,m-1,B}$. Also, let
$\iota$ denote the Binomial$(m-2,1/(2B))$ distribution. Conditioned on
$C^{\mu,m-1,B}=C^{\zeta,m-1,B}$, we have that $\mu^{m-1,B}$ and
$\zeta^{m-1,B}$ are equal as distributions, and so
$D_{TV}(\mu^{m-1,B},\zeta^{m-1,B})\leq D_{TV}(\mu^{\prime},\zeta^{\prime}).$
We use the following fact:
###### Fact C.1.
(see, e.g., Fact 2.4 of [GMRZ11]). Any binomial distribution $X$ with variance
equal to $\sigma^{2}$ satisfies $D_{TV}(X,X+1)\leq 2/\sigma$.
By definition,
$\mu^{\prime}=(1-1/(2B))\cdot\iota+1/(2B)\cdot\zeta^{\prime}.$
Since the variance of the Binomial$(m-2,1/(2B))$ distribution is
$(m-2)/(2B)\cdot(1-1/(2B))=m/(2B)(1-o(1)),$
applying Fact C.1 we have
$\displaystyle D_{TV}(\mu^{\prime},\zeta^{\prime})$ $\displaystyle=$
$\displaystyle
D_{TV}((1-1/(2B))\cdot\iota+(1/(2B))\cdot\zeta^{\prime},\zeta^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\cdot\|(1-1/(2B))\cdot\iota+(1/(2B))\cdot\zeta^{\prime}-\zeta^{\prime}\|_{1}$
$\displaystyle=$ $\displaystyle(1-1/(2B))\cdot D_{TV}(\iota,\zeta^{\prime})$
$\displaystyle\leq$ $\displaystyle\frac{2\sqrt{2B}}{\sqrt{m}}\cdot(1+o(1))$
$\displaystyle=$ $\displaystyle O\left(\sqrt{\frac{B}{m}}\right).$
It follows that the success probability of $\Pi^{\prime}$ is at least
$1-4\delta-O\left(\sqrt{\frac{B}{m}}\right)\geq 1-5\delta.$
Let $E$ be an indicator random variable for the event that $S\neq j$. Then
$H(E)=O((\log m)/m)$. Hence,
$\displaystyle I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j})$ $\displaystyle\geq$
$\displaystyle I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j},E)-O((\log m)/m)$
$\displaystyle\geq$ $\displaystyle(1-1/m)\cdot
I(X_{j},Y_{j};\Pi^{\prime}|Z_{j},P_{j},S\neq j)-O((\log m)/m)$
$\displaystyle=$ $\displaystyle\Omega(1/B^{2}),$
where we assume that $\Omega(1/B^{2})-O((\log m)/m)=\Omega(1/B^{2}).$
Hence, $I(X,Y;\Pi|Z,P)=\Omega(m/B^{2})$ when inputs $X$ and $Y$ are
distributed according to $\zeta^{m,B}$, and $\Pi$ succeeds with probability at
least $1-\delta$ on $X$ and $Y$ distributed according to $\sigma^{m,B}$.
|
arxiv-papers
| 2012-05-15T21:38:53 |
2024-09-04T02:49:30.966924
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eric Price and David P. Woodruff",
"submitter": "Eric Price",
"url": "https://arxiv.org/abs/1205.3518"
}
|
1205.3601
|
# A Symplectic Method to Generate Multivariate Normal Distributions
C. Baumgarten Paul Scherrer Institute, Switzerland
christian.baumgarten@psi.ch
###### Abstract
The AMAS group at the Paul Scherrer Institute developed an object oriented
library for high performance simulation of high intensity ion beam transport
with space charge HPC1 ; HPC2 . Such particle-in-cell (PIC) simulations
require a method to generate multivariate particle distributions as starting
conditions.
In a preceeding publications it has been shown that the generators of
symplectic transformations in two dimensions are a subset of the real Dirac
matrices (RDMs) and that few symplectic transformations are required to
transform a quadratic Hamiltonian into diagonal form rdm_paper ; geo_paper .
Here we argue that the use of RDMs is well suited for the generation of
multivariate normal distributions with arbitrary covariances. A direct and
simple argument supporting this claim is that this is the “natural” way how
such distributions are formed. The transport of charged particle beams may
serve as an example: An uncorrelated gaussian distribution of particles
starting at some initial position of the accelerator is subject to linear
deformations when passing through various beamline elements. These
deformations can be described by symplectic transformations.
Hence, if it is possible to derive the symplectic transformations that bring
up these covariances, it is also possible to produce arbitrary multivariate
normal distributions without Cholesky decomposition. The method allows the use
of arbitrary uncoupled distributions. The functional form of the coupled
multivariate distributions however depends in the general case on the type of
the used random number generator. Only gaussian generators always yield
gaussian multivariate distributions.
Hamiltonian mechanics, coupled oscillators, beam optics,statistics
###### pacs:
45.20.Jj, 05.45.Xt, 41.85.-p, 02.50.-r
## I Introduction
In Ref. geo_paper the author presented a so-called “decoupling” method that
is based on the systematic use the real Dirac matrices (RDMs) in coupled
linear optics. The RDMs are constructed from four pairwise anti-commuting
basic matrices with the “metric tensor” $g_{\mu\nu}=\mathrm{Diag}(-1,1,1,1)$,
formally written as:
$\gamma_{\mu}\,\gamma_{\nu}+\gamma_{\nu}\,\gamma_{\mu}=2\,g_{\mu\nu}\,.$ (1)
The remaining $12$ RDMs are constructed as products of the basic matrices as
described in the appendix.
The use of the RDMs enables to derive a straightforward method to transform
transport matrices, force matrices (“symplices”) and $\sigma$-matrices in such
a way that the transformed variables are independent, i.e. decoupled.
The reverse is required to generate multivariate normal distributions: A
transformation that transforms linear independent distributions of variables
in such a way that a given covariance matrix is generated. The idea therefore
is the following: Generate a set of independent normally distributed variables
with given variances and apply the inverse of the decoupling transformation
derived from the desired covariance matrix. This will couple the “independent”
variables in exactly the desired way. The presented scheme assumes an even
number of variables since it is based on canonical pairs, i.e. position
$q_{i}$ and momentum $p_{i}$ \- but it is always possible to ignore one of
those variables.
Since the method is based on pairs of canonical variables, the decoupling
scheme always treats two pairs of variables at a time, resulting in the use of
$4\times 4$-matrices. If more than four random variables are required, the
decoupling can be used iteratively in analogy to the Jacobi diagonalization
scheme for symmetric matrices geo_paper .
## II Coupled Linear Optics
In this section we give a brief summary of the major concept. Given the
following Hamiltonian function
$H={1\over 2}\,\psi^{T}\,{\bf A}\,\psi\,,$ (2)
where ${\bf A}$ is a symmetric matrix and $\psi$ is a state-vector or “spinor”
of the form $\psi=(q_{1},p_{1},q_{2},p_{2})^{T}$. The state vector hence
contains two pairs of canonical variables. The equations of motion (EQOM) then
have the familiar form
$\begin{array}[]{rcl}\dot{q}_{i}&=&{\partial H\over\partial p_{i}}\\\
\dot{p}_{i}&=&-{\partial H\over\partial q_{i}}\,,\end{array}$ (3)
or in vector notation:
$\begin{array}[]{rcl}\dot{\psi}&=&\gamma_{0}\,\nabla_{\psi}\,H\\\ &=&{\bf
F}\,\psi\\\ \end{array}$ (4)
where the force matrix ${\bf F}$ is given as ${\bf F}=\gamma_{0}\,{\bf A}$.
The matrix $\gamma_{0}$ is the symplectic unit matrix (sometimes labeled
${\cal J}$ or ${\cal S}$) and is identified with the real Dirac matrix
$\gamma_{0}$ (see appendix). We define the symmetric matrix of second moments
$\sigma$ containing the variances as diagonal and the covariances as off-
diagonal elements. The matrix ${\bf S}$ is simply defined as the product of
$\sigma$ with $\gamma_{0}$:
${\bf S}=\sigma\,\gamma_{0}\,.$ (5)
Both matrices, ${\bf F}$ and ${\bf S}$, fulfill the following equation (using
$\gamma_{0}^{T}=-\gamma_{0}$ and $\gamma_{0}^{2}=-{\bf 1}$):
${\bf F}^{T}=\gamma_{0}\,{\bf F}\,\gamma_{0}\,.$ (6)
Matrices that obey Eq. 6 have been named symplices, but they are also called
“infinitesimally symplectic” or “Hamiltonian” matrices Talman . Symplices
allow superposition, i.e. any sum of symplices is a symplex, but only the
product of anti-commuting symplices is a symplex rdm_paper .
Any real-valued $4\times 4$-matrix ${\bf M}$ can be written as a linear
combination of real Dirac matrices (RDM):
${\bf M}=\sum\limits_{k=0}^{15}\,m_{k}\,\gamma_{k}\,.$ (7)
The RDM-coefficients $m_{k}$ can be computed from the matrix ${\bf M}$ by:
$m_{k}={\mathrm{Tr}(\gamma_{k}^{2})\over 32}\,\mathrm{Tr}({\bf
M}\,\gamma_{k}+\gamma_{k}\,{\bf M})\,,$ (8)
where $\mathrm{Tr}({\bf X})$ is the trace of ${\bf X}$.
Hence the RDMs form a complete system of all real $4\times 4$-matrices, but
only ten RDMs fulfill Eq. 6 and are therefore symplices: The basic matrices
$\gamma_{0},\dots,\gamma_{3}$ and the six “bi-vectors”, i.e. the six possible
products of two basic matrices. The symplices are the generators of symplectic
transformations, i.e. the generators of the symplectic group.
As well-known, the Jacobi matrix of a canonical transformation is symplectic,
i.e. it fulfills the following equation Arnold ; Talman :
${\bf M}\,\gamma_{0}\,{\bf M}^{T}=\gamma_{0}\,.$ (9)
The EQOM have the general solution
$\psi(t)={\bf M}(t,t_{0})\,\psi(t_{0})\,,$ (10)
where ${\bf M}$ is a symplectic transfer matrix that is in case of constant
forces given by
${\bf M}(t,t_{0})=\exp{\left({\bf F}\,(t-t_{0})\right)}\,.$ (11)
Given now an (initial) set of $N$ normally distributed uncorrelated random
variables $\psi_{i}$, then the $\sigma$-matrix of these variables is given by
$\sigma={1\over
N}\,\sum\limits_{i=0}^{N-1}\,\psi_{i}\,\psi_{i}^{T}\equiv\langle\psi\,\psi^{T}\rangle\,,$
(12)
where the superscript “T” indicates the transpose, then the distribution at
time $t$ is given by:
$\sigma_{t}={1\over N}\,\sum\limits_{i=0}^{N-1}\,{\bf
M}\,\psi_{i}\,\psi_{i}^{T}\,{\bf M}^{T}={\bf M}\,\sigma_{0}\,{\bf M}^{T}\,.$
(13)
Hence with Eqn. (5) and (9) one has:
$\begin{array}[]{rcl}{\bf S}_{t}&=&-{\bf M}\,\sigma_{0}\,\gamma_{0}^{2}\,{\bf
M}^{T}\,\gamma_{0}\\\ &=&{\bf M}\,{\bf S}_{0}\,{\bf M}^{-1}\,.\end{array}$
(14)
That is - the transformation of ${\bf S}$ is a similarity-transformation with
a symplectic transformation matrix. The reverse transformation obviously is
${\bf S}_{0}={\bf M}^{-1}\,{\bf S}_{t}\,{\bf M}\,.$ (15)
Now we refer to the structural identity of the matrix ${\bf S}$ with the force
matrix ${\bf F}$. Both are symplices and since a transformation that decouples
${\bf F}$ has been shown to diagonalize the matrix ${\bf A}$ of the
Hamiltonian rdm_paper ; geo_paper , it is clear that the same method can be
used to diagonalize $\sigma$. The reverse of this transformation then
generates the desired distribution from an initially uncorrelated $\sigma$.
Instead of a Cholesky-decomposition we may therefore use a symplectic
similarity-transformation to generate the correlated distribution from an
initially uncorrelated distribution. In the context of charged particle
optics, the algorithm delivers even more useful information: the
transformation matrix ${\bf M}^{-1}$ is the transport matrix that is required
to generate an uncorrelated beam.
## III Symplectic Transformations and the Algorithm
The general form of a symplectic transformation matrix ${\bf R}_{b}$ is that
of a matrix exponential of a symplex $\gamma_{b}$ multiplied by a parameter
$\varepsilon$ representing either the angle or the “rapidity”:
$\begin{array}[]{rcl}{\bf
R}_{b}(\varepsilon)&=&\exp{(\gamma_{b}\,{\varepsilon\over 2})}={\bf
1}\,c+\gamma_{b}\,s\\\ {\bf
R}_{b}^{-1}(\varepsilon)&=&\exp{(-\gamma_{b}\,{\varepsilon\over 2})}={\bf
1}\,c-\gamma_{b}\,s\,,\end{array}$ (16)
where
$\begin{array}[]{rcl}c&=&\left\\{\begin{array}[]{lp{10mm}lcr}\cos{(\varepsilon/2)}&for&\gamma_{b}^{2}&=&-{\bf
1}\\\ \cosh{(\varepsilon/2)}&for&\gamma_{b}^{2}&=&{\bf 1}\\\
\end{array}\right.\\\
s&=&\left\\{\begin{array}[]{lp{10mm}lcr}\sin{(\varepsilon/2)}&for&\gamma_{b}^{2}&=&-{\bf
1}\\\ \sinh{(\varepsilon/2)}&for&\gamma_{b}^{2}&=&{\bf 1}\\\
\end{array}\right.\\\ \end{array}$ (17)
Transformations with $\gamma_{b}^{2}=-{\bf 1}$ are orthogonal transformations,
i.e. rotations, while those with $\gamma_{b}^{2}={\bf 1}$ are boosts.
The matrix ${\bf S}$ then is transformed according to:
${\bf S}\to{\bf R}\,{\bf S}\,{\bf R}^{-1}\,.$ (18)
The decoupling requires a sequence of transformations, so that the RDM-
coefficients of ${\bf S}$ have to be recomputed after each step.
Eqn. 8 may be used to compute the RDM-coefficients $s_{k}$ of the matrix ${\bf
S}$
${\bf S}=\sigma\,\gamma_{0}=\sum\limits_{i=0}^{9}\,s_{k}\,\gamma_{k}\,.$ (19)
Numerically it is faster to analyze directly the composition. For the choice
of RDMs used in Ref. rdm_paper ; geo_paper the RDM-coefficients of ${\bf S}$
as a function of $\sigma$ are given by:
$\begin{array}[]{rcl}s_{0}&=&(\sigma_{11}+\sigma_{22}+\sigma_{33}+\sigma_{44})/4\\\
s_{1}&=&(-\sigma_{11}+\sigma_{22}+\sigma_{33}-\sigma_{44})/4\\\
s_{2}&=&(\sigma_{13}-\sigma_{24})/2\\\ s_{3}&=&(\sigma_{12}+\sigma_{34})/2\\\
s_{4}&=&(\sigma_{12}-\sigma_{34})/2\\\ s_{5}&=&-(\sigma_{14}+\sigma_{23})/2\\\
s_{6}&=&(\sigma_{11}-\sigma_{22}+\sigma_{33}-\sigma_{44})/4\\\
s_{7}&=&(\sigma_{13}+\sigma_{24})/2\\\
s_{8}&=&(\sigma_{11}+\sigma_{22}-\sigma_{33}-\sigma_{44})/4\\\
s_{9}&=&(\sigma_{14}-\sigma_{23})/2\\\ \end{array}$ (20)
Now we use the following abbreviation using the notation of $3$-dimensional
vector algebra:
$\begin{array}[]{rcl}{\cal E}&=&s_{0}\\\ \vec{P}&=&(s_{1},s_{2},s_{3})^{T}\\\
\vec{E}&=&(s_{4},s_{5},s_{6})^{T}\\\
\vec{B}&=&(s_{7},s_{8},s_{9})^{T}\,,\end{array}$ (21)
and furthermore:
$\begin{array}[]{rclp{5mm}rcl}M_{r}&=&\vec{E}\,\vec{B}&&\vec{r}&\equiv&{\cal
E}\,\vec{P}+\vec{B}\times\vec{E}\\\
M_{g}&=&\vec{B}\,\vec{P}&&\vec{g}&\equiv&{\cal
E}\,\vec{E}+\vec{P}\times\vec{B}\\\
M_{b}&=&\vec{E}\,\vec{P}&&\vec{b}&\equiv&{\cal
E}\,\vec{B}+\vec{E}\times\vec{P}\\\ \end{array}$ (22)
The decoupling is done by a sequence of maximal six symplectic transformations
geo_paper . A transformation with $\varepsilon=0$ can be omitted. After each
transformation, the RDM-coefficients $s_{k}$ have to be updated and Eqns. (21)
and (LABEL:eq_aux_vecs) have to be re-evaluated:
1. 1.
${\bf R}_{0}(\varepsilon)$ with $\varepsilon=\arctan{({M_{g}\over M_{r}})}$.
2. 2.
${\bf R}_{7}(\varepsilon)$ with $\varepsilon=\arctan{({b_{z}\over b_{y}})}$.
3. 3.
${\bf R}_{9}(\varepsilon)$ with $\varepsilon=-\arctan{({b_{x}\over b_{y}})}$.
4. 4.
${\bf R}_{2}(\varepsilon)$ with $\varepsilon=\mathrm{artanh}{({M_{r}\over
b_{y}})}$.
5. 5.
${\bf R}_{0}(\varepsilon)$ with $\varepsilon={1\over
2}\,\arctan{({2\,M_{b}\over\vec{E}^{2}-\vec{P}^{2}})}$
6. 6.
${\bf R}_{8}(\varepsilon)$ with $\varepsilon=-\arctan{({P_{z}\over P_{x}})}$.
Given an initial covariance matrix $\sigma_{0}$, the sequence of computation
therefore is:
1. 1.
Compute the RDM-coefficients $s_{k}$ according to Eqn. (LABEL:eq_rdm_coeffs)
and the quantities defined in Eqns. (21) and (LABEL:eq_aux_vecs).
2. 2.
Compute the first (or next, resp.) transformation matrix ${\bf R}$.
3. 3.
Compute the product of the transformation matrices (and of the inverse) ${\bf
M}_{n+1}={\bf R}_{n+1}\,{\bf R}_{n}$.
4. 4.
Apply the first (or next, resp.) transformation ${\bf S}_{n+1}={\bf R}\,{\bf
S}_{n}\,{\bf R}^{-1}$.
5. 5.
Compute $\sigma_{n+1}=-{\bf S}_{n+1}\,\gamma_{0}$.
6. 6.
Continue with next transformation at step 1).
The six iterations yield the desired diagonal matrix $\sigma_{6}$ and the
matrices ${\bf M}_{6}$ and its inverse, so that
${\bf S}_{6}={\bf M}_{6}\,{\bf S}_{0}\,{\bf M}_{6}^{-1}\,.$ (23)
or:
$\sigma_{6}={\bf M}_{6}\,\sigma_{0}\,{\bf M}_{6}^{T}\,.$ (24)
The diagonal elements of $\sigma_{6}$ are the variances of the uncoupled
gaussian distribution. Given $\psi_{i}$ is the i-th uncoupled random state
vector, then ${\bf M}_{6}^{-1}\,\psi_{i}$ is the corresponding state vector
with the multivariate normal distribution.
## IV Example
Consider for instance the (arbitrary) matrix of second moments $\sigma_{0}$
$\left(\begin{array}[]{cccccc}5.8269&-0.0303&0.2292&0.0000&-0.0960&1.4897\\\
-0.0303&0.8851&0.0000&-0.0311&1.8053&-0.0015\\\
0.2292&0.0000&3.6058&-0.0235&0.0000&0.0000\\\
0.0000&-0.0311&-0.0235&0.6844&0.0000&0.0000\\\
-0.0960&1.8053&0.0000&0.0000&7.0607&-0.0224\\\
1.4897&-0.0015&0.0000&0.0000&-0.0224&0.7304\\\ \end{array}\right)$ (25)
Figure 1: Top: Correlations between several variables of the multivariate
normal distribution. Bottom: The resulting probability distributions for the
individual variables are again gaussian.
The diagonal matrix $\sigma_{6}$ is computed to be
$\left(\begin{array}[]{cccccc}1.9982&0&0&0&0&0\\\ 0&1.2984&0&0&0&0\\\
0&0&1.1116&0&0&0\\\ 0&0&0&2.2124&0&0\\\ 0&0&0&0&1.4029&0\\\
0&0&0&0&0&1.6637\\\ \end{array}\right)$ (26)
Now $10^{5}$ random vectors have been generated with a Gaussian random number
generator of unit variance. The vector elements have been scaled with
corresponding variances, given by the root of the diagonal elements of
$\sigma_{6}$ and then been multiplied (or transformed) with ${\bf M}^{-1}$
given by
$\left(\begin{array}[]{cccccc}-0.1727&-0.0330&0.0081&-1.6049&-0.0725&0.1893\\\
-0.0371&0.3392&0.7051&0.0093&0.3402&0.1034\\\
0.9474&0.1485&0.0008&-0.0613&-0.3429&0.9838\\\
-0.0573&0.5025&-0.0072&0.0001&-0.4733&-0.1464\\\
-0.1641&1.6387&0.3732&0.0202&1.4755&0.4320\\\
-0.3455&-0.0555&0.0042&-0.3515&-0.1204&0.3416\\\ \end{array}\right)$ (27)
Then the covariance matrix of the produced random vectors was evaluated. The
result is:
$\left(\begin{array}[]{cccccc}5.7946&-0.0247&0.2343&-0.0060&-0.1036&1.4825\\\
-0.0247&0.8917&0.0023&-0.0306&1.8200&0.0034\\\
0.2343&0.0023&3.5910&-0.0299&-0.0158&0.0033\\\
-0.0060&-0.0306&-0.0299&0.6849&-0.0000&-0.0012\\\
-0.1036&1.8200&-0.0158&-0.0000&7.0928&-0.0148\\\
1.4825&0.0034&0.0033&-0.0012&-0.0148&0.7294\\\ \end{array}\right)$ (28)
Fig. 1 shows some of the distributions as examples.
The same procedure can be done with any initial probability distribution and
the algorithm will produce the desired second moments. But the functional form
of the resulting distributions of the transformed variables will only be
similar to the initial distribution in the Gaussian case. Fig. 2 shows the
results for the same covariance matrix if the decoupled variables have a
uniform probability distribution, but same variances. The covariance matrix is
correctly reproduced.
Figure 2: Top: Correlations between several variables of the multivariate
flat distribution. Bottom: The resulting probability distributions for the
individual variables strongly depend on the correlations. The stronger the
correlations, the more gaussian the distribution will be (central limiting
theorem).
## V Conclusion
The method of symplectic decoupling of linearily coupled variables has been
applied to the problem of multivariate random distributions. It has been shown
that the use of sympleptic algebra has severe advantages: The same methods can
be applied to solve a variety of problems. The presented algorithm is
especially interesting for the generation of starting conditions of particle
tracking codes like - for example - OPAL HPC1 ; HPC2 .
In cases where the decoupled process is known to have a non-Gaussian
probability distribution and if the transport matrix ${\bf M}$ of a linear
transport system is known, it should be possible to derive unknown parameters
of the initial distribution by comparison with the computed expected
distribution. Fig. 2 shows that a flat distribution yields a clear
“signature”.
###### Acknowledgements.
The software used for the computation has been written in “C” and been
compiled with the GNU©-C++ compiler 3.4.6 on Scientific Linux. The CERN
library (PAW) was used to generate the figures.
## Appendix A The $\gamma$-Matrices
The real Dirac matrices used throughout this paper are:
$\begin{array}[]{rclp{4mm}rcl}\gamma_{0}&=&\left(\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{array}\right)&&\gamma_{1}&=&\left(\begin{array}[]{cccc}0&-1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{array}\right)\\\
\gamma_{2}&=&\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\
\end{array}\right)&&\gamma_{3}&=&\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
\gamma_{14}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3};&&\gamma_{15}&=&{\bf
1}\\\
\gamma_{4}&=&\gamma_{0}\,\gamma_{1};&&\gamma_{7}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{1}=\gamma_{2}\,\gamma_{3}\\\
\gamma_{5}&=&\gamma_{0}\,\gamma_{2};&&\gamma_{8}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{2}=\gamma_{3}\,\gamma_{1}\\\
\gamma_{6}&=&\gamma_{0}\,\gamma_{3};&&\gamma_{9}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{3}=\gamma_{1}\,\gamma_{2}\\\
\gamma_{10}&=&\gamma_{14}\,\gamma_{0}=\gamma_{1}\,\gamma_{2}\,\gamma_{3}&&\gamma_{11}&=&\gamma_{14}\,\gamma_{1}=\gamma_{0}\,\gamma_{2}\,\gamma_{3}\\\
\gamma_{12}&=&\gamma_{14}\,\gamma_{2}=\gamma_{0}\,\gamma_{3}\,\gamma_{1}&&\gamma_{13}&=&\gamma_{14}\,\gamma_{3}=\gamma_{0}\,\gamma_{1}\,\gamma_{2}\\\
\end{array}$ (29)
## References
## References
* (1) J. J. Yang, A. Adelmann, M. Humbel, M. Seidel, and T. J. Zhang, Phys. Rev. ST Accel. Beams 13, 064201 (2010).
* (2) Y. J. Bi, A. Adelmann, R. Dölling, M. Humbel, W. Joho, M. Seidel, and T. J. Zhang, Phys. Rev. ST Accel. Beams 14, 054402 (2011).
* (3) C. Baumgarten; Phys. Rev. ST Accel. Beams. 14, 114002 (2011).
* (4) C. Baumgarten; arXiv:1201.0907 (2012), submitted to Phys. Rev. ST Accel. Beams.
* (5) R. Talman: Geometric Mechanics; 2nd Ed., Wiley-VCH Weinheim, Germany, 2007.
* (6) V.I. Arnold: Mathematical Methods of Classical Mechanics; 2nd Ed., Springer, New York 2010.
|
arxiv-papers
| 2012-05-16T09:07:30 |
2024-09-04T02:49:30.977075
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian Baumgarten",
"submitter": "Christian Baumgarten",
"url": "https://arxiv.org/abs/1205.3601"
}
|
1205.3648
|
# 6-Body Central Configurations Formed by Tow Isosceles Triangles111This work
is partially supported by NSF of China and Youth found of Mianyang Normal
University.
Furong Zhao1,2 and Shiqing Zhang1
1Department of Mathematics, Sichuan University, Chengdu, 610064,P.R.China
2Department of Mathematics and Computer Science, Mianyang Normal University,
Mianyang, Sichuan,621000,P.R.China
Abstract: In this paper,we show the existence of a class of 6-body central
configurations with two isosceles triangles;which are congruent to each other
and keep some distance.We also study the necessary conditions about masses for
the bodies which can form a central configuration.
Keywords :6-body problems,central configurations,isosceles triangles.
MSC: 34C15,34C25.
## 1 Introduction and Main Results
The Newtonian N-body problem concerns the motion of N particles with masses
$m_{j}\in R^{+}$ and positions $q_{j}\in R^{3}$$(j=1,2,...,N)$ ,the motion is
governed by Newton’s second law and the Universal law:
$m_{j}\ddot{q}_{j}=\frac{\partial U(q)}{\partial{q}_{j}},$ (1.1)
where $q=(q_{1},q_{2},\cdots,q_{N})$ and $U(q)$ is Newtonian potential:
$U(q)=\sum_{1\leqslant j<k\leqslant N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|},$ (1.2)
Consider the space
$X=\\{q=(q_{1},q_{2},\cdots,q_{N})\in R^{3N}:\sum_{j=1}^{N}m_{j}q_{j}=0\\},$
(1.3)
i.e,suppose that the center of mass is fixed at the origin of the space.
Because the potential is singular when two particles have same position, it is
natural to assume that the configuration avoids the collision set
$\triangle=\\{q=(q_{1},\cdots,q_{N}):q_{j}=q_{k}$ for some $k\neq j\\}$.The
set $X\backslash\triangle$ is called the configuration space.
Definition 1.1([17,22]):A configuration $q=(q_{1},q_{2},\cdots,q_{N})\in
X\backslash\triangle$ is called a central configuration if there exists a
constant $\lambda$ such that
$\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant N.$ (1.4)
The value of constant $\lambda$ in (1.4) is uniquely determined by
$\lambda=\frac{U}{I},$ (1.5)
where
$I=\sum_{k=1}^{N}m_{k}|q_{k}|^{2}.$ (1.6)
Since the general solution of the N-body problem can’t be given, great
importance has been attached to search for particular solutions from the very
beginning. A homographic solution is that a configuration is preserved for all
time. Central configurations and homographic solutions are linked by the
Laplace theorem ([17,22]). Collapse orbits and parabolic orbits have relations
with the central configurations([15,16]).So finding central configurations
becomes very important. The main general open problem for the cental
configurations is due to Winter[22]and Smale[20]:Is the number of planar
central configurations finite for any choice of positive masses
$m_{1},...,m_{N}$?Hampton and Moeckel([6]) have proved this conjecture for
four any given positive masses.
In 1941, Wintner([22]) have studied regular polygon central configurations.
Moeckel ([11]),Zhang and Zhou([23]) have studied highly symmetrical central
configuration of Newtonian N-body problems.Llibre and Mello ([8]) have studied
a class of 6-body central configurations.
Based the above works,we find a classes of central configurations in the
6-body problems, for which three bodies are at the vertices of an isosceles
triangles , the others are located at the vertices of another isosceles
triangles and the two triangles are congruent to each other;
Related assumptions will be interpreted more precisely in the following.
Assume $m_{1},m_{2}$ and $m_{5}$ are located the vertices of a isosceles
triangles $\Delta_{1}$;$m_{3},m_{4}$ and $m_{6}$ are located at the vertices
of another isosceles triangles $\Delta_{2}$. $\Delta_{1}$ and $\Delta_{2}$ are
coplanar and are congruent to each other;$q_{1}-q_{2}$ is parallel to
$q_{4}-q_{3}$;$|q_{1}-q_{4}|<|q_{5}-q_{6}|$;$q_{5}$ and $q_{6}$ are located at
the common perpendicular bisector for $q_{1}q_{2}$ and $q_{3}q_{4}$.Without
loss of generality we can take a coordinate system such that
$q_{1}=(-1,y)$,$q_{2}=(-1,-y)$,$q_{3}=(1,-y)$,
$q_{4}=(1,y)$,$q_{5}=(-1-x,0)$,$q_{6}=(1+x,0)$.(See Fig).
$m_{1}$$m_{2}$$m_{5}$$m_{6}$$m_{3}$$m_{4}$$x$$y$2
We have:
Theorem1.1:If $m_{1},m_{2},m_{3},m_{4},m_{5}$ and $m_{6}$ form a central
configuration,then $m_{1}=m_{2}=m_{3}=m_{4}$ and $m_{5}=m_{6}$.
Theorem1.2:Assume $m_{1}=m_{2}=m_{3}=m_{4}=1$,$m_{5}=m_{6}=m$, then there
exists exists a non-empty open set $U\subset(1,+\infty),$ $\varphi(y)\in C(U)$
such that $\varphi(\sqrt{3})=1$ and $m=m(x,y)$=$m(\varphi(y),y)$, so that
$(q_{1},q_{2},q_{3},q_{4},q_{5},q_{6})$ form a central configuration.
Remark:When $x=1$ and $y=\sqrt{3}$, $q_{i}$ is the vertex of a regular 6-gons
$(i=1,\cdots,6)$.
## 2 The Proofs of Theorems
### 2.1 The Proof of Theorem 1.1
Note that
$\begin{split}\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k}=-\lambda m_{k}(q_{k}-0)\\\ =-\lambda
m_{k}(q_{k}-\frac{\sum_{j=1}^{N}m_{j}q_{j}}{M})=-m_{k}\frac{\lambda}{M}\sum_{j=1}^{N}m_{j}(q_{k}-q_{j})\end{split}$
(2.1)
where $M=\sum_{i=1}^{N}m_{i}$.
So (1.4) is also equivalent to
$\sum_{j=1,j\neq
k}^{N}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})=0$
(2.2)
By (2.2) we have
$\begin{split}0m_{1}+0m_{2}+2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{3}+2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{4}\\\
-x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.3)
$\begin{split}0m_{1}-2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{2}-2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{3}+0m_{4}\\\
-y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}-y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.4)
$\begin{split}0m_{1}+0m_{2}+2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{3}+2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{4}\\\
-x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.5)
$\begin{split}2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{1}+0m_{2}+0m_{3}+2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{4}\\\
+y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.6)
$\begin{split}-2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{1}-2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{2}+0m_{3}+0m_{4}\\\
-(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.7)
$\begin{split}2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{1}+0m_{2}+0m_{3}+2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{4}\\\
+y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.8)
$\begin{split}-2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{1}-2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{2}+0m_{3}+0m_{4}\\\
-(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.9)
$\begin{split}0m_{1}-2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{2}-2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{3}+0m_{4}\\\
-y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}-y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.10)
$\begin{split}x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})(m_{1}+m_{2})+\\\
(x+2)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})(m_{3}+m_{4})\\\
+0m_{5}+2(1+x)(\frac{1}{|2(1+x)|^{3}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.11)
$\begin{split}(x+2)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})(m_{1}+m_{2})\\\
+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})(m_{3}+m_{4})+\\\
+2(1+x)(\frac{1}{|2(1+x)|^{3}}-\frac{\lambda}{M})m_{5}+0m_{6}=0,\end{split}$
(2.12)
By(2.3),(2.5),(2.7) and (2.9),we have:
$\begin{split}(m_{3}-m_{4})(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})=0,\\\
(m_{1}-m_{2})(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})=0.\end{split}$
(2.13)
By(2.4),(2.6),(2.8) and (2.10),we have:
$\begin{split}(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})(m_{1}-m_{3})+(\frac{1}{|2y|^{3}}-\frac{1}{2^{3}})(m_{4}-m_{2})=0,\\\
(\frac{1}{|2y|^{3}}-\frac{1}{2^{3}})(m_{1}-m_{3})+(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})(m_{4}-m_{2})=0.\end{split}$
(2.14)
By(2.13) and (2.14),we have:
$m_{1}=m_{2}=m_{3}=m_{4}.$ (2.15)
By (2.4),(2.6),(2.11), (2.12) and (2.15),we have
$m_{5}=m_{6}.$ (2.16)
The proof of Theorem1.1 is completed.
### 2.2 The Proof of Theorem 1.2
Notice that $(q_{1},\cdots,q_{6})$ is a central configuration if and only if
$\sum_{j=1,j\neq
k}^{6}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant 6.$ (2.17)
Since the symmetries,(2.17) is equivalent to
$\sum_{j=1,j\neq
k}^{6}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},k=2,5.$ (2.18)
Now (2.18) is equivalent to
$\lambda=\frac{1}{4}+\frac{1}{4|1+y^{2}|^{3/2}}-\frac{xm}{|x^{2}+y^{2}|^{3/2}}+\frac{(2+x)m}{|x^{2}+y^{2}+4x+4|^{3/2}},$
(2.19)
$\lambda=\frac{1}{4y^{3}}+\frac{1}{4|1+y^{2}|^{3/2}}+\frac{m}{|x^{2}+y^{2}|^{3/2}}+\frac{m}{|x^{2}+y^{2}+4x+4|^{3/2}},$
(2.20)
$\lambda=\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}+\frac{m}{4|1+x|^{3}},$
(2.21)
(2.19) ,(2.20) and (2.21) are equivalent to
$(\frac{1+x}{|x^{2}+y^{2}|^{3/2}}-\frac{1+x}{|x^{2}+y^{2}+4x+4|^{3/2}})m=\frac{1}{4}(1-\frac{1}{y^{3}}),$
(2.22)
$\begin{split}(\frac{1}{|x^{2}+y^{2}|^{3/2}}+\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{1}{4|1+x|^{3}})m=\\\
\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}-\frac{1}{4y^{3}}-\frac{1}{4|1+y^{2}|^{3/2}},\end{split}$
(2.23)
By (2.22) we have
$m=m_{1}(x,y)=\frac{1}{4}(1-\frac{1}{y^{3}})\frac{|x^{2}+y^{2}|^{3/2}|x^{2}+y^{2}+4x+4|^{3/2}}{(1+x)(|x^{2}+y^{2}+4x+4|^{3/2}-|x^{2}+y^{2}|^{3/2})}$
(2.24)
$m_{1}(x,y)>0$ if and only if $y>1$.
By (2.23) we have
$\begin{split}m=m_{2}(x,y)=[\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}\\\
-\frac{1}{4y^{3}}-\frac{1}{4|1+y^{2}|^{3/2}}]\times[\frac{1}{|x^{2}+y^{2}|^{3/2}}+\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{1}{4|1+x|^{3}}]^{-1}\end{split}$
(2.25)
Then $(q_{1},\cdots,q_{6})$ is a central configuration if and only if
$m_{1}(x,y)=m_{2}(x,y)>0$ (2.26)
It is obvious that
$m_{1}(1,\sqrt{3})=m_{2}(1,\sqrt{3})=1.$ (2.27) $\frac{\partial
m_{1}(1,\sqrt{3})}{\partial x}=\frac{1}{4}.$ (2.28) $\frac{\partial
m_{2}(1,\sqrt{3})}{\partial
x}=\frac{1}{2}\frac{(9-16\sqrt{3})}{(27+4\sqrt{3})}\neq\frac{1}{4}.$ (2.29)
By implicit function theorem,there exists exists a non-empty open set $U$ and
$\varphi(y)\in C(U)$ such that,$\sqrt{3}\in U$ , $\varphi(\sqrt{3})=1$ and
$\forall y\in U$, $m_{1}(\varphi(y),y)=m_{2}(\varphi(y),y)$.
The proof of Theorem1.2 is completed.
## References
* [1] Abraham R.and Marsden J.E.,Foundation of Mechanics,2nd edition,Benjamin,New York,1978.
* [2] Albouy A.,The symmetric central configurations of four equal masses,Amer.Math.Soc,Providence,RI,1996,pp,131-135.
* [3] Albouy A., Fu Y. and Sun S.Z.,Symmetry of planar four-body convex central configurations,Proc.R.Soc.A 464(2008),1355-1365.
* [4] Diacu F.,The masses in a symmetric centered solution of the n-body problem,Proc.AMS 109(1990),1079-1085.
* [5] Hampton M.,Stacked central configurations:new examples in the planar five-body problem,Nonlinearity 18(2005),2299-2304.
* [6] Hampton M.,Moeckel R.,Finiteness of relative equilibria of the four-body problem.Invent.Math,163(2006)289-312.
* [7] Lei J.and Santoprete M.,Rosette central configurations,degenerate central configurations and bifurcations,Celetial Mechanics and Dynamical Astronomy 94(2006):271-287.
* [8] Llibre J.,Mello L.F.,Triple and quadruple nested central configurations for the planar n-body problem,Physica D 238(2009),563-571.
* [9] Long Y.,Admissible shapes of 4-body non-collinear relative equilibria,Adv.Nonlinear Stud.1(2003),495-509.
* [10] Moeckel R.,On central configurations,Math.Z.205(1990),499-517.
* [11] Moeckel R.,Simo C.,Bifurcation of spatial central configurations from planar ones,SIAM J.Math.Anal.26(1995),978-998.
* [12] Moulton,F.R.,The straight line solutions of the n-body problem,Annals of Math,Second Series,12(1910),1-17.
* [13] Perko L.M.and Walter E.L.,Regular polygon solutions of N-body problem,Proc.AMS 94(1985),301-309.
* [14] Saari,D.G.,Singularities and collions of Newtonian gravitational systems,Arch.Rational Mech.49(1973),311-320.
* [15] Saari,D.G.,On the role and properties of N body central configurations,Celestial Mechanics and Dynamical Astronomy 21(1980),9-20.
* [16] Saari,D.G.,On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem,J.Diff.Eqs. 41(1981), 27-43.
* [17] Saari D.G.,Collisions,Rings and Other Newtonian N-body Problems,AMS Providence,Rhode Island.2005.
* [18] Sekiguchi M.,Bifurcations of central configurations in the 2N+1 body problem, Celetial Mechanics and Dynamical Astronomy 90(2004),355-360.
* [19] Shi J.,Xie Z.,Classification of four-body central configurations with three equal masses,J.Math.Anal.Appl.363(2010),512-524.
* [20] Smale S.,Mathematical problems for the next century,Math. Intelligenceer 20(1998),141-145.
* [21] Smale S.,Topology and mechanics II,Inv.Math 11(1970),45-64.
* [22] Wintner A.,The analytical foundations of celestial mechanics, Princeton Univ. Press, 1941.
* [23] Zhang S.Q.and Zhou Q.,Periodic solution for planar 2N-body problems,Proc.AMS.131(2003),2161-2170.
|
arxiv-papers
| 2012-05-16T12:06:40 |
2024-09-04T02:49:30.983213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Furong Zhao and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1205.3648"
}
|
1205.3655
|
arxiv-papers
| 2012-05-16T12:26:51 |
2024-09-04T02:49:30.987150
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Asia Furones",
"submitter": "Asia Furones",
"url": "https://arxiv.org/abs/1205.3655"
}
|
|
1205.3844
|
11institutetext: P.D.Andriushchenko 22institutetext: Far Eastern Federal
University, School of Natural Sciences, 8 Sukhanova St., Vladivostok 690950,
Russia. 22email: pitandmind@gmail.com 33institutetext: K.V.Nefedev
44institutetext: Far Eastern Federal University, School of Natural Sciences, 8
Sukhanova St., Vladivostok 690950, Russia. 44email: knefedev@phys.dvgu.ru
# Magnetic phase transitions in the Ising model
P.D.Andriushchenko and K.V.Nefedev
###### Abstract
In this paper we consider an approach, which allows researching a processes of
order-disorder transition in various systems (with any distribution of the
exchange integrals signs) in the frame of Ising model. A new order parameters,
which can give a description of a phase transitions, are found. The common
definition of these order parameters is the mean value of percolation cluster
size. Percolation cluster includes spins with given energy. The transition
from absolute disorder to correlated phase could be studied with using of
percolation theory methods.
## 1 Introduction
Ordered systems, such as ferromagnets are studied well now. Physics of systems
with a complex type of exchange interaction is not so simple and evident. For
instance, the theoretical research of the paramagnetic-spin glass transition,
which started several decades ago, is still in process Vasin:2006 ;
Ginzburg:1989 . The theory of magnetic states and transitions from
paramagnetism to antiferromagnetism of various types (A, B, C, CE, G, and
others), for instance in manganites, is still in development Nagaev:1996 ;
Nagaev:2001 ; Gor'kov:2004 ; Shuai:2008 . It is well known that for such
systems the average magnetization cannot be used as an order parameter. At low
temperature the correlations between spins grow. This fact is proved by known
temperature dependence of specific heat and magnetic susceptibility behavior
and difference in the temperature behavior of the magnetization, which is
measured in ZFC and FC modes.
In this paper, we present the result of research of magnetic phase transitions
in the Ising model on a simple square lattice using the numerical simulation
methods. We worked with the following three models: with ferromagnetic
interactions, antiferromagnetic interactions and random distribution of
exchange integrals (spin-glass models).
## 2 Parallel algorithm for finding the equilibrium configuration
The parallel search scheme for the equilibrium configuration is shown in Fig.
1. The values of spins, their energies, as well as links of a square lattice
of the magnet were recorded in one-dimensional dynamic arrays for more
flexible allocation of memory. At the start of simulation the temperature is
set to be corresponding to paramagnetic state (in reduced units $T=4.5$, which
is higher than the Curie temperature for a square lattice, obtained by Onsager
$T_{c}=2.28$). The initial configuration, that corresponds to the random
distribution, of spin directions, is generated. Configuration data, the
temperature, the number of MC steps and other technical information are sent
out to all the computing processes by means of MPI technology.
Figure 1: Scheme of the parallel realization of Monte Carlo algorithm
Each process performs a Monte Carlo spin flip (approx. $10^{9}$ spin flips for
$1000\times 1000$ spin system) until the system comes into equilibrium, which
is defined by the energy of the system. The system came into equilibrium at
the temperature $T$ in case if the system s energy became less after certain
quantity of MC steps (for modeling the transition with decreasing temperature)
and will not change significantly during further modeling. The energy of
configurations at the given temperature are passed into the root thread, which
one also compares the obtained values. Root process selects the energy
corresponding to the extreme value (for modeling the transition with
decreasing temperature – minimum value). All variables are recorded to the
output files, and the temperature lowered at the defined value ($\Delta
T=0.1$). The thread, which number is defined by the root process and with the
lowest system energy, distributes the data about the selected equilibrium
distribution of spins and the new temperature value to the other threads. This
cycle is repeated until the temperature achieve zero. The average
magnetization is calculated by the difference between the spins directed
upward and downward. The total energy is calculated by the summarizing all
energies of the interacting spin pairs.
To count the number of nodes with minimal energy in the maximum cluster we can
use breadth-first search (BFS) algorithm with the creation of queue (which is
the graph traversal task).
## 3 Ordering and bond-percolation on the simple square lattice
The probability of any possible configuration is given by the Gibbs
distribution Landau:1980 . If we know the partition function for a system of
interacting spins, it allows us to calculate all possible average physical
values, which fully describe the state of the system under the given external
conditions. Currently, in research of the phenomenon of percolation numerical
methods is mainly used (Monte Carlo). They are widely used in statistical
physics Newman:1999 .
We only point out that in the Ising model with Hamiltonian
$H=-\frac{1}{2}\sum\limits_{ij}{J_{ij}S_{i}S_{j}}$ (1)
which takes into account the ferromagnetic interaction, exchange integral
$J_{ij}=1$ between each spin $S_{i}$ and its nearest neighbor $S_{j}$. For
antiferromagnetic interactions $J_{ij}=-1$.
Monte Carlo simulation with Metropolis algorithm allows to calculate the
average relative magnetization $<M>$ of the ferromagnetic system $1000\times
1000$ Ising spins (the number of Monte Carlo steps is $2*10^{10}$) on a simple
square lattice with $z=4$ nearest neighbors. Temperature behavior $<M>$ shown
in Fig. 2.
The same method was used to calculate the average size of the percolation
cluster $\gamma_{1}(T)$, which is defined as the ratio of the number of spins
in the ground state to the total number of spins. There is a coincidence
between the critical temperature of the magnetization $<M>$ and the order
parameter $\gamma_{1}(T)$. The law of variation and the critical exponents for
the temperature dependence of the assumed physical value $\gamma_{1}(T)$
(average in time and configuration) coincide with observed characteristics for
the average magnetization of ferromagnetic systems Onsager:1944 . The
difference in the temperature behavior within reviewed order parameters at
$T\leq T_{c}$, (higher growing rate of $<M>$ compared to $\gamma_{1}$), caused
by the fact that the percolation cluster does not contain all the spins in the
ground state, that means the difference in growing rate caused by low density
of the percolation cluster and extention of new phase clusters in its pores.
Ordering process is usually characterized by the formation of a set of new
phase nuclei. At $T=T_{c}$ only a part of small cluster are united in one most
size the percolation cluster. Thus, in this model, the order parameter $<M>$
describes the balance between the number of particles ”up” $N\uparrow$ and the
number of particles ”down” $N\downarrow$, and the proposed new order parameter
$\gamma_{1}(T)$ describes the process of growth of the percolation cluster.
Figure 2: Temperature dependence of the relative size of the percolation
cluster for Ising $\gamma_{1}(T)$ with $J=+1$ and $J=-1$. The behavior of the
relative number of spins $\gamma_{2}(T)$ in the maximal cluster (energy $E=-4$
and $E=-2$) and magnetization $<M(T)>$ (For antiferromagnetic $<M(T)>=0$ for
any temperature). All values are presented in reduced units. FM -
ferromagnetism, AFM - antiferromagnetism, SPM - superparamagnetism (clustered
FM), PM - paramagnetism.
The results of this research show that $\gamma_{1}(T)$ in antiferromagnetic
model (${J_{ij}=-1}$) have the same jump in the phase transition, as in the
ferromagnetic model (${J_{ij}=+1}$), as shown in Fig. 2. While the
magnetization $<M(T)>$ in antiferromagnetic systems is equal to zero at any
temperature, and therefore cannot serve as an order parameter. This fact is
due to the universality of order parameter $\gamma_{1}(T)$.
Besides, the function $\gamma_{2}(T)$ represents the relative size of the
maximal cluster, which one unites a spins with the negative interaction
energy. Fig. 2 shows, that the function $\gamma_{2}(T)$ allows us to determine
the transition temperature of the system from absolutely randomized
paramagnetic (PM) phase to correlated superparamagnetic (SPM).
## 4 Simulation of the transition from paramagnetism to spin glass
In 1975 S. Edwards and P. Anderson considered the lattice model of exchange-
coupled magnetic moments interacting so that the exchange integral is a random
function Edwards:1975 . In this spin glass the one half of pairs interacts
ferromagnetically, and second part interacts antiferromagnetically. The types
of interactions distributed randomly. The current research is based on the S.
Edwardson - P. Anderson model specified to frustration in every spin
$\sum\limits_{i=1}^{z=4}{J_{i}=0}$ (2)
in case of simple square lattice (the summation is over $z=4$ neighbors).
Figure 3: Temperature dependence of the relative size of the percolation
cluster of Ising spins $\gamma_{1}(T)$ for spin glass model. Behavior of the
relative number of spins $\gamma_{2}(T)$ in the maximal cluster (with energy
$E=-4$ and $E=-2$). All values are presented in reduced units.
Monte-Carlo simulation of transition processes in described above model of
spin glass lead to the existence of specific critical temperature $T_{f}$,
Fig. 3. In almost completely (99%) frustrated system of $1000\times 1000$ of
Ising spin glass, there is the function $\gamma_{2}(T)$ which one has abrupt
changes of value in transition region. Function $\gamma_{2}(T)$ is not equal
to $1$ even at $T=0$, because there is large number of frustrated spins in
excited state. This typical phenomenon for spin glass state leads to nonzero
magnetic capacity heat at zero value of the absolute temperature.
We suppose that in the limit of infinite number of particles this jump should
be even more evident.
## 5 Conclusion
The relative power of a percolation cluster could be used as a universal order
parameter for systems with direct exchange interaction between spins. This
parameter describes the short-range or of the long-range order, depending the
ways of cluster integration of spins over values of exchange interaction
energy. In the numerical experiments the possibility of phase separation of
paramagnetic and superparamagnetic regimes, paramagnetic and spin glass states
is showed. The results of given research can be summarized in the following
conclusion:
1. 1.
The law of temperature behavior of the new order parameter – function
$\gamma_{1}(T)$ coincides with the law of temperature behavior of the average
magnetization for a ferromagnet $<M>$. There is the coincidence critical
temperature of magnetization formation and critical temperature of percolation
threshold.
2. 2.
The function $\gamma_{1}(T)$ in the antiferromagnet undergoes a jump at the
point of the phase transition, and it’s behavior is the same as
$\gamma_{1}(T)$ in a ferromagnetic. It stands as the universality of the order
parameter.
3. 3.
The existence of function which has the jump at the spin glass-paramagnetic
transition region could give the solution of the phase transition problem in
PM-SG (PM-SPM). The $\gamma_{1}(T)=0$ at spin glass tells about the absence of
phase transition in this $2D$ lattice of Ising spins.
4. 4.
The proposed approach allows to unite the concepts of a phase transitions in
ordered and disordered systems, including systems with competing interactions
with the developed ideas in percolation theory.
5. 5.
The order parameter is universal for any magnetic system. It could be measured
experimentally using spectroscopy methods.
This approach can be extended to the case of a complex alternating-sign
exchange of long-range interaction.
The interesting questions are the research of 3D lattice spin glass state for
existence of phase transition and also the simulation ZFC and FC regimes.
This work was supported by grant No. 14.740.11.0289, No. 07.514.11.4013 and
No. 02.740.11.0549 from Ministry of Education and Science of the Russian
Federation.
## References
* (1) M. Vasin, “Description of the paramagnet-spin glass transition in the edwards-anderson model using critical-dynamics methods,” _Theoretical and Mathematical Physics_ , vol. 147, pp. 721–728, 2006.
* (2) S.L.Ginzburg, _Irreversible Phenomena in Spin Glasses_. Nauka; Moscow, 1989, in Russian.
* (3) E. L. Nagaev, “Lanthanum manganites and other giant-magnetoresistance magnetic conductors,” _Physics-Uspekhi_ , vol. 39, no. 8, pp. 781–805, 1996.
* (4) E. Nagaev, “Colossal-magnetoresistance materials: manganites and conventional ferromagnetic semiconductors,” _Physics Reports_ , vol. 346, no. 6, pp. 387 – 531, 2001.
* (5) L. P. Gor’kov and V. Z. Kresin, “Mixed-valence manganites: fundamentals and main properties,” _Physics Reports_ , vol. 400, no. 3, pp. 149 – 208, 2004\.
* (6) S. Dong, R. Yu, S. Yunoki, J.-M. Liu, and E. Dagotto, “Ferromagnetic tendency at the surface of ce-type charge-ordered manganites,” _Phys. Rev. B_ , vol. 78, p. 064414, Aug 2008.
* (7) L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskiı̆, _Statistical physics / by L.D. Landau and E.M. Lifshitz ; translated from the Russian by J.B. Sykes and M.J. Kearsley_ , 3rd ed. Oxford; New York : Pergamon Press, 1980, translation of Statisticheskai︠a︡ fizika.
* (8) M. E. J. Newman and G. T. Barkema, _Monte Carlo methods in statistical physics / M.E.J. Newman and G.T. Barkema_. Oxford : Clarendon Press, 1999\.
* (9) L. Onsager, “Crystal statistics. i. a two-dimensional model with an order-disorder transition,” _Phys. Rev._ , vol. 65, pp. 117–149, Feb 1944\.
* (10) S. F. Edwards and P. W. Anderson, “Theory of spin glasses,” _J. Phys. F_ , 1975.
|
arxiv-papers
| 2012-05-17T03:44:27 |
2024-09-04T02:49:30.994263
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P.D.Andriushchenko and K.V.Nefedev",
"submitter": "Peter Andriushchenko",
"url": "https://arxiv.org/abs/1205.3844"
}
|
1205.3875
|
# Instability and morphology of polymer solutions coating a fiber111Accepted
in Journal of Fluid Mechanics.
F. Boulogne, L. Pauchard, F. Giorgiutti-Dauphiné222Univ Pierre et Marie Curie-
Paris 6, Univ Paris-Sud, CNRS, F-91405, Lab FAST, Bat 502, Campus Univ, Orsay,
F-91405, France.
###### Abstract
We report an experimental study on the dynamics of a thin film of polymer
solution coating a vertical fiber. The liquid film has first a constant
thickness and then undergoes the Rayleigh-Plateau instability which leads to
the formation of sequences of drops, separated by a thin film, moving down at
a constant velocity. Different polymer solutions are used, i.e. xanthan
solutions and polyacrylamide (PAAm) solutions. These solutions both exhibit
shear-rate dependence of the viscosity, but for PAAm solutions, there are
strong normal stresses in addition of the shear-thinning effect. We
characterize experimentally and separately the effects of these two non-
Newtonian properties on the flow on the fiber. Thus, in the flat film observed
before the emergence of the drops, only shear-thinning effect plays a role and
tends to thin the film compared to the Newtonian case. The effect of the non-
Newtonian rheology on the Rayleigh-Plateau instability is then investigated
through the measurements of the growth rate and the wavelength of the
instability. Results are in good agreement with linear stability analysis for
a shear-thinning fluid. The effect of normal stress can be taken into account
by considering an effective surface tension which tends to decrease the growth
rate of the instability. Finally, the dependence of the morphology of the
drops with the normal stress is investigated and a simplified model including
the normal stress within the lubrication approximation provides good
quantitative results on the shape of the drops.
###### Contents
1. 1 Introduction
2. 2 Materials and characterizations
1. 2.1 Samples preparation
1. 2.1.1 Polyacrylamide
2. 2.1.2 xanthan
2. 2.2 Rheological characterization
1. 2.2.1 Polyacrylamide
2. 2.2.2 xanthan
3. 2.3 Experimental setup
3. 3 Flat film
4. 4 Instability growth rate
5. 5 Drop morphologies: normal stress effect
6. 6 Conclusion
## 1 Introduction
The fiber coating process is widespread in numerous industrial applications
such as the manufacture of glass, polymeric and optical fibers, conducting
cables or textile fibers. The application of a thin layer on these solid
substrates should ensure mechanical or optical properties of the final
deposit. Hence, it is of crucial interest to control the final thickness of
the liquid film. It has been well known since [27] that a cylindrical free
surface of a fluid is unstable under the action of the surface tension. Later,
[4] described in his monograph the patterns observed in a spider web with a
sticky fluid. He reproduced the experiment with castor oil and a quartz fiber,
and detailed the spatial variations of the film thickness. For a fiber drawn
out of a bath, [18] and later [30] provided a first view of the flowing regime
as a function of the capillary number and the Goucher number defined as the
ratio between the radius of the fiber and the capillary length. Subsequently,
different configurations have been studied to identify the different
mechanisms responsible for destabilization of the film and the dependence on
various parameters such as the radius of the fiber, the viscosity, the
inertial forces or the influence of surfactant on the growth rate of the
instability [16, 17, 25, 26, 6, 10, 11, 28, 21]. More recently, we have
reported a flow regime diagram which identifies, depending on the fiber radius
and the flow rate on the fiber, the dominant physical mechanisms [11, 12, 10].
It appears that for small fiber radii compared to the capillary length and low
flow rates, the liquid film is dominated by the surface tension, and the
instability mechanism is the Rayleigh-Plateau instability whose nature is
absolute. For higher values of the parameters (fiber radius and/or flow rate)
two other regimes have been discerned respectively dominated by gravity and
drag (the drag/dravity regime) or by inertia and drag (the drag/inertia
regime). The nature of the instability is then convective for these regimes.
Extensive theoretical studies have investigated the dynamics of the film [15,
25, 20]. Among the most recent studies, we can mention the works of [21] and
[8] on film thickness of the same order as the radius fiber and with
negligible inertia contribution. Their numerical results are compared with
experimental results [11] performed in the three kinds of regime depicted
above with a predictable deviation for the Drag/Inertia regime.
In most industrial situations, the coating fluid is a polymer material or a
complex fluid and exhibits non-Newtonian properties depending on miscellaneous
parameters such as concentration, structure or flexibility of polymers. Yield
stress, shear-thinning or elastic effects are some of the non-Newtonian
behaviors which can affect the structure of the flow, the appearance of the
instability or the morphology of the patterns. [9], in the case of a “dip-
coating” configuration where the fiber is drawn out of a bath of liquid,
observed that the film swells due to the presence of polymer in the solution.
Considering the normal stress and the lubrication approximation, they found an
analytical expression of the film thickness as a function of the withdrawal
velocity and the normal stress coefficient. In general for most of the cases,
when the instability is studied for non-Newtonian fluids it reveals different
classes of patterns and the presence of polymers can drastically change the
dynamic of the system, as in Faraday or Saffman-Taylor instabilities [22]. In
the case of the instability of a liquid jet [7, 29, 13, 1], the addition of
polymers causes the formation of a “beads-on-a-string” structure where
adjacent beads are joined by a thread which grows thinner and strongly delays
the detachment of droplets. In this configuration, the flow is subject to a
strong elongation: a velocity gradient exists in the direction of the flow due
to gravity forces. This additional resistance to breakup compared to a simple
fluid is due to large extensional stresses. Contrary to the case of a liquid
jet, there is no elongational viscosity in the case of the flow down a fiber
due to the no-slip condition on the fiber as detailed further.
A complication inherent in the use of these complex fluids is that they
exhibit different non-Newtonian properties with opposite effects. Notably,
most polymer solutions are both shear-thinning and present elastic effects. In
the context of fiber coating, and to obtain independently the role of the
shear-thinning effect and normal stress on the Rayleigh-Plateau instability,
we have performed experiments with two different polymer solutions: one with a
rod-like polymer (xanthan), exhibiting a pure shear-thinning effect; the other
with a flexible polymer (polyacrylamide abbreviated as PAAm) which exhibits
non-negligible normal stress along with shear-rate dependence of viscosity
similar to xanthan solutions.
Our experiments are all performed in a regime where inertial and gravitational
forces are negligible compared with capillary, elastic and viscous forces. The
Rayleigh-Plateau instability is then absolute [12] and the flow patterns
consist of drops, where fluid is partly trapped in a recirculation zone,
sliding down a very thin (smaller than $100$ $\mu$m) and quasistatic liquid
substrate. This droplike wave train emerges from a constant film thickness,
i.e. the flat film region. We investigate the role of the non-Newtonian
properties on such a flow, i.e, the flat film and the drop-like wave train
resulting from the Rayleigh-Plateau instability.
This paper is organized as follows. In section 2.1, we proceed to a
rheological characterization of the solutions. In section 2, we present the
experimental setup and visualisation techniques. In section 3, we look at the
flat film before the appearance of the instability and the shear-thinning
effect on the film thickness. In section 4, the experimental growth rate and
the wavelength of the instability are measured experimentally for xanthan
solutions (only shear-thinning effect) and for PAAm solutions exhibited strong
normal forces and similar viscosity shear-rate dependence. These experimental
data are then compared to the results of a linear stability analysis taking
into account a non constant viscosity with the shear-rate. In the last part,
section 5, we will treat the effect of normal stress on the morphology of the
drops and provide a simplified model to explain the dependence of the drop
shape with the normal stress.
## 2 Materials and characterizations
Before considering the non-Newtonian properties, certain conditions are
required: first, to avoid inertial forces, and second to ensure a perfect
wetting on the fiber. In a previous paper [11], we have presented a diagram of
the expected flow regime which details the dominant physical mechanisms in the
plane of the dimensionless numbers $R/l_{c}$ versus $h_{0}/R$ with $R$, the
radius of the fiber, $l_{c}$ the capillary number and $h_{0}$ the flat film
thickness. To ensure that the flow is dominated by capillary forces, with no
inertia, some conditions on the fiber radius, the surface tension and the
viscosity of the fluid must be fulfilled in agreement with the flow regime
diagram mentioned above. In that capillary region, the flow on the fiber
consists of drops sliding on a quasistatic thin film. To ensure a good wetting
on the fiber and consequently an axisymmetric pattern on the fiber, the
surface tension has to be lower than $40$ mN/m. Finally, these two conditions
are satisfied by using a fiber radius equal to $0.28$ mm and solutions
composed of a mixture of water, glycerol (to increase viscosity) and
surfactant to reach a surface tension close to $30$ mN/m.
Some previous experiments carried out with Newtonian fluids on the same
experiment [11], have shown that film thicknesses between $0.1$ and $1$ mm are
possible with velocities ranging from $1$ to $10$ cm/s and viscosities between
$50$ and $500$ mPa.s. This implies that the shear-rate range encountered in
our experiments is from $10$ to $1000$ s-1.
The liquids used are then semi-dilute solutions of polymers: xanthan and
Polyacrylamide (PAAm) purchased from Sigma-Aldrich. Both present shear-
thinning behavior but only the second one exhibits large elastic effects in
the shear-rate range considered in the experiments. To discern the effect of
elasticity, Boger fluids would have been theoretically more appropriate.
Nevertheless, in practice, the large quantities of liquid required for our
experiments and the possible degradation of the nylon fiber by the solvents
used in Boger fluids, are two reasons prohibiting their use.
### 2.1 Samples preparation
#### 2.1.1 Polyacrylamide
PAAm is a high molecular weight chain resulting in long flexible chains
($M\simeq 1.5\times 10^{7}$ g/mol). The samples are prepared very carefully to
get homogeneous solutions. First, a solvent is prepared by mixing $50$% of
purified water and $50$% of pure glycerol (all percentages presented in this
article are weight percentages). Six polymer concentrations were studied in
the range from $0.1\%$ to $0.6\%$. The surfactant selected to reduce the
surface tension is Triton X-100 (TX-100). This choice was motivated by its
mixing properties at high polymer concentrations and for the resulting low
surface tension [32]. The TX-100 concentration is $4.5$% (about $300$ times
the Critical Micelle Concentration, CMC, in water). For this high
concentration we assume that the surfactant mobility timescale is higher than
the timescale for the instability growth rate. Indeed, the time variation of
the surface tension due to diffusion of surfactant is $10^{-2}$ s for a TX-100
concentration $50$ times the CMC in water [14]. This time scale is of the same
order of magnitude as the characteristic growth rate of the instability:
$\frac{\eta R^{4}}{\gamma h^{3}}\sim\frac{0.1\times(0.6\times
10^{-3})^{4}}{30\times 10^{-3}(0.2\times 10^{-3})^{3}}=0.05$ s (using typical
values for the viscosity $\eta$, the fiber radius $R$, the surface tension
$\gamma$, and the film thickness $h$). Consequently, in the range of
concentrations considered in our experiments we assume that the interface is
rapidly saturated with surfactant molecules before the instability occurs. The
surface tension of the final solution is then $\gamma=32.3\pm 0.5$ mN/m. We
should note that no apparent rheological modifications are observed by varying
the TX-100 concentration.
#### 2.1.2 xanthan
xanthan is a rigid rod-like polymer ($M\simeq 5\times 10^{6}$ g/mol). A
preparation protocol similar to the PAAm solutions was used for xanthan
solutions. The solvent is slightly different: $60$% of glycerol and $40$% of
water. Since xanthan is a polyelectrolyte polymer, the resulting rheological
properties of this polymer are known to be modified by the addition of salt
[31]. Thus, different concentrations of NaCl allow adjustment of the
rheological properties. After the addition of TX-100, the surface tension of
the final solution is $\gamma=32.7\pm 0.8$ mN/m independent of the salt
concentration.
### 2.2 Rheological characterization
The rheology of polymer solutions was performed using an “Anton Paar”
rheometer with a cone and plate geometry. We have chosen a large cone (radius:
$49.988$ mm) with small angle (angle: $0.484^{\circ}$) in order to measure
precisely normal stress in a large range of shear-rates. The sample
temperature was fixed at $20.00\pm 0.05^{\circ}$C.
#### 2.2.1 Polyacrylamide
The evolution of the apparent shear viscosity, $\eta$, is plotted versus the
shear-rate, $\dot{\gamma}$, for two concentrations in figure 1. As usual,
$\eta$ decreases with $\dot{\gamma}$ and, at a given shear-rate, increases
with the polymer concentration. Such curves are typical for shear-thinning
fluids where a constant Newtonian low-shear viscosity is followed by a power-
law dependence before reaching the viscosity of the solvent at high shear-
rates. These measurements can be reasonably fitted by a four-parameter Carreau
model [24]:
$\eta=\eta_{\infty}+\frac{\eta_{0}-\eta_{\infty}}{\left(1+(\tau\dot{\gamma})^{2}\right)^{\frac{1-n}{2}}}$
(1)
where $\eta_{0}$ and $\eta_{\infty}$ are respectively the viscosity for the
zero-shear limit and infinite shear-rate, and $\tau$ denotes a characteristic
time scale that measures the scale at which the shear-thinning effect becomes
important. The exponent $n$ is the power of the following Ostwald power-law
equation:
$\eta=\beta\dot{\gamma}^{n-1}$ (2)
The zero-shear limit increases rapidly with the polymer concentration, that is
typical of entangled polymer solutions (see the inset of figure 1). Recently,
it has been shown [23] that in a good solvent the entangled concentration
$c_{e}$ for PAAm is about nine times the crossover concentration $c^{*}\simeq
0.2$ g/L. The temperature effect on the samples indicates that the viscosity
decreases by $10$% for a $5^{\circ}$C increase.
Figure 1: Variations of the shear viscosity $\eta$, of PAAm solutions vs.
shear-rate $\dot{\gamma}$, in a log-log scale. Each symbol refers to a
different polymer concentration. Plain curves correspond to data fits with a
Carreau model. The inset shows the zero shear-rate viscosity $\eta_{0}$, as a
function of the concentration in PAAm.
Figure 2: (a) Log-log plot of the normal stress, $N_{1}$, as a function of the
shear-rate, $\dot{\gamma}$, for PAAm solutions in water-glycerol ($50$% :
$50$%) solvent; $4.5$% TX-100 surfactant was added to the solutions. Plain
curves correspond to data fits using equation 3. (b)
$\frac{N_{1}}{\eta(\dot{\gamma})\dot{\gamma}}$ versus shear-rate,
$\dot{\gamma}$ , for PAAm solutions
The normal stress measurements are presented in figure 2 as a function of the
shear-rate for different PAAm concentrations over a wide range of shear-rates.
A significant increase of the normal stress with the shear-rate is observed in
accordance with:
$N_{1}=\psi_{1}\dot{\gamma}^{2}$ (3)
where $\psi_{1}$ is the first normal stress coefficient characterizing the
fluid [2]. Values are given in figure 2 for several Polyacrylamide
concentrations. In the inset graph, we have presented the data using a double
log-plot. It appears more clearly that, except for PAAm solutions at $0.6\%$,
there is a discrepancy between the data and the curve fit for low shear-rates
(lower than $200$ s-1), indicating dependence of the first normal stress
difference with the shear-rate. Thus the $0.6\%$ solution would be the best
candidate for studying the normal force effect in section 5. Normal stress
magnitude can be compared to viscous stress by estimating the ratio
$\frac{N_{1}}{\eta(\dot{\gamma})\dot{\gamma}}$ as a function of the shear-rate
2. This ratio increases with the shear-rate, highlighting the importance of
the normal stress, which starts to be dominant compared to the shear-thinning
effect for shear-rates larger than $100$ s-1. One should then expect a large
amount of normal stress in the drops, for which the shear-rate is always
larger than $100$ s-1.
#### 2.2.2 xanthan
Rheological measurements are typical of shear-thinning fluids where a constant
Newtonian low-shear viscosity is followed by a power-law dependence before
reaching the viscosity of the solvent at high shear-rates (figure 3).
Nevertheless, as shown in figure 3, the power-law behavior failed to fit the
experimental results in the whole range of shear-rates. Since xanthan is a
polyelectrolyte, the solution rheology and molecular configuration are greatly
affected by the solution’s ionic strength. Thus by adding $0.8$% NaCl to the
solution, the shear-thinning effect can be adjusted to $n=0.73$ in a
reasonable range of shear-rates from $5$ to $2000$ s-1. No significant normal
stress has been detected for xanthan below $4000$ s-1, a shear-rate which is
not expected to be reached in the experiment. Subsequently, we will
exclusively use salted xanthan solutions as pure shear-thinning solutions.
Figure 3: Viscosity $\eta$, as a function of shear-rate $\dot{\gamma}$, for
two solutions constituted by water-glycerol ($50$% : $50$%), $4.5$% TX-100 and
$0.4$% xanthan. Red cross correspond to free-salt solution and blue one are
for a salt concentration up to $0.8$%. Numbers indicate the slopes of the
Ostwald power-law model from fits over a $\dot{\gamma}$ range: $[10;2000]$
s-1.
### 2.3 Experimental setup
As depicted in figure 4, the fluid flows from an upper reservoir (diameter:
$14$ cm) down a nylon fiber (diameter: $0.56$ mm). The relative pressure
variation is about $0.001$% during one minute for the highest measured flow
rates. The flow rate is controlled by a valve composed of two axisymmetric
cones. The mass flow rate $Q$ is measured from the weight variation of a
collecting tank recorded by a computer-controlled scale. A transparent nozzle
guides the fluid on the fiber. Its verticality is crucial to obtaining an
axisymmetric flow and it is ensured by a mechanical device which enables very
accurate fiber displacements with a sensibility of $2.4$ arc sec. Two
perpendicular cameras with zoom lens help to control the axisymmetry of the
film flowing down the fiber.
Figure 4: (a) Experimental setup. Scheme showing a fluid flowing down on a
fiber from a upper tank. The flow rate is controlled by a valve and guided
with a nozzle. (b) Picture snapped by a high-speed camera with a telecentric
lens ($1\times$). The white bar length is $5$ mm. (c) Spatiotemporal evolution
of the film obtained by a vertical linear camera passing through top drops
(black lines). Letters A, B and C denote respectively the flat film, the
ordered and the disordered pattern regions.
As depicted in figure 4, the flow presents three regions along the fiber. A
meniscus is followed by a flat film with a constant thickness on a distance
called the healing length, which increases slightly with the flow rate [12].
Then, the Rayleigh-Plateau instability leads to the formation of a regular
pattern of beads flowing on a very thin and flat film. In this paper, we will
deal only with regime dominated by capillary forces, so we exclude low flow
rates (the dripping regime) and high flow rates where inertial flow dominates.
The flow regime is then absolute. The film thickness and the shape of the
drops are captured by a high-speed digital camera with a telecentric lens. The
interface position is detected in both space and time, so we are able to
measure the film thickness $h(z,t)$ with an accuracy of $0.02$ mm. A linear
camera provides spatiotemporal diagrams which deliver information on the
dynamic of the flow. A vertical pixel line passing through the peaks of the
drops is recorded and stored at constant time intervals. The resulting
spatiotemporal diagram produces the $(z,t)$ trajectories of the drops along
the fiber. A typical spatiotemporal diagram is shown on figure 4. The uniform
grey region, located at the upper part of the fiber is the place where the
film is flat (region A) and gives rise to a zone of regular stripes with a
constant wavelength and velocity for the drops (region B). Finally,
downstream, some coalescences between drops lead to the formation of a
disordered pattern (region C).
## 3 Flat film
In this section we focus on the region close to the inlet where the film
thickness is constant (flat film, grey uniform region, see figure 4). In the
case of Newtonian fluids, the thickness of the film, $h$, is given by the
classical Nusselt solution [12]. In the case of very thin films ($h\ll R$),
i.e. the planar case, there is a cubic relation between the flow rate on the
fiber and $h$. We define the cylindrical coordinates system $(r,\theta,z)$,
where $r$ is the radial coordinate (the fiber center is the origin), $\theta$
the azimuthal coordinate and $z$ the axial coordinate oriented downward in the
flow direction. In the case of a shear-thinning solution exhibiting normal
stress effects, for a steady axisymmetric flow, the stress balance in the
axial direction $z$ is written as:
$\frac{\partial\sigma_{zz}}{\partial
z}+\frac{1}{r}\frac{\partial(r\sigma_{rz})}{\partial r}=\frac{\partial
p}{\partial z}-\rho g$ (4)
for $R<r<R+h(z)$, where $\sigma$ denotes the stress tensor, $p$ is the
pressure field in the film, $\rho$ and $g$ are respectively the fluid density
and the gravitational acceleration. Since the first normal stress difference,
$N_{1}$, can be expressed as
$\sigma_{zz}-\sigma_{rr}=\psi_{1}\left(\frac{\partial v}{\partial
r}\right)^{2}$, with $v(r,z)$ the axial velocity, which varies along the film
thickness, equation (4) becomes
$\frac{\partial N_{1}}{\partial
z}+\frac{1}{r}\frac{\partial(r\sigma_{rz})}{\partial r}=\frac{\partial
p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho g$ (5)
As no spatial variations are detected in the flat film, $z$-invariance of the
velocity field implies that $\frac{\partial N_{1}}{\partial z}=0$. Thus,
normal stress has no effect on the flat film.
To calculate the velocity profile in the flowing film, we model the shear-
thinning effect by a power-law in accordance with expression 2. Thus (5)
becomes
$1+\frac{1}{1+\tilde{r}}\frac{d}{d\tilde{r}}\left((1+\tilde{r})\left(\frac{d\tilde{v}(\tilde{r})}{d\tilde{r}}\right)^{n}\right)=0$
(6)
where the following dimensionless variables are introduced:
$\tilde{r}=\frac{r-R}{R}$, $\tilde{h}=\frac{h}{R}$, $\tilde{v}=\frac{v}{V}$
and $V\equiv R\left(\frac{\rho gR}{\beta}\right)^{1/n}$.
The fluid velocity satisfies two boundary conditions: no-slip on the fiber
($\tilde{v}(\tilde{r}=0)=0$) and zero tangential stress at the liquid-air
interface ($\partial_{\tilde{r}}\tilde{v}(\tilde{r}=\tilde{h})=0$).
Considering this last boundary condition, equation (6) becomes:
$\frac{d\tilde{v}}{d\tilde{r}}=\left(\frac{1}{1+\tilde{r}}\left(\tilde{h}(1+\tilde{h}/2)-\tilde{r}(1+\tilde{r}/2)\right)\right)^{\frac{1}{n}}$
(7)
The flow rate per unit length $q=\frac{Q}{2\pi\rho R}$ is written in
dimensionless form as
$\tilde{q}=q\frac{1}{RV}=\int_{0}^{\tilde{h}}\tilde{v}(\tilde{r}+1)d\tilde{r}$
(8)
We note that for the Newtonian case $n=1$ and $\beta=\eta$, we recover the
analytical Nusselt solution $v=\frac{\rho
g}{4\eta}\left(2(R+h)^{2}\ln\left(\frac{r}{R}\right)-(r^{2}-R^{2})\right)$.
Equation (7) is solved using a Runge-Kutta algorithm (starting at
$\tilde{v}(\tilde{r}=0)=0$ in order to satisfy the boundary condition on the
fiber). First, the influence of the parameter $n$ is studied for a constant
flow rate $\tilde{q}=1$, and the integral of the equation 8 is estimated by
the trapezium rule. We choose two values for the $\tilde{h}$ parameter
($h_{\textrm{min}}$ and $h_{\textrm{max}}$) satisfying
$\tilde{q}(\tilde{h}_{\textrm{min}})\leq
1\leq\tilde{q}(\tilde{h}_{\textrm{max}})$ and we find the film thickness by a
bisection method for which the condition $\tilde{q}=1$ is satisfied at $0.1$%.
The results are shown in figure 5. The numerical solution for the Newtonian
case, $n=1$ is identical to the analytical Nusselt solution. For a constant
flow rate, an increase in the shear-thinning effect modifies the velocity
profile shape: the parabolic profile tends to be replaced by a plug-like
profile. This results in higher velocity gradient close to the fiber, whereas
close to the interface, the velocity gradient is almost zero. The film
thickness is always smaller than for Newtonian fluids. The film thickness is
plotted as a function of the flow rate on the film for a PAAm solution (0.4%,
n=0.71), in figure (6). We choose to present experimental data only for PAAm
solutions, since for xanthan solutions, the healing length is two or three
times smaller than for PAAm solutions. The good agreement between the
numerical solutions and our experimental data validates the choice of the
Oswald power-law model for the viscosity and also our assumption on the
negligible effect of surfactant on the zero-shear stress boundary condition at
the liquid-air interface.
Figure 5: Velocity profiles of the flowing film at a constant flow rate
($\tilde{q}=1$) for several $n$ values from a Newtonian fluid ($n=1$) to a
high shear-thinning effect ($n=0.4$). The inset is a close-up of the liquid-
air interface region. Figure 6: Numerical solution (solid line) and
experimental results for a concentration in PAAm equal to $0.4$%. The dashed
curve is the analytical solution for $h\ll R$ given by the equation 16 for
$n=0.73$ .
## 4 Instability growth rate
The impact of the shear-thinning and elastic effects on the growth rate of the
Rayleigh-Plateau instability is investigated through experiments with xanthan
(0.8% NaCl) and PAAm solutions (0.4%) in order to distinguish the role of each
non-Newtonian property. After the flat film region, some variations on the
film thickness are detected and a regular pattern of drops emerges due to the
Rayleigh-Plateau instability as shown in figure 4. The wavelength of the drop-
like pattern is plotted in figure 7 as a function of the distance to the
entrance nozzle. The wavelength increases (regime A in figure 7) until it
reaches a well-defined value (regime B in figure 7). Then, lower down, some
coalescence events can disrupt the regular pattern (regime C particularly in
figure 7a). The wavelengths of the regular pattern for non-Newtonian fluids
are somewhat higher than those expected with Newtonian fluids. Nevertheless,
in both cases, the classical Rayleigh Plateau wavelength fails to fit the
experimental data and is always smaller. The length of the regular pattern
depends on the flow rate (at high flow rates, coalescence events occur
earlier) but it is typically is of the order of seven centimeters for PAAm
solutions (figure 7a) and shorter, about four centimeters, for xanthan
solutions (figure 7b). In the former case, we can note that the axisymmetric
conformation is not the only case observed on the fiber: some non-axially
symmetric conformations can be observed with asymmetric drops. Such
conformations have been described by [5] as a roll-up transition and must be
avoided in our case.
Figure 7: Wavelength at different flow rates for (a) xanthan and (b) PAAm
solution. Letters A, B and C denote respectively the growth of the
instability, the ordered and the disordered pattern regions.
In order to characterize the instability growth rate, we record a stack of
images at a typical frame rate of 1000 images per second. Then, the position
of the film interface is detected over space and time: $h(z,t)$. Figure 8
shows the average film thickness over the time $<h(z,t)>_{t}$ and the extremal
film positions for a PAAm solution. It shows successively the meniscus, the
flat film and the onset of the instability which is marked by a strong
variation of the film thickness. The velocity of the interface is calculated
for each stack. Then, a point on the interface (chosen to become a point of
maximum height) is followed at this velocity using the set of data $h(z,t)$.
The resulting values of the normalized profile $(h-h_{0})/h_{0}$ as a function
of time for a typical experiment are plotted in the inset of figure 9 and
fitted by an exponential law $\frac{h-h_{0}}{h_{0}}=Ae^{\Omega t}$ in the
early linear stages. From this fit, the growth rate, $\Omega$, is extracted
and averaged over several other experiments; the results for xanthan and PAAm
solutions were reported in figure 9.
Figure 8: Average film thickness in time $<h>_{t}$ along the $z$ fiber axis
($Q=0.032$ g/s). Bars indicate the extreme values of the film thickness. The
flat film thickness is denoted by $h_{0}$. Figure 9: The growth rate,
$\Omega$, is plotted versus the flat film thickness, $h_{0}$. The experimental
data are represented by the dots and the equation 22 by curves. The inset
shows the growth of the film using the method described in section 4.
A first simplified attempt to obtain an expression for the growth rate
consists in a linear stability analysis. The fluid is assumed to exhibit pure
shear-thinning effects with $\eta(\dot{\gamma})=\beta\dot{\gamma}^{n-1}$, and
we assume very thin films such that $h \ll R$ (planar approximation). Thus, in
cartesian coordinates, the following momentum equation holds, in the
lubrication approximation:
$0=\Pi+\frac{\partial\eta(\dot{\gamma})\dot{\gamma}}{\partial r}$ (9)
where $\Pi$ is the pressure gradient given by:
$\Pi=\rho
g+\gamma\left(\frac{\partial_{z}h}{(R+h)^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)\simeq\rho
g+\gamma\left(\frac{\partial_{z}h}{R^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)$ (10)
GIven that there is no fluid slippage on the fiber and no stress on the
liquid-air interface, the velocity is calculated from the momentum equation,
so that
$v(r)=\frac{1}{1+1/n}\left(\frac{\Pi}{\beta}\right)^{1/n}\left[h^{1+1/n}-(h-r)^{1+1/n}\right]$
(11)
The flow rate per unit length, defined as $q_{p}=\int_{0}^{h}vdr$, is given
by:
$q_{p}=\left(\frac{\Pi}{\beta}\right)^{1/n}\frac{h^{2+1/n}}{2+1/n}$ (12)
and also satisfies the mass conservation equation:
$\frac{\partial h}{\partial t}+\frac{\partial q_{p}}{\partial z}=0$ (13)
We assume infinitesimal perturbations around the uniform film thickness,
$h_{0}$, and for the corresponding flow rate, $q_{p0}$, so that
$\displaystyle h(z,t)=h_{0}+h_{1}(z,t),\qquad$ (14) $\displaystyle
q_{p}=q_{p0}+q_{p1},\qquad$ (15)
This results in the following expressions
$\displaystyle q_{p0}$ $\displaystyle=$ $\displaystyle\left(\frac{\rho
g}{\beta}\right)^{1/n}\frac{h_{0}^{2+1/n}}{2+1/n}$ (16) $\displaystyle q_{p1}$
$\displaystyle=$ $\displaystyle q_{p0}\left[\frac{\gamma}{n\rho
g}\left(\frac{\partial_{z}h_{1}}{R^{2}}+\frac{\partial^{3}h_{1}}{\partial
z^{3}}\right)+\left(2+\frac{1}{n}\right)\frac{h_{1}}{h_{0}}\right]$ (17)
and in the linearized equation:
$-\frac{\partial h_{1}}{\partial t}=\frac{q_{p0}\gamma}{n\rho
g}\left(\frac{\partial_{z}^{2}h_{1}}{R^{2}}+\frac{\partial^{4}h_{1}}{\partial
z^{4}}\right)+\left(2+\frac{1}{n}\right)\frac{q_{p0}}{h_{0}}\frac{\partial
h_{1}}{\partial z}$ (19)
Developing the thickness perturbation as $h_{1}=Ae^{i(kz-\omega t)}$ leads to
the dispersion relation:
$\omega(k)=k\frac{q_{p0}}{h_{0}}\left(2+\frac{1}{n}\right)+i\frac{q_{p0}\gamma}{n\rho
g}\left(\frac{k^{2}}{R^{2}}-k^{4}\right)$ (20)
The maximum of $\mbox{Im}(\omega(k))$ gives:
$\Omega=\frac{q_{p0}\gamma}{4n\rho gR^{4}},\qquad h\ll R\\\ $ (21)
As shown in figure 6, the flow rate given by 16 does not reproduce the
experiment where the planar approximation ($h\ll R$) is not valid. In previous
works on Newtonian fluids, it has been shown [8] that for $h\sim R$, the
expression for the growth rate is similar except that $R$ should be replaced
by $R+h$. Moreover, we made the choice to use our numerical calculation
described in section 3 which provides $q_{\textrm{num}}(h)$ without assumption
on the film thickness. Finally, the addition of a large amount of surfactant
modifies the growth rate by a factor $4$, as described by [6] since
surfactants change the surface elasticity.
Taking into account these corrections, we obtain the expression
$\Omega=\frac{1}{4}\frac{q_{\textrm{num}}(h)\gamma}{4n\rho g(R+h)^{4}}\\\ $
(22)
which is plotted in figure 9 for both chemical systems.
Data for the xanthan solution are well described by 22. To validate our method
and to compare with a Newtonian fluid of similar surface tension
($\gamma=20.9$ mN/m) and viscosity ($\eta=0.965$ Pa.s), we have performed an
experiment with a silicon oil (with $\rho=96.5$ kg.m-3 and $h_{0}=0.55$ mm)
and measured a growth rate $\Omega$ equal to $10.9\pm 0.7$ s-1. Equation 22
for $n=1$ gives $\Omega$ equal to $11.1$ s-1. The small variations in growth
rate between the xanthan solution and the silicon oil are reasonable as the
value of the viscosity is of the same order. Concerning PAAm solutions, there
is significant deviation from the theory due to the normal stress of this
solution. A qualitative explanation of the role of normal stress can be
provided by the “hoop stress” effect [19]. A short description of this effect
can be made by considering the liquid surface as an infinite cylindrical shell
of thickness $e$ (figure 10). For a cylinder of radius $R+h$, the balance
between the internal pressure $P$ and the stretching stress
$\sigma_{\theta\theta}$, leads to $2(R+h)LP=2L\sigma_{\theta\theta}e$. The
internal pressure $P$ is generated by the normal stress $N_{1}$ in the bulk.
Interpreting the stretching force per unit length, $2\sigma_{\theta\theta}e$,
in term of surface tension $\gamma_{\psi_{1}}$, we obtain
$\gamma_{\psi_{1}}=-\psi_{1}\dot{\gamma}^{2}(R+h)$ (23)
An estimation for $\dot{\gamma}=100$ s-1 gives a $\gamma_{\psi_{1}}$ of about
$-10$ mN/m significantly lowering the effective surface tension. The growth
rate should therefore be estimated with the effective surface tension lower
than the fluid surface tension and the resulting curve for the growth rate
would be shifted and enable us to recover the experimental data.
Figure 10: Stretching stress, $\sigma_{\theta\theta}$, in a thin cylindrical
shell of radius $R+h$ and thickness $e\ll R+h$.
## 5 Drop morphologies: normal stress effect
This section is devoted to a comparison between the patterns of flowing films
of PAAm and xanthan, focusing particularly on the normal stress effect on the
shape of the drops. Such a comparison requires polymeric solutions having
similar shear-thinning properties. Further, $0.8$% NaCl was added to xanthan
($0.4$%) solutions to decrease the high shear-thinning effect (figure 3).
Optimal adjustment of the shear-thinning of PAAm solutions ($0.6$%) is
achieved as shown in the inset of figure 11. Thus, the difference between the
two solutions concerns only the presence or absence of normal stress.
The typical pattern observed on the fiber consists of an axisymmetric film of
constant thickness. Then the film breaks up spontaneously into a drop-like
wave train as described in the previous section.
For axisymmetric patterns, the superposition of drop profiles in a PAAm and
xanthan films is shown in figure 11. Clear differences can be noticed in the
profiles, notably the steepening of the drop front for the PAAm solution
compared to the xanthan solution. In both profiles, there is a clear asymmetry
between the front and back of the drops which is more accentuated for the
xanthan drop. This remark suggests that the shape of the drops is affected by
gravity. The apex heights of both drops are identical as well as the film
substrate between drops (the trailing edge for PAAm is longer than for
xanthan). These observations confirm the fact that the normal stress plays a
significant role in the thin regions, close to the tail and the front of the
drops, but exhibits no effect in the center of the drop (the thick region). To
quantify experimentally the swelling effect observed with PAAm solution, we
define the slope of the front $H/L$ as shown in figure 11. This parameter is
plotted in PAAm and xanthan solutions for different flowing rates in figure
12.
Figure 11: Drop shapes for two polymer solutions. The drop front of the
viscoelastic solution (dashed green line) is swollen compared with the pure
shear-thinning liquid (solid red line). Figure 12: Slopes of drop fronts for
xanthan and PAAm as a function of the capillary number Ca.
To highlight the swelling process, we consider a scaling law analysis,
starting from the stress balance 5. Since the film is not flat the
$z$-invariance is no longer valid. Exhibiting the contribution of the normal
stress difference $\sigma_{zz}-\sigma_{rr}$, and shear stress $\sigma_{rz}$,
equation 5 becomes
$\frac{\partial(\psi_{1}\left(\frac{\partial v}{\partial
r}\right)^{2})}{\partial
z}+\frac{1}{r}\frac{\partial(r\eta(\dot{\gamma})\dot{\gamma})}{\partial
r}=\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho
g$ (24)
The axial velocity $v(r,z)$, is determined using a series of the form [3]:
$v(r,z)=a_{0}(z)+a_{1}(z)r+a_{2}(z)r^{2}$ (25)
Functions $a_{0}(z)$, $a_{1}(z)$ et $a_{3}(z)$ are calculated using the
following three equations, two for the boundary conditions and the last one
for the mass conservation:
* •
Boundary condition at the interface with the fiber $v(r=R,z)=0$;
* •
Boundary condition at the liquid/air interface $\frac{\partial
v(r=R+h(z))}{\partial r}=0$;
* •
The mass conservation equation,
$\frac{\partial h}{\partial t}+\frac{\partial}{\partial
z}\int_{R}^{R+h(z,t)}v(r,z)dr=0$ (26)
Considering equation 26 in the reference frame of a drop moving at a velocity
$U$ and using the condition that the mean flow rate satisfies
$\overline{q}\sim Uh\underset{h\rightarrow 0}{\longrightarrow}0$, the axial
velocity is
$v(r,z)=\frac{3}{2}U\frac{(R-r)(r-2h(z)-R)}{2h^{2}(z)}$ (27)
Assuming a constant viscosity, $\eta$, the stress balance equation 24 to zero-
order in $r$ is
$3\eta
U\frac{h(z)-R}{2Rh^{2}(z)}-U^{2}\psi_{1}\frac{9h^{\prime}}{2h^{3}(z)}=\frac{\partial
p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho g$ (28)
The normal stress balance at the free surface of the film assumes that
$-P+\sigma_{rr}=\gamma\kappa$ with $\kappa$ the curvature of the interface. If
$L$ is the characteristic length in the axial direction and if $H$ is the
characteristic apex height of the drop, then the normal stress balance at the
free surface of the film, accounting for the curvature, is given by
$\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial
z}\sim-\gamma\left(\frac{\partial_{z}h}{R^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)$ (29)
Thus the right-hand side of equation 28 becomes:
$\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho
g\sim-\gamma\left(\frac{H}{L^{3}}+\frac{H}{LR^{2}}+l_{c}^{-2}\right)$ (30)
where $l_{c}=\sqrt{\frac{\gamma}{\rho g}}$ is the capillary length.
Experimental observations suggest that $\frac{H}{L^{3}}\ll\frac{H}{LR^{2}}$
and $\kappa^{2}\ll\frac{H}{LR^{2}}$. So, the scaling law analysis leads to the
following equation:
$\frac{H^{3}}{L^{3}}\sim\frac{R^{2}}{L^{2}}\textrm{Ca}\left(1-6\frac{\mathscr{L}}{L}\right)$
(31)
where $\textrm{Ca}=\frac{\eta U}{\gamma}$ is the capillary number and
$\mathscr{L}=\frac{\psi_{1}U}{\eta}$ is the normal stress characteristic
length.
This scaling law gives the slope of the front of a drop, $H/L$, as a function
of the viscoelastic properties of the polymeric solution. In particular, in
the case of polymeric solutions exhibiting normal stress, $\psi_{1}\neq 0$,
the expression 31 clearly shows that $H/L$ decreases. A comparison between the
experimental data and the results of the scaling analysis is presented in
figure 12 for different flow rates. There is a good agreement between the
experiment and the model which succeeds in highlighting the swelling effect on
the drop shape induced by the normal stress effect.
## 6 Conclusion
The effects of non-Newtonian properties of fluids have been investigated in
the case of a film flowing down a vertical fiber. The flow on the fiber can be
divided into three regions: (A) at the inlet, the film exhibits a uniform
thickness, i.e the flat film region; (B) the uniform film is progressively
replaced by a well-defined pattern of drops separated by a thin film, i.e the
Rayleigh-Plateau region; (C) the coalescence of drops disrupts the flow and
give rise to a disordered pattern. In order to disentangle the role of the
shear-thinning effect and of the normal stress, we have considered two kinds
of polymer solutions. The first consists of rigid rod-like polymers (xanthan),
exhibiting a strong shear-thinning behaviour but negligible elastic effects.
For the second solution, we used flexible polymers (PAAm) exhibiting strong
elastic effects and shear-thinning effects similar to those of xanthan under
certain physico-chemical conditions. Some adjustments have been made by
modifying both the polymer concentration and the physico-chemical properties
of the solutions to enhance or reduce one of the non-Newtonian properties :
shear-thinning or elastic effect.
Consequences of both effects have been investigated in the first two regions
of the flowing film. In the flat film region, due to the invariance of the
film thickness in the axial direction, only the shear-thinning effect is
effective. At a constant flow rate, our experiments demonstrate that, as a
consequence of the shear-thinning effect, the thickness of the film is always
smaller than in the case of Newtonian fluid. Our results clearly show the
influence of the shear-thinning effect on the velocity profile: a parabolic
profile in the Newtonian case tends to become a plug-like profile. Thus, an
increase of the shear-thinning effect yields a thinner, unperturbed film.
Further downstream on the fiber, the film undergoes the Rayleigh-Plateau
instability. The growth rate of the instability has been investigated
experimentally and theoretically using a linear stability analysis. Good
agreement is found between the experimental data for xanthan and the model.
For PAAm solutions, and to take into account the normal forces, we consider an
effective surface tension (lower than the fluid surface tension) which tends
to decrease the growth rate and to recover the experimental data. The
morphology of the patterns resulting from the instability depends on the non-
Newtonian properties. In particular, the drops formed with PAAm solutions
exhibit a swelling effect compared to drops observed with xanthan solution,
for a similar shear-thinning effect. We observe that the drop of fluid with
normal forces is less rounded compared with the case of a pure shear-thinning
drop. This swelling effect has been quantified by a scaling law analysis where
the slope of the drop front is expressed as a function of the normal stress.
As a conclusion, by considering two kinds of polymeric solutions with the same
shear-thinning effect, which differ from each other in the presence of normal
forces, we have succeeded in understanding the relationship between the
rheological properties and the destabilization of the flowing film on a fiber
as well as the morphology of the observed patterns. This should be helpful in
understanding what happens with more complex fluids, in particular fluids
which exhibit more elastic effects where the elasticity could prevent the
growth of the instability.
##### Acknowledgments:
We thank Fédération Paris VI (high-speed camera) and Triangle de la Physique
(rheometer apparatus). The authors thank Liyan Yu and Prof. John Hinch for
fruitful discussions. Also we thank Lionel Auffray, Rafael Pidoux and Alban
Aubertin for the experiment engineering and technical improvements.
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* [19] M.D. Graham. Interfacial hoop stress and instability of viscoelastic free surface flows. Physics of Fluids, 15:1702–1710, 2003.
* [20] S. Kalliadasis and H. Chang. Drop formation during coating of vertical fibres. Journal of Fluid Mechanics, 261:135–168, 1994.
* [21] I.L. Kliakhandler, S.H. Davis, and S.G. Bankoff. Viscous beads on vertical fibre. Journal of Fluid Mechanics, 429:381–390, 2001.
* [22] A. Lindner and C. Wagner. Viscoelastic surface instabilities. Comptes Rendus Physique, 10:712 – 727, 2009.
* [23] Y. Liu, Y. Jun, and V. Steinberg. Concentration dependence of the longest relaxation times of dilute and semi-dilute polymer solutions. Journal of Rheology, 53:1069–1085, 2009.
* [24] C. W. Macosko. Rheology Principles, Measurements, and Applications. Wiley Edition, 1994.
* [25] D. Quéré. Thin films flowing on vertical fibers. Europhysics Letters, 13:721, 1990.
* [26] D. Quéré. Fluid coating on a fiber. Annual Review of Fluid Mechanics, 31:347–384, 1999.
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|
arxiv-papers
| 2012-05-17T08:07:14 |
2024-09-04T02:49:31.000289
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne and Ludovic Pauchard and Fr\\'ed\\'erique\n Giorgiutti-Dauphin\\'e",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1205.3875"
}
|
1205.3887
|
# COMPARISON OF COMPRESSION SCHEMES FOR CLARA
P. H. Williams peter.williams@stfc.ac.uk J. K. Jones & J. W. McKenzie
STFC Daresbury Laboratory ASTeC & Cockcroft Institute UK
###### Abstract
CLARA (Compact Linear Advanced Research Accelerator) at Daresbury Laboratory
is proposed to be the UK’s national FEL test facility. The accelerator will be
a $\sim 250$ MeV electron linac capable of producing short, high brightness
electron bunches. The machine comprises a $2.5$ cell RF photocathode gun, one
$2$ m and three $5$ m normal conducting S-band ($2998$ MHz) accelerating
structures and a variable magnetic compression chicane. CLARA will be used as
a test bed for novel FEL configurations. We present a comparison of
acceleration and compression schemes for the candidate machine layout.
## 1 INTRODUCTION
The design approach adopted for CLARA is to build in flexibility of operation
and layout, enabling as wide an exploration of FEL schemes as possible. For a
full overview of the aims of the project and details of FEL schemes under
consideration see [1]. To this end a range of possible accelerator
configurations have been considered, a selection of this work is presented
here.
A major aim is to be able to test seeded FEL schemes. This places a stringent
requirement on the longitudinal properties of the electron bunches, namely
that the slice parameters should be nearly constant for a large proportion of
the full-width bunch length. In addition, the intention is that CLARA has the
ability to deliver high peak current bunches for SASE operation and ultra-
short pulse generation schemes, and low-emittance velocity compressed bunches.
This flexibility of delivering tailored pulse profiles will allow a direct
comparison of FEL schemes in one facility.
## 2 ENERGY AT MAGNETIC COMPRESSOR
A large proportion of the FEL schemes under consideration require small
correlated energy spread at the undulators, therefore when magnetic
compression is to be used the compressor must be situated at substantially
less than full energy. This ensures that the chirp needed at compression is
able to be adiabatically damped or suppressed through running subsequent
accelerating structures beyond crest. This requirement must be balanced
against the fact that compressing at low energy exacerbates space-charge
effects. To quantify this we use the laminarity parameter
$\rho_{L}\equiv\left(\frac{I/(2I_{A})}{\epsilon_{th}\gamma\gamma^{{}^{\prime}}\sqrt{1/4+\Omega^{2}}}\right)^{2},$
where $I$ is the current in the slice under consideration, $I_{A}$ is the
Alfven current, $\epsilon_{th}$ is the thermal emittance,
$\gamma^{{}^{\prime}}\equiv d\gamma/ds$ and $\Omega$ is a solenoidal focusing
field (zero in our case). When this parameter is greater than $1$, we should
consider space-charge effects in the bunch evolution. To inform this we select
two candidate configurations, one with magnetic compression at $70$ MeV and
one at $130$ MeV. We track a candidate $200$ pC bunch through both
configurations, setting the machine parameters attempt to produce a zero chirp
bunch of peak current $350$ A at $250$ MeV. Tracking is carried out with ASTRA
[2, 3] to the exit of the first linac section to include space-charge,
followed by ELEGANT [4] taking into account the effect of cavity wakefields,
longitudinal space-charge and coherent synchrotron radiation emittance
dilution. Fig. 1 shows the resultant laminarities and final bunch longitudinal
phase spaces.
---
|
Figure 1: (Upper) Laminarity (black / blue) and peak current (red / green) in
the $10\%$ of charge slice containing the peak current with compression at
$70$ MeV / $130$ MeV (the gun and first $2$ m linac module are not shown).
(Lower left) Long. phase space with compression at $130$ MeV. (Lower right)
Long. phase space with compression at $70$ MeV.
We see that in both cases space-charge should be considered, this will be
achieved by re-tracking the magnetically compressed bunches in ASTRA to
elucidate any deviations due to 3-d effects as compared to the purely 1-d
effects included in ELEGANT. However compression at $130$ MeV does not allow
us to subsequently de-chirp the bunch, note that we go no further than
$30^{\circ}$ beyond crest in the final accelerating structures in order to
avoid large jitter effects. As we wish the facility to be flexible we select a
nominal compressor energy of $70$ MeV, but we achieve this by reducing the
gradient in the accelerating structure before the compressor. This gives us
the option of compressing at higher energy in regimes where a de-chirped bunch
is not required. With the above considerations in mind we define engineering
specifications for the CLARA variable bunch compressor as shown in Table 1.
For flexibility, the compressor has a continuously variable $R_{56}$ and is
rated for maximum energy of $150$ MeV. The ability to set a straight through
path also allows investigation of purely velocity compressed bunches.
Table 1: Specification of variable bunch compressor. | Value | Unit
---|---|---
Energy at compressor | 70 - 150 | MeV
Min. : Max. bend angle | 0 : 200 | mrad
Bend magnetic length | 200 | mm
Max. bend field | 0.5 | T
Min. : Max. transverse offset | 0 : 300 | mm
Z-distance DIP-01/04 - DIP-02/03 | 1500 | mm
Z-distance DIP-02 - DIP-03 | 1000 | mm
Max. bellows extension | 260 | mm
Min. : Max. $R_{56}$ | 0 : -72 | mm
Max. $\sigma_{x}$ from $\delta_{E}$ ($\pm 3\sigma$) | 0 : 10 | mm
Max. $\sigma_{x}$ from $\beta_{x}$ ($\pm 3\sigma$) | 1.5 | mm
## 3 BUNCH FOR SEEDING SCHEMES
A seeded FEL scheme requires constant bunch parameters over a large proportion
of the bunch. This reduces the sensitivity to timing jitter between the seed
laser and electron bunch. Specifically, we require a constant peak current of
$350$ A over $300$ fs of the bunch, with zero chirp, constant emittances and
zero transverse offset. In order to achieve this we must cancel the curvature
that originates from RF acceleration. It is possible to do this purely
magnetically, although typically it is done with higher harmonic RF. As
harmonic RF entails additional expense we compare two schemes, a bunch
compressor with non-linear elements inserted, and a fourth harmonic X-band
cavity.
### 3.1 LINEARISATION VIA NONLINEAR CHICANE
The lower right plot of Fig. 1 shows residual curvature in the longitudinal
phase space. This can be flattened by changing the sign of the natural
$T_{566}$ term in the bunch compressor chicane. To achieve this sextupoles
were added to the chicane. The number, positions and strengths of these were
parameters of an optimisation. We impose the constraints that the $T_{566}<20$
cm, the derivative of dispersion with respect to energy and it’s derivative
with respect to $s$ should be zero on exit of the chicane, the projected
emittances should not exceed $1$ mm mrad and the sextupoles $k_{2}<2000$ m-2.
This constraint set was chosen by trial and error.
Figure 2: Results for an example optimised nonlinear bunch compressor. (1)
Chromatic amplitude functions (black, red) & chromatic derivative of
dispersion (blue). (2) $R_{56}$ (black) & $T_{566}$ (red). (3)
$\varepsilon_{N(x,y)}$ less dispersive contributions (black, red). (4)
Longitudinal phase space (blue - optimised, red - without sextupoles). (5)
Current profile ($20$ fs slices, blue - optimised, red - without sextupoles).
(6) $x-t$ phase space (blue - optimised, red - without sextupoles). (7)
Normalised slice emittances ($20$ fs slices): horizontal (blue - optimised,
red - without sextupoles) and vertical (green - optimised, orange - without
sextupoles). (8) Slice energy spread ($20$ fs slices) (blue - optimised, red -
without sextupoles)
Figure 2 shows the optimisation results. Flattening the longitudinal curvature
is relatively straightforward however the chromatic properties are easy to
spoil, resulting in increased projected and slice emittance. Up to six
sextupoles were tried with similar results. These nonlinear compressors have
also been studied under energy jitter and the bunch parameters found to vary
substantially.
### 3.2 LINEARISATION VIA HARMONIC RF
We insert a fourth harmonic $0.7$ m structure immediately prior to the
magnetic compressor. An optimisation [5] was then performed with variables
being the harmonic voltage and phase, the off crest phase of the preceding
linac and the angle of the compressor dipoles. Results for two candidate
tunings are shown in Fig. 3. The peak voltage on the linearising cavity is $7$
MV/m. It can be seen that the additional complication of a harmonic cavity is
justified by ability to predictably tailor longitudinal phase space.
Figure 3: Two candidate optimisations linearising with harmonic cavity. (1)
Optics: $\beta_{x,y}$ (black, red) & $\eta_{x}$. (2) $\varepsilon_{N(x,y)}$
less dispersive contributions (black, red). (3) Longitudinal phase space (blue
- optimised for $200$ fs flat top, red - optimised for $300$ fs flat top). (4)
Current profile ($40$ fs slices, optimised for $200$ fs flat top, red -
optimised for $300$ fs flat top). (5) Normalised slice emittances ($40$ fs
slices): horizontal / vertical (blue / red - optimised for $200$ fs flat top,
green / orange - optimised for $300$ fs flat top) (6) Slice energy spread
($20$ fs slices) optimised for $200$ fs flat top, red - optimised for $300$ fs
flat top.
## 4 VELOCITY BUNCHING
An alternative to magnetic compression is to use velocity bunching in the low
energy section of the accelerator. The first $2$ m linac section is set to the
zero crossing to impart a time-velocity chirp along the bunch. The bunch then
compresses in the following drift space. The second linac section is
positioned at the waist of the bunch length evolution after $3$ m of drift to
rapidly accelerate the beam and capture the short bunch length. Solenoids are
required around the bunching section to control the transverse beam size and
prevent emittance degradation. ASTRA was used to track until the end of the
second linac module followed by ELEGANT. The quadrupoles between the first and
second linac sections are switched off in order to keep the beam axially
symmetric, and the bunch compressor set to zero angle. An evolutionary
algorithm was used to optimise the beamline for both bunch length and
transverse emittance. We present two tunings with $100$ pC bunch charge.
Figure 4: Velocity bunched beam tuned for low emittance. Figure 5: Velocity
bunched beam tuned for peak current.
Fig. 4 shows a bunch with similar peak current and current profile to the non-
linearised magnetically compressed bunch of Fig. 2-5. This is achieved at half
the total bunch charge, with lower slice energy spread, but higher slice
emittance. In Fig. 4 we show that a similar peak current to the non-linearised
magnetically compressed bunch of Fig. 2-5 is easily achieved with smaller
slice energy spread but higher slice emittance. Fig. 5 shows a beam tuned for
peak current at the exit of the second linac module. The peak current then
degrades along the accelerator. This bunch has the capabilities to provide
single-spike SASE FEL operation.
## 5 CONCLUSIONS
This initial study has established an accelerator layout for CLARA that is
inherently flexible in the pulse profiles it is capable of producing. We have
shown this by simulating bunches suitable for seeded and SASE FEL operation.
Further work will entail jitter tolerance analysis of the presented
configurations.
## References
* [1] J. A. Clarke et. al., TUPPP066, these proceedings
* [2] K. Flöttmann, http://www.desy.de/~mpyflo.
* [3] J. W. McKenzie & B. L .Militsyn, THPC132, IPAC 11.
* [4] M. Borland, Advanced Photon Source, LS-287 (2000).
* [5] R. Luus & T. H. I. Jaakola, AIChE Journal 19, 760 (1973).
|
arxiv-papers
| 2012-05-17T09:17:54 |
2024-09-04T02:49:31.008644
|
{
"license": "Public Domain",
"authors": "Peter H. Williams, James K. Jones and Julian W. McKenzie",
"submitter": "Peter Williams",
"url": "https://arxiv.org/abs/1205.3887"
}
|
1205.3923
|
The following article appeared in J. Vac. Sci. Technol. B 30, 03D112 (2012)
and may be found at http://link.aip.org/link/?jvb/30/03D112.
# Valley and spin polarization from graphene line defect scattering
Daniel Gunlycke Naval Research Laboratory, Washington, D.C. 20375, USA
Carter T. White Naval Research Laboratory, Washington, D.C. 20375, USA
###### Abstract
Quantum transport calculations describing electron scattering off an extended
line defect in graphene are presented. The calculations include potentials
from local magnetic moments recently predicted to exist on sites adjacent to
the line defect. The transmission probability is derived and expressed as a
function of valley, spin, and angle of incidence of an electron at the Fermi
level being scattered. It is shown that the previously predicted valley
polarization in a beam of transmitted electrons is not significantly
influenced by the presence of the magnetic moments. These moments, however, do
introduce some spin polarization, in addition to the valley polarization,
albeit no more than about 20%.
## I Introduction
Figure 1: Extended line defect in graphene. (a) An electronic Bloch wave
approaching the line defect at an angle of incidence $\alpha$ is being
scattered. This scattering is influenced by local magnetic moments on the
sites (between green arrows) next to the line defect sites (between blue
arrows). (b) The semi-infinite sheet of graphene to the left of the line
defect transformed into momentum space along the line defect. The chain has
alternating couplings, $\gamma$ and $\gamma^{\prime}$, making the self energy
$\Delta$ representing the influence of the sites away from the end point
depend on whether the end site is of type A (green) or B (blue). (c) Primitive
cell of the graphene line defect. The sites are paired to form the two chains
$\nu=0,1$.
The success of electronics rests on the ability to control electron motion.
Such control is typically achieved by varying the electrostatic potential in a
semiconductor with a suitable band gap. This straightforward way to control
electron motion has proven tremendously successful and is the main reason for
the considerable effort by the graphene community devoted to graphene
nanoribbonsKlei94_1 ; Fuji96_1 ; Han07_1 and bilayer graphene in the presence
of an electric field.Lu06_1 ; Guin06_1 ; McCa06_2 ; Zhan09_1 Another more
subtle way to control electron motion is though scattering off deliberate
defects in the material.Yazy10a ; Gunl11_1 This approach could add new
functionality and ultimately prove to be the future for electronics. Graphene
is a promising material for controlled electron scattering for several
reasons: (i) it has a well-defined structure,Wall47_1 ; Novo05_1 (ii) owing
to its sp2 hybridization, it has $\pi$-orbitals near the Fermi level that can
form extended states,Wall47_1 (iii) it is a semi-metal with only a limited
number of scattering channels available near the Fermi level,Wall47_1 and
(iv) it offers an electron mobilityBolo08_1 ; Du08_1 ; Orli08_1 high enough
to support ballistic transport in the micron range.Bolo08_1
Herein, we consider electron scattering off the extended line defect in
graphene illustrated in Fig. 1(a). This structure both preserves the sp2
hybridization of carbon and is precisely defined. It is not a hypothetical
structure, but one that has already been observed in experiments.Lahi10 An
arbitrary electron scattering off the line defect occupies a state that away
from the line defect approaches asymptotically one of graphene, and if the
energy of this electron is near the graphene Fermi level, it can in addition
to its energy be identified by its direction of motion, its valley, and its
spin. Herein, a tight-binding model is used to derive the transmission
probability of an arbitrary incident electron. The model includes a potential
to describe ferromagnetically aligned local magnetic moments that has been
shown to be present in the line defect structure.Whit12_1 These moments break
the spin-degeneracy, otherwise present, causing there to be spin polarization
among the electrons of a transmitted beam. This spin polarization is found to
be rather small, limiting its use. More important is that the local magnetic
moments do not appear to degrade the predicted valley polarization of the
transmitted electrons near the Fermi level.Gunl11_1 Therefore, the graphene
line defect remain an illustrative example of a system where the valley degree
of freedom can be exploited instead of or alongside the spin degree of freedom
for applications in quantum information processing.
The next section develops the theoretical formalism to describe the electron
scattering off the line defect. This theory is applied in Sec. III to the
scattering of electrons near and at the Fermi level. Conclusions drawn from
the results are presented in Sec. IV, including symmetry argument explaining
the large valley polarization for electrons scattering at high angles of
incidence.
## II Scattering formalism
Our objective is to obtain the transmission probability for an arbitrary
right-moving electron approaching the line defect. The scattering calculations
are founded on a tight-binding model with a basis set consisting of
orthonormal $\pi$-orbitals.Wall47_1 Herein, we consider nearest-neighbor
interactions with a hopping parameter $\gamma=-2.6$ eV. Longer-range
interactionsGunl11_1 and distortionsJian11 do not qualitatively change the
results and has therefore been ignored for presentational clarity.
The scattering problem is solved in steps. First, we recognize that the
structure in Fig. 1(a) can be viewed as a set of line defect sites connected
to two semi-infinite graphene sheets. Next, we solve for the self energy
representing all interactions within one of the semi-infinite sheets. This can
be achieved by exploiting translational symmetry along the $y$-direction. Once
the self energy has been derived, we can then calculate the retarded Green
function on the line defect sites, which is needed to obtain the sought after
transmission probability.
To keep the notation tidy, much of the formalism below is presented in
dimensionless units, which can be recognized by their diacritic tildes. Energy
units are scaled with the absolute value of the hopping parameter $\gamma$.
Other dimensionless units are also introduced below, as needed.
### II.1 Semi-infinite graphene
The influence of sites away from the line defect is captured by a self energy
defined at the edges of the semi-infinite sheets of graphene on each side of
the line defect. To derive this self energy, we first recognize that a semi-
infinite sheet of graphene with a zigzag edge has translational symmetry in
the direction along the edge with a period equal to the graphene lattice
constant $a$. If a Bloch wave is considered with an arbitrary wave vector
$k_{y}$ along the direction of translational symmetry, we can transform the
semi-infinite graphene sheet into a semi-infinite linear chain with
alternating couplings $\gamma$ and $\gamma^{\prime}\equiv
2\gamma\cos\tilde{k}_{y}$, where we have defined $\tilde{k}_{y}\equiv
k_{y}a/2$. See Fig. 1(b). Self-energy recurrence relations can be generated
through a process of replacing the sites away from the end site with a self
energy, adding a site to the chain, and recalculate the self energy at the new
end site. In this case, the recurrence relations become
$\displaystyle\tilde{\Delta}^{A}$
$\displaystyle=4\cos^{2}\tilde{k}_{y}\left(\tilde{E}-\tilde{\Delta}^{B}\right)^{-1}$
(1) $\displaystyle\tilde{\Delta}^{B}$
$\displaystyle=\left(\tilde{E}-\tilde{\Delta}^{A}\right)^{-1},$ (2)
where $\tilde{E}$ is the energy of the Bloch wave measured relative to the
energy at Fermi level and $\tilde{\Delta}^{\lambda}$ is the self energy for an
end site of type $\lambda\in\\{A,B\\}$. The end site type is related to the
sublattice of the semi-infinite graphene sheet to which the edge sites belong.
The retarded solution to the self-energy recurrence relations can be expressed
as
$\displaystyle\tilde{\Delta}^{A}$
$\displaystyle=\frac{1}{\tilde{E}}\Big{[}4\cos^{2}\tilde{k}_{y}+2\cos\tilde{k}_{y}\,e^{i\tilde{k}_{x}}\Big{]}$
(3) $\displaystyle\tilde{\Delta}^{B}$
$\displaystyle=\frac{1}{\tilde{E}}\Big{[}1+2\cos\tilde{k}_{y}\,e^{i\tilde{k}_{x}}\Big{]},$
(4)
where
$\displaystyle\tilde{k}_{x}$
$\displaystyle=\pi+\operatorname{sgn}\tilde{E}\left[\pi-\operatorname{Re}\left\\{\arccos\frac{\tilde{E}^{2}-1-4\cos^{2}\tilde{k}_{y}}{4\cos\tilde{k}_{y}}\right\\}\right]$
$\displaystyle+i\left|\operatorname{Im}\left\\{\arccos\frac{\tilde{E}^{2}-1-4\cos^{2}\tilde{k}_{y}}{4\cos\tilde{k}_{y}}\right\\}\right|.$
(5)
The local moments on the sites adjacent to the line defect sites (between the
green arrows) can be modeled through a spin-dependent onsite potential
$\tilde{\varepsilon}_{\sigma}$, where $\sigma\in\\{-1,1\\}$ is the spin.
Assuming the Hubbard parameter $\tilde{U}=1.06$ [Gunl07_4, ] and a difference
of the average spin population $\langle n_{\sigma}\rangle$ per semi-infinite
chain of $\langle n_{1}\rangle-\langle n_{-1}\rangle=1/6$, we approximate the
onsite potential to be $\tilde{\varepsilon}_{\sigma}=\tilde{U}\big{[}\langle
n_{-\sigma}\rangle-\langle n_{\sigma}\rangle\big{]}/2\approx\mp 0.09$ in
dimensionless energy units for spin $\sigma=\pm 1$, respectively.Whit12_1 The
local moments affect the self energy with the end point at the line defect,
which is given by
$\tilde{\Delta}=\left(\tilde{E}-\tilde{\varepsilon}_{\sigma}-\tilde{\Delta}^{A}\right)^{-1}.$
(6)
There is translation symmetry, not only in the semi-infinite graphene sheets,
but also in the full line defect structure in Fig. 1(a). The line defect
structure has a period $2a$ along the line defect, i.e., twice the period of
the semi-infinite graphene sheet. The primitive cell of the line defect
structure is shown in Fig. 1(c), where the sites have been divided into
bottom, $\nu=0$, and top, $\nu=1$, sites. Rather than using the real space
basis $|\nu\rangle$, it is more convenient to use a basis $|n\rangle$, in
which the self energy is diagonal. To find this basis, we first recognize that
the wave vector of the line defect structure must be conserved in the
scattering process. As a result, the Bloch wave in the semi-infinite graphene
sheet with wave vector $k_{y}$ can only couple to one other Bloch wave in the
scattering process, the one with wave vector $k_{y}+\pi/a$. From the phase
relationship between equivalent sites with $\nu=0,1$ imposed by the
translational symmetry of the semi-infinite graphene sheet, we obtain
$|n\rangle=\frac{1}{\sqrt{2}}\sum_{\nu}e^{i(2\tilde{k}_{y}+n\pi)\nu}|\nu\rangle,$
(7)
where $n=0,1$. As the states $|n\rangle$ are eigenstates of the self-energy
operator $\tilde{\Sigma}$, we have
$\langle
n|\tilde{\Sigma}|n^{\prime}\rangle=\tilde{\Sigma}_{n}\delta_{nn^{\prime}},$
(8)
where $\tilde{\Sigma}_{n}$ is the self energy $\tilde{\Delta}$ in Eq. (6)
calculated for the Bloch wave with wave vector $k_{y}+n\pi/a$.
In calculating the transmission probability, we also need the elements of the
broadening operator $\tilde{\Gamma}\equiv
i\left(\tilde{\Sigma}-\tilde{\Sigma}^{\dagger}\right)$. From this definition,
we see that $|n\rangle$ are also eigenstates of $\tilde{\Gamma}$, yielding the
elements
$\langle
n|\tilde{\Gamma}|n^{\prime}\rangle=-2\operatorname{Im}\tilde{\Sigma}_{n}\,\delta_{nn^{\prime}}.$
(9)
### II.2 Line Defect
Let us focus on the center two sites in the primitive cell in Fig. 1(c)
forming the line defect. With the coupling of these sites to all other sites
in the primitive cell already accounted for through the self energy, the only
coupling in the Hamiltonian is the coupling $\gamma$ between the two sites
with $\nu=0$ and $\nu=1$. In the basis set defined by Eq. (7), the Hamiltonian
elements are
$\langle
n|\tilde{H}|n^{\prime}\rangle=-\frac{1}{2}\left[e^{i(2\tilde{k}_{y}+n^{\prime}\pi)}+e^{-i(2\tilde{k}_{y}+n\pi)}\right].$
(10)
Given the self energy and the Hamiltonian, we can calculate the retarded Green
function operator
$\tilde{G}=\big{(}\tilde{E}I-\tilde{H}-2\tilde{\Sigma}\big{)}^{-1}$, where $I$
is the unit operator. The factor 2 in front of the self energy operator
reflects the connections of the line defect to the two semi-infinite graphene
sheets. Using Eq. (8) and Eq. (10), we find the elements of the retarded Green
function operator, which can be expressed as
$\displaystyle\langle n|\tilde{G}|n^{\prime}\rangle$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{\tilde{E}-\cos(2\tilde{k}_{y}+n\pi)-2\tilde{\Sigma}_{1-n}}{\det\big{(}\tilde{E}I-\tilde{H}-2\tilde{\Sigma}\big{)}}&\quad\mathrm{for}~{}n=n^{\prime},\vspace{0.05
in}\\\
\frac{i\sin(2\tilde{k}_{y}+n\pi)}{\det\big{(}\tilde{E}I-\tilde{H}-2\tilde{\Sigma}\big{)}}&\quad\mathrm{for}~{}n\neq
n^{\prime},\end{array}\right.$ (13)
where
$\displaystyle\det\big{(}\tilde{E}I-\tilde{H}-2\tilde{\Sigma}\big{)}=\,$
$\displaystyle\big{(}\tilde{E}-2\tilde{\Sigma}_{0}\big{)}\big{(}\tilde{E}-2\tilde{\Sigma}_{1}\big{)}-1$
$\displaystyle+2\cos
2\tilde{k}_{y}\big{(}\tilde{\Sigma}_{0}-\tilde{\Sigma}_{1}\big{)}.$ (14)
Using the elements of the broadening operator in Eq. (9) and the Green
function operator in Eq. (14), we find that the probability that a state
$|n\rangle$ transmits through the line defect into state $|n^{\prime}\rangle$
is given by
$\displaystyle T_{n\rightarrow n^{\prime}}$ $\displaystyle=\langle
n|\tilde{\Gamma}|n\rangle\langle n|\tilde{G}|n^{\prime}\rangle\langle
n^{\prime}|\tilde{\Gamma}|n^{\prime}\rangle\langle
n^{\prime}|\tilde{G}^{\dagger}|n\rangle$
$\displaystyle=4\operatorname{Im}\tilde{\Sigma}_{n}\operatorname{Im}\tilde{\Sigma}_{n^{\prime}}\Big{|}\langle
n|\tilde{G}|n^{\prime}\rangle\Big{|}^{2}.$ (15)
## III Electrons at the Fermi level
Because the scattering process conserves energy and the wave vector associated
with the translation symmetry along the line defect, it is natural to develop
the scattering formalism based on these parameters. Although these parameters
together with the index $n$ form a parameter space covering all possible
asymptotic graphene states, there are other more intuitive representations. An
arbitrary graphene state is typically described by a spin, a band index, and a
two-dimensional wave vector $\vec{k}=\left(k_{x},k_{y}\right)$. Rather than
using the conventional hexagonal first Brillouin zone in graphene, it is
herein more convenient to use an alternative reciprocal primitive cell bounded
by $k_{x}\in\left[0,4\pi/\sqrt{3}a\right[$ and
$k_{y}\in\left[-\pi/a,\pi/a\right[$. Assuming an extended right-moving
asymptotic graphene state, the wave vector component $k_{x}$ is determined
from Eq. (5) with $\tilde{k}_{x}\equiv\sqrt{3}k_{x}a/2$. The graphene band
structure, given byWall47_1
$\tilde{E}=\eta\sqrt{1+4\cos^{2}\tilde{k}_{y}+4\cos\tilde{k}_{y}\cos\tilde{k}_{x}},$
(16)
where $\eta=\pm 1$ refers to the conduction and valence bands, is shown in
Fig. 2(a).
Figure 2: Energy of the asymptotic states given by the graphene band
structure. (a) The band structure presented using a rectangular reciprocal
primitive cell, in which the the two valley $\tau=\pm 1$ are shown as blue and
red, respectively. The energy is set to zero at the Fermi level. (b) Graphene
band structure folded to be commensurate with the line defect structure. Bands
farthest and closest to the Fermi level have $n=0$ and $n=1$, respectively.
To make the asymptotic graphene state commensurate with the line defect
structure, we fold the graphene band structure as shown in Fig. 2(b), where in
this case $k_{y}\in\left[-\pi/2a,\pi/2a\right[$. The bands furthest and
closest to the Fermi level, located where the Dirac cones meet, correspond to
$n=0$ and $n=1$, respectively.
Let us focus our attention to the asymptotic states of most interest, i.e.
those near the Fermi level. For energies with $|\tilde{E}|<\sqrt{2}-1$, which
include all energies within an eV from the Fermi level,
$\operatorname{Im}\tilde{\Sigma}_{0}=0$, i.e. all extended states are in the
$n=1$ band. In this energy regime, elastic scattering does not permit any
interband scattering at the line defect, and thus the transmission probability
$T_{\tau,\sigma}$, given by $T_{1\rightarrow 1}$ in Eq. (15), is
$T_{\tau,\sigma}=\frac{4\Big{(}\operatorname{Im}\tilde{\Sigma}_{1}\Big{)}^{2}\left(\tilde{E}+\cos
2\tilde{k}_{y}-2\tilde{\Sigma}_{0}\right)^{2}}{\left|\big{(}\tilde{E}-2\tilde{\Sigma}_{0}\big{)}\big{(}\tilde{E}-2\tilde{\Sigma}_{1}\big{)}-1+2\cos
2\tilde{k}_{y}\big{(}\tilde{\Sigma}_{0}-\tilde{\Sigma}_{1}\big{)}\right|^{2}}.$
(17)
The expression above is an exact result that can be evaluated numerically. To
make further analytical progress, we focus on the low energy limit. In doing
so, we first introduce the graphene wave vector $\vec{q}=(q_{x},q_{y})$, where
$q_{x}=k_{x}-2\pi/\sqrt{3}a$ and $q_{y}=k_{y}+2\pi\tau/3a$, centered at valley
$\tau\in\\{-1,1\\}$. In our low-energy regime, we can use $\tau$, $\sigma$,
and $\vec{q}$ to describe the asymptotic graphene state. To lowest order in
$q\equiv|\vec{q}|$, the energy dispersion in Eq. (16) is $E=\eta\hbar v_{F}q$,
where $v_{F}=\sqrt{3}|\gamma|a/2\hbar$ is the Fermi velocity. Next, we
introduce the angle of incidence $\alpha$ shown in Fig. 1(a). From the group
velocity relation $\tan\alpha=\big{(}\partial E/\partial
q_{y}\big{)}\Big{/}\big{(}\partial E/\partial q_{x}\big{)}=q_{y}\big{/}q_{x}$
and the assumption of a right-moving state, which gives
$\operatorname{sgn}q_{x}=\eta$, we obtain $q_{x}=(E/\hbar v_{F})\cos\alpha$
and $q_{y}=(E/\hbar v_{F})\sin\alpha$. Therefore, the asymptotic graphene
state can be expressed uniquely by the energy $E$, valley $\tau$, spin
$\sigma$, and angle of incidence $\alpha$ of the incident electron. After
expressing the transmission probability in Eq. (17) using these quantities, we
find the zero energy limit
$T_{\tau,\sigma}(\alpha)=\left|\frac{\operatorname{Im}\tilde{\Sigma}_{1}}{1+\tilde{\Sigma}_{1}}\right|^{2}=\frac{1}{1+\left[\frac{1-\sin\tau\alpha+\tilde{\varepsilon}_{\sigma}\big{(}1-2\sin\tau\alpha\big{)}+\tilde{\varepsilon}_{\sigma}^{2}}{\cos\tau\alpha}\right]^{2}}$
(18)
From this expression, shown in Fig. 3, we see that
$T_{\tau,\sigma}(\alpha)=T_{\sigma}(\tau\alpha)$, which implies that changing
valley has the same effect as changing the sign of the angle of incidence.
Figure 3: Transmission probability of an incident electron at the Fermi level
in valley $\tau$ and with spin $\sigma$, approaching the line defect at the
angle of incidence $\alpha$.
By letting $\partial T_{\sigma}\big{/}\partial\tau\alpha=0$, we find that the
transmission has a stationary point at
$\tau\alpha=\arcsin\left(\frac{1+2\tilde{\varepsilon}_{\sigma}}{1+\tilde{\varepsilon}_{\sigma}+\tilde{\varepsilon}_{\sigma}^{2}}\right)^{\sigma}.$
(19)
Inserted into Eq. (18), this relation gives the maximum transmission
$T_{\sigma}^{\mathrm{max}}=\left\\{\begin{array}[]{cc}\left(1-2\tilde{\varepsilon}_{\sigma}-\tilde{\varepsilon}_{\sigma}^{2}+2\tilde{\varepsilon}_{\sigma}^{3}+\tilde{\varepsilon}_{\sigma}^{4}\right)^{-1}&\quad\sigma=+1,\vspace{0.05
in}\\\ 1&\quad\sigma=-1,\end{array}\right.$ (20)
This maximum at the stationary point can be seen in Fig. 3.
If, rather than a single electron, a beam of electrons is sent towards the
line defect, the scattered electrons would be both valley- and spin-polarized.
Figure 4: Spin and valley polarization of the transmitted portion of a beam of
electrons at the Fermi level after scattering off the line defect at the angle
of incidence $\alpha$.
The former valley polarization, defined as
$\mathcal{P}_{\mathrm{v}}\equiv\frac{\sum_{\sigma}T_{1,\sigma}-\sum_{\sigma}T_{-1,\sigma}}{\sum_{\tau\sigma}T_{\tau,\sigma}},$
(21)
is shown in Fig. 4. This polarization is very close to
$\mathcal{P}_{\mathrm{v}}=\sin\alpha$ predicted previously in the absence of
the local magnetic moments.Gunl11_1 Similarly, we can define the spin
polarization of the beam of transmitted electrons, as
$\mathcal{P}_{\mathrm{s}}\equiv\frac{\sum_{\tau}T_{\tau,1}-\sum_{\tau}T_{\tau,-1}}{\sum_{\tau\sigma}T_{\tau,\sigma}}.$
(22)
Although there is some spin polarization, as can be seen in Fig. 4, this
polarization is small compared to the valley polarization.
## IV Conclusions
Graphene is a promising material for controlling electron motion through
scattering off well-defined defects. Herein, we have shown that electrons near
the Fermi level scattered off an observed extended line defect can be both
valley- and spin-polarized. The spin polarization, arising from local magnetic
moments on sites adjacent to the line defect, is found to be less than 20%.
The valley polarization, on the other hand, can reach near 100%.
The valley filtering taking place at the line defect, which is very different
from other suggested methods for obtaining valley polarization,Ryce07 ; Tkac09
; Zhai10 ; Wu11 can be understood from its reflection symmetry. Consider an
asymptotic graphene state near the Fermi level, which could be expressed as
$|\Phi_{\tau}\rangle=\left(|A\rangle+ie^{-i\theta}|B\rangle\right)/\sqrt{2}$,
where $|A\rangle$ and $|B\rangle$ refer to the two graphene sublattices and
$\theta$ is a pseudospin angle providing the phase relationship between the
two sublattices. The only asymptotic graphene states also eigenstates of the
reflection operator, which maps the $A$ sublattice on one side of the line
defect onto the $B$ sublattice on the opposite side, and vice versa, are those
with $\theta=\pm\pi/2$. These states are symmetric and antisymmetric,
respectively. Antisymmetric states must have a node at the line defect, making
the coupling across the line defect, and concomitantly the transmission,
small. Although, the potential describing the local magnetic moments is a
source of scattering, the transmission through the symmetric states is
generally good. As a result, the line defect allows Bloch waves with
$\theta=\pi/2$ to transmit, while blocking those with $\theta=-\pi/2$.
The asymptotic graphene state $|\Phi_{\tau}\rangle$ is, in general, not an
eigenstate of the reflection operator and has $\theta\neq\pm\pi/2$. It can,
however, always be written as a superposition of the symmetric and
antisymmetric states. If the transmission through the symmetric and
antisymmetric states are 1 and 0, respectively, the transmission probability
can be obtained from the modulus square of the symmetric component of the
incident graphene state, i.e. $(1+\sin\theta)/2$. From the graphene
eigenstates, it can also be shown that the pseudospin angle
$\theta=\tau\alpha$, yielding a transmission probability equal to that in Eq.
(18) in the absence of the potential describing the local magnetic
moments.Gunl11_1
To summarize, the valley filtering is a consequence of the imbalance between
the transmission probabilities for the symmetric and antisymmetric components
of the incident graphene state. This imbalance originates from the symmetry of
the line defect structure. As neither the introduction of longer-range
interactions,Gunl11_1 distortion,Jian11 or potentials from the presence of
local magnetic moments, considered herein, can undo the imbalance, we conclude
that the valley filtering is a robust property of the graphene line defect for
high angles of incidence.
###### Acknowledgements.
The authors acknowledge support from the U.S. Office of Naval Research,
directly and through the U.S. Naval Research Laboratory.
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* (12) K. I. Bolotin, K. J. Sikes, Z. Jiang, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008)
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|
arxiv-papers
| 2012-05-17T13:09:36 |
2024-09-04T02:49:31.014142
|
{
"license": "Public Domain",
"authors": "Daniel Gunlycke and Carter T. White",
"submitter": "Daniel Gunlycke",
"url": "https://arxiv.org/abs/1205.3923"
}
|
1205.3967
|
# Critical Properties of the Kitaev-Heisenberg Model
Craig C. Price Natalia B. Perkins Department of Physics, University of
Wisconsin, 1150 University Ave., Madison, Wisconsin 53706, USA
###### Abstract
We study the critical properties of the Kitaev-Heisenberg (KH) model on the
honeycomb lattice at finite temperatures that might describe the physics of
the quasi two-dimensional (2D) compounds, Na2IrO3 and Li2IrO3. The model
undergoes two phase transitions as a function of temperature. At low
temperature, thermal fluctuations induce magnetic long-range order by the
order-by-disorder mechanism. This magnetically ordered state with a
spontaneously broken $Z_{6}$ symmetry persists up to a certain critical
temperature. We find that there is an intermediate phase between the low-
temperature, ordered phase and the high-temperature, disordered phase. Finite-
sized scaling analysis suggests that the intermediate phase is a critical
Kosterlitz-Thouless (KT) phase with continuously variable exponents. We argue
that the intermediate phase has been observed above the low-temperature,
magnetically ordered phase in Na2IrO3, and also likely exists in Li2IrO3.
Introduction. The Ir-based transition metal oxides, in which the orbital
degeneracy is accompanied by a strong relativistic spin-orbit coupling (SOC),
have recently attracted a lot of theoretical and experimental attention
nakatsuji06 ; okamoto07 ; kim09 ; gegenwart10 ; gegenwart12 ; liu11 ; choi12 ;
takagi11 . This is because the strong SOC creates a different, and frequently
novel, set of magnetic and orbital states due to the unusual anisotropic
exchange interactions between localized moments which are in turn determined
by the combination of spin and lattice symmetries. The spin-orbital models
that describe the low-energy physics of iridium systems often include
anisotropic terms that do not reduce to the conventional easy-plane and easy-
axis anisotropies because they involve the products of different components of
multiple spin operators. These terms are responsible for exotic Mott-
insulating states kim09 , topological insulators Shitade09 ; Pesin10 , spin-
orbital liquid states nakatsuji06 ; okamoto07 , and non-trivial long-range
magnetic orders kim09 ; gegenwart10 ; liu11 .
A prominent example of such an anisotropic spin-orbital model is the KH model
on the honeycomb lattice jackeli09 ; jackeli10 which likely describes the
low-energy physics of the quasi 2D compounds, Na2IrO3 and Li2IrO3. In these
compounds, Ir4+ ions are in a low spin $5d^{5}$ configuration and form weakly
coupled hexagonal layers gegenwart10 ; liu11 ; takagi11 . Due to strong SOC,
the atomic ground state is a doublet where the spin and orbital angular
momenta of Ir4+ ions are coupled into $J_{\rm eff}=1/2$. It was suggested
jackeli09 ; jackeli10 that the interactions between these effective moments
can be described by a spin Hamiltonian containing two competing nearest
neighbor (NN) interactions: an isotropic antiferromagnetic (AF) Heisenberg
exchange interaction and a highly anisotropic ferromagnetic (FM) Kitaev
exchange interaction kitaev06 . This competition can be described with the
parameter, $0\leq\alpha\leq 1$, which sets the relative strength of these two
interactions. At $\alpha=0$, the coupling corresponds to the AF Heisenberg
interaction, and at $\alpha=1$, it corresponds to the Kitaev interaction.
This model immediately attracted a lot of attention; several theoretical
studies were published in the last few years jackeli10 ; jiang11 ; reuther11 ;
trousselet11 on both the ground state and its properties at finite
temperature. The ground state phase diagram of the KH model exhibits three
distinct phases: the AF Néel phase for small $\alpha\in(0.,0.4)$, the stripy
AF phase for intermediate $\alpha\in(0.4,0.86)$, and the disordered spin-
liquid phase at large $\alpha\in(0.86,1.)$. While the phase transition between
the Néel and the stripy phase appears to be discontinuous, numerical studies
including density matrix renormalization group jiang11 and exact
diagonalization results jackeli10 suggest that the transition between the
spin liquid and the stripy state is continuous or weakly first-order.
Additionally, quantum fluctuations select all of the magnetically ordered
phases to have the order parameter point along one of the cubic axes.
In this Letter, we discuss finite temperature properties of the KH model on
the honeycomb lattice. A first step in this direction was made in Ref.
reuther11 , where the critical ordering scale for the magnetically ordered
states was analyzed using a pseudofermion functional renormalization group
approach. Here we present numerical results obtained using Monte Carlo (MC)
simulations. We study the classical KH model because the corresponding quantum
model has a sign problem precluding quantum MC analysis and also because the
existence of long-range order at low temperatures in Na2IrO3 and in Li2IrO3
indicates that quantum fluctuations are not dominating in these materials
gegenwart10 ; gegenwart12 ; liu11 ; choi12 ; takagi11 .
We show that the thermal fluctuations of classical spins give rise to two
distinct temperature dependent effects. At low temperature they predominantly
act as the source of the order-by-disorder phenomenon and select collinear
magnetic order where the spins are oriented along one of the cubic directions.
There are six possible ordered states, one of which is spontaneously chosen by
the system. At high temperatures, when $T$ is larger than any energy scale in
the system, the fluctuations destroy any order putting the KH model into a
three dimensional paramagnetic state. The main goal of our study is to see how
these two phases are connected.
We argue that the classical KH model effectively behaves like a six-state
clock model jose77 ; isakov03 ; chern12 ; ortiz12 and that it undergoes two
continuous phase transitions as a function of temperature separating three
phases: a low-T ordered phase, an intermediate critical phase, and a high-T
disordered phase. The critical phase has an emergent, continuous $U(1)$
symmetry which is fully analogous to the low-T phase of the XY model, a well-
known KT phase of critical points with floating exponents and algebraic
correlations. Here we present numerical data only for $\alpha=0.25$ and
$\alpha=0.75$ since these values likely characterize the ratio between the AF
Heisenberg interaction and the Kitaev interaction in Na2IrO3 and Li2IrO3.
However, we note that recent inelastic neutron scattering measurements on
Na2IrO3 have shown that the KH model alone is insufficient to describe the
magnetic properties of this compound choi12 . It has been demonstrated that it
is essential to include substantial further-neighbor exchanges to describe
both the zigzag ground state and the excitation spectrum in Na2IrO3. The full
finite-temperature phase diagram for the KH model with second and third
neighbor exchange interactions will be published elsewhere unpublished .
Figure 1: Four possible magnetic configurations: (a) the FM ordering; (b) the
two-sublattice, AF Néel order; (c) the stripy order; (d) the zigzag order.
Open and filled circles correspond to up and down directions of spins.
The Model. The classical version of the KH model which describes the
interactions among the $J=1/2$ degrees of freedom of Ir4+ ions reads as
$\displaystyle\mathcal{H}=-J_{K}\sum_{\langle
ij\rangle_{\gamma}}S_{i}^{\gamma}S_{j}^{\gamma}+J_{H}\sum_{\langle
ij\rangle}{\bf S}_{i}{\bf S}_{j}~{}.$ (1)
where the spin quantization axes are taken along the cubic axes of the IrO6
octahedra. $\gamma=x,y,z$ denotes the three bonds of the honeycomb lattice.
The exchange constants, $J_{K}=2\alpha$ and $J_{H}=1-\alpha$, correspond to
the Kitaev and Heisenberg interactions which can be derived from a
multiorbital Hubbard Hamiltonian jackeli10 .
Figure 2: Histograms of the order parameter $m_{N(S)}$, obtained for the
system with 2*84*84 spins in the ordered phase, (a) and (e), in the
intermediate phase, (b)-(c) and (f)-(g), and in the disordered phase, (d) and
(h). Histograms (a)-(d) are computed for $\alpha=0.25$, and (e)-(h) are for
$\alpha=0.75$. The histograms are presented on the complex plane (Re
$|m_{N(S)}|$, Im $|m_{N(S)}|$).
Order by Disorder. The symmetry of the KH model combines the cubic symmetry of
both the spin and the lattice space. It consists of simultaneous permutations
between the $x,y,z$ spin components and a $C_{3}$-rotation of the lattice
which defines a discrete symmetry. The classical ground state has a higher
symmetry than that of the Hamiltonian – the ground state energy does not
change under a simultaneous rotation of all spins. Since this applies only to
the ground state,the KH model has only an accidental continuous rotation
symmetry. Its actual symmetry is discrete; at zero temperature, the ”pseudo”
SU(2) symmetry is broken by quantum fluctuations that restore the underlying
cubic symmetry of the model jackeli10 . The magnetically ordered phase is
gapped with a spin gap that corresponds to the finite energy cost of deviating
the order parameter from one of the cubic axes. We show in the following that
thermal fluctuations of classical spins at finite T also select a collinear
spin configuration whose order parameter points along one of the cubic axes.
Parameters of the Simulations. We have carried out classical MC simulations of
the model (1) using the standard Metropolis algorithm. In our MC simulations,
we treat the spins as three-dimensional (3D) vectors, ${\bf
S}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$, of unit magnitude with
$(S_{i}^{x})^{2}+(S_{i}^{y})^{2}+(S_{i}^{z})^{2}=1$ at every site. At each
temperature, more than $10^{7}$ MC sweeps were performed. Of these, $5*10^{5}$
were used to equilibrate the system, and afterwards only 1 out of every 5
sweeps was used to calculate the averages of physical quantities. We present
all energies in the units of $J_{H}$ and assume $k_{B}=1$. The calculations
were carried out on several finite systems with size $2*L*L$ that are spanned
by the primitive vectors of a triangular lattice ${\bf
a}_{1}=(1/2,\sqrt{3}/2)$ and ${\bf a}_{2}=(1,0)$ with a 2-point basis using
periodic boundary conditions.
Results. To study the possible phases of the model (1), we introduce four
magnetic configurations (Fig. 1): a FM order, a simple two-sublattice AF Néel
order, a stripy order, and a zigzag spin order. The classical energies of
these states can be easily computed: $E_{cl}^{\mathcal{M}}=3-5\alpha$,
$E_{cl}^{\mathcal{Z}}=-3\alpha+1$, $E_{cl}^{\mathcal{S}}=-\alpha-1$, and
$E_{cl}^{\mathcal{N}}=5\alpha-3$ for the FM, the zigzag, the stripy and the
Néel phases, respectively. For $0\leq\alpha<1$, the classical ground state is
either the Néel AF with the vector order parameter
$\mathcal{N}=\frac{1}{N}\sum_{i}({\bf S}_{iA}-{\bf S}_{iB})$ or the stripy
phase described by $\mathcal{S}=\frac{1}{N}\sum_{i=n}({\bf S}_{iA}-{\bf
S}_{iB}+{\bf S}_{iC}-{\bf S}_{iD})$. Here, $A,B$ and $A,B,C,D$ denote either
two or four sublattices that respectively characterize the Néel AF and stripy
order. The classical phase transition between them occurs at $\alpha=1/3$. At
$\alpha=1$, the FM, stripy, and zigzag phases all have the same classical
energy. However, the classical degeneracy of this point, which corresponds to
the pure Kitaev model, is much higher. This limit has been thoroughly studied
by Baskaran et al. baskaran08 .
Figure 3: The log-log plots of the order parameter $m_{N(S)}$ as a function
of system size $L$ at various temperatures. The solid curves indicate the
linear behavior that corresponds to a power-law dependence, $m_{N(S)}\sim
L^{-\eta/2}$, cooresponding to the intermediate critical phase. The dashed
curves show deviation away from the linear behavior outside the critical
phase.
Figure 4: A snapshot of the coarse-grained order parameter $\langle
m_{N}\rangle$ at $T=0.168$. The vortex-like topological excitations are
evident.
To make an analogy to the six-state clock model, we map the order parameter
describing the magnetically ordered phase of the KH model onto a 2D complex
order parameter, $m_{N(S)}=\sum_{i=1}^{6}|m_{i,N(S)}|e^{\imath\theta_{i}}$,
such that the six possible ordered states are characterized by $\theta_{i}=\pi
n_{i}/3$, $n_{i}=0,..5$ chern12 . The mapping is exact only well within the
ordered state since there is no guarantee that the thermal fluctuations of the
order parameter will actually have a 2D character given that the spin degrees
of freedom are three-dimensional. Depending on the strength of the spin
stiffness in different directions, the long-range low-T magnetic order can be
destroyed in one of several ways. If the stiffness of thermal fluctuations
along the circle is softer than the stiffness of fluctuations in the direction
transverse to the circle, the long-range order may be destroyed by a
discontinuous first-order transition, by two continuous phase transitions with
an intermediate partially ordered phase, or by two KT phase transitions with
an intermediate critical phase jose77 ; isakov03 ; chern12 ; ortiz12 . In the
last scenario, the critical phase is destroyed by topological excitations in
the form of discrete vortices whose existence is directly related to the
emergence of a continuous symmetry; the high-T transition will first bring the
system into a disordered phase where fluctuations are primarily 2D, and the
crossover to the 3D paramagnet occurs at even higher temperatures.
In Fig. 2 we present the results of the histogram method for the complex order
parameter. At low temperatures, Figs. 2 (a) and (e), a sixfold degeneracy
present in the ordered phase is seen. For both $\alpha=0.25$ and
$\alpha=0.75$, the six states which have the highest weight in the histogram
are where the order parameter $m_{N(S)}$ points along one of the cubic axes.
In Figs. 2 (b) and (f), when the temperature increases beyond a certain
critical temperature, a continuous $U(1)$ symmetry emerges signaling both the
disappearance of the sixfold anisotropy and the appearance of the critical
phase. The formation of vortices can be seen in Fig. 4 where we present a
snapshot of the coarse-grained order parameter $\langle m_{N}\rangle$ at
$T=0.168$. Upon a further increase in temperature, the amplitude of the order
parameter decreases (Figs. 2 (c) and (g)) until it shrinks to zero indicating
the transition to the paramagnetic phase (Figs. 2 (d) and (h)).
To better understand the properties of the intermediate phase and to confirm
its critical nature, we performed the finite-size scaling analysis appropriate
for KT transitions challa86 . The full finite-size scaling analysis is rather
involved and will be reported elsewhere unpublished . Here we present only the
scaling behavior of the order parameter. At the KT transition, the order
parameter exhibits the power law dependence on system size, $m\sim
L^{-\eta/2}$. As each point of the intermediate critical phase can be
understood as a critical point, the power law behavior of the order parameter
should hold throughout the entire phase. We found that the boundaries of the
critical phase are characterized by critical exponents close to $1/9$ and
$1/4$ for the lower and upper boundaries at $T_{c_{1}}$ and $T_{c_{2}}$, which
is in agreement with critical exponents for the six-state clock model obtained
by the renormalization group analysis jose77 . Fig. 3 shows the log-log plots
of the order parameter $m_{N(S)}$ as a function of system size for different
temperatures. For $\alpha=0.25$, the data points in Fig. 3 a) show a linear
behavior in the temperature interval between $T_{c_{1}}\simeq 0.152$ and
$T_{c_{2}}\simeq 0.162$, in which there are several critical lines
characterized by $\eta$ between $1/9$ and $1/4$. For $\alpha=0.75$, we have
detected the critical phase in the temperature interval between
$T_{c_{1}}\simeq 0.125$ and $T_{c_{2}}\simeq 0.127$.
Figure 5: The Binder cumulant as a function of temperature for (a)
$\alpha=0.25$ and (b) $\alpha=0.75$. From the crossing points of different
Binder’s curves, we estimate $T_{c_{1}}=0.152$ and $T_{c_{1}}=0.124$ for
$\alpha=0.25$ and $\alpha=0.75$, respectively.
The lower transition temperature $T_{c_{1}}$ can be independently determined
using fourth-order Binder cumulant (Figs. 5 (a) and (b)). The Binder cumulant
has a scaling dimension of zero; thus the crossing point of the cumulants for
different lattice sizes provides a reliable estimate for the value of the
critical temperature $T_{c_{1}}$ at which the long range order is destroyed.
The crossing points for $\alpha=0.25$ and $\alpha=0.75$ are $T_{c_{1}}=0.152$
and $T_{c_{1}}=0.124$, respectively. They are in good agreement with estimates
obtained from the log-log plots in Fig. 3.
In Figs. 6 (a) and (b) we present the temperature dependence of the specific
heat, $C=(\langle E^{2}\rangle-\langle E\rangle^{2})/NT^{2}$. While the low-T
transition, seen as small peak at temperatures $T_{c_{1}}=0.152$ and $0.1247$
for $\alpha=0.25$ and $0.75$, respectively, is in a good agreement with our
previous estimates, the features corresponding to the high-T transition
$T_{c_{2}}$ are barely distinguished by eye. This is not surprising as the
high-T transition is a usual KT transition at which the specific heat does not
diverge at the critical point last . It is also likely that the high-T KT
transition might be shadowed by the crossover to the 3D paramagnet, which is
seen in Fig. 6 as a very broad hump at higher-T.
Figure 6: Specific heat $C$ as a function of temperature for (a)
$\alpha=0.25$ and (b) $\alpha=0.75$.
Our findings for the specific heat show a lot of similarities between the
experimental data obtained on the Na2IrO3 and Li2IrO3 compounds by Refs.
gegenwart10 ; gegenwart12 and takagi11 . In Na2IrO3, both the lambda-like
anomaly at the Néel ordering temperature, $T_{N}=15$ K, and a broad tail which
extends into higher temperatures are seen in the specific heat measurements
gegenwart10 . The latter suggests the presence of short-range order above the
bulk 3D ordering that can be understood by our proposed scenario of the
critical phase.
Let us estimate the temperatures of the KT transitions and the width of the
critical phase in Na2IrO3 based on our results obtained for the KH model with
$\alpha=0.25$. On the mean field level, the exchange on the NN bonds may be
estimated from the classical energy, $J_{1}\simeq(3-5\alpha)/3$, in the Néel
phase. From the recent neutron scattering experiment choi12 , the NN exchange
in Na2IrO3 was estimated to be $J_{1}=4.17$ meV. In the bulk of our paper, all
energies were measured in the units of $J_{H}$, and thus we estimate $J_{1}$
to be equal to 12.7 meV. This gives the prediction for the critical
temperature to be $T_{c_{1}}=16.8$ K, which is very close to the experimental
value $T_{N}=15$ K gegenwart10 ; gegenwart12 . Our estimate for the upper
boundary of the critical phase is $T_{c_{2}}=17.7$ K which makes the predicted
critical phase very narrow. We note here that the critical phase survives in
the extended KH model with included further-neighbor exchange couplings choi12
; gegenwart12 ; mazin12 which are essential for comparison with experiment.
However, in order to determine the upper boundary of the critical phase
additional extensive numerical simulations must be performed.
Acknowledgements. The authors are particularly thankful to C. Batista, G.-W.
Chern, G. Jackeli, and Y. Kato for stimulating discussions and many helpful
suggestions. We are grateful to H. Takagi and T. Takayama for sharing with us
unpublished data on Na2IrO3 and Li2IrO3. N.P. acknowledges the support from
NSF grant DMR-1005932. N.P. also thanks the hospitality of the visitors
program at MPIPKS, where the part of the work has been done.
## References
* (1) S. Nakatsuji et al., Phys. Rev. Lett. 96, 087204 (2006).
* (2) Y. Okamoto et al., Phys. Rev. Lett. 99, 137207 (2007).
* (3) B. J. Kim et al., Science 323, 1329 (2009).
* (4) Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).
* (5) Yogesh Singh et al., Phys. Rev. Lett. 108, 127203 (2012)
* (6) X. Liu et al., Phys. Rev. B 83, 220403(R) (2011).
* (7) S. K. Choi et al., Phys. Rev. Lett. 108, 127204 (2012).
* (8) H. Takagi, unpublished.
* (9) G.-W. Chern and N. B. Perkins, Phys. Rev. B 80, 180409(R) (2009).
* (10) A. Shitade et al., Phys. Rev. Lett. 102, 256403 (2009).
* (11) D.Pesin, L.Balents, Nature Physics 6, 376 (2010).
* (12) G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
* (13) J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010).
* (14) A. Kitaev, Ann. Phys. 321, 2 (2006).
* (15) H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and Simon Trebst Phys. Rev. B 83, 245104 (2011).
* (16) J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84, 100406 (2011).
* (17) F. Trousselet, G. Khaliullin, P. Horsch, Phys. Rev. B 84, 054409 (2011).
* (18) J.V. José, L.P. Kadanoff, S. Kirkpatrick S and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
* (19) S. V. Isakov and R. Moessner, Phys. Rev. B68, 104409 (2003).
* (20) G.-W. Chern, O. Tchernyshyov, arXiv:1109.0275.
* (21) G. Ortiz, E. Cobanera, Z. Nussinov, Nuclear Physics B 854, 780 (2012).
* (22) G. Baskaran, D. Sen, and R. Shankar, Phys. Rev. B 78, 115116 (2008).
* (23) C.Price and N.B.Perkins, unpublished.
* (24) M.S.S. Challa and D.P. Landau, Phys. Rev. B 33, 437 (1986).
* (25) I. I. Mazin et al., arXiv:1205.0434
* (26) Fabien Aletet al., Phys. Rev. E 74, 041124 (2006).
|
arxiv-papers
| 2012-05-17T16:04:08 |
2024-09-04T02:49:31.022218
|
{
"license": "Public Domain",
"authors": "Craig Price and Natalia B. Perkins",
"submitter": "Natalia Perkins",
"url": "https://arxiv.org/abs/1205.3967"
}
|
1205.4015
|
# The Four Dimensional Helicity Scheme Beyond One Loop
William B. Kilgore Physics Department, Brookhaven National Laboratory, Upton,
New York 11973, USA.
[kilgore@bnl.gov]
###### Abstract
I describe a procedure by which one can transform scattering amplitudes
computed in the four dimensional helicity scheme into properly renormalized
amplitudes in the ’t Hooft-Veltman scheme. I describe a new renormalization
program, based upon that of the dimensional reduction scheme and explain how
to remove both finite and infrared-singular contributions of the evanescent
degrees of freedom to the scattering amplitude.
## I Introduction
The Four Dimensional Helicity (FDH) scheme Bern and Kosower (1992); Bern et
al. (2002) is widely used for computing QCD corrections at next-to-leading
order in perturbation theory. It is particularly convenient for use with the
helicity method and the techniques of generalized unitarity. Unfortunately, as
I have recently shown Kilgore (2011), the FDH is not a unitary regularization
scheme. The standard renormalization prescription Bern et al. (2002) fails to
remove all of the ultraviolet poles, leading to incorrect results at two loops
and beyond. Thus the FDH cannot be viewed as a regularization scheme in which
one can compute scattering amplitudes. Instead, it should be looked upon as a
shortcut for obtaining scattering amplitudes in a unitary regularization
scheme. Indeed, this is how the FDH has always been used at one-loop; final
results have always been presented in the ’t Hooft-Veltman (HV) scheme ’t
Hooft and Veltman (1972) using the prescription of Kunszt, et al. Kunszt et
al. (1994) to transform the FDH scheme result, but it was not clear whether
this conversion was necessary or merely expedient, allowing one to match onto
standard definitions of the running coupling, etc.
It is now certain that one must convert the results of a calculation in the
FDH scheme into results in a properly defined scheme. A first step in this
direction was taken by Boughezal, et al. Boughezal et al. (2011), who put
forward a prescription for constructing the correct counterterms for
renormalization. For inclusive calculations, performed using the optical
theorem, like those considered in Refs. Kilgore (2011); Boughezal et al.
(2011), such a prescription is sufficient. Experiments, however, measure
differential cross sections, and the power of the FDH scheme is that it
facilitates the calculation of loop-level amplitudes, giving access to the
differential information they contain. To make use of the full amplitude, one
must control of both the infrared and ultraviolet structure.
In this paper, I will exploit the close relationship between the FDH and the
dimensional reduction (DRED) Siegel (1979) schemes to develop a prescription
for transforming FDH scheme amplitudes, which may be easier to compute using
unitarity methods, into HV scheme amplitudes that can actually be used in
calculations. The plan of the paper is: In Section II I will review the
regularization schemes that will be used; in Section III I will review the
infrared structure of QCD amplitudes; in Section IV I will define the FDH
scheme in terms of the DR scheme, compute the anomalous dimensions that
control the ultraviolet and infrared structure of DR scheme amplitudes through
two loops and specify the procedure for transforming FDH scheme results into
HV scheme amplitudes.
## II Regularization schemes
All of the schemes that I will be working with are variations on dimensional
regularization ’t Hooft and Veltman (1972), which specifies that loop-momenta
are to treated as $D_{m}=4-2\,{\varepsilon}$ dimensional. In dimensional
regularization, the singularities (both ultraviolet and infrared) that appear
in four-dimensional calculations are transformed into poles in the parameter
${\varepsilon}$. The ultraviolet poles are removed through renormalization,
while the infrared poles cancel when one performs “sufficiently inclusive”
calculations.
### II.1 The ’t Hooft-Veltman and conventional dimensional regularization
schemes
In the original dimensional regularization scheme ’t Hooft and Veltman (1972),
the HV scheme, observed states are treated as four-dimensional, while internal
states (both their momenta and their spin degrees of freedom) are treated as
$D_{m}$ dimensional. Internal states include states that circulate inside of
loop diagrams as well as nominally external states that have infrared overlaps
with other nominally external states. It turns out that one can treat internal
fermions as having exactly two degrees freedom, just as they have in four
dimensions, even though their momenta are $D_{m}$ dimensional, but massless
internal gauge bosons must have $(D_{m}-2)$ spin degrees of freedom, while
massive internal gauge bosons have $(D_{m}-1)$.
The conventional dimensional regularization (CDR) scheme Collins (1984) is
closely related to the HV scheme. In the CDR scheme, all states and momenta,
both internal and observed, are taken to be $D_{m}$ dimensional. This often
turns out to be computationally more convenient, especially in infrared
sensitive theories like QCD, since one set of rules governs all interactions.
Because the HV and CDR schemes handle ultraviolet singularities in the same
manner, their behavior under the renormalization group, anomalous dimensions,
running coupling, etc., are identical.
In the HV and CDR schemes, internal momenta are taken to be
$D_{m}=4-2\,{\varepsilon}$ dimensional. In general, ${\varepsilon}$ is a
complex number and it’s exact value is unimportant, but taking ${\varepsilon}$
to be real and positive (negative) is preferred by ultraviolet (infrared)
power-counting arguments. It is important, however, that the
$D_{m}$-dimensional vector space in which momenta take values is larger than
the standard four-dimensional space-time. This means that the standard four-
dimensional metric tensor $\eta^{\mu\nu}$ spans a smaller space than the
$D_{m}$ dimensional metric tensor, and the four-dimensional Dirac matrices
$\gamma^{0,1,2,3}$ form a subset of the full $\gamma^{\mu}$. These
considerations are of particular importance when considering chiral objects
involving $\gamma_{5}$ and the Levi-Civita tensor, but cannot be neglected
when, as in the HV scheme, one restricts observed states to be strictly four-
dimensional.
### II.2 The dimensional reduction Scheme
The DRED scheme was devised for application to supersymmetric theories. In
supersymmetry, it is essential that the number of bosonic degrees of freedom
is exactly equal to the number of fermionic degrees of freedom. In the DRED
scheme, the continuation to $D_{m}$ dimensions is taken as a compactification
from four dimensions. Thus, while space-time is taken to be four-dimensional
and particles have the standard number of degrees of freedom, momenta are
regularized dimensionally and span a $D_{m}$ dimensional vector space which is
smaller than four-dimensional space-time.
Because the Ward Identity only applies in the $D_{m}$ dimensional vector space
in which momenta are defined, the extra $2\,{\varepsilon}$ spin degrees of
freedom of gauge bosons are not protected by the Ward Identity and must
renormalize differently than the $2-2\,{\varepsilon}$ degrees of freedom that
are protected. In supersymmetric theories, the supersymmetry provides the
missing part of the Ward Identity which demands that the $2\,{\varepsilon}$
spin degrees of freedom be treated as gauge bosons. In non-supersymmetric
theories, however, they must be considered to be distinct particles, with
distinct couplings and renormalization properties. These extra degrees of
freedom are referred to as “${\varepsilon}$-scalars” or as “evanescent”
degrees of freedom.
Since the evanescent degrees of freedom are independent of the gauge bosons,
their self-couplings and their coupling to fermions are independent of the
gauge coupling and of one another. The quartic self-coupling splits into
multiple independent terms; if the gauge theory is $SU(2)$, there are two
independent quartic self-couplings, in $SU(3)$, there are three independent
quartic self-couplings, and if the gauge theory is $SU(N);N\geq 4$, there are
four independent quartic self-couplings Jack et al. (1994a). These new
couplings run differently from the gauge coupling under the renormalization
group and cannot consistently be identified with it.
Notwithstanding its semantic appeal, the insistence on a proper
compactification, so that $D_{m}\subset 4$ in the DRED scheme, is problematic
when dealing with chiral theories Siegel (1980). Chirality is a four-
dimensional concept and one cannot consistently define chiral operators in a
vector space with fewer than four dimensions. One way around this is to adopt
a hierarchy of vector spaces $D_{s}\supset D_{m}\supset 4$ (where
$D_{m}=4-2\,{\varepsilon}$ and $D_{s}$ is assigned the value $D_{s}=4$), as in
the FDH scheme (described below). In such a scheme, chiral operators can be
defined in the four-dimensional subspace of $D_{m}$, just as they are in the
HV/CDR schemes. Stöckinger and Signer Stöckinger (2005); Signer and Stöckinger
(2009) have long advocated that this is the proper definition of the DRED
scheme. Aside from the treatment of chiral operators, there are no important
computational distinctions between $D_{m}\supset 4$ and $D_{m}\subset 4$. In
this paper, I will adopt the $D_{m}\supset 4$ convention and refer to this
variation of dimensional reduction as the DR scheme.
### II.3 The four dimensional helicity Scheme
In the four-dimensional helicity scheme, one again defines a vector space of
dimensionality $D_{m}\supset 4$ (again $D_{m}=4-2\,{\varepsilon}$), in which
loop momenta take values, and a still larger vector space $D_{s}\supset
D_{m}$, ($D_{s}=4$), in which internal spin degrees of freedom take values.
Note that the relative numerical values of $D_{s}$, $D_{m}$ and $4$ are not
important. What is important is that as vector spaces, $D_{s}\supset
D_{m}\supset 4$.
The FDH scheme, like the HV scheme, treats observed states as four-
dimensional, except, as in inclusive calculations, where there are infrared
overlaps among external states. When infrared overlaps occur, external states
are taken to be $D_{s}$ dimensional.
As in the DRED scheme, spin degrees of freedom take values in a vector space
that is larger than that in which momenta take values. It would seem,
therefore, that the same remarks regarding the Ward Identity and the
conclusion that the $D_{x}=D_{s}-D_{m}$ dimensional components of the gauge
fields and their couplings must be considered as distinct from the $D_{m}$
dimensional gauge fields and couplings would apply.
That is not, however, how the FDH scheme has been used. All field components
in the $D_{s}$ dimensional space are treated as gauge fields and no
distinction is made between the couplings. The reason for doing this is to
facilitate the use of helicity amplitudes in conjunction with unitarity
methods, the idea being to “sew together” (four dimensional) tree-level
helicity amplitudes into loop-level amplitudes. While helicity methods can be
used in the CDR scheme Kosower (1991), they are most transparently and
compactly represented using four-dimensional external states. Thus, the FDH
scheme demands that the gluons circulating through loop amplitudes have the
same number of spin degrees of freedom as the external gluons of helicity
amplitudes.
Unfortunately, this framework fails to subtract all of the ultraviolet poles
Kilgore (2011) and generates incorrect results. The evanescent couplings and
degrees of freedom need to be renormalized separately from their gauge boson
counterparts, but there is no mechanism within the FDH for doing so. The
errors, however, are only of order ${\cal O}({\varepsilon}^{1})$ in NLO
calculations (which is the level at which the FDH has been used in practical
calculations to date) and therefore do not adversely affect those results. At
NNLO the errors would be of order ${\cal O}({\varepsilon}^{0})$ and at N3LO
and beyond the errors would be singular in ${\varepsilon}$.
## III The infrared structure of QCD amplitudes
The infrared structure of QCD amplitudes is governed by a set of anomalous
dimensions which allow one to predict, for any amplitude, the complete
infrared structure Catani (1998); Sterman and Tejeda-Yeomans (2003). These
anomalous dimensions are known completely, in both the massless and massive
cases for one and two loop amplitudes, and their properties beyond the two-
loop level are being actively studied Aybat et al. (2006a, b); Mitov et al.
(2009); Becher and Neubert (2009a); Gardi and Magnea (2009a); Becher and
Neubert (2009b, c); Gardi and Magnea (2009b); Dixon et al. (2010); Mitov et
al. (2010).
For a general $n$ parton scattering process, the set of partons is labeled by
${\bf f}=\\{f_{i}\\}_{i=1\dots n}$. In the formulation of Refs. Sterman and
Tejeda-Yeomans (2003); Aybat et al. (2006a, b), a renormalized amplitude may
be factorized into three functions: the jet function ${\cal J}_{\bf f}$, which
describes the collinear dynamics of the external partons that participate in
the collision; the soft function ${\bf S_{f}}$, which describes soft exchanges
between the external partons; and the hard-scattering function $\left|H_{\bf
f}\right\rangle$, which describes the short-distance scattering process,
$\left|{\cal M}_{\bf
f}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)\right\rangle={\cal
J_{\bf f}}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)\ {\bf
S_{f}}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)\
\left|H_{\bf
f}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2})\right)\right\rangle\,.$
(1)
The notation indicates that $\left|H_{\bf f}\right\rangle$ is a vector and
${\bf S_{f}}$ is a matrix in color space Catani and Seymour (1996, 1997);
Catani (1998). As with any factorization, there is considerable freedom to
move terms about from one function to the others. It is convenient Aybat et
al. (2006a, b) to define the jet and soft functions, ${\cal J}_{\bf f}$ and
${\bf S_{f}}$, so that they contain all of the infrared poles but only contain
infrared poles, while all infrared finite terms, including those at higher-
order in ${\varepsilon}$, are absorbed into $\left|H_{\bf f}\right\rangle$.
### III.1 The jet function in the HV/CDR schemes
The jet function ${\cal J}_{\bf f}$ is found to be the product of individual
jet functions ${\cal J}_{f_{i}}$ for each of the external partons,
${\cal J}_{\bf
f}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)=\prod_{i\in{\bf{f}}}\ {\cal
J}_{i}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)\,.$ (2)
Each individual jet function is naturally defined in terms of the anomalous
dimensions of the Sudakov form factor Sterman and Tejeda-Yeomans (2003),
$\begin{split}\ln{\cal J}_{i}^{\rm
CDR}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)&=-{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(1)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(1)}({\varepsilon})\right]\\\
&\quad+{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}\left\\{\frac{{\beta_{0}^{{\overline{{\rm
MS}}}}}}{8}\frac{1}{{\varepsilon}^{2}}\left[\frac{3}{4\,{\varepsilon}}\gamma_{K\,i}^{(1)}+{\cal
G}_{i}^{(1)}({\varepsilon})\right]-\frac{1}{8}\left[\frac{\gamma_{K\,i}^{(2)}}{4\,{\varepsilon}^{2}}+\frac{{\cal
G}_{i}^{(2)}({\varepsilon})}{{\varepsilon}}\right]\right\\}+\dots\,,\end{split}$
(3)
where
$\begin{split}\gamma_{K\,i}^{(1)}&=2\,C_{i},\quad\gamma_{K\,i}^{(2)}=C_{i}\,K=C_{i}\left[C_{A}\left(\frac{67}{18}-\zeta_{2}\right)-\frac{10}{9}T_{f}\,N_{f}\right],\quad
C_{q}\equiv C_{F},\quad C_{g}\equiv C_{A},\\\ {\cal
G}_{q}^{(1)}&=\frac{3}{2}C_{F}+\frac{{\varepsilon}}{2}C_{F}\left(8-\zeta_{2}\right),\qquad{\cal
G}_{g}^{(1)}=2\,{\beta_{0}^{{\overline{{\rm
MS}}}}}-\frac{{\varepsilon}}{2}C_{A}\,\zeta_{2},\\\ {\cal
G}_{q}^{(2)}&=C_{F}^{2}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{2545}{432}+\frac{11}{12}\zeta_{2}-\frac{13}{4}\zeta_{3}\right)-C_{F}\,T_{f}\,N_{f}\left(\frac{209}{108}+\frac{1}{3}\zeta_{2}\right),\\\
{\cal G}_{g}^{(2)}&=4\,{\beta_{1}^{{\overline{{\rm
MS}}}}}+C_{A}^{2}\left(\frac{10}{27}-\frac{11}{12}\zeta_{2}-\frac{1}{4}\zeta_{3}\right)+C_{A}\,T_{f}\,N_{f}\left(\frac{13}{27}+\frac{1}{3}\zeta_{2}\right)+\frac{1}{2}C_{F}\,T_{f}\,N_{f}\,.\end{split}$
(4)
Although ${\cal G}_{i}$ and $\gamma_{K\,i}$ are defined through the Sudakov
form factor, they can be extracted from fixed-order calculations Gonsalves
(1983); Kramer and Lampe (1987); Matsuura and van Neerven (1988); Matsuura et
al. (1989); Harlander (2000); Moch et al. (2005a, b). $\gamma_{K\,i}$ is the
cusp anomalous dimension and represents a pure pole term. The ${\cal G}_{i}$
anomalous dimensions contain terms at higher order in ${\varepsilon}$, but I
only keep terms in the expansion that contribute poles to $\ln\left({\cal
J}_{i}\right)$. $C_{F}=(N_{c}^{2}-1)/(2\,N_{c})$ denotes the Casimir operator
of the fundamental representation of SU($N_{c}$), while $C_{A}=N_{c}$ denotes
the Casimir of the adjoint representation. $N_{f}$ is the number of quark
flavors and $T_{f}=1/2$ is the normalization of the QCD charge of the
fundamental representation. $\zeta_{n}=\sum_{k=1}^{\infty}1/k^{n}$ represents
the Riemann zeta-function of integer argument $n$.
### III.2 The soft function in the HV/CDR schemes
The soft function is determined entirely by the soft anomalous dimension
matrix ${\bm{\Gamma}}_{S_{f}}$,
$\begin{split}{\bf S_{f}}^{\rm
CDR}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)&=1+\frac{1}{2\,{\varepsilon}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{8\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(1)}\times{\bm{\Gamma}}_{S_{f}}^{(1)}\\\
&\qquad-\frac{{\beta_{0}^{{\overline{{\rm
MS}}}}}}{4\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{4\,{\varepsilon}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(2)}+\dots\,.\end{split}$ (5)
In the color-space notation of Refs. Catani and Seymour (1996, 1997); Catani
(1998), the soft anomalous dimension is given by Aybat et al. (2006a, b)
${\bm{\Gamma}}_{S_{f}}^{(1)}=\frac{1}{2}\,\sum_{i\in{\bf f}}\ \sum_{j\neq
i}{\bf T}_{i}\cdot{\bf
T}_{j}\,\ln\left(\frac{\mu^{2}}{-s_{ij}}\right),\qquad{\bm{\Gamma}}_{S_{f}}^{(2)}=\frac{K}{2}{\bm{\Gamma}}_{S_{f}}^{(1)}\,,$
(6)
where $K=C_{A}\left(67/18-\zeta_{2}\right)-10\,T_{f}\,N_{f}/9$ is the same
constant that relates the one- and two-loop cusp anomalous dimensions. The
${\bf T}_{i}$ are the color generators in the representation of parton $i$
(multiplied by $(-1)$ for incoming quarks and gluons and outgoing anti-
quarks).
## IV The FDH scheme at two loops
The failure of the FDH scheme as a unitary regularization scheme does not mean
that it is of no value in computing higher-order corrections beyond the next-
to-leading order. Even at NLO, the FDH scheme has always been used as a means
of obtaining scattering amplitudes in the HV scheme. There is no reason for
that to change at two loops. The only difference is that one must recognize
that the FDH scheme result is not a physical scattering amplitude, but only an
intermediate step toward obtaining one.
In formulating a prescription for converting FDH scheme amplitudes into HV
scheme amplitudes, the first problem to address, of course, is that of
renormalization. One solution to the renormalization problem, dubbed
“dimensional reconstruction,” has been proposed by Boughezal, et al. Boughezal
et al. (2011). The idea behind dimensional reconstruction is that if one knows
the the one-loop behavior of an amplitude with arbitrary (integer) numbers of
extra spin dimensions (momenta are always $D_{m}$ dimensional) then the
correct two-loop amplitude can be determined from the renormalization
constants at different integer spin dimensions. Note that it is a basic
assumption of dimensional reconstruction that when one is computing a two-loop
amplitude, the tree-level and one-loop terms that contribute via
renormalization are essentially trivial, and that there is no appreciable cost
to performing extra one-loop calculations if doing so saves effort on the two-
loop piece. The transformations that I will develop will also subscribe to
this viewpoint.
While dimensional reconstruction is a completely valid approach to the
renormalization problem of the FDH scheme, it does have some drawbacks. One
drawback is that it appears that one must determine new renormalization
constants for each process at each order of perturbation theory. This is quite
different from working within a renormalizable theory, where the
renormalization constants can be determined in advance through the study of
corrections to 1PI Green functions. A more serious drawback is that
dimensional reconstruction does not address the infrared structure of
amplitudes computed in the FDH scheme.
It is certain that the infrared structure of FDH scheme amplitudes is not
equal to that of HV scheme amplitudes. It is also clear from optical theorem
calculations Kilgore (2011); Boughezal et al. (2011) that once the
renormalization problem is fixed, one could proceed with FDH scheme
calculations because the infrared overlaps will sort themselves out. For
differential calculations, one needs to know the soft and collinear
factorization properties of FDH scheme amplitudes in order to implement a
subtraction scheme, but this has already been worked out Bern et al. (1998,
1999); Kosower and Uwer (1999). The problem is that all of the FDH scheme
amplitudes, real and virtual, contain errors, though the structure of the
errors is such that, after renormalization, they cancel in the inclusive sum.
Even if one were willing to live with such circumstances, one would still want
to match onto standard definitions of the running coupling and would have to
face the fact that parton distribution functions are only available in the CDR
scheme. A far better choice is to transform the result to a framework like the
HV scheme that is known to be unitary and correct and which can be easily
connected to the parton distribution functions.
### IV.1 The connection between the FDH and DR schemes
In order to develop a rigorous set of rules for transforming FDH amplitudes,
it is necessary to define the FDH scheme in terms of a renormalizable scheme.
One can do this by exploiting the close connection between the FDH and DR
schemes. When formulating the QCD Lagrangians in these schemes, one starts
with the standard Yang-Mills Lagrangian and then extends the fields into
$D_{s}$-dimensions. In the FDH scheme, one proceeds directly to the
development of Feynman rules involving the $D_{s}$-dimensional metric tensor
and Dirac matrices Bern and Kosower (1992); Bern et al. (2002). In the DR
scheme, however, one first splits the gluon field into two independent
components, the $D_{m}$-dimensional gauge field and the $D_{x}$-dimensional
evanescent field Jack et al. (1994a, b); Harlander et al. (2006). The metric
tensor and Dirac matrices also decompose into orthogonal components. Those new
terms in the Lagrangian that do not involve gauge fields are assigned new,
independent couplings. The evanescent-quark-antiquark coupling is given the
value $g_{e}$ ($g_{e}^{2}=4\,\pi\,\alpha_{e}$) and the quartic evanescent
boson couplings are given values $\eta_{i\,,i=1,2,3}$, where $\eta_{1}$
represents the quartic interaction that has the same color flow as the quartic
gluon coupling, while $\eta_{2,3}$ represent the non-QCD-like interactions.
Thus, all of the DR scheme interactions are contained in those of the FDH
scheme, they are simply not labeled by independent couplings and evanescent
Lorentz structures.The only exception to this statement concerns the quartic
evanescent boson couplings. Because the evanescent bosons are not protected by
gauge symmetry, new quartic interactions, with new color-flows among the
evanescent bosons, are generated by higher-order corrections which must be
renormalized independently of the QCD-like quartic coupling that appears in
the classical Lagrangian. In recognition of the fact that such terms will
occur, they are usually assigned independent couplings and added to the
effective DR Lagrangian. The FDH scheme doesn’t have such couplings, but this
does not present a problem. The extra quartic terms introduced to the DR
Lagrangian clean up the renormalization procedure, but there is no reason that
the couplings assigned to these terms could not be chosen such that they do
not contribute to a DR scattering process until radiative corrections to the
QCD-like interactions demand that they appear.
### IV.2 The connection between the DR and CDR schemes
From the formulation of the Lagrangians, one can also draw a connection
between the structure of the amplitudes in the DR and CDR schemes. In
particular, the DR scheme Lagrangian contains all of the interactions that the
CDR scheme Lagrangian does, plus a host of interactions involving the
evanescent bosons. This means that the amplitudes in the DR scheme can be
partitioned into a part that is identical to the CDR scheme amplitude and a
part that involves the exchange of one or more evanescent bosons. One need not
consider the case of external evanescent bosons since the DR scheme
renormalization program ensures that such terms contribute to the S-matrix at
order ${\varepsilon}$ Capper et al. (1980); Jack et al. (1994a). The DR scheme
sub-amplitude that involves evanescent exchanges will necessarily include a
spin-sum over the evanescent degrees of freedom, with the result that this
sub-amplitude will be weighted by a factor of $D_{x}=2\,{\varepsilon}$. The
only way that a term from the evanescent sub-amplitude can make a finite (or
singular) contribution to the full amplitude is if it is weighted by
ultraviolet or infrared poles. Thus, the full evanescent contribution to an
amplitude up to order ${\varepsilon}^{0}$ is part of the universal
(ultraviolet or infrared) structure of the amplitude, and is controlled by
anomalous dimensions. This means that the evanescent contribution to an
$n$-loop amplitude (that is the part that is different from the CDR amplitude)
can be determined entirely in terms of ultraviolet counterterms, jet and soft
functions and lower-order ($0$ to $(n-1)$-loop) hard-scattering functions.
Thus, with a proper rearrangement of terms (the ${\widehat{{\rm DR}}}$ scheme
defined below), at any order $n$ the hard-scattering functions in the two
schemes are related by
$\left|H_{\bf f}^{(n)}\right\rangle_{{\widehat{{\rm DR}}}}=\left|H_{\bf
f}^{(n)}\right\rangle_{{\rm HV}}+{\cal O}({\varepsilon}).$ (7)
### IV.3 A new definition of the FDH scheme
Clearly, if one can draw a close connection between the FDH and DR schemes,
one should be able to develop a prescription for the direct transformation of
an amplitude computed in the FDH scheme to one that is computed in the HV
scheme. From the above considerations, it is quite simple to state the
connection.
The four-dimensional helicity scheme is the DR scheme with two extra
conditions:
1. 1.
External states are taken to be four dimensional.
2. 2.
The evanescent couplings ($\alpha_{e}$ and $\eta_{1}$) are identified with
$\alpha_{s}$.
The first condition asserts the same distinction between the FDH and DR
schemes as exists between the HV and CDR schemes. The restriction to four-
dimensional external states does not affect the anomalous dimensions of the
theory. The ultraviolet counterterms and the jet and soft functions are
unchanged. The only changes are to the exact form of the finite hard-
scattering matrix elements. The four-dimensional condition also forbids the
appearance of external evanescent states. As mentioned before, the
renormalization program of the DR scheme ensures that evanescent external
states can only contribute to the S-matrix at order ${\varepsilon}$ or higher,
so this restriction is of no consequence.
The second condition is the one that violates unitarity and renders the FDH
non-renormalizable. The evanescent couplings need to be renormalized
differently than the QCD coupling, but there is no means of doing so once the
couplings have been identified. Therefore, the FDH can only be used to compute
bare (unrenormalized) loop amplitudes.
In the DR scheme, on the other hand, one can determine the correct ultraviolet
counterterms, and the infrared counterterms needed to remove the evanescent
contribution, leaving the HV scheme result. By computing these counterterms in
the DR scheme and then identifying the couplings, one obtains the counterterms
needed to shift from the FDH to the HV scheme.
### IV.4 Ultraviolet counterterms for the FDH
When working within massless QCD, it is only necessary to renormalize the
couplings. It is common in dimensional reduction to determine ultraviolet
counterterms using modified minimal subtraction (this is known as the
${\overline{{\rm DR}}}$ scheme), dropping evanescent terms, even if they
contain ultraviolet poles, because the factor of $D_{x}$ renders them finite.
This procedure means that the renormalized coupling in the ${\overline{{\rm
DR}}}$ scheme, ${\alpha_{s}^{{\overline{{\rm DR}}}}}$ differs from the
standard coupling ${\alpha_{s}^{{\overline{{\rm MS}}}}}$ that appears in
HV/CDR calculations by a finite renormalization. This finite renormalization
corresponds precisely to the $D_{x}/{\varepsilon}$ terms that were dropped
from the $\beta$-function. My goal is to remove all evanescent contributions,
so I will include $(D_{x}/{\varepsilon})^{n}$ terms in my definitions of the
$\beta$-functions and anomalous dimensions. To distinguish it from the
${\overline{{\rm DR}}}$ scheme, I will call this the ${\widehat{{\rm DR}}}$
scheme.
Because there are so many independent couplings in the DR scheme, and because
they mix under renormalization, the simple $\beta_{0,1,2,\ldots}$ labeling of
the ${\overline{{\rm MS}}}$ scheme is insufficient. Instead, I write,
$\begin{split}{{\beta}^{{\widehat{{\rm
DR}}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}+\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}\frac{\partial
Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}{\partial{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}\,{{\beta}_{e}^{{\widehat{{\rm
DR}}}}}+\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}\frac{\partial
Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}{\partial{\eta_{i}^{{\widehat{{\rm
DR}}}}}}\,{{\beta}_{\eta_{i}}^{{\widehat{{\rm
DR}}}}}\right)\left(1+\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}\frac{\partial
Z_{{\alpha_{s}^{{\widehat{{\rm DR}}}}}}}{\partial{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}\right)^{-1}\\\ &=-{\varepsilon}\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}-\sum_{i,j,k,l,m}\,{{\beta}_{ijklm}^{{\widehat{{\rm
DR}}}}}\,{\left(\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{i}\,{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{j}\,{\left(\frac{{\eta_{{1}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{k}\,{\left(\frac{{\eta_{{2}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{l}\,{\left(\frac{{\eta_{{3}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{m}\,.\end{split}$ (8)
Similar equations yield
$\begin{split}{{\beta}_{e}^{{\widehat{{\rm
DR}}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}&=-{\varepsilon}\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}-\sum_{i,j,k,l,m}\,{{\beta}_{e,\,ijklm}^{{\widehat{{\rm
DR}}}}}\,{\left(\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{i}\,{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{j}\,{\left(\frac{{\eta_{{1}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{k}\,{\left(\frac{{\eta_{{2}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{l}\,{\left(\frac{{\eta_{{3}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{m}\,,\\\ {{\beta}_{\eta_{s}}^{{\widehat{{\rm
DR}}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\eta_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}&=-{\varepsilon}\,\frac{{\eta_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}-\sum_{i,j,k,l,m}\,{{\beta}_{s,\,ijklm}^{{\widehat{{\rm
DR}}}}}\,{\left(\frac{{\alpha_{s}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{i}\,{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{j}\,{\left(\frac{{\eta_{{1}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{k}\,{\left(\frac{{\eta_{{2}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{l}\,{\left(\frac{{\eta_{{3}}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{m}\,.\end{split}$ (9)
The values of the coefficients through three loops (for
${{\beta}^{{\widehat{{\rm DR}}}}}$ and ${{\beta}_{e}^{{\widehat{{\rm DR}}}}}$)
are given in Appendix A. Note that with the rearrangement of the evanescent
contributions, the terms in ${{\beta}^{{\widehat{{\rm DR}}}}}$ that are not
proportional to $D_{x}$ are identical to the coefficients of the
$\beta$-function in the ${\overline{{\rm MS}}}$ scheme. This indicates that
the renormalized coupling of the ${\widehat{{\rm DR}}}$ scheme coincides with
that of the ${\overline{{\rm MS}}}$ scheme.
The ultraviolet counterterms for FDH amplitudes are computed as follows.
First, one computes the lower loop amplitudes in the DR scheme and then
expands the bare couplings in terms of the renormalized couplings using the
$\beta$-functions of the ${\widehat{{\rm DR}}}$ scheme. Finally, the
evanescent couplings are identified with the QCD coupling and the factors of
$D_{x}$ are evaluated ($D_{x}=2\,{\varepsilon}$).
$\left|{\cal M}(\alpha_{s})\right\rangle^{\rm CT}_{{\rm
FDH}}=\left.\left|{\cal M}(\alpha_{s},\alpha_{e},\eta_{1})\right\rangle^{\rm
CT}_{{\widehat{{\rm
DR}}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}$ (10)
This will remove all of the ultraviolet terms, including the evanescent terms
that appear to be finite because of the factor of $D_{x}$.
### IV.5 The infrared structure of the DR scheme
The next step is to remove the unwanted evanescent component of the infrared
structure of FDH scheme amplitudes. As with the ultraviolet counterterms, the
terms to be removed can be identified by studying the structure of DR scheme
amplitudes. The basic form of the infrared structure in the DR scheme is the
same as in HV/CDR, but the anomalous dimensions receive evanescent
corrections. In addition, there are new ${\cal G}$ anomalous dimensions that
depend on the evanescent couplings. Through two-loops, the corrections and new
anomalous dimensions depend only on the fermion-evanescent coupling, not the
quartic evanescent couplings. Furthermore, because the evanescent couplings
are not gauge couplings, there are no new counterparts to the cusp or soft
anomalous dimensions, which are associated with the exchange of gauge bosons.
I have determined the values of the infrared anomalous dimensions in the DR
scheme by the direct calculation of two-loop amplitudes. I first determine the
anomalous dimensions for external quarks from the Drell-Yan amplitude. I then
obtain the anomalous dimensions for external gluons from the
$q\overline{q}\to\ g\gamma$ amplitude Anastasiou et al. (2001, 2002); Glover
and Tejeda-Yeomans (2003). In principle, it would be easier to extract the
gluon jet function by calculating the amplitude for $g\,g\to\ H$, but the
Higgs - gluon coupling is governed by a set of effective operators generated
by integrating out the top quark. This system, involving operator mixing and
higher-order corrections to the Wilson coefficients, has been studied to high
order in the CDR scheme Chetyrkin et al. (1997, 1998), but not in the non-
supersymmetric DR scheme.
The calculations of the infrared anomalous dimensions as well as the wave-
function and vertex corrections used to extract the $\beta$-functions were all
calculated within the same framework. The Feynman diagrams were generated with
QGRAF Nogueira (1993) and the symbolic algebra program FORM Vermaseren (2000)
was used to implement the Feynman rules and perform algebraic manipulations to
reduce the result to a set of Feynman integrals and their coefficients. The
method of Ref. Davydychev et al. (1998) was used to reduce the calculation of
the vertex corrections to propagator integrals. The full set of Feynman
integrals was reduced to master integrals using the program REDUZE-2 von
Manteuffel and Studerus (2012). REDUZE-2 offers significant improvements over
the previous version Studerus (2010) and was particularly effective at
reducing the non-planar double-box integrals that contribute to the
$q\overline{q}\to\ g\gamma$ amplitude. All of the master integrals needed for
these calculations are known in the literature Chetyrkin et al. (1980);
Kazakov (1984); Gehrmann et al. (2005); Smirnov (1999); Anastasiou et al.
(2000a); Tausk (1999); Anastasiou et al. (2000b).
The jet function in the DR scheme takes the form,
$\begin{split}\ln\widehat{\cal J}_{i}^{\rm
DR}\left(\alpha_{s}(\mu^{2}),\alpha_{e}(\mu^{2}),{\varepsilon}\right)=&-{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\hat{\gamma}_{K\,i}^{(1)}+\frac{1}{4\,{\varepsilon}}\widehat{\cal
G}_{i}^{(1)}({\varepsilon})\right]-{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}\frac{\widehat{\cal
G}_{i,e}^{(0,1)}({\varepsilon})}{4\,{\varepsilon}}\\\
&+{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}\left[\frac{{{\beta}_{20}^{{\widehat{{\rm
DR}}}}}}{8}\frac{1}{{\varepsilon}^{2}}\left(\frac{3}{4\,{\varepsilon}}\hat{\gamma}_{K\,i}^{(1)}+\widehat{\cal
G}_{i}^{(1)}({\varepsilon})\right)-\frac{1}{8}\left(\frac{\hat{\gamma}_{K\,i}^{(2)}}{4\,{\varepsilon}^{2}}+\frac{\widehat{\cal
G}_{i}^{(2)}({\varepsilon})}{{\varepsilon}}\right)\right]\\\
&+{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}\frac{1}{8}\left[\frac{{{\beta}_{e,\,1\,1}^{{\widehat{{\rm
DR}}}}}\,\widehat{\cal
G}_{i,e}^{(0,1)}({\varepsilon})}{{\varepsilon}^{2}}-\frac{\widehat{\cal
G}_{i,e}^{(1,1)}({\varepsilon})}{{\varepsilon}}\right]\\\
&+{\left(\frac{{\alpha_{e}^{{\widehat{{\rm
DR}}}}}}{\pi}\right)}^{2}\frac{1}{8}\left[\frac{{{\beta}_{e,\,0\,2}^{{\widehat{{\rm
DR}}}}}\,\widehat{\cal
G}_{i,e}^{(0,1)}({\varepsilon})}{{\varepsilon}^{2}}-\frac{\widehat{\cal
G}_{i,e}^{(0,2)}({\varepsilon})}{{\varepsilon}}\right]+\dots\,,\end{split}$
(11)
where the anomalous dimensions in the ${\widehat{{\rm DR}}}$ scheme are
$\begin{split}\hat{\gamma}_{K\,i}^{(1)}&=2\,C_{i},\quad\hat{\gamma}_{K\,i}^{(2)}=C_{i}\,\hat{K}=C_{i}\left[C_{A}\left(\frac{67}{18}-\zeta_{2}\right)-\frac{10}{9}T_{f}\,N_{f}-\frac{2}{9}D_{x}\,C_{A}\right],\quad
C_{q}\equiv C_{F},\quad C_{g}\equiv C_{A},\\\ \widehat{\cal
G}_{q}^{(1)}&=\frac{3}{2}C_{F}+\frac{{\varepsilon}}{2}C_{F}\left(8-\zeta_{2}\right),\hskip
133.0pt\widehat{\cal G}_{g}^{(1)}=2\,{{\beta}_{20}^{{\widehat{{\rm
DR}}}}}-\frac{{\varepsilon}}{2}C_{A}\,\zeta_{2},\\\ \widehat{\cal
G}_{q,e}^{(0,1)}&=-\frac{1}{4}D_{x}\,C_{F}\,,\hskip 195.0pt\widehat{\cal
G}_{g,e}^{(0,1)}=0\,,\\\ \widehat{\cal
G}_{q}^{(2)}&=C_{F}^{2}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\,\zeta_{3}\right)+C_{A}\,C_{F}\left(\frac{2545}{432}+\frac{11}{12}\zeta_{2}-\frac{13}{4}\zeta_{3}\right)-C_{F}\,T_{f}\,N_{f}\left(\frac{209}{108}+\frac{1}{3}\zeta_{2}\right)\\\
&-D_{x}\,C_{A}\,C_{F}\left(\frac{311}{864}+\frac{1}{24}\zeta_{2}\right)\,,\\\
\widehat{\cal G}_{g}^{(2)}&=4\,{{\beta}_{30}^{{\widehat{{\rm
DR}}}}}+C_{A}^{2}\left(\frac{10}{27}-\frac{11}{12}\zeta_{2}-\frac{1}{4}\zeta_{3}\right)+C_{A}\,T_{f}\,N_{f}\left(\frac{13}{27}+\frac{1}{3}\zeta_{2}\right)+\frac{1}{2}C_{F}\,T_{f}\,N_{f}\\\
&+D_{x}\,C_{A}^{2}\left(\frac{7}{54}+\frac{1}{24}\zeta_{2}\right)\,,\\\
\widehat{\cal
G}_{q,e}^{(1,1)}&=D_{x}\left(-\frac{11}{16}C_{A}\,C_{F}+\frac{1}{4}C_{F}^{2}+\frac{1}{4}C_{F}^{2}\,\zeta_{2}\right)\,,\hskip
95.0pt\widehat{\cal G}_{g,e}^{(1,1)}=2\,{{\beta}_{21}^{{\widehat{{\rm
DR}}}}}\,,\\\ \widehat{\cal
G}_{q,e}^{(0,2)}&=\frac{3}{16}D_{x}\,C_{F}\,T_{f}\,N_{f}\,,\hskip
177.0pt\widehat{\cal G}_{g,e}^{(0,2)}=0\,,\\\ {{\beta}_{20}^{{\widehat{{\rm
DR}}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}-\frac{1}{24}D_{x}\,C_{A}\,,\\\
{{\beta}_{30}^{{\widehat{{\rm
DR}}}}}&=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}-\frac{7}{48}D_{x}\,C_{A}^{2}\,,\hskip
60.0pt{{\beta}_{21}^{{\widehat{{\rm
DR}}}}}=\frac{1}{16}D_{x}\,C_{F}\,N_{f}\,,\\\
{{\beta}_{e,\,0\,2}^{{\widehat{{\rm
DR}}}}}&=\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}-\frac{1}{4}D_{x}\,\left(C_{A}-C_{F}\right)\,,\hskip
77.0pt{{\beta}_{e,\,1\,1}^{{\widehat{{\rm
DR}}}}}=\frac{3}{2}C_{F}\,.\end{split}$ (12)
Note that the QCD coupling is ${\alpha_{s}^{{\overline{{\rm MS}}}}}$, the same
coupling used in HV/CDR calculations. Since I extract the anomalous dimensions
from amplitude calculations, I cannot separate the order ${\varepsilon}$ part
of the one-loop $\widehat{\cal G}$ anomalous dimensions, which contributes at
two-loops when multiplied by a $\beta$-function coefficient, from the pure
two-loop $\widehat{\cal G}$ anomalous dimensions. This merely constitutes a
rearrangement of terms and does not affect the prediction of the infrared
structure.
The soft function changes very little in going to the DR scheme. This is
because evanescent exchanges do not add new soft anomalous dimensions, they
only add corrections to the existing terms.
$\begin{split}\widehat{\bf S_{f}}^{\rm
DR}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)&=1+\frac{1}{2\,{\varepsilon}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{8\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}\times\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}\\\
&\qquad-\frac{{{\beta}_{20}^{{\widehat{{\rm
DR}}}}}}{4\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{4\,{\varepsilon}}{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}^{2}\widehat{\bm{\Gamma}}_{S_{f}}^{(2)}\,,\end{split}$
(13) $\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}=\frac{1}{2}\,\sum_{i\in{\bf f}}\
\sum_{j\neq i}{\bf T}_{i}\cdot{\bf
T}_{j}\,\ln\left(\frac{\mu^{2}}{-s_{ij}}\right),\qquad\widehat{\bm{\Gamma}}_{S_{f}}^{(2)}=\frac{\hat{K}}{2}\widehat{\bm{\Gamma}}_{S_{f}}^{(1)}\,,$
(14)
where
$\hat{K}=C_{A}\left(67/18-\zeta_{2}\right)-10/9\,T_{f}\,N_{f}-2/9\,D_{x}\,C_{A}$
is again the same constant that relates the one- and two-loop cusp anomalous
dimensions, this time in the ${\widehat{{\rm DR}}}$ scheme.
### IV.6 Transforming FDH amplitudes into HV amplitudes
I have now assembled all of the pieces needed to convert bare amplitudes
computed in the FDH scheme into renormalized amplitudes in the HV scheme. To
obtain an $n$-loop amplitude in the HV scheme, one needs
1. 1.
The bare $n$-loop amplitude in the FDH scheme.
2. 2.
The renormalized $m$-loop amplitudes ($m\in\\{0,\dots,n-1\\}$) to order
${\varepsilon}^{2\,(n-m)}$ in the HV scheme.
3. 3.
The jet and soft functions to order $n$ in the HV scheme.
4. 4.
The renormalized $m$-loop amplitudes ($m\in\\{0,\dots,n-1\\}$) to order
${\varepsilon}^{2\,(n-m)}$ in the ${\widehat{{\rm DR}}}$ scheme.
5. 5.
The jet and soft functions to order $n$ in the ${\widehat{{\rm DR}}}$ scheme.
Note that computing the $n$-loop squared amplitude to order
${\varepsilon}^{0}$ already required the higher-order in ${\varepsilon}$
contributions to the lower-loop amplitudes in the HV scheme. The conversion
procedure requires them in the ${\widehat{{\rm DR}}}$ scheme as well.
The first step is to expand Eq. (1) by orders of $\alpha_{s}$,
$\begin{split}\left|{\cal M}^{(n)}\right\rangle_{\rm
HV}&=\sum_{i=0}^{n}\,\left[{\cal J}\otimes{\bf S}\right]^{(i)}\left|{\cal
H}^{(n-i)}\right\rangle_{\rm HV}\\\ \left|{\cal
M}^{(n)}\right\rangle_{\widehat{{\rm
DR}}}&=\sum_{i=0}^{n}\,\left[\widehat{\cal J}\otimes\widehat{\bf
S}\right]^{(i)}\left|{\cal H}^{(n-i)}\right\rangle_{\widehat{{\rm DR}}}\\\
\end{split}$ (15)
I now define the “renormalized” FDH scheme amplitude as
$\left|{\cal M}^{(n)}\right\rangle_{\rm FDH}=\left|{\cal
M}^{(n)}\right\rangle_{\rm FDH}^{\rm Bare}+\left|{\cal
M}^{(n)}\right\rangle^{\rm CT}_{{\rm FDH}}=\left.\left|{\cal
M}^{(n)}\right\rangle_{\widehat{{\rm
DR}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}\,.$ (16)
From this I find that
$\left.\left|{\cal H}^{(n)}\right\rangle_{\widehat{{\rm
DR}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}=\left|{\cal M}^{(n)}\right\rangle_{\rm
FDH}-\sum_{i=1}^{n}\,\left[\widehat{\cal J}\otimes\widehat{\bf
S}\right]^{(i)}\left.\left|{\cal H}^{(n-i)}\right\rangle_{\widehat{{\rm
DR}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}\,.$ (17)
Finally, using Eq. (7), I obtain
$\left|{\cal H}^{(n)}\right\rangle_{\rm HV}=\left|{\cal
M}^{(n)}\right\rangle_{\rm FDH}^{\rm Bare}+\left|{\cal
M}^{(n)}\right\rangle^{\rm CT}_{{\rm FDH}}-\sum_{i=1}^{n}\,\left[\widehat{\cal
J}\otimes\widehat{\bf S}\right]^{(i)}\left.\left|{\cal
H}^{(n-i)}\right\rangle_{\widehat{{\rm
DR}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}+{\cal O}({\varepsilon})\,.$ (18)
The infrared structure of the HV scheme amplitude can be extracted from
$\left|{\cal M}^{(n)}\right\rangle_{\rm FDH}^{\rm Bare}$ in a similar way or
constructed directly in terms of the lower order hard scattering matrix
elements and the jet and soft functions.
Let me now write out explicitly the transformation of a one-loop bare
amplitude in the FDH scheme, involving $n_{q}$ quarks and anti-quarks and
$n_{g}$ gluons, into a renormalized one-loop amplitude in the HV scheme.
Starting with
$\left|{\cal H}^{(1)}\right\rangle_{\rm HV}=\left|{\cal
M}^{(1)}\right\rangle_{\rm FDH}^{\rm Bare}+\left|{\cal
M}^{(1)}\right\rangle^{\rm CT}_{{\rm FDH}}-\left[\widehat{\cal J}+\widehat{\bf
S}\right]^{(1)}\left.\left|{\cal H}^{(0)}\right\rangle_{\widehat{{\rm
DR}}}\right|_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}+{\cal O}({\varepsilon})\,,$ (19)
I add in the infrared parts of the HV amplitude (note that the one-loop soft
functions of the HV and ${\widehat{{\rm DR}}}$ scheme are identical) to obtain
$\begin{split}\left|{\cal M}^{(1)}\right\rangle_{\rm HV}&=\left|{\cal
M}^{(1)}\right\rangle_{\rm FDH}^{\rm
Bare}-{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\left(\frac{n_{q}+n_{g}-2}{2\,{\varepsilon}}{{\beta}_{20}^{{\widehat{{\rm
DR}}}}}\right)\left|{\cal H}^{(0)}\right\rangle_{{\rm HV}}\\\ &\hskip
90.0pt+\left({\cal J}^{(1)}-\widehat{\cal
J}^{(1)}\right)_{\genfrac{}{}{0.0pt}{}{\alpha_{e},\eta_{1}\to\alpha_{s}}{D_{x}\to
2\,{\varepsilon}\hfill}}\left|{\cal H}^{(0)}\right\rangle_{\rm HV}+{\cal
O}({\varepsilon})\\\ &=\left|{\cal M}^{(1)}\right\rangle_{\rm FDH}^{\rm
Bare}-{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\frac{n_{q}+n_{g}-2}{2\,{\varepsilon}}{\beta_{0}^{{\overline{{\rm
MS}}}}}\left|{\cal H}^{(0)}\right\rangle_{{\rm HV}}\\\
&+{\left(\frac{\alpha_{s}^{{\overline{{\rm
MS}}}}}{\pi}\right)}\left(\frac{n_{q}+n_{g}-2}{24}C_{A}-\frac{n_{q}}{8}C_{F}-\frac{n_{g}}{24}C_{A}\right)\left|{\cal
H}^{(0)}\right\rangle_{\rm HV}+{\cal O}({\varepsilon})\end{split}$ (20)
The first line is just the bare one-loop amplitude with standard
${\overline{{\rm MS}}}$ ultraviolet counterterm, while the second line is the
finite shift, broken into ultraviolet, infrared $n_{q}$ and infrared $n_{g}$
pieces, identified by Kunszt, et al. Kunszt et al. (1994). Beyond one loop,
the transformations are not so simple and involve the structure of the
amplitudes in addition to the identities of the external states.
## V Conclusion
In this paper, I have described a procedure for transforming bare loop
amplitudes computed in the four dimensional helicity scheme into renormalized
amplitudes in the ’t Hooft-Veltman scheme. One of the simplifying features of
the FDH, the treatment of the evanescent states as if they were gluons,
renders the scheme non-renormalizable. Nevertheless, the FDH can be defined in
terms of a renormalizable scheme, a variant of the dimensional reduction
scheme. Through this connection to the DR scheme, I have shown that the
differences between amplitudes calculated in the FDH scheme and the HV scheme
(up to order ${\varepsilon}^{-}$) are either ultraviolet or infrared in origin
and are therefore part of the universal structure of the amplitude which is
controlled by anomalous dimensions. By computing these anomalous dimensions in
the ${\widehat{{\rm DR}}}$ scheme, defined above, through two loops, I provide
concrete formulæ for the transformation of the amplitudes.
The utility of such transformations lies in the close connection between the
FDH scheme and the techniques of generalized unitarity and the helicity
method. These techniques are a natural fit for the FDH scheme, but the results
need to be transformed into a renormalizable scheme so that they can be used
in practical calculations. With the procedures described in this paper, such
transformations can be performed.
#### Acknowledgments:
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.
## Appendix A ${\widehat{{\rm DR}}}$ Scheme $\beta$-functions
The non-vanishing coefficients for ${{\beta}^{{\widehat{{\rm DR}}}}}$ through
three loops are:
$\begin{split}{{\beta}_{20}^{{\widehat{{\rm
DR}}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}-\frac{1}{24}D_{x}\,C_{A}\,,\\\
{{\beta}_{30}^{{\widehat{{\rm
DR}}}}}&=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}-\frac{7}{48}D_{x}\,C_{A}^{2}\,,\hskip
60.0pt{{\beta}_{21}^{{\widehat{{\rm DR}}}}}=\frac{1}{16}D_{x}\,C_{F}\,N_{f}\\\
{{\beta}_{40}^{{\widehat{{\rm
DR}}}}}&=\frac{2857}{3456}C_{A}^{3}-\frac{1415}{3456}C_{A}^{2}\,N_{f}-\frac{205}{1152}C_{A}\,C_{F}\,N_{f}+\frac{1}{64}C_{F}^{2}\,N_{f}+\frac{79}{3456}C_{A}\,N_{f}^{2}+\frac{11}{576}C_{F}\,N_{f}^{2}\\\
&+D_{x}\left(-\frac{2749}{6912}C_{A}^{3}+\frac{13}{432}C_{A}^{2}\,N_{f}+\frac{23}{2304}C_{A}\,C_{F}\,N_{f}\right)+\frac{145}{13824}D_{x}^{2}\,C_{A}^{3}\\\
{{\beta}_{31}^{{\widehat{{\rm
DR}}}}}&=D_{x}\left(\frac{5}{256}C_{A}^{2}\,N_{f}+\frac{7}{32}C_{A}\,C_{F}\,N_{f}+\frac{3}{128}C_{F}^{2}\,N_{f}\right)\\\
{{\beta}_{22}^{{\widehat{{\rm
DR}}}}}&=D_{x}\left(-\frac{1}{64}C_{A}^{2}\,N_{f}+\frac{7}{128}C_{A}\,C_{F}\,N_{f}-\frac{3}{64}\,C_{F}^{2}\,N_{f}+\frac{1}{256}C_{A}\,N_{f}^{2}-\frac{7}{256}C_{F}\,N_{f}^{2}\right)\\\
&+D_{x}^{2}\left(\frac{1}{256}C_{A}^{2}\,N_{f}-\frac{5}{256}C_{A}\,C_{F}\,N_{f}\right)\\\
{{\beta}_{30100}^{{\widehat{{\rm
DR}}}}}&=\frac{27}{512}D_{x}\left(1-D_{x}\right)\,,\hskip
30.0pt{{\beta}_{30010}^{{\widehat{{\rm
DR}}}}}=-\frac{45}{126}D_{x}\left(2+D_{x}\right)\,,\hskip
30.0pt{{\beta}_{30001}^{{\widehat{{\rm
DR}}}}}=-\frac{9}{256}D_{x}\left(1-D_{x}\right)\\\
{{\beta}_{20200}^{{\widehat{{\rm
DR}}}}}&=-\frac{81}{512}D_{x}\left(1-D_{x}\right)\,,\hskip
30.0pt{{\beta}_{20101}^{{\widehat{{\rm
DR}}}}}=\frac{27}{128}D_{x}\left(1-D_{x}\right)\,,\\\
{{\beta}_{20020}^{{\widehat{{\rm
DR}}}}}&=\frac{45}{64}D_{x}\left(2+D_{x}\right)\,,\hskip
30.0pt{{\beta}_{20002}^{{\widehat{{\rm
DR}}}}}=-\frac{63}{256}D_{x}\left(1-D_{x}\right)\,,\end{split}$ (21)
where I omit the last three indices if they all vanish.
The coefficients of ${{\beta}_{e}^{{\widehat{{\rm DR}}}}}$ through two loops
are:
$\begin{split}{{\beta}_{e,\,0\,2}^{{\widehat{{\rm
DR}}}}}&=\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}-\frac{1}{4}D_{x}\,\left(C_{A}-C_{F}\right)\,,\qquad{{\beta}_{e,\,1\,1}^{{\widehat{{\rm
DR}}}}}=\frac{3}{2}C_{F}\,,\\\\[5.0pt] {{\beta}_{e,\,0\,3}^{{\widehat{{\rm
DR}}}}}&=\frac{3}{8}\,C_{A}^{2}-\frac{5}{4}\,C_{A}\,C_{F}+C_{F}^{2}-\frac{3}{16}\,C_{A}\,N_{f}+\frac{3}{8}\,C_{F}\,N_{f}+D_{x}\left(-\frac{1}{2}\,C_{A}^{2}+\frac{3}{2}\,C_{A}\,C_{F}-C_{F}^{2}+\frac{3}{32}\,C_{A}\,N_{f}\right)\\\
&+D_{x}^{2}\left(\frac{3}{32}\,C_{A}^{2}-\frac{1}{4}\,C_{A}\,C_{F}+\frac{9}{64}\,C_{F}^{2}\right)\,,\\\
{{\beta}_{e,\,1\,2}^{{\widehat{{\rm
DR}}}}}&=-\frac{3}{8}\,C_{A}^{2}+\frac{7}{4}\,C_{A}\,C_{F}-2\,C_{F}^{2}-\frac{5}{16}\,C_{F}\,N_{f}+D_{x}\left(-\frac{11}{16}\,C_{A}\,C_{F}+\frac{1}{2}\,C_{F}^{2}\right)\,,\\\
{{\beta}_{e,\,2\,1}^{{\widehat{{\rm
DR}}}}}&=-\frac{7}{64}\,C_{A}^{2}+\frac{61}{48}\,C_{A}\,C_{F}+\frac{3}{16}\,C_{F}^{2}+\frac{1}{16}\,C_{A}\,N_{f}-\frac{5}{24}\,C_{F}\,N_{f}+D_{x}\left(\frac{1}{64}\,C_{A}^{2}-\frac{11}{96}\,C_{A}\,C_{F}\right)\,,\\\\[5.0pt]
{{\beta}_{e,\,0\,2100}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{8}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,0\,2010}^{{\widehat{{\rm
DR}}}}}=\frac{5}{8}\left(2+D_{x}\right)\,,\qquad{{\beta}_{e,\,0\,2001}^{{\widehat{{\rm
DR}}}}}=\frac{3}{4}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,1200}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{27}{64}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,0\,1020}^{{\widehat{{\rm
DR}}}}}=-\frac{15}{8}\left(2+D_{x}\right)\,,\qquad{{\beta}_{e,\,0\,1002}^{{\widehat{{\rm
DR}}}}}=\frac{21}{32}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,1101}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{16}\left(1-D_{x}\right)\,,\\\ \end{split}$ (22)
The three-loop coefficients that do not involve the quartic couplings are:
$\begin{split}{{\beta}_{e,\,0\,4}^{{\widehat{{\rm
DR}}}}}&=\frac{9}{16}\,C_{A}^{3}\,\zeta_{3}-C_{A}^{2}\,C_{F}\,\left(\frac{5}{16}+\frac{69}{16}\,\zeta_{3}\right)+C_{A}\,C_{F}^{2}\,\left(\frac{5}{4}+\frac{15}{2}\,\zeta_{3}\right)-C_{F}^{3}\,\left(\frac{5}{4}+\frac{9}{4}\,\zeta_{3}\right)\\\
&+C_{A}^{2}\,N_{f}\,\left(\frac{3}{128}-\frac{9}{32}\,\zeta_{3}\right)-C_{A}\,C_{F}\,N_{f}\,\left(\frac{15}{32}-\frac{51}{32}\,\zeta_{3}\right)+C_{F}^{2}\,N_{f}\,\left(\frac{27}{32}-\frac{33}{16}\,\zeta_{3}\right)+N_{f}^{2}\,\left(\frac{1}{256}\,C_{A}-\frac{1}{128}\,C_{F}\right)\\\
&+D_{x}\left[-C_{A}^{3}\,\left(\frac{7}{32}+\frac{3}{8}\,\zeta_{3}\right)+C_{A}^{2}\,C_{F}\,\left(\frac{91}{64}+\frac{135}{32}\,\zeta_{3}\right)-C_{A}\,C_{F}^{2}\,\left(\frac{13}{4}+\frac{249}{32}\,\zeta_{3}\right)+C_{F}^{3}\,\left(\frac{41}{16}+\frac{27}{16}\,\zeta_{3}\right)\right.\\\
&\left.+C_{A}^{2}\,N_{f}\,\left(\frac{21}{128}+\frac{3}{64}\,\zeta_{3}\right)-C_{A}\,C_{F}\,N_{f}\,\left(\frac{37}{256}+\frac{33}{64}\,\zeta_{3}\right)-C_{F}^{2}\,N_{f}\,\left(\frac{47}{128}-\frac{27}{32}\,\zeta_{3}\right)-N_{f}^{2}\,\left(\frac{1}{512}\,C_{A}+\frac{3}{64}\,C_{F}\right)\right]\\\
&+D_{x}^{2}\left[+\frac{9}{64}\,C_{A}^{3}-C_{A}^{2}\,C_{F}\,\left(\frac{35}{64}+\frac{69}{64}\,\zeta_{3}\right)+C_{A}\,C_{F}^{2}\,\left(\frac{461}{512}+\frac{147}{64}\,\zeta_{3}\right)-C_{F}^{3}\,\left(\frac{189}{256}+\frac{9}{32}\,\zeta_{3}\right)\right.\\\
&\left.-C_{A}^{2}\,N_{f}\,\left(\frac{29}{512}-\frac{3}{128}\,\zeta_{3}\right)+C_{A}\,C_{F}\,N_{f}\,\left(\frac{49}{512}-\frac{9}{128}\,\zeta_{3}\right)-C_{F}^{2}\,N_{f}\,\left(\frac{43}{1024}-\frac{3}{64}\,\zeta_{3}\right)\right]\\\
&+D_{x}^{3}\left[-C_{A}^{3}\,\left(\frac{1}{32}-\frac{3}{128}\,\zeta_{3}\right)+\frac{33}{256}\,C_{A}^{2}\,C_{F}-C_{A}\,C_{F}^{2}\,\left(\frac{189}{1024}+\frac{9}{128}\,\zeta_{3}\right)+C_{F}^{3}\,\left(\frac{109}{1024}-\frac{3}{64}\,\zeta_{3}\right)\right]\,,\\\
{{\beta}_{e,\,1\,3}^{{\widehat{{\rm
DR}}}}}&=-C_{A}^{3}\,\left(\frac{25}{64}-\frac{3}{4}\,\zeta_{3}\right)+C_{A}^{2}\,C_{F}\,\left(\frac{85}{32}-\frac{15}{4}\,\zeta_{3}\right)-C_{A}\,C_{F}^{2}\,\left(\frac{11}{2}-6\,\zeta_{3}\right)+C_{F}^{3}\,\left(\frac{7}{2}-3\,\zeta_{3}\right)\\\
&+C_{A}^{2}\,N_{f}\,\left(\frac{7}{32}-\frac{3}{8}\,\zeta_{3}\right)-C_{A}\,C_{F}\,N_{f}\,\left(\frac{27}{32}-\frac{9}{8}\,\zeta_{3}\right)+C_{F}^{2}\,N_{f}\,\left(\frac{13}{16}-\frac{3}{4}\,\zeta_{3}\right)+\frac{3}{64}\,C_{A}\,N_{f}^{2}\\\
&+D_{x}\left[-C_{A}^{3}\,\left(\frac{13}{32}+\frac{3}{4}\,\zeta_{3}\right)+C_{A}^{2}\,C_{F}\left(1+\frac{63}{16}\,\zeta_{3}\right)+C_{A}\,C_{F}^{2}\,\left(\frac{5}{64}-\frac{105}{16}\,\zeta_{3}\right)-C_{F}^{3}\,\left(\frac{29}{32}-\frac{27}{8}\,\zeta_{3}\right)\right.\\\
&\left.+C_{A}^{2}\,N_{f}\,\left(\frac{1}{128}+\frac{3}{16}\,\zeta_{3}\right)+C_{A}\,C_{F}\,N_{f}\,\left(\frac{51}{128}-\frac{9}{16}\,\zeta_{3}\right)-C_{F}^{2}\,N_{f}\,\left(\frac{25}{128}-\frac{3}{8}\,\zeta_{3}\right)\right]\\\
&+D_{x}^{2}\left[+C_{A}^{3}\,\left(\frac{13}{128}+\frac{3}{16}\,\zeta_{3}\right)-C_{A}^{2}\,C_{F}\,\left(\frac{25}{128}+\frac{33}{32}\,\zeta_{3}\right)-C_{A}\,C_{F}^{2}\,\left(\frac{3}{128}-\frac{57}{32}\,\zeta_{3}\right)+C_{F}^{3}\,\left(\frac{1}{8}-\frac{15}{16}\,\zeta_{3}\right)\right]\,,\\\
{{\beta}_{e,\,2\,2}^{{\widehat{{\rm
DR}}}}}&=C_{A}^{3}\,\left(\frac{121}{512}-\frac{45}{16}\,\zeta_{3}\right)-C_{A}^{2}\,C_{F}\,\left(\frac{167}{256}-\frac{207}{16}\,\zeta_{3}\right)+C_{A}\,C_{F}^{2}\,\left(\frac{131}{128}-18\,\zeta_{3}\right)-C_{F}^{3}\,\left(\frac{85}{64}-\frac{27}{4}\,\zeta_{3}\right)\\\
&-C_{A}^{2}\,N_{f}\,\left(\frac{899}{1024}-\frac{45}{32}\,\zeta_{3}\right)+C_{A}\,C_{F}\,N_{f}\,\left(\frac{273}{128}-\frac{171}{32}\,\zeta_{3}\right)-C_{F}^{2}\,N_{f}\,\left(\frac{641}{256}-\frac{99}{16}\,\zeta_{3}\right)-N_{f}^{2}\left(\frac{1}{256}\,C_{A}-\frac{1}{16}\,C_{F}\right)\\\
&+D_{x}\left[-C_{A}^{3}\,\left(\frac{4355}{1024}-\frac{45}{32}\,\zeta_{3}\right)+C_{A}^{2}\,C_{F}\,\left(\frac{21071}{1024}-\frac{99}{16}\,\zeta_{3}\right)-C_{A}\,C_{F}^{2}\,\left(\frac{3381}{128}-\frac{261}{32}\,\zeta_{3}\right)\right.\\\
&\left.+C_{F}^{3}\,\left(\frac{13}{256}-\frac{45}{16}\,\zeta_{3}\right)+\frac{1}{1024}\,C_{A}^{2}\,N_{f}+\frac{15}{64}\,C_{A}\,C_{F}\,N_{f}+\frac{1}{16}\,C_{F}^{2}\,N_{f}\right]\\\
&+D_{x}^{2}\left[-\frac{1}{1024}\,C_{A}^{3}+\frac{83}{1024}\,C_{A}^{2}\,C_{F}-\frac{33}{512}\,C_{A}\,C_{F}^{2}\right]\,,\\\
{{\beta}_{e,\,3\,1}^{{\widehat{{\rm
DR}}}}}&=-\frac{3025}{4608}\,C_{A}^{3}+\frac{12601}{3456}\,C_{A}^{2}\,C_{F}-\frac{453}{128}\,C_{A}\,C_{F}^{2}+\frac{129}{64}\,C_{F}^{3}+\frac{475}{2304}\,C_{A}^{2}\,N_{f}-C_{A}\,C_{F}\,N_{f}\,\left(\frac{151}{1728}+\frac{3}{4}\,\zeta_{3}\right)\\\
&-C_{F}^{2}\,N_{f}\,\left(\frac{23}{32}-\frac{3}{4}\,\zeta_{3}\right)-\frac{5}{576}\,C_{A}\,N_{f}^{2}-\frac{35}{864}\,C_{F}\,N_{f}^{2}\\\
&+D_{x}\left[+\frac{643}{9216}\,C_{A}^{3}-\frac{883}{1728}\,C_{A}^{2}\,C_{F}-\frac{5}{256}\,C_{A}\,C_{F}^{2}-\frac{1}{144}\,C_{A}^{2}\,N_{f}-\frac{19}{864}\,C_{A}\,C_{F}\,N_{f}\right]\\\
&+D_{x}^{2}\left[-\frac{11}{9216}\,C_{A}^{3}-\frac{5}{13824}\,C_{A}^{2}\,C_{F}\right]\,,\end{split}$
(23)
while the three-loop coefficients that do involve the quartic interactions
are:
$\begin{split}{{\beta}_{e,\,0\,3100}^{{\widehat{{\rm
DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{64}\left(1-7\,D_{x}+6\,D_{x}^{2}\right)+\frac{135}{128}N_{f}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,3010}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{5}{64}\left(8-18\,D_{x}-11\,D_{x}^{2}\right)-\frac{75}{128}\,N_{f}\left(2+D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,3001}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{3}{64}\left(2-19\,D_{x}+17\,D_{x}^{2}\right)-\frac{45}{64}\,N_{f}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,1\,2100}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{51}{8}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,1\,2010}^{{\widehat{{\rm
DR}}}}}=\frac{85}{24}\left(2+D_{x}\right)\,,\qquad{{\beta}_{e,\,1\,2001}^{{\widehat{{\rm
DR}}}}}=\frac{17}{4}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,2\,1100}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{801}{1024}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,2\,1010}^{{\widehat{{\rm
DR}}}}}=\frac{375}{256}\left(2+D_{x}\right)\,,\qquad{{\beta}_{e,\,2\,1001}^{{\widehat{{\rm
DR}}}}}=\frac{507}{512}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,2200}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{3}{1024}\left(422-553\,D_{x}+131\,D_{x}^{2}\right)-\frac{405}{1024}\,N_{f}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,2020}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{5}{384}\left(652+136\,D_{x}-95\,D_{x}^{2}\right)+\frac{225}{128}\,N_{f}\left(2+D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,2002}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{1}{1536}\left(394-731\,D_{x}+337\,D_{x}^{2}\right)-\frac{315}{512}\,N_{f}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,2110}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{55}{32}\left(2-D_{x}-D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,0\,2101}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{1}{256}\left(622-773\,D_{x}+151\,D_{x}^{2}\right)+\frac{135}{256}\,N_{f}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,2011}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{205}{96}\left(2-D_{x}-D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,1\,1200}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{405}{128}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,1\,1020}^{{\widehat{{\rm
DR}}}}}=-\frac{225}{16}\left(2+D_{x}\right)\,,\\\
{{\beta}_{e,\,1\,1002}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{315}{64}\left(1-D_{x}\right)\,,\qquad{{\beta}_{e,\,1\,1101}^{{\widehat{{\rm
DR}}}}}=-\frac{135}{32}\left(1-D_{x}\right)\,,\\\
{{\beta}_{e,\,0\,1300}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{27}{1024}\left(11-10\,D_{x}-D_{x}^{2}\right)\,,\qquad{{\beta}_{e,\,0\,1210}^{{\widehat{{\rm
DR}}}}}=-\frac{135}{256}\left(2-D_{x}-D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,0\,1201}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{27}{512}\left(11-10\,D_{x}-D_{x}^{2}\right)\,,\qquad{{\beta}_{e,\,0\,1120}^{{\widehat{{\rm
DR}}}}}=-\frac{45}{64}\left(2-D_{x}-D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,0\,1111}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{45}{32}\left(2-D_{x}-D_{x}^{2}\right)\,,\qquad{{\beta}_{e,\,0\,1102}^{{\widehat{{\rm
DR}}}}}=\frac{9}{256}\left(14-25\,D_{x}+11\,D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,0\,1030}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{5}{4}\left(16+10\,D_{x}+D_{x}^{2}\right)\,,\qquad{{\beta}_{e,\,0\,1021}^{{\widehat{{\rm
DR}}}}}=\frac{105}{64}\left(2-D_{x}-D_{x}^{2}\right)\,,\\\
{{\beta}_{e,\,0\,1012}^{{\widehat{{\rm DR}}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{105}{64}\left(2-D_{x}-D_{x}^{2}\right)\,,\qquad{{\beta}_{e,\,0\,1003}^{{\widehat{{\rm
DR}}}}}=-\frac{7}{256}\left(14-25\,D_{x}+11\,D_{x}^{2}\right)\,,\end{split}$
(24)
A consistent description of ${{\beta}^{{\widehat{{\rm DR}}}}}$ and
${{\beta}_{e,\,\,}^{{\widehat{{\rm DR}}}}}$ through three loops only requires
knowledge of the ${{\beta}_{\eta_{i}}^{{\widehat{{\rm DR}}}}}$’s through one
loop. These coefficients are:
$\begin{split}{{\beta}_{\eta_{1},\,20000}^{{\widehat{{\rm
DR}}}}}&=-\frac{3}{8}\,,\qquad{{\beta}_{\eta_{1},\,02000}^{{\widehat{{\rm
DR}}}}}=\frac{1}{3}N_{f}\,,\qquad{{\beta}_{\eta_{1},\,10100}^{{\widehat{{\rm
DR}}}}}=\frac{9}{2}\,,\qquad{{\beta}_{\eta_{1},\,01100}^{{\widehat{{\rm
DR}}}}}=-\frac{1}{2}N_{f}\,,\\\ {{\beta}_{\eta_{1},\,00200}^{{\widehat{{\rm
DR}}}}}&=-\frac{11}{8}-\frac{1}{8}\,D_{x}\,,\qquad{{\beta}_{\eta_{1},\,00110}^{{\widehat{{\rm
DR}}}}}=-2-D_{x}\,,\qquad{{\beta}_{\eta_{1},\,00101}^{{\widehat{{\rm
DR}}}}}=\frac{7}{2}-\frac{1}{2}\,D_{x}\,,\\\
{{\beta}_{\eta_{2},\,20000}^{{\widehat{{\rm
DR}}}}}&=-\frac{9}{16}\,,\qquad{{\beta}_{\eta_{2},\,02000}^{{\widehat{{\rm
DR}}}}}=\frac{1}{24}N_{f}\,,\qquad{{\beta}_{\eta_{2},\,10010}^{{\widehat{{\rm
DR}}}}}=\frac{9}{2}\,,\qquad{{\beta}_{\eta_{2},\,01010}^{{\widehat{{\rm
DR}}}}}=-\frac{1}{2}N_{f}\,,\\\ {{\beta}_{\eta_{2},\,00200}^{{\widehat{{\rm
DR}}}}}&=\frac{3}{16}\left(1-D_{x}\right)\,,\qquad{{\beta}_{\eta_{2},\,00110}^{{\widehat{{\rm
DR}}}}}=\frac{1}{2}\left(1-D_{x}\right)\,,\qquad{{\beta}_{\eta_{2},\,00101}^{{\widehat{{\rm
DR}}}}}=-\frac{1}{2}\left(1-D_{x}\right)\,,\\\
{{\beta}_{\eta_{2},\,00020}^{{\widehat{{\rm
DR}}}}}&=-\frac{32}{3}-\frac{4}{3}\,D_{x}\,,\qquad{{\beta}_{\eta_{2},\,00011}^{{\widehat{{\rm
DR}}}}}=-\frac{7}{6}\left(1-D_{x}\right)\,,\qquad{{\beta}_{\eta_{2},\,00002}^{{\widehat{{\rm
DR}}}}}=\frac{7}{12}\left(1-D_{x}\right)\,,\\\
{{\beta}_{\eta_{3},\,10001}^{{\widehat{{\rm
DR}}}}}&=\frac{9}{2}\,,\qquad{{\beta}_{\eta_{3},\,01001}^{{\widehat{{\rm
DR}}}}}=-\frac{1}{2}N_{f}\,,\qquad{{\beta}_{\eta_{3},\,00110}^{{\widehat{{\rm
DR}}}}}=2+D_{x}\,,\qquad{{\beta}_{\eta_{3},\,00101}^{{\widehat{{\rm
DR}}}}}=\frac{5}{2}-D_{x}\,,\\\ {{\beta}_{\eta_{3},\,00020}^{{\widehat{{\rm
DR}}}}}&=\frac{5}{3}\left(2+D_{x}\right)\,,\qquad{{\beta}_{\eta_{3},\,00011}^{{\widehat{{\rm
DR}}}}}=-\frac{10}{3}\left(2+D_{x}\right)\,,\qquad{{\beta}_{\eta_{3},\,00002}^{{\widehat{{\rm
DR}}}}}=-\frac{7}{6}+\frac{11}{12}\,D_{x}\,,\qquad\end{split}$ (25)
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|
arxiv-papers
| 2012-05-17T19:37:00 |
2024-09-04T02:49:31.029551
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "William B. Kilgore",
"submitter": "William Kilgore",
"url": "https://arxiv.org/abs/1205.4015"
}
|
1205.4041
|
# Synchrotron Spectral Curvature from 22 MHz to 23 GHz
A. Kogut11affiliation: Code 665, Goddard Space Flight Center, Greenbelt, MD
20771 Alan.J.Kogut@nasa.gov
###### Abstract
We combine surveys of the radio sky at frequencies 22 MHz to 1.4 GHz with data
from the ARCADE-2 instrument at frequencies 3 to 10 GHz to characterize the
frequency spectrum of diffuse synchrotron emission in the Galaxy. The radio
spectrum steepens with frequency from 22 MHz to 10 GHz. The projected spectral
index at 23 GHz derived from the low-frequency data agrees well with
independent measurements using only data at frequencies 23 GHz and above.
Comparing the spectral index at 23 GHz to the value from previously published
analyses allows extension of the model to higher frequencies. The combined
data are consistent with a power-law index $\beta=-2.64\pm 0.03$ at 0.31 GHz,
steepening by an amount $\Delta\beta=0.07$ every octave in frequency.
Comparison of the radio data to models including the cosmic ray energy
spectrum suggests that any break in the synchrotron spectrum must occur at
frequencies above 23 GHz.
###### Subject headings:
radio continuum: general, radiation mechanisms: non-thermal
††slugcomment: Accepted by The Astrophysical Journal
## 1\. Introduction
Synchrotron emission from relativistic cosmic ray electrons accelerated in the
Galactic magnetic field dominates the diffuse radio continuum at frequencies
below 1 GHz. It is an important foreground contaminant for measurements of the
cosmic microwave background radiation, and also serves to probe the Galactic
magnetic field and cosmic ray distributions. Measurements of the synchrotron
frequency spectrum are thus of interest to several areas in astrophysics.
An isotropic distribution of relativistic electrons at a single energy
$E=\gamma mc^{2}$ propagating in a uniform magnetic field $B$ has emissivity
$\epsilon(\nu)=\frac{\sqrt{3}e^{3}}{mc^{2}}B\sin\alpha F(x)~{},$ (1)
where $\alpha$ is the pitch angle between the magnetic field and the line of
sight, and
$F(x)=x\int^{\infty}_{x}K_{5/3}(x^{\prime})dx^{\prime}$ (2)
is defined in terms of the modified Bessel function of order $5/3$ with
variable $x=\nu/\nu_{c}$ and
$\nu_{c}=\frac{3}{4\pi}\frac{e}{mc}\gamma^{2}B\sin\alpha$ (3)
(Schwinger, 1949; Westfold et al., 1959; Oster, 1961). For a power-law
distribution of electron energies $N(E)~{}\propto~{}E^{p}$ propagating in a
uniform magnetic field, the synchrotron emission is also a power law,
$T_{A}(\nu)\propto\nu^{\beta}$ (4)
where $T_{A}$ is antenna temperature, $\nu$ is the radiation frequency, and
$\beta=\frac{p-3}{2}$ (5)
(Rybicki & Lightman, 1979).
Measurements of the synchrotron spectral index provide important input for
models of cosmic ray propagation. Solar modulation reduces the local cosmic
ray electron density for electron energies below a few GeV so that synchrotron
emission provides the most direct probe of low-energy cosmic rays.
Measurements of the cosmic ray spectrum above a few GeV in turn inform models
of the high-frequency synchrotron spectrum. Energy losses from cosmic ray
propagation steepen the cosmic ray spectrum, increasing $p$ toward higher
energies. The observed steepening from $p\sim-2.6$ at 5 GeV to $p\sim-3.2$ at
50 GeV predicts a corresponding steepening in the synchrotron spectrum from
$\beta\sim-2.8$ at 1 GHz to $\beta\sim-3.1$ at 100 GHz (Strong, Moskalenko, &
Ptuskin, 2007).
Comparison of the cosmic ray spectra to the predicted synchrotron spectrum is
complicated by confusion from competing radio emission sources. The diffuse
radio continuum is a superposition of the cosmic microwave background,
synchrotron emission, free-free emission from the warm ionized interstellar
medium, and emission from interstellar dust. A number of authors have
attempted to disentangle the various emission sources to determine the
synchrotron spectral index (for a recent review see Appendix A of Strong,
Orlando, & Jaffe (2011)). Despite some discrepant results, the general trend
shows a steepening of the synchrotron spectrum from $\beta\sim-2.5$ at 22 MHz
to $\beta\sim-3.0$ above 23 GHz, in rough agreement with the observed cosmic
ray spectra.
Several factors contribute to the observed scatter in estimates of the
synchrotron spectral index. Most estimates, particularly those below 23 GHz,
assume a power-law spectrum for synchrotron and do not explicitly model
spectral steepening. Comparisons between closely-separated frequencies more
accurately reflect the local synchrotron spectrum, but have larger
uncertainties from competing emission sources or measurement offsets. Analyses
with broader frequency coverage reduce foreground and offset uncertainties but
average over any spectral steepening.
Two additional effects are important for analyses including data from the
Wilkinson Microwave Anisotropy Probe (WMAP) at frequencies 23 to 94 GHz. A
growing body of evidence suggests that a substantial fraction of the diffuse
continuum near 23 GHz consists of electric dipole radiation from a population
of small, rapidly spinning dust grains (Kogut et al., 1996; de Oliveira-Costa
et al., 1997, 2004; Miville-Deschênes et al., 2008; Dobler & Finkbeiner, 2008;
Ysard, Miville-Deschênes, & Verstraete, 2010; Kogut et al., 2011; Gold et al.,
2011; Planck collaboration, 2011). Spinning dust emission is expected to peak
at frequencies near 20 GHz (Draine & Lazarian, 1998; Ali-Haïmoud et al., 2009;
Hoang, Draine, & Lazarian, 2010; Ysard & Verstraete, 2010). Analyses that
ignore this component to attribute the observed emission only to synchrotron
radiation tend to over-predict the synchrotron amplitude at frequencies near
23 GHz, biasing the derived spectral index to flatter values when comparing to
lower frequencies and steeper values when comparing to higher frequencies.
A second systematic error can result from improper treatment of offsets in the
data. Measurements from radio surveys at frequencies below 20 GHz generally
include the absolute intensity (zero level) of the sky. The WMAP differential
radiometers are insensitive to any constant (monopole) intensity on the sky;
the zero level of the WMAP sky maps is set so that the map intensity in the
Galactic polar caps matches a cosecant fit to the mid-latitude sky (Bennett et
al., 2003; Hinshaw et al., 2009). Analyses that directly compare low-frequency
radio surveys to the WMAP data without subtracting a monopole component from
the radio data will miss the corresponding emission in the WMAP bands, biasing
the derived spectral index to steeper values.
Figure 1.— Toy model showing effect of improper zero level subtraction on the
derived spectral index between 408 MHz and 23 GHz.
The zero level bias can be significant. The Haslam et al. (1981) survey at 408
MHz is commonly used to model synchrotron emission. The North Galactic pole
has measured temperature $19\pm 3$ K at 408 MHz, while a $\csc|b|$ fit to the
same 408 MHz map predicts a polar contribution of only $5.1\pm 0.6$ K. Similar
results apply to the Southern hemisphere, where the measured polar cap
temperature of $21\pm 3$ K significantly exceeds the value $4.0\pm 0.5$ K
obtained from a $\csc|b|$ fit. Only 2.7 K of the difference can be attributed
to emission from the cosmic microwave background, leaving a large residual.
Figure 1 illustrates the bias induced by including this residual at 408 MHz
but excluding it from the WMAP data. We take the 408 MHz map, remove the 2.7 K
CMB monopole, and scale the remaining (radio) emission to 23 GHz using a
power-law index $\beta=-2.7$ to produce a toy model of the radio sky at 23
GHz. Following the WMAP processing, we then remove a monopole from the scaled
map so that the map temperature in the south polar cap matches the $\csc|b|$
fit. We then compute the bias in apparent spectral index by comparing the 408
MHz map to the scaled 23 GHz map before and after removing the scaled
monopole. Figure 1 shows the bias in spectral index at 23 GHz, binned by
Galactic latitude. Dis-similar treatment of the map zero level creates a
spatially varying bias $\Delta\beta\approx 0.15$, comparable to the total
spectral steepening predicted by the measured cosmic ray spectra. The bias is
largest in regions where the sky brightness is faintest, at high latitudes or
away from the Galactic center.
Table 1Sky Surveys Used for Synchrotron Analysis Frequency | Calibration | Offset | Relative
---|---|---|---
(GHz) | Uncertainty | Uncertainty (K) | Uncertaintya
0.022 | 0.05 | 5000 | 0.15
0.045 | 0.10 | 250 | 0.11
0.408 | 0.10 | 3.0 | 0.17
1.420 | 0.05 | 0.5 | 0.63
3.20 | 0.001 | 0.011 | 0.10
3.41 | 0.001 | 0.006 | 0.07
7.98 | 0.001 | 0.036 | 0.89
8.33 | 0.001 | 0.042 | 2.64
9.72 | 0.001 | 0.003 | 0.34
10.49 | 0.001 | 0.002 | 0.27
aQuadrature sum of calibration and offset uncertainties,
divided by the mid-latitude sky temperature.
Measurement uncertainties in the absolute level of the sky brightness can also
introduce bias in estimates of the synchrotron spectral index. Many of the
low-frequency surveys have uncertainty in the measured zero level approaching
30% of the polar cap brightness. As with the toy model above, such measurement
errors introduce spatially dependent biases that are largest where the sky
brightness is faintest. Minimizing uncertainty in the derived synchrotron
spectrum requires a combination of measurements with good sky coverage and
good control of offset uncertainty at frequencies where competing emission
sources are faint. No such ideal data set yet exists. In this paper, we model
the synchrotron spectral index and curvature using low-frequency radio surveys
with high sky coverage but large offset uncertainty, combined with higher-
frequency measurements with limited sky coverage but still useful offset
uncertainty.
## 2\. Sky Maps
We model synchrotron emission using radio surveys at 22 MHz (Roger et al.,
1999), 45 MHz (Maeda et al., 1999; Alvarez et al., 1997), 408 MHz (Haslam et
al., 1981), and 1420 MHz (Reich, Testori, & Reich, 2001; Reich & Reich, 1986).
These surveys have full or nearly-full sky coverage at frequencies where
Galactic radio emission is significant, with gain and zero-level systematics
controlled at the 10–20% level. We supplement the radio surveys with sky maps
from the Absolute Radiometer for Cosmology, Astrophysics, and Diffuse Emission
(ARCADE 2) instrument111 The ARCADE data are available at the Legacy Archive
for Microwave Background Data Analysis, http://lambda.gsfc.nasa.gov at 3, 8,
and 10 GHz (Kogut et al., 2011). The ARCADE 2 data observe both the Galactic
plane and mid-latitude regions ($|b|<40\arcdeg$) with sufficient control of
zero-level uncertainty to constrain the synchrotron curvature relative to the
lower-frequency radio surveys.
Table 1 summarizes the input sky maps. The increase in the offset uncertainty
at low frequency is compensated by a corresponding increase in sky brightness.
The final column shows the relative measurement uncertainty for a mid-latitude
region, defined as the ratio of the combined offset and calibration
uncertainty to the measured brightness at $(l,b)=(17\arcdeg,-35\arcdeg)$ after
removing the CMB monopole. The selected maps provide roughly uniform relative
sensitivity to synchrotron emission over 2.5 decades of frequency.
We convert all maps to units of antenna temperature and subtract the CMB
monopole at (thermodynamic) temperature 2.725 K from the measured sky
temperatures. We then convolve each map to the 11$\fdg$6 angular resolution of
the ARCADE 2 instrument. At frequencies of 10 GHz and below, both thermal dust
emission and spinning dust emission are negligible. Free-free emission,
however, can still be appreciable. We correct the convolved maps by scaling
the WMAP 7-year maximum entropy free-free model (Gold et al., 2011) to each
frequency using spectral index $-2.15$, convolving the scaled model to
11$\fdg$6 angular resolution, and subtracting the convolved model from each
sky survey. The resulting maps are dominated by synchrotron emission.
Figure 2.— Sky coverage for this analysis. The plot shows the 408 MHz sky
survey convolved to 11$\fdg$6 angular resolution. Pixels common to all 10
radio surveys are shown in color. The sky coverage is limited by the ARCADE 2
observations but includes the Galactic plane, mid-latitude sky, and portions
of the North Galactic Spur (radio Loop I).
## 3\. Analysis
The input sky maps define a data set $T(\hat{n},\nu)$ sampled at discrete
pixel directions $\hat{n}$ and 10 discrete frequencies $\nu$ ranging from 22
MHz to 10 GHz. We restrict the analysis to the 8% of the sky observed at all
10 frequencies. Figure 2 shows the resulting sky coverage. Within the common
sky coverage, we model synchrotron emission as a modified power law
$T(\hat{n},\nu)=A(\hat{n})\left(\frac{\nu}{\nu_{0}}\right)^{\beta+C\ln(\nu/\nu_{0})}$
(6)
with spectral index $\beta$ and curvature $C$ defined with respect to
reference frequency $\nu_{0}$ = 310 MHz. The adopted value for $\nu_{0}$
minimizes covariance between the fitted amplitude $A$ and spectral index
$\beta$, simplifying extrapolation to other frequencies.
For each pixel $\hat{n}$ we define a 10 by 10 data covariance matrix $M$ with
diagonal elements determined by the instrument noise, calibration and offset
uncertainty. The input maps are not all linearly independent. Measurements at
22 MHz used the 408 MHz map to determine the declination dependence of the
gain. We estimate the resulting correlation of spatial structure in the two
maps at 50%. The ARCADE 2 maps have independent instrument noise but share a
fraction of the offset uncertainty related to absolute thermometry uncertainty
and ground glint (Singal et al., 2011). All 10 maps share a common model for
free-free emission. We conservatively estimate the uncertainty in the free-
free correction at 30% of the free-free amplitude; however, the results do not
change significantly as the model free-free amplitude is varied by as much as
50%. Off-diagonal elements in $M$ include these effects.
Figure 3.— Sky temperatures and best-fit model (solid line) for a 4$\arcdeg$
diameter patch on the Galactic plane centered on the the brightest pixels in
the ARCADE 2 sky coverage. For clarity, the data are plotted relative to the
best-fit power-law model ($\beta=-2.56$) with zero curvature (dashed line).
All fits include the non-trivial covariance between individual data points.
The data are well described by a model with spectral index $\beta=-2.60\pm
0.04$ and curvature $C=-0.081\pm 0.028$. Figure 4.— Sky maps of best-fit
spectral parameters. Left to right: synchrotron amplitude, spectral index, and
curvature evaluated at reference frequency 310 MHz. The top panels show
results from the 10-frequency fit, while the bottom panels include the
spectral constraint at 23 GHz.
For each pixel, a least-squares minimization determines the best-fit
parameters $A$, $\beta$, and $C$. Figure 3 shows the measured temperature and
best-fit model for the brightest Galactic plane region
$(l,b)=(52\arcdeg,0\arcdeg)$ within the common sky coverage. The data show
evidence for spectral curvature, with best-fit values $\beta=-2.60\pm 0.04$
and $C=-0.081\pm 0.028$ evaluated at $\nu_{0}=310$ MHz. The spectral curvature
$C$ for this region is significant at approximately 3 standard deviations
compared to the baseline model with $C=0$.
We may extrapolate the spectral models to compare the results at frequencies
10 GHz and below to independent determinations of the spectral index using
WMAP data at frequencies 23 GHz and above. We compute the antenna temperature
of the modeled spectra to derive the effective power-law index for frequencies
near 23 GHz. Note that this is not equivalent to evaluating Eq. 6 at $\nu=23$
GHz, which would yield the scaling from 310 MHz to 23 GHz but not the power-
law index at 23 GHz. The mean for all 258 pixels in the common sky coverage is
$\beta_{23}=-3.02$ with standard deviation $0.22$.
The extrapolated value compares well with independent determinations of the
spectral index above 23 GHz. Kogut et al. (2004) analyze WMAP polarization
data to derive synchrotron spectral index $\beta=-3.2\pm 0.1$ averaged over
the full sky. Dunkley et al. (2009) use a Bayesian analysis of polarization
data and find the mean spectral index $\beta=-3.03\pm 0.04$ with pixel-to-
pixel standard deviation $0.25$ over the high-latitude sky. Gold et al. (2011)
use template fitting techniques to derive spectral index $\beta=-3.13$ between
23 and 33 GHz.
The mean spectral index derived from the 10 low-frequency radio surveys agrees
with the value derived from WMAP data at higher frequencies. Much of the
scatter in the extrapolated spectral indices results from pixels at high
latitude where the emission is faintest. The extrapolated index in these
pixels can reach unphysical values. We reduce the scatter in the fitted
spectral parameters by applying additional constraints using WMAP data at 23
GHz and above. The simplest such constraint, adding WMAP temperature data to
the multi-frequency fit, is problematic. Not only would the procedure need to
include additional free parameters to account for emission from thermal dust
or spinning dust (both negligible at lower frequencies), but each of the low-
frequency maps would require a correction to remove the monopole contribution
missing from the WMAP data. Although the WMAP zero level is clearly defined by
a $\csc|b|$ fit to mid-latitude data, the astrophysical interpretation of a
similar procedure applied to low-frequency radio surveys is less clear. The
coldest region of the radio sky is not at the Galactic poles, but at mid-
latitudes above the Galactic anti-center. Subtraction of too large a monopole
can leave unphysical negative residuals. Limited sky coverage exacerbates this
problem.
Figure 5.— Spectral index evaluated at 310 MHz. The dashed line shows the
distribution from the 10-frequency radio data while the solid line includes
the prior at 23 GHz.
We avoid these problems by using a constraint based on the spectral index
derived solely from WMAP data. For each pixel, we use the radio data (Table 1)
to fit the synchrotron amplitude $A(\hat{n})$ over a 2-dimensional grid in the
spectral parameters $\beta$ and $C$ (Eq. 6). At each grid point, we compute
the $\chi^{2}$ value $R^{T}M^{-1}R$ where $M^{-1}$ is the inverse covariance
matrix and $R$ is the difference vector between the measured and modeled
temperatures. We then use the spectral parameters $\beta$ and $C$ to evaluate
the power-law index at 23 GHz and compare the resulting value to a prior. We
use the difference between the extrapolated spectral index and the prior to
augment the $\chi^{2}$ at each grid point,
$\chi^{2}\rightarrow\chi^{2}+\left(\frac{\beta_{23}-\beta_{p}}{\sigma_{p}}\right)^{2},$
(7)
where $\beta_{23}$ is the model spectral index evaluated at 23 GHz, and
$\beta_{p}\pm\sigma_{p}~{}=~{}-3.1\pm 0.1$ is the prior at 23 GHz. The minimum
$\chi^{2}$ over the entire grid then defines the best-fit model at that pixel.
This allows inclusion of the spectral information derived from frequencies
above 23 GHz without confusion from either additional emission components
(thermal or spinning dust) above 23 GHz or the missing zero level in the WMAP
data.
Figure 6.— Spectral curvature at 310 MHz. The dashed line shows the
distribution from the 10-frequency radio data while the solid line includes
the prior at 23 GHz. The mean curvature $C=-0.052$ corresponds to a steepening
of the local spectral index by an amount $\Delta\beta=0.07$ every octave in
frequency.
Figure 4 shows the best-fit spectral parameter maps, while Figures 5 and 6
show the distribution of the best-fit values for the spectral index and
curvature. Including the constraint at 23 GHz, the best-fit spectral index has
mean value $-2.64$ and standard deviation $0.03$ at reference frequency
$\nu_{0}=310$ MHz. The best-fit curvature has mean value $-0.052$ with
standard deviation $0.005$. The corresponding spectral index at 23 GHz is
$\langle\beta_{23}\rangle=-3.09$ with standard deviation 0.05. Inclusion of
the prior at 23 GHz does not induce a significant shift in the mean for the
extrapolated spectral index, but does significantly reduce the pixel-to-pixel
scatter.
Comparison of the parameter distributions with and without the spectral
constraint at 23 GHz demonstrates that the addition of the spectral constraint
mainly affects the fitted curvature values (Figs. 5 and 6). The adopted value
for the spectral constraint comes from independent analysis of WMAP data and
is weighted toward regions of higher synchrotron intensity. We test whether
this creates a bias in the fitted curvature values by splitting the observed
sky coverage into two subsets of equal area, defined by the brightest and
faintest 50% of the fitted amplitudes at 310 MHz. Within each subset, we
compare the mean and standard deviation of the fitted curvature derived from
the 10-frequency fit or the enhanced fit including the constraint at 23 GHz.
The “bright” subset shows no shift in the mean spectral curvature when the 23
GHz constraint is added (although the scatter is significantly reduced). The
‘faint” subset shows both a reduction in scatter and a modest shift in the
mean value, with curvature parameter steepening from -0.034 to -0.049 when the
23 GHz constraint is included. This shift is less than one standard deviation:
given the limited sky coverage, the available radio data do not yet provide
significant evidence for spatial variation in the synchrotron curvature.
## 4\. Discussion
Radio data show statistically significant steepening of the synchrotron
spectrum from 22 MHz to 10 GHz. The nearly uniform relative uncertainty of the
selected data minimizes dependence of the fitted parameters on offset or
calibration errors at any one frequency. We test whether the best-fit
parameters are particularly sensitive to any one input map by repeating the
analysis after dropping one or two maps from the fit. We select either one
radio survey (22 MHz, 45 MHz, 408 MHz, or 1420 MHz) or a pair of ARCADE
frequency channels (3 GHz, 8 GHz, or 10 GHz) and repeat the fit after deleting
the corresponding elements from the data vector $T$ and covariance matrix $M$.
The resulting shift in either the spectral index or curvature parameters is
smaller than the pixel-to-pixel standard deviation using all 10 frequency
channels. Systematic errors in the offset or temperature calibration do not
appear to dominate the multi-frequency analysis.
The steepening of the synchrotron spectrum is broadly consistent with models
of cosmic ray propagation in the Galactic magnetic field. Jaffe et al. (2011)
combine radio observations with the galprop222http://galprop.stanford.edu
cosmic ray propagation code to model synchrotron emission on the Galactic
plane. They find a power-law index $-2.8<\beta<-2.74$ from 408 MHz to 2.3 GHz
and $-2.98<\beta<-2.91$ from 2.3 GHz to 23 GHz. The corresponding values for
the spectral steepening model presented here are $\beta=-2.76$ from 408 MHz to
2.3 GHz and $\beta=-2.97$ from 2.3 GHz to 23 GHz, in good agreement with the
cosmic ray model.
We may use the best-fit values in each pixel to predict the synchrotron
spectrum at higher frequencies where the emission is fainter and competing
sources stronger. Previous attempts to disentangle competing emission from
free-free, synchrotron, thermal dust, and spinning dust emission using WMAP
data have suffered from degeneracy between the synchrotron and spinning dust
emission, both of which are falling at frequencies above 33 GHz (see, e.g.,
the discussion in Gold et al. (2011)). Extending the synchrotron curvature
observed at lower frequencies into the millimeter band reduces confusion
between the spinning dust and synchrotron spectra and may facilitate
characterization of both the spatial distribution and frequency spectrum of
spinning dust emission in the interstellar medium.
Table 2Local Power-Law Spectral Index Frequency | Power-Law
---|---
(GHz) | Index
0.022 | -2.36
0.045 | -2.44
0.408 | -2.67
3.3 | -2.89
23 | -3.09
33 | -3.13
41 | -3.15
61 | -3.19
94 | -3.24
Table 2 shows the local power-law index $T\propto\nu^{\beta}$ at selected
frequency bands. The modeled spectrum steepens by $\Delta\beta=0.07$ every
octave in frequency, from $\beta=-2.67$ at 408 MHz to $\beta=-3.24$ at 94 GHz.
Note, however, that the spectral steepening observed at low frequencies can
not continue indefinitely. _Fermi_ measurements of the cosmic ray energy
spectrum are consistent with a single power law from energy 7 GeV to 1 TeV
(Abdo et al., 2009; Ackermann et al., 2010, 2012). If anything, the _Fermi_
data suggest a modest flattening of the cosmic ray energy spectrum at higher
energies, which would induce a positive curvature to the synchrotron spectrum
at frequencies above 23 GHz.
Figure 7.— Comparison of the synchrotron spectral index as a function of
frequency for different models. The break in the spectral index for the
Strong, Orlando, & Jaffe (2011) cosmic ray model at frequencies of a few GHz
is not reproduced by the ARCADE 2 observations at 3–10 GHz, which prefer
models with nearly constant spectral curvature.
Strong, Orlando, & Jaffe (2011) combine radio data at 22, 45, 150, 408, and
1420 MHz with WMAP data at 23 through 94 GHz and Fermi Large Area Telescope
cosmic ray measurements and the galprop code to estimate the magnetic field
intensity and synchrotron spectrum from 22 MHz to 94 GHz. Figure 7 compares
the resulting synchrotron spectral index (diffusion model with injection index
1.3) to the curvature model from this paper. The cosmic ray model analyzes a
limited latitude range $10\arcdeg<|b|<50\arcdeg$, which we follow in Figure 7
by excluding pixels at latitudes $|b|<10\arcdeg$. The two methods agree for
frequencies below 408 MHz (where they share common radio data) but differ at
higher frequencies. The cosmic ray model shows an increased synchrotron
curvature from 408 MHz to a few GHz, followed by a spectral break to near-
constant index $\beta\sim-3.1$ at higher frequencies. The ARCADE 2 data at
3–10 GHz do not reproduce these features, but are instead consistent with
constant spectral curvature from 22 MHz to 10 GHz (Fig 3).
Both models agree at frequencies near 23 GHz. The spectral index at 23 GHz
derived from the 10-frequency radio data without the 23 GHz prior is
consistent with independent measurements of the index above 23 GHz and with
the full radio fit including the 23 GHz prior. The radio data, taken alone, do
not support a break in the synchrotron spectrum at GHz frequencies. Comparison
of the radio fit to the cosmic ray model suggests that any spectral break in
the synchrotron spectrum must occur at frequencies above 23 GHz. Direct
confirmation of the synchrotron spectrum above 23 GHz remains a challenge.
## 5\. Conclusions
Radio data are consistent with a synchrotron spectrum that steepens with
frequency from 22 MHz to 10 GHz. Direct comparison of low-frequency radio
surveys with the WMAP data at 23 to 94 GHz is complicated both by the presence
of additional emission components at higher frequencies and by the subtraction
of a substantial monopole component of sky emission by the differential WMAP
instrument. The synchrotron spectral index at 23 GHz, derived using only
lower-frequency radio surveys, is consistent with the value derived
independently using only data at higher frequencies. We extend the radio data
by comparing the extrapolated index at 23 GHz to a prior based on higher-
frequency data. The combined data have mean spectral index $\beta=-2.64\pm
0.03$ and curvature $C=-0.052\pm 0.005$ at reference frequency 0.31 GHz. The
measured spectrum steepens by an amount $\Delta\beta=0.07$ every octave in
frequency. Comparison of the radio data to models including the cosmic ray
energy spectrum suggests that any break in the synchrotron spectrum must occur
at frequencies above 23 GHz.
This research is based upon work supported by the National Aeronautics and
Space Administration through the Science Mission Directorate under the
Astronomy and Physics Research and Analysis suborbital program.
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|
arxiv-papers
| 2012-05-17T20:30:52 |
2024-09-04T02:49:31.040276
|
{
"license": "Public Domain",
"authors": "A. Kogut",
"submitter": "Alan Kogut",
"url": "https://arxiv.org/abs/1205.4041"
}
|
1205.4192
|
A NEW APPROACH TO MODIFIED $q$-BERNSTEIN POLYNOMIALS FOR FUNCTIONS OF TWO
VARIABLES WITH THEIR GENERATING AND INTERPOLATION FUNCTIONS
Mehmet ACIKGOZ and Serkan ARACI
University of Gaziantep, Faculty of Arts and Science, Department of
Mathematics, 27310 Gaziantep, Turkey
acikgoz@gantep.edu.tr; mtsrkn@hotmail.com
Abstract
The aim of this paper is to give a new approach to modified $q$-Bernstein
polynomials for functions of two variables. By using these type polynomials,
we derive recurrence formulas and some new interesting identities related to
the second kind Stirling numbers and generalized Bernoulli polynomials.
Moreover, we give the generating function and interpolation function of these
modified $q$-Bernstein polynomials of two variables and also give the
derivatives of these polynomials and their generating function.
2000 Mathematics Subject Classification 11M06, 11B68, 11S40, 11S80, 28B99,
41A50.
Key Words and Phrases Generating function, Bernstein polynomial of two
variables, Bernstein operator of two variables, Shift difference operator,
$q$-difference operator, Second kind Stirling numbers, Generalized Bernoulli
polynomials, Mellin transformation, Interpolation function.
## 1\. Introduction, Definitions and Notations
In approximation theory, the Bernstein polynomials, named after their creater
S. N. Bernstein in 1912, have been studied by many researchers for a long
time. But nothing about generating function of Bernstein polynomials were
known in the literature. Recently, Simsek and Acikgoz, ([17]), constructed a
new generating function of ($q$-) Bernstein type polynomials based on the
$q$-analysis. They gave some new relations related to ($q$-) Bernstein type
polynomials, Hermite polynomials, Bernoulli polynomials of higher order and
the second kind Stirling numbers. By applying Mellin transformation to this
generating function they defined the interpolation function of ($q$-)
Bernstein type polynomials. They gave some relations and identities on these
polynomials. They constructed the generating function for classical Bernstein
polynomials, and for Bernstein polynomials for functions of two variables and
gave their properties (see [1], [2], [3], for details).
Throughout this paper, we use some notations like $\mathbb{N},$
$\mathbb{N}_{0}$ and $D,$ where $\mathbb{N}$ denotes the set of natural
numbers,
$\mathbb{N}_{0}:=\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$ and
$D=\left[0,1\right]$.
Let $C\left(D\times D\right)$ denotes the set of continuous functions on $D$.
For $f\in C\left(D\times D\right)$
$\displaystyle\mathbf{B}_{n,m}\left(f;x,y\right)$ $\displaystyle:$
$\displaystyle=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}f\left(\frac{k}{n},\frac{j}{m}\right)\binom{n}{k}\binom{m}{j}x^{k}y^{j}\left(1-x\right)^{n-k}\left(1-y\right)^{m-j}$
(1.1) $\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}f\left(\frac{k}{n},\frac{j}{m}\right)B_{k,j;n,m}\left(x,y\right)$
where $\binom{n}{k}=\frac{n\left(n-1\right)\cdots\left(n-k+1\right)}{k!}.$
Here $\mathbf{B}_{n,m}\left(f;x,y\right)$ is called the Bernstein operator of
two variables of order $n+m$ for $f$. For $k,j,n,m\in\mathbb{N}_{0}$, the
Bernstein polynomial of two variables of degree $n+m$ is defined by
$B_{k,j;n,m}\left(x,y\right)=\binom{n}{k}\binom{m}{j}x^{k}y^{j}\left(1-x\right)^{n-k}\left(1-y\right)^{m-j},$
(1.2)
where $x\in D$ and $y\in D$. Thus, throughout this work, we will assume that
$x\in D$ and $y\in D$. Then, we easily see the following
$B_{k,j;n,m}\left(x,y\right)=B_{k,n}\left(x,y\right)B_{j,m}\left(x,y\right)$
(1.3)
and they form a partition of unity; that is;
$\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}B_{k,j;n,m}\left(x,y\right)=1$
(1.4)
and by using the definition of Bernstein polynomials for functions of two
variables, it is not difficult to prove the property given above as
$\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}B_{k,n}\left(x,y\right)B_{j,m}\left(x,y\right)=1.$
(1.5)
Some Bernstein polynomials of two variables are given below:
$B_{0,0;1,0}\left(x,y\right)=\left(1-x\right),\text{
}B_{0,0;0,1}\left(x,y\right)=\left(1-y\right),\text{
}B_{0,0;1,1}\left(x,y\right)=\left(1-x\right)\left(1-y\right),$
$B_{0,1;1,1}\left(x,y\right)=y\left(1-x\right),\text{
}B_{1,0;1,1}\left(x,y\right)=x\left(1-y\right),\text{
}B_{1,1;1,1}\left(x,y\right)=xy\text{.}$
Also, $B_{k,j;n,m}\left(x,y\right)=0$ for $k>n$ or $j>m$, because
$\binom{n}{k}=0$ or $\binom{m}{j}=0.~{}$There are $nm+n+m+1,\ n+m$-th degree
Bernstein polynomials (see [3] and [6] for details).
Some researchers have used the Bernstein polynomials of two variables in
approximation theory (See [5], [6]). But no result was known anything about
the generating function of these polynomials. Note that for
$k,j,n,m\in\mathbb{N}_{0}$, we have
$\displaystyle\frac{\left(tx\right)^{k}\left(ty\right)^{j}e^{2t}}{k!j!e^{t\left(x+y\right)}}$
$\displaystyle=$
$\displaystyle\frac{t^{k}x^{k}t^{j}y^{j}}{k!j!}e^{t\left(1-x\right)}e^{t\left(1-y\right)}$
$\displaystyle=$
$\displaystyle\frac{x^{k}}{k!}\left(t^{k}\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{\left(1-x\right)^{n}}{n!}t^{n}\right)\frac{y^{j}}{j!}\left(t^{j}\mathop{\displaystyle\sum}\limits_{m=0}^{\infty}\frac{\left(1-y\right)^{m}}{m!}t^{m}\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{n=k}^{\infty}\mathop{\displaystyle\sum}\limits_{m=j}^{\infty}B_{k,j;n,m}\left(x,y\right)\frac{t^{n}}{n!}\frac{t^{m}}{m!}$
From the above, we obtain the generating function for
$B_{k,j;n,m}\left(x,y\right)$ as follows:
$F_{k,j}\left(t;x,y\right)=\frac{\left(tx\right)^{k}\left(ty\right)^{j}e^{t\left(2-\left(x+y\right)\right)}}{k!j!}=\mathop{\displaystyle\sum}\limits_{n=k}^{\infty}\mathop{\displaystyle\sum}\limits_{m=j}^{\infty}B_{k,j;n,m}\left(x,y\right)\frac{t^{n}}{n!}\frac{t^{m}}{m!},$
(1.6)
where $k,j,n,m\in\mathbb{N}_{0}$. We notice that,
$B_{k,j;n,m}\left(x,y\right)=\left\\{\begin{array}[]{cccccc}\binom{n}{k}\binom{m}{j}x^{k}y^{j}\left(1-x\right)^{n-k}\left(1-y\right)^{m-j}&,&\text{if}&n\geq
k&\text{and}&m\geq j\\\ 0&,&\text{if}&n<k&\text{or}&m<j\end{array}\right.$
for $n,k,m,j\in\mathbb{N}_{0}$ (for details, see [2]).
Let $q\in\left(0,1\right)$. Then, $q$-integer of $x$ by
$[x]_{q}:=\frac{1-q^{x}}{1-q}$ and
$[x]_{-q}:=\frac{1-\left(-q\right)^{x}}{1+q}$ ( See [7]-[17] for details).
Note that $\underset{q\rightarrow 1}{\lim}[x]_{q}=x$. [7] motivated the
authors to write this paper and we have extended the results given in that
paper to modified $q$-Bernstein polynomials of two variables.
## 2\. The Modified $q$-Bernstein Polynomials for Functions of two Variables
For $0\leq k\leq n$ and $0\leq j\leq m$, the $q$-Bernstein polynomials of
degree $n+m$ are defined by
$B_{k,j;n,m}\left(x,y;q\right)=\left\\{\begin{array}[]{cccccc}\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}&,&\text{if}&n\geq
k&\text{and}&m\geq j\\\ 0&,&\text{if}&n<k&\text{or}&m<j\end{array}\right..$
(2.1)
For $q\in\left(0,1\right)$, consider the $q$-extension of (1.6) as follows:
$\displaystyle F_{k,j}\left(t,q;x,y\right)$ $\displaystyle=$
$\displaystyle\frac{\left(t[x]_{q}\right)^{k}\left(t[y]_{q}\right)^{j}}{k!j!}e^{t\left([1-x]_{q}+[1-y]_{q}\right)}$
(2.2) $\displaystyle=$
$\displaystyle\frac{[x]_{q}^{k}}{k!}\left(\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{[1-x]_{q}^{n}}{n!}t^{n+k}\right)\frac{[y]_{q}^{j}}{j!}\left(\mathop{\displaystyle\sum}\limits_{m=0}^{\infty}\frac{[1-y]_{q}^{m}}{m!}t^{m+j}\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{n=k}^{\infty}\mathop{\displaystyle\sum}\limits_{m=j}^{\infty}\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}\frac{t^{n}}{n!}\frac{t^{m}}{m!}$
where $k,j,n,m\in\mathbb{N}_{0}$. Note that $\underset{q\rightarrow
1}{\lim}F_{k,j}\left(t,q:x,y\right)=F_{k,j}\left(t;x,y\right).$
###### Definition 1.
The modified $q$-Bernstein polynomials for functions of two variables is
defined by means of the following generating function:
$F_{k,j}\left(t,q;x,y\right)=\frac{\left(t[x]_{q}\right)^{k}\left(t[y]_{q}\right)^{j}}{k!j!}e^{t\left([1-x]_{q}+[1-y]_{q}\right)}=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\mathop{\displaystyle\sum}\limits_{m=0}^{\infty}B_{k,j;n,m}\left(x,y;q\right)\frac{t^{n}}{n!}\frac{t^{m}}{m!}$
(2.3)
where $k,j,n,m\in\mathbb{N}_{0}$.
By comparing the coefficients of (2.2) and (2.3), we obtain a formula for
modified $q$-Bernstein polynomials of two variables given in the following
theorem:
###### Theorem 1.
For $k,j,n,m\in\mathbb{N}_{0}$, then, we have
$B_{k,j;n,m}\left(x,y;q\right)=\left\\{\begin{array}[]{cccccc}\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}&,&\text{if}&n\geq
k&\text{and}&m\geq j\\\ 0&,&\text{if}&n<k&\text{or}&m<j\end{array}\right..$
(2.4)
###### Theorem 2.
(Recurrence Formula for $B_{k,j;n,m}\left(x,y;q\right)$) For
$k,j,n,m\in\mathbb{N}_{0}$, we have
$\displaystyle B_{k,j;n,m}\left(x,y;q\right)$ $\displaystyle=$
$\displaystyle[1-x]_{q}[1-y]_{q}B_{k,j;n-1,m-1}\left(x,y;q\right)+[1-x]_{q}[y]_{q}B_{k,j-1;n-1,m-1}\left(x,y;q\right)$
$\displaystyle+[x]_{q}[1-y]_{q}B_{k-1,j;n-1,m-1}\left(x,y;q\right)+[x]_{q}[y]_{q}B_{k-1,j-1;n-1,m-1}\left(x,y;q\right).$
###### Proof.
By using the definition of Bernstein polynomials for functions of two
variables defined by (2.4), we have
$\displaystyle B_{k,j;n,m}\left(x,y;q\right)$ $\displaystyle=$
$\displaystyle\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle\left[\binom{n-1}{k}+\binom{n-1}{k-1}\right]\left[\binom{m-1}{j}+\binom{m-1}{j-1}\right][x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle[1-x]_{q}[1-y]_{q}B_{k,j;n-1,m-1}\left(x,y;q\right)+[1-x]_{q}[y]_{q}B_{k,j-1;n-1,m-1}\left(x,y;q\right)$
$\displaystyle+[x]_{q}[1-y]_{q}B_{k-1,j;n-1,m-1}\left(x,y;q\right)+[x]_{q}[y]_{q}B_{k-1,j-1;n-1,m-1}\left(x,y;q\right).$
###### Theorem 3.
For $k,j,n,m\in\mathbb{N}_{0}$, we get
$B_{n-k,m-j;n,m}\left(1-x,1-y;q\right)=B_{k,j;n,m}\left(x,y;q\right)$ (2.5)
and
$\mathbf{B}_{n,m}\left(1:x,y,q\right)=\left(1+\left(1-q\right)[x]_{q}[1-x]_{q}\right)^{n}\times\left(1+\left(1-q\right)[y]_{q}[1-y]_{q}\right)^{m}.$
###### Proof.
Let $f$ be a continuous function of two variables on $D\times D$. Then the
modified $q$-Bernstein operator of order $n+m$ for $f$ is defined by
$\mathbf{B}_{n,m}\left(f:x,y,q\right)=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}f\left(\frac{k}{n},\frac{j}{m}\right)B_{k,j;n,m}\left(x,y;q\right)$
(2.6)
where $0\leq x\leq 1,$ $0\leq y\leq 1$, $n,m\in\mathbb{N}.$ From Theorem 1 and
the definition of modified $q$-Bernstein operator given by (2.6) for
$f\left(x,y\right)=xy$, we have
$\displaystyle\mathbf{B}_{n,m}\left(f:x,y,q\right)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}f\left(\frac{k}{n},\frac{j}{m}\right)\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[1-x]_{q}^{n-k}[y]_{q}^{k}[1-y]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle[x]_{q}\left(1-[1-x]_{q}[x]_{q}\left(q-1\right)\right)^{n-1}\times[y]_{q}\left(1-[1-y]_{q}[y]_{q}\left(q-1\right)\right)^{m-1}$
$\displaystyle=$ $\displaystyle
f\left([x]_{q},[y]_{q}\right)\left(1+\left(1-q\right)[x]_{q}[1-x]_{q}\right)^{n-1}\times\left(1+\left(1-q\right)[y]_{q}[1-y]_{q}\right)^{m-1}$
From Theorem 1, we have
$\displaystyle\mathbf{B}_{n,m}\left(1:x,y,q\right)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=0}^{n}\mathop{\displaystyle\sum}\limits_{j=0}^{m}B_{k,j;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=0}^{n}B_{k,n}\left(x,q\right)\mathop{\displaystyle\sum}\limits_{j=0}^{m}B_{j,m}\left(y,q\right)$
$\displaystyle=$
$\displaystyle\left(1+\left(1-q\right)[x]_{q}[1-x]_{q}\right)^{n}\left(1+\left(1-q\right)[y]_{q}[1-y]_{q}\right)^{m}.$
The modified $q$-Bernstein polynomials of two variables are symmetric
polynomials:
$\displaystyle B_{n-k,m-j;n,m}\left(1-x,1-y;q\right)$ $\displaystyle=$
$\displaystyle\binom{n}{n-k}[x]_{q}^{k}[1-x]_{q}^{n-k}\binom{n}{m-j}[y]_{q}^{j}[1-y]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle\binom{n}{k}[1-x]_{q}^{k}[x]_{q}^{n-k}\binom{n}{j}[1-y]_{q}^{j}[y]_{q}^{m-j}$
$\displaystyle=$ $\displaystyle B_{k,j;n,m}\left(x,y;q\right).$
by replacing $k$ by $n-k$ and $j$ by $m-j$.
###### Theorem 4.
For $\xi,\rho\in\mathbb{C}$, and for $n,m\in\mathbb{N}$, then, we procure
$B_{k,j;n,m}\left(x,y;q\right)=-\frac{n!m!}{4\pi^{2}}\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{\left(\left[x\right]_{q}\xi\right)^{k}\left(\left[y\right]_{q}\rho\right)^{j}}{k!j!}e^{\left(\left[1-x\right]_{q}\xi+\left[1-y\right]_{q}\rho\right)}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}$
(2.7)
where $C$ is a circle around the origin and integration is in the positive
direction.
###### Proof.
By using the definition of the modified $q$-Bernstein polynomials of two
variables and the basic theory of complex analysis including Laurent series
that
$\displaystyle\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{\left(\left[x\right]_{q}\xi\right)^{k}\left(\left[y\right]_{q}\rho\right)^{j}}{k!j!}e^{\left(\left[1-x\right]_{q}\xi+\left[1-y\right]_{q}\rho\right)}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{l=0}^{\infty}\mathop{\displaystyle\sum}\limits_{r=0}^{\infty}\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{B_{k,l}\left(x,q\right)\xi^{l}}{l!}\frac{B_{j,r}\left(y,q\right)\rho^{r}}{r!}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}$
$\displaystyle=$ $\displaystyle\left(2\pi
i\right)^{2}\left(\frac{B_{k,j;n,m}\left(x,y;q\right)}{n!m!}\right)\text{.}$
By using (2.7 ) and (4), we obtain
$\frac{n!m!}{\left(2\pi
i\right)^{2}}\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{\left(\left[x\right]_{q}\xi\right)^{k}}{k!}\frac{\left(\left[y\right]_{q}\rho\right)^{j}}{j!}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}=B_{k,j;nm}\left(x,y;q\right)$
and
$\displaystyle\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{\left(\left[x\right]_{q}\xi\right)^{k}}{k!}\frac{\left(\left[y\right]_{q}\rho\right)^{j}}{j!}e^{\left(\left[1-x\right]_{q}\xi+\left[1-y\right]_{q}\rho\right)}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}$
$\displaystyle=$ $\displaystyle\left(2\pi
i\right)^{2}\left(\frac{\left[x\right]_{q}^{k}\left[y\right]_{q}^{j}\left[1-x\right]_{q}^{n-k}\left[1-y\right]_{q}^{m-j}}{k!j!\left(n-k\right)!\left(m-j\right)!}\right)\text{.}$
We also obtain from (2.5) and (4) that
$\displaystyle\frac{n!m!}{\left(2\pi
i\right)^{2}}\mathop{\displaystyle\oint}\limits_{C}\mathop{\displaystyle\oint}\limits_{C}\frac{\left(\left[x\right]_{q}\xi\right)^{k}\left(\left[y\right]_{q}\rho\right)^{j}}{k!j!}e^{\left(\left[1-x\right]_{q}\xi+\left[1-y\right]_{q}\rho\right)}\frac{d\xi}{\xi^{n+1}}\frac{d\rho}{\rho^{m+1}}$
$\displaystyle=$
$\displaystyle\binom{n}{k}\binom{m}{j}\left[x\right]_{q}^{k}\left[1-x\right]_{q}^{n-k}\left[y\right]_{q}^{j}\left[1-y\right]_{q}^{m-j}.$
Therefore we see that from (4) and (4) that
$B_{k,j;n,m}\left(x,y;q\right)=\binom{n}{k}\binom{m}{j}\left[x\right]_{q}^{k}\left[1-x\right]_{q}^{n-k}\left[y\right]_{q}^{j}\left[1-y\right]_{q}^{m-j}.$
###### Theorem 5.
(The Derivative Formula for $B_{k,j;n,m}\left(x,y;q\right)$) For
$k,j,n,m\in\mathbb{N}$, then, we derive the following
$\displaystyle\frac{\partial^{2}}{\partial x\partial
y}\left(B_{k,j;n,m}\left(x,y;q\right)\right)$ $\displaystyle=$ $\displaystyle
nm(q^{x+y}B_{k-1,j-1;n-1,m-1}\left(x,y;q\right)-q^{x-y+1}B_{k-1,j;n-1,m-1}\left(x,y;q\right)$
$\displaystyle-q^{1-x+y}B_{k,j-1;n-1,m-1}\left(x,y;q\right)+q^{2-\left(x+y\right)}B_{k,j;n-1,m-1}\left(x,y;q\right))\frac{\ln^{2}q}{\left(q-1\right)^{2}}.$
###### Proof.
Using the definition of modified $q$-Bernstein polynomials for functions of
two variables and the property (1.3), we have
$\frac{\partial^{2}}{\partial x\partial
y}\left(B_{k,j;n,m}\left(x,y;q\right)\right)=\frac{\partial^{2}}{\partial
x\partial
y}\left(B_{k,n}\left(x;q\right)B_{j,m}\left(y;q\right)\right)=\frac{d}{dx}\left(B_{k,n}\left(x;q\right)\right)\frac{d}{dy}\left(B_{j,m}\left(y;q\right)\right)$
and after some calculations, the proof is complete.
Therefore, we can write the modified $q$-Bernstein polynomials for functions
of two variables as a linear combination of polynomials of higher order as
follows:
###### Theorem 6.
For $k,j,n,m\in\mathbb{N}_{0}$, we have
$\displaystyle\left(1+\left(1-q\right)[x]_{q}\left[1-x\right]_{q}\right)\left(1+\left(1-q\right)[y]_{q}\left[1-y\right]_{q}\right)B_{k,j;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\left(\frac{n-k+1}{n+1}\right)\left(\frac{m-j+1}{m+1}\right)B_{k,j;n+1,m+1}\left(x,y;q\right)+\left(\frac{n-k+1}{n+1}\right)\left(\frac{j+1}{m+1}\right)B_{k,j+1;n+1,m+1}\left(x,y;q\right)$
$\displaystyle+\left(\frac{k+1}{n+1}\right)\left(\frac{m-j+1}{m+1}\right)B_{k+1,j;n+1,m+1}\left(x,y;q\right)+\left(\frac{k+1}{n+1}\right)\left(\frac{j+1}{m+1}\right)B_{k+1,j+1;n+1,m+1}\left(x,y;q\right).$
###### Proof.
It follows after expanding the series and some algebraic operations.
###### Theorem 7.
For $k,j,n,m\in\mathbb{N}_{0}$, we have
$B_{k,j;n,m}\left(x,y;q\right)=\left(\frac{n-k+1}{k}\right)\left(\frac{m-j+1}{j}\right)\left(\frac{[x]_{q}[y]_{q}}{[1-x]_{q}[1-y]_{q}}\right)B_{k-1,j-1;n,m}\left(x,y;q\right).$
###### Proof.
To prove this theorem, we start with the right hand side:
$\displaystyle\left(\frac{n-k+1}{k}\right)\left(\frac{m-j+1}{j}\right)\left(\frac{[x]_{q}[y]_{q}}{[1-x]_{q}[1-y]_{q}}\right)B_{k-1,j-1;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\frac{n!}{\left(n-k\right)!k!}.\frac{m!}{\left(m-j\right)!j!}\left(\frac{[x]_{q}[y]_{q}}{[1-x]_{q}[1-y]_{q}}\right)[x]_{q}^{k-1}[y]_{q}^{j-1}[1-x]_{q}^{n-k+1}[1-y]_{q}^{m-j+1}$
$\displaystyle=$
$\displaystyle\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}[1-x]_{q}^{n-k}[1-y]_{q}^{m-j}=B_{k,j;n,m}\left(x,y;q\right).$
###### Theorem 8.
For $k,j,n,m\in\mathbb{N}_{0}$, we obtain
$B_{k,j;n,m}\left(x,y;q\right)=\mathop{\displaystyle\sum}\limits_{l=k}^{n}\mathop{\displaystyle\sum}\limits_{r=j}^{m}\binom{n}{l}\binom{l}{k}\binom{m}{j}\binom{j}{r}\left(-1\right)^{l-k+r-j}q^{\left(l-k\right)\left(1-x\right)+\left(r-j\right)\left(1-y\right)}[x]_{q}^{l}[y]_{q}^{r}.$
###### Proof.
From the definition of modified $q$-Bernstein polynomials of two variables and
binomial theorem with $k,j,n,m\in\mathbb{N}_{0}$, we have
$\displaystyle B_{k,j;n,m}\left(x,y;q\right)$ $\displaystyle=$
$\displaystyle\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[1-x]_{q}^{n-k}[y]_{q}^{j}[1-y]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle\binom{n}{k}\binom{m}{j}[x]_{q}^{k}[y]_{q}^{j}\left(1-q^{1-x}[x]\right)^{n-k}\left(1-q^{1-y}[y]\right)^{m-j}$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{l=k}^{n}\mathop{\displaystyle\sum}\limits_{r=j}^{m}\binom{n}{l}\binom{l}{k}\binom{m}{j}\binom{j}{r}\left(-1\right)^{l-k+r-j}q^{\left(l-k\right)\left(1-x\right)+\left(r-j\right)\left(1-y\right)}[x]_{q}^{l}[y]_{q}^{r}.$
###### Theorem 9.
The following identity
$\left([x]_{q}[y]_{q}\right)^{l}=\frac{1}{\left(\left[1-x\right]_{q}+\left[x\right]_{q}\right)^{n-l}\left(\left[1-y\right]_{q}+\left[y\right]_{q}\right)^{m-l}}\mathop{\displaystyle\sum}\limits_{k=l}^{n}\mathop{\displaystyle\sum}\limits_{j=l}^{m}\frac{\binom{k}{l}\binom{j}{l}}{\binom{n}{l}\binom{m}{l}}B_{k,j;n,m}\left(x,y;q\right)$
is true.
###### Proof.
We easily see that from the property of the modified $q$-Bernstein polynomials
of two variables that
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=1}^{n}\mathop{\displaystyle\sum}\limits_{j=1}^{m}\frac{kj}{nm}B_{k,j;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=1}^{n}\mathop{\displaystyle\sum}\limits_{j=1}^{m}\binom{n-1}{k-1}\binom{m-1}{j-1}\left[x\right]_{q}^{k}\left[y\right]_{q}^{j}\left[1-x\right]_{q}^{n-k}\left[1-y\right]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle\left[x\right]_{q}\left[y\right]_{q}\left(\left[x\right]_{q}+\left[1-x\right]_{q}\right)^{n-1}\left(\left[y\right]_{q}+\left[1-y\right]_{q}\right)^{m-1}$
and that
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=2}^{n}\mathop{\displaystyle\sum}\limits_{j=2}^{m}\frac{\binom{k}{2}\binom{j}{2}}{\binom{n}{2}\binom{m}{2}}B_{k,j;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=2}^{n}\mathop{\displaystyle\sum}\limits_{j=2}^{m}\binom{n-2}{k-2}\binom{m-2}{j-2}\left[x\right]_{q}^{k}\left[y\right]_{q}^{j}\left[1-x\right]_{q}^{n-k}\left[1-y\right]_{q}^{m-j}$
$\displaystyle=$
$\displaystyle\left[x\right]_{q}^{2}\left[y\right]_{q}^{2}\left(\left[x\right]_{q}+\left[1-x\right]_{q}\right)^{n-2}\left(\left[y\right]_{q}+\left[1-y\right]_{q}\right)^{m-2}$
Continuing this way, we have
$\mathop{\displaystyle\sum}\limits_{k=l}^{n}\mathop{\displaystyle\sum}\limits_{j=l}^{m}\frac{\binom{k}{l}\binom{j}{l}}{\binom{n}{l}\binom{m}{l}}B_{k,j;n,m}\left(x,y;q\right)=\left[x\right]_{q}^{l}\left[y\right]_{q}^{l}\left(\left[x\right]_{q}+\left[1-x\right]_{q}\right)^{n-l}\left(\left[y\right]_{q}+\left[1-y\right]_{q}\right)^{m-l}$
and after some algebraic operations, we obtain the desired result.
We see that from the theorem above, it is possible to write
$\left([x]_{q}[y]_{q}\right)^{k}$ as a linear combination of the two variables
modified $q$-Bernstein polynomials.
For $k\in\mathbb{N}_{0}$, the Bernoulli polynomials of degree $k$ are defined
by
$\left(\frac{t}{e^{t}-1}\right)^{k}e^{xt}=\underset{k-times}{\underbrace{\left(\frac{t}{e^{t}-1}\right)\times\cdots\times\left(\frac{t}{e^{t}-1}\right)}}e^{xt}=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}B_{n}^{(k)}\left(x\right)\frac{t^{n}}{n!},$
and $B_{n}^{\left(k\right)}=B_{n}^{(k)}\left(0\right)$ are called the $n$-th
Bernoulli numbers of order $k$. It is well known that the second kind Stirling
numbers are defined by
$\frac{\left(e^{t}-1\right)^{k}}{k!}:=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}S\left(n,k\right)\frac{t^{n}}{n!}$
(2.11)
for $k\in\mathbb{N}$ (see [7]). By using the above relations we can give the
following theorem:
###### Theorem 10.
For $k,j,n,m\in\mathbb{N}_{0}$, we have
$\displaystyle B_{k,j;n,m}\left(x,y;q\right)$ $\displaystyle=$
$\displaystyle[x]_{q}^{k}[y]_{q}^{j}\mathop{\displaystyle\sum}\limits_{l=0}^{n}\mathop{\displaystyle\sum}\limits_{r=0}^{m}\binom{n}{l}\binom{m}{r}$
$\displaystyle\times
B_{l}^{\left(k\right)}\left(\left[1-x\right]_{q}\right)B_{r}^{\left(j\right)}\left(\left[1-y\right]_{q}\right)S\left(n-l,k\right)S\left(m-r,j\right).$
###### Proof.
By using the generating function of modified $q$-Bernstein polynomials of two
variables, we have
$\displaystyle\frac{\left(t[x]_{q}\right)^{k}\left(t[y]_{q}\right)^{j}}{k!j!}e^{t\left([1-x]_{q}+[1-y]_{q}\right)}=[x]_{q}^{k}[y]_{q}^{j}\left(\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}S\left(n,k\right)\frac{t^{n}}{n!}\right)\left(\mathop{\displaystyle\sum}\limits_{m=0}^{\infty}S\left(m,j\right)\frac{t^{m}}{m!}\right)$
$\displaystyle\times\left(\mathop{\displaystyle\sum}\limits_{l=0}^{\infty}B_{l}^{\left(k\right)}\left([1-x]_{q}\right)\frac{t^{l}}{l!}\right)\left(\mathop{\displaystyle\sum}\limits_{r=0}^{\infty}B_{r}^{\left(j\right)}\left([1-y]_{q}\right)\frac{t^{r}}{r!}\right)$
$\displaystyle=$ $\displaystyle\mathop{\displaystyle\sum}\limits_{n\geq
k}\mathop{\displaystyle\sum}\limits_{m\geq
j}B_{k,j;n,m}\left(x,y;q\right)\frac{t^{n}}{n!}\frac{t^{m}}{m!}$
by using the Cauchy product. By comparing last two relations, we have the
desired result.
Let $\Delta$ be the shift difference operator defined by $\Delta
f\left(x\right)=f\left(x+1\right)-f\left(x\right)$. By using the iterative
method we have
$\Delta^{n}f\left(0\right)=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\binom{n}{k}\left(-1\right)^{n-k}f\left(k\right),$
(2.12)
for $n\in\mathbb{N}$.
$\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}S\left(n,k\right)\frac{t^{n}}{n!}=\frac{1}{k!}\mathop{\displaystyle\sum}\limits_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}e^{lt}=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\left\\{\frac{1}{k!}\mathop{\displaystyle\sum}\limits_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}l^{n}\right\\}\frac{t^{n}}{n!}=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{\Delta^{k}0^{n}}{k!}\frac{t^{n}}{n!}.$
By comparing the coefficients on both sides above, we have
$S\left(n,k\right)=\frac{\Delta^{k}0^{n}}{k!}$ (2.13)
for $n,k\in\mathbb{N}_{0}$. By using the equations (2.11) and (2.12), we
obtain the following relation
$\displaystyle B_{k,j;n,m}\left(x,y;q\right)$ $\displaystyle=$
$\displaystyle[x]_{q}^{k}[y]_{q}^{j}\mathop{\displaystyle\sum}\limits_{l=0}^{n}\mathop{\displaystyle\sum}\limits_{r=0}^{m}\binom{n}{l}\binom{m}{r}$
(2.14) $\displaystyle\times
B_{l}^{\left(k\right)}\left([1-x]_{q}\right)B_{r}^{\left(j\right)}\left([1-y]_{q}\right)\frac{\Delta^{k}0^{n-l}}{k!}\frac{\Delta^{j}0^{m-r}}{j!}$
which is the relation of the $q$-Bernstein polynomials of two variables in
terms of Bernoulli polynomials of order $k$ and second Stirling numbers with
shift difference operator.
Let $\left(Eh\right)\left(x\right)=h\left(x+1\right)$ be the shift operator.
Then the $q$-difference operator is defined by
$\Delta_{q}^{n}=\mathop{\displaystyle\prod}\limits_{j=0}^{n-1}\left(E-q^{j}I\right)$
(2.15)
where $I$ is and identity operator ( See [7] ).
For $f\in C[0,1]$ and $n\in\mathbb{N}$, we have
$\Delta_{q}^{n}f\left(0\right)=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\binom{n}{k}_{q}\left(-1\right)^{k}q^{\binom{n}{2}}f\left(n-k\right),$
(2.16)
where$\binom{n}{k}_{q}$ is called the Gaussian binomial coefficients, which
are defined by
$\binom{n}{k}_{q}=\frac{[x]_{q}[x-1]_{q}\cdots[x-k+1]_{q}}{[k]_{q}!}.$ (2.17)
###### Theorem 11.
For $n,m,l,r\in\mathbb{N}_{0}$, we have
$\displaystyle\frac{1}{\left(\left[x\right]_{q}+\left[1-x\right]_{q}\right)^{n-l}\left(\left[y\right]_{q}+\left[1-y\right]_{q}\right)^{m-l}}\mathop{\displaystyle\sum}\limits_{k=l}^{n}\mathop{\displaystyle\sum}\limits_{j=l}^{m}\frac{\binom{k}{l}\binom{j}{l}}{\binom{n}{l}\binom{m}{l}}B_{k,j;n,m}\left(x,y;q\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{k=0}^{l}\mathop{\displaystyle\sum}\limits_{j=0}^{l}q^{\binom{k}{2}+\binom{j}{2}}\binom{x}{k}\binom{y}{j}\left[k\right]_{q}!\left[j\right]_{q}!S\left(l,k;q\right)S\left(l,j;q\right).$
###### Proof.
Let $F_{q}\left(t\right)$ be the generating function of the $q$-extension of
the second kind Stirling numbers as follows:
$F_{q}\left(t\right):=\frac{q^{-\binom{k}{2}}}{[k]_{q}!}\mathop{\displaystyle\sum}\limits_{j=0}^{k}\left(-1\right)^{k-j}\binom{k}{j}_{q}q^{\binom{k-j}{2}}e^{[i]_{q}t}=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}S\left(n,k;q\right)\frac{t^{n}}{n!}$
From the above, we have
$S\left(n,k;q\right)=\frac{q^{-\binom{k}{2}}}{[k]_{q}!}\mathop{\displaystyle\sum}\limits_{j=0}^{k}\left(-1\right)^{j}q^{\binom{j}{2}}\binom{k}{j}_{q}[k-j]_{q}^{n}=\frac{q^{-\binom{k}{2}}}{[k]_{q}!}\Delta_{q}^{k}0^{n}$
where $[k]_{q}!=[k]_{q}[k-1]_{q}\cdots[2]_{q}[1]_{q}.$ It is easy to see that
$[x]_{q}^{n}=\mathop{\displaystyle\sum}\limits_{k=0}^{n}q^{\binom{k}{2}}\binom{x}{k}_{q}[k]_{q}!S\left(n,k;q\right)$
(2.18)
by similar way
$[y]_{q}^{j}=\mathop{\displaystyle\sum}\limits_{r=0}^{j}q^{\binom{r}{2}}\binom{y}{r}_{q}[r]_{q}!S\left(j,r;q\right).$
(2.19)
We have above equality. Then, we obtain the desired result in Theorem from the
equations (2.18), (2.19) and Theorem 7.
## 3\. Interpolation Function of Modified q-Bernstein Polynomial for
Functions of Two Variables
For $s\in\mathbb{C}$, and $x\neq 1$, $y\neq 1$, by applying the Mellin
transformation to generating function of Bernstein polynomials of two
variables, we get
$\displaystyle S_{q}\left(s,k,j;x,y\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(s\right)}\mathop{\displaystyle\int}\limits_{0}^{\infty}F_{k,j}\left(-t,q;x,y\right)t^{s-k-j-1}dt$
(3.1) $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k+j}[x]_{q}^{k}[y]_{q}^{j}}{k!j!}\left([1-x]_{q}+[1-y]_{q}\right)^{-s}.$
By using the equation (3.1), we define the interpolation function of the
polynomials $B_{k,j;n,m}\left(x,y;q\right)$ as follows:
###### Definition 2.
Let $s\in\mathbb{C}$ and $x\neq 1$, $y\neq 1,$ we define
$S_{q}\left(s,k,j;x,y\right)=\frac{[x]_{q}^{k}[y]_{q}^{j}}{k!j!}\left(-1\right)^{k+j}\left([1-x]_{q}+[1-y]_{q}\right)^{-s}.$
(3.2)
By using (3.2), we have $S_{q}\left(s,k,j;x,y\right)\rightarrow
S\left(s,k,j;x,y\right)$ as $q\rightarrow 1.$ Thus one has
$S\left(s,k,j;x,y\right)=\frac{\left(-1\right)^{k+j}}{k!j!}x^{k}y^{j}\left(2-\left(x+y\right)\right)^{-s}.$
(3.3)
By substituting $x=1$ and $y=1$ into the above, we have
$S\left(s,k,j;x,y\right)=\infty$.
We now evaluate the $m$th $s$-derivatives of $S\left(s,k,j;x,y\right)$ as
follows:
$\frac{\partial^{m}}{\partial
s^{m}}S\left(s,k,j;x,y\right)=\log^{m}\left(\frac{1}{2-\left(x+y\right)}\right)S\left(s,k,j;x,y\right)$
(3.4)
where $x\neq 1$ and $y\neq 1.$
## References
* [1] Acikgoz, M., and Aracı, S., On the generating function of the Bernstein polynomials, Numerical Analysis and Applied Mathematics, International conference 2010, pp. 1141-1143.
* [2] Acikgoz, M., and Aracı, S., New generating function of Bernstein type polynomials for two variables, Numerical Analysis and Applied Mathematics, International conference 2010, pp. 1133-1136.
* [3] Acikgoz, M., and Aracı, S., A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics Modelling and Simulation,2010, vol. 1, no. 1(2), ISSN 1913-8342, pp. 10-14.
* [4] Acikgoz, M., and Simsek, M., On multiple interpolation functions of the Nörlund-type $q$-Euler polynomials, Abstr. Appl. Anal. 2009, Art. ID 382574, 14 pp.
* [5] Buyukyazici, İ., and İbikli, E., Bernstein polynomials of two variable functions, Graduate School of Natural and Applied Sciences, Department of Mathematics, 1999, 49 pages, Ankara, Turkey.
* [6] Buyukyazici, İ., and İbikli, E., The approximation properties of generalized Bernstein polynomials of two variables, Applied Math. and Comput. 156 (2004) 367-380.
* [7] Kim, T., Jang, L.-C., and Yi, H., Note on the modified $q$-Bernstein polynomials, Discrete Dyanmics in Nature and Society, Volume 2010 (2010), Article ID 706483, 12 pages.
* [8] Kim, T., A note $q$-Bernstein polynomials, Russ. J. Math. Phys. 18(2011), page 41-50.
* [9] Kim, T., Choi, J. and Kim, Y. H., Some identities on the $q$-Bernstein polynomials, $q$-Stirling numbers and $q$-Bernoulli numbers, Adv. Stud. Contemp. Math. 20(2010), page 335-341.
* [10] Kim, T., Choi, J. and Kim, Y. H., $q$-Bernstein Polynomials Associated with $q$-Stirling Numbers and Carlitz’s $q$-Bernoulli Numbers, Abstract and Applied Analysis, Article ID 150975, 11 pages.
* [11] Kim, T., Choi, J., Kim, Y. H. and Ryoo, C. S., On the fermionic $p$-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, J. Inequal. Appl. 2010 (2010), Art ID 864247, 12 pages.
* [12] Kim, T., Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4, page 484–491.
* [13] Ryoo, C. S., A note on the weighted $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), page 47-54.
* [14] Oruc, H., and Phillips, G. M., A generalization of the Bernstein polynomials, Proceedings of the Edinburgh Mathematical society (1999) 42, 403-413.
* [15] Ostrovska, S., On the $q$-Bernstein polynomials, Adv. Stud. Contemp. Math. 11 (2) (2005), 193-204.
* [16] Phillips, G. M., A survey of results on the $q$-Bernstein polynomials, IMA Journal of Numerical Analysis Advance Access published online on June 23, (2009), 1-12, doi:10.1093/imanum/drn088.
* [17] Simsek, Y., and Acikgoz, M., A new generating function of $q$-Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, volume 2010, Article ID 769095, 12 pages, doi: 10.1155/2010/769095.01-313.
|
arxiv-papers
| 2012-05-18T16:30:56 |
2024-09-04T02:49:31.049858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehmet Acikgoz and Serkan Araci",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1205.4192"
}
|
1205.4203
|
# Orbitron. Part I.
Stable orbital motion of
magnetic dipole
in the field of permanent magnets
Stanislav S. Zub∗
###### Contents
1. 1 Introduction
2. 2 Hamiltonian formalism for magnetic dipole in the axisymmetrical magnetic field
3. 3 Mathematical model of Orbitron
4. 4 Example of stable orbital motion
1. 4.1 Relative equilibrium
2. 4.2 Choice of supporting point
3. 4.3 Necessary condition of stability and Lagrangian coefficients
4. 4.4 Allowable variations
5. 4.5 Basic quadratic form
6. 4.6 Conditions of positive definiteness of the basic quadratic form
7. 4.7 Physical meaning of positive definiteness conditions
5. 5 Numeral simulation
6. 6 Summary
7. 7 References
11footnotetext: Faculty of Cybernetics. Taras Shevchenko National University
of Kyiv.
Glushkov boul., 2, corps 6. UA-03680, Ukraine. E-mail: stah@univ.kiev.ua
## 1 Introduction
The problem of magnetic configurations stability has a long history. In 1600
W.Gilbert published a treatise On the Magnet, Magnetic Bodies, and the Great
Magnet of the Earth where he proposed that magnets can form the noncontact
stable systems.
Since then a number of world known scientists, for example, Newton, Earnshaw,
Heisenberg, Kapitsa, Braunbeck, Tamm, Ginzburg made their contributions to the
study of this problem.
The problem of magnetic equilibrium stability can be naturally divided into
two tasks of static and dynamic equilibrium.
Unfortunately, in both of these scientific fields a number of prejudices and
errors appeared which are not entirely solved until now.
Concerning the static equilibrium, this resulted in unjustified transference
of the conclusion from Earnshaw theorem about systems instability in
electrostatics into the field of magnetic phenomena. Partly this error was
eliminated in the light of magnetic levitation experiments conducted by
Braunbeck and Kapitsa-Arkad’ev (i.e. in combination both of magnetic and
gravity forces).
Studies [1] as well as dissertation [2] were devoted to solving the problem of
static equilibrium of bodies, which interact only via magnetic force.
Particular prejudices have also penetrated the problem of dynamic stability
equilibrium in magnetic systems.
At the dawn of the nuclear age, in the years when research of an atomic
nucleus has been thriving, magnetic interactions were considered as a possible
mechanism of keeping particles in the nucleus. In 1941- 1947 Tamm and Ginzburg
have shown that in the case of two interacting magnetic dipoles the orbital
motion is impossible, due to the particles falling down the center both in
classical and in quantum mechanics [3]. In physics this fact was called
problem $1/r^{3}$ and together with Earnshaw theorem, extended to the case of
magnetostatics, resulted in opinion of global instability of the magnetic
systems for a long time.
However even after these results famous physicists, J.Schwinger [4] for
example, had their interest in the magnetic model of matter.
New splash of interest to the problem of dynamic equilibrium in the magnetic
systems resulted in creation by Roy Harrigan a levitron in 1983. Internet
provided wide possibilities for popularization of this unusual toy, and also
other experiments with magnetic bodies [5].
Against this background the importance of theoretical and experimental
research conducted by V. Kozorez almost ten years before, in 1974 [6,7] seemed
to attract unfairly small interest among the world physics community.
V. Kozorez tried to develope the known idea of the solution of the magnetic
systems stability problem based on the consideration that magnetic particles
are extended presented by Heisenberg in the twenties of last century.
He succeeded in building the experimental prototype where a small magnet
accomplished quasiorbital motion up to 6 minutes in duration (this prototype,
in contrast to levitron was not patented).
As we have already noted in [8,9] his theoretical research had rather
estimating character, because adequate mathematical apparatus [10-13] for
studying the stability of such systems has not been developed yet.
In particular, the condition of stability that he has got for the system
analogous to the system considered in this article is only one of the three
sufficient conditions for stability. This condition gives the exact expression
which reflects the well known fact, that on a considerable distance any
magnetic system presents a dipole.
In respect of the experiment per se, from the philosophical point of view an
experiment as such or computational modeling in principle cannot prove the
dynamic stability, but can only give certain reasons in support of the
stability [14].
For the first time the strictly analytical proof of an orbital motion
stability in magnetic systems is given in [8,9]. Analytical conditions for the
stability of the system formed by two magnetic dumbbells have a complicated
form and values of parameters for the systems stability area were determined
numerically.
Therefore it makes sense to consider a simpler magnetic system that we call
Orbitron for convenience. Here we analytically prove not only the existence of
stable orbits but also stability conditions which have simple physical meaning
for this system.
In this system a movable body is a small permanent magnet with an axial
symmetry. Its interaction with magnetic field is described by magnetic dipole
approximation, and its motion obeys the laws of rigid body motion.
This differs from the model accepted in work [15]. We do not use the analogy
originated from the attempts of classical description of such quantum-
mechanical parameter as particle spin.
Thus one of tasks of this article is to provide the motion equations of such
magnetic rigid body for the systems like Kozorezs prototype and levitron.
Here, for the mathematical model we call Orbitron, we give accurate analytical
prove of the existence of stable orbital motion of a small magnetized rigid
body, described as a magnetic dipole based on the theorem from [12]. The
sufficient conditions of stability in this model have a simple form allowing
clear physical interpretation.
## 2 Hamiltonian formalism for magnetic dipole in the axisymmetrical magnetic
field
Lets consider the Hamiltonian dynamics of small rigid body in the
axisymmetrical magnetic field assuming magnetic dipole approximation. Such
field can be created by cylindrical magnets, solenoids, current-carrying rings
and other objects with axial symmetry along $z$ axis.
The variant of such formalism can be obtained from the formalism of work [16]
by the limiting process $m_{1}\longrightarrow\leavevmode\nobreak\ \infty$.
Thus we consider the first body immobile and $z$ axis oriented. Then its
dynamical variables disappear from consideration or become model parameters.
Therefore the group of symmetry of the task converges to $SO(1)$.
For constructing Hamiltonian dynamics based on Poisson structures it is
necessary to specify a Poisson manifold and also kinetic and potential energy
of the system.
Poisson manifold of Orbitron is the direct product of Euclidean spaces
$P=R^{3}_{x}\times R^{3}_{p}\times R^{3}_{\nu}\times R^{3}_{n}$ $None$
with Poisson brackets for correspondent generatrix.
Generatrix for Orbitron will be: $x_{i}$ \- dipole coordinates; $p_{i}$ \- its
components of momentum (orbital motion); $n_{i}$ \- components of dipole
intrinsic moment of momentum; $\nu_{i}$ \- components of directing unit vector
of dipoles axis of symmetry.
Nonzero Poisson brackets between generatrix on $P$ look like
$\begin{cases}\\{x_{i},p_{j}\\}=\delta_{ij};\\\
\\{n_{i},\nu_{j}\\}=\varepsilon_{ijk}\nu_{k};\quad\\{n_{i},n_{j}\\}=\varepsilon_{ijk}n_{k}.\end{cases}$
$None$
Casimir functions of this Poisson structure that are easily checked will be
$\vec{\nu}{\ }^{2}=1$ and $(\vec{\nu},\vec{n})=const$.
System Hamiltonian we write down in the form:
$h=T+U(r,c^{{}^{\prime}},c^{{}^{\prime\prime}},c^{{}^{\prime\prime\prime}}),$
$None$
where
$U=-(\vec{\mu}\cdot\vec{B})$ $None$
$\vec{B}(\vec{r})=B_{r}(r,c^{\prime})\vec{e}_{r}+B_{z}(r,c^{\prime})\vec{e}_{z},$
$None$ $\begin{cases}r=|\vec{r}|;\\\ \vec{e}_{r}=\vec{r}/|\vec{r}|;\\\
c^{\prime}=\vec{e}_{z}\cdot\vec{e}_{r}=x_{3}/r;\\\
c^{\prime\prime}=\vec{\nu}\cdot\vec{e}_{r};\\\
c^{\prime\prime\prime}=\vec{e}_{z}\cdot\vec{\nu}=\nu_{3}.\end{cases}$ $None$
As usual, kinetic energy of movable body (dipole) consists of kinetic energy
of both translational and rotational motions [16,8].
$T(p^{2},\vec{n}^{2})=\frac{1}{2M}p^{2}+\frac{\alpha}{2}\vec{n}^{2},$
where $M$ – dipole mass; $\alpha=\frac{1}{I_{\bot}}$ (as well as before we
suppose, that $I_{1}=I_{2}=I_{\bot}$, where $I_{1},I_{2},I_{3}$ – intrinsic
moments of the bodys inertia).
Get the system of motion equations for magnetic dipole in axisymmetrical
magnetic field:
$\begin{cases}\dot{\vec{r}}=\vec{p}/M;\\\
\dot{\vec{p}}=-{\partial}_{r}U\vec{e}_{r}-\frac{1}{r}({\partial}_{c^{{}^{\prime}}}UP_{\bot}^{e}(\vec{e}_{z})+{\partial}_{c^{{}^{\prime\prime}}}UP_{\bot}^{e}(\vec{\nu}));\\\
\dot{\vec{\nu}}=\alpha(\vec{n}\times\vec{\nu});\\\
\dot{\vec{n}}=-\vec{\nu}\times(\vec{e}_{r}\partial_{c^{{}^{\prime\prime}}}+\vec{e}_{z}\partial_{c^{{}^{\prime\prime\prime}}})U,\end{cases}$
$None$
where $P_{\bot}^{e}$ – projector on the plane perpendicular to the vector
$\vec{e}_{r}$, i.e.
$P_{\bot}^{e}(\vec{e}_{z})=\vec{e}_{z}-c^{{}^{\prime}}\vec{e}_{r}$ and
$P_{\bot}^{e}(\vec{\nu})=\vec{\nu}-c^{{}^{\prime\prime}}\vec{e}_{r}$.
Expressions for the force and the force momentum acting on a dipole in an
external magnetic field are well known. We can show that the second and fourth
equations of the system (7) can be presented in classical representation.
Concerning the second equation in the system (7), it has been obtained from
the standard expression of Hamiltonian formalism
$\dot{p_{i}}=\\{p_{i},H\\}=\\{p_{i},U\\}=\partial_{r}U\\{p_{i},r\\}+\partial_{c^{{}^{\prime}}}U\\{p_{i},c^{{}^{\prime}})\\}+\partial_{c^{{}^{\prime\prime}}}U\\{p_{i},c^{{}^{\prime\prime}})\\}$
$None$
For potential energy in form (4) we obtain a classic expression of the force
$\dot{\vec{p}}=\\{\vec{p},H\\}=\\{\vec{p},U\\}=-\nabla
U=\nabla(\vec{\mu}\cdot\vec{B})$ $None$
Regarding the fourth equation in the system (7), the potential energy of a
dipole is described by formula (4) in the axisymmetrical magnetic field which
is described by formula (5), therefore we obtain
$(\vec{e}_{r}\partial_{c^{{}^{\prime\prime}}}+\vec{e}_{z}\partial_{c^{{}^{\prime\prime\prime}}})U=-\mu\vec{B}$
$None$
Then we get the last equation in the system (7) in usual classical
representation
$\dot{\vec{n}}=\mu\vec{\nu}\times\vec{B}=\vec{\mu}\times\vec{B}$ $None$
Now the system of equations (7) can be written in the form:
$\begin{cases}\dot{\vec{r}}=\vec{p}/M;\\\
\dot{\vec{p}}=\nabla(\vec{\mu}\cdot\vec{B});\\\
\dot{\vec{\mu}}=(\vec{n}\times\vec{\mu})/I_{\bot};\\\
\dot{\vec{n}}=\vec{\mu}\times\vec{B},\end{cases}$ $None$
A few remarks are necessary regarding the systems of motion equations (7,7a).
1\. Both systems are correct in the quasi-stationary electromagnetic field
approximation [17,18]. This approximation is characterized by the possibility
to neglect the finiteness of electromagnetic disturbances propagation speed
and displacement current in the range of the system and calculate magnetic
fields using formulas of magnetostatics.
2\. The system of equations (7) uses the concept of magnetic potential energy,
which is incident to long-range action conception in classic mechanics. As
just was mentioned, this is possible in quasi-stationary approximation. The
chosen form of the potential energy, as in formula (3), describes not only
dipoles but also wide enough class of axisymmetrical magnetic bodies.
3\. The system (7a) corresponds to the concept of short-range interactions in
the electromagnetic field theory. Therefore these equations are obviously
valid not only for the axisymmetrical magnetic field but also describe the
motion of a dipole in an arbitrary external magnetic field.
4\. We consider a magnetic dipole, as a small magnetized rigid body with axial
symmetry as in levitron for example. Equation (1) in work [15] in this case
could not replace the third and fourth equations of the system (7a). This
distinguishes our mathematical model from that accepted in work [15].
## 3 Mathematical model of Orbitron
Not all axisymmetrical magnetic fields can create the possibility for a stable
orbital motion of a magnetic dipole. For example, the field of magnetic-dipole
type results in problem $1/r^{3}$ mentioned in introduction. Therefore, it may
be useful to use Heisenbergs hypothesis about the possibility of stable
magnetic configurations with magnetic extended bodies (see also [7]).
Here we offer the following model of Orbitron.
Put two magnetic unlike poles on axis $z$ at points $\pm h$. These poles
create the axisymmetrical magnetic field in which a magnetic dipole is moving.
We assume that stable orbital motion of the system is possible under certain
parameters.
It is important to give some explanation here. Equations of magnetostatics do
not suppose the existence of isolated magnetic charges. However, the field
outside a thin solenoid, for example (the same for the thin cylindrical
magnet) will coincide with high accuracy with the field of two poles [19]. On
the other hand, the field inside the solenoid not only does not coincide with
charges field but also opposite in sign, so that the flow through the
unbounded surface embracing only one pole is equal to zero, as required by
magnetostatics equations.
It is assumed that the dipole moves a sufficient distance from the poles of
the magnet, which is the source of the field, and the model of two magnetic
charges describes the field with high accuracy.
Thus, magnetic field in the system has the form of sum of the coulomb fields
of two charges $\pm\kappa$:
$\vec{B}(\vec{r})=\sum_{\varepsilon=\pm
1}\vec{B}_{\varepsilon}(\vec{r}),\qquad\vec{B}_{\varepsilon}=\frac{\mu_{0}}{4\pi}\varepsilon\kappa\frac{\vec{r}-\varepsilon
h\vec{e}_{z}}{|\vec{r}-\varepsilon h\vec{e}_{z}|^{3}}.$ $None$
where each of the fields $\vec{B}_{\varepsilon}$, and consequently the total
field can be presented by formula (5).
Then for the potential energy of a dipole in the magnetic field we get the
expression
$U(r,c^{\prime},c^{\prime\prime},c^{\prime\prime\prime})=-\frac{\lambda_{0}}{4\pi}\sum_{\varepsilon=\pm
1}\varepsilon U_{\varepsilon}(r,c^{\ {}^{\prime}},c^{\ {}^{\prime\prime}},c^{\
{}^{\prime\prime\prime}}),\quad\lambda_{0}=\mu_{0}\kappa\mu$ $None$
where
$U_{\varepsilon}(r,c^{\ {}^{\prime}},c^{\ {}^{\prime\prime}},c^{\
{}^{\prime\prime\prime}})=\frac{rc^{\ {}^{\prime\prime}}-\varepsilon hc^{\
{}^{\prime\prime\prime}}}{R_{\varepsilon}(r,c^{\ {}^{\prime}})^{3}}$ $None$
and
$R_{\varepsilon}(r,c^{\ {}^{\prime}})=\left(r^{2}-2\varepsilon hrc^{\
{}^{\prime}}+h^{2}\right)^{1/2}$ $None$
Lets show the first derivatives of the function $U_{\varepsilon}$:
$\partial_{r}U_{\varepsilon}=\frac{c^{{}^{\prime\prime}}}{R_{\varepsilon}(r,c^{\
{}^{\prime}})^{3}}-\frac{3\left(rc^{{}^{\prime\prime}}-\varepsilon
hc^{{}^{\prime\prime\prime}}\right)\left(r-\varepsilon
hc^{{}^{\prime}}\right)}{R_{\varepsilon}(r,c^{\ {}^{\prime}})^{5}}$ $None$
$\partial_{c^{{}^{\prime}}}U_{\varepsilon}=\frac{3\left(rc^{{}^{\prime\prime}}-\varepsilon
hc^{{}^{\prime\prime\prime}}\right)\varepsilon hr}{R_{\varepsilon}(r,c^{\
{}^{\prime}})^{5}}$ $None$
$\partial_{c^{{}^{\prime\prime}}}U_{\varepsilon}=\frac{r}{R_{\varepsilon}(r,c^{\
{}^{\prime}})^{3}}$ $None$
$\partial_{c^{{}^{\prime\prime\prime}}}U_{\varepsilon}=-\frac{\varepsilon
h}{R_{\varepsilon}(r,c^{\ {}^{\prime}})^{3}}$ $None$
## 4 Example of stable orbital motion
The main aim of this work (i.e. Part I) is to prove the existence of stable
orbital motion in the systems of bodies, which interact only by magnetic
forces. The example of such system is described in a section 3, and example of
a stable orbit will be the circular orbit in plane $z=0$.
### 4.1 Relative equilibrium
A special role in orbital motions stability of Hamiltonian systems plays the
so-called relative equilibrium [11,12], i.e. such trajectories of the dynamic
system which simultaneously are one-parameter sub-groups of the systems
invariance group.
As it has been already mentioned, the invariance group of Orbitron is $SO(1)$.
Every one-parameter sub-group of this group is characterized by the intrinsic
rotational angular velocity $\vec{\omega}=\omega\vec{e}_{z}$. For our problem
the rate of change of any physical value $\vec{v}$ along the orbit of the sub-
group will be set by the formula $\dot{\vec{v}}=\vec{\omega}\times\vec{v}$.
Therefore, for the relative equilibrium to exist the following relationships
must hold
$\begin{cases}\dot{\vec{r}}=\omega(\vec{e}_{z}\times\vec{r});\\\
\dot{\vec{p}}=\omega(\vec{e}_{z}\times\vec{p});\\\
\dot{\vec{\nu}}=\omega(\vec{e}_{z}\times\vec{\nu});\\\
\dot{\vec{n}}=\omega(\vec{e}_{z}\times\vec{n}).\end{cases}$ $None$
We show that a dynamic orbit for which these relationships are satisfied
exists. Examine an orbit, spatially located in the $z=0$ plane. Also suppose
that $\vec{\nu}\parallel\vec{e_{z}}$ and $\vec{n}\parallel\vec{e_{z}}$. Then
$c^{{}^{\prime}}=c^{{}^{\prime\prime}}=0$, $c^{{}^{\prime\prime\prime}}=\pm 1$
along the whole trajectory and
${\partial}_{c^{{}^{\prime}}}U={\partial}_{c^{{}^{\prime\prime}}}U={\partial}_{c^{{}^{\prime\prime\prime}}}U=0$
as follows from formulas (11,15-17).
So, the third and the fourth equations of the system (7) then hold
identically, the first and the second are reduced to the second order
equation:
$M\ddot{\vec{r}}+\left(\frac{\partial_{r}U}{r}\right)_{|r=r_{0}}\vec{r}=0$
$None$
On condition that $(\partial_{r}U)_{|r=r_{0}}>0$ equation (19) has solution
corresponding to the motion on the circumference with radius $r_{0}$ and
frequency, which is determined by relationship
$\left(\frac{\partial_{r}U}{r}\right)_{|r=r_{0}}=\omega^{2}M$ $None$
Thus, one can prove that the reduced orbit indeed is a relative equilibrium.
Theorem 4.8. in [12] is a suitable instrument for investigating stability of
relative equilibria on Poisson manifolds. Important advantage of group
theoretical methods is that the functional space of investigation of
trajectories is substituted by investigation of finite-dimensional vector
space of dynamic variables variations in a fixed point on the trajectory. Thus
the investigation approach for stability is very similar to the study of a
functions conditional extremum by Lagrange multiplier method.
### 4.2 Choice of supporting point
Lets set the point on an orbit of relative equilibrium
$z_{e}=\begin{cases}\vec{x}_{0}=r_{0}\vec{e}_{1};\\\
\vec{p}_{0}=p_{0}\vec{e}_{2};\\\ \vec{\nu}=-\vec{e}_{3};\\\
\vec{n}=n_{0}\vec{e}_{3};\end{cases}$ $None$
Notice that we do not fix the sign of the mechanical moment $n_{0}$, it can be
arbitrary. As for a sign of $p_{0}$, for a positive angular velocity its value
will be positive.
Lets show that in supporting point the following relationships hold
$\partial_{c^{\prime}}U_{|z_{e}}=0;\qquad\partial_{c^{\prime\prime}}U_{|z_{e}}=0;$
$None$
Since in supporting point
$\begin{cases}c^{\prime}=0;\\\ c^{\prime\prime}=0;\\\
c^{\prime\prime\prime}=-1;\end{cases}$ $None$
therefore
$R_{\varepsilon}(r,c^{\prime})_{|z_{e}}=(r^{2}+h^{2})^{1/2}$ $None$
From the expressions of potential energy derivatives (15-16)
$(\partial_{c^{\prime}}U_{\varepsilon})_{|z_{e}}=\frac{3h^{2}r}{(r^{2}+h^{2})^{5/2}},\qquad(\partial_{c^{\prime\prime}}U_{\varepsilon})_{|z_{e}}=\frac{r}{(r^{2}+h^{2})^{3/2}}$
$None$
notice that both expressions do not depend on $\varepsilon$, meaning that in
sum on $\varepsilon$ (with $\varepsilon$\- multiplier) they will give 0.
### 4.3 Necessary condition of stability and Lagrangian coefficients
As motion integrals we will take
$\begin{cases}j_{3}=x_{1}p_{2}-x_{2}p_{1}+n_{3};\\\
C_{1}=\frac{\lambda_{1}}{2}\vec{\nu}^{2};\\\
C_{2}=\lambda_{2}(\vec{\nu},\vec{n});\end{cases}$ $None$
where 1st line represents a third conserved quantity of a body total angular
momentum, and the other two are Casimir functions of the system.
Write out the correspondent differentials in $z_{e}$ point
$\begin{cases}(\boldsymbol{d}j_{3})_{|z_{e}}=p_{0}\boldsymbol{d}x_{1}+r_{0}\boldsymbol{d}p_{2}+\boldsymbol{d}n_{3};\\\
(\boldsymbol{d}C_{1})_{|z_{e}}=-\lambda_{1}\boldsymbol{d}\nu_{3};\\\
(\boldsymbol{d}C_{2})_{|z_{e}}=\lambda_{2}(n_{0}\boldsymbol{d}\nu_{3}-\boldsymbol{d}n_{3});\end{cases}$
$None$
Efficiency function (adjoined Hamiltonian) looks like
$\tilde{H}=T+U-\omega j_{3}+\lambda_{1}C_{1}+\lambda_{2}C_{2}$ $None$
The necessary condition of stability in theorem 4.8. [12] requires the
differential of efficiency function to be equal to zero in a supporting point,
i.e. $\boldsymbol{d}\tilde{H}_{|_{z_{e}}}=0$.
For the differential of potential energy we have
$\boldsymbol{d}U_{|z_{e}}=\partial_{r}U\boldsymbol{d}x_{1}+\partial_{c^{\prime\prime\prime}}U\boldsymbol{d}\nu_{3}$
$None$
For the differential of kinetic energy we have
$\boldsymbol{d}T_{|z_{e}}=\frac{p_{0}}{M}\boldsymbol{d}p_{2}+\alpha
n_{0}\boldsymbol{d}n_{3};$ $None$
Collecting the differentials of efficiency function, we get
$\boldsymbol{d}\tilde{H}_{|z_{e}}=(\partial_{r}U_{|z_{e}}-\omega
p_{0})\boldsymbol{d}x^{1}+\left(\frac{p_{0}}{M}-\omega
r_{0}\right)\boldsymbol{d}p_{2}$ $None$
$+(\partial_{c^{{}^{\prime\prime\prime}}}U_{|z_{e}}-\lambda_{1}+\lambda_{2}n_{0})\boldsymbol{d}\nu^{3}+(\alpha
n_{0}-\omega-\lambda_{2})\boldsymbol{d}n_{3}$
Equating $\boldsymbol{d}\tilde{H}_{|z_{e}}=0$, we derive the following
expression for Lagrange multipliers
$\begin{cases}p_{0}/M=\omega r_{0};\\\ \omega
p_{0}=\partial_{r}U_{|z_{e}}=\frac{3Kr_{0}}{R^{2}};\\\ \lambda_{2}=\alpha
n_{0}-\omega;\\\
\lambda_{1}=\partial_{c^{\prime\prime\prime}}U_{|z_{e}}+\lambda_{2}n_{0}=K+n_{0}(\alpha
n_{0}-\omega),\end{cases}$ $None$
where
$K=\partial_{c^{\prime\prime\prime}}U_{|z_{e}}=\frac{\lambda_{0}h}{2\pi
R^{3}}$ $None$
the first equation in (32) is an ordinary relationship between linear and
angular velocity during circular orbital motion.
second equation in (32) represents the equality of centrifugal (on the left)
and centripetal (on the right) forces.
From this two expressions we get the relationship for angular velocity,
namely:
$M\omega^{2}=\frac{1}{r_{0}}\partial_{r}U_{|z_{e}}=\frac{3K}{R^{2}}$ $None$
### 4.4 Allowable variations
For the application of the sufficient condition of stability in the theorem
4.8. in [12] it is necessary to extract a linear subspace of allowable
variations.
Lets consider the variations of the dynamic variables annihilating the
differentials in formula (27).
From the second line in (27) it follows, that $\delta\nu^{3}=0$, then it
ensues from the third line, that $\delta n^{3}=0$.
Thus, we obtain
$\begin{cases}\delta\nu^{3}=0;\\\ \delta n_{3}=0;\\\ \delta
p_{2}=-\frac{p_{0}}{r_{0}}\delta x_{1};\end{cases}$ $None$
Hence it ensues that the variations in the form
$\delta x^{1},\delta x^{2},\delta x^{3};\quad\delta p_{1},\delta
p_{3};\quad\delta\nu^{1},\delta\nu^{2};\quad\delta n_{1},\delta n_{2}$ $None$
can be considered as independent variations, furthermore, we must exclude from
this subspace the direction which is tangent to the orbit
It ensues from formula (18), that this direction (in $z_{e}$ point) is
determined as
$\begin{cases}\delta\vec{x}=r_{0}\vec{e}_{2};\\\
\delta\vec{p}=-p_{0}\vec{e}_{1};\\\ \delta\vec{\nu}=0;\\\
\delta\vec{n}=0.\end{cases}$ $None$
In order to eliminate the variation (37), we impose another additional
condition on variations, and then we get the constraints
$\begin{cases}\delta\nu^{3}=0;\\\ \delta n_{3}=0;\\\ \delta
p_{1}=\frac{p_{0}}{r_{0}}\delta x_{2};\\\ \delta
p_{2}=-\frac{p_{0}}{r_{0}}\delta x_{1};\end{cases}$ $None$
and an independent set of variations will be
$\delta x^{1},\delta x^{2},\delta x^{3};\quad\delta
p_{3};\quad\delta\nu^{1},\delta\nu^{2};\quad\delta n_{1},\delta n_{2};$ $None$
### 4.5 Basic quadratic form
Sufficient condition for a minimum consists in positive definiteness of
quadratic form of type $\boldsymbol{d}^{2}\tilde{H}_{|_{z_{e}}}(\delta
z,\delta z^{{}^{\prime}})$, where variation vectors $\delta z$, $\delta
z^{{}^{\prime}}$ must be expressed through independent variations (39) taking
into account the constraints (38). Quadratic form defined in independent
variations we denote by $Q$.
Calculations of the efficiency function hessian (adjoined Hamiltonian) and
basic quadratic form in independent variations were performed in Maple.
For better structuring of the expressions indefinite Lagrange multipliers are
hidden at the first stage.
After insignificant transposition of columns (and corresponding lines with the
same number) the matrix of basic quadratic form acquires a form
$\begin{bmatrix}Q_{11}&0&0&0&0&0&0&0\\\ 0&Q_{22}&0&0&0&0&0&0\\\
0&0&Q_{44}&0&0&0&0&0\\\ 0&0&0&Q_{33}&Q_{35}&0&0&0\\\
0&0&0&Q_{35}&Q_{55}&Q_{57}&0&0\\\ 0&0&0&0&Q_{57}&Q_{77}&0&0\\\
0&0&0&0&0&0&Q_{66}&Q_{68}\\\ 0&0&0&0&0&0&Q_{68}&Q_{88}\end{bmatrix}$ $None$
Lets write out non zero elements from matrix of quadratic form (per line)
$Q_{11}=3\left(\frac{h^{2}-4r_{0}^{2}}{R^{2}}\frac{K}{R^{2}}+M\omega^{2}\right)$
$None$ $Q_{22}=3\left(\frac{K}{R^{2}}+M\omega^{2}\right)$ $None$
$Q_{44}=\frac{1}{M}$ $None$
$Q_{33}=\frac{3r_{0}^{2}-2h^{2}}{R^{2}}\frac{3K}{R^{2}},\qquad
Q_{35}=-\frac{3Kr_{0}}{R^{2}}$ $None$ $Q_{55}=\lambda_{1},\quad
Q_{57}=\lambda_{2}$ $None$ $Q_{77}=\alpha$ $None$
$Q_{66}=\lambda_{1}=Q_{55},\quad Q_{68}=\lambda_{2}=Q_{57}$ $None$
$Q_{88}=\alpha=Q_{77}$ $None$
Substituting $M\omega^{2}$ for expression (34) in $Q_{11},Q_{22}$, we obtain
$Q_{11}=\frac{3K}{R^{2}}\frac{4h^{2}-r_{0}^{2}}{R^{2}}$ $None$
$Q_{22}=\frac{12K}{R^{2}}$ $None$
### 4.6 Conditions of positive definiteness of the basic quadratic form
For matrix $Q$ to be positive definite it is foremost necessary that all
diagonal elements of the matrix are positive. $Q_{22},Q_{44},Q_{77},Q_{88}$
are scienter positive. Remaining conditions are as follows
$\begin{cases}0<Q_{11}=\frac{3K}{R^{2}}\frac{4h^{2}-r_{0}^{2}}{R^{2}};\\\
0<Q_{33}=\frac{3K}{R^{2}}\frac{3r_{0}^{2}-2h^{2}}{R^{2}};\\\
0<Q_{55}=Q_{66}=\lambda_{1}=K+n_{0}(\alpha n_{0}-\omega);\\\ \end{cases}$
$None$
The first two conditions result in purely geometrical limitations
$\left(Q_{11}>0\right)\&\left(Q_{33}>0\right)\longrightarrow\left(\sqrt{\frac{2}{3}}<\frac{r_{0}}{h}\right)\&\left(\frac{r_{0}}{h}<2\right)$
$None$
These conditions of positive definiteness of the matrix $Q$ have to be
supplemented now by the conditions of positive definiteness of two submatrices
of $3\times 3$ and $2\times 2$, namely
$\begin{bmatrix}Q_{33}&Q_{35}&0\\\ Q_{35}&Q_{55}&Q_{57}\\\
0&Q_{57}&Q_{77}\end{bmatrix}$ $None$
and
$\begin{bmatrix}Q_{66}&Q_{68}\\\ Q_{68}&Q_{88}\end{bmatrix}$ $None$
Taking into account the above mentioned considerations it is sufficient for
the matrix (51) to check the condition of positiveness of its determinant of
$Q_{66}Q_{88}-Q_{68}^{2}$. Thus, additionally to the conditions (49) the
following condition is added
$0<Q_{66}Q_{88}-Q_{68}^{2}=\alpha K+\omega(\alpha n_{0}-\omega)$ $None$
Now we investigate the conditions of positive definiteness of the matrix (50).
The first condition $Q_{33}>0$ we have considered already.
Thus the additional conditions of positive definiteness of matrix (50) are
reduced to positiveness of two determinants
$0<Q_{33}Q_{5,5}-Q_{35}^{2}$ $None$
and
$0<Q_{33}Q_{55}Q_{77}-Q_{33}Q_{57}^{2}-Q_{77}Q_{35}^{2}$ $None$
Condition (54) can be also written in form
$Q_{77}(Q_{33}Q_{55}-Q_{35}^{2})>Q_{33}Q_{57}^{2}$ $None$
and, since $Q_{77}>0$ then (54) transforms into (54b) which replaces condition
(53) as it accounts for it
$Q_{33}Q_{55}-Q_{35}^{2}>\frac{Q_{33}}{Q_{77}}Q_{57}^{2}$ $None$
So, condition (53) is superfluous and it is necessary to study only condition
(54). Remind that
$\begin{cases}Q_{33}=\frac{3K}{R^{2}}\frac{3r_{0}^{2}-2h^{2}}{R^{2}};\\\
Q_{35}=-3\frac{Kr_{0}}{R^{2}};\\\ Q_{55}=K+n_{0}(\alpha
n_{0}-\omega)=Q_{66};\\\ Q_{57}=\alpha n_{0}-\omega=Q_{68};\\\
Q_{77}=\alpha=Q_{88};\end{cases}$ $None$
Now write down (54) in form
$Q_{55}Q_{77}-Q_{57}^{2}>\frac{Q_{77}Q_{35}^{2}}{Q_{33}}$ $None$
as supposed $Q_{33}>0$.
Taking into account formulas (55), condition (54c) is equivalent to
$Q_{66}Q_{88}-Q_{68}^{2}>\frac{Q_{77}Q_{35}^{2}}{Q_{33}}>0$ $None$
Thus, condition (52) is a consequence of condition (54) and $Q_{33}>0$ from
(48). It means that condition (52) can be omitted.
We have the following reduced number of conditions for matrix $Q$ positive
definiteness:
$\begin{cases}0<Q_{11}=\frac{3K}{R^{2}}\frac{4h^{2}-r_{0}^{2}}{R^{2}};\\\
0<Q_{33}=\frac{3K}{R^{2}}\frac{3r_{0}^{2}-2h^{2}}{R^{2}};\\\
0<Q_{55}=Q_{66}=\lambda_{1}=K+n_{0}(\alpha n_{0}-\omega);\\\
0<Q_{33}Q_{55}Q_{77}-Q_{33}Q_{57}^{2}-Q_{77}Q_{35}^{2}\end{cases}$ $None$
We investigate condition (54) in form
$0<Q_{33}(Q_{55}Q_{77}-Q_{57}^{2})-Q_{77}Q_{35}^{2}$
$=Q_{33}(Q_{66}Q_{88}-Q_{68}^{2})-Q_{77}Q_{35}^{2}$
$=\frac{3K}{R^{4}}[(3r_{0}^{2}-2h^{2})\omega(\alpha n_{0}-\omega)-2\alpha
Kh^{2}]$
That is
$\frac{\omega}{\alpha}(\alpha n_{0}-\omega)>K\frac{2h^{2}}{3r_{0}^{2}-2h^{2}}$
So, condition (54) is equivalent
$\frac{\omega}{\alpha}(\alpha
n_{0}-\omega)>\frac{K}{\frac{3}{2}(\frac{r_{0}}{h})^{2}-1}$ $None$
In particular, as $Q_{33}>0$, we have $\alpha n_{0}-\omega>0$, and it means
that conditions $Q_{55}=Q_{66}=\lambda_{1}>0$ are fulfilled a priory and can
be omitted.
Therefore, conditions
$\begin{cases}\sqrt{\frac{2}{3}}<\frac{r_{0}}{h}<2;\\\
\frac{\omega}{\alpha}(\alpha
n_{0}-\omega)>\frac{K}{\frac{3}{2}(\frac{r_{0}}{h})^{2}-1}\end{cases}$ $None$
define positive definiteness of form $Q$.
### 4.7 Physical meaning of positive definiteness conditions
In the previous section we have established the conditions of the systems
parameters which provide positive definiteness of basic quadratic form,
namely:
$\begin{cases}\sqrt{\frac{2}{3}}<\frac{r_{0}}{h}<2;\\\
\frac{\omega}{\alpha}(\alpha
n_{0}-\omega)>\frac{K}{\frac{3}{2}(\frac{r_{0}}{h})^{2}-1}\end{cases}$ $None$
The first condition in (59) is purely geometrical and determines a possible
range for a radius of the orbit representing a relative equilibrium (21).
The second condition is dynamic and determines lower boundary for the
intrinsic moment of momentum of the body. In particular, it means that a body
must be sufficiently rapidly revolved. Therefore it is worth to solve this
inequality relative to $n_{0}$.
Using relationships (34), we obtain
$n_{0}>\frac{\omega}{\alpha}+\frac{1}{3}\frac{1+(\frac{h}{r_{0}})^{2}}{\frac{3}{2}(\frac{r_{0}}{h})^{2}-1}(\omega
Mr_{0}^{2})$ $None$
Value of $\frac{\omega}{\alpha}$ corresponds to the intrinsic moment which a
body would have if it were to revolve with angular velocity $\omega$ athwart
to the own axis of symmetry.
Value of $\omega(Mr_{0}^{2})$ is simply the orbital moment of momentum
$(L_{z})_{|z_{e}}$.
For condition (60) one can give such physical meaning: intrinsic moment of
body rotation must be of the same or higher order of its orbital moment.
Indeed, multiplier before $\omega(Mr_{0}^{2})$ is a geometrical factor which
is $\frac{r_{0}}{h}=1.0$ equal to $\sim 1.33$, and at $\frac{r_{0}}{h}=1.5$
equal to $\sim 0.2$.
In these estimations the first term in the right part of expression (60) can
be neglected.
## 5 Numeral simulation
From a mathematical point of view the stability conditions obtained here allow
a wide range of parameters values of the problem to exist, however not all of
them can be physically realized. From a physical perspective the values of
parameters are limited by the properties of present materials.
In addition it seems difficult to realize in practice high rate of rotations,
especially as far as it concerns intrinsic angular velocity of movable
magnetic body (dipole).
Therefore it appears necessary to specify such values of parameters which can
be realized in an experiment.
For the magnets, made from $Nd-Fe-B$, we have the following characteristics:
$\rho=7.4\cdot 10^{3}(kg/m^{3})$ density and $B_{r}=0.25(T)$ – remaining
induction. Then it is easy to obtain magnetic charge of the poles
$\kappa=17.6(A\cdot m)$. Distance between the poles is $L=2h=0.1(m)$.
A movable magnet we choose in a form of cylinder (disk) with the diameter of
$d=0.014(m)$ and height $l=0.006(m)$. Then disk magnetic moment
$\mu=0.18(A\cdot m^{2})$.
As a result for the orbit with the radius $r_{0}=1.5h=0.075(m)$ we obtain the
angular velocity of the orbital motion $\omega=1.54(rad/sec)$, with minimum
angular velocity of disk intrinsic rotation in this case is
$\Omega=72.8(rad/sec)$. Such values of angular velocity appear fully
reasonable.
Using the indicated values of Orbitron parameters the numeral modeling of
orbital motion was conducted under the deviations of initial values of dynamic
variables from the values, which correspond to relative equilibrium within
$1\%$ error. 1000 castings which showed the stability of orbital motion were
accomplished by Monte Carlo method (i.e. by random selection of the initial
values in the vinicity). We will elucidate this in more detail in the next
parts of this work.
## 6 Summary
The main aim of this work was to give constructive proof of stable orbital
motions existence in the systems of bodies, which interact only by magnetic
forces.
For this purpose it is enough to analytically prove the existence of stability
for one orbit in comparatively simple system described by equations which do
not contradict the laws of electrodynamics and classical mechanics.
We named such a system Orbitron, found its parameters which can be physically
realized and conducted the Monte Carlo numeral modeling.
To be continued $\dots$
## 7 References
1. 1.
Zub S. S. in Proceedings of the Int. Conference on Magnetically Levitated
Systems and Linear Drivers (MAGLEV’2002), Lausanne, Switzerland, 2002, eConf
CPP02105 (2002).
2. 2.
Zub S. S. Influence of superconductive elements topology on the stability of
the free body equilibrium, (Ukrainian) / S.S. Zub // Synopsis of Ph.D.
Dissertation, Institute of Cybernetics, National Academy of Sciences of
Ukraine, Kiev, 24 p. 2005.
3. 3.
Ginzburg V. L. Mezotrons Theory and Nuclear Forces / V.L. Ginzburg // –
Phys.-Uspekhi. –1947. – Vol. 31., issues 2. – P. 174 – 209.
4. 4.
Schwinger J. A Magnetic Model of Matter, Science 165 (No. 3895), 757 (1969).
5. 5.
Harrigan R. M. Levitation device, U.S. Patent 382245, May 3, 1983.
6. 6.
Kozoriz V. V. About a problem of two magnets / V.V. Kozorez // Bull. of the
Ac. of Sc. of USSR, Mech. of a Rigid Body. – 1974. – N3. – P. 29 – 34.
7. 7.
Kozoriz V. V. Dynamic Systems of Free Magnetically Interacting Bodies,
(Russian) / V.V. Kozoriz // Naukova Dumka, – Kyiv, 1981. – 139 p.
8. 8.
Zub S. Research into Orbital Motion Stability in System of Two Magnetically
Interacting Bodies, [math-ph/1701], arXiv:1101.3237
9. 9.
Zub S. S. Research into orbital motion stability in system of two magnetically
interacting bodies / S.S. Zub // Visnyk Taras Shevchenko KNU. — Physics and
Mathematics. – 2011. Vol. 2. – P. 176 – 184.
10. 10.
Marsden J. E. Introduction to Mechanics and Symmetry / Jerrold E. Marsden,
Tudor S. Ratiu // Cambridge University Press, – London, 1998. – 549 p.
11. 11.
Marsden J. E. Lectures on Mechanics. – London : Cambridge University Press,
1992. – 254 p.
12. 12.
Ortega J-P., Ratiu T. S. Non-linear stability of singular relative periodic
orbits in Hamiltonian systems with symmetry // J. Geom. Phys. – 1999. – 32. –
P. 160 –188.
13. 13.
Marsden J. E. Hamiltonian reduction by stages / Marsden J.E., Misiolek G.,
Ortega J.P. et al. // Springer, – Berlin, 2007. – 519 p.
14. 14.
Zub S. S. Hamiltonian formalism for magnetic interaction of free bodies / S.S.
Zub, S.I.Lyashko // J. Num. Appl. Math. – 2012. issues 2(102). –P. 49 – 62.
15. 15.
Simon M. D. Spin stabilized magnetic levitation / M.D. Simon, L.O. Heflinger,
and S.L. Ridgway // Am. J. Phys., 65, 286292 (1997).
16. 16.
Zub S. Mathematical model of magnetically interacting rigid bodies //
PoS(ACAT08)116. – 2009. – 5 p.
17. 17.
Tamm I. E. Fundamentals of the theory of electricity / I.E. Tamm // Mir,
(1979).
18. 18.
Landau L. D. Electrodynamics of continous media / L.D. Landau, E.M. Lifshitz
// Pergamon, (1960).
19. 19.
Smythe W. R., Static and Dynamic Electricity / W.R. Smythe // McGraw-Hill, New
York (1939).
|
arxiv-papers
| 2012-05-18T17:26:04 |
2024-09-04T02:49:31.056998
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stanislav S. Zub",
"submitter": "Stanislav Zub",
"url": "https://arxiv.org/abs/1205.4203"
}
|
1205.4272
|
# Mechanical control of a microrod-resonator optical frequency comb
Scott B. Papp scott.papp@nist.gov Pascal Del’Haye Scott A. Diddams National
Institute of Standards and Technology, Boulder, Colorado 80305, USA
###### Abstract
Robust control and stabilization of optical frequency combs enables an
extraordinary range of scientific and technological applications, including
frequency metrology at extreme levels of precision Rosenband2008 ; Jiang2011 ,
novel spectroscopy of quantum gases Ni2008a and of molecules from visible
wavelengths to the far infrared Diddams2007 , searches for exoplanets
Steinmetz2008 ; Li2008 ; Ycas2012 , and photonic waveform synthesis Jiang2007
; Fortier2011 . Here we report on the stabilization of a microresonator-based
optical comb (microcomb) by way of mechanical actuation. This represents an
important step in the development of microcomb technology, which offers a
pathway toward fully-integrated comb systems. Residual fluctuations of our
32.6 GHz microcomb line spacing reach a record stability level of $5\times
10^{-15}$ for 1 s averaging, thereby highlighting the potential of microcombs
to support modern optical frequency standards. Furthermore, measurements of
the line spacing with respect to an independent frequency reference reveal the
effective stabilization of different spectral slices of the comb with a $<$0.5
mHz variation among 140 comb lines spanning 4.5 THz. These experiments were
performed with newly-developed microrod resonators, which were fabricated
using a CO2-laser-machining technique.
Femtosecond-laser optical frequency combs have revolutionized frequency
metrology and precision timekeeping by providing a dense set of absolute
reference lines spanning more than an octave. These sources exhibit sub-
femtosecond timing jitter corresponding to, for example, an ultralow phase
noise of $<100$ $\mu$rad on the 10 GHz harmonic of the repetition frequency
(line spacing) Fortier2011 ; Diddams2004 . Achieving this remarkable
performance depends jointly on low intrinsic comb noise and on frequency
control of the comb to duplicate the high stability of fixed optical
references across the entire output spectrum.
Recently, a new class of frequency combs has emerged based on monolithic
microresonators Kippenberg2011 , henceforth denoted microcombs. These devices
have the potential to significantly reduce the bulk, cost, and complexity of
conventional laser combs. Such factors stand in the way of next-generation
applications that will require high-performance optical clocks for experiments
outside the lab, or even in space Schiller2009 . Here the comb generation
relies on parametric conversion provided simply by third-order nonlinear
optical effects and is enabled by advances in the quality factor $Q$ and the
mode volume of microresonators. These devices require only a single
continuous-wave laser source, but the achievable frequency span of the comb
depends on low dispersion, making material properties critical. To date
microcombs have been explored with a number of microresonator technologies,
including microtoroids DelHaye2007 , crystalline resonators Savchenkov2008 ;
Grudinin2009 ; Chembo2010 , microrings Levy2010 ; Razzari2010 , fiber cavities
Braje2009 , machined disks Papp2011 , and wedge resonators Lee2011a . Unique
comb spectra have been demonstrated featuring octave spans DelHaye2011 ;
Okawachi2011 and a wide range of line spacings Li2012 . And some aspects of
the microcomb frequency-domain behavior have been explained Savchenkov2008 ;
Savchenkov2008a ; DelHaye2008 ; Papp2011 .
Figure 1: Fabrication of fused-quartz microrod resonators by use of CO2 laser
machining. (a) A rotating fused-quartz rod is illuminated with a focussed CO2
laser that selectively removes material. By applying the laser at different
positions along the rod’s axis, a microresonator is produced. (b) Image of a
microrod with $Q=5\times 10^{8}$, 2 mm diameter, and $\sim$ 100 $\mu$m
thickness. (c) Fabrication of resonators with variable diameter. Starting at
top left and counting clockwise, the resonator diameters are 0.58 mm, 0.36 mm,
0.71 mm, 1.2 mm, 1.5 mm, and 1.0 mm. (d) Optical spectrum from the device in
(b), which has a modulated envelope characteristic of parametric combs
Chembo2010 ; Chembo2010a ; Papp2011 ; Matsko2012 . The span of this comb is
sufficient to access the D1 and D2 transitions of atomic Rb, following second-
harmonic generation.
Microcombs present an interesting challenge for frequency stabilization, as
first pointed out by Del’Haye et al. in Ref. DelHaye2008 . Specifically, the
center frequency of a microcomb spectrum is matched to the pump laser, and the
line spacing must be controlled by changing the resonator’s physical
properties. Future metrology applications of microcombs will require
stabilization of the line spacing with respect to fixed optical and microwave
frequency standards. Hence the key factors for stabilization are a line
spacing in the measurable 10’s to 100 GHz range, low intrinsic fluctuations,
and the capability for fast modulation. Additionally, a threshold power for
comb generation in the milliWatt range, and the potential for integration with
chip-based photonic circuits would enable portable applications.
Here we report a new microcomb platform for achieving these goals. We have
developed a CO2-laser-machining technique that yields microrod optical
resonators with a $Q\gtrsim 5\times 10^{8}$, a user-defined diameter, and a
small effective mode area. The resonant optical frequencies of these devices
can be rapidly controlled by using mechanical forces that alter the
resonator’s shape. With such microresonators, we create a comb spectrum with
32.6-GHz-spaced lines spanning from 1510 nm to 1620 nm. Our work introduces
wideband mechanical control of the microcomb line spacing and its
stabilization with respect to microwave standards. We have improved by more
than a factor of 200 the residual line-spacing fluctuations beyond all
previous microcomb work. And, for the first time, we demonstrate the potential
of microcombs to support optical frequency references that feature fractional
stability in the 10-15 range.
Figure 2: Mechanical control mechanism for microresonator combs. (a) Control
apparatus. A PZT compresses the rod containing our microrod resonator to
adjust its mode frequencies. (b, c) Response of optical resonance (b) and
microcomb line spacing (c) with PZT drive frequency. (d) A three-hour record
of the microcomb line spacing ($\Delta\nu$) under free running and stabilized
conditions. The line spacing set point $f_{s}$ is 32.5671 GHz. A portion of
the stabilized record is vertically scaled by $10^{5}$ to make the level of
fluctuations visible.
To create microrod resonators for comb generation, we use a CO2 laser to
simultaneously shape and polish a fused-quartz rod (Fig 1). At 10.6 $\mu$m
fused quartz is highly absorptive, such that melting and evaporation of the
material is easily accomplished with a focussed $<5$ W CO2 laser beam.
Moreover, the thermal conductivity of fused quartz is low, which enables
localized heating to beyond the melting point of $\approx 1600$ ∘C. Figure 1a
illustrates the basic procedure for resonator fabrication. A 2 mm diameter
fused-quartz rod is rotated in a ball-bearing spindle and the CO2 laser is
directed normal to the axis of the rod. The basic shape of a spheroidal
resonator is created by iteratively applying 5 s laser pulses at locations
laterally separated by 0.3 mm. This process also limits re-deposition of
material on the resonator surface. At constant CO2 power, the machining self
terminates when the volume of fused quartz that reaches the evaporation
temperature is removed. The resonator fabrication procedure we developed has
several unique features, including a $<1$ minute run time, built-in polishing
of the resonator surface to support ultrahigh $Q$, and a resonator yield of
nearly 100 % with $Q$ in the 1–6 $\times 10^{8}$ range. A video demonstration
is included in the Supplementary Information.
We discovered that self termination of the CO2 process enables control over
the resonator diameter with $\pm$10 $\mu$m precision. Before machining a
resonator, we can arbitrarily reduce the quartz rod diameter by positioning
the CO2 laser beam slightly above (or below) the rod and repeatedly moving it
back and forth along the rod’s axis. All the quartz material subjected to
sufficiently high laser power is removed. Moreover, this procedure results in
a smooth surface with respect to the fixed position of the laser beam. Figure
1c shows fused-quartz rods that were turned down in this manner, and we have
fabricated microrods that produce combs with line spacing ranging from 33 GHz
to 150 GHz. A video of the diameter reduction process is also available in the
Supplementary Information. The image at right in Fig. 1c demonstrates our
capability to fabricate microrods of varying diameter on a single fused-quartz
sample. This feature will enable future experiments that require precise
control of resonator free spectral range, such as accessing narrowband
Brillouin gain Li2012 or matching the line spacing of microcombs to the
ground-state hyperfine transition frequencies of atoms.
To generate microcomb spectra (Fig. 1d), we pump a microrod with light coupled
via a tapered optical fiber Cai2000 ; Spillane2003 . The pump laser, a tunable
semiconductor laser operating near 1560 nm, is amplified in erbium fiber and
then spectrally filtered to remove ASE noise; 280 mW of light is available at
the input to the tapered fiber. The microrod is passively locked to the pump
laser via thermal bistability Carmon2004 . This allows us to stabilize the
microcomb center to an auxiliary laser, which in turn is frequency doubled and
referenced to a rubidium D2 transition at 780 nm Ye1996 . The Rb atoms provide
an absolute fractional stability of $\sim 10^{-11}$ at 1 s, but the 1 s
residual noise of $<10^{-17}$ between the microcomb pump and the auxiliary
laser indicates that much more stable references can be employed in the
future. The spectrum of our comb, which spans $\sim 100$ nm, reaches the
corresponding wavelength (1590 nm) of Rb D1 lines at 795 nm. This opens the
possibility for all-optical stabilization of the comb center and mode spacing
using Rb transitions; a future paper will explore this idea. In this Letter,
we focus on characterization and stabilization of the 32.6 GHz line spacing,
which is measured by way of direct photodetection. After conversion to
baseband (described below), the line-spacing signal is analyzed with respect
to ultralow phase- and frequency-noise hydrogen-maser oscillators, which
feature an $\approx 10^{-13}$ at 1 s fractional frequency stability. An
important measure of comb performance is the intrinsic frequency jitter of the
line spacing. Figure 2d shows a two-hour record of the free running microcomb
line spacing. The 1 s Allan deviation for 100 s increments of this data, taken
under typical laboratory conditions, ranges from $2\times 10^{-8}$ to
$10^{-7}$.
Figure 3: Microcomb line spacing stability. (a) Schematic of our system with
independent paths for microcomb generation and line spacing stabilization
(green box), and “out-of-loop” analysis (gray box). The entire optical path is
in fiber, including a programmable optical filter used to study the line
spacing stability for portions of the comb (Fig. 4). Frequency references
$f_{1,2}$ are used for baseband conversion, and the signals $S_{1,2}$ are
measured. (b) Line-spacing Allan deviation versus averaging time for:
(triangles) stabilized residual, (points) stabilized absolute, and (open
circles) free running. The gray line shows the Allen deviation of frequency
reference $f_{2}$.
Here we introduce a mechanism for control of the comb’s line-spacing noise via
a mechanical force applied along the axis of the fused-quartz rod. Mechanical
control offers significant advantages including low-power operation, simple
integration with bulk resonators, and response potentially much faster than
resonator thermal conduction. An image of our setup for line spacing control
is shown in Fig. 2a. A piezoelectric (PZT) element is used to compress the
fused-quartz rod, resulting in axial expansion and tuning of the resonator’s
mode structure. In Fig. 2b and c, we characterize the magnitude and phase
modulation response of a resonator mode and the line spacing of our comb,
respectively. For a pump power well below the thermal bistability point, we
monitor the resonance frequency of a mode as the PZT voltage is varied; see
Fig. 2b. The PZT adjusts the mode frequency by 5 MHz/V below a mechanical
resonance of the system at 25 kHz. This response is less than what is expected
($P_{PZT}\nu/E\times 2\rm{mm}$), given the Young’s modulus $E$ and Poisson
ratio $\nu$ for fused quartz, and the $\sim 1$ MPa/V PZT stroke. The
discrepancy is likely explained by a poor mechanical connection. The line
spacing of the comb also tunes with PZT voltage up to 25 kHz; however, the
resonator thermal locking mechanism reduces the magnitude response at low
frequency. A near-zero phase delay between the modulation and the PZT-induced
response indicates the passive nature of the thermal lock, and it satisfies a
basic requirement for providing useful feedback. The PZT enables stabilization
of the line spacing, which is evident starting at 120 min in Fig. 2d. Compared
to the free-running case, its drift has been reduced by a factor of $\sim
10^{6}$.
We analyze the line spacing in detail to understand a microcomb’s potential
for replicating in each comb line the stability of state-of-the-art frequency
references. Figure 3a shows the important elements of our apparatus. Following
generation, the microcomb spectrum is delivered to two systems for independent
stabilization and analysis. In both these paths the 32.6 GHz comb line
spacings ($\Delta\nu_{1}$ and $\Delta\nu_{2}$) are photodetected, amplified,
and converted to the baseband signals $S_{1}$ and $S_{2}$. Importantly,
$S_{1,2}$ carry the fluctuations of both the line spacing and the microwave
references ($f_{1}$ and $f_{2}$), which are locked to independent maser
signals. The Allan deviation and phase-noise spectra of signals $S_{1,2}$ are
recorded separately by use of a commercial phase noise analyzer, which is
referenced to maser 1 (2) for residual (absolute) measurements. By initiating
a phase-locked loop using $S_{1}$ and the PZT, we stabilize $\Delta\nu_{1}$
with respect to maser 1. At an averaging time of 1 s, the $5\times 10^{-15}$
residual fluctuations of $\Delta\nu_{1}$ (green triangles in Fig. 3b) are far
below the stability of maser 1. This signifies that the microcomb closely
follows the reference frequency $f_{1}$ and attains its stability.
Furthermore, our analysis system tests the microcomb’s ability to characterize
independent microwave frequencies such as $f_{2}$. In Fig. 3b the solid line
shows the Allan deviation of $f_{2}$ from a separate measurement, and the
filled points show the combined fluctuations of $f_{2}$ and $\Delta\nu_{2}$.
These data confirm the expectation from our residual measurements that the
absolute stability of $\Delta\nu_{2}$ is significantly better than $1.5\times
10^{-12}$ at 1 s. The consistent $1/\rm{time}$ averaging behavior observed in
both our residual and absolute measurements is evidence of the phase-locked
stabilization. In contrast, the open circles in Fig. 3b show the free-running
line-spacing drift that increases with time.
Figure 4: Line-spacing equidistance and stability for different spectral
slices of the comb. (a) Microcomb optical spectrum about the pump laser
frequency $\nu_{0}$ prior to filtering. (b) Measurements of the frequency
difference ($\Delta\nu_{1}-\Delta\nu_{2}$) between the whole comb and a
spectral slice. The shaded region indicates the frequency range for one
measurement. (c) For different spectral slices, the points show the 1 s Allan
deviation of $\Delta\nu_{2}$, and the triangles characterize residual
fluctuations between $\Delta\nu_{1}$ and $\Delta\nu_{2}$. For the open circle
and open triangle data points, only a 0.3 THz range about $\nu_{0}$ is
blocked.
The $S_{1}$ signal used for line-spacing stabilization is a composite of all
the comb lines, and its largest contributions naturally come from the most
intense pairs. Hence, an uneven distribution of comb optical power, along with
the complicated nonlinear comb generation process, opens the possibility of
degraded line-spacing stabilization for different spectral slices of the comb.
To quantify these effects, we probe the line-spacing frequency and its
stability with our comb analysis system. By use of the 1535 nm to 1565 nm
(C-band) programmable optical filter with 10 GHz resolution shown in Fig. 3a,
we obtain an arbitrary selection of comb lines. Figure 4b shows measurements
of the difference in line spacing ($\Delta\nu_{1}-\Delta\nu_{2}$) between the
entire comb and various portions of it. Here the horizontal bars indicate the
range of optical frequencies present in the filtered $\Delta\nu_{2}$ signal,
and $\Delta\nu_{1}$ is determined by the set point of our phase-locked loop.
(The residual offset between maser 1 and locked $S_{1}$ is $<1$ $\mu$Hz.) For
reference, the shaded area indicates the comb lines studied in a single
measurement. The weighted mean of all data is -0.4 mHz (on the 32.6 GHz line
spacing) from the anticipated null, which is consistent with their
uncertainties and with our knowledge of the offset between the maser-
referenced $f_{1}$ and $f_{2}$ signals. Moreover, a fit of the slope in Fig.
4b demonstrates that the line spacing does not change by more than the
$5\times 10^{-15}$ standard error over a 4.5 THz span of the comb.
The line-spacing stability of the spectral slices also characterizes the PZT
stabilization. Figure 4c shows the 1 s Allan deviation associated with each
400 s long frequency difference measurement. The stability of $\Delta\nu_{2}$
throughout the C-band portion of the comb is $1.5\times 10^{-12}$, a value
dominated by frequency reference $f_{2}$. It appears that the mechanisms
responsible for line-spacing noise act similarly to different components of
the comb, and our PZT control can effectively counter them. To understand the
residual stability of $\Delta\nu_{2}$ that is possible apart from the noise of
$f_{2}$, we reconfigure our system to use $f_{1}$ for baseband conversion of
both $\Delta\nu_{1}$ and the optically-filtered $\Delta\nu_{2}$. In this case,
common $f_{1}$ noise contributions are suppressed when the $S_{1,2}$ signals
are presented to our noise analyzer. What remains is: uncontrolled jitter
between the spectral slices and the whole comb, and the noise associated with
the independent optical and electrical measurement paths. The level of these
residual fluctuations is mostly below $10^{-14}$ at 1 s; see the red triangles
in Fig. 4c. This demonstrates that future microcomb experiments could take
advantage of frequency references even more stable than a maser.
Figure 5: Spectrum of line spacing fluctuations $S_{\Delta\nu_{1}}$. The black
and green lines show the free-running and stabilized line-spacing spectral
density, respectively. The broad resonance at 600 kHz in the green line
coincides with a mechanical resonance of the fused quartz rod, which we
speculate is weakly excited via the PZT. The red line indicates the
contribution from reference $f_{1}$. The gray and blue lines show the
predicted contributions from pump frequency and intensity noise, respectively.
To understand the pathway for future improvements in line spacing stability,
we characterize the free-running noise spectrum of $\Delta\nu_{1}$; see the
black curve in Fig. 5. Our servo electronics reduce the frequency noise
spectrum by up to $10^{5}$ within the 25 kHz bandwidth permitted by the PZT,
and the spectrum after stabilization is shown by the green curve in Fig. 5.
Achieving further reduction in $S_{\Delta\nu_{1}}$ will depend on improvements
among the feedback mechanism and the underlying source of the noise. Here we
focus on the latter. In our current system, the primary contribution to
$S_{\Delta\nu_{1}}$ is pump-frequency noise that maps onto the line spacing
via a mostly constant relationship $\gamma_{f}=10$ Hz${}_{\Delta\nu_{1}}$/kHz.
This calibration was performed by modulating the pump frequency and recording
the associated modulation in $\Delta\nu_{1}$. By measuring the spectral
density of pump-frequency noise and scaling it by $\gamma_{f}$, we obtain the
gray curve in Fig. 5. We also characterized the degree that pump intensity
noise contributes to $S_{\Delta\nu_{1}}$. In this case the mapping
relationship is $\gamma_{P}=2$ kHz/mW, and it leads to the blue curve in Fig.
5. It’s surprising that intensity noise does not contribute more significantly
to $S_{\Delta\nu_{1}}$, especially in light of previous data Papp2011 . Still,
our characterization of $S_{\Delta\nu_{1}}$ suggests a lower noise pump laser
should be used in future experiments. In particular with a factor of 10
improvement, a residual frequency noise of $<100$ $\mu$Hz/$\sqrt{\rm{Hz}}$ at
a 10 Hz offset from the 32.6 GHz line-spacing signal would be possible. Access
to microwave signals with such high spectral purity would enable interesting
scientific and technological applications Fortier2011 . Noise contributions
from our measurement system also appear in $S_{\Delta\nu_{1}}$. The red line
shows the spectrum of $f_{1}$, which is generated by a high-performance
commercial synthesizer. This highlights the promise of microcomb technology,
which here we demonstrate is already capable of producing signals commensurate
with those of widely-used microwave signal generators.
In conclusion, we have introduced new techniques for fabricating
microresonators with $Q\gtrsim 5\times 10^{8}$, and for controlling the line
spacing of parametric frequency combs created with them. These resonators are
exceptionally simple to create, and we have presented a deterministic
procedure for varying their diameter. Furthermore, we have reported a detailed
study of microcomb line-spacing stabilization using piezoelectric mechanical
control. The achieved levels of absolute and residual fluctuations are
respectively factors of 10 and 200 beyond all previous results DelHaye2008 .
This type of mechanical line-spacing control can easily be introduced into a
variety of microcomb generators based on, for example, crystalline resonators
Savchenkov2008 or integrated silicon nitride devices Okawachi2011 . Our work
has demonstrated microcomb residual noise that is capable of supporting modern
frequency references beyond the $10^{-13}$ at 1 s level associated with
traditional microwave oscillator technology. Future work will focus on
increasing the frequency span of the comb.
We thank Chris Oates and Gabe Ycas for their comments on this manuscript. This
work is supported by the DARPA QuASAR program and NIST. This paper is a
contribution of NIST and is not subject to copyright in the United States. SP
acknowledges support from the National Research Council.
## References
* (1) T. Rosenband et al., Science 319, 1808 (2008).
* (2) Y. Y. Jiang et al., Nat Photon 5, 158 (2011).
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|
arxiv-papers
| 2012-05-18T21:59:14 |
2024-09-04T02:49:31.066936
|
{
"license": "Public Domain",
"authors": "Scott B. Papp, Pascal Del'Haye, Scott A. Diddams",
"submitter": "Scott Papp",
"url": "https://arxiv.org/abs/1205.4272"
}
|
1205.4345
|
Involving copula functions in Conditional Tail Expectation
Brahim Brahimi111E-mail addresses:brah.brahim@gmail.com, Tel.:+213-7 73 54 60
63; fax:+213-33 74 77 88.
Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria
Abstract
Our goal in this paper is to propose an alternative risk measure which takes
into account the fluctuations of losses and possible correlations between
random variables. This new notion of risk measures, that we call Copula
Conditional Tail Expectation describes the expected amount of risk that can be
experienced given that a potential bivariate risk exceeds a bivariate
threshold value, and provides an important measure for right-tail risk. An
application to real financial data is given.
Keywords: Conditional tail expectation; Positive quadrant dependence; Copulas;
Dependence measure; Risk management; Market models.
AMS 2010 Subject Classification: 62P05; 62H20; 91B26; 91B30.
## 1\. Introduction
In actuarial science literature a several risk measures have been proposed,
namely: the Value-at-Risk (VaR), the expected shortfall or the conditional
tail expectation (CTE), the distorted risk measures (DRM), and recently the
copula distorted risk measure (CDRM) as risk measure which takes into account
the fluctuations dependence between random variables (rv). See Brahimi et al.
(2010).
The CTE in risk analysis represents the conditional expected loss given that
the loss exceeds its VaR and provides an important measure for right-tail
risk. In this paper we will always consider random variables with finite mean.
For a real number $s\ $in $\left(0,1\right),$ the CTE of a risk $X$ is given
by
$\mathbb{CTE}\left(s\right):=\mathbb{E}\left[\left.X\right|X>VaR_{X}\left(s\right)\right],$
(1.1)
where $VaR_{X}\left(s\right):=\inf\left\\{x:F\left(x\right)\geq s\right\\}$ is
the quantile of order $s$ pertaining to distribution function (df) $F.$
One of the strategy of an Insurance companies is to set aside amounts of
capital from which it can draw from in the event that premium revenues become
insufficient to pay out claims. Of course, determining these amounts is not a
simple calculation. It has to determine the best risk measure that can be used
to determine the amount of loss to cover with a high degree of confidence.
Suppose now that we deal with a couple of random losses $(X_{1},X_{2}).$ It’s
clear that the CTE of $X_{1}$ is unrelated with $X_{2}.$ If we had to control
the overflow of the two risks $X_{1}$ and $X_{2}$ at the same time, CTE does
not answer the problem, then we need another formulation of CTE which takes
into account the excess of the two risks $X_{1}$ and $X_{2}.$ Then we deal
with the amount
$\mathbb{E}\left[\left.X_{1}\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right].$
(1.2)
If the couple of random losses $(X_{1},X_{2})$ are independents rv’s then the
amount (1.2) defined only the CTE of $X_{1}.$ Therefore the case of
independence is not important.
In the recent years dependence is beginning to play an important role in the
world of risk. The increasing complexity of insurance and financial activities
products has led to increased actuarial and financial interest in the modeling
of dependent risks. While independence can be defined in only one way, but
dependence can be formulated in an infinite ways. Therefore, the assumption of
independence it makes the treatment easy. Nevertheless, in applications
dependence is the rule and independence is the exception.
The copulas is a function that completely describes the dependence structure,
it contains all the information to link the marginal distributions to their
joint distribution. To obtain a valid multivariate distribution function, we
combines several marginal distribution functions, or a different
distributional families, with any copula function. Using Sklar’s theorem
(Sklar, 1959), we can construct a bivariate distributions with arbitrary
marginal distributions. Thus, for the purposes of statistical modeling, it is
desirable to have a large collection of copulas at one’s disposal. A great
many examples of copulas can be found in the literature, most are members of
families with one or more real parameters. For a formal treatment of copulas
and their properties, see the monographs by Hutchinson and Lai (1990),
Dall’Aglio et al. (1991), Joe (1997), the conference proceedings edited by
Benes̆ and S̆tĕpán (1997), Cuadras et al. (2002), Dhaene et al. (2003) and the
textbook of Nelsen (2006).
Recently in finance, insurance and risk management has emphasized the
importance of positive or negative quadrant dependence notions (PQD or NQD)
introduced by Lehmann (1966), in different areas of applied probability and
statistics, as an example, see; Dhaene and Goovaerts (1997), Denuit et al.
(2001). Two rv’s are said to be PQD when the probability that they are
simultaneously large (or small) is at least as great as it would be were they
are independent. In terms of copula, if their copula is greater than their
product, i.e., $C(u_{1},u_{2})>u_{1}u_{2}$ or, simply $C>C^{\perp},$ where
$C^{\perp}$ denotes the product copula. For the sake of brevity, we will
restrict ourselves to concepts of positive dependence.
The main idea of this paper is to use the information of dependence between
PQD or NQD risks to quantifying insurance losses and measuring financial risk
assessments, we propose a risk measure defined by:
$\mathbb{CCTE}_{X_{1}}\left(s;t\right):=\mathbb{E}\left[\left.X_{1}\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right].$
We will call this new risk measure by the Copula Conditional Tail Expectation
(CCTE), like a risk measure which measure the conditional expectation given
the two dependents losses exceeds $VaR_{X_{1}}\left(s\right)$ and
$VaR_{X_{2}}\left(t\right)$ for $0<s,t<1$ and usually with $s,t>0.9.$ Again,
CCTE satisfies all the desirable properties of a coherent risk measure
(Artzner et al., 1999). The notion of copula in risk measure filed has
recently been considered by several authors, see for instance Embrechts et al.
(2003a), Di Clemente and Romano (2004), Dalla Valle (2009), Brahimi et al.
(2010) and the references therein.
This risk measures can give a good quantifying of losses when we have a
combined dependents risks, this dependence can influence in the losses of
interested risks. Therefore, quantify the riskiness of our position is useful
to decide if it acceptable or not. For this reason we use the all informations
a bout this interest risk. The dependence of our risk with other risks is one
of important information that we must take it in consideration.
This paper is organized as follows. In section 2, we give an explicit
formulations of the new notion copula conditional tail expectation risk
measure in bivariate case. The relationship of this new concept and tail
dependence measure, given in section 3. In section 4 we presents an
illustration examples to explain how to use the new CCTE measure. Application
in real financial data is given in section 5. Concluding notes are given in
Section 6. Proofs are relegated to the Appendix.
## 2\. Copula conditional tail expectation
A risk measure quantifies the risk exposure in a way that is meaningful for
the problem at hand. The most commonly used risk measure in finance and
insurance are: VaR and CTE (also known as Tail-VaR or expected shortfall). The
risk measure is simply the loss size for which there is a small (e.g. $1\%$)
probability of exceeding. For some time, it has been recognized that this
measure suffers from serious deficiencies if losses are not normally
distributed.
According to Artzner et al. (1999) and Wirch and Hardy (1999), the conditional
tail expectation of a random variable $X_{1}$ at its
$VaR_{X_{1}}\left(s\right)$ is defined by:
$\mathbb{CTE}_{X_{1}}\left(s\right)=\frac{1}{1-F_{X_{1}}(VaR_{X_{1}}\left(s\right))}\int_{VaR_{X_{1}}\left(s\right)}^{\infty}xdF_{X_{1}}(x),$
where $F_{X_{1}}$ is the df of $X_{1}$.
Since $X_{1}$ is continuous, then $F_{X_{1}}(VaR_{X_{1}}\left(s\right))=s,$ it
follows that for all $0<s<1$
$\mathbb{CTE}_{X_{1}}\left(s\right)=\frac{1}{1-s}\int_{s}^{1}VaR_{X_{1}}\left(u\right)du.$
(2.3)
The CTE can be larger that the VaR measure for the same value of level $s$
described above since it can be thought of as the sum of the quantile
$VaR_{X_{1}}\left(s\right)$ and the expected excess loss. Tail-VaR is a
coherent measure in the sense of Artzner et al. (1999). For the application of
this kind of coherent risk measures we refer to the papers Artzner et al.
(1999) and Wirch and Hardy (1999).
Thus the CTE is nothing, see Overbeck and Sokolova (2008), but the
mathematical transcription of the concept of ”average loss in the worst
$100(1-s)\%$ case”, defining by $\upsilon=VaR_{X_{1}}(s)$ a critical loss
threshold corresponding to some confidence level $s,$
$\mathbb{CTE}_{X_{1}}(s)$ provides a cushion against the mean value of losses
exceeding the critical threshold $\upsilon.$
Now, assume that $X_{1}$ and $X_{2}$ are dependent with joint df $H$ and
continuous margins $F_{X_{i}},$ $i=1,2,$ respectively. Through this paper we
calls $X_{1}$ the target risk and $X_{2}$ the associated risk. In this case,
the problem becomes different and its resolution requires more than the usual
background.
Our contribution is to introduce the copula notion to provide more flexibility
to the CTE of risk of rv’s in terms of loss and dependence structure. For
comprehensive details on copulas one may consult the textbook of Nelsen
(2006).
According to Sklar’s Theorem Sklar (1959), there exists a unique copula
$C:\left[0,1\right]^{d}\rightarrow\left[0,1\right]$ such that
$H\left(x_{1},x_{2}\right)=C\left(F_{1}\left(x_{1}\right),F_{2}\left(x_{2}\right)\right).$
(2.4)
The formula of CTE focuses only on the average of loss. For this you should
think of an other formula to be more inclusive, this formula must take in
consideration the dependence structure and the behavior of margin tails. These
two aspects have an important influence when quantifying risks. On the other
hand if the correlation factor is neglected, the calculation of the CTE
follows from formula (2.3), which only focuses on the target risk.
Now by considering the correlation between the target and the associated
risks, we define a new notion of CTE called Copula Conditional Tail
Expectation (CCTE) given in (1.2), this notion led to give a risk measurement
focused in the target risk and the link between target and associated risk.
Let’s denote the survival functions by
$\overline{F}_{i}(x_{i})=\mathbb{P}(X_{i}>x_{i}),$ $i=1,2,$ and the joint
survival function by
$\overline{H}(x_{1},x_{2})=\mathbb{P}(X_{1}>x_{1},X_{2}>x_{2}).$ The function
$\overline{C}$ which couples $\overline{H}$ to $\overline{F}_{i},$ $i=1,2$ via
$\overline{H}(x_{1},x_{2})=\overline{C}(\overline{F}_{1}(x_{1}),\overline{F}_{2}(x_{2}))$
is called the survival copula of $\left(X_{1},X_{2}\right).$ Furthermore,
$\overline{C}$ is a copula, and
$\overline{C}(u_{1},u_{2})=u_{1}+u_{2}-1+C(1-u_{1},1-u_{2}),$ (2.5)
where $C$ is the (ordinary) copula of $X_{1}$ and $X_{2}.$ For more details on
the survival copula function see, Section 2.6 in Nelsen (2006).
The CCTE of the target risk $X_{1}$ with respect to the associated risk
$X_{2}$ is given in the following proposition.
###### Proposition 2.1.
Let $\left(X_{1},X_{2}\right)$ a bivariate rv with joint df represented by the
copula $C.$ Assume that $X_{1}$ have a finite mean and df $F_{X_{1}}.$ Then
for all $s$ and $t$ in $\left(0,1\right)$ the copula conditional tail expected
of $X_{1}$ with respect to the bivariate thresholds $(s,t)$ is given by
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)\left(1-C_{u}(u,t)\right)du}{\overline{C}\left(1-s,1-t\right)},$
(2.6)
where $F_{X_{1}}^{-1}$ is the quantile function of $F_{X_{1}}$and
$C_{u}(u,v):=\partial C(u,v)/\partial u.$
This notion does not depend on the df of the associated risk, but it depend
only by the copula function and the df of target risk.
By definition of PQD risks we have that $C(u,v)>uv,$ then it easy to check
that
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)\leq\frac{\mathbb{CTE}_{X_{1}}\left(s\right)}{\left(1-t\right)}\text{
for }s,t<1,$
next, in Section 4, we will proved that the risk when we consider the
correlation between PQD risks is greater than in the case of a single one.
That means, for all $s\leq t$ and $s,$ $t$ in $\left(0,1\right)$ then
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)\geq\mathbb{CTE}_{X_{1}}\left(s\right).$
(2.7)
Notice that in the NQD rv’s we have the reverse inequality of (2.7) and the
CCTE coincide with CTE measures in the non-dependence case, i.e. the copula
$C=C^{\bot}.$
## 3\. CCTE and tail dependence
The concept of tail dependence is an asymptotic measure of the dependence
between two random variables in the tail of their joint distribution function.
Specifically, tail dependence is the probability that a random variable
$X_{1}$ and $X_{2}$ takes a values in the extreme tail of its distributions
simultaneously, for example we consider $X_{1}$ and $X_{2}$ which measure
bankruptcy for two companies and both companies simultaneously go bankrupt.
We describes the joint upper tail dependence of the random variables $X_{1}$
and $X_{2}:$
$\lim_{\begin{subarray}{c}t\rightarrow 1^{-}\\\ s\rightarrow
1^{-}\end{subarray}}\mathbb{P}\left(\left.X_{1}>F_{X_{1}}^{-1}\left(s\right)\right|X_{2}>F_{X_{2}}^{-1}\left(t\right)\right)$
However, it can be seen as a good indicator of systemic risk (for $s=t$). If
we considering the tail dependence as a dependence measure in the extreme
tails of the joint distribution, it is possible for two rv’s to be dependent,
but for there to be no dependence in the tail of the distributions, this is
the case described for example by the Gaussian copula, hyperbolic copula or
Farlie-Gumbel-Morgenstern copula (tail dependence is zero). Furthermore, the
Clayton copula puts the entire tail dependence in the lower tail unlike Gumbel
copula in the upper tail and the Student copula behave identically in the
lower as in the upper tail. However, it is not suitable to model extreme
negative outcomes similarly as with extreme positive outcomes.
###### Remark 3.1.
Negative outcomes can be treated in the same way that the extremes positive
outcomes by replacing their copula by the survival copula.
The tail dependence can be also expressed through copula
$\lambda_{U}=\lim_{u\rightarrow 1^{-}}\frac{1-2u+C\left(u,u\right)}{1-u}\text{
and }\lambda_{L}=\lim_{u\rightarrow 0^{+}}\frac{C\left(u,u\right)}{u}.$
Now, let’s denote by
$\tilde{\lambda}_{U}\left(u,v\right):=\frac{1-u-v+C\left(u,v\right)}{1-v}\text{
and }\tilde{\lambda}_{L}\left(u,v\right):=\frac{C\left(u,v\right)}{v}.$
Note that $\lim_{u,v\rightarrow
1^{-}}\tilde{\lambda}_{U}\left(u,v\right)=\lambda_{U}$ and
$\lim_{u,v\rightarrow 0^{+}}\tilde{\lambda}_{L}\left(u,v\right)=\lambda_{L}.$
We can rewrite CCTE of according to $\tilde{\lambda}_{U}$ as
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)\left(1-C_{u}(u,t)\right)du}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)},$
this has no impact on the limiting value at $0$ for PQD risks. Then we have
$\lim_{\begin{subarray}{c}s\rightarrow 1^{-}\\\ t\rightarrow
1^{-}\end{subarray}}\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)=0.$
From Theorem 2.2.7 in (Nelsen, 2006, page 13) we have $0\leq C_{u}(u,t)\leq 1$
for such $u$ and $t,$ then
$\left|\mathbb{CCTE}_{X_{1}}\left(s;t\right)\right|\leq\left|\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)}\right|\leq\left|\frac{\mathbb{E}\left(X_{1}\right)}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)}\right|.$
In the next section, we give an example to describe the impact of the upper
tail dependence nearly $1$ and the lower tail dependence near $0$ in CCTE, and
we discuss the relationship between the properties of the dependence of copula
model with upper and lower tail dependence and how to derive the degree of
risk in each case.
## 4\. Illustration examples
### 4.1. CCTE via Farlie-Gumbel-Morgenstern Copulas
One of the most important parametric family of copulas is the Farlie-Gumbel-
Morgenstern (FGM) family defined as
$C_{\theta}^{FGM}(u,v)=uv+\theta uv(1-u)(1-v),\ \ \ u,v\in[0,1],$ (4.8)
where $\theta\in[-1,1].$ The family was discussed by Morgenstern (1956),
Gumbel (1958) and Farlie (1960).
The copula given in (4.8) is PQD for $\theta\in(0,1]$ and NQD for
$\theta\in[-1,0).$ In practical applications this copula has been shown to be
somewhat limited, for copula dependence parameter
$\theta\in\left[-1,1\right],$ Spearman’s correlation
$\rho\in\left[-1/3,1/3\right]$ and Kendall’s $\tau\in\left[-2/9,2/9\right],$
for more details on copulas see, for example, Nelsen (2006).
Members of the FGM family are symmetric, i.e.,
$C_{\theta}^{FGM}(u,v)=C_{\theta}^{FGM}(v,u)$ for all $(u,v)$ in
$\left[0,1\right]^{2}$ and have the lower and upper tail dependence
coefficients equal to $0.$
A pair $\left(X,Y\right)$ of rv’s is said to be exchangeable if the vectors
$\left(X,Y\right)$ and $\left(Y,X\right)$ are identically distributed. Note
that, in applications, exchangeability may not always be a realistic
assumption. For identically distributed continuous random variables,
exchangeability is equivalent to the symmetry of the FGM copula.
For practical purposes we consider a copula families with only positive
dependence. Furthermore, risk models are often designed to model positive
dependence, since in some sense it is the “dangerous” dependence: assets (or
risks) move in the same direction in periods of extreme events, see Embrechts
et al. (2003b).
Consider the bivariate loss PQD rv’s $(X_{i},Y),$ $i=1,2,3,$ having continuous
marginal df’s $F_{X_{i}}(x)$ and $F_{Y}(y)$ and joint df $H_{X_{i},Y}(x,y)$
represented by FGM copula of parameters $\theta_{i}$, respectively for
$i=1,2,3$
$H_{X_{i},Y}(x,y)=C_{\theta_{i}}^{FGM}(F_{X_{i}}\left(x\right),F_{Y}\left(y\right)).$
The marginal survival functions $\overline{F}_{X_{i}}(x),$ $i=1,2,3$ and
$\overline{F}_{Y}(y)$ are given by
$\overline{F}_{X_{i}}\left(x\right)=\left\\{\begin{tabular}[]{ll}$\left(1+x\right)^{-\alpha},$&$x\geq
0,$\\\ $1,$&$x<0,$\end{tabular}\ \ \ \right.\text{ and }\ \
\overline{F}_{Y}\left(y\right)=\left\\{\begin{tabular}[]{ll}$\left(1+y\right)^{-\alpha},$&$y\geq
0,$\\\ $1,$&$y<0.$\end{tabular}\ \ \ \ \ \ \ \right.$ (4.9)
where $\alpha>0$ called the Pareto index, the case $\alpha\in(1,2)$ means that
$X_{i}$ have a heavy-tailed distributions. So that $X_{i}$ and $Y$ have
identical Pareto df’s.
For each couple $\left(X_{i},Y\right),$ $i=1,2,3,$ we propose
$\theta_{1}=0.01,$ $\theta_{2}=0.5$ and $\theta_{3}=1,$ respectively. The
choice of parameters $\theta_{i},i=1,2,3$ correspond respectively to the weak,
medium and the high dependence.
In this example, the target risks are $X_{i}$ and the associated risk is $Y.$
The $\mathbb{CTE}$’s and the VaR’s of $X_{i}$ are the same and are given
respectively by
$\mathbb{CTE}_{X_{i}}\left(s\right)=\frac{\alpha\left(1-s\right)^{-1/\alpha}}{\alpha-1}$
(4.10)
and
$VaR_{X_{i}}\left(s\right)=(1-s)^{-1/\alpha},$ (4.11)
for $i=1,2,3.$
We have that
$\overline{C}\left(1-s,1-t\right)=(1-s)(1-t)\left(st\theta_{i}+1\right).$
(4.12)
Now, we calculate
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{s}^{1}(1-u)^{-1/\alpha}\left(1-t\right)\left(2tu\theta_{i}-t\theta_{i}+1\right)du$
by substitution (4.12) we get
$\mathbb{CCTE}_{X_{i}}\left(s;t\right)=\frac{\alpha\left(2\alpha+t\theta_{i}-2st\theta_{i}+2st\alpha\theta_{i}-1\right)}{\left(2\alpha^{2}-3\alpha+1\right)\left(st\theta_{i}+1\right)}\left(1-s\right)^{-1/\alpha}.$
(4.13)
We have in Table 4.1 and Figures 4.1 the comparison of the riskiness of
$X_{1},$ $X_{2}$ and $X_{3}.$ Recall that, the $\mathbb{CTE}$’s risk measure
of $X_{i}$ at level $s$ are the same in all cases. Note that $\mathbb{CCTE}$
coincide with $\mathbb{CTE}$ in the independence case $(\theta_{1}=0).$ The
$\mathbb{CCTE}$ of the loss $X_{3}$ is riskier than $X_{2}$ and $X_{1}$ but
not very significant, in the 6th column of Table 4.1, the relative difference
between $64.7946$ and $64.633$ is only about $0.025\%$. This is due to that
FGM copula does not take into account the dependence in upper and lower tail
$(\lambda_{L}=\lambda_{U}=0).$ In this case we can not clearly confirm which
is the risk the more dangerous.
${\small s}$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.6415}$ | ${\small 5.5013}$ | ${\small 6.9144}$ | ${\small 9.8192}$ | ${\small 21.5443}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.9247}$ | ${\small 16.5039}$ | ${\small 20.7433}$ | ${\small 29.4577}$ | ${\small 64.6330}$
${\small t}$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=0.01}$
${\small 0.9000}$ | ${\small 13.9309}$ | ${\small 13.9311}$ | ${\small 13.9312}$ | ${\small 13.9314}$ | ${\small 13.9316}$
${\small 0.9225}$ | ${\small 16.5096}$ | ${\small 16.5097}$ | ${\small 16.5099}$ | ${\small 16.5100}$ | ${\small 16.5101}$
${\small 0.9450}$ | ${\small 20.7484}$ | ${\small 20.7485}$ | ${\small 20.7487}$ | ${\small 20.7488}$ | ${\small 20.7489}$
${\small 0.9675}$ | ${\small 29.4619}$ | ${\small 29.4620}$ | ${\small 29.4621}$ | ${\small 29.4623}$ | ${\small 29.4624}$
${\small 0.9900}$ | ${\small 64.6359}$ | ${\small 64.6359}$ | ${\small 64.6360}$ | ${\small 64.6361}$ | ${\small 64.6362}$
${\small t}$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ \ {\small\theta=0.5}$
${\small 0.9000}$ | ${\small 14.1477}$ | ${\small 14.1517}$ | ${\small 14.1555}$ | ${\small 14.1594}$ | ${\small 14.1631}$
${\small 0.9225}$ | ${\small 16.7072}$ | ${\small 16.7108}$ | ${\small 16.7143}$ | ${\small 16.7178}$ | ${\small 16.7212}$
${\small 0.9450}$ | ${\small 20.9234}$ | ${\small 20.9266}$ | ${\small 20.9297}$ | ${\small 20.9327}$ | ${\small 20.9357}$
${\small 0.9675}$ | ${\small 29.6077}$ | ${\small 29.6103}$ | ${\small 29.6129}$ | ${\small 29.6154}$ | ${\small 29.6179}$
${\small 0.9900}$ | ${\small 64.7336}$ | ${\small 64.7353}$ | ${\small 64.7370}$ | ${\small 64.7387}$ | ${\small 64.7404}$
${\small t}$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=1}$
${\small 0.9000}$ | ${\small 14.2709}$ | ${\small 14.2756}$ | ${\small 14.2803}$ | ${\small 14.2848}$ | ${\small 14.2892}$
${\small 0.9225}$ | ${\small 16.8183}$ | ${\small 16.8226}$ | ${\small 16.8267}$ | ${\small 16.8308}$ | ${\small 16.8348}$
${\small 0.9450}$ | ${\small 21.0208}$ | ${\small 21.0245}$ | ${\small 21.0281}$ | ${\small 21.0316}$ | ${\small 21.0351}$
${\small 0.9675}$ | ${\small 29.6880}$ | ${\small 29.6910}$ | ${\small 29.6940}$ | ${\small 29.6969}$ | ${\small 29.6997}$
${\small 0.9900}$ | ${\small 64.7868}$ | ${\small 64.7888}$ | ${\small 64.7908}$ | ${\small 64.7927}$ | ${\small 64.7946}$
Table 4.1. Risk measures of dependent pareto(1.5) rv’s with FGM copula. Figure
4.1. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD pareto
(1.5) rv’s with FGM copula and $0.9\leq s=t\leq 0.99$
### 4.2. CCTE via Archimedean Copulas
A bivariate copula is said to be Archimedean (see, Genest and MacKay, 1986) if
it can be expressed by
$C(u_{1},u_{2})=\psi^{[-1]}\left(\psi(u_{1})+\psi(u_{2})\right),$
where $\psi,$ called the generator of $C,$ is a continuous strictly decreasing
convex function from $\left[0,1\right]$ to $[0,\infty]$ such that $\psi(1)=0$
with $\psi^{[-1]}$ denotes the pseudo-inverse of $\psi,$ that is
$\psi^{[-1]}\left(t\right)=\left\\{\begin{tabular}[]{lll}$\psi^{-1}\left(t\right),$&for&$t\in\left[0,\psi\left(0\right)\right],$\\\
$0,$&for&$t\geq\psi\left(0\right).$\end{tabular}\ \right.$
When $\psi(0)=\infty,$ the generator $\psi$ and $C$ are said to be strict and
therefore $\psi^{[-1]}=\psi^{-1}.$ All notions of positive dependence that
appeared in the literature, including the weakest one of PQD as defined by
Lehmann (1966), require the generator to be strict.
Archimedean copulas are widely used in applications due to their simple form,
a variety of dependence structures and other “nice” properties. For example,
in the Actuarial field: the idea arose indirectly in Clayton (1978) and was
developed in Oakes (1982), Cook and Johnson (1981). A survey of Actuarial
applications is in Frees and Valdez (998).
For an Archimedean copula, the Kendall’s tau can be evaluated directly from
the generator of the copula, as shown in Genest and MacKay (1986)
$\tau=4{\displaystyle\int_{0}^{1}}\frac{\psi\left(u\right)}{\psi^{\prime}\left(u\right)}du+1.$
(4.14)
where $\psi^{\prime}\left(u\right)$ exists a.e., since the generator is
convex. This is another “nice” feature of Archimedean copulas. As for tail
dependency, as shown in (Joe, 1997, page 105) the coefficient of upper tail
dependency is
$\lambda_{U}=2-2\lim_{s\rightarrow
0^{+}}\frac{\psi^{-1\prime}\left(2s\right)}{\psi^{-1\prime}\left(s\right)}$
and the coefficient of lower tail dependency is
$\lambda_{L}=2\lim_{s\rightarrow+\infty}\frac{\psi^{-1\prime}\left(2s\right)}{\psi^{-1\prime}\left(s\right)}.$
A collection of twenty-two one-parameter families of Archimedean copulas can
be found in Table 4.1 of Nelsen (2006).
Notice that in the case of Archimedean copula the copula conditional tail
expectation has not an explicit formula, so we give by the following
Proposition the expression of CCTE in terms of generator.
###### Proposition 4.1.
Let $C$ be an Archimedean copula absolutely continuous with generator $\psi,$
the CCTE of the target risk in terms of generator with respect to the
bivariate thresholds $(s,t),$ $0<s,t<1,$ is given by
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\left(1-s\right)\mathbb{CTE}_{X_{1}}\left(s\right)-\int_{s}^{1}\frac{\psi^{\prime}(u)F_{X_{1}}^{-1}\left(u\right)}{\psi^{\prime}\left(C\left(u,t\right)\right)}du\right).$
(4.15)
Note that in practice we can easily fit copula-based models with the maximum
likelihood method or with estimate the dependence parameter by the
relationship between Kendall’s tau of the data and the generator of the
Archimedean copula given in (4.14) under the specified copula model.
In the following section we give same examples to explain how to calculate and
compare the CCTE with other risk measure such VaR and CTE.
#### 4.2.1. CCTE via Gumbel Copula
The Gumbel family has been introduced by Gumbel (1960). Since it has been
discussed in Hougaard (1986), it is also known as the Gumbel-Hougaard family.
The Gumbel copula is an asymmetric Archimedean copula. This copula is given by
$C_{\theta}^{G}\left(u,v\right)=\exp\left\\{-\left[\left(-\ln
u\right)^{\theta}+\left(-\ln v\right)^{\theta}\right]^{1/\theta}\right\\},$
its generator is
$\psi_{\theta}\left(t\right)=\left(-\ln t\right)^{\theta}.$
The dependence parameter is restricted to the interval $[1,\infty).$ It
follows that the Gumbel family can represent independence and “positive”
dependence only, since the lower and upper bound for its parameter correspond
to the product copula and the upper Fréchet bound. The Gumbel copula families
is often used for modeling heavy dependencies in right tail. It exhibits
strong upper tail dependence $\lambda_{U}=2-2^{1/\theta}$ and relatively weak
lower tail dependence $\lambda_{L}=0.$ If outcomes are known to be strongly
correlated at high values but less correlated at low values, then the Gumbel
copula will be an appropriate choice.
We give the CCTE of rv’s $X_{i},$ $i=1,2,3$ in terms of Gumbel copula by
$\displaystyle\mathbb{CCTE}_{X_{i}}\left(s;t\right)$
$\displaystyle=\frac{1}{\overline{C}_{\theta_{i}}^{G}\left(1-s,1-t\right)}\left(\frac{\alpha\left(1-s\right)^{1-1/\alpha}}{\alpha-1}\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\left.-\int_{s}^{1}u^{-1}\left(1-u\right)^{-1/\alpha}\left(-\ln
u\right)^{\theta_{i}-1}C_{\theta_{i}}^{G}\left(u,t\right)\left(-\ln\left(C_{\theta_{i}}^{G}\left(u,t\right)\right)\right)^{1-\theta_{i}}du\right),$
(4.16)
where
$\overline{C}_{\theta_{i}}^{G}\left(s,t\right)=s+t-1+C_{\theta_{i}}^{G}(1-s,1-t).$
The CTE’s and VaR’s of $X_{i}$ is the same and it’s given respectively by
(4.10) and (4.11), for $i=1,2,3.$
By the relationship between Kendall’s tau $\tau$ and the Gumbel copula
parameter $\theta_{i}$ given by:
$\tau=\left(\theta_{i}-1\right)/\theta_{i},$
we select the values of $\theta_{i}$ corresponding respectively to a weak, a
moderate and a strong positive association witch summarized in Table 4.2.
$\lambda_{U}$ | $\theta_{i}$ | $\tau$
---|---|---
$0.013$ | $1.01$ | $0.009$
$0.585$ | $2$ | $0.500$
$0.928$ | $10$ | $0.900$
Table 4.2. Upper tail, Kendall’s tau and Gumbel copula parameters used in
calculate of risk measures.
Table 4.3 and Figure 4.2 shows that the loss $X_{1}$ is considerably riskier
than $X_{2}$ and $X_{3},$ in the 6th column of Table 4.3, the relative
difference between $112.1868$ and $69.6017$ is about $61.184\%.$
${\small s}$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.641}$ | ${\small 5.501}$ | ${\small 6.914}$ | ${\small 9.819}$ | ${\small 21.544}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.924}$ | ${\small 16.503}$ | ${\small 20.743}$ | ${\small 29.457}$ | ${\small 64.633}$
${\small t}$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=1.01}$
${\small 0.9000}$ | ${\small 15.937}$ | ${\small 16.485}$ | ${\small 17.410}$ | ${\small 19.365}$ | ${\small 25.007}$
${\small 0.9225}$ | ${\small 18.879}$ | ${\small 19.528}$ | ${\small 20.625}$ | ${\small 22.948}$ | ${\small 33.690}$
${\small 0.9450}$ | ${\small 23.699}$ | ${\small 24.507}$ | ${\small 25.873}$ | ${\small 28.760}$ | ${\small 40.588}$
${\small 0.9675}$ | ${\small 33.556}$ | ${\small 34.667}$ | ${\small 36.534}$ | ${\small 40.454}$ | ${\small 56.275}$
${\small 0.9900}$ | ${\small 72.992}$ | ${\small 75.133}$ | ${\small 78.645}$ | ${\small 85.726}$ | ${\small 112.18}$
${\small t}$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ {\small\ \theta=2}$
${\small 0.9000}$ | ${\small 18.158}$ | ${\small 19.769}$ | ${\small 22.691}$ | ${\small 28.950}$ | ${\small 52.929}$
${\small 0.9225}$ | ${\small 20.209}$ | ${\small 21.653}$ | ${\small 24.338}$ | ${\small 30.607}$ | ${\small 53.742}$
${\small 0.9450}$ | ${\small 23.842}$ | ${\small 25.059}$ | ${\small 27.383}$ | ${\small 33.070}$ | ${\small 55.276}$
${\small 0.9675}$ | ${\small 31.849}$ | ${\small 32.766}$ | ${\small 34.543}$ | ${\small 39.128}$ | ${\small 59.207}$
${\small 0.9900}$ | ${\small 66.087}$ | ${\small 66.606}$ | ${\small 67.583}$ | ${\small 70.074}$ | ${\small 86.385}$
${\small t}$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=10}$
${\small 0.9000}$ | ${\small 13.765}$ | ${\small 16.694}$ | ${\small 23.338}$ | ${\small 39.483}$ | ${\small 128.31}$
${\small 0.9225}$ | ${\small 15.612}$ | ${\small 16.626}$ | ${\small 21.902}$ | ${\small 36.924}$ | ${\small 120.00}$
${\small 0.9450}$ | ${\small 19.378}$ | ${\small 19.446}$ | ${\small 20.821}$ | ${\small 32.807}$ | ${\small 106.54}$
${\small 0.9675}$ | ${\small 29.457}$ | ${\small 29.458}$ | ${\small 29.480}$ | ${\small 31.692}$ | ${\small 95.737}$
${\small 0.9900}$ | ${\small 64.633}$ | ${\small 65.034}$ | ${\small 66.412}$ | ${\small 67.753}$ | ${\small 69.601}$
Table 4.3. Risk measures of PQD pareto (1.5) rv’s with Gumbel copula. Figure
4.2. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD pareto
(1.5) rv’s with Gumbel copula and $0.9\leq s=t\leq 0.99.$
By definition of our risk measurement, the risks concern the study is
necessarily comonotonic, then to have a good decision we must select a copula
model with upper tail dependence, we show in next example that the dependence
models with no upper tails dependence do not helps us to take a decision.
#### 4.2.2. CCTE via Clayton Copula
In the following example, we consider the bivariate Clayton copula which is a
member of the class of Archimedean copula, with the dependence parameter
$\theta$ in $\left.\left[-1,\infty\right)\right\backslash\left\\{0\right\\}$.
The Clayton family was first proposed by Clayton (1978) and studied by Oakes,
(1982; 1986), Cox and Oakes, (1981; 1986). The Clayton copula has been used to
study correlated risks, it has the form
$C_{\theta}^{C}(u,v):=\left[\max\left(u^{-\theta}+v^{-\theta}-1,0\right)\right]^{-1/\theta}.$
(4.17)
For $\theta>0$ the copulas are strict and the copula expression simplifies to
$C_{\theta}^{C}(u,v)=\left(u^{-\theta}+v^{-\theta}-1\right)^{-1/\theta}.$
(4.18)
Asymmetric tail dependence is prevalent if the probability of joint extreme
negative realizations differs from that of joint extreme positive
realizations. it can be seen that the Clayton copula assigns a higher
probability to joint extreme negative events than to joint extreme positive
events. The Clayton copula is said to display lower tail dependence
$\lambda_{L}=2^{-1/\theta},$ while it displays zero upper tail dependence
$\lambda_{U}=0,$ for $\theta\geq 0.$ The converse can be said about the Gumbel
copula (displaying upper but zero lower tail dependence). The margins become
independent as $\theta$ approaches to zero, while for
$\theta\rightarrow\infty,$ the Clayton copula arrives at the comonotonicity
copula. For $\theta=-1$ we obtain the Fréchet-Hoeffding lower bound and the
copula attains the Fréchet upper bound as $\theta$ approaches to infinity.
Clayton copula is the best suited for applications in which two outcomes are
likely to experience low values together, since the dependence is strong in
the lower tail and weak in the upper tail.
We take the same example as in the Subsection 4.1, we may now represents the
joint df’s $H_{i},$ $i=1,2,3,$ respectively by the Clayton copulas
$C_{\theta_{i}}^{C}$ given in (4.18) to have an idea about the effects of
lower tail dependence on our risk measurement.
The relationship between Kendall’s tau $\tau$ and the Clayton copula is given
by
$\tau=\theta_{i}/\left(\theta_{i}+2\right),$ (4.19)
we select a different dependents parameters corresponding to several levels of
positive dependency summarized in Table 4.4 for a weak, a moderate and a
strong positive association, to calculate and compare the CCTE’s of
$X_{i},i=1,2,3.$
$\lambda_{L}$ | $\theta_{i}$ | $\tau$
---|---|---
$0.250$ | $0.5$ | $0.200$
$0.707$ | $2$ | $0.500$
$0.943$ | $12$ | $0.857$
Table 4.4. Lower tail, Kendall’s tau and Clayton copula parameters used in
calculate of risk measures.
The CCTE of the rv’s $X_{i}$ with respect to the bivariate thresholds $(s,t)$
is given by
$\mathbb{CCTE}_{X_{i}}\left(s;t\right)=\frac{1}{\overline{C}_{\theta_{i}}^{C}\left(1-s,1-t\right)}\left(\frac{\alpha\left(1-s\right)^{-1/\alpha+1}}{\left(\alpha-1\right)}-{\displaystyle\int_{s}^{1}}\frac{\left(t^{-\theta_{i}}+u^{-\theta_{i}}-1\right)^{-1-1/\theta_{i}}}{\left(1-u\right)^{1/\alpha}u^{\theta_{i}+1}}du\right).$
(4.20)
The differences as reported in Table 4.5 and Figure 4.3 do not look very
significant, in the 6th column of Table 4.5, the relative difference between
$66.3802$ and $64.6330$ is only about $1.027\%.$ The differences is not found
also when $t$ is small compared to $s,$
$\mathbb{CCTE}_{X_{1}}\left(0.99;0.01\right)=64.6332$ and
$\mathbb{CCTE}_{X_{3}}\left(0.99;0.01\right)=64.6329$ the difference is about
$1\%.$
$s$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.641}$ | ${\small 5.501}$ | ${\small 6.914}$ | ${\small 9.819}$ | ${\small 21.544}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.924}$ | ${\small 16.503}$ | ${\small 20.743}$ | ${\small 29.457}$ | ${\small 64.633}$
$t$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=0.5}$
${\small 0.9000}$ | ${\small 14.088}$ | ${\small 14.092}$ | ${\small 14.096}$ | ${\small 14.101}$ | ${\small 14.105}$
${\small 0.9225}$ | ${\small 16.652}$ | ${\small 16.656}$ | ${\small 16.660}$ | ${\small 16.664}$ | ${\small 16.667}$
${\small 0.9450}$ | ${\small 20.874}$ | ${\small 20.878}$ | ${\small 20.881}$ | ${\small 20.884}$ | ${\small 20.888}$
${\small 0.9675}$ | ${\small 29.566}$ | ${\small 29.569}$ | ${\small 29.572}$ | ${\small 29.575}$ | ${\small 29.577}$
${\small 0.9900}$ | ${\small 64.706}$ | ${\small 64.707}$ | ${\small 64.709}$ | ${\small 64.711}$ | ${\small 64.713}$
$t$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ \ {\small\theta=2}$
${\small 0.9000}$ | ${\small 14.500}$ | ${\small 14.536}$ | ${\small 14.572}$ | ${\small 14.610}$ | ${\small 14.648}$
${\small 0.9225}$ | ${\small 17.023}$ | ${\small 17.056}$ | ${\small 17.089}$ | ${\small 17.123}$ | ${\small 17.159}$
${\small 0.9450}$ | ${\small 21.199}$ | ${\small 21.227}$ | ${\small 21.257}$ | ${\small 21.288}$ | ${\small 21.319}$
${\small 0.9675}$ | ${\small 29.833}$ | ${\small 29.857}$ | ${\small 29.882}$ | ${\small 29.907}$ | ${\small 29.934}$
${\small 0.9900}$ | ${\small 64.882}$ | ${\small 64.898}$ | ${\small 64.915}$ | ${\small 64.932}$ | ${\small 64.950}$
$t$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=12}$
${\small 0.9000}$ | ${\small 15.605}$ | ${\small 16.118}$ | ${\small 16.743}$ | ${\small 17.494}$ | ${\small 18.383}$
${\small 0.9225}$ | ${\small 17.913}$ | ${\small 18.366}$ | ${\small 18.930}$ | ${\small 19.618}$ | ${\small 20.447}$
${\small 0.9450}$ | ${\small 21.888}$ | ${\small 22.274}$ | ${\small 22.762}$ | ${\small 23.371}$ | ${\small 24.119}$
${\small 0.9675}$ | ${\small 30.331}$ | ${\small 30.637}$ | ${\small 31.033}$ | ${\small 31.536}$ | ${\small 32.169}$
${\small 0.9900}$ | ${\small 65.169}$ | ${\small 65.363}$ | ${\small 65.619}$ | ${\small 65.951}$ | ${\small 66.380}$
Table 4.5. Risk measures of dependent pareto (1.5) rv’s with Clayton copula.
Figure 4.3. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD
pareto (1.5) rv’s with Clayton copula and $0.9\leq s=t\leq 0.99.$
## 5\. Application
The relationship between the parameter of an Archimedean copula and Kendall’s
tau has allowed us to calculate the value of this parameter assuming a well
precise Archimedean copula e.g., Gumbel copula. Once endowed with the
parameter value, we are able to compute any joint probability between the
stock indices.
For instance we analyzed $500$ observations from four European stock indices
return series calculated by $\log\left(X_{t+1}/X_{t}\right)$ for the period
July 1991 to June 1993 (see, Figure 5.4 ), available in ”QRM and datasets
packages” of R software, it contains the daily closing prices of major
European stock indices: Germany DAX (Ibis), Switzerland SMI, France CAC and UK
FTSE. The data are sampled in business time, i.e., weekends and holidays are
omitted. Table 5.6 summaries the Kendall’s tau between the four Market Index
returns.
Figure 5.4. Scatterplots of $500$ pseudo-observations drawn from a four European stock indices returns. Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small 1}$ | ${\small 0.4052}$ | ${\small 0.4374}$ | ${\small 0.3706}$
SMI | ${\small 0.4052}$ | ${\small 1}$ | ${\small 0.3791}$ | ${\small 0.3924}$
CAC | ${\small 0.4374}$ | ${\small 0.3791}$ | ${\small 1}$ | ${\small 0.4076}$
FTSE | ${\small 0.3706}$ | ${\small 0.3924}$ | ${\small 0.4076}$ | ${\small 1}$
Table 5.6. Kendall’s tau matrix estimates from four European stock indices
returns.
By assuming that Gumbel copula represents our four dependence structures, we
obtain the fitted dependence parameters of the six bivariate joint df’s,
presented in Table 5.7.
Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small\infty}$ | ${\small 1.6815}$ | ${\small 1.7777}$ | ${\small 1.5888}$
SMI | ${\small 1.6815}$ | ${\small\infty}$ | ${\small 1.6106}$ | ${\small 1.6459}$
CAC | ${\small 1.7777}$ | ${\small 1.6106}$ | ${\small\infty}$ | ${\small 1.6880}$
FTSE | ${\small 1.5888}$ | ${\small 1.6459}$ | ${\small 1.6880}$ | ${\small\infty}$
Table 5.7. Fitted copula parameter correspoding to Kendall’s tau, Gumbel
copula.
The $\alpha$-stable distribution offers a reasonable improvement to the
alternative distributions, each stable distribution
$S_{\alpha}(\sigma;\beta;\mu)$ has the stability index $\alpha$ that can be
treated as the main parameter, when we make an investment decision, skewness
parameter $\beta$, in the range $[-1,1]$, scale parameter $\sigma$ and shift
parameter $\mu.$ In models that use financial data, it is generally assumed
that $\alpha\in(1,2].$ By using the ”fBasics” package in R software, based on
the maximum likelihood estimators to fit the parameters of a df’s of the four
Market Index returns, the results are summarized in Table 5.8.
| DAX | SMI | CAC | FTSE
---|---|---|---|---
${\small\alpha}$ | ${\small 1.6420}$ | ${\small 1.8480}$ | ${\small 1.6930}$ | ${\small 1.8740}$
${\small\beta}$ | ${\small 0.1470}$ | ${\small 0.1100}$ | ${\small 0.0380}$ | ${\small 0.9500}$
${\small\sigma}$ | ${\small 0.0046}$ | ${\small 0.0045}$ | ${\small 0.0006}$ | ${\small 0.0053}$
${\small\mu}$ | ${\small-0.0001}$ | ${\small 0.0006}$ | ${\small-0.0001}$ | ${\small-0.0005}$
Table 5.8. Maximum likelihood fit of four-parameters stable distribution to
four European stock indices retuns data.
The $\alpha$-stable distribution has Pareto-type tails, it’s like a power
function, i.e., $F$ is regularly varying (at infinity) with index
$\left(-\alpha\right),$ meaning that
$\overline{F}\left(x\right)=x^{-\alpha}L\left(x\right)$ as $x$ becomes large,
where $L>0$ is a slowly varying function, which can be interpreted as slower
than any power function (see, Resnick; 1987 and Seneta; 1976 for a technical
treatment of regular variation). By using the Equations (4.16) for the Gumbel
copula fitting, we calculate for a fixed levels $s=t=0.95$ the CCTE’s risk
measures for the all cases, the results are summarized in Table 5.9.
Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small-}$ | ${\small 21.5009}$ | ${\small 21.0786}$ | ${\small 21.9731}$
SMI | ${\small 14.2812}$ | ${\small-}$ | ${\small 14.4703}$ | ${\small 14.3737}$
CAC | ${\small 18.8362}$ | ${\small 19.4915}$ | ${\small-}$ | ${\small 19.1671}$
FTSE | ${\small 13.9075}$ | ${\small 13.7593}$ | ${\small 13.6576}$ | ${\small-}$
Table 5.9. CCTE’s Risk measures for $s=0.99$ and $t=0.99$ with Gumbel copula
(left panel) and Clayton copula (right panel).
The smallest values in Table 5.9 gives the lowest risk. So, the less risky
couples $(X,Y)$ are: (DAX, CAC), (SMI, DAX), (CAC, DAX) and (FTSE, CAC), where
$X$ is the target risk and $Y$ is the associated risk.
## 6\. Conclusion notes
This paper discussed a new risk measure called copula conditional tail
expectation. This measure aid to understanding the relationships among
multivariate assets and to help us significantly about how best to position
our investments and improve our financial risk protection.
Tables 4.3 show that the copula conditional tail expectation measure become
smaller as the dependency increase. However, CTE and VaR are neither
increasing nor decreasing as the correlation increase. Therefore, the
dependency information helps us to minimize the risk.
Acknowledgements. The author is indebted to an anonymous referee for their
careful reading and suggestions for improvements.
## 7\. Appendix
###### Proof of Proposition 2.1.
By conditional probability is easily to obtain
$\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)=\frac{\mathbb{P}\left(x\geq
X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)}{\mathbb{P}\left(X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)}\vskip
6.0pt plus 2.0pt minus 2.0pt$
On the other hand, we have
$\displaystyle\mathbb{P}\left(x\geq
X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)$
$\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s\right)-\mathbb{P}\left(F_{2}\left(X_{2}\right)\leq
t\right)+\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right),$
$\displaystyle\mathbb{P}\left(X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)\
$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(F_{1}\left(X_{1}\right)\leq
s\right)-\mathbb{P}\left(F_{2}\left(X_{2}\right)\leq
t\right)+\mathbb{P}\left(F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right)$ $\displaystyle=$ $\displaystyle
1-s-t+C\left(s,t\right)$ $\displaystyle=$
$\displaystyle\overline{C}\left(1-s,1-t\right),$
$\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s\right)=s-VaR_{X_{1}}\left(x\right)$
and
$\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right)=C\left(s,t\right)-C(VaR_{X_{1}}(x),t).$
Then
$\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)=1+\frac{VaR_{X_{1}}\left(x\right)-C(VaR_{X_{1}}(x),t)}{\overline{C}\left(1-s,1-t\right)}$
Then the CCTE is given by
$\displaystyle\mathbb{CCTE}_{X_{1}}\left(s,t\right)$ $\displaystyle=$
$\displaystyle\int_{X_{1}>VaR_{X_{1}}\left(s\right)}xd\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{VaR_{X_{1}}\left(s\right)}^{\infty}xd\left(VaR_{X_{1}}\left(x\right)-C(VaR_{X_{1}}(x),t)\right).$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)d\left(u-C(u,t)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du-\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)dC(u,t)\right)$
This close the proof of Proposition 2.1.
###### Proof of Proposition 4.1.
Let’s denote by
$C_{u}\left(u,v\right):=\frac{\partial C\left(u,v\right)}{\partial u}$
then by (2.6), we have
$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du-\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)C_{u}(u,t)du\right).$
So, $C$ is Archimedean copula, then
$C_{u}\left(u,v\right)=\frac{\psi^{\prime}(u)}{\psi^{\prime}\left(C\left(u,v\right)\right)},$
(7.21)
Finely, we get (4.15) by substitution of
$\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du=\left(1-s\right)\mathbb{CTE}_{X_{1}}\left(s\right)$
and (7.21) in (2.6).
## References
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|
arxiv-papers
| 2012-05-19T16:18:33 |
2024-09-04T02:49:31.074982
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Brahim Brahimi",
"submitter": "Brahimi Brahim",
"url": "https://arxiv.org/abs/1205.4345"
}
|
1205.4371
|
# Lagrangian tools to monitor transport and mixing in the ocean
S. V. Prants, M. V. Budyansky and M. Yu. Uleysky E-mail: prants@poi.dvo.ru,
http://dynalab.poi.dvo.ru
(Pacific Oceanological Institute of the Russian Academy of Sciences,
Vladivostok, 690041, Russia)
###### Abstract
We apply the Lagrangian approach to study surface transport and mixing in the
ocean. New tools have been developed to track the motion of water masses,
their origin and fate and to quantify transport and mixing. To illustrate the
methods used we compute the Lagrangian synoptic maps a comparatively small
marine bay, the Peter the Great Bay in the Japan Sea near Vladivostok city
(Russia), and in a comparatively large region in the North Pacific, the
Kuroshio Extension system. In the first case we use velocity data from a Japan
Sea circulation numerical model and in the second one the velocity data are
derived from satellite altimeter measurements of anomalies of the sea height
distributed by AVISO.
Keywords: Mixing; Eddy; Lagrangian synoptic map; Marine bay; Kuroshio
Extension.
## 1 Introduction
The ocean presents a variety of dynamical phenomena with different space
scales ranging from millemeters to a few thousand of kilometers. Despite of
that, large-scale coherent structures are easlily visible, say, at satellite
images of the sea color and surface temperature and can be identified by means
of in-situ measurements. The striking examples are the major western boundary
oceanic currents, the Gulf Stream in the Atlantic and the Kuroshio in the
Pacific. They are “rivers” with the warm water in the ocean with the width on
the order of 100–200 km and the maximal speed of current at the surface of 2
m/s. Such currents separate waters with different physical, chemical and
biological characteristics. The other examples are mesoscale (with the size of
a few hundred of kilometers) and submesoscale (a few tenth of kilometers)
eddies that can transport water over hundreds and even thousands of kilometers
and can survive for months before breaking down. Being coherent features, they
do not contain the same waters but exchange them with the surrounding ocean,
the process known as mixing.
Lagrangian and dynamical systems methods have been developed to study large-
scale transport and mixing in the ocean [4, 5, 3, 2, 6, 7, 12, 11]. The main
purposes of those studies are to track the fluid motion, to elucidate and
quantify transport and mixing processes. Simply speaking, we would like to
know where these or those waters come from, what is their fate and how they
mix in this or that region. In the Lagrangian approach one integrates
trajectories for a large number of synthetic particles advected by an Eulerian
velocity field
$\frac{d\vec{r}}{dt}=\vec{v}(\vec{r},t).$ (1)
The velocity field, $\vec{v}(\vec{r},t)$, is supposed to be known
analytically, numerically or estimated from satellite altimetry. While in the
Eulerian approach we get frozen snapshots of data, Lagrangian diagnostics
enable to quantify spatio-time variability of the velocity field. It has been
established theoretically and experimentally that even a simple deterministic
velocity field may cause practically unpredictable particle trajectories, the
phenomenon known as chaotic advection [8, 5]. The real oceanic flows are not,
of course, deterministic and regular, but if the Eulerian correlation time is
large as compared to the Lagrangian one, the problem may be treated in the
framework of chaotic advection concept.
It is important to separate chaotic and turbulent mixing in the ocean. The
process of chaotic advection provides transport and mixing with the
characteristic scales on the order of a few tenths or even hundreds of
kilometers, whereas turbulence works at smaller scales. At a comparatively
large scale, turbulent mixing is homogeneous whereas the chaotic one is not.
Typical patterns of chaotic advection consist of large-scale convoluted curves
visible in some surface-temperature and color satellite images. The effect of
turbulent mixing is in small-scale fluctuations superimposed on the large-
scale convoluted curves. If the velocity field on comparatively large scales
is quasicoherent in space and quasiregular in time but the motion of tracers
is mainly irregular, one deals with chaotic mixing. Turbulent mixing means
that the velocity field is irregular in space and time at the same scales at
which the tracer’s motion is irregular.
In this paper we report on our recent results on developing Lagrangian tools
to monitor surface transport and mixing in the ocean. We propose with this aim
new Lagrangian criteria that enable to track and quantify the water exchange
processes and reveal the underlying physical mechanisms. As an output, we
compute different Lagrangian synoptic maps of the regions under study for a
given period of year and analyze them. The methos is illustrated with a
comparatively small marine bay, the Peter the Great Bay in the Japan Sea near
Vladivostok (Russia), and a comparatively large region in the North Pacific,
the Kuroshio Extension system. In the first case we have used velocity data
from a Japan Sea eddy-resolved circulation numerical model with the fine
resolution of 2.5 km, in the second one — satellite altimetric velocity data
with the coarse resolution of the order of 35 km.
## 2 Lagrangian and dynamical systems methods to study transport and mixing
in the ocean
Motion of a fluid particle in a two-dimensional flow is the trajectory of a
dynamical system with given initial conditions governed by the velocity field
computed either by solving the corresponding master equations or as the output
of a numerical ocean model or derived from a measurement
$\frac{dx}{dt}=u(x,y,t),\quad\frac{dy}{dt}=v(x,y,t),$ (2)
where $(x,y)$ is the location of the particle, $u$ and $v$ are the zonal and
meridional components of its velocity. Even if the Eulerian velocity field is
fully deterministic, the particle’s trajectories may be very complicated and
practically unpredictable. It means that a distance between two initially
nearby particles grows exponentially in time
$\|\delta{\mathbf{r}}(t)\|=\|\delta{\mathbf{r}}(0)\|\,e^{\lambda t},$ (3)
where $\lambda$ is a positive number, known as the Lyapunov exponent, which
characterizes asymptotically the average rate of the particle dispersion, and
$\|\cdot\|$ is a norm of the vector $\mathbf{r}=(x,y)$. It immediately follows
from (3) that we are unable to forecast the fate of the particles beyond the
so-called predictability horizon
$T_{p}\simeq\frac{1}{\lambda}\ln\frac{\|\Delta\|}{\|\Delta(0)\|},$ (4)
where $\|\Delta\|$ is the confidence interval of the particle location and
$\|\Delta(0)\|$ is a practically inevitable inaccuracy in specifying the
initial location. The deterministic dynamical system (2) with a positive
maximal Lyapunov exponent for almost all vectors $\delta\mathbf{r}(0)$ (in the
sense of nonzero measure) is called chaotic. It should be stressed that the
dependence of the predictability horizon $T_{p}$ on the lack of our knowledge
of exact location is logarithmic, i. e., it is much weaker than on the measure
of dynamical instability quantified by $\lambda$. Simply speaking, with any
reasonable degree of accuracy on specifying initial conditions there is a time
interval beyond which the forecast is impossible, and that time may be rather
short for chaotic systems.
Since the phase plane of the two-dimensional dynamical system (2) is the
physical space for fluid particles, many abstract mathematical objects from
dynamical systems theory (stationary points, KAM tori, stable and unstable
manifolds, periodic and chaotic orbits, etc.) are material surfaces, curves
and points in fluid flows. It is well known that besides “trivial” elliptic
fixed points, the motion around which is stable, there are hyperbolic fixed
points which organize fluid motion in their neighbourhood in a specific way.
In a steady flow the hyperbolic points are typically connected by the
separatrices which are their stable and unstable invariant manifolds. In a
time-periodic flow the hyperbolic points are replaced by the corresponding
hyperbolic trajectories with associated invariant manifolds which in general
intersect transversally resulting in a complex manifold structure known as a
heteroclinic tangle. The fluid motion in these regions is so complicated that
it may be strictly called chaotic, the phenomenon known as chaotic advection
[8, 5]. Adjacent fluid particles in such tangles rapidly diverge providing
very effective mechanism for mixing.
Stable and unstable manifolds are important organizing structures in the flow
because they attract and repel fluid particles (not belonging to them) at an
exponential rate and partition the flow into regions with different types of
motion. Invariant manifold in a two-dimensional flow is a material line, i.
e., it is composed of the same fluid particles in course of time. By
definition stable ($W_{s}$) and unstable ($W_{u}$) manifolds of a hyperbolic
trajectory $\gamma(t)$ are material lines consisting of a set of points
through which at time moment $t$ pass trajectories asymptotical to $\gamma(t)$
at $t\to\infty$ ($W_{s}$) and $t\to-\infty$ ($W_{u}$). They are complicated
curves infinite in time and space that act as boundaries to fluid transport.
The real oceanic flows are not, of course, strictly time-periodic. However, in
aperiodic flows there exist under some mild conditions hyperbolic points and
trajectories of a transient nature. In aperiodic flows it is possible to
identify aperiodically moving hyperbolic points with stable and unstable
effective manifolds [4, 3]. Unlike the manifolds in steady and periodic flows,
defined in the infinite time limit, the “effective” manifolds of aperiodic
hyperbolic trajectories have a finite lifetime. The point is that they play
the same role in organizing oceanic flows as do invariant manifolds in simpler
flows. The effective manifolds in course of their life undergo stretching and
folding at progressively small scales and intersect each other in the
homoclinic points in the vicinity of which fluid particles move chaotically.
Trajectories of initially close fluid particles diverge rapidly in these
regions, and particles from other regions appear there. It is the mechanism
for effective transport and mixing of water masses in the ocean. Moreover,
stable and unstable effective manifolds constitute Lagrangian transport
barriers between different regions because they are material invariant curves
that cannot be crossed by purely advective processes.
The stable and unstable manifolds of influencial hyperbolic trajectories are
so important because (1) they form a kind of a sceleton in oceanic flows, (2)
they divide a flow in dynamically different regions, (3) they are in charge of
forming an inhomogeneous mixing with spirals, filaments and intrusions, (4)
they are transport barriers separating water masses with different
characteristics. Stable manifolds act as repellers for surrounding waters but
unstable ones are attractors. That is why unstable manifolds may be rich in
nutrients being oceanic “dining rooms”.
There is a quantity, the finite-time Lyapunov exponents (FTLE), that enables
to detect and visualize stable and unstable manifolds in complex velocity
fields. The FTLE is the finite-time average of the maximal separation rate for
a pair of neighbouring advected particles which is given by [9]
$\lambda(\mathbf{r}(t))\equiv\frac{1}{\tau}\ln\sigma(G(t)),$ (5)
where $\tau$ is an integration time, $\sigma(G(t))$ the largest singular value
of the evolution matrix for linearized advection equations. Scalar field of
the FTLE is Eulerian but the very quantity is a Lagrangian one that measures
an integrated separation between trajectories. Ridges (curves of the local
maxima) of the FTLE field visualize stable manifolds when integrating
advection equations forward in time and unstable ones when integrating them
backward in time.
## 3 Transport and mixing in marine bays
When studing transport and mixing in marine bays, it is important to know
which waters enter the bay under study, which ones quit the bay, by which
transport corridors they do that and how the different waters mix in the bay
interior. The Lagrangian approach, allowing to compute the origin and fate of
different waters, is the most suitable for that. Transport and mixing in
marine bays is more inhomogeneous as compared with those processes in open
basins because of a complicated structure of currents and eddies of different
scales, strong tides and presence of river estuaries. In this section we apply
Largangian tools to characrerize horizontal subsurface transport and mixing in
the Peter the Great Bay near Vladivostok city (Russia). That is the largest
bay in the Japan Sea with a few shallow-water smaller bays and estuaries of
three major rivers with a wide shelf and steep continental slope. The water
exchange between the bay and the open sea is governed mainly by a cyclonic
circulation over the deep central basin and the Primorskoye current flowing to
the southwest along the continental slope of the Primorsky Krai (Russia). We
have used velocity data from the MHI ocean circulation model [10] which is a
set of 3D primitive equations in $Z$-coordinate system with 10 quasi-isopycnal
layers and the resolution of 2.5 km.
To characterize the water exchange between the Peter the Great Bay and the
open sea we compute the FTLE map and the exit-time map (Fig. 1). A large
number of synthetic particles have been uniformly distributed over the region
with [$130^{\circ}12^{\prime}:133^{\circ}12^{\prime}$] E and
[$41^{\circ}42^{\prime}:43^{\circ}19^{\prime}$] N. In Fig. 1a we compute the
FTLE, $\lambda$, by the method proposed in Ref. [9]. The advection equations
(2) have been integrated forward in time for 54 days in the August and
September of a typical year. The gray shades code the magnitude of $\lambda$.
The value $\lambda=0.085$, at which the distance between neighbouring
particles increases in 100 times, is chosed to be a threshold. The regions
with $\lambda<0.085$ are supposed to be regular, the ones with $\lambda>0.085$
— chaotic. The black ridges with $\lambda\gg 0.085$ visualize stable manifolds
of influencial hyperbolic trajectories in the region. Spiral-like structures
reveal eddies of different scales, the white and light-grey zones are the
stagnation regions or shear currents. The sandwich-like structures are signs
of the most intense mixing. The synoptic Lyapunov map in Fig. 1a shows the
scalar filed of this quantity in geographic coordinates which are initial
positions of the synthetic particles. This map along with the Lyapunov map,
computed backward in time (not shown), demonstrates with a high resolution the
complicated character of transport and mixing in the Peter the Great Bay.
The exit-time map is shown in Fig. 1b. The color in the map codes the time,
$T$, particles (initially distributed over the same region) need to reach the
open sea or the coastline. In fact, we compute the trajectories till they
reach the 3 km band along the coastline. The white wide band along the coast
in Fig. 1b demonstrates the Primorskoye current along which particles quickly
leave the bay to the southwest. The large white corridor in the central part
of the region selected separates the Peter the Great Bay from the open sea.
Black color marks the particles that did not leave the bay for the computation
time, 54 days. The stagnation zones are situted, as expected, in the smaller
bays, the Amursky and the Ussyrisky ones, which are visible as black spots on
the both sides of the peninsula in the north. The exit-time map reveals the
complicated process of chaotic mixing in the central part of the bay with the
spiral-like anticyclonic eddy (with the center at $132^{\circ}45^{\prime}$ E
and $42^{\circ}40^{\prime}$ E) and gives a valuable information about origin
and fate of waters.
Figure 1: (a) The Lyapunov map in the Peter the Great Bay and the surrounding
region of the Japan Sea. (b) The exit-time map in the same region.
To get an information about the character of motion of different waters, their
drift, rotation and oscillation, we compute the new Lagrangian synoptic maps:
rotation and mixing maps, transport and visitor maps. We compute for a large
number of particles the number of cyclonic, $\eta_{c}$, and anticyclonic,
$\eta_{a}$, rotations and their difference $\eta$. The typical kinds of
particle’s motion are the following: 1) simple drift or linear displacement if
$\eta_{c}$, $\eta_{a}$, $|\eta|$ $<\eta_{\rm cr}=5$, where $\eta_{\rm cr}$ is
a threshold value of the rotation number; 2) rotation, if $|\eta|>\eta_{\rm
cr}=5$; 3) oscillation, if $\eta_{c}$, $\eta_{a}>\eta_{\rm cr}=5$ but
$|\eta|<\eta_{\rm cr}$. In the rotation map in Fig. 2a white and black colors
mean cyclonic, $\eta_{c}$, and anticyclonic, $\eta_{a}$, rotations,
respectively, computed for the same period of time, 54 days. Grey color codes
the particle with predominant displacements or oscillations. The map
demonstrates clearly the same spiral-like anticyclonic eddy as in Fig. 1 and
the large-scale filaments with foldings typical to chaotic advection in the
ocean.
To characterize the chaotic mixing more clearly we compute along with the
rotation numbers the FTLE $\lambda$. If $\lambda>\lambda_{\rm cr}=0.85$ and
$\eta_{a}>\eta_{\rm cr}=5$, we will speak about unstable rotations in the
corresponding region. If $\lambda>\lambda_{\rm cr}=0.85$ but
$\eta_{a}<\eta_{\rm cr}=5$ one deals with unstable linear displacement of the
corresponding particles. The mixing map in Fig. 2b shows by color regions with
different dynamical properties specified by the rotation numbers and the
maximal Lyapunov exponent. White color marks the regions with regular
oscillations and/or predominant displacements. The spots of particles, placed
in those regions, move as whole being deformed slightly. The white grey color
— the regions with unstable displacements which are peripheries of the
anticyclonic eddies and their filaments. The spots, placed in those regions,
are elongated strongly. The dark grey color — the regions with unstable
oscillator motion with the particles rotating for 54 days in the cyclonic and
then in the anticyclonic directions. The black color corresponds to the
unstable rotation that manifests itself in narrow filaments and spiral-like
structures in anticyclones.
Figure 2: (a) Rotation and (b) mixing maps in the Peter the Great Bay.
In order to find frontal zones and transport pathways we propose to compute
the transport maps showing the final positions of particles when integrating
the advection equations (2) forward and backward in time (see Figs. 3a and 3b,
respectively). In other words, the equations (2) have been solved for each of
the million particles initially distributed over the region selected for 54
days forward and backward in time. In the first case we get the particle’s
fate map (Fig. 3a) with the black (white) particles leaving the bay through
the eastern (western) border. The grey particles are those that did not leave
the bay for the computation time. When integrating the equations (2) backward
in time, we get the particle’s origin map with the black (white) particles
entering the bay through the eastern (western) border and the resident
particles shown in grey. The frontal zone, separating the waters with
different fate and origin, consistes of smooth, meandered and spiral-like
fragments.
Figure 3: Transport maps in the Peter the Great Bay. (a) The particle’s fate
and (b) origin maps.
## 4 Transport and mixing in the Kuroshio Extension region
The Kuroshio Extension prolongs the Kuroshio Current when the latter separates
from the continental shelf at about $30^{\circ}$ N. It flows eastward from
this point as a strong unstable meandering jet constituting a front separating
the warm subtropical and cold subpolar waters of the North Pacific Ocean. It
is a region with one of the most intense air–sea heat exchange and the highest
eddy kinetic energy level strongly affecting climate. Transport of water
masses is of cruicial importance and may cause heating and freshing of waters
with a great impact on the weather and living organisms.
The surface ocean currents used in this section are derived from satellite
altimeter measurements of sea height (http://www.aviso.oceanobs.com). The
velocity data covers the period from 1992 to 2011 with weekly data on a
$1/3^{\circ}$ Mercator grid. In our study we focus on the region between
$30^{\circ}$ and $45^{\circ}$ N and between $130^{\circ}$ and $165^{\circ}$ E.
Bicubical spatial interpolation and third order Lagrangian polinomials in time
have been used to provide accurate numerical results. Lagrangian synoptic
maps, manifolds and chaotic advection structures in general are determined by
the large-scale advection field, which is appropriately captured by altimetry.
Thus, computation of particle’s trajectories statistically is not especially
sensitive to imperfections of the velocity field caused by the interpolation
and measurement imperfections.
In Fig. 4a we demonstrate the displacement map for the region computed for 45
days after the beginning of the incident at the Fukushima Daiichi nuclear
power plant. The shades of gray depict the magnitude of the displacement of a
tracer $D=\sqrt{(x_{f}-x_{0})^{2}+(y_{f}-y_{0})^{2}}$, from its initial
position, $(x_{0},y_{0})$, to a final one $(x_{f},y_{f})$. The Kuroshio
Current is well pronounced including meanders and intrusions, its extension,
and mesoscale eddies. Two light-colored eddy patches are of particular
interest. Their centers are approximately at the latitude of the Fukushima
plant and at longitudes $153^{\circ}$ E (the mushroom-like dipolar eddy) and
$161^{\circ}$ E (the circular eddy), both eddies being surrounded by dark-
colored necklaces having a relatively high magnitude of $D$. This pattern
exemplifies the ring birth process due to the meandering of the Kuroshio
current and subsequent detachment of eddies from the main jet.
Figure 4: (a) Displacement map in the Kuroshio Extension region. The color
codes the magnitude of displacement $D$ in minutes. The Fukushima Daiichi
nuclear power plant is marked with the sign of radioactivity. (b) Velocity
field of the region on 1 January, 2010. Circles and crosses are instantaneous
elliptic and hyperbolic points, respectively.
To get a picture of an “instantaneous ” state of the region we show in Fig. 4b
the surface velocity field computed on the fixed day, 1 January, 2010\. The
main meandering jet is depicted by the black arrows corresponding to
comparatively large velocities. The grey arrows around the jet with smaller
velocities reveal a number of cyclonic and anticyclonic eddies on both flanks
of the jet. We have computed instantaneous elliptic and hyperbolic points of
the flow and showed them by circles and crosses, respectively. The elliptic
points are situated mainly in the centers of the eddies, whereas the
hyperbolic ones are in the regions between the eddies with different polarity
and in the periphery of isolated eddies. The hyperbolic points are especially
important because they may be connected by instantaneous stable and unstable
manifolds dividing the flow into regions with cardinally different dynamics.
We plan to show in this section that the Lagrangian diagnostics is well
suitable to describe the mesoscale and submesoclace features of the complex
picture of mixing in the Kuroshio Extension region. The altimetric velocity
data we used covers the period from 1 January to 3 July, 2010. We focus on a
vortex pair on the jet’s southern flank, consisting of the anticyclone (AC)
and cyclone (C). The pair manifests itself on the Lagrangian maps in Fig. 5
computed for a large number of synthetic particles seeded over the region
considered. All the maps visualize the northern hat-like AC with the axes of
150 and 100 km and the southern circular C with the diameter 150 km. In Fig.
5a the color codes the meridional displacement, $D_{y}$, of particles on the
60th day of integration. The spiral structure of the C is well developed with
the spiral untwisting counter-clockwise, whereas it is less pronounced for the
AC with the spiral untwisting clockwise. The character of the water motion in
the C and AC is also different and becomes evident after computing the number
of particle’s rotation around the vortex centers. It follows from Fig. 5b that
water in the AC core circulates with approximately the same angular velocity,
whereas this quantity decreases from the center of the C to its periphery (pay
attention to the ring-like structure of the C). In order to visualize the
stable manifolds of the hyperbolic trajectories around the vortex pair, we
compute in Fig. 5c the FTLE, $\lambda$, and displacements, $D$, of the
particles. The shades of grey in this figure modulate different combinations
and magnitudes of $\lambda$ and $D$ with respect to some chosen “critical”
values: $\lambda_{\rm cr}$, corresponding to divergence of initially close
particles over 100 km, and $D_{\rm cr}=100$ km. The black convoluted curves in
the figure between the eddies, around each of them and around the very pair
delineate the corresponding $W_{s}$ manifolds.
Figure 5: Lagrangian maps of the AC–C vortex pair in which the color codes:
(a) the meridional displacement of synthetic particles, $D_{y}$, on the 60th
day of integration, (b) the number of their rotation around the vortex centers
on the 15th day and (c) their Lyapunov exponents, $\lambda$, and, $D$,
displacements on the 45th day with the following legenda: white means
$\lambda<\lambda_{\rm cr}$, $D\geq D_{\rm cr}$, grey — $\lambda<\lambda_{\rm
cr}$, $D<D_{\rm cr}$ and black — $\lambda\geq\lambda_{\rm cr}$, $D\geq D_{\rm
cr}$.
To give a detailed description of the structure of each eddy in the vortex
pair we apply the method of particle’s scattering elaborated in Ref. [1]. We
cross both the eddies by a material line and compute rotation number $\eta$,
and the maximal FTLE on initial particle’s latitude $y_{0}$. The scattering
plot in Fig. 6a demonstrates that the waters in the C core really rotate with
different angular velocities decreasing from the center to its periphery.
Rotation in the AC core is much more homogeneous. Moreover, the waters in the
C rotate in two times faster than in the AC. The scattering plot
$\lambda(y_{0})$ in Fig. 6b demonstrates smooth segments in the cores of C and
AC and irregular oscillations in their periphery. It simply means that the
water in the cores moves more or less coherently whereas the motion in the
eddy’s peripheries is erratic due to numerous intersections of stable and
unstable manifolds. Computation of the dependence of the time of exit of the
particles $T$, belonging to the material line, on $y_{0}$ (not shown) confirms
that waters prefer to quit the C more or less periodically by portions. Each
portion is represented by a $\cup$-like segment of the $T(y_{0})$ function
which consists of a large number of particles with approximately the same time
of exit and the same rotation number $\eta$. In difference from the C,
particles quit the AC core practically at the same time. In other words, the
particles quit the C by portions along spiral-like transport pathways, whereas
the periphery of the AC exchanges water with the surrounding but its core
moves coherently as a whole for a time.
Figure 6: The scattering plots for the vortex pair on the 30th day of
integration. (a) Number of times, $\eta$, the particles rotate around the
vortex centers vs initial particle’s latitude position $y_{0}$, (b) the
corresponding maximal FTLE vs $y_{0}$.
In conclusion we demonstrate in Fig. 7 how frequently fluid particles, chosen
in the cores of the C and AC, visit for 180 days different places in the
Kuroshio Extension region. It is evident that the C was absorbed by the main
jet in a short time and then its waters travelled eratically within the jet
with a few excursions to its northern and southern flanks. It is interesting
that in course of time C waters have formed the new cyclonic eddy nearby
$(x_{0}=155^{\circ}$ E, $y_{0}=32^{\circ}$ N). In contrast to the C, the AC
waters have walked eratically on the southern flank of the jet in a restricted
region within $x_{0}=[140^{\circ}:150^{\circ}]$ E,
$y_{0}=[28^{\circ}:35^{\circ}]$ N.
Figure 7: Visitor maps for (a) the cyclone and (b) anticyclone show how
frequently fluid particles from the corresponding eddy’s cores visit for 180
days different places in the Kuroshio Extension region.
## 5 Conclusion
The Lagrangian approach has been shown to be very useful to gain new
information on chaotic transport and mixing in the ocean. We have elaborated
new Lagrangian diagnostic tools to visualize and quantify those processes: the
time of exit of fluid particles off a selected box, their displacements, the
number of their cyclonic and anticyclonic rotations and the number of times
they visit different places in the region. Along with the Lyapunov maps, the
corresponding high-resolution Lagrangian synoptic maps of those quantities,
computed by solving advection equations forward and backward in time for
different periods of the year, are new diagnostic and prognostic products
characterizing the state of the ocean. The technique developed can be applied
to the global ocean and its basins.
In this paper we have focused on a comparatively small marine bay, the Peter
the Great Bay in the Japan Sea near Vladivostok (Russia), and on a
comparatively large region in the North Pacific, the Kuroshio Extension
system. In the bay study in summer and autumn periods, we have used the
velocity data from a Japan Sea eddy-resolved circulation numerical model with
the resolution of 2.5 km. It has been shown that the Lyapunov and exit-time
maps, the rotation, mixing and transport maps allowed to quantify and specify
movement of water masses, their mixing and the degree of its chaoticity in the
bay. Those high-resolution maps allowed to visualize transport pathways by
which waters exit and enter the bay.
As to the Kuroshio Extension, we have used the velocity data derived from
satellite altimeter measurements of sea height with the corresponding
interpolation. The main attention has been paid to study structure, transport
and mixing of a vortex pair with strongly interacting cyclonic and
anticyclonic eddies. Such dipoles occur frequently in that region. We have
computed Lagrangian synoptic maps for the time of exit of particles, the
number of changes of the sign of zonal and meridional velocities, and for
other quantities. Along with the Lyapunov map, they have been shown to be able
to reveal the vortex structure and its evolution, meso- and submesoscale
filaments, repelling material lines, hyperbolic and non-hyperbolic regions in
the sea. In particular, we have found that the eddies have a prominent spiral-
like structure resembling the spiral patterns at satellite images in that
region.
The work was supported by the Program “Fundamental Problems of Nonlinear
Dynamics” of the Russian Academy of Sciences, by the Russian Foundation for
Basic Research (projects nos. 09-05-98520 and 11-01-12057) and by the
Prezidium of the Far-Eastern Branch of the RAS.
## References
* [1] M. V. Budyansky, M. Yu. Uleysky, and S. V. Prants, JETP 99 (2004), no. 5, 1018–1027.
* [2] Francesco d’Ovidio, Jordi Isern-Fontanet, Cristóbal López, Emilio Hernández-García, and Emilio García-Ladona, Deep Sea Research Part I: Oceanographic Research Papers 56 (2009), no. 1, 15–31.
* [3] G. Haller, Physics of Fluids 14 (2002), no. 6, 1851–1861.
* [4] G. Haller and A.C. Poje, Physica D: Nonlinear Phenomena 119 (1998), no. 3-4, 352–380.
* [5] K.V. Koshel and S.V. Prants, Physics Uspekhi 49 (2006), 1151–1178.
* [6] Francois Lekien, Chad Coulliette, Arthur J. Mariano, Edward H. Ryan, Lynn K. Shay, George Haller, and Jerry Marsden, Physica D: Nonlinear Phenomena 210 (2005), no. 1-2, 1–20.
* [7] Ana M. Mancho, Des Small, and Stephen Wiggins, Physics Reports 437 (2006), no. 3-4, 55–124.
* [8] J.M. Ottino, _The kinematics of mixing: Stretching, chaos, and transport_ , Cambridge University Press, Cambridge, U.K., 1989.
* [9] S. V. Prants, M. V. Budyansky, V. I. Ponomarev, and M. Yu. Uleysky, Ocean modelling 38 (2011), no. 1-2, 114–125.
* [10] S. V. Prants, V. I. Ponomarev, M. V. Budyansky, M. Yu. Uleysky, and P. A. Fayman, Izvestiya, Atmospheric and Oceanic Physics (in press).
* [11] Emilie Tew Kai, Vincent Rossi, Joel Sudre, Henri Weimerskirch, Cristobal Lopez, Emilio Hernandez-Garcia, Francis Marsac, and Veronique Garçon, PNAS 106 (2009), no. 20, 8245–8250.
* [12] Darryn W. Waugh, Edward R. Abraham, and Melissa M. Bowen, Journal of Physical Oceanography 36 (2006), no. 3, 526–542.
|
arxiv-papers
| 2012-05-20T02:21:41 |
2024-09-04T02:49:31.085668
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants, M. V. Budyansky and M. Yu. Uleysky",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1205.4371"
}
|
1205.4372
|
CHAOTIC WALKING AND FRACTAL SCATTERING OF ATOMS
---
IN A TILTED OPTICAL LATTICE
S.V. Prants, V.O. Vitkovsky
Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
690041 Vladivostok, Russia, URL: dynalab.poi.dvo.ru
∗Corresponding author e-mail: prants@poi.dvo.ru
###### Abstract
Chaotic walking of cold atoms in a tilted optical lattice, created by two
counter propagating running waves with an additional external field, is
demonstrated theoretically and numerically in the semiclassical and
Hamiltonian approximations. The effect consists in random-like changing the
direction of atomic motion in a rigid lattice under the influence of a
constant force due to a specific behavior of the atomic dipole-moment
component that changes abruptly in a random-like manner while atoms cross
standing-wave nodes. Chaotic walking generates a fractal-like scattering of
atoms that manifests itself in a self-similar structure of the scattering
function in the atom-field detuning, position and momentum spaces. The
probability distribution function of the scattering time is shown to decay in
a non-exponential way with a power-law tail.
Keywords: cold atom, tilted potential, chaos, fractal
## 1 Introduction
The mechanical action of light upon neutral atoms placed in a laser standing
wave is at the heart of laser cooling, trapping, and Bose-Einstein
condensation [1]. Numerous applications of the mechanical action of light
include isotope separation, atomic interferometry, atomic lithography and
epitaxy, atomic-beam deflection and splitting, manipulating translational and
internal atomic states, measurement of atomic positions, etc. Atoms and ions
in an optical lattice, formed by a laser standing wave, are perspective
objects for implementation of quantum information processing and quantum
computing. Advances in cooling and trapping of atoms, tailoring optical
potentials of a desired form and dimension, controlling the level of
dissipation and noise are now enabling the direct experiments with single
atoms to study fundamental principles of quantum physics, quantum chaos,
decoherence, and quantum-classical correspondence.
Nonlinear dynamics of cold atoms in optical lattices is a fastly growing
branch of atomic physics. There are a number of theoretical works and
impressive experiments on quantum chaos, dynamical localization, chaos-
assisted tunneling, Lévy flights, etc. (for reviews see [2, 3]). To suppress
spontaneous emission and provide a coherent quantum dynamics one usually works
far from the optical resonance. Adiabatic elimination of the excited state
amplitude leads to an effective Hamiltonian for the center-of-mass motion,
whose 3/2 degree-of-freedom classical analogue has a mixed phase space with
regular islands embedded in a chaotic sea. New possibilities are opened if one
works near the optical resonance and take the internal atomic dynamics into
account. A single atom in a standing-wave laser field may be semiclassically
treated as a nonlinear dynamical system with coupled internal (electronic) and
external (mechanical) degrees of freedom [4, 5, 6]. In the semiclassical and
Hamiltonian limits (when one treats atoms as point-like particles and neglects
spontaneous emission and other losses of energy), a number of nonlinear
dynamical effects have been analytically and numerically demonstrated with
this system: chaotic Rabi oscillations [4, 5, 6], Hamiltonian chaotic atomic
transport and dynamical fractals [7, 8, 9, 10], Lévy flights and anomalous
diffusion [6, 11, 12]. These effects are caused by local instability of the CM
motion in a laser field. A set of atomic trajectories under certain conditions
becomes exponentially sensitive to small variations in initial quantum
internal and classical external states or/and in the control parameters,
mainly, the atom-laser detuning. Hamiltonian evolution is a smooth process
that is well described in a semiclassical approximation by the coupled
Hamilton-Schrödinger equations. A detailed theory of Hamiltonian and
dissipative chaotic transport of atoms in a laser standing wave has been
developed in Refs. [10] and [13, 14], respectively.
Additional possibilities to manipulate the atomic transport are created by
applying an external force to the standing-wave optical potential. It is
obvious that for cold atoms in a vertical optical lattice it is necessary to
account for the Earth’s acceleration. It is possible as well to create
horizontal accelerated optical lattices by adding a constant force whose
magnitude along the optical axis can be easily varied. The problem of atomic
motion in a tilted optical potential is closely related to the old problem of
electron motion in a a periodic crystal with dc or ac forces applied. The
analogue of well known Bloch oscillations with cold atoms has been
experimentally found in Ref.[15, 16].
In the present paper we apply the ideas and methods, elaborated in the field
of nonlinear dynamics of cold atoms, to study theoretically and numerically
motion of point-like atoms in a tilted optical lattice. It will be shown that
varying only one parameter, the detuning between the frequencies of a working
atomic transition and the laser field, one can explore a variety of regimes of
atom motion, including chaotic walking, dynamical fractals and chaotic
scattering.
## 2 Chaotic and regular regimes of motion of atoms in a tilted potential
In the one-dimensional case, the Hamiltonian of a two-level atom in a
standing-wave laser field and an additional external field can be written in
the frame rotating with the laser frequency $\omega_{f}$ as follows:
$\begin{gathered}\hat{H}=\frac{P^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\hbar\Omega_{0}\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}X}+FX,\end{gathered}$
(1)
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $X$ and $P$ are the classical atomic position and
momentum, $\omega_{a}$ and $\Omega_{0}$ are the atomic transition and maximal
Rabi frequencies, respectively. $F$ stands for the static force induced by
external field.
In the semiclassical approximation, where the transversal atomic momentum $p$
is supposed to be, in average, much larger than the photon one $\hbar k_{f}$,
atom with quantized internal dynamics is treated as a point-like particle to
be described by the Bloch–Hamilton equations of motion without relaxation
terms
$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\sin
x-\kappa,\quad\dot{u}=\Delta v,\\\ \dot{v}=-\Delta u+2z\cos
x,\quad\dot{z}=-2v\cos x,\end{gathered}$ (2)
where $u$ and $v$ are synchronized (with the laser field) and quadrature
components of the atomic electric dipole moment, respectively, and $z$ is the
atomic population inversion. Equations (2) are written in the dimensionless
form with $x\equiv k_{f}X$ and $p\equiv P/\hbar k_{f}$ to be classical atomic
center-of-mass position and momentum, respectively. Dot denotes
differentiation with respect to the dimensionless time $\tau\equiv\Omega t$.
The set (2) has the three control parameters
$\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega,\quad\Delta\equiv(\omega_{f}-\omega_{a})/\Omega,\quad\kappa\equiv
F/\hbar k_{f}\Omega,$ (3)
which are the normalized recoil frequency, $\omega_{r}$, atom-field detuning,
$\Delta$, and applied force $\kappa$. The system has two integrals of motion,
namely the total energy
$H\equiv\frac{\omega_{r}}{2}p^{2}+\kappa x-u\cos x-\frac{\Delta}{2}z,$ (4)
and the length of the Bloch vector $u^{2}+v^{2}+z^{2}=1$.
The external force is directed in the negative direction of the optical axis
$x$. So, if the initial atomic momentum, $p_{0}$, is chosen to be in the
negative direction, the force will simply accelerate the corresponding atoms.
If $p_{0}>0$, then one may expect much more complicated motion.
Equations (2) constitute a nonlinear Hamiltonian autonomous system with two
and half degrees of freedom. Owing to two integrals of motion, phase points
move on a three-dimensional hypersurface with a given energy value $H$. In
general, motion in a three-dimensional phase space in characterized by a
positive Lyapunov exponent, a negative exponent equal in magnitude to the
positive one, and zero exponent. The maximal Lyapunov exponent characterizes
the mean rate of the exponential divergence of initially close trajectories
and serves as a quantitative measure of dynamical chaos in the system. Because
of a transient character of chaos, we have computed the finite-time Lyapunov
exponent $\lambda$ by the method developed in Refs. [17, 18]. It has been
found that at the fixed value, $\omega_{r}=10^{-3}$, of the recoil frequency
$\lambda$ is positive in the following ranges of values of the other control
parameters: the detuning $-0.5<\Delta<0.5$ and the force $-0.25<\kappa<0.6$.
Therefore, we expect chaotic motion of atoms with the parameter’s values in
those ranges.
In numerical experiments throughout the paper we suppose that two-level atoms
are initially prepared in the ground states, $u_{0}=v_{0}=0,z_{0}=-1$, with
$x_{0}=0$ and fixed values of the two control parameters, the normalized
recoil frequency, $\omega_{r}=10^{-3}$, and the external force $\kappa=0.01$.
The atom-field detuning, $\Delta$, can be changed in a wide range. It will be
shown in this paper that atoms may perform chaotic walking, the new type of
motion in absolutely deterministic environment where atoms can change the
direction of motion alternating between flying through the standing wave and
being trapped in its potential wells. We would like to stress that the local
instability produces chaotic center-of-mass motion in a rigid optical lattice
without any modulation of its parameters.
In Fig. 1 we illustrate different regimes of the center-of-mass atomic motion
along the optical axis with the initial atomic momentum chosen to be
$p_{0}=10$. It is simply the motion on the phase plane $x-p$. A typical
picture with chaotic walking is shown in Fig. 1a with the value of the
detuning, $\Delta=0.15$, at which the maximal Lyapunov exponent, $\lambda$, is
positive. The atom starts to move in the positive $x$-direction, changes soon
the direction of motion a few times, acquiring irregularly positive and
negative values of the momentum, and suddenly begins to move in the positive
$x$-direction for a comparatively long time. Then it is decelerated, turns
back and flies in the negative direction. After that it changes the direction
of motion many times demonstrating what we call “chaotic walking”.
For comparison, we show in Fig. 1b the phase-plane motion with the larger
detuning $\Delta=1$ (and at the same other conditions) at which the maximal
Lyapunov exponent is not positive. The atom moves initially in the positive
$x$-direction, accelerating and decelerating alternatively. Soon it changes
the direction of motion and moves permanently in the negative direction. The
motion is regular with a slight modulation of the momentum. The motion at
exact resonance, $\Delta=0$, is even more simple (Fig. 1c).
What is the ultimate reason of chaotic walking? For an optical lattice without
an external force, it has been found in Ref. [10] that instability is caused
by the specific behavior of the Bloch-vector component of a moving atom, $u$,
whose shallow oscillations between the standing-wave nodes are interrupted by
sudden jumps with different amplitudes while atom crosses each node of the
wave. It looks like a random like shots happened in a fully deterministic
environment. The reason of chaotic walking in a tilted potential is the same.
It follows from the second equation in the set (2) that those jumps result in
sudden changes of the atomic momentum while crossing nodes. If the value of
the atomic energy is close to the separatrix one, the atom after the
corresponding jump-like change in $p$ can either overcome the potential
barrier and leave a potential well or it will be trapped by the well, or it
will move as before crossing the node. The evolution of all the Bloch
components in the regime of chaotic walking is shown in Fig. 2. For
comparison, we show in Figs. 3 and 4 their evolution in the regular regimes,
far off the resonance and at exact resonance, respectively.
## 3 Fractal scattering of atoms in a tilted potential
Different types of fractal-like structures may arise in chaotic Hamiltonian
systems (see reviews [19, 20]). It is known from many studies in celestial
mechanics [21], fluid dynamics [22, 23], atomic physics [25, 7, 12, 10],
cavity quantum electrodynamics [8, 9], underwater acoustics [24] and other
disciplines [26] that under certain conditions the motion inside an
interaction region may have features that are typical for dynamical chaos,
(homoclinic and heteroclinic tangles, fractals, strange invariant sets,
positive finite-time Lyapunov exponents, etc.) although the particle’s
trajectories are not chaotic in a rigorous sense because chaos is defined as
an irregular motion over infinite time.
Let us place atoms one by one at the point $x_{0}=0$ with the same value of
the initial momentum $p_{0}=10$ but change slightly the value of the detuning
$\Delta$. All the other initial conditions and the control parameters are
supposed to be the same for all the atoms. We fix the time moment $T$ when
each atom crosses the point $x=0$ moving in the negative direction. The exit
time function $T(\Delta)$ in Fig. 5 demonstrates the complicated structure
with smooth intervals alternating with wildly oscillating peaks that cannot be
resolved in principle, no matter how large the magnification factor. The
panels (b) and (c) in Fig. 5 are successive 50 times magnifications of the
detuning intervals shown in the panel (a). Further magnifications reveal a
self-similar fractal-like structure that is typical for Hamiltonian systems
with chaotic scattering. The exit time $T$ increases with increasing the
magnification factor. The same picture is observed when computing the exit
time function in the position and momentum spaces. It is a clear demonstration
of a fractal-like behavior of chaotically walking atoms.
It is established in theory of one and half degree-of-freedom systems that
transient Hamiltonian chaos in the interaction region occurs due to existence
of, at least, one non-attractive chaotic invariant set consisting of an
infinite number of localized unstable periodic orbits and aperiodic orbits.
This set possesses stable and unstable manifolds extending into the regions of
regular motion. The particles with the initial positions close to the stable
manifold follow the chaotic-set trajectories for a comparatively long time,
then deviate from them, and leave the interaction region along the unstable
manifold. In a typical Hamiltonian system there exists an infinite number of
trajectories of zero measure with infinite exit time which belong to that
chaotic invariant set. Our system with two and half degrees of freedom is a
much more complicated one, and it is practically impossible to reveal the
corresponding chaotic invariant set with its stable and unstable manifolds.
However, the mechanism of chaotic scattering and fractal-like structures
should be the same.
The statistics of exit times $T$ is shown in Fig. 6 in a semilogarithmic and
logarithmic scales. The probability distribution function (PDF) in this figure
gives the probability for an atom to have a given value of $T$. The bold
straight line in Fig. 6a implies that the PDF is exponential in its middle
part, $P\sim\exp(-\alpha T)$, with the exponent $\alpha=-0.000270722$.
However, the tail of the PDF is not exponential. To prove that we plot the
function in the logarithmic scale in Fig. 6b and compute the slope at the
tail. It has no sense to calculate the slope at the very tail because of a
small number of events with very large values of $T$. The bold straight line
implies that the PDF is a power-law one, $P\sim T^{-\gamma}$, with the
coefficient $\gamma=-2.53086$. It is interesting that the slope at the PDF
tail around the value $-2.5$ is rather typical for many chaotic Hamiltonian
systems [27, 28]. The reason of that is unclear.
In hyperbolic chaotic systems the PDFs should decay exponentially because the
phase space of such systems is homogeneous, and all the trajectories are
unstable. It is not the case even with one and half degree-of-freedom systems
with inhomogeneous phase space, where exist so-called stability islands
embedded in a stochastic sea, because the borders of those islands are
“sticky”. It means that a typical chaotic trajectory, wandering in the
stochastic sea, approaches the island’s borders and “stick” to them for a long
time. By that reason, the corresponding PDFs are not exponential but power-law
ones at their tails. PDFs with power-law decay imply that the corresponding
quantity, the exit time in our case, is scale invariant i.e., there is no a
single dominant scale in the process. Geometrically it means that chaotic
trajectories for such a process are self-similar.
## 4 Conclusion
It is shown that point-like atoms in a tilted optical potential with a
constant external force applied can move chaotically changing the direction of
motion in a random-like way. The existence of chaos is confirmed by direct
computation of the maximal finite-time Lyapunov exponent of the equations of
motion that is shown to be positive in a range of the atom-laser detuning and
the applied-force strength. The ultimate reason of chaotic walking is the
specific behavior of the Bloch-vector component of a moving atom, $u$, whose
shallow oscillations between the standing-wave nodes are interrupted by sudden
jumps with different amplitudes while atom crosses each node of the wave. It
is demonstrated numerically that such a behavior arises exactly at those
values of the detuning for which the Lyapunov exponent is positive and atoms
move chaotically. We illustrate different regimes of the center-of-mass motion
simply varying the detuning. It is an easy way to manipulate the atomic
transport in tilted optical lattices.
Treating motion of atoms in a tilted optical lattice as a scattering problem,
we show that the scattering of atoms under conditions of chaotic walking is
chaotic and typical for Hamiltonian systems. Fixing the time moment $T$ when
atoms with slightly different values of the detuning, momentum or initial
position cross a fixed point ($x=0$), we show that the corresponding
scattering functions demonstrate the complicated structure that cannot be
resolved in principle, no matter how large the magnification factor. Owing to
that the probability to have a given value of $T$ is not exponential but
decays at its tail by a power law.
## Acknowledgments
This work was supported by the Integration grant from the Far-Eastern and
Siberian branches of the Russian Academy of Sciences (12-II-0-02-001), and by
the Program “Fundamental Problems of Nonlinear Dynamics in Mathematics and
Physics”.
Figure 1: Motion of a cold atom in a deterministic tilted optical lattice as
it looks on the phase plane $x-p$. (a) Chaotic walking at $\Delta=0.15$,
$\kappa=0.01$, $\omega_{r}=10^{-3}$. (b) Regular motion at $\Delta=1$ and with
the same other conditions. (c) Regular motion at the resonance, $\Delta=0$,
with the same other conditions.
Figure 2: Evolution of the atomic Bloch components in the regime of chaotic
walking ($\Delta=0.15$).
Figure 3: The same as in Fig. 2 but far from the resonance ($\Delta=1$).
Figure 4: The same as in Fig. 2 but at the resonance ($\Delta=0$).
Figure 5: Atomic dynamical fractal. Self-similar dependence of the exit time,
$T$, with given initial position, $x_{0}=0$ and momentum $p_{0}=10$, on the
detuning. The successive magnifications are shown.
Figure 6: The probability distribution function for exit times $T$ in (a)
semilogarithmic scale (exponential decay in the middle part with the exponent
$\alpha=-0.000270722$) and (b) logarithmic scale (power-law decay at the tail
with the coefficient $\gamma=-2.53086$).
## References
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* [3] S.V. Prants, Hamiltonian chaos with a cold atom in an optical lattice. In book: Hamiltonian Chaos beyond the KAM Theory. (Editors:A.C.J. Luo and N. Ibragimov) (Springer Verlag and Beijing: Higher Education Press, Berlin, 2010), 193-223.
* [4] S. V. Prants and L.E. Kon’kov, JETP Letters, 73, 1801 (2001) [Pis’ma ZhETF, 73, 200 (2001)].
* [5] S.V. Prants and V.Yu. Sirotkin, Phys. Rev. A, 64, 033412 (2001).
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|
arxiv-papers
| 2012-05-20T02:23:30 |
2024-09-04T02:49:31.092663
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.V. Prants, V.O. Vitkovsky",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1205.4372"
}
|
1205.4374
|
11institutetext: Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya st., 690041 Vladivostok, Russia, URL: dynalab.poi.dvo.ru
Quantum chaos; semiclassical methods Mechanical effects of light on atoms,
molecules, and ions Nonclassical interferometry, subwavelength lithography
# Chaotic scattering of atoms at a standing laser wave
S. V. Prants
###### Abstract
Atoms, propagating across a detuned standing laser wave, can be scattered in a
chaotic way even in the absence of spontaneous emission and any modulation of
the laser field. Spontaneous emission masks the effect in some degree, but the
Monte Carlo simulation shows that it can be observed in real experiments by
the absorption imaging method or depositing atoms on a substrate. The effect
of chaotic scattering is explained by a specific behavior of the dipole
moments of atoms crossing the field nodes and is shown to depend strongly on
the value of the atom-laser detuning.
###### pacs:
05.45.Mt
###### pacs:
37.10.Vz
###### pacs:
42.50.St
## 1 Introduction
The deflection of an atomic beam at a laser standing wave (SW) is explained by
the dipole forces which are well described by the classical atom-field
interaction model [1, 2]. The ability of a SW to diffract, focus or splitting
an atomic beam [3] has been used for a variety of applications including atom
microscopy, interferometry, isotope separation, and optical lithography [4, 5,
6]. On the other hand, cold atoms are ideal candidates to test fundamental
principles of quantum physics including the phenomenon of dynamical chaos at
the microscopic level that is known as a kind of random-like motion in a
deterministic environment [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Dynamical
chaos is characterized by exponential sensitivity of trajectories in the phase
space to small variations in initial conditions and/or control parameters.
It has been proposed in Ref. [7] to study quantum chaos and the corresponding
effect of dynamical localization placing cold atoms in a far-detuned SW with a
periodic kick-like modulation of positions of the SW nodes. A number of
experiments [9] have been carried out in accordance with this proposal. At
large detunings, the atoms are not excited being quantum analogues of
classical kick rotors. Since those experiments on atom optics realization of
the $\delta$-kicked quantum rotor, cold atoms provide new grounds for
experiments and theory on quantum chaos.
It has been shown in Ref. [10] that even a single-pulse far-detuned SW can
induce chaos in atomic motion. For sufficiently large detuning, the excited
state amplitude can be adiabatically eliminated [7], leading to a Hamiltonian
with an external degree of freedom only. The corresponding equations of motion
for an externally modulated nonlinear pendulum constitute the well-known model
with one and half degree of freedom that can be chaotic under some conditions.
The other possibility is to induce chaos in spin degree of freedom of atoms
periodically kicked by applying short magnetic field pulses [11]. That is the
one and half degree of freedom model of a kicked top.
In difference from those and other papers on the related topic, we consider
the physical situation with comparatively small detunings and should take into
account a coupling between external and internal atomic dynamics, leading to a
model with three degrees of freedom. It will be shown in the present paper
that in this case one needs no modulation or any other perturbation of the SW
to induce chaotic internal and external dynamics of atoms crossing the SW
laser field.
Near the atom-field resonance, when the interaction between the internal and
external atomic degrees of freedom is intense, there is a possibility to
create conditions for chaotic behavior without any kicking and modulation [15,
16, 17]. If so, it is open the way to test the novel regime of atomic motion
caused by the peculiarities of the dipole force in the strong coupling regime.
In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle with the Hamilton–Schrödinger equations of
motion constituting a nonlinear dynamical system [15, 16, 17, 18]. In a
certain range of the atom-field detunings, a set of atomic trajectories
becomes exponentially sensitive to small variations in initial quantum
internal and classical external states or/and in the control parameters.
Hamiltonian evolution is a smooth process that is well described in a
semiclassical approximation by the Hamilton-Schrödinger equations. The problem
becomes much more complicated because of spontaneous emission of atoms with a
specific shot quantum noise acting in a dynamical system which is chaotic in
the absence of noise. A number of nonlinear Hamiltonian and dissipative
effects have been found numerically and analytically near the resonance
including chaotic Rabi oscillations, chaotic atomic transport, dynamical
fractals, and Lévy flights [15, 16, 17, 18, 19, 20, 21, 22].
The main aim of the paper is to demonstrate theoretically and numerically that
the new type of atomic diffraction at a rigid SW without any modulation,
chaotic atomic scattering, can be observed in a real experiment. The scheme of
such an experiment is shown in Fig. 1 with a beam of atoms crossing a SW laser
field. One either measures a spatial atomic distribution after the interaction
by the absorption imaging technique or measures an atomic distribution on a
silicon substrate in the far-field zone. The results are expected to be
different depending on the value of the atom-field detuning. The distribution
is expected to be comparatively narrow at those values of the detuning at
which atomic scattering is regular (r.s. distribution in Fig. 1) or wide at
the detuning values providing chaotic scattering of atoms due to their chaotic
walking along the SW (c.s. distribution in Fig. 1).
Figure 1: Schematic representation of the proposed experiment on regular
(r.s.) and chaotic (c.s.) atomic scattering at a Gaussian standing laser wave.
## 2 Results
### 2.1 Main equations and the regimes of atomic motion
A beam of two-level atoms in the $z$ direction crosses a SW laser field with
optical axis in the $x$ direction (Fig. 1). The laser field amplitude has the
Gaussian profile $\exp[-(z-z_{0})^{2}/r^{2}]$ with $r$ being the $e^{-2}$
radius at the laser beam waist. The characteristic length of the atom-field
interaction is supposed to be $3r$ because the light intensity at $z=z_{0}\pm
1.5r$ is two orders of magnitude smaller than the peak value. The longitudinal
velocity of atoms, $v_{z}$, is much larger than their transversal velocity
$v_{x}$ and is supposed to be constant. Thus, the spatial laser profile may be
replaced by the temporal one. The Hamiltonian of an atom in the 1D SW field
can be written in the frame rotating with the laser frequency $\omega_{f}$ as
follows:
$\begin{gathered}\hat{H}=\frac{P^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\\\
\hbar\Omega_{0}\exp[-(t-\frac{3}{2}\sigma_{t})^{2}/\sigma^{2}_{t}]\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}X}-\frac{i\hbar\Gamma}{2}\hat{\sigma}_{+}\hat{\sigma}_{-},\end{gathered}$
(1)
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $X$ and $P$ are the classical atomic position and
momentum, $\Gamma$, $\omega_{a}$, and $\Omega_{0}$ are the decay rate, the
atomic transition and maximal Rabi frequencies, respectively, and
$\sigma_{t}\equiv r/v_{z}$, i.e., $3\sigma_{t}$ is the transit time. The
wavefunction for the electronic degree of freedom is
${|\Psi(t)\closeket}=a(t){|2\closeket}+b(t){|1\closeket}$, where $a\equiv
A+i\alpha$ and $b\equiv B+i\beta$ are the probability amplitudes to find the
atom in the excited, ${|2\closeket}$, and ground, ${|1\closeket}$, states,
respectively.
Figure 2: Finite-time Lyapunov exponent $\lambda$ vs atom-field detuning
$\Delta$ (in units of the Rabi frequency $\Omega_{0}$) and initial atomic
transversal momentum $p_{0}$ (in units of the photon momentum $\hbar k_{f}$)
at the normalized recoil frequency $\omega_{r}=10^{-3}$ and $\gamma=0$.
In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle (the transversal atomic momentum $p$ is
supposed to be, in average, much larger than the photon one, $\hbar k_{f}$)
with the equations of motion written for the real and imaginary parts of the
probability amplitudes
$\begin{gathered}\dot{x}=\omega_{r}p,\,\dot{p}=-2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}](AB+\alpha\beta)\sin
x,\\\ \dot{A}=\frac{1}{2}(\omega_{r}p^{2}-\Delta)\alpha-\frac{1}{2}\gamma
A-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\beta\cos x,\\\
\dot{\alpha}=-\frac{1}{2}(\omega_{r}p^{2}-\Delta)A-\frac{1}{2}\gamma\alpha+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]B\cos
x,\\\
\dot{B}=\frac{1}{2}(\omega_{r}p^{2}+\Delta)\beta-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\alpha\cos
x,\\\
\dot{\beta}=-\frac{1}{2}(\omega_{r}p^{2}+\Delta)B+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]A\cos
x,\end{gathered}$ (2)
where $x\equiv k_{f}X$ and $p\equiv P/\hbar k_{f}$ are atomic center-of-mass
position and transversal momentum, respectively and dot denotes
differentiation with respect to the dimensionless time
$\tau\equiv\Omega_{0}t$. The recoil frequency, $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega_{0}\ll 1$, the atom-laser detuning,
$\Delta\equiv(\omega_{f}-\omega_{a})/\Omega_{0}$, the decay rate
$\gamma=\Gamma/\Omega_{0}$, and the characteristic interaction time,
$\sigma_{\tau}\equiv r\Omega_{0}/v_{z}$, are the control parameters.
Figure 3: Evolution ($\tau$ is in units of $\Omega_{0}^{-1}$) of the atomic
dipole-moment component $u=2(AB+\alpha\beta)$ in (a) the chaotic
($\Delta=0.2$) and (b) regular ($\Delta=1$) regimes of atomic motion.
Without the decay, $\gamma=0$, Eqs. (2) constitute the nonlinear Hamiltonian
dynamical system with three degrees of freedom describing an atom moving in
the six-dimensional phase space. The simple way to know how complicated this
motion may be is to compute the quantitative measure of chaos, maximal
Lyapunov exponent characterizing the mean rate of the exponential divergence
of initially close trajectories. Because of a transient character of chaos we
compute the finite-time Lyapunov exponent $\lambda$, i.e. the value of the
exponent at the moment when atoms leave the interaction region. The result of
computation with Eqs. (2) at the given value of the recoil frequency,
$\omega_{r}=10^{-3}$, and zero decay rate in dependence on the detuning
$\Delta$ and the initial atomic transversal momentum $p_{0}$ is shown in Fig.
2. Color codes the magnitude of the finite-time Lyapunov exponent. In white
regions the values of $\lambda$ are almost zero, and the internal and
translational motion is regular in the corresponding ranges of $\Delta$ and
$p_{0}$. In shadowed regions positive values of $\lambda$ imply unstable
motion.
Figure 4: Trajectories in the real space for 50 atoms without spontaneous
emission. Hamiltonian (a) chaotic ($\Delta=0.2$) and (b) regular scattering
($\Delta=1$). The atomic position $x$ is in units of the optical wavelength.
Figure 5: The same as Fig. 4 but in the momentum space.
The scheme of the proposed experiment in Fig. 1 resembles the scattering
problem with particles entering an interaction region along completely regular
trajectories and leaving it along asymptotically regular trajectories [23, 24,
25, 26, 27, 28]. However, in difference from the standard chaotic scattering
with a nonattractive fractal invariant set existing over an infinite time,
this process may be interpreted as a transient chaos or a finite-time chaotic
scattering.
There are three possible chaotic regimes of the center-of-mass motion along
the SW optical axis [17, 18]. In dependence on the initial conditions and the
values of the control parameters, atoms may oscillate chaotically in wells of
the optical potential or move ballistically over its hills with chaotic
variations of their velocity. Chaotic motion becomes possible in a narrow
range of the detuning values, $0<|\Delta|<1$. At $\Delta=0$, the synchronized
electric-dipole component, $u=2(AB+\alpha\beta)$ becomes a constant. That
implies the additional integral of motion in the Hamiltonian version of
Eqs.(2) and the regular motion with $\lambda=0$. Far from the resonance, at
$|\Delta|>1$, the motion is again regular both in the trapping and flight
modes. That speculation is confirmed by the Lyapunov map in Fig. 2.
Moreover, there is a specific type of motion, chaotic walking in a
deterministic optical potential, when atoms can change the direction of motion
alternating between flying through the SW and being trapped in its potential
wells. We would like to stress that the local instability produces chaotic
center-of-mass motion in a rigid SW without any modulation of its parameters
in difference from the case with periodically kicked and far detuned optical
lattices [7, 10, 9, 11]. The trivial time dependence in the Hamiltonian (2)
cannot produce chaotic motion, it simply accounts for a modulation of the
interaction of atoms with a Gaussian laser beam. Even if the atoms would cross
an absolutely homogeneous (in the $z$-direction) laser beam there would be
under appropriate conditions chaotic atomic center-of-mass motion in the
transversal $x$-direction.
Chaotic walking occurs due to the specific behavior of the Bloch-vector
component, $u$, of a moving atom whose shallow oscillations between the SW
nodes are interrupted by sudden jumps with different amplitudes while atom
crosses each node [18]. We illustrate in Fig. 3 the behavior of the $u$
component with different values of the detuning $\Delta=0.2$ and $\Delta=1$ at
which the atomic motion in accordance with the $\lambda$-map in Fig. 2 is
chaotic and regular, respectively. The time of the atomic interaction with the
SW field is estimated to be $3\sigma_{\tau}=1200$. So, the jumps of the $u$
variable (if any) disappear after that time in Fig. 3. It follows from the
second equation in the set (2) that jumps in the variable
$u=2(AB+\alpha\beta)$ result in jumps of the atomic momentum while crossing a
node of the SW. If the value of the atomic energy is close to a separatrix
one, the atom after the corresponding jump-like change in $p$ can either
overcome the potential barrier and leave a potential well or it will be
trapped by the well, or it will move as before. The jump-like behavior of $u$
is the ultimate reason of chaotic atomic walking along a deterministic SW.
It is easy to estimate the range of initial momenta at which atoms are
expected to change their direction of motion or move ballistically. At small
detunings, $|\Delta|\ll 1$, the total energy consists of the kinetic one,
$K=\omega_{r}p^{2}/2$, and the potential one, $U=u\cos x$. If
$K(\tau=0)>|U_{\rm max}|=1$, then the atom will move ballistically. This
occurs if the initial atomic momentum, $p_{0}$, satisfies to the condition
$p_{0}>\sqrt{2/\omega_{r}}>44$. If the initial conditions are chosen to give
$0\leq K(\tau=0)+U(\tau=0)\leq 1$, the atoms with $0\lesssim p_{0}\lesssim 44$
are expected to perform a chaotic walking at positive $\Delta$.
### 2.2 Hamiltonian chaotic scattering
Figure 6: Scattering of 50 spontaneously emitting atoms at the SW with the
decay rate $\gamma=0.05$ and the same other conditions as in Fig. 4. (a)
Chaotic ($\Delta=0.2$) and (b) regular ($\Delta=1$) regimes.
Figure 7: The distributions of $10^{4}$ lithium atoms (a) without and (b) with
spontaneous emission at $\tau=1000$ under the conditions of chaotic scattering
at $\Delta=0.2$ (bold curves) and regular scattering at $\Delta=1$ (dashed
curves).
Let us consider a spatially uniform and previously focused beam of atoms
crossing a Gaussiam laser beam. The position and momentum distributions of
atoms are measured after interaction with the SW field. We predict that those
distributions would be much broader at those values of $\Delta$ at which one
expects chaotic walking to occur. Firstly, we perform simulation with a
negligible probability of spontaneous emission. To be concrete let us take
calcium atoms with the working intercombination transition
$4^{1}S_{0}-4^{3}P_{1}$ at $\lambda_{a}=657.5$ nm, the recoil frequency
$\nu_{\rm rec}\simeq 10$ KHz, and the lifetime of the excited state $T_{\rm
sp}=0.4$ ms. Taking the maximal Rabi frequency to be $\Omega_{0}/2\pi=2\cdot
10^{7}$ Hz, the radius of the laser beam $r=0.3$ cm, and the mean longitudinal
velocity $v_{z}=10^{3}$ m/s, the interaction time is estimated to be $0.9$ ms.
The normalized recoil frequency is $\omega_{r}=4\pi\nu_{\rm
rec}/\Omega_{0}=10^{-3}$ and $\sigma_{\tau}=400$.
We numerically solve the equations of motion (2) with $\gamma=0$ at two values
of $\Delta$ corresponding to chaotic and regular regimes of the center-of-mass
motion. In accordance with the Lyapunov map in Fig. 2, behavior of the Bloch
component $u$ in Fig. 3, and the simple estimates given above, we expect the
chaotic scattering of atoms at $\Delta=0.2$ and their regular motion at
$\Delta=1$. Trajectories in the real and momentum spaces for 50 atoms with the
same initial momentum, $p_{0}=10$, and initial positions in the range
$-\pi/10\leq x\leq\pi/10$ are shown in Figs. 4 and 5, respectively, at the
fixed value of the recoil frequency $\omega_{r}=10^{-3}$. The upper panels in
Figs. 4 and 5 illustrate the broad distributions of the atoms in the $x$ and
$p$ spaces in the regime of chaotic scattering that contrasts strictly with
those obtained in the case of the regular scattering at $\Delta=1$ (Figs. 4b
and 5b). In order to create narrow atomic beams, one may use a pair of light
masks. The first SW with a red large detuning ($\Delta<0$) splits the initial
atomic beam into a number of narrow beams with the widths much smaller than
the optical wavelength which then cross the second SW. The method for creating
narrow wave packets in the nonadiabatic regime of scattering has been proposed
in Ref.[29].
### 2.3 Dissipative chaotic scattering and simulation of a real experiment
We have illustrated in Figs. 4 and 5 the Hamiltonian chaotic scattering that
may occur in the absence of any losses. To simulate trajectories of
spontaneously emitting atoms we use the standard stochastic wave-function
technique (see, for example, [30, 31, 32]) for solving Eqs. (2). The
integration time is divided into a large number of small time intervals
$\delta\tau$. At the end of the first one $\tau=\tau_{1}$ the probability of
spontaneous emission,
$s_{1}=\gamma\delta\tau|a_{\tau_{1}}|^{2}/(|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2})$,
is computed and compared with a random number $\varepsilon$ from the interval
$[0,1]$. If $s_{1}<\varepsilon_{1}$, then one prolongs the integration but
renormalizes the state vector in the end of the first interval at
$\tau=\tau_{1}^{+}$:
$a_{\tau_{1}^{+}}=a_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$
and
$b_{\tau_{1}^{+}}=b_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$.
If $s_{1}\geq\varepsilon_{1}$, then the atom emits a spontaneous photon and
performs the jump to the ground state at $\tau=\tau_{1}$ with
$A_{\tau_{1}}=\alpha_{\tau_{1}}=\beta_{\tau_{1}}=0$, $B_{\tau_{1}}=1$. Its
momentum in the $x$ direction changes for a random number from the interval
$[0,1]$ due to the photon recoil effect, and the next time step commences.
We simulate lithium atoms with the relevant transition $2S_{1/2}-2P_{3/2}$,
the corresponding wavelength $\lambda_{a}=670.7$ nm, recoil frequency
$\nu_{\rm rec}=63$ KHz, and the decay time $T_{\rm sp}=2.73\cdot 10^{-8}$ s.
With the maximal Rabi frequency $\Omega_{0}/2\pi\simeq 126$ MHz and the radius
of the laser beam $r=0.05$ cm one gets $\omega_{r}=10^{-3}$,
$\sigma_{\tau}=400$, and $\gamma=0.05$. Simulated trajectories in the real
space for 50 spontaneously emitting atoms under the same conditions as in Fig.
4 are shown in Fig. 6. Even though deterministic Hamiltonian chaos is masked
by random events of spontaneous emission, nevertheless the spatial and
momentum (not shown) distributions are much broader at those values of
$\Delta$ at which the Hamiltonian center-of-mass motion is chaotic. Namely the
chaotic Hamiltonian walking is eventually responsible for divergency of atomic
beams in the real and momentum spaces.
To simulate a real experiment we consider a beam of $10^{4}$ lithium atoms
with the initial Gaussian distribution (the rms $\sigma_{x}=\sigma_{p}=2$ and
the average values, $x_{0}=0$, and momentum, $p_{0}=10$) and compute their
distribution at a fixed moment of time. In Fig. 7a we compare the atomic
position distributions at $\tau=1000$ for the chaotic scattering at
$\Delta=0.2$ (bold curve) and the regular scattering at $\Delta=1$ (dashed
curve) when neglecting spontaneous emission. The difference is evident. In the
regime of the chaotic scattering at $\Delta=0.2$ atoms are distributed more or
less homogeneously over a large distance of 8 wavelengths along the $x$-axis
whereas they form a few peaks in a much more narrow interval under the
conditions of the regular scattering at $\Delta=1$. Figure 7b demonstrates the
distributions of spontaneously emitting atoms at the normalized decay rate
$\gamma=0.05$ under the same conditions as in Fig. 7a. The regularly scattered
atoms at $\Delta=1$ (dashed curve) form the contrast atomic relief with the
bifurcated peaks around the first few SW nodes at $x=\pm 1/4$, $x=\pm 3/4$ and
$x=5/4$. The distribution of chaotically scattered atoms at $\Delta=0.2$ (bold
curve) has the peaks without any bifurcation at $x=\pm 1/4$ and $x=3/4$ with a
smaller number of atoms in each one. Moreover, this distribution is less
contrast as compared to the previous one. Thus, we predict that under the
conditions of chaotic scattering there should appear less contrast and more
broadened atomic reliefs as compared to the case of regular scattering because
a large number of atoms are expected to be deposited between the nodes as a
result of chaotic walking along the SW axis. The effect is expected to be more
prominent under the coherent evolution but it seems to be observable with
spontaneously emitting atoms as well.
We predict that experiments on the scattering of atomic beams at a SW laser
field can directly image chaotic walking of atoms along the SW. In a real
experiment the final spatial distribution can be recorded via fluorescence or
absorption imaging on a CCD, commonly used methods in atom optics experiments
yielding information on the number of atoms and the cloud’s spatial size. The
other possibility is a nanofabrication where the atoms after the interaction
with the SW are deposited on a silicon substrate in a high vacuum chamber. In
this case the spatial distribution can be analyzed with an atomic force
microscope. As to the momentum distribution, it can be measured, for example,
by a time-of-flight technique. The modern tools of atom optics enable to
create narrow initial atomic distributions in position and momentum, reduce
coupling to the environment and technical noise, create one-dimensional
optical potentials, and to measure spatial and momentum distributions with
high sensitivity and accuracy [9].
## 3 Conclusion
We have simulated the new type of atomic diffraction at a SW without any
modulation of its parameters and shown that it can be observed in real
experiments. That would be the prove of existence of the novel type of atomic
motion, chaotic walking in a deterministic environment. The effect could be
used in optical nanolithography to fabricate complex atomic structures on
substrates.
## 4 Acknowledgments
The work was supported by the Integration grant from the Far-Eastern and
Siberian branches of the Russian Academy of Sciences (12-II-0-02-001), and by
the Program “Fundamental Problems of Nonlinear Dynamics in Mathematics and
Physics”. I thank L.E. Konkov and M.Yu. Uleysky for the help in preparing some
figures.
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|
arxiv-papers
| 2012-05-20T02:34:18 |
2024-09-04T02:49:31.099232
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1205.4374"
}
|
1205.4429
|
# $L^{1}$–Stability of Vortex Sheets and Entropy Waves
in Steady Supersonic Euler Flows over Lipschitz Walls
Gui-Qiang Chen Vaibhav Kukreja Gui-Qiang G. Chen, Mathematical Institute,
University of Oxford, Oxford, OX1 3LB, UK; Department of Mathematics,
Northwestern University, Evanston, IL 60208, USA; School of Mathematical
Sciences, Fudan University, Shanghai 200433, China chengq@maths.ox.ac.uk
Vaibhav Kukreja, Instituto de Matem$\acute{\text{a}}$tica Pura e Aplicada
(IMPA), Rio de Janeiro, Brazil; Department of Mathematics, Northwestern
University, Evanston, IL 60208, USA vaibhav@impa.br;
vkukreja@math.northwestern.edu
(Date: August 27, 2024)
###### Abstract.
We establish the well-posedness of compressible vortex sheets and entropy
waves in two-dimensional steady supersonic Euler flows over Lipschitz walls
under a $BV$ boundary perturbation. In particular, when the total variation of
the incoming flow perturbation around the background strong vortex
sheet/entropy wave is small, we prove that the two-dimensional steady
supersonic Euler flows containing a strong vortex sheet/entropy wave past a
Lipschitz wall are $L^{1}$–stable. Both the Lipschitz wall (whose boundary
slope function has small total variation) and incoming flow perturb the
background strong vortex sheet/entropy wave. The weak waves are reflected
after nonlinear waves interact with the strong vortex sheet/entropy wave and
the wall boundary. Using the wave-front tracking method, the existence of
solutions in $BV$ over Lipschitz walls is first shown, when the incoming flow
perturbation of the background strong vortex sheet/entropy wave has small
total variation. Then we establish the $L^{1}$–contraction of the solutions
with respect to the incoming flows. To achieve this, a Lyapunov functional,
equivalent to the $L^{1}$–distance between two solutions containing strong
vortex sheets/entropy waves, is carefully constructed to include the nonlinear
waves generated both by the wall boundary and from the incoming flow. This
functional is then shown to decrease in the flow direction, leading to the
$L^{1}$–stability, as well as the uniqueness, of the solutions. Furthermore,
the uniqueness of solutions extends to a larger class of viscosity solutions.
###### Key words and phrases:
Full Euler equations, entropy waves, compressible vortex sheets,
$L^{1}$–stability, steady flows, supersonic Euler flow, Riemann solutions,
Lipschitz wall, $BV$ perturbation, Glimm’s functional, nonlinear interaction,
global existence
###### 2010 Mathematics Subject Classification:
Primary: 35B35, 35B40, 76J20, 35L65, 85A05, 35A05
## 1\. Introduction
We study the well-posedness of two-dimensional steady supersonic Euler flows
past a curved Lipschitz wall containing strong vortex sheets/entropy waves in
the $L^{1}$–norm. The inviscid compressible flows are governed by the two-
dimensional steady Euler system:
$\begin{cases}(\rho u)_{x}+(\rho v)_{y}=0,\\\ (\rho u^{2}+p)_{x}+(\rho
uv)_{y}=0,\\\ (\rho uv)_{x}+(\rho v^{2}+p)_{y}=0,\\\ (\rho
u(E+\frac{p}{\rho}))_{x}+(\rho v(E+\frac{p}{\rho}))_{y}=0,\end{cases}$ (1.1)
with $(u,v)$, $p$, $\rho$, and $E$ representing the fluid velocity, scalar
pressure, density, and total energy, respectively. Furthermore, the total
energy $E$ is explicitly given by
$E=\frac{1}{2}(u^{2}+v^{2})+e(\rho,p),$
where the internal energy $e$ can be written as a function of $(\rho,p)$
defined through the thermodynamical relations. The temperature $T$ and entropy
$S$ are the other two thermodynamic variables.
In the case of an ideal gas, the pressure p and internal energy e can be
expressed as
$p=R\rho T,\hskip 14.22636pte=c_{\nu}T$ (1.2)
with the adiabatic index $\gamma$ given by
$\gamma=1+\frac{R}{c_{\nu}}>1.$ (1.3)
In particular, in terms of the density $\rho$ and entropy S, we have
$p=p(\rho,S)=\kappa\rho^{\gamma}e^{S/c_{\nu}},\qquad
e=\frac{\kappa}{\gamma-1}\rho^{\gamma-1}e^{S/c_{\nu}}=\frac{RT}{\gamma-1},$
(1.4)
The constants $R,c_{\nu}$, and $\kappa$ in the above relations are all greater
than zero.
When the entropy $S$ = constant, the flow is called isentropic. In this case,
the pressure $p$ can be written as a function of the density $\rho$,
$p=p(\rho)$, and the flow is governed by the isentropic Euler equations:
$\begin{cases}(\rho u)_{x}+(\rho v)_{y}=0,\\\ (\rho u^{2}+p)_{x}+(\rho
uv)_{y}=0,\\\ (\rho uv)_{x}+(\rho v^{2}+p)_{y}=0.\end{cases}$ (1.5)
Then, by scaling, the pressure-density relation is
$p(\rho)=\frac{\rho^{\gamma}}{\gamma}.$ (1.6)
The adiabatic exponent $\gamma>1$ corresponds to the isentropic polytropic
gas. The limiting case $\gamma=1$ corresponds to the isothermal flow. Define
$c=\sqrt{p_{\rho}(\rho,S)}$
as the sonic speed. For polytropic gases, the sonic speed is $c=\sqrt{\gamma
p/\rho}$.
The flow type is classified by the Mach number
$M=\frac{\sqrt{u^{2}+v^{2}}}{c^{2}}$. When $M>1$, system (1.1) or (1.5)
governs a supersonic flow (i.e., $u^{2}+v^{2}>c^{2}$), which has all real
eigenvalues and is hyperbolic. For $M<1$, system (1.1) or (1.5) governs a
subsonic flow (i.e., $u^{2}+v^{2}<c^{2}$), which has complex eigenvalues and
is elliptic-hyperbolic mixed and composite. When $M=1$, the flow is called
sonic.
We are interested in whether compressible vortex sheets/entropy waves in
supersonic flow over the Lipschitz wall are always stable under the $BV$
perturbation of the incoming flow. Multidimensional steady supersonic Euler
flows are important in many physical applications (cf. Courant-Friedrichs
[12]). In particular, when the upstream flow is a uniform steady flow above
the plane wall in $x<0$ all the time, the flow downstream above a Lipschitz
wall in $x>0$ is governed by a steady Euler flow after a sufficiently long
time. Moreover, compressible vortex sheets and entropy waves occur
ubiquitously in nature and are fundamental waves. Furthermore, since steady
Euler flows are asymptotic states and may be global attractors of the
corresponding unsteady Euler flows, it is important to establish the existence
of steady Euler flows and understand their qualitative behavior to shed light
on the long-time asymptotic behavior of the unsteady compressible Euler flows,
one of the most fundamental problems in mathematical fluid dynamics which is
still wide open.
We observe that the stability of contact discontinuities for the Cauchy
problem for strictly hyperbolic systems in one space dimension under a $BV$
perturbation has been studied by Sabl$\acute{\text{e}}$-Tougeron [25] and
Corli–Sabl$\acute{\text{e}}$-Tougeron [13]. In particular, the reflection
coefficients, such as $K_{11}$ here, are required to be less than one, which
is the stability condition for the mixed problem in the strip $\\{(t,x):t\geq
0,-1<x<1\\}$ in the earlier works; see, e.g., Sabl$\acute{\text{e}}$-Tougeron
[25]. Working with the non-isentropic Euler system (1.1) and a uniform
upstream flow, Chen-Zhang-Zhu [11] first proved the global existence in $BV$
of supersonic Euler flows containing a strong vortex sheet/entropy wave under
the $BV$ perturbation of the Lipschitz wall by using the Glimm scheme. The
essential difference between system (1.1) as analyzed in [11] (and in Sections
2–7 here) and strictly hyperbolic systems as considered in [13, 25] is that
two of the four characteristic eigenvalues coincide and have two linearly
independent eigenvectors which determine precisely the compressible vortex
sheets and entropy waves so that two independent parameters are required to
describe them, respectively.
In this paper, for completeness, we first show, via the wave-front tracking
method, the existence of solutions to the problem when a small $BV$
perturbation is added to the uniform incoming flow. Then the $L^{1}$–stability
of entropy solutions containing strong vortex sheets/entropy waves is
established. As corollaries of these results, the estimates on the uniformly
Lipschitz semigroup $\mathscr{S}$ of entropy solutions generated by the wave-
front tracking approximations are obtained, and the uniqueness of weak
solutions containing strong vortex sheets/entropy waves is established in a
larger set of solutions, namely the class of viscosity solutions.
In the following, we focus mainly on the problem in the region $\Omega$ over
the Lipschitz wall for the supersonic Euler flows $U(u,v,p,\rho)$ governed by
system (1.1), given that the corresponding problem for the isentropic system
(1.5) is simpler to analyze. The subsequent figure provides the schematic
diagram for the problem we study:
Figure 1.1. Stability of the compressible vortex sheet/entropy wave in
supersonic flow
The boundary and initial data in the problem are as follows:
* (i)
There is a Lipschitz function $g\in{\rm Lip}(\mathbb{R}_{+};\mathbb{R})$ such
that
$g(0)=g^{\prime}(0+)=0,\hskip
8.53581pt\displaystyle\lim_{x\to\infty}\text{arctan}(g^{\prime}(x+))=0,\hskip
8.53581ptg^{\prime}\in BV(\mathbb{R}_{+};\mathbb{R})$
and
${\rm TV}(g^{\prime}(\cdot))<\varepsilon\quad\hskip 14.22636pt\text{for some
constant }\varepsilon>0.$
Denote $\Omega\mathrel{\mathop{:}}=\\{(x,y):y>g(x),x\geq 0\\}$,
$\Gamma\mathrel{\mathop{:}}=\\{(x,y):y=g(x),x\geq 0\\}$, and
$\textbf{n}(x\pm)$ =
$\frac{(-g^{\prime}(x\pm),1)}{\sqrt{(g^{\prime}(x\pm))^{2}+1}}$ as the outer
normal vectors to $\Gamma$ at the respective points $x\pm$ (cf. Fig. 1.1).
* (ii)
The incoming flow $U=\overline{U}(y):=U_{0}^{b}+\widetilde{U_{0}}$ at $x=0$ is
composed of two parts:
1. (a)
The upstream flow $U_{0}^{b}$ consists of one straight vortex sheet/entropy
wave $y=y_{0}^{\ast}>0$ and two constant vectors $U^{-}_{0}=U_{-}$, when
$0<y<y_{0}^{\ast}$, and $U^{+}_{0}=U_{+}$, when $y>y_{0}^{\ast}>0$, satisfying
$v_{-}=v_{+}=0,\hskip 14.22636ptu_{\pm}>c_{\pm}>0,$
where $c_{\pm}=\sqrt{\gamma p_{\pm}/\rho_{\pm}}$ is the sonic speed of state
$U_{\pm}$.
2. (b)
The $BV$ perturbation
$\widetilde{U_{0}}=(\tilde{u}_{0},\tilde{v}_{0},\tilde{p}_{0},\tilde{\rho}_{0})(y)\in
L^{1}\cap BV(\mathbb{R};\mathbb{R}^{4})$ at $x=0$ so that ${\rm
TV}(\widetilde{U_{0}})\ll 1$.
Then we consider the following initial-boundary value problem for system
(1.1):
$\displaystyle\textbf{Cauchy Condition}:\qquad\quad$ $\displaystyle
U|_{x=0}=\overline{U}(y)=U_{0}^{b}+\widetilde{U_{0}};$ (1.7)
$\displaystyle\textbf{Boundary Condition}:\quad\quad$
$\displaystyle(u,v)\cdot\textbf{n}=0\qquad\text{ on }\Gamma.$ (1.8)
Definition 1.1 (Admissible entropy solutions). A $BV$ function $U=U(x,y)$ is
said to be an entropy solution of the initial-boundary value problem (1.1) and
(1.7)–(1.8) if and only if the following conditions hold:
* (i)
$U$ is a weak solution of (2.1) and satisfies
$U|_{x=0}=\overline{U}(y)\quad\mbox{and}\quad(u,v)\cdot\textbf{n}|_{y=g(x)}=0\,\,\,\text{
in the trace sense;}$
* (ii)
$U$ satisfies the steady entropy Clausius inequality:
$(\rho uS)_{x}+(\rho vS)_{y}\geq 0$ (1.9)
in the distributional sense in $\Omega$ including the Lipschitz wall boundary.
One of the essential developments within this paper is to develop suitable
methods to deal with the challenges caused by the nonstrictly hyperbolicity of
the system and the Lipschitz wall boundary, in comparison with the previous
progress with the strictly hyperbolic systems of conservation laws,
particularly to the analysis of the Cauchy problem. For supersonic Euler flow
with a strong shock-front emanating from the wedge vertex, Chen-Li [10] worked
out the issue for a Lipschitz wedge boundary. We now discuss some main
differences in our work here from the Cauchy problem and the resulting key
difficulties. We remark that, in the case of the Cauchy problem concerning
only $weak$ waves, the decrease of the Lyapunov functional and the
$L^{1}$–stability of the solutions were obtained through the cancellation of
distances on both sides of waves. In the presence of a strong shock, for the
$L^{1}$–stability of solutions of the Cauchy problem for strictly hyperbolic
systems of conservation laws, the Lyapunov functional was found to decrease by
employing the strength of the strong shock to control the strengths of weak
waves of the other families (e.g., see Lewicka-Trivisa [22]). In contrast with
our Lipschitz wall problem, which is a problem of initial-boundary value type,
there is no such cancellation by the boundary as only one-side is possible
near it. Furthermore, no strong vortex sheets/entropy waves (characteristic
discontinuities) nor strong shocks are present to handle the strength of the
weak waves of the other families, and the terms in the estimates for the first
and fourth family carry different signs. As such, it is difficult to say
whether the functional can be made to decrease for our case of strong vortex
sheets and entropy waves with multiplicity of eigenvalues. One of the key
steps to resolve this is to use the physical feature of the boundary condition
that the flow of two solutions near the boundary must run in parallel (also
see [10]). This observation helps us to obtain additional quantitative
relations near the boundary. Then, applying suitable weights and adjustments
in the coefficients of the Lyapunov functional and using the cancellation
between the different families, the functional is found to decrease in the
flow direction.
The rest of the paper is organized as follows. In Section 2, we recall some
fundamental properties of the two-dimensional steady Euler system (1.1) and
discuss related nonlinear waves and wave interaction estimates. In Section 3,
the wave-front tracking algorithm is discussed, working in the presence of
strong vortex sheets/entropy waves, the suitable interaction potential
$\mathcal{Q}$ is constructed, including the effect of the Lipschitz wall, and
the existence of entropy solutions in $BV$ is established for the initial-
boundary value problem. In Section 4, we construct the Lyapunov functional
$\mathit{\Phi}$ (equivalent to the $L^{1}$–distance between two entropy
solutions $U$ and $V$) to include the nonlinear waves produced by the wall
boundary vertices. Then, in Section 5, the monotone decrease of the functional
$\mathit{\Phi}$ is established in the flow direction, leading to the
$L^{1}$–stability of the solutions containing strong vortex sheets/entropy
waves. Using the estimates established in Sections 3–5, in Section 6, we
obtain the existence of a Lipschitz semigroup of solutions generated by a
wave-front tracking approximation, as well as some estimates on the uniformly
Lipschitz semigroup $\mathscr{S}$ produced by the limit of wave-front tracking
approximations. Moreover, the uniqueness of solutions with strong vortex
sheets/entropy waves is obtained in the larger class of viscosity solutions.
## 2\. Adiabatic Euler equations: Nonlinear waves and wave interactions
In this section, we first present some basic properties of the steady Euler
system (1.1). Then related nonlinear waves and interaction estimates are
discussed, which will be employed in the later sections.
Consider the following vector functions of the solution $U$:
$W(U)=(\rho u,\rho u^{2}+p,\rho uv,\rho u(h+\frac{u^{2}+v^{2}}{2}))^{\top},$
$H(U)=(\rho v,\rho uv,\rho v^{2}+p,\rho v(h+\frac{u^{2}+v^{2}}{2}))^{\top},$
where $h=\frac{\gamma p}{(\gamma-1)\rho}$. Then the steady Euler equations in
(1.1) can be expressed in the following conservative form:
$W(U)_{x}+H(U)_{y}=0,\hskip 14.22636ptU=(u,v,p,\rho)^{\top}$ (2.1)
When $U(x,y)$ is a smooth solution, system (2.1) is equivalent to
$\nabla_{U}W(U)U_{x}+\nabla_{U}H(U)U_{y}=0.$ (2.2)
Then the roots of the fourth degree polynomial
${\rm det}(\lambda\nabla_{U}W(U)-\nabla_{U}H(U)),$ (2.3)
are the eigenvalues of (2.1); that is, the solutions of the equation
$(v-\lambda u)^{2}\big{(}(v-\lambda u)^{2}-c^{2}(1+\lambda^{2})\big{)}=0,$
(2.4)
where $c=\sqrt{\frac{\gamma p}{\rho}}$ is the sonic speed. For supersonic
flows (i.e. $u^{2}+v^{2}>c^{2}$), system (2.1) is hyperbolic. Specifically,
when $u>c$, system (2.1) has four real eigenvalues in the $x$-direction:
$\displaystyle\lambda_{d}$ $\displaystyle=$
$\displaystyle\frac{uv+(-1)^{d}c\sqrt{u^{2}+v^{2}-c^{2}}}{u^{2}-c^{2}},\qquad
d=1,4;$ $\displaystyle\lambda_{k}$ $\displaystyle=$
$\displaystyle\frac{v}{u},\qquad k=2,3,$ (2.5)
with the four corresponding linearly independent eigenvectors given by
$\displaystyle\textbf{r}_{d}$ $\displaystyle=$
$\displaystyle\kappa_{d}(-\lambda_{d},1,\rho(\lambda_{d}u-v),\frac{\rho(\lambda_{d}u-v)}{c^{2}})^{\top},\qquad
d=1,4,$ $\displaystyle\textbf{r}_{2}$ $\displaystyle=$
$\displaystyle(u,v,0,0)^{\top},\hskip
14.22636pt\textbf{r}_{3}=(0,0,0,\rho)^{\top},$ (2.6)
where $\kappa_{d}$ the re-normalization factors such that
$\textbf{r}_{d}\cdot\nabla\lambda_{d}=1$, given that the $d$th-characteristic
fields, $d=1,4$, are genuinely nonlinear. The second and third linearly
degenerate characteristic fields satisfy
$\textbf{r}_{k}\cdot\nabla\lambda_{k}=0$, $k=2,3$, which correspond to vortex
sheets and entropy waves, respectively.
The wave curves in the phase space are now described. The Rankine-Hugoniot
jump conditions for (2.1) are
$\sigma\left[W(u)\right]=\left[H(u)\right],$ (2.7)
and the discontinuity propagates with the speed $\sigma$.
There are two different waves associated with the fields
$\lambda_{k}=\frac{v_{0}}{u_{0}},k=2,3$, with the corresponding linearly
independent right eigenvectors $\textbf{r}_{k},k=2,3$, in (2.6):
Vortex sheets:
$C_{2}(U_{0}):\hskip 8.53581pt\sigma=\frac{v}{u}=\frac{v_{0}}{u_{0}},\hskip
11.38109ptp=p_{0},\hskip 8.53581ptS=S_{0},\hskip 8.53581ptu^{2}+v^{2}\neq
u_{0}^{2}+v^{2}_{0},$ (2.8)
Entropy waves:
$C_{3}(U_{0}):\hskip 8.53581pt\sigma=\frac{v}{u}=\frac{v_{0}}{u_{0}},\hskip
11.38109ptp=p_{0},\hskip 8.53581pt(u,v)=(u_{0},v_{0}),\hskip 8.53581ptS\neq
S_{0}.$ (2.9)
Albeit the two contact discontinuities, the vortex sheet and the entropy wave,
above match as a single discontinuity in the physical $(x,y)$–plane, two
independent parameters are needed to describe them in the phase space
$U=(u,v,p,\rho)$ since there are two linearly independent eigenvectors
corresponding to the repeated eigenvalues
$\lambda_{2}=\lambda_{3}=\frac{v}{u}$ of the linearly degenerate
characteristics fields.
The nonlinear waves associated with $\lambda_{d},d=1,4$, are shock waves and
rarefaction waves. The shock waves have their speeds of propagation given by
$\sigma=\sigma_{d}\mathrel{\mathop{:}}=\frac{u_{0}v_{0}+(-1)^{d}\bar{c}_{0}\sqrt{u_{0}^{2}+v_{0}^{2}-\bar{c}_{0}^{2}}}{u^{2}-\bar{c}_{0}^{2}},\qquad
d=1,4,$
where $\bar{c}_{0}^{2}=\frac{c^{2}_{0}}{b_{0}}\frac{\rho}{\rho_{0}}$ and
$b_{0}=\frac{\gamma+1}{2}-\frac{\gamma-1}{2}\frac{\rho}{\rho_{0}}$.
Substituting $\sigma_{d},$ $d=1,4$, into (2.7), the $d$-Hugoniot curve
$S_{d}(U_{0})$ through the state $U_{0}$ is
$S_{d}(U_{0}):\hskip 8.53581pt[p]=\frac{c^{2}_{0}}{b_{0}}[\rho],\hskip
8.53581pt[u]=-\sigma_{d}[v],\hskip
8.53581pt\rho_{0}(\sigma_{d}u_{0}-v_{0})[v]=[p],\qquad d=1,4.$
Written as $S_{d}^{+}(U_{0})$, $d=1,4$, the half curves of $S_{d}(U_{0})$ for
$\rho>\rho_{0}$ in the phase space are said to be the shock curves on which
any state forms a shock with the below state $U_{0}$ in the $(x,y)$–plane
respecting the entropy condition (1.9). Furthermore, for each $d=1$ or $d=4$,
the curves $S^{+}_{d}(U_{0})$ and $R^{-}_{d}(U_{0})$ at the state $U_{0}$ have
the same curvature.
If $U$ is a piecewise smooth solution (see also [11]), then any of the
following conditions below is equivalent to the entropy inequality (1.9) in
Definition 1.1 for a shock wave:
* (i)
The physical entropy condition: The density increases across the shock in the
flow direction,
$\rho_{\text{back}}<\rho_{\text{front}}.$ (2.10)
* (ii)
The Lax entropy condition: On the $d$th-shock, the shock speed $\sigma_{d}$
satisfies
$\displaystyle\lambda_{d}(\text{back})<\sigma_{d}<\lambda_{d}(\text{front})\hskip
8.53581pt\mbox{for}\,\,\,d=1,4,$ (2.11)
$\displaystyle\sigma_{1}<\lambda_{2,3}(\text{back}),\hskip
14.22636pt\lambda_{2,3}(\text{front})<\sigma_{4}.$ (2.12)
The rarefaction wave curves $R_{l}^{-}(U_{0})$ through the state $U_{0}$ in
the state space are given by
$R^{-}_{d}:\hskip 8.53581ptdp=c^{2}d\rho,\hskip
8.53581ptdu=-\lambda_{d}dv,\hskip 8.53581pt\rho(\lambda_{d}u-v)dv=dp\hskip
8.53581pt\text{ for }\rho<\rho_{0},\qquad d=1,4.$ (2.13)
We next discuss several essential properties of the nonlinear waves and
related wave interaction estimates in Lemmas 2.1–2.7 below. These facts will
be used in the later sections. We also refer the reader to Chen-Zhang-Zhu [11]
for further details.
_2.1. Riemann Problems and Riemann Solutions_
We focus on the related Riemann problems and their solutions in this section,
which serve as the building blocks for the front tracking algorithm for the
initial-boundary value problem (2.1) and (1.7)–(1.8).
_Riemann problem of lateral-type._ We note that the straight-sided wall
problem is the case when problem (2.1) and (1.7)–(1.8) is considered with the
boundary $g\equiv 0$. It can be seen that, when the angle between the
straight-sided wall and the flow direction of the incoming flow is zero,
problem (2.1) and (1.7)–(1.8) has an entropy solution made up of two constants
states $U_{-}=(u_{-},0,p_{-},\rho_{-})$ and $U_{+}=(u_{+},0,p_{+},\rho_{+})$,
satisfying $u_{\pm}>c_{\pm}>0$ in the subdomains $\Omega_{+}$ and $\Omega_{-}$
of $\Omega$ separated by a straight vortex sheet/entropy wave. These are
precisely the states $U_{-}$ and $U_{+}$ below and above the large vortex
sheet/entropy wave. The principal aim of this paper is to establish the
$L^{1}$–well-posedness for problem (2.1) and (1.7)–(1.8) for the solutions
near the background solution containing a strong vortex sheet/entropy wave
$\\{U_{-},U_{+}\\}$ with $g\equiv 0$.
It has been observed in [12] that, if the angle between the flow direction of
the front state and the wall at a boundary vertex is smaller than $\pi$ and
larger than the extreme angle determined by the incoming flow state and
$\gamma\geq 1$, then a unique $4$-shock is generated, separating the front-
state from the supersonic back-state. If the angle between the flow direction
of the front-state and the wall at a boundary vertex is larger than $\pi$ and
less than the extreme angle, then a $4$-rarefaction wave is produced,
emanating from the vertex. These waves are easily seen through the shock polar
analysis (cf. [11, 12]). This signifies that, when the angle between the flow
direction of the front-state and the wall at a boundary vertex is close to
$\pi$, the lateral Riemann problem can be uniquely solved. For further
details, see Lemma 2.3 and [11]. For an indepth discussion, we also refer to
Courant–Friedrichs [12].
_Riemann problem involving only weak waves._ Consider the subsequent initial
value problem with piecewise constant initial data:
$\begin{cases}W(U)_{x}+H(U)_{y}=0,\\\\[5.69054pt]
U|_{x=x_{0}}=\underline{U}=\begin{cases}U_{a},\hskip 5.69054pty>y_{0},\\\
U_{b},\hskip 5.69054pty<y_{0},\end{cases}\end{cases}$ (2.14)
with the constant states $U_{a}$ and $U_{b}$ denoting the above state and
below state with respect to the line $y=y_{0}$, respectively. Then there is
$\varepsilon>0$ so that, for any states $U_{b},U_{a}$ in the neighborhood
$\textit{O}_{\varepsilon}(U_{+})$ of $U_{+}$, or $U_{b},U_{a}$ in the
neighborhood $\textit{O}_{\varepsilon}(U_{-})$ of $U_{-}$, the initial value
problem (2.14) has a unique admissible solution consisting of four waves,
consisting of shocks, rarefaction waves, vortex sheets and/or entropy waves.
_Riemann problem involving the strong vortex sheets/entropy waves._ From now
on, the notation
$\\{U_{b},U_{a}\\}=\left(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\right)$
will be used to write
$U_{a}={\Phi}\left(\alpha_{4},\alpha_{3},\alpha_{2},\alpha_{1};U_{b}\right)$
as the solution of the Riemann problem, where ${\Phi}\in C^{2}$, and
$\alpha_{j}$ is the strength of the $j$-wave. For any wave with $U_{b}\in
O_{\varepsilon}(U_{-})$ and $U_{a}\in O_{\varepsilon}(U_{+})$, we also use
$\\{U_{b},U_{a}\\}=\left(0,\sigma_{2},\sigma_{3},0\right)$ to denote the
strong vortex sheet/entropy wave that connects $U_{b}$ and $U_{a}$ with
strength $\left(\sigma_{2},\sigma_{3}\right)$. That is,
$U_{m}=\Phi(\sigma_{2};U_{b})\mathrel{\mathop{:}}=\left(u_{b}e^{\sigma_{2}},v_{b}e^{\sigma_{2}},p_{b},\rho_{b}\right),\quad
U_{a}=\Phi(\sigma_{3};U_{m})\mathrel{\mathop{:}}=\left(u_{m},v_{m},p_{m},\rho_{m}e^{\sigma_{3}}\right).$
Particularly, we observe that
$U_{+}=(u_{+},0,p_{+},\rho_{+})=(u_{-}e^{\sigma_{20}},0,p_{-},\rho_{-}e^{\sigma_{30}}).$
We write
$G\left(\sigma_{3},\sigma_{2};U_{b}\right)=\Phi_{3}(\sigma_{3};\Phi_{2}(\sigma_{2};U_{b}))$
for any $U_{b}\in O_{\varepsilon}(U_{-})$. Then we have
Lemma 2.1. _The vector function G( $\sigma_{3},\sigma_{2};U_{b}$) satisfies_
$G_{\sigma_{2}}\left(\sigma_{3},\sigma_{2};U_{b}\right)=\left(u_{b}e^{\sigma_{2}},v_{b}e^{\sigma_{2}},0,0\right),\hskip
11.38109ptG_{\sigma_{3}}\left(\sigma_{3},\sigma_{2};U_{b}\right)=(0,0,0,\rho_{b}e^{\sigma_{3}}),$
(2.15)
_and_
$\nabla_{U}G(\sigma_{3},\sigma_{2};U_{b})={\rm
diag}(e^{\sigma_{2}},e^{\sigma_{2}},1,e^{\sigma_{3}}).$ (2.16)
_Furthermore, for the plane vortex sheet and entropy wave with the lower state
$U_{-}=(u_{-},0,p_{-},\rho_{-})$, upper state
$U_{+}=(u_{+},0,p_{+},\rho_{+})$, and strength $(\sigma_{20},\sigma_{30})$_,
${\rm
det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))>0.$
(2.17)
These can be easily obtained from direct calculations and are thus omitted.
The properties in (2.15)–(2.17) above play a fundamental role in achieving the
necessary estimates on the strengths of reflected weak waves in the
interaction between the strong vortex sheet/entropy wave and weak waves (see
the proofs for Lemmas 2.4–2.7).
_2.2. Wave Interactions and Reflection Estimates._ In the following, several
essential estimates are provided on wave interactions and reflections. For
their proofs and all the related details, we refer to [11].
Weak wave interactions estimates. For the weak wave interaction away from both
the strong vortex sheet/entropy wave and the wall boundary in the regions
$\Omega_{+}$ or $\Omega_{-}$, we have the following estimate:
Lemma 2.2. _Assume that_
$U_{b},U_{m},U_{a}\in O_{\varepsilon}(U_{+}),\quad\mbox{or}\quad
U_{b},U_{m},U_{a}\in O_{\varepsilon}(U_{-}),$
_are three states with
$\\{U_{b},U_{m}\\}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$,
$\\{U_{m},U_{a}\\}=(\beta_{1},\beta_{2},\beta_{3},\beta_{4})$. Then
$\\{U_{b},U_{a}\\}=(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$ with_
$\gamma_{i}=\alpha_{i}+\beta_{i}+O(1)\Delta(\alpha,\beta),$ (2.18)
_where
$\Delta(\alpha,\beta)=(|\alpha_{4}|+|\alpha_{3}|+|\alpha_{2}|)|\beta_{1}|+|\alpha_{4}|(|\beta_{2}|+|\beta_{3}|)+\sum_{j=1,4}\Delta_{j}(\alpha,\beta)$
and_
$\Delta_{j}(\alpha,\beta)=\begin{cases}0,\hskip 8.53581pt&\alpha_{j}\geq
0,\beta_{j}\geq 0,\\\ |\alpha_{j}||\beta_{j}|,\hskip
5.69054pt&otherwise.\end{cases}$ (2.19)
Estimates on the boundary perturbation of weak waves and the reflection of
weak waves on the boundary. We write $\\{C_{l}(a_{l},b_{l})\\}^{\infty}_{l=0}$
for the points $\\{(a_{l},b_{l})\\}^{\infty}_{l=0}$ in the $(x,y)$–plane with
$0<a_{l}<a_{l+1}$. Define
$\displaystyle\left\\{\begin{array}[]{ll}\theta_{{l},{l+1}}=\text{arctan}\left(\frac{b_{l+1}-b_{l}}{a_{l+1}-a_{l}}\right),\hskip
5.69054pt\theta_{l}=\theta_{{l},{l+1}}-\theta_{{l-1},{l}},\hskip
5.69054pt\theta_{-1,0}=0,\\\\[5.69054pt]
\Omega_{l+1}=\\{(x,y):x\in\left[a_{l},a_{l+1}\right],y>b_{l}+(x-a_{l})\text{tan}(\theta_{{l},{l+1}})\\},\\\\[5.69054pt]
\Gamma_{l+1}=\\{(x,y):x\in\left(a_{l},a_{l+1}\right),y=b_{l}+(x-a_{l})\text{tan}(\theta_{{l},{l+1}})\\},\end{array}\right.$
(2.23)
and the outer normal vector to $\Gamma_{l}$:
$\textbf{n}_{l+1}=\frac{(b_{l+1}-b_{l},a_{l}-a_{l+1})}{\sqrt{(b_{l+1}-b_{l})^{2}+(a_{l+1}-a_{l})^{2}}}=(\text{sin}(\theta_{l},\theta_{l+1}),-\text{cos}(\theta_{l},\theta_{l+1})).$
(2.24)
With the constant state $\underline{U}$, consider the following initial-
boundary value problem:
$\begin{cases}(2.1)\hskip 62.59596pt&\text{in}\hskip 2.84526pt\Omega_{l+1},\\\
U|_{x=a_{l}}=\underline{U},\\\ (u,v)\cdot\textbf{n}_{l+1}=0\hskip
8.53581pt&\text{on}\hskip 2.84526pt\Gamma_{l+1}.\end{cases}$ (2.25)
Lemma 2.3. _Suppose
$\left\\{U_{m},U_{a}\right\\}=(\beta_{1},\beta_{2},\beta_{3},0)$ and
$\left\\{U_{l},U_{m}\right\\}=(0,0,0,\alpha_{4})$ with_
$(u_{l},v_{l})\cdot\textbf{n}_{l}=0.$
_Then there exists a unique solution $U_{l+1}$ of problem (2.25) such that
$\left\\{U_{l+1},U_{a}\right\\}=(0,0,0,\delta_{4})$ and
$U_{l+1}\cdot(\textbf{n}_{l+1},0,0)=0$. Moreover,_
$\delta_{4}=\alpha_{4}+K_{b1}\beta_{1}+K_{b2}\beta_{2}+K_{b3}\beta_{3}+K_{b0}\theta_{l},$
(2.26)
_where $K_{b1}$, $K_{b2}$, $K_{b3}$, and $K_{b0}$ are $C^{2}$-functions of
$\beta_{3}$, $\beta_{2}$, $\beta_{1}$, $\alpha_{4}$, $\theta_{l+1}$, and
$U_{a}$ satisfying_
$K_{b1}|_{\left\\{\theta_{l}=\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,U_{a}=U_{-}\right\\}}=1,\hskip
28.45274ptK_{bi}|_{\left\\{\theta_{l}=\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,U_{a}=U_{-}\right\\}}=0,\,\,i=2,3,$
(2.27)
_and $K_{b0}$ is bounded. In particular, $K_{b0}<0$ at the origin._
This lemma has two purposes. The first is to estimate the weak waves generated
by the vertices on the Lipschitz wall boundary. This boundedness will be used
to control the boundary perturbation; see (3.2) in the construction of the
wave interaction potential $\mathcal{Q}(x)$. The second is to estimate the
strength of the reflected wave $\delta_{4}$ with respect to the incident wave
$\alpha_{1}$. Property (2.27) of the coefficients will play an important role
to control the reflected waves.
Estimates on the interaction between the strong vortex sheet/entropy wave and
weak waves from below. Estimate (2.28) below plays a key role in ensuring the
$L^{1}$–stability of entropy solutions, especially for the existence of the
constants $w^{b}_{1}$ and $w^{b}_{4}$ in Lemma 5.1 (see below). This estimate
also ensures the existence of $K^{\ast}\in$ ($K_{11}$, 1) in the construction
of the wave interaction potential $\mathcal{Q}(x)$ in (3.2).
Lemma 2.4. _Let $U_{b},U_{m}\in O_{\varepsilon}(U_{-})$ and $U_{a}\in
O_{\varepsilon}(U_{+})$ with_
$\\{U_{b},U_{m}\\}=(0,\alpha_{2},\alpha_{3},\alpha_{4}),\hskip
19.91692pt\\{U_{m},U_{a}\\}=(\beta_{1},\sigma_{2},\sigma_{3},0).$
_Then there exists a unique
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ such that
the Riemann problem (2.14) admits an admissible solution that consists of a
weak $1-$wave of strength $\delta_{1}$, a strong vortex sheet of strength
$\sigma_{2}$, a strong entropy wave of strength $\sigma_{3}$, and a weak
$4-$wave of strength $\delta_{4}$:_
$\\{U_{b},U_{a}\\}=(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4}).$
_Moreover,_
$\displaystyle\delta_{1}=\beta_{1}+K_{11}\alpha_{4}+O(1)\Delta^{\prime},\hskip
14.22636pt\delta_{4}=K_{14}\alpha_{4}+O(1)\Delta^{\prime},$
$\displaystyle\sigma_{2}^{\prime}=\sigma_{2}+\alpha_{2}+K_{12}\alpha_{4}+O(1)\Delta^{\prime},\hskip
14.22636pt\sigma_{3}^{\prime}=\sigma_{3}+\alpha_{3}+K_{13}\alpha_{4}+O(1)\Delta^{\prime},$
$\displaystyle|K_{11}|_{\left\\{\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}=\left|\frac{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{4}(U_{-})}{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}\right|<1,$
(2.28)
_where $\Delta^{\prime}=|\beta_{1}|(|\alpha_{2}|+|\alpha_{3}|)$, and
$\sum_{j=2}^{4}|K_{1j}|$ is bounded._
Lemma 2.5. _The coefficient
$|K_{14}|_{\left\\{\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}$
in the strength $\delta_{4}$ of a weak 4-wave in Lemma 2.4 remains bounded
away from zero._
Proof. By Lemma 2.4, we can find a unique solution
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ as a
$C^{2}$-function of $\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},$
$\sigma_{2},\sigma_{3}$, and $U_{b}$ to
$\Phi_{4}(\delta_{4};G(\sigma^{\prime}_{3},\sigma^{\prime}_{2};\Phi_{1}(\delta;U_{b})))=G(\sigma_{2},\sigma_{3};\Phi_{1}(\beta_{1},\Phi(\alpha_{4},\alpha_{3},\alpha_{2},0;U_{b}))).$
(2.29)
That is,
$\sigma^{\prime}_{i}=\sigma^{\prime}_{i}(\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3}),\hskip
5.69054pti=2,3;\hskip
14.22636pt\delta_{j}=\delta_{j}(\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3}),\hskip
5.69054ptj=1,4,$
where we have omitted $U_{b}$ for simplicity.
From [11], we know that
$K_{1j}=\int\limits_{0}^{1}\
\partial_{\alpha_{4}}\delta_{j}(\alpha_{2},\alpha_{3},\theta\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3})\,d\theta,\hskip
14.22636ptj=1,4.$
Differentiate (2.29) with respect to $\alpha_{4}$, and let
$\beta_{1}=\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}=\sigma_{20}$, and
$\sigma_{3}=\sigma_{30}$. We obtain
$\displaystyle\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{4}(U_{-})$
$\displaystyle=$
$\displaystyle\partial_{\alpha_{4}}\delta_{4}\,\textbf{r}_{4}(U_{+})+\partial_{\alpha_{4}}\sigma^{\prime}_{3}\,G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-})$
$\displaystyle+\hskip
2.84526pt\partial_{\alpha_{4}}\sigma^{\prime}_{2}\,G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-})+\partial_{\alpha_{4}}\delta_{1}\,\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}).$
By Lemma 2.1, we have
$|\partial_{\alpha_{4}}\delta_{4}|$
$\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{4}(U_{-}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{1}(U_{-})\kappa_{4}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{2\sigma_{20}+\sigma_{30}}(\lambda_{4}(U_{-})-\lambda_{1}(U_{-}))}{\kappa_{1}(U_{-})\kappa_{4}(U_{+})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{2\kappa_{4}(U_{-})e^{\sigma_{20}}\lambda_{4}(U_{-})}{\kappa_{4}(U_{+})(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-}))}\right|>0.$
This completes the proof.
_Estimates on the interaction between the strong vortex sheet/entropy wave and
weak waves from above._ We have
Lemma 2.6. _Let $U_{b}\in O_{\varepsilon}(U_{-})$ and $U_{m},U_{a}\in
O_{\varepsilon}(U_{+})$ with_
$\\{U_{b},U_{m}\\}=(0,\sigma_{2},\sigma_{3},\alpha_{4}),\hskip
19.91692pt\\{U_{m},U_{a}\\}=(\beta_{1},\beta_{2},\beta_{3},0).$
_Then there exists a unique
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ such that
the Riemann problem (2.14) admits an admissible solution that consists of a
weak $1-$wave of strength $\delta_{1}$, a strong vortex sheet of strength
$\sigma_{2}$, a strong entropy wave of strength $\sigma_{3}$, and a weak
$4-$wave of strength $\delta_{4}$:_
$\\{U_{b},U_{a}\\}=(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4}).$
_Moreover,_
$\delta_{1}=K_{21}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},\hskip
14.22636pt\sigma_{2}^{\prime}=\sigma_{2}+\beta_{2}+K_{22}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},$
$\sigma_{3}^{\prime}=\sigma_{3}+\beta_{3}+K_{23}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},\hskip
14.22636pt\delta_{4}=\alpha_{4}+K_{24}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},$
_where $\sum_{j=1}^{4}|K_{2j}|$ is bounded and
$\Delta^{{}^{\prime\prime}}=|\alpha_{4}|(|\beta_{2}|+|\beta_{3}|).$_
The constant $K_{21}$ here is used in the definition of weighted strength
$b_{\alpha}$ of weak waves in (3.1).
Lemma 2.7. _The coefficient
$|K_{21}|_{\left\\{\beta_{3}=\beta_{2}=\beta_{1}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}$
in the strength $\delta_{1}$ of a weak 1-wave in Lemma 2.6 remains bounded
away from zero, while the reflection coefficient $|K_{24}|<1$._
Proof. By Lemma 2.6, we can find a unique solution
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ as a
$C^{2}$-function of $\alpha_{2},\alpha_{3},\alpha_{4}$, $\beta_{1}$,
$\sigma_{2}$, $\sigma_{3}$, and $U_{b}$ to
$\Phi_{4}(\delta_{4};G(\sigma^{\prime}_{3},\sigma^{\prime}_{2};\Phi_{1}(\delta;U_{b})))=\Phi(0,\beta_{3},\beta_{2},\beta_{1};\Phi_{4}(\alpha_{4};G(\sigma_{3},\sigma_{2};U_{b}))).$
(2.30)
That is,
$\sigma^{\prime}_{i}=\sigma^{\prime}_{i}(\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3}),\hskip
5.69054pti=2,3;\hskip
14.22636pt\delta_{j}=\delta_{j}(\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3}),\hskip
5.69054ptj=1,4,$
where we have omitted $U_{b}$ for simplicity.
From [11], we know that
$K_{2j}=\int\limits_{0}^{1}\
\partial_{\beta_{1}}\partial_{j}(\theta\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3})\,d\theta,\hskip
8.53581ptj=1,4.$
Differentiate (2.30) with respect to $\beta_{1}$ and let
$\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,\sigma_{2}=\sigma_{20}$, and
$\sigma_{3}=\sigma_{30}$. We obtain
$\displaystyle\textbf{r}_{1}(U_{+})$ $\displaystyle=$
$\displaystyle\partial_{\beta_{1}}\delta_{4}\,\textbf{r}_{4}(U_{+})+\partial_{\beta_{1}}\sigma^{\prime}_{3}\,G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-})$
$\displaystyle+\hskip
2.84526pt\partial_{\beta_{1}}\sigma^{\prime}_{2}\,G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-})+\partial_{\beta_{1}}\delta_{1}\,\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}).$
By Lemma 2.1, we have
$\displaystyle|\partial_{\beta_{1}}\delta_{1}|$ $\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\textbf{r}_{1}(U_{+}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{4}(U_{+})\kappa_{1}(U_{+})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{+})e^{\sigma_{20}+\sigma_{30}}\big{)})}{\kappa_{4}(U_{+})\kappa_{1}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-})\big{)}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{2\kappa_{1}(U_{+})\lambda_{4}(U_{+})e^{\sigma_{20}+\sigma_{30}}}{\kappa_{1}(U_{-})(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-}))}\right|>0.$
However, for the reflection coefficient $|K_{24}|$, we have
$\displaystyle|\partial_{\beta_{1}}\delta_{4}|$ $\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\textbf{r}_{1}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{1}(U_{-})\kappa_{1}(U_{+})\rho_{-}u_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}-\rho_{-}u_{-}\lambda_{1}(U_{-})+e^{\sigma_{20}}\rho_{+}u_{+}\lambda_{1}(U_{+})\big{)}}{\kappa_{4}(U_{+})\kappa_{1}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-})\big{)}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{-\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}\right|<1,$
where $|K_{24}|$ is not necessarily bounded away from zero, but is less than
one.
## 3\. The Wave-Front Tracking Algorithm and Global Existence of Weak
Solutions
We start off here with a brief description of the wave-front tracking method
to be employed throughout in Sections 4–7 and then establish the existence of
entropy solutions when the perturbation of the incoming flow has small total
variation at $x=0$.
The main scheme in the method of wave-front tracking is to construct
approximate solutions within a class of piecewise constant functions. At the
beginning, we approximate the initial data by a piecewise constant function.
Then we solve the resulting Riemann problems exactly with the exception of the
rarefaction waves, which are replaced by rarefaction fans with many small
wave-fronts of equal strengths. The outgoing fronts are continued up to the
first time when two waves collide and a new Riemann problem is solved. In this
process, one has to modify the algorithm and introduce a simplified Riemann
solver in order to keep the number of wave-fronts finite for all $x\geq 0$ in
the flow direction. We refer the reader to Bressan [4, 6] and Baiti-Jenssen
[2] for related references.
_3.1. The Riemann Solvers_. As seen in Section 2, the solution to the Riemann
problem $\\{U_{b},U_{a}\\}$ is a self-similar solution given by at most five
states separated by shocks, vortex sheets, entropy waves, or rarefaction
waves. To connect the state $U_{a}$ to $U_{b}$, there exist $C^{2}$–curves
$\eta\rightarrow\varphi(\eta)(U)$ with arc length parametrization such that
$U_{b}=\varphi(\eta)(U_{a}):=\Upsilon_{4}(\eta_{4})\circ\cdots\circ\Upsilon_{1}(\eta_{1})(U_{a})$
for some $\eta=(\eta_{1},\ldots,\eta_{4})$, and
$U_{j}=\Upsilon_{j}(\eta_{j})\circ\cdots\circ\Upsilon_{1}(\eta_{1})(U_{a})$,
$j=1\ldots 3$.
Next, we discuss the construction of front tracking approximations for our
initial-boundary value problem. Let $\vartheta$ denote the initial
approximation parameter. For given initial data $\overline{U}$ and with
$\vartheta>0$, consider $\overline{U}^{\vartheta}$ a sequence of piecewise
constant functions approximating $\overline{U}$ in the $L^{1}$–norm, and the
wall boundary is also approximated as described in Section 2. Set
$\mathcal{Z}_{\vartheta}$ to be the total number of jumps in the initial data
$\overline{U}^{\vartheta}$ and the tangential angle function of the wall
boundary. Let $\delta_{\vartheta}>0$ be a parameter so that a rarafaction wave
is replaced by a step function whose “steps” are no further apart than
$\delta_{\vartheta}$. The discontinuity between two steps is set to propagate
with a speed equal to the Rankine-Hugoniot speed of the jump connecting the
states corresponding to the two steps. At any time, the simplified Riemann
solver (defined below) is employed, the constant $\hat{\lambda}$ denotes the
speed of the generated non-physical wave, which is strictly greater than all
the wave speeds of system (2.1). Note that the strength of the non-physical
wave is the error generated when the simplified Riemann solver is applied.
Accurate Riemann solver. The accurate Riemann solver (ARS) is the exact
solution to the Riemann problem, with the condition that every rarefaction
wave $\\{w,R_{d}(w)(\alpha)\\}$, $d=1,4$, is divided into equal parts and
replaced by a piecewise constant rarefaction fan of several new wave-fronts of
equal strength.
Simplified Riemann solver. When only weak waves are involved, the simplified
Riemann solver (SRS) here is the same as the one described in [2, 6]. That is,
all new waves are put together in a single non-physical front, travelling
faster than all characteristic speeds. In the case of a weak wave interacting
with the strong vortex sheet/entropy wave, the purpose of (SRS) is to ignore
the strength of the weak wave, while preserving the strength of the strong
vortex sheet/entropy wave, and to place the error in the non-physical wave in
the following manner:
Case 1 : A weak wave $\\{U_{-},U_{1}\\}$ collides with the strong vortex
sheet/entropy wave $\\{U_{1},U_{+}\\}$ from below. The Riemann problem
$\\{U_{-},U_{+}\\}$ is solved as follows:
$\left\\{\begin{array}[]{ll}U_{-}&\quad\mbox{for
$\frac{y}{x}<\chi(U_{1},U_{+})$},\\\ U_{2}&\quad\mbox{for
$\chi(U_{1},U_{+})<\frac{y}{x}<\hat{\lambda}$},\\\ U_{+}&\quad\mbox{for
$\frac{y}{x}>\hat{\lambda}$},\end{array}\right.$
with $\chi(U_{1},U_{+})$ as the speed of the strong vortex sheet/entropy wave,
and the state $U_{2}$ is solved in a way that $\\{U_{-},U_{2}\\}$ is the
strong vortex sheet/entropy wave starting from $U_{-}$ and
$\chi(U_{1},U_{+})=\chi(U_{-},U_{2})$. Hence, we find that (SRS) keeps the
same strength of the strong vortex sheet/entropy wave, and the error appears
in the non-physical fronts.
Case 2: A weak wave $\\{U_{2},U_{+}\\}$ collides with the strong vortex
sheet/entropy wave $\\{U_{-},U_{2}\\}$ from above. The Riemann problem
$\\{U_{-},U_{+}\\}$ is solved as follows:
$\left\\{\begin{array}[]{ll}U_{-}&\quad\mbox{for
$\frac{y}{x}<\chi(U_{-},U_{2})$},\\\ U_{2}&\quad\mbox{for
$\chi(U_{-},U_{2})<\frac{y}{x}<\hat{\lambda}$},\\\ U_{+}&\quad\mbox{for
$\frac{y}{x}>\hat{\lambda}$},\end{array}\right.$
with $\chi(U_{-},U_{2})$ denoting the speed of the strong vortex sheet/entropy
wave.
_3.2. Construction of Wave Front Tracking Approximations_
Given $\vartheta$, the corresponding front tracking approximate solution
$U^{\vartheta}(x,y)$ is built up as follows. At $x=0$, all the Riemann
problems in $\overline{U}^{\vartheta}$ are solved by using the accurate
Riemann solver. Furthermore, one can change the speed of one of the incoming
fronts so that, at any time $x>0$, there is at most one collision involving
only two incoming fronts. Of course, this adjustment of speed can be chosen
arbitrarily small. Let $\omega_{\vartheta}$ be a fixed small parameter with
$\omega_{\vartheta}\rightarrow 0$, as $\vartheta\rightarrow 0$, which will be
determined later. For convenience, the index $j$ in $\alpha_{j}$ will be
dropped henceforward, and we will write $\alpha_{j}$ as $\alpha$ when there is
no ambiguity involved; the same applies for $\beta$; and we will moreover
employ the same notation $\alpha$ as a wave and its strength as before.
_Case 1: Two weak waves with strengths $\alpha$ and $\beta$ interact at some
$x>0$_. The Riemann problem produced by this collision is solved in the
following way:
* •
If $|\alpha\beta|>\omega_{\vartheta}$ and the two waves are physical, then the
accurate Riemann solver is employed.
* •
If $|\alpha\beta|<\omega_{\vartheta}$ and the two waves are physical, or there
is a non-physical wave, then the simplified Riemann solver is employed.
_Case 2: A weak wave $\alpha$ interacts with the strong vortex sheet/entropy
wave and one weak wave at some $x>0$_. The Riemann problem produced by this
collision is solved in the following way:
* •
If $|\alpha|>\omega_{\vartheta}$ and the weak wave is physical, then the
accurate Riemann solver is applied.
* •
If $|\alpha|<\omega_{\vartheta}$ and the weak wave is physical, or this wave
is non-physical, then the simplified Riemann solver is applied.
_Case 3: The flow perturbation due to the Lipschitz wall boundary_.
* •
When the change of the angle of the boundary is larger than
$\omega_{\vartheta}$ and the weak wave is physical, then the accurate Riemann
solver is employed to solve the lateral Riemann problem.
* •
If the change of the angle of the boundary is less than $\omega_{\vartheta}$,
then this perturbation is ignored.
_Case 4: The physical wave collides with the boundary_. The accurate Riemann
solver is employed to solve the lateral Riemann problem.
_Case 5: The non-physical wave collides with the boundary_. We can allow these
waves to cross the boundary.
_3.3. Glimm’s Functional and Wave Interaction Potential_
The goal in this subsection is to construct the suitable Glimm-type functional
and the associated wave interaction potential $\mathcal{Q}$ for our initial-
boundary value problem. This involves a careful incorporation of the
additional nonlinear waves generated from the wall boundary vertices.
Definition 3.1 (_Approaching waves_). (i) Two weak fronts $\alpha$ and
$\beta$, located at points $y_{\alpha}<y_{\beta}$ and of the characteristic
families $j_{\alpha}$, $j_{\beta}$ $\in$ $\\{1,\ldots,4\\}$, respectively, are
said to be approaching each other if the following two conditions are
concurrently satisfied:
* •
$y_{\alpha}$ and $y_{\beta}$ are both in one of the two intervals into which
$\mathbb{R}$ is partitioned by the location of the strong vortex sheet/entropy
wave. That is, both waves are either in $\Omega_{-}$ or $\Omega_{+}$;
* •
Either $j_{\alpha}>j_{\beta}$ or else $j_{\alpha}$ = $j_{\beta}$ and at least
one of them is a genuinely nonlinear shock.
In this case, we write $(\alpha,\beta)$ $\in$ $\mathcal{A}$.
(ii) We say that a weak wave $\alpha$ of the characteristic family
$j_{\alpha}$ is approaching the strong vortex sheet/entropy wave if either
$\alpha\in\Omega_{-}$ and $j_{\alpha}=4$, or $\alpha\in\Omega_{+}$ and
$j_{\alpha}=1$. We then write $\alpha\in\mathcal{A}_{v/e}$.
(iii) We say that a weak wave $\alpha$ of the characteristic family
$j_{\alpha}$ is approaching the boundary if $\alpha\in\Omega_{-}$ and
$j_{\alpha}=1$. We then write $\alpha\in\mathcal{A}_{b}$.
Define the total (weighted) strength of weak waves in $U^{\vartheta}(x,\cdot)$
as
$\mathcal{V}(x)=\sum_{\alpha}|b_{\alpha}|.$
Here, for a weak wave $\alpha$ of the $j$-family, its weighted strength is
defined as
$b_{\alpha}=\begin{cases}k_{+}\alpha&\quad\text{if $\alpha\in\Omega_{+}$ and
$j_{\alpha}=1$},\\\ \alpha&\quad\text{if $\alpha\in\Omega_{-}$},\end{cases}$
(3.1)
where $k_{+}=\frac{{2}K_{21}}{K^{*}}$ and the coefficient $K_{21}$ as given in
Lemma 2.6.
Next, the wave interaction potential $\mathcal{Q}(x)$ is defined as
$\displaystyle\mathcal{Q}(x)$ $\displaystyle=$ $\displaystyle
C^{\ast}\sum_{(\alpha,\beta)\in\mathcal{A}}|b_{\alpha}b_{\beta}|+K^{\ast}\sum_{\alpha\in\mathcal{A}_{v/e}}|b_{\alpha}|+\sum_{\beta\in\mathcal{A}_{b}}|b_{\beta}|+\widetilde{K_{b0}}\sum_{a_{l}>x}|\omega_{l}|$
(3.2) $\displaystyle=$
$\displaystyle\mathcal{Q}_{\mathcal{A}}+\mathcal{Q}_{v/e}+\mathcal{Q}_{b}+\mathcal{Q}_{\Theta}.$
Here the constants $K^{\ast}\in$ ($K_{11}$, 1) and
$\widetilde{K_{b0}}>K_{b0}$, while $C^{\ast}$ is a constant to be specified
later. To control the total variation of the new waves produced by the
boundary vertices, $\mathcal{Q}_{\Theta}$ in our wave interaction potential
$\mathcal{Q}(x)$ is an added term, compared to that for the Cauchy problem.
_The Glimm-type functional $\mathcal{G}$ is defined as follows_
$\mathcal{G}(x)=\mathcal{V}(x)+\kappa\mathcal{Q}(x)+|U^{\diamond}(x)-U^{+}_{0}|+|U_{\diamond}(x)-U^{-}_{0}|,$
(3.3)
_where the states $U_{\diamond}(x)$ and $U^{\diamond}(x)$ are the below state
and the above state of the strong vortex sheet/entropy wave respectively at
“time” x, $U^{-}_{0}$ and $U^{+}_{0}$ are the below and above state of the
strong vortex sheet/entropy wave respectively at $x=0$, and $\kappa$ is a
large positive constant to be determined later._
We remark that $\mathcal{V}$, $\mathcal{Q}$, and $\mathcal{G}$ remain
unchanged between any pair of subsequent interaction times. However, we will
demonstrate that, across an interaction “time” $x$, both $\mathcal{Q}$ and
$\mathcal{G}$ decrease.
Proposition 3.1. _Assume that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(g^{\prime}\left(\cdot\right))$ is sufficiently small. Then
V$\left(x\right)$ will remain sufficiently small for all $x>0$. Moreover, the
quantity ${\rm TV}(U^{\vartheta}\left(x,\cdot\right))$ has a uniform bound for
any $\vartheta>0$_.
Proof. With the Glimm-type functional $\mathcal{G}$, consider
$\Delta\mathcal{G}(x)=\mathcal{G}(x^{+})-\mathcal{G}(x^{-}),$
where $x^{-}$ and $x^{+}$ denote the “times” before and after the interaction
“time” $x>0$, respectively.
_Case 1: Two weak waves $\alpha$ and $\beta$ collide_. Then the states
$U^{\diamond}\left(x\right)$ and $U_{\diamond}\left(x\right)$ do not alter
across this interaction “time” $x>0$. Hence, we have
$\displaystyle\Delta\mathcal{G}(x)$ $\displaystyle=$
$\displaystyle\mathcal{V}(x^{+})-\mathcal{V}(x^{-})+\kappa\left(\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})\right)$
$\displaystyle\leq$
$\displaystyle\mathcal{B}_{1}|b_{\alpha}b_{\beta}|+\kappa\left(-C^{\ast}|b_{\alpha}b_{\beta}|+C^{\ast}|b_{\alpha}b_{\beta}|V(x^{-})+\mathcal{B}_{0}|b_{\alpha}b_{\beta}|\right),$
where $\mathcal{B}_{0}$ and $\mathcal{B}_{1}$ are constants independent of
$\vartheta$.
_Case 2: A weak wave $\alpha$ of the 1-family interacts with the boundary_.
$\Delta\mathcal{G}(x)=K_{b1}\alpha-\alpha+\kappa\left(C^{*}K_{b1}V(x^{-})\alpha+K^{\ast}K_{b1}\alpha-\alpha\right),$
where $K^{\ast}K_{b1}<1$.
_Case 3: A new 4-wave $\alpha$ produced by the Lipschitz wall boundary_.
$\Delta\mathcal{G}(x)=K_{b0}\theta_{l}+\kappa\left(C^{\ast}K_{b0}\theta_{l}V(x^{-})+K^{*}K_{b0}\theta_{l}-\widetilde{K_{b0}}\theta_{l}\right),$
where $K_{b0}<\widetilde{K_{b0}}$ is large.
In the following two cases, the states $U_{\diamond}(x)$ and $U^{\diamond}(x)$
change across this interaction “time” $x>0$.
_Case 4: A weak wave $\alpha$ of the 4-family collides with the strong vortex
sheet/entropy wave from below_.
$\displaystyle\Delta\mathcal{G}(x)$ $\displaystyle=$
$\displaystyle\mathcal{V}(x^{+})-\mathcal{V}(x^{-})+|U^{\diamond}(x^{+})-U^{\diamond}(x^{-})|$
$\displaystyle+|U_{\diamond}(x^{+})-U_{\diamond}(x^{-})|+\kappa\left(\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})\right)$
$\displaystyle=$
$\displaystyle\sum^{4}_{j=1}K_{1j}\alpha-\alpha+\kappa\Big{(}C^{\ast}\big{(}K_{11}V(x^{-})\alpha+K_{14}V(x^{-})\alpha\big{)}-K^{\ast}\alpha+K_{11}\alpha\Big{)}.$
_Case 5: A weak wave $\alpha$ of the 1-family collides with the strong vortex
sheet/entropy wave from above_.
$\Delta\mathcal{G}(x)=\sum^{4}_{j=1}K_{2j}\alpha-
b_{\alpha}+\kappa\Big{(}C^{\ast}\big{(}K_{21}V(x^{-})\alpha+K_{24}V(x^{-})\alpha\big{)}-K^{\ast}b_{\alpha}+K_{21}\alpha\Big{)}.$
In the cases above, $K_{11}<K^{\ast}<1$, $b_{\alpha}>{2}K_{21}|\alpha|$ in
connection with the weight $k_{+}$, and the constant
$C^{\ast}>\mathcal{B}_{0}>0$ is large.
Next, we establish that the total (weighted) strength of waves in
$U^{\vartheta}(x,\cdot)$ remain sufficiently small for all $x>0$ if it is
sufficiently small at $x=0$. More precisely,
$\mathcal{V}(x)\ll 1\qquad\text{ for all }x>0.$
This can be proved as follows:
_(i) “Time” $x_{1}>0$ is the first interaction_. Given that
$\mathcal{V}\left(x^{-}_{1}\right)=\mathcal{V}(0)\leq{\rm
TV}(\widetilde{U_{0}}(\cdot))\ll 1$ and
$\sum_{\l=0}^{\infty}\theta_{l}\leq{\rm TV}(g^{\prime}(\cdot))\ll 1$ in Cases
1–5 above, we conclude that, for $\kappa$ sufficiently large and
$\omega_{\vartheta}$ small enough,
$\Delta\mathcal{G}(x_{1})\leq
0,\quad\text{i.e.,}\quad\mathcal{G}(x_{1}^{+})\leq\mathcal{G}(x_{1}^{-})=\mathcal{G}(0).$
Therefore,
$\displaystyle\mathcal{V}(x_{1}^{+})$ $\displaystyle\leq$
$\displaystyle\mathcal{G}(x_{1}^{+})\leq\mathcal{G}(0)\leq\mathcal{V}(0)+\kappa\mathcal{Q}(0)$
$\displaystyle=$
$\displaystyle\mathcal{V}(0)+\kappa\Big{(}C^{\ast}\mathcal{V}^{2}(0)+\mathcal{V}(0)+\widetilde{K_{b0}}\sum_{l=0}^{\infty}\theta_{l}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\mathcal{V}(0)+\sum_{l=0}^{\infty}\theta_{l}\Big{)}\ll 1.$
_(ii) $\mathcal{V}(x_{m}^{-})\ll 1$ and
$\mathcal{G}(x_{m}^{+})\leq\mathcal{G}(x_{m}^{-})$ for any $m<n$_. Then, for
the next interaction “time” $x_{n}$, similar to Case 1, we also conclude
$\Delta\mathcal{G}(x_{n})\leq
0,\quad\text{i.e.,}\quad\mathcal{G}(x_{n}^{+})\leq\mathcal{G}(x_{n}^{-})=\mathcal{G}(x_{n-1}^{+}).$
Therefore, all together, we obtain
$\displaystyle\mathcal{V}(x_{n}^{+})+|U^{\diamond}(x_{n}^{+})-U_{0}^{+}|+|U_{\diamond}(x_{n}^{+})-U_{0}^{-}|\qquad\qquad$
$\displaystyle\qquad\leq\mathcal{G}(x_{n}^{+})\leq\mathcal{G}(x_{n}^{-})=\mathcal{G}(x_{n-1}^{+})\leq\ldots\leq\mathcal{G}(0)$
$\displaystyle\qquad=\mathcal{V}(0)+\kappa\mathcal{Q}(0)$
$\displaystyle\qquad=\mathcal{V}(0)+\kappa\Big{(}C^{\ast}\mathcal{V}^{2}(0)+\mathcal{V}(0)+\widetilde{K_{b0}}\sum_{l=0}^{\infty}\theta_{l}\Big{)}$
$\displaystyle\qquad\leq
C\Big{(}\mathcal{V}(0)+\sum_{l=0}^{\infty}\theta_{l}\Big{)}\ll 1.$
This implies that $\mathcal{V}(x)\ll 1$ for all $x>0$, since $C$ is
independent of $x$.
Furthermore, the total variation of $U^{\vartheta}(x,\cdot)$ is uniformly
bounded. More precisely, we conclude that
${\rm TV}\\{U^{\vartheta}(x,\cdot)\\}\approx
V(x)+|U^{\diamond}(x)-U_{0}^{+}|+|U_{\diamond}(x)-U_{0}^{-}|+|\sigma_{20}|+|\sigma_{30}|=\mathcal{O}(1).$
(3.4)
This completes the proof.
In order to have a front tracking approximate solution
$U^{\vartheta}(x,\cdot)$ defined for any time $x>0$, along with a uniform
bound on the total variation, we also need to have that the number of wave-
fronts in $U^{\vartheta}(x,\cdot)$ is finite. This is given by the subsequent
lemma.
Lemma 3.2. _For any fixed $\vartheta>0$ small enough, the number of wave-
fronts in $U^{\vartheta}\left(x,y\right)$ is finite and the approximate
solutions $U^{\vartheta}\left(x,y\right)$ are defined for all $x>0$. Moreover,
for any $x>0$, the total strength of the all non-physical waves is of order
$\mathcal{O}(1)\left(\delta_{\vartheta}+\omega_{\vartheta}\right)$_.
Proof. We first note the total interaction potential $\mathcal{Q}(x)$ remains
unchanged when there is no interaction and decreases across an interaction
“time” $x>0$ as discussed in Cases 1–5 in Proposition 3.1. Furthermore, from
Cases 1–5 and the subsequent analysis above, we have concluded that
$\mathcal{V}(x)\ll 1$. Hence, one can fix some number $\nu\in(0,1)$ such that
$\displaystyle\Delta\mathcal{Q}(x)$ $\displaystyle=$
$\displaystyle\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})$ (3.8) $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}-\nu|b_{\alpha}b_{\beta}|&\mbox{ if
both waves $\alpha$ and $\beta$ are weak,}\\\ -\nu|b_{\alpha}|&\mbox{ if the
weak wave $\alpha$ hits the strong vortex sheet/entropy wave,}\\\
-\nu|\theta_{l}|&\mbox{ if the angle of the boundary
changes.}\end{array}\right.$
Now, following an argument similar to the one given in [2], we reach the
following conclusions. Note that initially $\mathcal{Q}(0)$ is bounded and $Q$
decreases thereafter for each case. Moreover, in the case where the
interaction potential between the incoming waves or the change of the angle of
the boundary is larger than $\omega_{\vartheta}$, $Q$ decreases by at least
$\nu\omega_{\vartheta}$ in these interactions, as implied by the bounds given
in (3.8). Following the wave-front tracking method in our problem, new
physical waves can be only produced by such interactions. Furthermore, when
the weak wave $\alpha$ of 1-family collides with the wall boundary, we solve
the lateral Riemann problem and have shown earlier that, after this
interaction, there is only a reflected wave of 4-family with the reflection
coefficient $1$. Hence, before and after this interaction, the number of the
waves keeps the same, and this implies that the number of the waves is finite.
Finally, because non-physical waves are generated only when physical waves
collide, we can also conclude that the number of non-physical wave fronts are
finite; and, provided that two waves can only collide once, the number of
interactions is also finite. Consequently, it follows that the approximate
solutions $U^{\vartheta}(x,\cdot)$ are defined for all times $x>0$. The
similar argument allows us to conclude that the total strength of all non-
physical wave fronts at any $x$ is of order
$\mathcal{O}(1)(\delta_{\vartheta}+\omega_{\vartheta})$. This completes the
proof Lemma 3.2.
Following the line of arguments given in [2, 4] for the wave-front tracking
algorithm and Lemma 3.1 above, we finish this section with the following
theorem for the global existence of entropy solutions to the initial-boundary
value problem (1.1) and (1.7)–(1.8).
Theorem 3.1. _Suppose that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(g^{\prime}(\cdot))$ is small enough. Then, for the initial-boundary value
problem (1.1) and (1.7)–(1.8), there exists a global weak solution in BV
satisfying the steady Clausius entropy inequality (1.9)._
## 4\. The Lyapunov Functional for the ${\bf L}^{1}$–Distance between Two
Solutions
To show that the front tracking approximations, constructed for the existence
analysis in Section 3, converge to a unique limit, we estimate the distance
between any two $\vartheta$-approximate $U$ and $V$ of problem (1.1) and
(1.7)–(1.8) of initial-boundary value type. To this end, we develop the
Lyapunov functional $\Phi(U,V)$, equivalent to the $L^{1}$–distance:
${C}^{-1}\,\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\mathit{\Phi}(U,V)\leq{C}\,\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$
and prove that the functional $\Phi(U,V)$ is almost decreasing along pairs of
solutions,
$\mathit{\Phi}\left(U(x_{2},\cdot),V(x_{2},\cdot)\right)-\mathit{\Phi}\left(U(x_{1},\cdot),V(x_{1},\cdot)\right)\leq
C\vartheta(x_{2}-x_{1}),\hskip 14.22636pt\text{for all }x_{2}>x_{1}>0,$
for some constant ${C}>0$. Here $U$ and $V$ are two approximate solutions
constructed via the wave-front tracking method, and the small approximation
parameter $\vartheta$ is responsible for controlling the subsequent errors:
* •
Errors in the approximation of the initial data and the boundary.
* •
Errors in the speeds of shock, vortex sheet, entropy wave, and rarefaction
fronts.
* •
The total strength of all non-physical fronts.
* •
The maximum strength of rarefaction fronts.
Along the line of arguments presented in [9, 22, 24], with “time” $x$ fixed,
at each $y$, one connects the state $U(y)$ with $V(y)$ in the state space by
going along the Hugoniot curves $S_{1},C_{2},C_{3}$, and $S_{4}$. Depending on
the location of the strong vortex sheet/entropy wave in $U(y)$ and $V(y)$, the
distance between $U(y)$ and $V(y)$ is estimated along discontinuity waves in
possibly different “directions”, determining the strength of the $j$-Hugoniot
wave $h_{j}(y)$ in the following way:
* •
Suppose that $U(y)$ and $V(y)$ are both in $\Omega_{-}$ and $\Omega_{+}$. Then
one begins at the state $U(y)$ and moves along the Hugoniot curves to reach
the state $V(y)$.
* •
Suppose that $U(y)$ is in $\Omega_{-}$ and $V(y)$ is in $\Omega_{+}$. Then one
begins at the state $U(y)$ and moves along the Hugoniot curves to reach the
state $V(y)$.
* •
Suppose that $V(y)$ is in $\Omega_{-}$ and $U(y)$ is in $\Omega_{+}$. Then one
begins at the state $V(y)$ and moves along the Hugoniot curves to reach the
state $U(y)$.
Define the $L^{1}$–weighted strengths of the waves in the solution of the
Riemann problem $\left(U(y),V(y)\right)$ or $\left(V(y),U(y)\right)$ as
follows:
$\displaystyle q_{j}(y)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}w^{b}_{j}\cdot h_{j}(y)&\mbox{
whenever $U(y)$ and $V(y)$ are both in $\Omega_{-}$,}\\\ w^{m}_{j}\cdot
h_{j}(y)&\mbox{ whenever $U(y)$ and $V(y)$ are both in different domains,}\\\
w^{a}_{j}\cdot h_{j}(y)&\mbox{ whenever $U(y)$ and $V(y)$ are both in
$\Omega_{+}$,}\end{array}\right.$ (4.4)
with the constants $w^{b}_{j}$, $w^{m}_{j}$, and $w^{a}_{j}$ above to be
specified later on, based on the estimates of wave interactions and
reflections in Lemmas 2.2–2.7.
We define the following Lyapunov functional,
$\mathit{\Phi}(U,V)=\sum_{j=1}^{4}\int\limits_{g(x)}^{\infty}\
|q_{j}(y)|W_{j}(y)\,dy,$ (4.5)
where the weights are given by
$W_{j}(y)=1+\kappa_{1}A_{j}(y)+\kappa_{2}\left(\mathcal{Q}(U)+\mathcal{Q}(V)\right).$
(4.6)
The constants $\kappa_{1}$ and $\kappa_{2}$ are to be determined later. Here
$\mathcal{Q}$ denotes the total wave interaction potential incorporating the
boundary effect as defined in (3.2), and $A_{j}(y)$ denotes the total strength
of waves in $U$ and $V$, which approach the $j$-wave $q_{j}(y)$, defined in
the following manner (for $y$ where there is no jump in $U$ or $V$):
$A_{j}(y)=F_{j}(y)+G_{j}(y)+\left\\{\begin{array}[]{ll}H_{j}(y)&\mbox{ if
$j$-wave $q_{j}(y)$ is small and the $j$-field is genuinely nonlinear,}\\\
0&\mbox{ if $j$ = 2, 3 and $q_{j}(y)$ = $B$ is large.}\end{array}\right.$
(4.7)
Next, we define the following global weights $G_{j}$:
$G_{j}(y)=$ | $U,V$ are both in $\Omega_{-}$ | $U,V$ are in distinct regions | $U,V$ are both in $\Omega_{+}$
---|---|---|---
$G_{1}(y)$ | 4B | 2B | 4B
$G_{2,3}(y)$ | 0 | 0 | 0
$G_{4}(y)$ | 4B | 2B | 2B
Under the assumption that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(\widetilde{V_{0}}(\cdot))+{\rm TV}(g^{\prime}(\cdot))$ is small enough
with $U(x,\cdot)$, $V(x,\cdot)$ $\in{\rm BV}\cap L^{1}$, one concludes
$\displaystyle\mathcal{M}^{-1}\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\sum_{j=1}^{4}\int\limits^{\infty}\limits_{g(x)}|q_{j}(y)|\,dy\leq\mathcal{M}\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$
$\displaystyle 1\leq W_{j}(y)\leq\mathcal{M},\hskip 5.69054ptj=1,\ldots,4,$
where the constant $\mathcal{M}$ is independent of $\vartheta$ and “time” $x$.
Here _we define the strength of any large wave of the $2$\- or
$3$-characteristic family to equal to some fixed number B (bigger than all
strengths of small waves),_ and the concepts “small” and “large” mean the
waves that connect the states in the same or in the distinct domains
$\Omega^{-}$ and $\Omega^{+}$, respectively.
The summands in (4.7) are defined as follows,
$\displaystyle F_{j}(y)$ $\displaystyle=$
$\displaystyle\Biggl{(}\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}\setminus\mathcal{SC}\\\
y_{\alpha}<y,j<k_{\alpha}\leq
4\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}\setminus\mathcal{SC}\\\
y_{\alpha}>y,1\leq k_{\alpha}<j\end{subarray}}\Biggr{)}|\alpha|,$
$\displaystyle H_{j}(y)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}(\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(U)\setminus\mathcal{SC},y_{\alpha}<y,k_{\alpha}=j\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(V)\setminus\mathcal{SC},y_{\alpha}>y,k_{\alpha}=j\end{subarray}})|\alpha|&\quad\mbox{
if $q_{j}(y)<0$,}\\\
(\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(V)\setminus\mathcal{SC},y_{\alpha}<y,k_{\alpha}=j\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(U)\setminus\mathcal{SC},y_{\alpha}>y,k_{\alpha}=j\end{subarray}})|\alpha|&\quad\mbox{
if $q_{j}(y)>0$,}\end{array}\right.$ (4.10)
where, at each $x$, $\alpha$ stands for the (non-weighted) strength of the
wave $\alpha\in\mathcal{J}$, located at the point $y_{\alpha}$ and belonging
to the characteristic family $k_{\alpha}$;
$\mathcal{J}=\mathcal{J}(U)\cup\mathcal{J}(V)$,
$\mathcal{SC}=\mathcal{SC}(U)\cup\mathcal{SC}(V)$ is the set of all waves (in
$U$ and $V$) and the set of all large (strong) characteristic discontinuities
(in $U$ and $V$) respectively.
Consequently, there holds
${C}^{-1}\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\mathit{\Phi}(U,V)\leq{C}\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$
(4.11)
for any $x\geq 0$ with the constant ${C}>0$ depending only on the quantities
independent of $x$: the strength of the strong vortex sheet/entropy wave and
${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm TV}(\widetilde{V_{0}}(\cdot))+{\rm
TV}(g^{\prime}(\cdot))$.
We now analyze the evolution of the Lyapunov functional $\mathit{\Phi}$ in the
flow direction $x>0$. For $j=1,\ldots,4$, we call $\lambda_{j}(y)$ the speed
of the $j$-wave $q_{j}(y)$ (along the Hugoniot curve in the phase space).
Then, at a “time” $x>0$ which is not the interaction time of the waves either
in $U(x)=U(x,\cdot)$ or $V(x)=V(x,\cdot)$, an explicit computation gives
$\displaystyle{d\over dx}\mathit{\Phi}\left(U(x),V(x)\right)$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}\left(\lvert
q_{j}(y_{\alpha}^{-})\rvert W_{j}(y_{\alpha}^{-})-\lvert
q_{j}(y_{\alpha}^{+})\rvert
W_{j}(y_{\alpha}^{+})\right)\dot{y}_{\alpha}+\sum_{j=1}^{4}\lvert
q_{\j}(b)\rvert W_{j}(b)\dot{y}_{b}$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}\left(\lvert
q_{j}(y_{\alpha}^{-})\rvert
W_{j}(y_{\alpha}^{-})\left(\dot{y}_{\alpha}-\lambda_{j}(y_{\alpha}^{-})\right)-\lvert
q_{j}(y_{\alpha}^{+})\rvert
W_{j}(y_{\alpha}^{+})\left(\dot{y}_{\alpha}-\lambda_{j}(y_{\alpha}^{+})\right)\right)$
$\displaystyle\quad+\sum_{j=1}^{4}\lvert q_{j}(b)\rvert
W_{j}(b)\left(\dot{y}_{b}+\lambda_{j}(b)\right),$ (4.12)
where $\dot{y}_{\alpha}$ denotes the speed of the Hugoniot wave
$\alpha\in\mathcal{J}$, $b=g(x)^{+}$ stands for the points close to the
boundary, and $\dot{y}_{b}$ is the slope of the boundary.
We present the notation
$\displaystyle E_{\alpha,j}$ $\displaystyle=\lvert q_{j}^{+}\rvert
W_{j}^{+}\left(\lambda_{j}^{+}-\dot{y}_{\alpha}\right)-\lvert q_{j}^{-}\rvert
W_{j}^{-}\left(\lambda_{j}^{-}-\dot{y}_{\alpha}\right),$ (4.13) $\displaystyle
E_{b,j}$ $\displaystyle=\lvert q_{j}(b)\rvert
W_{j}(b)\left(\dot{y}_{b}+\lambda_{j}(b)\right),$ (4.14)
where $q_{j}^{\pm}$ = $q_{j}(y^{\pm}_{\alpha})$,
$W^{\pm}_{j}=W_{j}(y^{\pm}_{\alpha})$, and $\lambda^{\pm}_{j}$ =
$\lambda_{j}(y^{\pm}_{\alpha})$.
Then (4.12) can be written as
${d\over
dx}\mathit{\Phi}\left(U(x),V(x)\right)=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}E_{\alpha,j}+\sum_{j=1}^{4}E_{b,j}.$
(4.15)
Our central aim is to prove the bounds:
$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq\mathcal{O}(1)\vartheta\lvert\alpha\rvert\quad\text{
when $\alpha$ is a weak wave in $\mathcal{J}$},$ (4.16)
$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq\mathcal{O}(1)\lvert\alpha\rvert\quad\text{
when $\alpha$ is a non-physical wave in $\mathcal{J}$,}$ (4.17)
$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq 0\quad\text{ when $\alpha$ is a
strong vortex sheet/entropy wave in $\mathcal{J}$,}$ (4.18)
$\displaystyle\sum_{j=1}^{4}E_{b,j}\leq 0\quad\text{ near the boundary},$
(4.19)
where the quantities denoted by the Landau symbol $\mathcal{O}$(1) are
independent of the constants $\kappa_{1}$ and $\kappa_{2}$.
From (4.16)–(4.19) together with the uniform bound on the total strengths of
waves (3.4), we obtain
${d\over dx}\mathit{\Phi}\left(U(x),V(x)\right)\leq\mathcal{O}(1)\vartheta.$
(4.20)
Integration of (4.20) over the interval $\left[0,x\right]$ yields
$\mathit{\Phi}\left(U(x),V(x)\right)\leq\mathit{\Phi}\left(U(0),V(0)\right)+\mathcal{O}(1)\vartheta
x.$ (4.21)
We remark that, at each interaction “time” $x$ when two fronts of $U$ or two
fronts of $V$ interact, by the Glimm interaction estimates, all the weight
functions $W_{j}(y)$ decrease, if the constant $\kappa_{2}$ in the Lyapunov
functional is taken to be sufficiently large. Furthermore, due to the self-
similar property of the Riemann solutions, $\mathit{\Phi}$ decreases at this
“time”.
In the next section, we establish the bounds (4.16)–(4.19), particularly
(4.18) and (4.19), when $\alpha$ is a strong vortex sheet/entropy wave in
$\mathcal{J}$ and near the Lipschitz wall boundary, respectively.
## 5\. The $L^{1}$–Stability Estimates
For the case of the non-physical waves in $\mathcal{J}$, as well as the case
that the weak wave
$\alpha\in\mathcal{J}\mathrel{\mathop{:}}=\mathcal{J}(U)\cup\mathcal{J}(V)$,
which appears when $U$ and $V$ are both in $\Omega_{-}$ or $\Omega_{+}$,
estimates (4.16) and (4.17) are shown similarly based on the arguments in
Bressan-Liu-Yang [9], provided that $\frac{2|B|}{|\sigma_{20}|+|\sigma_{30}|}$
is sufficiently small and $\kappa_{1}$ is sufficiently large. In what follows,
we focus only on the other two cases, namely (4.18) and (4.19).
Case 1: The first strong vortex sheet/entropy wave $\alpha$ in $U$ or $V$ is
crossed. Then, by Lemma 2.4, we have the estimates:
$\displaystyle h^{+}_{1}$ $\displaystyle=$ $\displaystyle
h^{-}_{1}+K_{11}h^{-}_{4},$ (5.1) $\displaystyle h^{+}_{4}$ $\displaystyle=$
$\displaystyle K_{14}h^{-}_{4}.$ (5.2)
Moreover, the essential estimate $|K_{11}|<1$ given in Lemma 2.4 ensures the
existence of desired weights $w^{b}_{1}$ and $w^{b}_{4}$ in the following way.
Lemma 5.1. _There exist $w^{b}_{1}$, $w^{b}_{4}$, and $\gamma_{b}$ satisfying_
$\displaystyle\frac{w^{b}_{4}}{w^{b}_{1}}<1,$ (5.3)
$\displaystyle\frac{w^{b}_{1}}{w^{b}_{4}}K_{11}\left|\frac{\lambda_{1}^{-}-\lambda_{2,3}}{\lambda_{4}^{-}-\lambda_{2,3}}\right|<\gamma_{b}<1.$
(5.4)
With Lemma 5.1, we estimate $E_{j}$ for $j=1,\ldots,4$, starting with $E_{1}$:
By (5.1) and (5.4),
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q_{1}^{-}|(\lambda_{1}^{-}-\dot{y}_{\alpha})(W_{1}^{+}-W_{1}^{-})+W^{+}_{1}\left(|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle=$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+W^{+}_{1}\left(|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+W^{+}_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+w^{b}_{1}K_{11}|h^{-}_{4}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+2B\kappa_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+\gamma_{b}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|,$
where
$W^{+}_{1}=W_{1}(y_{\alpha}^{+})=2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{1}}=F_{1}(y_{\alpha}^{+})+H_{1}(y_{\alpha}^{+})$ here is the
total strength of all the weak waves in $U$ and $V$ which approach the
$1$-wave $q_{1}^{+}=q_{1}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is
from the weight $G_{1}(y_{\alpha}^{+})$.
For $j=2,3$, since $W^{+}_{j}=W^{-}_{j}$, (4.12) reduces to
$\displaystyle
E_{j}=W^{-}_{j}\big{(}|q_{j}^{+}|(\lambda_{j}^{+}-\dot{y}_{\alpha})-|q_{j}^{-}|(\lambda_{j}^{-}-\dot{y}_{\alpha})\big{)}\leq\mathcal{O}(1)B\Big{(}\vartheta+\sum_{i\neq\\{{2,3\\}}}|q_{i}^{-}|+|q_{k}^{-}|\Big{)},$
where $k\neq\\{j,1,4\\}$.
For $j$ = 4,
$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q_{4}^{-}|(\lambda_{4}^{-}-\dot{y}_{\alpha})(W_{4}^{+}-W_{4}^{-})+W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})+\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha}),$
where
$W^{+}_{4}=W_{4}(y_{\alpha}^{+})=2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{4}}=F_{4}(y_{\alpha}^{+})+H_{4}(y_{\alpha}^{+})$ is the total
strength of all the weak waves in $U$ and $V$ which approach the $4$-wave
$q_{4}^{+}=q_{4}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is from the
weight $G_{4}(y_{\alpha}^{+})$.
For the weighted $L^{1}$–strength $q_{j}(y)$ in (4.4), we choose $w^{b}_{1}$
small enough relatively to $w^{m}_{1}$ and $w^{b}_{4}$ large enough relatively
to $w^{m}_{4}$, choose $\kappa_{1}$ large enough and the total variation of
$U$ and $V$ so small, and use (5.1)–(5.2) to obtain
$\displaystyle\sum^{4}_{j=1}E_{j}$ $\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+\gamma_{b}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle+{}(\kappa_{1}A_{W^{+}_{1}}+M)|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}w^{m}_{4}|K_{14}p^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{-}|+\left(|q_{2}^{-}|+|q_{3}^{-}|\right)\Big{)}$
$\displaystyle=$
$\displaystyle-(1-\gamma_{b})2B\kappa_{1}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)\big{(}w^{m}_{4}|K_{14}h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\big{)}$
$\displaystyle+2B\kappa_{1}w^{b}_{1}(|h^{+}_{1}|+|h^{-}_{1}|)|\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+2B\kappa_{1}w^{m}_{4}|K_{14}h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-2B\kappa_{1}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+{}\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{-}|+\left(|q_{2}^{-}|+|q_{3}^{-}|\right)\Big{)}$
$\displaystyle\leq$ $\displaystyle 0.$
Case 2: The weak wave $\alpha$ between the two strong vortex sheets/entropy
waves in $U$ and $V$ is crossed. For $j=1$, we have
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q^{\pm}_{1}|(W^{+}_{1}-W^{-}_{1})(\lambda^{\pm}_{1}-\dot{y}_{\alpha})+W^{\mp}_{1}\left(|q^{+}_{1}|(\lambda^{+}_{1}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{1}||\alpha||\lambda^{\pm}_{1}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{1}|-|q^{-}_{1}|)(\lambda^{+}_{1}-\dot{y}_{\alpha})+|q^{-}_{1}|(\lambda^{+}_{1}-\lambda^{-}_{1})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{1}||\alpha||\lambda^{\pm}_{1}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{1}|-|q^{-}_{1}|)(\lambda^{+}_{1}-\dot{y}_{\alpha})+\mathcal{O}(1)|q^{-}_{1}||\alpha|\right).$
For the cases when $j=2,3$, we have
$\displaystyle E_{j}$ $\displaystyle=$ $\displaystyle
B\left(\big{(}W^{+}_{j}-W^{-}_{j}\big{)}\big{(}\lambda^{\pm}_{j}-\dot{y}_{\alpha}\big{)}+W^{\mp}_{j}\big{(}\lambda^{\pm}_{j}-\lambda^{\mp}_{j}\big{)}\right)$
$\displaystyle\leq$ $\displaystyle
B\left(-\kappa_{1}|\alpha||\lambda^{+}_{j}-\dot{y}_{\alpha}|+\mathcal{O}(1)|\alpha|\right).$
For $j$ = 4, we have
$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q^{\pm}_{4}|(W^{+}_{4}-W^{-}_{4})(\lambda^{\pm}_{4}-\dot{y}_{\alpha})+W^{\mp}_{4}\left(|q^{+}_{4}|(\lambda^{+}_{4}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{4}||\alpha||\lambda^{\pm}_{4}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{4}|-|q^{-}_{4}|)(\lambda^{+}_{4}-\dot{y}_{\alpha})+|q^{-}_{4}|(\lambda^{+}_{4}-\lambda^{-}_{4})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{4}||\alpha||\lambda^{\pm}_{4}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{4}|-|q^{-}_{4}|)(\lambda^{+}_{4}-\dot{y}_{\alpha})+\mathcal{O}(1)|q^{-}_{4}||\alpha|\right).$
Then we have
$\sum_{j=1}^{4}E_{j}\leq\kappa_{1}\mathcal{O}(1)\Big{(}-|\alpha|+|\alpha|\sum_{k\neq\\{2,3\\}}|q_{k}^{+}|+|q_{k}^{-}|+\sum_{k\neq\\{2,3\\}}|q_{k}^{+}|-|q_{k}^{-}|\Big{)}+\mathcal{O}(1)|\alpha|.$
Since
$\big{|}|q_{k}^{+}|-|q_{k}^{-}|\big{|}\leq|q_{k}^{+}-q_{k}^{-}|\leq\mathcal{O}(1)|\alpha|$
when $k\neq\\{2,3\\}$, we obtain
$\sum_{j=1}^{4}E_{j}\leq 0$
if all the weights $w^{m}_{j}$ are small enough and $\kappa_{1}$ is
sufficiently large.
Notice that the choice of the upper or lower superscripts depends on the
family number $k_{\alpha}$.
Case 3: The second strong vortex sheet/entropy wave $\alpha$ in $U$ or $V$ is
crossed. For this case, by Lemma 2.6, we have
$\displaystyle h^{-}_{1}$ $\displaystyle=$ $\displaystyle K_{21}h^{+}_{1},$
(5.5) $\displaystyle h^{-}_{4}$ $\displaystyle=$ $\displaystyle
h^{+}_{4}+K_{24}h^{+}_{1}.$ (5.6)
Moreover, the essential estimate $|K_{24}|<1$ in Lemma 2.6 ensures the
existence of desired weights $w^{a}_{1}$ and $w^{a}_{4}$ in the following
manner.
Lemma 5.2. _There exist $w^{a}_{1}$, $w^{a}_{4}$, and $\gamma_{a}$ satisfying_
$\frac{w^{a}_{4}}{w^{a}_{1}}\left|\frac{\lambda_{4}^{+}-\lambda_{2,3}}{\lambda_{1}^{+}-\lambda_{2,3}}\right|K_{24}<\gamma_{a}<1.$
(5.7)
With Lemma 5.2, we estimate $E_{j}$ for $j=1,\ldots,4$ as follows: By (5.5),
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q_{1}^{-}|(\lambda_{1}^{-}-\dot{y}_{\alpha})(W_{1}^{+}-W_{1}^{-})+W^{+}_{1}\left(|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+(4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})$
$\displaystyle{}+w^{m}_{1}|K_{21}h^{+}_{1}|(4B\kappa_{1}+A_{W^{+}_{1}}+M)|\lambda_{1}^{-}-\dot{y}_{\alpha}|,$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+w^{m}_{1}|K_{21}h^{+}_{1}|(4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|\lambda_{1}^{-}-\dot{y}_{\alpha}|$
$\displaystyle-
w_{1}^{a}|h^{+}_{1}|(2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|\lambda_{1}^{+}-\dot{y}_{\alpha}|-2B\kappa_{1}w_{1}^{a}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|,$
where
$W^{+}_{1}=W_{1}(y_{\alpha}^{+})=4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M$, and
$M=1+\kappa_{2}\big{(}\mathcal{Q}(U)+\mathcal{Q}(V)\big{)}$ is a positive
constant. The term $A_{W^{+}_{1}}=F_{1}(y_{\alpha}^{+})+H_{1}(y_{\alpha}^{+})$
here is the total strength of all the weak waves in $U$ and $V$ which approach
the $1$-wave $q_{1}^{+}=q_{1}(y_{\alpha}^{+})$, and the term $4B\kappa_{1}$ is
from the weight $G_{1}(y_{\alpha}^{+})$.
For $j=2,3$, since $W^{+}_{j}=W^{-}_{j}$, (4.12) reduces to
$\displaystyle E_{j}$ $\displaystyle=$ $\displaystyle
W^{+}_{j}\big{(}|q_{j}^{+}|(\lambda_{j}^{+}-\dot{y}_{\alpha})-|q_{j}^{-}|(\lambda_{j}^{-}-\dot{y}_{\alpha})\big{)}$
$\displaystyle\leq$
$\displaystyle\mathcal{O}(1)B\Big{(}\vartheta+\sum_{i\neq\\{{2,3\\}}}|q_{i}^{+}|+|q_{k}^{+}|\Big{)},$
where $k\neq\\{j,1,4\\}$.
By (5.6) and (5.7),
$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q_{4}^{-}|(\lambda_{4}^{-}-\dot{y}_{\alpha})(W_{4}^{+}-W_{4}^{-})+W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle=$ $\displaystyle
W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$ $\displaystyle
W^{+}_{4}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+w^{a}_{4}K_{24}|h^{+}_{1}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+\gamma_{a}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha}),$
where
$W^{+}_{4}=W_{4}(y_{\alpha}^{+})=2\kappa_{1}B+\kappa_{1}A_{W^{+}_{4}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{4}}=F_{4}(y_{\alpha}^{+})+H_{4}(y_{\alpha}^{+})$ here is the
total strength of all the weak waves in $U$ and $V$ which approach the
$4$-wave $q_{4}^{+}=q_{4}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is
from the weight $G_{4}(y_{\alpha}^{+})$.
For the weighted $L^{1}$–strength $q_{j}(y)$ in (4.1), when $w^{a}_{4}$ is
small enough relatively to $w^{m}_{4}$ and $w^{a}_{1}$ is large enough
relatively to $w^{m}_{1}$, $\kappa_{1}$ is large enough, applying (5.5) and
(5.6), the total variation of $u$ and $v$ is so small that
$\displaystyle\sum^{4}_{j=1}E_{j}$ $\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+\gamma_{a}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle+{}(\kappa_{1}A_{W^{+}_{4}}+M)|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+\big{(}4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M\big{)}w^{m}_{1}|K_{21}h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M\big{)}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{+}|+\left(|q_{2}^{+}|+|q_{3}^{+}|\right)\Big{)}$
$\displaystyle=$
$\displaystyle-(1-\gamma_{a})2B\kappa_{1}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)\big{(}w^{m}_{1}|K_{21}h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\big{)}$
$\displaystyle+2B\kappa_{1}w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-2B\kappa_{1}w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+2B\kappa_{1}(-w^{m}_{1}|K_{21}h^{+}_{1}|+2w^{m}_{1}|K_{21}h^{+}_{1}|)|\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)w^{a}_{4}|h^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+{}\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{+}|+\left(|q_{2}^{+}|+|q_{3}^{+}|\right)\Big{)}$
$\displaystyle\leq$ $\displaystyle 0,$
which yields (4.16).
Case 4: Close to the Lipschitz wall boundary. This case differs from the
Cauchy problem. Here we will use the particular property of the boundary
condition (1.8): The flows of $U$ and $V$ are tangent to the Lipschitz wall,
implying that they must be parallel with each other along the boundary. Then a
piecewise constant weak solution is constructed only along the Hugoniot curves
determined by the Riemann data $U(b)$ and $V(b)$, the states of solutions $U$
and $V$, respectively, close to the boundary.
Proposition 5.2. _Suppose that U $\left(b\right)$ =
$(\breve{u},\breve{v},\breve{p},\breve{\rho})$ and V$\left(b\right)$ =
$(\tilde{u},\tilde{v},\tilde{p},\tilde{\rho})$ are states in a small
neighborhood $O_{\varepsilon}(U_{-})$ of $U_{-}$ satisfying
$\frac{\breve{v}}{\breve{u}}=\frac{\tilde{v}}{\tilde{u}}=\dot{z}_{b}$, and
$\breve{v},\tilde{v}\approx 0$. Denote by $h_{j}(b)$ the strength of the
$j^{th}$ shock in the Riemann problem determined by U$\left(b\right)$ and
V$\left(b\right)$, and by $\lambda_{j}$ the corresponding
$j^{th}$-characteristic speed. Then_
$\displaystyle|\lambda_{j}-\dot{z}_{b}|\sim|h_{1}(b)|,\hskip 5.69054ptj=2,3,$
(5.8)
$\displaystyle|h_{4}(b)|\leq|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+|h_{1}(b)|\mathcal{O}(1)|\dot{z}_{b}|,$
(5.9) $\displaystyle|h_{1}(b)|=\bar{\mathcal{O}}(1)|h_{4}(b)|,\hskip
14.22636pt\frac{1}{2}<\tilde{\mathcal{O}}(1)<\frac{3}{2},$ (5.10)
_where $\dot{z}_{b}$ is the slope of the Lipschitz wall._
Proof. We do the proof by analyzing the following two cases.
_Case 1: $h_{1}(b)=0$ and $h_{4}(b)=0$ that corresponds to the case
$\breve{p}=\tilde{p}$_. Starting at the state $U_{b}$, we move along the
Hugoniot curves of the second and third families to reach $V_{b}$. Note that
these two families are the contact Hugoniot curves, and so $\lambda_{2}$ and
$\lambda_{3}$ are constant along the Hugoniot curves. Given that
$\lambda_{2,3}=\frac{v}{u}$, $\textbf{r}_{2}=(1,\frac{v}{u},0,0)^{\top}$, and
$\textbf{r}_{3}=(0,0,0,\rho)^{\top}$, the quantity $\frac{v}{u}$ remains
unchanged as the initial value $\frac{v(U_{b})}{u(U_{b})}$, i.e.,
$\dot{z}_{b}$ in this process by the boundary condition (1.8). Therefore, we
conclude that $\lambda_{2,3}=\dot{z}_{b}$, equivalently,
$\dot{z}_{b}-\lambda_{2,3}=0.$
_Case 2: $h_{1}(b)\neq 0$ that corresponds to $\breve{p}\neq\tilde{p}$_.
Starting at the state $U(b)$, we move along the 1-Hugoniot curve to reach
$U_{1}$, then possibly move along the 2-contact Hugoniot curve to reach
$U_{2}$, the 3-Hugoniot curve to reach $U_{3}$, and the 4-Hugoniot curve to
reach $V(b)$.
To make clear some essential relations among the strengths
$h_{1}(b),h_{2}(b),h_{3}(b),\text{and }h_{4}(b)$, we project $(u,v,p,\rho)$
onto the $(u,v)$–plane. Let $\textbf{r}_{1}|_{u}$ be the projection of
$\textbf{r}_{1}$ onto the $u$-axis, $\textbf{r}_{2}|_{(u,v)}$ be the
projection of $\textbf{r}_{2}$ onto the $(u,v)$–plane; and so on. At the
background state $U_{-}$, there holds
$\textbf{r}_{1}|_{u}=-\textbf{r}_{4}|_{u},\hskip
5.69054pt\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v},\hskip
5.69054pt\textbf{r}_{1}|_{(p,\rho)}=-\textbf{r}_{4}|_{(p,\rho)},\hskip
5.69054pt$ $\textbf{r}_{2}=\textbf{r}_{2}|_{(u,v)},\hskip
5.69054pt\textbf{r}_{3}|_{(u,v)}=0.$
We first note that $h_{4}(b)\neq 0$. Given that
$\textbf{r}_{1}|_{(u,v)}=k_{1}(-\lambda_{1},1)^{\top}$ along with finite
characteristic speeds $\lambda_{1}$ and $\dot{z}_{b}\approx 0$, there always
holds $\dot{z}_{b}<-\frac{1}{\lambda_{1}}$ near the state $U_{-}$. So we can
conclude that, in the $(u,v)$–plane, the derivative $\frac{dv}{du}$ along the
1-curve is always larger than $\dot{z}_{b}$. This implies that
$\frac{v(U_{1})}{u(U_{1})}\neq\frac{v(U_{b})}{u(U_{b})}$. Meanwhile, we have
$\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\frac{v(U_{3})}{u(U_{3})}$
and $\frac{v(V_{b})}{u(V_{b})}=\frac{v(U_{b})}{u(U_{b})}$. Hence,
$\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\frac{v(U_{3})}{u(U_{3})}\neq\frac{v(V_{b})}{u(V_{b})}.$
Thus, we conclude that there is some distance along the 4-Hugoniot curve to
reach $V_{b}$. Therefore, $h_{4}\neq 0$.
Next, we present an essential estimate to bound $\left|h_{4}\right|$ more
precisely in terms of $\left|h_{4}\right|$. To that end, define the signed
length of $(U_{1}-U_{b})|_{(u,v)}$ and $(V_{b}-U_{3})|_{(u,v)}$ by $d_{1}$ and
$d_{4}$ on the $(u,v)$–plane:
$\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\|(U_{1}-U_{b})|_{(u,v)}\|&\mbox{ if
$h_{1}>0$,}\\\\[5.69054pt] -\|(U_{1}-U_{b})|_{(u,v)}\|&\mbox{ if
$h_{1}<0$,}\\\ \end{array}\right.$
and
$\displaystyle d_{4}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\|(V_{b}-U_{3})|_{(u,v)}\|&\mbox{ if
$h_{4}>0$,}\\\\[5.69054pt] -\|(V_{b}-U_{3})|_{(u,v)}\|&\mbox{ if
$h_{4}<0$.}\\\ \end{array}\right.$
Secondly, we note that
$|\lambda_{2}-\dot{z}_{b}|=\mathcal{O}(1)|d_{1}|=\mathcal{O}(1)|h_{1}(b)|.$
Moreover, since
$\lambda_{2}=\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\lambda_{3}$,
we can similarly conclude
$|\lambda_{3}-\dot{z}_{b}|=\mathcal{O}(1)|d_{1}|=\mathcal{O}(1)|h_{1}(b)|.$
Using the following projections on the $(u,v)$–plane,
$\textbf{r}_{1}|_{u}=-\textbf{r}_{4}|_{u},\quad\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v},$
$\textbf{r}_{2}=\textbf{r}_{2}|_{(u,v)},\quad\textbf{r}_{3}|_{(u,v)}=0.$
Thirdly, we note that
$-d_{4}=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+\tilde{d},$
where $\tilde{d}\cos\varphi_{1}=d_{1}\cos\varphi_{2}$, $\varphi_{1}$ denotes
the angle between $(1,\dot{z}_{b})$ and $\textbf{r}_{4}|_{(u,v)}$,
$\varphi_{2}$ denotes the angle between $\textbf{r}_{1}|_{(u,v)}$ and
$(1,\dot{z}_{b})$, $\varphi_{1}=\varphi_{2}+2\alpha$ for $\alpha={\rm
arctan}(\dot{z}_{b})$, and
$\displaystyle\tilde{d}$ $\displaystyle=$ $\displaystyle
d_{1}\frac{\cos\varphi_{2}}{\cos\varphi_{1}}=d_{1}\frac{\cos(\varphi_{1}-2\alpha)}{\cos\varphi_{1}}=d_{1}\frac{\cos\varphi_{1}\cos(2\alpha)+\sin\varphi_{1}\sin(2\alpha)}{\cos\varphi_{1}}$
$\displaystyle=$ $\displaystyle
d_{1}\big{(}\text{cos}(2\alpha)+\mathcal{O}(1)\text{sin}(2\alpha)\big{)}=d_{1}\big{(}1+\mathcal{O}(1)\alpha\big{)}=d_{1}\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$
Hence, there holds
$-d_{4}=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+d_{1}\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$
At $U_{-}$, there also holds $\textbf{r}_{1}|_{(u,p,\rho)}$ =
-$\textbf{r}_{4}|_{(u,p,\rho)}$ and $\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v}$,
which implies
$\frac{d_{1}}{h_{1}}=\frac{d_{4}}{h_{4}}.$
Hence we note the following key estimate:
$-h_{4}(b)=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+h_{1}(b)\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$
(5.15)
Estimate (5.15) now implies
$\displaystyle|h_{4}(b)|$ $\displaystyle\leq$
$\displaystyle|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+|h_{1}(b)|\mathcal{O}(1)|\dot{z}_{b}|$
$\displaystyle\leq$
$\displaystyle|h_{1}(b)|+\mathcal{O}(1)\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|,$
yielding
$|h_{1}(b)|=\tilde{\mathcal{O}}(1)|h_{4}(b)|\hskip 17.07164pt\text{with
}\frac{1}{2}<\bar{\mathcal{O}}(1)<\frac{3}{2},$
given that $|h_{2}(b)|+|\dot{z}_{b}|$ is always small enough, and this is
guaranteed by the sufficiently small total variation of the initial
perturbation $\widetilde{U_{0}}$ and the perturbation of the boundary. $\Box$
We note that the requirement
$\frac{\bar{v}}{\bar{u}}=\frac{\hat{v}}{\hat{u}}=\dot{z}_{b}$ in Proposition
5.2 is just the boundary condition (1.8) because $\dot{z}_{b}$ here is the
slope of the Lipschtiz wall.
Applying Proposition 5.2 now yields
$\displaystyle E_{b,1}$ $\displaystyle=$
$\displaystyle|q_{1}(b)|W_{1}(b)(\dot{z}_{b}+\lambda_{1})$ $\displaystyle=$
$\displaystyle-4B\kappa_{1}w_{1}^{b}|h_{1}(b)|\,|\lambda_{1}|+\mathcal{O}(1)|h_{1}(b)|$
$\displaystyle=$
$\displaystyle-4B\kappa_{1}w_{1}^{b}|h_{1}(b)||\lambda_{1}|+\mathcal{O}(1)|h_{4}(b)|,$
$\displaystyle E_{b,j}$ $\displaystyle=$
$\displaystyle|q_{j}(b)|W_{j}(b)(\dot{z}_{b}+\lambda_{j})=|q_{j}(b)|W_{j}(b)(\dot{z}_{b}-\lambda_{j})+2\lambda_{j}|q_{j}(b)|W_{j}(b)$
$\displaystyle=$
$\displaystyle\mathcal{O}(1)w^{b}_{j}|h_{j}(b)|(\dot{z}_{b}-\lambda_{j})+\mathcal{O}(1)\lambda_{j}w^{b}_{j}|h_{j}(b)|$
$\displaystyle=$
$\displaystyle\mathcal{O}(1)h_{1}(b)=\mathcal{O}(1)h_{4}(b),\hskip
17.07164ptj=2,3,$ $\displaystyle E_{b,4}$ $\displaystyle=$
$\displaystyle|q_{4}(b)|W_{4}(b)(\dot{z}_{b}+\lambda_{4})$ $\displaystyle=$
$\displaystyle
4B\kappa_{1}w^{b}_{4}|h_{4}(b)||\lambda_{1}|+\mathcal{O}(1)|h_{4}(b)|$
$\displaystyle\leq$ $\displaystyle
4B\kappa_{1}|\lambda_{1}|w^{b}_{4}\big{(}|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+\mathcal{O}(1)|h_{1}(b)||\dot{z}_{b}|\big{)}+\mathcal{O}(1)|h_{4}(b)|.$
Using Lemma 5.1, we can choose $w^{b}_{1}$ and $w^{b}_{4}$ such that
$w^{b}_{4}<w^{b}_{1}.$
Then, with the total variation of the incoming flow perturbation and the
boundary perturbation small enough and $\kappa_{1}$ large enough, one has
$\displaystyle\sum_{j=1}^{4}E_{b,j}$ $\displaystyle=$ $\displaystyle
4B\kappa_{1}(w^{b}_{4}-w^{b}_{1})|h_{1}(b)||\lambda_{1}|$
$\displaystyle+\mathcal{O}(1)B\kappa_{1}|\lambda_{1}|h^{b}_{4}\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|+\mathcal{O}(1)|h_{4}(b)|$
$\displaystyle\leq$
$\displaystyle\tilde{\mathcal{O}}(1)4B\kappa_{1}(w^{b}_{4}-w^{b}_{1})|h_{4}(b)|\,|\lambda_{1}|$
$\displaystyle+\mathcal{O}(1)B\kappa_{1}|\lambda_{1}|w^{b}_{4}\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|+\mathcal{O}(1)|h_{4}(b)|\leq
0,$
provided that $|h_{2}(b)|+|\dot{z}_{b}|$ is sufficiently small. This is
guaranteed since the total variation of the incoming flow perturbation and the
boundary perturbation are sufficiently small.
## 6\. Existence of a Semigroup of Solutions
As a corollary of the essential estimates in Sections 3–5, we can now
establish the existence of the semigroup $\mathscr{S}$ generated by the wave-
front tracking method, as well as the Lipschitz continuity of $\mathscr{S}$.
Proposition 6.1. _Suppose that
$TV(\widetilde{U_{0}}(\cdot))+TV(g^{\prime}(\cdot))$ is small enough. Then the
map_
$(\overline{U}(\cdot),x)\mapsto
U^{\vartheta}(x,\cdot)\mathrel{\mathop{:}}=\mathscr{S}^{\vartheta}_{x}(\overline{U}(\cdot))$
_produced by the wave-front tracking algorithm is a uniformly Lipschitz
continuous semigroup satisfying the properties_ :
* (i)
$\mathscr{S}^{\vartheta}_{0}\overline{U}=\overline{U},\quad\mathscr{S}^{\vartheta}_{x_{1}}\mathscr{S}^{\vartheta}_{x_{2}}\overline{U}=\mathscr{S}^{\vartheta}_{x_{1}+x_{2}}\overline{U}$,
for all $x_{1},x_{2}\geq 0$;
* (ii)
$\lVert\mathscr{S}^{\vartheta}_{x}\overline{U}-\mathscr{S}^{\vartheta}_{x}\overline{V}\rVert_{L^{1}}\leq
C\lVert\overline{U}-\overline{V}\rVert_{L^{1}}+C\vartheta x$, for all $x\geq
0$.
Proof. Since $\mathscr{S}^{\vartheta}$ is generated by the wave-front tracking
algorithm, property (i) is immediate. Next, property (ii) is proved as
follows. Take a pair of front tracking $\vartheta$-approximate solutions
$U^{\vartheta}$ and $V^{\vartheta}$ of (1.1) and (1.7)–(1.8) with
$\overline{U}(\cdot)$ and $\overline{V}(\cdot)$ as the initial data,
respectively. Using (4.11) and (4.21), at any $x\geq 0$, we have
$\displaystyle\lVert U^{\vartheta}(x)-V^{\vartheta}(x)\rVert_{L^{1}}$
$\displaystyle\leq$ $\displaystyle{C}\Phi(U^{\vartheta}(x),V^{\vartheta}(x))$
$\displaystyle\leq$
$\displaystyle{C}\Phi(U^{\vartheta}(0),V^{\vartheta}(0))+C\nu x$
$\displaystyle\leq$ $\displaystyle
C\lVert\overline{U}-\overline{V}\rVert_{L^{1}}+C\vartheta x.$
Hence, the $\vartheta$-semigroup is Lipschitz continuous.
Definition 6.1. For a given $\nu_{0}>0$, we define the domain:
$\mathcal{D}=cl\begin{cases}\mbox{the set consisting of points
}U:\mathbb{R}\mapsto\mathbb{R}^{4}\\\ \mbox{such that there exists one point
}y^{i}\in\mathbb{R}\mbox{ and}\\\
\tilde{U}(y)=\begin{cases}U_{-},\qquad&g(x)\leq y\leq y^{i},\\\
U_{+},\qquad&y^{i}<y,\end{cases}\\\ \mbox{so that }U-\tilde{U}\in
L^{1}(\mathbb{R};\mathbb{R}^{4})\mbox{ and }{\rm
TV}(U-\tilde{U})\leq\nu_{0}.\end{cases}$
Remark 6.1. Given a solution $U(x,y)$ to the initial-boundary value problem of
(1.1) and (1.7)–(1.8), we note that, if
$U^{x}(y)\mathrel{\mathop{:}}=U(x,y)\in\mathcal{D}$ at any fixed $x\geq 0$,
then $y^{i}>g(0)=0$ at $x=0$ and $y^{i}>g(x)$ for $x>0$ as a strong vortex
sheet/entropy wave is present.
The semigroup $\mathscr{S}$ generated by the wave-front tracking algorithm is
provided by the next theorem.
Theorem 6.1. _Suppose that
$TV(\widetilde{U_{0}}(\cdot))+TV(g^{\prime}(\cdot))$ is small enough. Then, in
the $L^{1}$–norm, $\mathscr{S}^{\vartheta}$ produced by the wave-front
tracking algorithm is a Cauchy sequence. Denote this unique limit by
$\mathscr{S}$ such that, for any $x>0$,
$\mathscr{S}_{x}(\overline{U})=\lim_{\vartheta\to
0}\mathscr{S}^{\vartheta}_{x}(\overline{U})$. Then the map
$\mathscr{S}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$ is a uniformly
Lipschtiz semigroup in $L^{1}$._ _In particular, the entropy solution to the
initial-boundary problem ( 1.1) and (1.7)–(1.8) constructed by the wave-front
tracking algorithm is unique and $L^{1}$ stable_.
Based on the essential estimates in Sections 3–5, the proof of Theorem 6.1 can
be shown in the same way as the argument given in [7]. Also see Chen-Li [10].
## 7\. Uniqueness of entropy solutions in the viscosity class
In this section, as an immediate consequence of the estimates obtained in
Sections 4–6, we find that the semigroup $\mathscr{S}$ produced by the wave-
front tracking method is the only standard Riemann semigroup (SRS) in the
sense of Definition 7.1 given below. In other words, the semigroup defined by
the wave-front tracking method is the canonical trajectory of the standard
Riemann semigroup (SRS). This yields the uniqueness of entropy solutions in a
broader class of viscosity solutions as introduced by Bressan in [5].
Furthermore, it coincides with the semigroup trajectory generated by the wave-
front tracking method.
Definition 7.1. Problem (1.1) and (1.7)–(1.8) is said to have a standard
Riemann semigroup (SRS) if, for some small $\nu_{0}$, there exist a continuous
mapping $\mathscr{R}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$ and a
constant $L$ satisfying the following properties:
* (i)
(Semigroup property):
$\mathscr{R}_{0}\overline{U}=\overline{U},\quad\mathscr{R}_{x_{1}}\mathscr{R}_{x_{2}}\overline{U}=\mathscr{R}_{x_{1}+x_{2}}\overline{U}$;
* (ii)
(Lipschitz continuity):
$\lVert\mathscr{R}_{x}\overline{U}-\mathscr{R}_{x}\overline{V}\rVert_{L^{1}}\leq
L\lVert\overline{U}-\overline{V}\rVert_{L^{1}}$;
* (iii)
(Consistency with the Riemann solver): _Given piecewise constant initial data
$\overline{U}\in\mathcal{D}$, then, for all $x\in[0,\nu_{0}]$, the function
$U(x,\cdot)=\mathscr{S}_{x}\overline{U}$ coincides with the solution of (1.1)
and (1.7)–(1.8) obtained by piecing together the standard Riemann solutions
and the lateral Riemann solutions._
Following the argument in [5], we employ the estimates obtained in Sections
4–6 to conclude
Theorem 7.1. _Suppose that problem ( 1.1) and (1.7)–(1.8) has a standard
Riemann semigroup $\mathscr{R}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$.
Consider the semigroup $\mathscr{S}$ produced by the wave-front tracking
method, that is, $\mathscr{S}_{x}(\overline{U})=\lim_{\vartheta\rightarrow
0}\mathscr{S}^{\vartheta}_{x}(\overline{U})$. Assume
$\overline{U}\in\mathcal{D}$. Then, for all $x>0$,
$\mathscr{R}_{x}\overline{U}=\mathscr{S}_{x}\overline{U}$_. _Furthermore, a
continuous map $U:[0,X]\mapsto\mathcal{D}$ is a viscosity solution of problem
(1.1) and (1.7)–(1.8) defined in [5] if and only if_
$U(x,\cdot)=\mathscr{R}_{x}\overline{U}\hskip 17.07164pt\textit{for any
}x\in[0,T].$ (7.1)
_In particular, a continuous map $U:[0,X]\mapsto\mathcal{D}$ is a viscosity
solution if and only if_
$U(x,\cdot)=\mathscr{S}_{x}\overline{U}\hskip 17.07164pt\textit{for any
}x\in[0,T].$ (7.2)
The proof here follows a similar argument to the one presented in [5]. The
only difference is that there is a strong vortex sheets/entropy waves in our
problem. Nonetheless, one can proceed with the proof by considering the
convergence of the wave-front tracking method which is shown in Section 3.
Remark 7.1. In the simpler cases of the isentropic or isothermal Euler flow
(1.5), as well as the potential flow, as far as the $L^{1}$–stability problem
is of concern, we realize the same results as for the full Euler system (1.1).
Acknowledgements: The research of Gui-Qiang Chen was supported in part by the
National Science Foundation under Grants DMS-0935967 and DMS-0807551, the UK
EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE
(EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal
Society–Wolfson Research Merit Award (UK). The research of Vaibhav Kukreja was
supported in part by the National Science Foundation under Grants DMS-0935967
and DMS-0807551, the UK EPSRC Science and Innovation Award to the Oxford
Centre for Nonlinear PDE (EP/E035027/1).
## References
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* [3] S. Biachini and A. Bressan, Vanishing viscosity of nonlinear hyperbolic systems, Ann. of Math. 161 (1) (2005), 223–342.
* [4] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking, J. Math. Anal. Appl. 170 (1992), 414–432.
* [5] A. Bressan, The unique limit of the Glimm scheme, Arch. Ration. Mech. Anal. 130 (1995), 205–230.
* [6] A. Bressan, Hyperbolic Systems of Conservations Laws: The One-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000.
* [7] A. Bressan and R. M. Colombo, The semigroup of $2\times 2$ conservation laws, Indiana Univ. Math. J. 44 (1995), 677–725.
* [8] A. Bressan, G. Crasta, and R. M. Colombo, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc. 146 (694), 2000.
* [9] A. Bressan, T.-P. Liu, and T. Yang, $L^{1}$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), 1–22.
* [10] G.-Q. Chen and T.-H. Li, Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge, J. Differential Equations, 244 (2008), 1521–1550.
* [11] G.-Q. Chen, Y.-Q. Zhang, and D.-W. Zhu, Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls, SIAM J. Math. Anal. 38 (2007), 1660–1693.
* [12] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.
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* [15] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Second ed., Springer-Verlag, Berlin, 2005.
* [16] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 20 (1976), 187–212.
* [17] J. Glimm, Solution in the large for nonlinear systems of conservation laws, Comm. Pure Appl. Math. 18 (1965), 697–715.
* [18] H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002.
* [19] Ph. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Birkh$\ddot{\text{a}}$user-Verlag, Basel, 2002.
* [20] M. Lewicka, $L^{1}$ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (2000), 1515–1537.
* [21] M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (2001), 1094–1116.
* [22] M. Lewicka and K. Trivisa, On the $L^{1}$ well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179 (2002), 133–177.
* [23] T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135–148.
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* [25] M. Sabl$\acute{\text{e}}$-Tougeron, M$\acute{\text{e}}$thode de Glimm et probl$\grave{\text{e}}$me mixte, Ann. Inst. H. Poincar$\acute{\text{e}}$ Anal. Nonlin$\acute{\text{e}}$aire, 10 (1993), 423–443.
|
arxiv-papers
| 2012-05-20T16:12:05 |
2024-09-04T02:49:31.107640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Gui-Qiang G. Chen and Vaibhav Kukreja",
"submitter": "Gui-Qiang G. Chen",
"url": "https://arxiv.org/abs/1205.4429"
}
|
1205.4431
|
# Large Social Networks can be Targeted for Viral Marketing with Small Seed
Sets
Paulo Shakarian and Damon Paulo Network Science Center and
Department of Electrical Engineering and Computer Science
United States Military Academy
West Point, New York 10996
Email: paulo[at]shakarian.net, damon.paulo[at]usma.edu
###### Abstract
In a “tipping” model, each node in a social network, representing an
individual, adopts a behavior if a certain number of his incoming neighbors
previously held that property. A key problem for viral marketers is to
determine an initial “seed” set in a network such that if given a property
then the entire network adopts the behavior. Here we introduce a method for
quickly finding seed sets that scales to very large networks. Our approach
finds a set of nodes that guarantees spreading to the entire network under the
tipping model. After experimentally evaluating $31$ real-world networks, we
found that our approach often finds such sets that are several orders of
magnitude smaller than the population size. Our approach also scales well - on
a Friendster social network consisting of $5.6$ million nodes and $28$ million
edges we found a seed sets in under $3.6$ hours. We also find that highly
clustered local neighborhoods and dense network-wide community structure
together suppress the ability of a trend to spread under the tipping model.
## I Introduction
A much studied model in network science, tipping[10, 20, 11] (a.k.a.
deterministic linear threshold[12]) is often associated with “seed” or
“target” set selection, [7] (a.k.a. the maximum influence problem). In this
problem we have a social network in the form of a directed graph and
thresholds for each individual. Based on this data, the desired output is the
smallest possible set of individuals such that, if initially activated, the
entire population will adopt the new behavior (a seed set). This problem is
NP-Complete [12, 9]. Although approximation algorithms have been proposed,
[15, 7, 3, 8] none seem to scale to very large data sets. Here, inspired by
shell decomposition, [5, 13, 2] we present a method guaranteed to find a set
of nodes that causes the entire population to activate - but is not
necessarily of minimal size. We then evaluate the algorithm on $31$ large
real-world social networks and show that it often finds very small seed sets
(often several orders of magnitude smaller than the population size). We also
show that the size of a seed set is related to Louvain modularity and average
clustering coefficient. Therefore, we find that dense community structure and
tight-knit local neighborhoods together inhibit the spreading of trends under
the tipping model.
The rest of the paper is organized as follows. In Section II, we provide
formal definitions of the tipping model. This is followed by the presentation
of our new algorithm in Section III. We then describe our experimental results
in Section IV. Finally, we provide an overview of related work in Section V.
## II Technical Preliminaries
Throughout this paper we assume the existence of a social network, $G=(V,E)$,
where $V$ is a set of vertices and $E$ is a set of directed edges. We will use
the notation $n$ and $m$ for the cardinality of $V$ and $E$ respectively. For
a given node $v_{i}\in V$, the set of incoming neighbors is $\eta^{in}_{i}$,
and the set of outgoing neighbors is $\eta^{out}_{i}$. The cardinalities of
these sets (and hence the in and out degrees of node $v_{i}$) are
$d^{in}_{i},d^{out}_{i}$ respectively. We now define a threshold function that
for each node returns the fraction of incoming neighbors that must be
activated for it to become activate as well.
###### Definition 1 (Threshold Function)
We define the threshold function as mapping from V to $(0,1]$. Formally:
$\theta:V\rightarrow(0,1]$.
For the number of neighbors that must be active, we will use the shorthand
$k_{i}$. Hence, for each $v_{i}$, $k_{i}=\lceil\theta(v_{i})\cdot
d^{in}_{i}\rceil$. We now define an activation function that, given an initial
set of active nodes, returns a set of active nodes after one time step.
###### Definition 2 (Activation Function)
Given a threshold function, $\theta$, an activation function $A_{\theta}$ maps
subsets of V to subsets of V, where for some $V^{\prime}\subseteq V$,
$A_{\theta}(V^{\prime})=V^{\prime}\cup\\{v_{i}\in V\ s.t.\ |\eta^{in}_{i}\cap
V^{\prime}|\geq k_{i}\\}$ (1)
We now define multiple applications of the activation function.
###### Definition 3 (Multiple Applications of the Activation Function)
Given a natural number $i>0$, set $V^{\prime}\subseteq V$, and threshold
function, $\theta$, we define the multiple applications of the activation
function, ${A^{i}_{\theta}}(V^{\prime})$, as follows:
$A^{i}_{\theta}(V^{\prime})=\begin{cases}A_{\theta}(V^{\prime})&\text{if
$i=1$}\\\
A_{\theta}(A^{i-1}_{\theta}(V^{\prime}))&\text{otherwise}\end{cases}$ (2)
Clearly, when $A^{i}_{\theta}(V^{\prime})=A^{i-1}_{\theta}(V^{\prime})$ the
process has converged. Further, this occurs in no more than $n$ steps (as, in
each step, at least one new node must be activated). Based on this idea, we
define the function $\Gamma$ which returns the set of all nodes activated upon
the convergence of the activation function.
###### Definition 4 ($\Gamma$ Function)
Let j be the least value such that
$A^{j}_{\theta}(V^{\prime})=A^{j-1}_{\theta}(V^{\prime})$. We define the
function $\Gamma_{\theta}:2^{V}\rightarrow 2^{V}$ as follows.
$\mathbf{\Gamma_{\theta}}(V^{\prime})=A^{j}_{\theta}(V^{\prime})$ (3)
We now have all the pieces to introduce our problem - finding the minimal
number of nodes that are initially active to ensure that the entire set $V$
becomes active.
###### Definition 5 (The MIN-SEED Problem)
The MIN-SEED Problem is defined as follows: given a threshold function,
$\theta$, return $V^{\prime}\subseteq V\ s.t.\ \Gamma_{\theta}(V^{\prime})=V$,
and there does not exist $V^{\prime\prime}\subseteq V$ where
$|V^{\prime\prime}|<|V^{\prime}|$ and $\Gamma_{\theta}(V^{\prime\prime})=V$.
The following theorem is from the literature [12, 9] and tells us that the
MIN-SEED problem is NP-complete.
###### Theorem 1 (Complexity of MIN-SEED [12, 9])
MIN-SEED in NP-Complete.
## III Algorithm
To deal with the intractability of the MIN-SEED problem, we design an
algorithm that finds a non-trivial subset of nodes that causes the entire
graph to activate, but we do not guarantee that the resulting set will be of
minimal size. The algorithm is based on the idea of shell decomposition often
cited in physics literature [21, 5, 13, 2] but modified to ensure that the
resulting set will lead to all nodes being activated. The algorithm,
TIP_DECOMP is presented in this section.
Algorithm 1 TIP_DECOMP
0: Threshold function, $\theta$ and directed social network $G=(V,E)$
0: $V^{\prime}$
1: For each vertex $v_{i}$, compute $k_{i}$.
2: For each vertex $v_{i},\ dist_{i}=d_{i}^{in}-k_{i}$.
3: FLAG = TRUE.
4: while FLAG do
5: Let $v_{i}$ be the element of $v$ where $dist_{i}$ is minimal.
6: if $dist_{i}=\infty$ then
7: FLAG = FALSE.
8: else
9: Remove $v_{i}$ from $G$ and for each $v_{j}$ in $\eta_{i}^{out}$, if
$dist_{j}>0$, set $dist_{j}=dist_{j}-1$. Otherwise set $dist_{j}=\infty$.
10: end if
11: end while
12: return All nodes left in $G$.
Intuitively, the algorithm proceeds as follows (Figure 1). Given network
$G=(V,E)$ where each node $v_{i}$ has threshold
$k_{i}=\lceil\theta(v_{i})\cdot d^{in}_{i}\rceil$, at each iteration, pick the
node for which $d^{in}_{i}-k_{i}$ is the least but positive (or $0$) and
remove it. Once there are no nodes for which $d^{in}_{i}-k_{i}$ is positive
(or $0$), the algorithm outputs the remaining nodes in the network.
Figure 1: Example of our algorithm for a simple network depicted in box A. We
use a threshold value set to $50\%$ of the node degree. Next to each node
label (lower-case letter) is the value for $d^{in}_{i}-k_{i}$ (where
$k_{i}=\lceil\frac{d^{in}_{i}}{2}\rceil$). In the first four iterations, nodes
e, f, h, and i are removed resulting in the network in box B. This is followed
by the removal of node j resulting in the network in box C. In the next two
iterations, nodes a and b are removed (boxes D-E respectively). Finally, node
c is removed (box F). The nodes of the final network, consisting of d and g,
have negetive values for $d_{i}-\theta_{i}$ and become the output of the
algorithm.
Now, we prove that the resulting set of nodes is guaranteed to cause all nodes
in the graph to activate under the tipping model. This proof follows from the
fact that any node removed is activated by the remaining nodes in the network.
###### Theorem 2
If all nodes in $V^{\prime}\ \subseteq\ V$ returned by TIP_DECOMP are
initially active, then every node in $V$ will eventually be activated, too.
###### Proof:
Let $w$ be the total number of nodes removed by TIP_DECOMP, where $v_{1}$ is
the last node removed and $v_{w}$ is the first node removed. We prove the
theorem by induction on $w$ as follows. We use $P(w)$ to denote the inductive
hypothesis which states that all nodes from $v_{1}$ to $v_{w}$ are active. In
the base case, $P(1)$ trivially holds as we are guaranteed that from set
$V^{\prime}$ there are at least $k_{1}$ edges to $v_{1}$ (or it would not be
removed). For the inductive step, assuming $P(w)$ is true, when $v_{w+1}$ was
removed from the graph $dist_{w+1}\geq 0$ which means that $d_{w+1}^{in}\geq
k_{w+1}$. All nodes in $\eta^{in}_{w+1}$ at the time when $v_{w+1}$ was
removed are now active, so $v_{w+1}$ will now be activated - which completes
the proof. ∎
We also note that by using the appropriate data structure (we used a binomial
heap in our implementation), for a network of $n$ nodes and $m$ edges, this
algorithm can run in time $O(m\log n)$.
###### Proposition 1
The complexity of TIP_DECOMP is $O(m\cdot log(n))$.
## IV Results
All experiments were run on a computer equipped with an Intel X5677 Xeon
Processor operating at 3.46 GHz with a 12 MB Cache. The machine was running
Red Hat Enterprise Linux version 6.1 and equipped with 70 GB of physical
memory. TIP_DECOMP was written using Python 2.6.6 in 200 lines of code that
leveraged the NetworkX library available from http://networkx.lanl.gov/. The
code used a binomial heap library written by Björn B. Brandenburg available
from http://www.cs.unc.edu/$\sim$bbb/. All statistics presented in this
section were calculated using R 2.13.1.
### IV-A Datasets
In total, we examined $31$ networks: nine academic collaboration networks,
three e-mail networks, and $19$ networks extracted from social-media sites.
The sites included included general-purpose social-media (similar to Facebook
or MySpace) as well as special-purpose sites (i.e. focused on sharing of
blogs, photos, or video).
All datasets used in this paper were obtained from one of four sources: the
ASU Social Computing Data Repository, [23] the Stanford Network Analysis
Project, [14] the University of Michigan, [17] and Universitat Rovira i
Virgili.[1] All networks considered were symmetric – i.e. if a directed edge
from vertex $v$ to $v^{\prime}$ exists, there is also an edge from vertex
$v^{\prime}$ to $v$. Tables I (A-C) show some of the pertinent qualities of
these networks. The networks are categorized by the results (explained later
in this section). In what follows, we provide their real-world context.
### IV-B Category A
* •
BlogCatalog is a social blog directory that allows users to share blogs with
friends. [23] The first two samples of this site, BlogCatalog1 and 2, were
taken in Jul. 2009 and June 2010 respectively. The third sample, BlogCatalog3
was uploaded to ASU’s Social Computing Data Repository in Aug. 2010.
* •
Buzznet is a social media network designed for sharing photographs, journals,
and videos. [23] It was extracted in Nov. 2010.
* •
Douban is a Chinese social medial website designed to provide user reviews
and recommendations. [23] It was extracted in Dec. 2010.
* •
Flickr is a social media website that allows users to share photographs. [23]
It was uploaded to ASU’s Social Computing Data Repository in Aug. 2010.
* •
Flixster is a social media website that allows users to share reviews and
other information about cinema. [23] It was extracted in Dec. 2010.
* •
FourSquare is a location-based social media site. [23] It was extracted in
Dec. 2010.
* •
Frienster is a general-purpose social-networking site. [23] It was extracted
in Nov. 2010.
* •
Last.Fm is a music-centered social media site. [23] It was extracted in Dec.
2010.
* •
LiveJournal is a site designed to allow users to share their blogs. [23] It
was extracted in Jul. 2010.
* •
Livemocha is touted as the “world’s largest language community.” [23] It was
extracted in Dec. 2010.
* •
WikiTalk is a network of individuals who set and received messages while
editing WikiPedia pages. [14] It was extracted in Jan. 2008.
### IV-C Category B
* •
Delicious is a social bookmarking site, designed to allow users to share web
bookmarks with their friends. [23] It was extracted in Dec. 2010.
* •
Digg is a social news website that allows users to share stories with friends.
[23] It was extracted in Dec. 2010.
* •
EU E-Mail is an e-mail network extracted from a large European Union research
institution. [14] It is based on e-mail traffic from Oct. 2003 to May 2005.
* •
Hyves is a popular general-purpose Dutch social networking site. [23] It was
extracted in Dec. 2010.
* •
Yelp is a social networking site that allows users to share product reviews.
[23] It was extracted in Nov. 2010.
### IV-D Category C
* •
CA-AstroPh is a an academic collaboration network for Astro Physics from Jan.
1993 - Apr. 2003. [14]
* •
CA-CondMat is an academic collaboration network for Condense Matter Physics.
Samples from 1999 (CondMat99), 2003 (CondMat03), and 2005 (CondMat05) were
obtained from the University of Michigan. [17] A second sample from 2003
(CondMat03a) was obtained from Stanford University. [14]
* •
CA-GrQc is a an academic collaboration network for General Relativity and
Quantum Cosmology from Jan. 1993 - Apr. 2003. [14]
* •
CA-HepPh is a an academic collaboration network for High Energy Physics -
Phenomenology from Jan. 1993 - Apr. 2003. [14]
* •
CA-HepTh is a an academic collaboration network for High Energy Physics -
Theory from Jan. 1993 - Apr. 2003. [14]
* •
CA-NetSci is a an academic collaboration network for Network Science from May
2006.
* •
Enron E-Mail is an e-mail network from the Enron corporation made public by
the Federal Energy Regulatory Commission during its investigation. [14]
* •
URV E-Mail is an e-mail network based on communications of members of the
University Rovira i Virgili (Tarragona). [1] It was extracted in 2003.
* •
YouTube is a video-sharing website that allows users to establish friendship
links. [23] The first sample (YouTube1) was extracted in Dec. 2008. The second
sample (YouTube2) was uploaded to ASU’s Social Computing Data Repository in
Aug. 2010.
TABLE I: Information on the networks in Categories A, B, and C.
### IV-E Runtime
First, we examined the runtime of the algorithm (see Figure 2). Our
experiments aligned well with our time complexity result (Proposition 1). For
example, a network extracted from the Dutch social-media site Hyves consisting
of $1.4$ million nodes and $5.5$ million directed edges was processed by our
algorithm in at most $12.2$ minutes. The often-cited LiveJournal dataset
consisting of $2.2$ million nodes and $25.6$ million directed edges was
processed in no more than $66$ minutes - a short time for an NP-hard
combinatorial problem on a large-sized input.
Figure 2: $m\ln n$ vs. runtime in seconds (log scale, $m$ is number of edges,
$n$ is number of nodes). The relationship is linear with $R^{2}=0.9015$,
$p=2.2\cdot 10^{-16}$.
### IV-F Seed Size
For each network, we performed $10$ “integer” trials. In these trials, we set
$\theta(v_{i})=\min(d^{in}_{i},k)$ where $k$ was kept constant among all
vertices for each trial and set at an integer in the interval $[1,10]$. We
evaluated the ability of a network to promote spreading under the tipping
model based on the size of the set of nodes returned by our algorithm (as a
percentage of total nodes). For purposes of discussion, we have grouped our
networks into three categories based on results (Figure 3 and Table II). In
general, online social networks had the smallest seed sets - $13$ networks of
this type had an average seed set size less than $2\%$ of the population. We
also noticed, that for most networks, there was a linear realtion between
threshold value and seed size.
Figure 3: Threshold value (assigned as an integer in the interval $[1,10]$)
vs. size of initial seed set as returned by our algorithm in our three
identified categories of networks (categories A-C are depicted in panels A-C
respectively). Average seed sizes were under $2\%$ for Categorty A, $2-10\%$
for Category B and over $10\%$ for Category C. The relationship, in general,
was linear for categories A and B and lograthimic for C. CA-NetSci had the
largest Louvain Modularity and clustering coefficient of all the networks.
This likely explains why that particular network seems to inhibit spreading.
Category A can be thought of as social networks highly susceptible to
influence - as a very small fraction of individuals initially having a
behavior can lead to adoption by the entire population. In our ten trials, the
average seed size was under $2\%$ for each of these $13$ networks. All were
extracted from social media websites. For some of the lower threshold levels,
the size of the set of seed nodes was particularly small. For a threshold of
three we had $11$ of the Category A networks with a seed size less than
$0.5\%$ of the population. For a threshold of four, we had nine networks
meeting that criteria.
Networks in Category B are susceptible to influence with a relatively small
set of initial nodes - but not to the extent of those in Category A. They had
an average initial seed size greater than $2\%$ but less than $10\%$. Members
in this group included two general purpose social media networks, two
specialty social media networks, and an e-mail network.
Category C consisted of networks that seemed to hamper diffusion in the
tipping model, having an average initial seed size greater than $10\%$. This
category included all of the academic collaboration networks, two of the email
networks, and two networks derived from friendship links on YouTube.
### IV-G Seed Size as a Function of Community Structure
In this section, we view the results of our heuristic algorithm as a
measurement of how well a given network promotes spreading. Here, we use this
measurement to gain insight into which structural aspects make a network more
likely to be “tipped.” We compared our results with two network-wide measures
characterizing community structure. First, clustering coefficient ($C$) is
defined for a node as the fraction of neighbor pairs that share an edge -
making a triangle. For the undirected case, we define this concept formally
below.
###### Definition 6 (Clustering Coefficient)
Let $r$ be the number of edges between nodes with which $v_{i}$ has an edge
and $d_{i}$ be the degree of $v_{i}$. The clustering coefficient,
$C_{i}=\dfrac{2r}{d_{i}(d_{i}-1)}$.
Intuitively, a node with high $C_{i}$ tends to have more pairs of friends that
are also mutual friends. We use the average clustering coefficient as a
network-wide measure of this local property.
Second, we consider modularity ($M$) defined by Newman and Girvan. [16]. For a
partition of a network, $M$ is a real number in $[-1,1]$ that measures the
density of edges within partitions compared to the density of edges between
partitions. We present a formal definition for an undirected network below.
###### Definition 7 (Modularity [16])
Modularity, $M=\dfrac{1}{2m}\sum_{i,j\in
V}[1-\dfrac{d_{i}d_{j}}{2m}]\delta(c_{i},c_{j})$, where $m$ is the number of
undirected edges, $d_{i}$ is node degree, $c_{i}$ is the community to which
$v_{i}$ belongs and $\delta(x,y)=1$ if $x=y$ and $0$ otherwise.
The modularity of an optimal network partition can be used to measure the
quality of its community structure. Though modularity-maximization is NP-hard,
the approximation algorithm of Blondel et al. [4] (a.k.a. the “Louvain
algorithm”) has been shown to produce near-optimal partitions.111Louvain
modularity was computed using the implementation available from CRANS at
http://perso.crans.org/aynaud/communities/. We call the modularity associated
with this algorithm the “Louvain modularity.” Unlike the $C$, which describes
local properties, $M$ is descriptive of the community level. For the $31$
networks we considered, $M$ and $C$ appear uncorrelated ($R^{2}=0.0538$,
$p=0.2092$).
We plotted the initial seed set size ($S$) (from our algorithm - averaged over
the $10$ threshold settings) as a function of $M$ and $C$ (Figure 4a) and
uncovered a correlation (planar fit, $R^{2}=0.8666$, $p=5.666\cdot 10^{-13}$,
see Figure 4 A). The majority of networks in Category C (less susceptible to
spreading) were characterized by relatively large $M$ and $C$ (Category C
includes the top nine networks w.r.t. $C$ and top five w.r.t. $M$). Hence,
networks with dense, segregated, and close-knit communities (large $M$ and
$C$) suppress spreading. Likewise, those with low $M$ and $C$ tended to
promote spreading. Also, we note that there were networks that promoted
spreading with dense and segregated communities, yet were less clustered (i.e.
Category A networks Friendster and LiveJournal both have $M\geq 0.65$ and
$C\leq 0.13$). Further, some networks with a moderately large clustering
coefficient were also in Category A (two networks extracted from BlogCatalog
had $C\geq 0.46$) but had a relatively less dense community structure (for
those two networks $M\leq 0.33$).
Figure 4: (A) Louvain modularity ($M$) and average clustering coefficient
($C$) vs. the average seed size ($S$). The planar fit depicted is
$S=43.374\cdot M+33.794\cdot C-24.940$ with $R^{2}=0.8666$, $p=5.666\cdot
10^{-13}$. (B) Same plot at (A) except the averages are over the 12
percentage-based threshold values. The planar fit depicted is $S=18.105\cdot
M+17.257\cdot C-10.388$ with $R^{2}=0.816$, $p=5.117\cdot 10^{-11}$.
We also studied the effects on spreading when the threshold values would be
assigned as a certain fraction of the node’s in-degree. [11, 22] This results
in heterogeneous $\theta_{i}$’s for the nodes. We performed $12$ trials for
each network. Thresholds for each trial were based on the product of in-degree
and a fraction in the interval $[0.05,0.60]$ (multiples of $0.05$). The
results (Figure 5 and Table II) were analogous to our integer tests. We also
compared the averages over these trials with $M$ and $C$ and obtained similar
results as with the other trials (Figure 4 B).
Figure 5: Threshold value (assigned as a fraction of node in-degree as a
multiple of $0.05$ in the interval $[0.05,0.60]$) vs. size of initial seed set
as returned by our algorithm in our three identified categories of networks
(categories A-C are depicted in panels A-C respectively, categories are the
same as in Figure 1). Average seed sizes were under $5\%$ for Categorty A,
$1-7\%$ for Category B and over $3\%$ for Category C. In general, the
relationship between threshold and initial seed size for networks in all
categories was exponential. TABLE II: Regression analysis and network-wide
measures for the networks in Categories A, B, and C.
## V Related Work
Tipping models first became popular by the works of [10] and [20] where it was
presented primarily in a social context. Since then, several variants have
been introduced in the literature including the non-deterministic version of
[12] (described later in this section) and a generalized version of [11]. In
this paper we focused on the deterministic version. In [22], the authors look
at deterministic tipping where each node is activated upon a percentage of
neighbors being activated. Dryer and Roberts [9] introduce the MIN-SEED
problem, study its complexity, and describe several of its properties w.r.t.
certain special cases of graphs/networks. The hardness of approximation for
this problem is described in [7]. The work of [3] presents an algorithm for
target-set selection whose complexity is determined by the tree-width of the
graph - though it provides no experiments or evidence that the algorithm can
scale for large datasets. The recent work of [18] prove a non-trivial upper
bound on the smallest seed set.
Our algorithm is based on the idea of shell-decomposition that currently is
prevalent in physics literature. In this process, which was introduced in
[21], vertices (and their adjacent edges) are iteratively pruned from the
network until a network “core” is produced. In the most common case, for some
value $k$, nodes whose degree is less than $k$ are pruned (in order of degree)
until no more nodes can be removed. This process was used to model the
Internet in [5] and find key spreaders under the SIR epidemic model in [13].
More recently, a “heterogeneous” version of decomposition was introduced in
[2] \- in which each node is pruned according to a certain parameter - and the
process is studied in that work based on a probability distribution of nodes
with certain values for this parameter.
### V-A Notes on Non-Deterministic Tipping
We also note that an alternate version of the model where the thresholds are
assigned randomly has inspired approximation schemes for the corresponding
version of the seed set problem.[12, 15, 8] Work in this area focused on
finding a seed set of a certain size that maximizes of the expected number of
adopters. The main finding by Kempe et al., the classic work for this model,
was to prove that the expected number of adopters was submodular - which
allowed for a greedy approximation scheme. In this algorithm, at each
iteration, the node which allows for the greatest increase in the expected
number of adopters is selected. The approximation guarantee obtained (less
than $0.63$ of optimal) is contingent upon an approximation guarantee for
determining the expected number of adopters - which was later proved to be
$\\#P$-hard. [8] Though finding a such a guarantee is still an open question,
work on counting-complexity problems such as that of Dan Roth [19] indicate
that a non-trivial approximation ratio is unlikely. Further, the simulation
operation is often expensive - causing the overall time complexity to be
$O(x\cdot n^{2})$ where $x$ is the number of runs per simulation and $n$ is
the number of nodes (typically, $x>n$). In order to avoid simulation, various
heuristics have been proposed, but these typically rely on the computation of
geodesics - an $O(n^{3})$ operation - which is also more expensive than our
approach.
Additionally, the approximation argument for the non-deterministic case does
not directly apply to the original (deterministic) model presented in this
paper. A simple counter-example shows that sub-modularity does not hold here.
Sub-modularity (diminishing returns) is the property leveraged by Kempe et al.
in their approximation result.
### V-B Note on an Upper Bound of the Initial Seed Set
Very recently, we were made aware of research by Daniel Reichman that proves
an upper bound on the minimal size of a seed set for the special case of
undirected networks with homogeneous threshold values. [18] The proof is
constructive and yields an algorithm that mirrors our approach (although
Reicshman’s algorithm applies only to that special case). We note that our
work and the work of Reichman were developed independently. We also note that
Reichman performs no experimental evaluation of the algorithm.
Given undirected network $G$ where each node $v_{i}$ has degree $d_{i}$ and
the threshold value for all nodes is $k$, Reichman proves that the size of the
minimal seed set can be bounded by $\sum_{i}\min\\{1,\frac{k}{d_{i}+1}\\}$.
For our integer tests, we compared our results to Reichman’s bound. Our seed
sets were considerably smaller - often by an order of magnitude or more. See
Figure 6 for details.
Figure 6: Integer threshold values vs. the seed size divided by Reichman’s
upper bound [18] the three categories of networks (categories A-C are depicted
in panels A-C respectively). Note that in nearly every trial, our algorithm
produced an initial seed set significantly smaller than the bound - in many
cases by an order of magnitude or more.
## VI Conclusion
As recent empirical work on tipping indicates that it can occur in real social
networks,[6, 24] our results are encouraging for viral marketers. Even if we
assume relatively large threshold values, small initial seed sizes can often
be found using our fast algorithm - even for large datasets. For example, with
the FourSquare online social network, under majority threshold ($50\%$ of
incoming neighbors previously adopted), a viral marketeer could expect a
$297$-fold return on investment. As results of this type seem to hold for many
online social networks, our algorithm seems to hold promise for those wishing
to “go viral.”
## Acknowledgments
We would like to thank Gaylen Wong (USMA) for his technical support.
Additionally, we would like to thank (in no particular order) Albert-László
Barabási (NEU), Sameet Sreenivasan (RPI), Boleslaw Szymanski (RPI), John James
(USMA), and Chris Arney (USMA) for their discussions relating to this work.
Finally, we would also like to thank Megan Kearl, Javier Ivan Parra, and Reza
Zafarani of ASU for their help with some of the datasets. The authors are
supported under by the Army Research Office (project 2GDATXR042) and the
Office of the Secretary of Defense (project F1AF262025G001). The opinions in
this paper are those of the authors and do not necessarily reflect the
opinions of the funders, the U.S. Military Academy, or the U.S. Army.
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|
arxiv-papers
| 2012-05-20T16:28:29 |
2024-09-04T02:49:31.120225
|
{
"license": "Public Domain",
"authors": "Paulo Shakarian and Damon Paulo",
"submitter": "Paulo Shakarian",
"url": "https://arxiv.org/abs/1205.4431"
}
|
1205.4506
|
# Entanglement creation with negative index metamaterials
Michael Siomau1, Ali A. Kamli1, Sergey A. Moiseev2 and Barry C. Sanders3
1Physics Department, Jazan University, P.O. Box 114, 45142 Jazan, Kingdom of
Saudi Arabia 2Kazan Physical-Technical Institute of Russian Academy of
Science, 420029 Kazan, Russia 3Institute for Quantum Information Science,
University of Calgary, Alberta T2N 1N4, Canada
###### Abstract
We propose a scheme for creating of a maximally entangled state comprising two
field quanta. In our scheme, two weak light fields, which are initially
prepared in either coherent or polarization states, interact with a composite
medium near an interface between a dielectric and a negative index
metamaterial. Such interaction leads to a large Kerr nonlinearity, reduction
of the group velocity of the light and significant confinement of the light
fields while simultaneously avoiding amplitude losses of the incoming
radiation. All these considerations make our scheme efficient.
###### pacs:
03.67.Bg, 42.50.Dv, 42.50.Gy, 03.67.Hk
Entanglement is an essential feature of quantum theory, which manifests
impressive advantages of recently established quantum technologies over their
classical counterpart Horodecki:09 ; Guehne:09 . Variety of physical systems,
starting from atoms, electrons and photons Amico:08 and ending with
sophisticated molecules and even living organisms Romero:10 , can exhibit
quantum entanglement. Among these systems, photons, the quanta of
electromagnetic field, take privileged position because of their exceptional
properties. Photons can be easily generated and measured and can carry
information over long distances being resistant to detrimental effect of
decoherence Braunstein:05 . At the same time, photons do not directly interact
with each other, what makes difficult to prepare them in entangled states.
A conventional way to create entanglement of light quanta is through their
interaction with nonlinear (Kerr) medium, which have intensity-dependent
refractive index Sanders:11 . In natural medium, however, Kerr nonlinearity is
very small. Therefore, to achieve significant entanglement of photons, one has
to increase both intensity of the field pulses and interaction time with the
medium. Such actions may not be always possible in practice, because of
diffraction and finite size of the medium, and are very unlikely from the
viewpoint of applications Braunstein:05 ; Kok:07 , where weak light fields
(i.e. of energy of a single photon) are desired.
In 2000, a pioneering proposal to “design” material exhibiting strong
nonlinear interaction at the single-photon level was made by Lukin and
Imamoğlu Lukin:00 . This idea stimulated a number of theoretical
investigations Petrosyan:02 ; Wang:06 ; Yavuz:10 as well as experiments
Kang:03 ; Chen:06 ; Li:08 , to name just a few. However, despite these
remarkable results in achieving large Kerr nonlinearity, efficient creation of
entanglement at the low energy limit remains challenging in many aspects
Shapiro:06 ; Gea:10 .
Figure 1: (Color online) Two weak light pulses create two surface polaritons
near the interface between the dielectric $z>0$ and the metamaterial $z<0$.
While electromagnetically induced transparency is established for both
polaritons simultaneously, they will propagate along the interface with small
group velocities $\upsilon_{g}<c$ and interact nonlinearly with each other.
In this work we suggest a scheme in which initially uncorrelated states of
light field become entangled due to their interaction with a medium near an
interface between a dielectric and a negative index metamaterial. The medium
of interest consists of a dielectric (which has a layer of thickness $z_{0}$
doped with six level atoms Petrosyan:02 ) and a metamaterial placed together,
as shown in Fig. 1. Due to interaction with the medium, an incident light beam
creates a spatially confined surface polariton Maier:07 which propagates
along the interface with substantially reduced group velocity
$\upsilon_{g}<c$. Although in natural mediums a surface polariton undergos
slashing amplitude loss, specific design of the medium makes possible to
suppress losses significantly in a narrow frequency bandwidth of the incoming
light Kamli:08 . Placing the layer of six level atoms near the interface
allows us to establish double electromagnetically induced transparency
Fleisch:05 (i.e. for two incoming pulses simultaneously) and, at the same
time, create large Kerr nonlinearity Petrosyan:02 . All mentioned factors
contribute to an efficient nonlinear interaction between the two surface
polaritons in the medium. Such interaction makes possible a mutual $\pi$-phase
shift between the polaritons which leads to entanglement of the light fields.
Ignoring presence of the atomic layer near the interface, the process of
interaction between the light fields and the medium can be considered from the
viewpoint of classical electrodynamics. Macroscopic properties of the material
can be characterized with electric permittivity $\varepsilon$ and magnetic
permeability $\mu$. For a dielectric, these parameters are strictly positive,
while both of them may be simultaneously negative for a metamaterial Smith:04
. Let us assume that the dielectric has constant homogeneous parameters
$\varepsilon_{1}$ and $\mu_{1}$, while parameters $\varepsilon_{2}$ and
$\mu_{2}$ are frequency-dependent for the metamaterial and are given by
Kamli:08
$\displaystyle\varepsilon_{2}(\omega)$ $\displaystyle=$
$\displaystyle\varepsilon_{\infty}-\frac{\omega_{e}^{2}}{\omega\,(\omega+i\,\gamma_{e})}\,,$
$\displaystyle\mu_{2}(\omega)$ $\displaystyle=$
$\displaystyle\mu_{\infty}-\frac{\omega_{m}^{2}}{\omega\,(\omega+i\,\gamma_{m})}\,,$
(1)
where $\omega_{e}$ and $\omega_{m}$ are electric and magnetic plasma
frequencies, $\gamma_{e}$ and $\gamma_{m}$ are corresponding (empiric) decay
rates and $\varepsilon_{\infty}$ and $\mu_{\infty}$ are background constants
Maier:07 . Here we have chosen the simplest (Drude-like) model for magnetic
permeability $\mu_{2}(\omega)$. This model is known to be adequate in the
optical region Maier:07 ; Merlin:09 , although more sophisticated models can
be taken into consideration Kamli:10 .
Electromagnetic field of the surface polaritons can be found form Maxwell
equations with boundary conditions for $\varepsilon_{i}$ and $\mu_{i}$
$(i=1,2)$. Since the permittivity and the permeability (Entanglement creation
with negative index metamaterials) may be simultaneously negative, both
transverse magnetic (TM) and transverse electric (TE) polarizations of the
electromagnetic field may exist in the medium. Natural mediums, in contrast,
support only TM polarization Maier:07 . To be specific, we shall later focus
on the TM waves. The wave vector of the electromagnetic field in the medium is
given by the dispersion relation as follows
$K(\omega)\,=\,\frac{\omega}{c}\>\sqrt{\varepsilon_{1}\,\varepsilon_{2}\,\frac{\varepsilon_{2}\,\mu_{1}-\varepsilon_{1}\,\mu_{2}}{\varepsilon_{2}^{2}-\varepsilon_{1}^{2}}}\,.$
(2)
The real part of this expression gives dispersion of the field, while its
imaginary part stands for absorbtion loss.
It has been shown that absorbtion loss can be completely suppressed in a
narrow frequency bandwidth due to destructive interference of electric and
magnetic absorbtion responses of the medium Kamli:08 . For example, taking
$\varepsilon_{1}=1.3$ and $\mu_{1}=1$ and $\omega_{e}=1.37\times 10^{16}{\rm
s}^{-1}$, $\gamma_{e}=2.73\times 10^{13}{\rm s}^{-1}$ (as for Ag) and assuming
$\omega_{m}=10^{15}{\rm s}^{-1}$, $\gamma_{m}=10^{12}{\rm s}^{-1}$,
$\varepsilon_{\infty}=5$ and $\mu_{\infty}=5$, one can see that absorbtion
loss ${\rm Im}[K(\omega)]$ vanishes for $\omega_{0}\approx 4.4\times
10^{14}s^{-1}=440$ THz, what corresponds to red light of visible spectrum. It
is important to note that metamaterials with negative refractive index have
been observed in the red region of visible spectrum Smith:04 . More details
about the dispersion relation (2) and possible parameters of the medium can be
found in Ref. Kamli:08 and references therein.
We are now at the position to consider the interaction between the surface
polariton light fields and the medium quantum mechanically. Electric field of
each of the surface polaritons can be quantized near the surface and written
in the plane wave expansion as Scully:97
$\textbf{E}(\textbf{r},t)=\int{\rm d}k\left[{\bf
E}_{0}(k)\,a(k)\,e^{ikx-\omega t}+{\rm H.c.}\right]\,.$ (3)
Here we introduced $k\equiv{\rm Re}[K(\omega)]$, taking into account that the
wave vector is approximated by its real part $k(\omega)\approx{\rm
Re}[K(\omega)]$ in the low loss frequency range. The amplitude ${\bf
E}_{0}(k)$ can be found from the requirement that it should obey the field
Hamiltonian ${\rm H}_{F}=1/2\int d^{3}r\,[\tilde{\varepsilon}\,<|{\bf
E}|^{2}>+\tilde{\mu}<|{\bf H}|^{2}>]$ in a dispersive lossless medium Kamli:10
. It is important to note that our quantization procedure is applicable only
in a narrow frequency bandwidth where the losses are low. In general case, the
quantization of electromagnetic field in dispersive and absorptive media is a
much more complicated task Bhat:06 ; Chen:10 ; Ginzburg:12 .
Figure 2: (Color online) Configuration of a six level atom for creating double
electromagnetically induced transparency for two weak fields $E_{a}$ and
$E_{b}$. Here, the fields $E_{a}$ and $E_{b}$ are coupled with transitions
$\left|2\right\rangle\rightarrow\left|4\right\rangle$ and
$\left|2\right\rangle\rightarrow\left|6\right\rangle$ resonantly and with
transitions $\left|3\right\rangle\rightarrow\left|5\right\rangle$ and
$\left|1\right\rangle\rightarrow\left|5\right\rangle$ off resonantly with
detuning $\Delta$. Two classical control fields drive transitions
$\left|1\right\rangle\rightarrow\left|4\right\rangle$ and
$\left|3\right\rangle\rightarrow\left|6\right\rangle$ Petrosyan:02 .
The quantized polariton fields exhibit a remarkable property of confinement
along $z$ direction. The confinement can be quantified as $\xi(\omega)=1/{\rm
Re}[k^{\perp}(\omega)]$, where
$k^{\perp}(\omega)=\sqrt{K^{2}(\omega)-\omega^{2}\varepsilon_{1}\,\mu_{1}/c^{2}}$
is the normal component of the real part of the wave vector (2). This property
of the quantized fields ensure interaction of the polaritons with the six
level atoms embedded into the dielectric and defines effective thickness of
the atomic layer $\xi(\omega)\approx z_{0}$ in practice. The reason of
injection the six level atoms near the dielectric-metamaterial interface, is
that such atomic system has been shown to cause the effect of double
electromagnetically induced transparency simultaneously exhibiting symmetric
nonlinearity for two incoming pulses Petrosyan:02 . The energy levels of the
system are shown schematically in Fig. 2. The model of a six level atom can be
practically realized on the rubidium isotope ${}^{87}{\rm Rb}$ Petrosyan:02 ,
for example.
The interaction of the surface polaritons and the six level atoms can be
modeled with electric dipole hamiltonian ${\rm H}_{ED}=-\sum{\bf
d}_{i}\cdot{\bf E}({\bf r}_{i})$, where ${\bf E}({\bf r}_{i})$ is the electric
field (3) of the surface polaritons, ${\bf r}_{i}$ the position of the atoms
and the summation is to be done over all atoms in the interaction volume
Kamli:10 .
Because of the symmetry of the atomic levels structure the two surface
polaritons propagate in the medium with equal group velocities
$\upsilon_{a}=\upsilon_{b}\equiv\upsilon_{g}$ Petrosyan:02 . Dynamics of the
surface polariton field operators can be obtained in Heisenberg picture by
solving corresponding set of equations
$\left(\frac{1}{\upsilon_{g}}\,\frac{\partial}{\partial
t}+\frac{\partial}{\partial
x}\right)\,\textbf{E}_{n}(\textbf{r},t)=i\,\chi\,I_{m}\,\textbf{E}_{n}(\textbf{r},t)\,,$
(4)
where the adiabatic approximation has been used to ignore time derivatives of
higher order Kamli:10 . Here $n,m=a,b\,(n\neq m)$,
$I_{m}=|\textbf{E}_{m}(\textbf{r},t)|^{2}$ and $\chi$ is Kerr coefficient
given by
$\chi=\frac{2\pi
nz_{0}\,f[(k_{a}+k_{b}-k_{c})z_{0}]}{\hbar^{4}\upsilon_{g}|\Omega_{c}|^{2}\,\Delta}\,<|{\bf
d}_{24}{\bf E}_{a}|^{2}|{\bf d}_{26}{\bf E}_{b}|^{2}>\,,$ (5)
where $n$ is atomic density, $z_{0}$ is thickness of the atomic layer,
$f[x]\equiv(e^{-x}\sinh{x})/x$, $k_{a}$ and $k_{b}$ are the real parts of the
polaritons wave numbers, $k_{c}$ and $\Omega_{c}$ are the wave number and the
Rabi frequency of the driving field, $\upsilon_{g}$ is group velocity of the
polaritons ignoring the atomic layer, $\Delta$ stands for spectral detuning,
${\bf d}_{24}$ and ${\bf d}_{26}$ give atomic dipole moments of the
corresponding transitions, ${\bf E}_{a}$ and ${\bf E}_{b}$ are electric field
operators of the polaritons and $<...>$ denotes averaging over orientation of
the dipole moments. Typical atomic density in a gas is $2\times 10^{14}{\rm
cm}^{-3}$. To establish double electromagnetically induced transparency in
${}^{87}{\rm Rb}$, Rabi frequency of the control field is to be
$\Omega_{c}=1{\rm MHz}$, the transition wavelength is $780{\rm nm}$, the
detuning is $\Delta=1.4{\rm MHz}$ and the dipole moments are about $5ea_{0}$,
where $e$ is the electron charge $a_{0}$ is the Bohr radius. Assuming the
thickness of the atomic layer $z_{0}=2\mu{\rm m}$, we obtain Kerr nonlinearity
as displayed in Fig. 3.
Figure 3: (Color online) Kerr coefficient $\chi\times 10^{4}$ (dashed blue)
and corresponding mutual phase shift $\phi$ in units $\pi$ (solid red) as
functions of the field frequency $\omega/\omega_{0}$.
Although the Kerr nonlinearity $\chi$ is of order of $10^{-4}$, a significant
mutual phase shift of order of unity can be achieved between the surface
polaritons. The mutual phase shift is given by
$\phi=\chi\,\omega\,L\,/\upsilon_{g}$, where $\omega$ is the light frequency,
$L$ is the length of interaction in the medium and $\upsilon_{g}$ is the group
velocity of the light in the medium ignoring the layer of the five level
atoms. For chosen parameters of the medium $\upsilon_{g}\approx 0.4c$ and
assuming $L=1{\rm mm}$, the mutual phase shift is shown in Fig. 3. The surface
polaritons receive a mutual phase shift of order of $\pi$ at frequency
$\omega_{\pi}\approx 1.24\,\omega_{0}=545\,{\rm THz}$ (green light) which is
close to the no-loss frequency $\omega_{0}$.
Here we would like to point out that, because of the symmetry of the levels of
the six level atom, the refractive index of the medium is exactly the same for
two surface polaritons. That is why the polaritons propagate in the medium
with equal group velocities and experience identical nonlinearity.
Alternatively, five level atoms Wang:06 can be placed near dielectric-
metamaterial interface Moiseev:10 . In this case, two surface polaritons
propagate in the medium with different group velocities and experience
different nonlinearity in the double electromagnetically induced transparency
regime. Latter scheme is best suitable to achieve a uniform cross-phase
modulation Marzlin:10 .
The mutual (symmetric) phase shift can be used to create entanglement between
initially uncorrelated field modes. For single-mode incident fields, the
interaction of the surface polaritons in the (Kerr) medium can be described
with the help of an effective Hamiltonian
${\rm H}_{\rm eff}\,=\,\hbar\,\chi\,a^{\dagger}a\,b^{\dagger}b\,,$ (6)
where $a^{\dagger},\,b^{\dagger}$ and $a,\,b$ are creation and annihilation
operators of the field modes of the two polaritons. Time evolution of these
operators is initiated by the unitary transformation
$U(t)=\exp(-i\,\phi\,a^{\dagger}a\,b^{\dagger}b)$ and is given by
$a(t)=e^{-i\phi\,b^{\dagger}b}a(0)\,,\hskip
11.38092ptb(t)=e^{-i\phi\,a^{\dagger}a}b(0)\,.$ (7)
If the initial states of the incident fields are uncorrelated single-mode
coherent states $\left|\alpha\right\rangle$ and $\left|\beta\right\rangle$
Scully:97 , the final state $\left|\psi(t)\right\rangle_{ab}$ after the
interaction can be written in the Fock basis as Paternostro:03
$\left|\psi(t)\right\rangle_{ab}=e^{-\frac{|\beta|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\beta^{n}}{\sqrt{n!}}\,\left|\alpha\,e^{-i\phi
n}\right\rangle_{a}\otimes\left|n\right\rangle_{b}\,,$ (8)
where the dynamics of the creation and annihilation operators (7) has been
taken into account. Assuming $\phi=\pi$ and decomposing the sum in Eq. (8)
over odd and even values of the index $n$, we obtain the following form of the
final state
$\left|\psi\right\rangle^{f}_{ab}=\frac{1}{2}\left(\left|\alpha\right\rangle_{a}\left(\left|\beta\right\rangle_{b}+\left|-\beta\right\rangle_{b}\right)+\left|-\alpha\right\rangle_{a}\left(\left|\beta\right\rangle_{b}-\left|-\beta\right\rangle_{b}\right)\right)\,.$
This state is local unitary equivalent to the entangled state
$\left(\left|\alpha\right\rangle_{a}\left|\beta\right\rangle_{b}+\left|-\alpha\right\rangle_{a}\left|-\beta\right\rangle_{b}\right)/\sqrt{N}$
Paternostro:03 , where $N=2-2\exp(-2|\alpha|^{2}-2|\beta|^{2})$, which is
known to preserve exactly one e(ntangled )bit of quantum information Enk:01 .
In contrast to our assumption above, the initial states
$\left|\psi\right\rangle_{a}$ and $\left|\psi\right\rangle_{b}$ of the
incident field can be also assumed to be polarization states of photons. In
this case, entanglement of the field modes can be achieved, for example, with
the help of Nemoto-Munro protocol Nemoto:04 . To understand this protocol
better, let us assume that the incident field $a$ is prepared into a
superposition of vacuum and a single photon states and is given by
$\left|\psi\right\rangle_{a}=c_{0}\left|0\right\rangle+c_{1}\left|1\right\rangle$
in Fock basis. The second incident field is prepared into a coherent state
$\left|\alpha\right\rangle_{b}$. When the two fields interact with the media,
the resulting state of the fields is given by
$U(t)\,\left|\psi\right\rangle_{a}\left|\alpha\right\rangle_{b}=c_{0}\left|0\right\rangle\left|\alpha\right\rangle_{b}+c_{1}\left|1\right\rangle\left|\alpha\,e^{i\phi}\right\rangle_{b}\,.$
(9)
The state of the field $a$ is unaffected by the interaction, while the state
of the field $b$ receives a phase shift, which is proportional to the number
of photons on the the state $\left|\psi\right\rangle_{a}$.
Assume, we have two polarization qubits to become entangled. The qubits are
initially prepared in single-photon superposition states
$\left|\psi\right\rangle_{a}=c_{0}\left|H\right\rangle+c_{1}\left|V\right\rangle$
and
$\left|\psi\right\rangle_{b}=d_{0}\left|H\right\rangle+d_{1}\left|H\right\rangle$,
where $\left|V\right\rangle$ and $\left|H\right\rangle$ are polarization
degrees of freedom. These qubits are split individually on polarizing beam
splitters into spatial modes and interact with an additional probe beam (which
is in a coherent state $\left|\alpha\right\rangle_{p}$) in the Kerr medium.
The resulting state of the three beams is given by
$\displaystyle\left|\psi\right\rangle_{abc}$ $\displaystyle=$
$\displaystyle\left(c_{0}d_{0}\left|HH\right\rangle+c_{1}d_{1}\left|VV\right\rangle\right)\left|\alpha\right\rangle_{p}$
(10) $\displaystyle+\,c_{0}d_{1}\left|HV\right\rangle\left|\alpha
e^{i\phi}\right\rangle_{p}+c_{1}d_{0}\left|VH\right\rangle\left|\alpha
e^{-i\phi}\right\rangle.$
The first term in this expression does not receive any phase shift, while the
the second and the third terms receive opposite sign shifts. This makes
possible to transform the three-party state into entangled (Bell) bipartite
states by performing a homodyne measurement on the probe. The measurement
results into either
$c_{0}d_{0}\left|HH\right\rangle+c_{1}d_{1}\left|VV\right\rangle$ or
$c_{0}d_{1}\left|HV\right\rangle+c_{1}d_{0}\left|VH\right\rangle$ states,
which are both maximally entangled states of qubits for
$c_{0}=c_{1}=d_{0}=d_{1}=1/\sqrt{2}$. It is also important to note that the
described above Nemoto-Munro protocol allows us to construct entangling
Controlled-NOT gate Kok:07 with large Kerr nonlinearity, opening prominent
possibility to use metamaterials in quantum computing.
Presented scheme for entanglement creation with negative index metamaterials
may also find its applications in quantum communication and quantum
teleportation with both coherent Braunstein:05 ; Enk:01 and polarization
states Kok:07 . Moreover, Kerr nonlinearity, created with the described
medium, can be used to generate multimode entangled coherent states Enk:03
and multiphoton Greenberger-Horne-Zeilinger states Jin:07 .
We also would like to outline that in the present discussion we restricted
ourselves with TM polarization of surface polaritons. As we mentioned before,
both TM and TE polarizations may exist on the dielectric-metamaterial
interface. These polarizations may be used for information encoding on a par
with encoding in quantum states of the field quanta. Another attractive idea
is to use a trade-off between confinement and losses of the surface polaritons
Kamli:10 ; Moiseev:10 . This trade-off may be used to establish two regimes,
corresponding to “manipulation” and “low-loss transmission”, which are highly
desired in quantum computation Ladd:10 . Both mentioned possibilities will be
the subject of further investigations.
In conclusion, we presented the scheme for entanglement creation with the
composite medium consisting of the dielectric and the negative index
metamaterial. The surface polaritons, which are created by the incident light
in the medium, propagate along the dielectric-metamaterial interface with
substantially reduced group velocity, exhibiting property of spatial
confinement and with suppressed amplitude losses. Placing a layer of six level
atoms near the interface allowed us to establish symmetric nonlinear
interaction between the surface polaritons, which can be utilized to create
entanglement between initially uncorrelated coherent or polarization states of
light.
## References
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|
arxiv-papers
| 2012-05-21T07:37:09 |
2024-09-04T02:49:31.128334
|
{
"license": "Public Domain",
"authors": "Michael Siomau, Ali A. Kamli, Sergey A. Moiseev and Barry C. Sanders",
"submitter": "Michael Siomau",
"url": "https://arxiv.org/abs/1205.4506"
}
|
1205.4548
|
arxiv-papers
| 2012-05-21T10:14:19 |
2024-09-04T02:49:31.134722
|
{
"license": "Public Domain",
"authors": "Vincent Dubost, Tristan Cren, Fran\\c{c}ois Debontridder, Dimitri\n Roditchev, Cristian Vaju, Vincent Guiot, Laurent Cario, Beno\\^it Corraze,\n Etienne Janod",
"submitter": "Tristan Cren",
"url": "https://arxiv.org/abs/1205.4548"
}
|
|
1205.4579
|
# Theoretical study of magnetic domain walls through a cobalt nanocontact
László Balogh Department of Theoretical Physics, Budapest University of
Technology and Economics, H-1111 Budapest, Hungary Condensed Matter Research
Group of the Hungarian Academy of Sciences, Budapest University of Technology
and Economics, H-1111 Budapest, Hungary Krisztián Palotás Department of
Theoretical Physics, Budapest University of Technology and Economics, H-1111
Budapest, Hungary László Udvardi udvardi@phy.bme.hu Department of
Theoretical Physics, Budapest University of Technology and Economics, H-1111
Budapest, Hungary Condensed Matter Research Group of the Hungarian Academy of
Sciences, Budapest University of Technology and Economics, H-1111 Budapest,
Hungary László Szunyogh Department of Theoretical Physics, Budapest
University of Technology and Economics, H-1111 Budapest, Hungary Condensed
Matter Research Group of the Hungarian Academy of Sciences, Budapest
University of Technology and Economics, H-1111 Budapest, Hungary Ulrich Nowak
Department of Physics, University of Konstanz, 78457 Konstanz, Germany
(August 27, 2024)
###### Abstract
To calculate the magnetic ground state of nanoparticles we present a self-
consistent first principles method in terms of a fully relativistic embedded
cluster multiple scattering Green’s function technique. Based on the
derivatives of the band energy, a Newton-Raphson algorithm is used to find the
ground state configuration. The method is applied to a cobalt nanocontact that
turned out to show a cycloidal domain wall configuration between oppositely
magnetized leads. We found that a wall of cycloidal spin-structure is about 30
meV lower in energy than the one of helical spin-structure. A detailed
analysis revealed that the uniaxial on-site anisotropy of the central atom is
mainly responsible to this energy difference. This high uniaxial anisotropy
energy is accompanied by a huge enhancement and anisotropy of the orbital
magnetic moment of the central atom. By varying the magnetic orientation at
the central atom, we identified the term related to exchange couplings (Weiss-
field term), various on-site anisotropy terms, and also those due to higher
order spin-interactions.
## I Introduction
As magnetic storage devices approach a physical limit of a single atom, the
investigation of nanoclusters has become one of the most important subjects in
magnetism. Recent developments in nanotechnology permit the construction of
clusters with well-controlled structures and enable the measurement of various
magnetic properties at the atomic scale. Probing the Kondo resonance in terms
of low temperature scanning tunneling spectroscopy Heinrich et al. Heinrich
_et al._ (2004) determined the spin-flip energy of a single manganese atom on
a nonmagnetic substrate, while Wahl et al. Wahl _et al._ (2007) were able to
estimate the exchange coupling between Co atoms on a Cu(001) surface. Atomic
scale contacts can be fabricated by using electromigrated break junctions
where the size of a macroscopic contact between two leads can be reduced down
to a single atom. Néel et al. Néel _et al._ (2007) studied the transition
from the tunneling to the contact regime by moving the STM tip closer to the
surface adatom, and an enhanced Kondo temperature was found. In conjunction
with the Kondo effect, Calvo et al. Calvo _et al._ (2009) found a Fano
resonance for ferromagnetic point contacts indicating that the reduced
coordination can dramatically effect the magnetic behavior of nanoclusters.
Experiments on atomic-sized contacts of ferromagnetic metals generated by
mechanically controllable break junction (MCBJ) revealed magnetoresistance
(MR) effects of unprecedented size. Chopra and Hua (2002); Viret _et al._
(2002, 2006) There are various mechanisms to this huge MR discussed in the
literature: depending, e.g., on the micromagnetic order of the sample
controlled by the size of the applied field, atomically enhanced anisotropic
MR (AAMR), giant MR (GMR), tunnel MR (TMR) or ballistic MR (BMR) effects can
be established. Egle _et al._ (2010) In particular, based on ab initio
calculations, the AAMR has been shown to emerge in wire-like transition metal
nanocontacts and related to the giant orbital moment formed at the central
atom. Autès _et al._ (2008)
Ab-initio calculations on magnetic nanostructures are useful for a clear
interpretation of experimental results and to attain better understanding of
the underlying physical phenomena. Several methods to determine complex
magnetic ground states of nanoparticles from first principles are based on a
fully unconstrained local spin-density approximation (LSDA) implemented within
the full-potential linearized augmented plane-wave (FLAPW) methodKurz _et
al._ (2001) or the projector augmented-wave (PAW) method.Hobbs _et al._
(2000) Unconstrained non-collinear magnetic calculations are also performed
within a tight-binding approach,Robles and Nordström (2006) using the tight-
binding linearized muffin-tin orbital (TB-LMTO) method Bergman _et al._
(2007a, b) or the Korringa-Kohn-Rostoker (KKR) method.Yavorsky and Mertig
(2006) Spin-orbit coupling (SOC) has an important role in the formation of
different magnetic states via magnetocrystalline anisotropy and Dzyaloshinsky-
Moriya (DM) interactions. Bode _et al._ (2007) SOC is usually treated as
perturbation or by directly solving the Dirac equation. The latter concept is
applied in studies relying on ab-initio spin-dynamics in terms of a
constrained LSDA by means of a fully relativistic KKR method. Újfalussy _et
al._ (2004); Lazarovits _et al._ (2004); Stocks _et al._ (2007)
In bulk ferromagnets the formation of a domain wall is governed by a
competition between the exchange and anisotropy energies Bloch (1932) and the
typical interface between the magnetic domains is the Bloch wall where the
magnetization remains perpendicular to the axis of the wall. In thin films
with easy plane anisotropy, a Néel wall is formed with atomic magnetic moments
lying in the plane of the film, however, DM interactions can give rise to
domain walls with out-of-plane magnetization and well-defined rotational
sense. Heide (2006); Heide _et al._ (2008) In a geometrically constrained
system the structure of a domain wall is mainly determined by the geometry
irrespective of the exchange and anisotropy energies.Bruno (1999) Thermal
effects play an additional role and can lead to new types of domain walls
beyond the usual restriction of constant magnetization magnitude.Kazantseva
_et al._ (2005) However, for a deeper understanding of the magnetic properties
of nanocontacts, models based on first princples calulations are of pronounced
importance.
In the present work, a domain wall through a point-contact between (001)
surfaces of fcc Co is studied, where the magnetizations are aligned in the
(110) and the ($\overline{1}\overline{1}$0) directions in the leads. It should
be noted that Co exhibits a hcp structure in bulk, however, as thin film it
often displays a fcc-related geometry. We apply a fully relativistic embedded
cluster Green’s function technique based on the KKR method (EC-KKR).Lazarovits
_et al._ (2002) Using gradients and second derivatives of the band energy
related to the transverse magnetization, a self-consistent Newton-Raphson
method is developed to find the ground state configuration of the domain wall.
An enhancement of the magnetic anisotropy energy has been established
theoretically in atomic scale junctions even for elements that are nonmagnetic
in bulk. Thiess _et al._ (2010) In agreement with this finding, our results
reveal that the central atom with the lowest coordination number has the main
contribution to the magnetic anisotropy of the contact. To highlight the
relationship between the obtained cycloidal domain wall configuration and the
magnetic anisotropy, the orientational dependence of the band energy of the
point-contact is analyzed in details.
## II Computational details
Our model of the atomic-sized point contact has been built from Co atoms
forming two identical pyramids facing each other between (001) interfaces of
fcc Co as it is shown in Fig. 1(a). The distance between the central atom and
its neighbors was chosen identical to the fcc nearest neighbor distance, $a$,
of 2.506 Å. Note that this geometrical model is the same as the one labelled
by C2 in Ref. Autès _et al._ , 2008, except that they studied a break-
junction between bcc Fe surfaces. In order to mimic the contraction and
expansion of the contact, the normal to plane distances in the vicinity of the
central atom have been scaled by a factor, hereinafter denoted by $x$, between
0.85 and 1.15, see Fig. 1(a). A host system assembled of two oppositely
magnetized semi-infinite Co leads and separated by 7 layers of empty spheres
(vacuum) is considered. The embedded cluster in the EC-KKR calculations
consisted of 29 ($9+4+1+1+1+4+9$) Co atoms forming the contact by substituting
empty spheres in the vacuum layers, $16+16$ Co atoms from the Co surfaces
adjacent to the contact, and we also included 80 empty spheres in the vicinity
of the Co atoms in the contact to let the electron density relax around the
cluster, see Fig. 1(b).
Figure 1: (Color online) (a) The geometry of the contact viewed from the
$(1\overline{1}0)$ direction. The leads are depicted as dark (blue)
rectangles, the cobalt atoms forming the contact are represented by gray
(orange) circles, $a$ denotes the nearest neighbor distance in the fcc
structure. The length of the contact is tuned via $x=$ 0.85, 0.90, 0.95, 1.00,
1.05, 1.10, and 1.15. Note that only the marked distances were scaled. (b)
Sketch of the embedded cluster. Dark (blue) circles: selected atoms of the
cobalt leads, gray (orange) circles: cobalt atoms in the nanocontact, and
empty circles: empty spheres around the contact. The directions of
magnetization in the leads are marked by dark (blue) arrows.
First, the electronic structure of the host was calculated in terms of the
fully relativistic screened KKR method applying the surface Green’s function
technique.Szunyogh _et al._ (1994a, b) Then the electronic structure of the
contact has been determined within the EC-KKR method,Lazarovits _et al._
(2002) in which the scattering path operator (SPO), corresponding to a finite
cluster, $\mathcal{C}$, embedded into a host system can be obtained from the
following equation,
$\bm{\tau}_{\mathcal{C}}(\varepsilon)=\left(\mathbf{t}_{\mathcal{C}}^{-1}(\varepsilon)-\mathbf{t}_{\text{h}}^{-1}(\varepsilon)+\bm{\tau}_{\text{h}}^{-1}(\varepsilon)\right)^{-1},$
(1)
where $\mathbf{t}_{\text{h}}(\varepsilon)$ and
$\bm{\tau}_{\text{h}}(\varepsilon)$ denote the single-site scattering matrix
and the SPO matrix for the host confined to the sites in $\mathcal{C}$,
respectively, while $\mathbf{t}_{\mathcal{C}}$ denotes the single-site
scattering matrices of the embedded atoms. The calculations for both the host
and the cluster were performed within the local spin-density approximation
(LSDA),Vosko _et al._ (1980) by using the atomic sphere approximation (ASA)
and $\ell_{\text{max}}=2$ for the angular momentum expansion.
A fully unconstrained extension of the relativistic EC-KKR method is used to
find the magnetic configuration of the point contact. The evolution of the
atomic magnetic moments is treated in a semi-classical manner similar to
molecular dynamics, whereby, in spirit of the magnetic force theorem,Jansen
(1999) the driving force is calculated as the derivative of the band energy,
$E_{\text{b}}=\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\left(\varepsilon-\varepsilon_{\text{F}}\right)n(\varepsilon)\,\mathrm{d}\varepsilon=-\int\limits_{-\infty}^{\varepsilon_{\text{F}}}N(\varepsilon)\,\mathrm{d}\varepsilon,$
(2)
with respect to the transverse change of the exchange field, where
$\varepsilon_{\text{F}}$ is the Fermi energy, while $n(\varepsilon)$ and
$N(\varepsilon)$ stand for the density of states (DOS) and for the integrated
DOS, respectively. In the multiple scattering formalism the exchange field
enters the electronic structure via the single-site scattering matrix,
$t_{i}$. The first and higher order changes of the $t_{i}$ matrices as well as
the derivatives of the band energy can straightforwardly be calculated in the
local frame of reference introduced at all sites of the cluster, where the
direction vector $\bm{\sigma}_{i}$ of the magnetization at site $i$, and the
two transverse vectors, $\mathbf{e}_{i1}$ and $\mathbf{e}_{i2}$, form a right
handed coordinate system as shown in Fig. 2. The first and second order change
of the single site scattering matrix at site $i$ with respect to rotations by
$\Delta\phi_{i\alpha}$ around the transverse axes $\mathbf{e}_{i\alpha}$ can
be given by the following commutator formulas,
$\displaystyle\Delta t_{i}^{(1)}$
$\displaystyle=i[\mathbf{e}_{i\alpha}\mathbf{J},t_{i}]\Delta\phi_{i\alpha},$
(3) $\displaystyle\Delta t_{i}^{(2)}$
$\displaystyle=-[\mathbf{e}_{i\alpha}\mathbf{J},[\mathbf{e}_{i\beta}\mathbf{J},t_{i}]]\Delta\phi_{i\alpha}\Delta\phi_{i\beta},$
(4)
where $\mathbf{J}$ is the matrix representation of the total angular momentum
operator and $\alpha,\beta\in\\{1,2\\}$. Following Ref. Udvardi _et al._ ,
2003, the first and second derivatives of the band energy can then be
expressed as
$\displaystyle\frac{\partial E_{\text{b}}}{\partial\phi_{i\alpha}}$
$\displaystyle=\frac{1}{\pi}\,\mathrm{Re}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ii}\left[\mathbf{e}_{i\alpha}\mathbf{J},m_{i}\right]\right\\}\,\mathrm{d}\varepsilon,$
(5)
$\displaystyle\frac{\partial^{2}E_{\text{b}}}{\partial\phi_{i\alpha}\partial\phi_{j\beta}}$
$\displaystyle=-\frac{1}{\pi}\mathrm{Im}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ij}[\mathbf{e}_{j\beta}\mathbf{J},m_{j}]\tau_{ji}[\mathbf{e}_{i\alpha}\mathbf{J},m_{i}]\right\\}\,\mathrm{d}\varepsilon$
$\displaystyle+\delta_{ij}\frac{1}{\pi}\mathrm{Im}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ii}[\mathbf{e}_{i\alpha}\mathbf{J},[\mathbf{e}_{i\beta}\mathbf{J},m_{i}]]\right\\}\,\mathrm{d}\varepsilon,$
(6)
where $m_{i}=t_{i}^{-1}$ and $\tau_{ij}$ is the block of the SPO matrix
between sites $i$ and $j$. Note that for brevity we dropped the energy
arguments of the corresponding matrices in Eqs. (3–II). In the spirit of a
gradient minimization, rotating the exchange field by a small amount around
the torque vector at each sites,
$\mathbf{T}_{i}=\mathbf{e}_{i1}\frac{\partial
E_{\text{b}}}{\partial\phi_{i1}}+\mathbf{e}_{i2}\frac{\partial
E_{\text{b}}}{\partial\phi_{i2}},$ (7)
the magnetic configuration gets closer to the local minimum of the energy,
however, the convergence is very slow. In order to speed up this procedure, a
Newton-Raphson iteration scheme has been applied, where the inverse of the
second derivative tensor, also referred to as the Hessian, Eq. (II), is used
to estimate the angle of rotations around the torque vector given by Eq. (7).
The eigenvalues of the Hessian also provide information about the stability of
the a configuration with zero torque: if the Hessian is a positive or negative
definite matrix then the given configuration is stable or unstable state of
equilibrium, respectively. Once the Newton-Raphson iteration has converged,
new effective potentials and exchange fields are generated and the procedure
is repeated until the effective potential converged and the torque in Eq. (7)
is decreased below a predefined value of, typically, $10^{-4}$ meV.
Figure 2: (Color online) Sketch of the local frame of reference: the unit
vector $\bm{\sigma}_{i}$ is parallel to the magnetization at site $i$, while
the unit vectors $\mathbf{e}_{i1}$ and $\mathbf{e}_{i2}$ point into the
transverse directions. Rotations around these axes by $\phi_{i1}$ and
$\phi_{i2}$ are also indicated.
The starting magnetic configuration for the above optimization procedure has
been determined by Monte Carlo simulated annealing based on a simple isotropic
Heisenberg model, $\mathcal{H}=\frac{1}{2}\sum_{i\neq
j}J_{ij}\bm{\sigma}_{i}\bm{\sigma}_{j}$, where $J_{ij}$ is the isotropic
exchange coupling between sites $i$ and $j$. The coupling coefficients between
the atomic moments were calculated by using the torque method proposed by
Liechtenstein et al. Liechtenstein _et al._ (1987) The exchange couplings
have been calculated in a ferromagnetic spin-configuration parallel to the
(100) direction.
In order to avoid the difficulties arising from the continuous degeneracy of
the spin-states in a Heisenberg model, the magnetization on the central atom
was fixed normal to the bulk magnetization. Considering the inversion symmetry
of the point-contact, only the (1$\overline{1}$0) and the (001) directions are
consistent with the (constrained) magnetic ground-state of the system. In the
first case, the magnetic moments at all sites (layers) remain within the (001)
plane, i.e., normal to the axis of the point-contact, therefore, in the
following this spin-configuration will be termed as a helical domain wall. In
the second case, all the spin moments are confined to the (1$\overline{1}$0)
plane, thus, we shall call this case the cycloidal domain wall. Note that the
helical and cycloidal spin-configurations closely resemble the Bloch and Néel
types of domain walls well-known in bulk and thin-film magnets, respectively.
Since, however, these types of domain walls are distinct through the
magnetostatic energy, to avoid confusion we skipped using the traditional
terminology.
## III Results and discussions
### III.1 Domain wall configurations
Self-consistent potentials and exchange fields have been first determined for
both the cycloidal and the helical domain walls and the Newton-Raphson
iterations were started from both initial configurations. Interestingly, when
starting from a helical spin-configuration, the gradients, Eq. (5), were
initially zero, but the Hessian had a negative eigenvalue indicating that the
helical spin-configuration belonged to a saddle point of the energy surface.
Throwing the system off this saddle point, the Newton-Raphson iterations
converged to the cycloidal spin-configuration. Thus, independent of the
starting configuration, the magnetic state of the nanojunction converged to
the cycloidal wall structure for the stretching range considered. In Fig. 3
the ground-state cycloidal wall configuration is displayed for $x=1$.
Figure 3: (Color online) The cycloidal spin-configuration obtained for the
unstretched contact ($x=1$). The lengths of the arrows, indicated also with
color coding, are proportional to the size of the spin magnetic moments.
At sites within the same geometrical layer, we obtained fairly similar
orientations for the magnetic moments, therefore, the shape of the domain wall
can well be characterized by orientations determined as an average within
layers. In Fig. 4 such a profile is shown for $x=1$ in terms of polar angles,
$\vartheta(z)$. Remarkably, the well-known analytical form,
$\vartheta(z)=-\frac{\pi}{2}\tanh(2z/d_{\text{w}})$ could be well fitted
defining, thus, the width of the domain wall, $d_{\text{w}}$. This fit is also
shown in Fig. 4. We note that following Ref. Kazantseva _et al._ , 2005 the
analytical form of a constrained wall profile should be better described by
Jacobian sine functions. However, testing this alternative approach resulted
in to a relative deviation of less than 0.5 % in the fitted domain wall
thicknesses.
Figure 4: Polar angles averaged within a layer of cobalt atoms in the contact
with $x=1$ as the function of the distance from the central atom (in units of
the fcc nearest neighbor distance, $a$). The solid curve displays the fit,
$\vartheta(z)=-\frac{\pi}{2}\tanh(2z/d_{\text{w}})$. Figure 5: Width of the
domain walls through the point contact as a function of the stretching factor,
$x$. Note that $d_{\text{w}}(1.00)=2.34\,a$, where $a$ is the fcc nearest
neighbor distance. The solid line stands for the identity function.
The change of the width of the domain walls against the length of the point
contact is shown in Fig. 5. For a clear interpretation, the width of the walls
is normalized to the width of the domain wall for $x=1$. As obvious from this
figure, $d_{\text{w}}(x)\approx x\,d_{\text{w}}(1.00)$ demonstrating that the
width of the domain walls follows the length of the point contact. In case of
Fe20Ni80 thin films it has been experimentally found that the constrained
geometry can reduce the width of the Néel wall.Jubert _et al._ (2004) The
effect is more pronounced in ultrathin films of few atomic layers where the
width of the domain wall can be as small as few nanometers in the vicinity of
a step edge.Pietzsch _et al._ (2000) The effect of the reduced dimensionality
is even more obvious in the case of a point contact. Since the exchange energy
gain for the few atoms of the contact is small compared to the increase of the
exchange energy of the leads, the domain wall can not penetrate into the
substrates and the wall is confined to the contact. The same conclusion has
been drawn by BrunoBruno (1999) based on a theoretical study of a continuous
model of domain walls in a confined geometry.
### III.2 Magnetic moments
The low coordination in thin films and in nanostructures is often accompanied
by the enhancement of the atomic spin and orbital moments. In Fig. 6 the
calculated values of the local spin and orbital moments are given in a point
contact with cycloidal wall configuration and stretching factor, $x=1$. Since
the orbital moment is found almost parallel to the spin-moment at each site,
we presented the projection of the orbital moment to the local spin
quantization axis. Since the contact has a mirror symmetry with respect to the
horizontal plane including the central atom, therefore, the moments in only
one half of the contact are displayed. Our data fit nicely to the observation
reported in Refs. Šipr _et al._ , 2007 and Błoński and Hafner, 2009 that the
spin and orbital moments at sites with lower coordination number are larger
then at sites with larger coordination number. This is, in particular, true
for the central atom with coordination number of only two where the values of
the spin and orbital moments are even larger than those obtained for small
clusters on Pt(111) and Au(111) surfaces.Šipr _et al._ (2007); Błoński and
Hafner (2009); Lazarovits _et al._ (2003)
$\mu_{\text{spin}}\leavevmode\nobreak\ (\mu_{\text{B}})$
$\mu_{\text{orb}}\leavevmode\nobreak\ (\mu_{\text{B}})$
Figure 6: Calculated atomic magnetic moments ($\mu_{B}$) in half of the
nanocontact for the stretching factor, $x=1$. In the upper and lower panels
shown are the spin and orbital moments, $\mu_{\text{spin}}$ and
$\mu_{\text{orb}}$, respectively. For comparison, the spin-moments at the Co
surface and in the bulk are $1.82\,\mu_{\text{B}}$ and
$1.67\,\mu_{\text{B}}/\text{atom}$, while the corresponding values of the
orbital moments are $0.14\,\mu_{\text{B}}$ and $0.08\,\mu_{\text{B}}$.
Fig. 7 shows the spin and orbital moments of the central atom as a function of
the stretching ratio $x$, for both the cycloidal and the helical spin-
configurations in the point-contact. Clearly, the spin moments are fairly
insensitive to the domain wall configuration: this can easily be understood as
the relative spin-directions are nearly the same in the two types of domain
walls. Also, there is only a moderate change of the spin moment in the range
of $2.35\,\mu_{\text{B}}\leq\mu_{\text{spin}}\leq 2.49\,\mu_{\text{B}}$ for
the stretching ratios under consideration. These values compare well to
$\mu_{\text{spin}}=2.15\,\mu_{\text{B}}$ and
$\mu_{\text{spin}}=2.26\,\mu_{\text{B}}$ calculated for a single Co adatom on
Pt and Au(111) surfaces in Refs. Šipr _et al._ , 2007 and Błoński and Hafner,
2009, respectively.
The dependence of the orbital moment of the central atom on the stretching is
more pronounced than that of the spin moment: in case of a cycloidal and a
helical wall it increases from about $1\,\mu_{B}$ to $2\,\mu_{B}$ and from
$0.3\,\mu_{B}$ to $1.5\,\mu_{B}$, respectively. Similar high values of
$\mu_{\text{orb}}$ for the central atom of a wire-like Fe point-contact were
reported in Ref. Autès _et al._ , 2008 and attributed to localized atomic-
like electronic states treated within a full Hartree-Fock scheme. It should be
mentioned that for a more reliable description of highly localized states, the
plain LSDA we used in our calculations should be extended with, e.g., the
local self-interaction correction, LSDA+SIC Lüders _et al._ (2005) or the
dynamical mean field theory, LSDA+DMFT. Kotliar _et al._ (2006)
Apparently, the orbital moment of the central atom is systematically larger in
a cycloidal wall than in a helical wall. This can be understood since these
orbital moments correspond to different directions: in case of a cycloidal
wall it points along the (001) directions, while, for a helical wall, along
the (1$\overline{1}$0) direction. Such a huge anisotropy of the orbital moment
at the central atom has also been observed in Ref. Autès _et al._ , 2008.
According to Bruno’s theory Bruno (1989) this large orbital momentum
anisotropy is related to a large magnetic anisotropy energy featuring the
(001) direction as easy axis, which clearly corroborates our result for the
preference of a cycloidal domain wall.
Figure 7: The spin- and orbital moments of the central atom as a function of
the stretching. Spin moments are displayed by open symbols, orbital moments
are displayed by filled symbols as calculated in the cycloidal wall (CW,
squares) and in the helical wall (HW, triangles) configurations.
### III.3 Rotational energy of the domain wall
The cycloidal and helical spin-configurations of the point contact can be
transformed into each other in term of a simultaneous rotation of the spin-
directions around the axis parallel to the magnetization of the leads. The
energy along the path of this global rotation, termed as the rotational energy
of the domain wall, was calculated using the magnetic force theorem, namely,
from the band energy of the system by rotating the orientation of the exchange
field at each atomic site around the (110) axis and keeping frozen the
effective potentials and fields as obtained for the ground state cycloidal
wall configuration. For the case of the unstretched configuration the results
are plotted in Fig. 8. The two minima and maxima of the band energy belong to
the two-fold degenerate cycloidal and helical domain wall configurations. The
height of the energy barrier between the two ground state cycloidal spin-
configurations is 32.0 meV. Similar behavior has been found for the whole
stretching range of the point contact. The energy differences between the two
types of domain wall as a function of the stretching ratio are displayed by
diamonds in Fig. 9.
Figure 8: The band energy of the nanocontact with $x=1.00$ while rotating the
exchange field at each atomic sites simultaneously around the (110) axis. By
rotating all the spins by $90^{\circ}$ the system goes over from the cycloidal
wall (CW) into the helical wall (HW). The dashed line denotes the leading
Fourier component of the band energy, $-15.2\;[\mathrm{meV}]\cos(2\theta)$,
see Eq. (8). Note that we shifted the zero level of the energy to the constant
term, $K_{0}$.
Due to time reversal symmetry, the magnetic anisotropy energy has a
periodicity of $\pi$, but it does not comply with a usual $\cos^{2}(\theta)$
dependence. To explore this deviation we performed the Fourier expansion,
$E_{\text{b}}(\theta)=K_{0}+\sum_{k=2,4,\dots}^{\infty}{K_{k}\cos(k\theta)}\,,$
(8)
for the contacts with different stretching. Note that because of the inversion
symmetry of the contact $E_{\text{b}}(\theta)=E_{\text{b}}(\pi-\theta)$
applies, therefore, the $\sin(k\theta)$ ($k=2,4,\dots)$ terms do not appear in
the expansion, Eq. (8). We summarized the Fourier coefficients, $K_{k}$, in
Table 1. We found that in each case the term $K_{2}\cos(2\theta)$ adds the
largest weight to the rotational energy of the domain wall. The next term
$K_{4}\cos(4\theta)$ is quite significant for $x\geq 0.95$, but it drops for
smaller stretching. Interestingly, in the stretching range of $x\leq 0.90$ the
$k=6$ term overweights the one of $k=4$, whereas in the complementary range
the $k=6$ term is negligible. It should be mentioned that the $k\geq 8$ terms
of the Fourier expansion have practically vanishing weight.
Table 1: The $k=2$, $4$ and $6$ Fourier coefficients (in units of meV) of the rotational energy of the point-contact, Eq. (8), as a function of the stretching parameter, $x$. $x$ | $k=2$ | $k=4$ | $k=6$
---|---|---|---
$0.85$ | $-6.3$ | $0.15$ | $0.397$
$0.90$ | $-10.0$ | $0.36$ | $0.499$
$0.95$ | $-13.6$ | $1.40$ | $0.298$
$1.00$ | $-15.2$ | $2.17$ | $-0.040$
$1.05$ | $-15.1$ | $2.35$ | $-0.122$
$1.10$ | $-14.4$ | $2.24$ | $-0.083$
$1.15$ | $-13.2$ | $1.92$ | $0.025$
### III.4 Magnetic anisotropy of the central atom
As we have seen in Sec. III.2, the central atom of the contact exhibits a huge
orbital moment anisotropy that should be accompanied by a large magnetic
anisotropy energy. For that reason, we analyze the band energy of the point-
contact, $E_{\text{b}}(\bm{\sigma})$, with $\bm{\sigma}$ denoting the spin-
orientation at the central atom, whereas the spin-orientations of all the
other sites in the contact are kept fixed as obtained in the ground-state
cycloidal wall configuration.
Our analysis is based on an expansion of $E_{\text{b}}(\bm{\sigma})$ in terms
of (real) spherical harmonics, $R_{\ell}^{m}(\bm{\sigma})$,
$E_{\text{b}}(\bm{\sigma})=\sum_{\ell,m}K_{\ell}^{m}R_{\ell}^{m}(\bm{\sigma})\,,$
(9)
with the angular momentum indices, $\ell=0,1,2,\dots$ and $-\ell\leq
m\leq\ell$. Similar to the rotational energy of the domain wall, we used the
magnetic force theorem to evaluate $E_{\text{b}}(\bm{\sigma})$, but here we
employed Lloyd’s formula, Lloyd (1967) since it accurately accounts for the
change of the band energy of the whole point-contact with respect to the
change of the spin-orientation at the central site. For the expansion, the
integration over $\bm{\sigma}$ was performed using a 51 points Gaussian
quadrature along the $z$-direction and a uniform mesh of 100 points in the
azimuth angle, resulting in a spherical grid of 5100 points. The obtained
coefficients are summarized in Table 2 up to $\ell=4$ and for all the
stretching ratios under consideration. Only the non-vanishing coefficients are
presented, for clarity, together with the definition of the corresponding
spherical harmonics, $R_{\ell}^{m}(\bm{\sigma})$.
Table 2: Expansion coefficients $K_{\ell}^{m}$ (in units of meV) of the band energy of the contact, see Eq. (9), according to real spherical harmonics $R_{\ell}^{m}$ up to $\ell=4$. $\ell$ | $m$ | $R_{\ell}^{m}\vphantom{\sqrt{\frac{1}{2}}}$ | $x=0.85$ | $x=0.90$ | $x=0.95$ | $x=1.00$ | $x=1.05$ | $x=1.10$ | $x=1.15$
---|---|---|---|---|---|---|---|---|---
1 | 0 | $\frac{1}{2}\sqrt{\frac{3}{\pi}}z$ | $-240$ | $-247$ | $-235$ | $-212$ | $-192$ | $-176$ | $-159$
2 | 0 | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\left(3z^{2}-1\right)$ | $-25.3$ | $-30.0$ | $-33.2$ | $-32.4$ | $-30.9$ | $-28.4$ | $-25.6$
2 | 2 | $\frac{1}{4}\sqrt{\frac{15}{\pi}}\left(x^{2}-y^{2}\right)$ | $4.30$ | $2.54$ | $1.39$ | $0.51$ | $-0.29$ | $-0.92$ | $-1.36$
3 | 0 | $\frac{1}{4}\sqrt{\frac{7}{\pi}}\left(5z^{3}-3z\right)$ | $4.12$ | $3.06$ | $1.63$ | $0.71$ | $-0.28$ | $-1.43$ | $-2.67$
3 | 2 | $\frac{1}{4}\sqrt{\frac{105}{\pi}}\left(x^{2}-y^{2}\right)z$ | $-0.199$ | $-0.093$ | $0.004$ | $0.108$ | $0.196$ | $0.267$ | $0.293$
4 | 0 | $\frac{3}{16}\sqrt{\frac{1}{\pi}}\left(35z^{4}-30z^{2}+3\right)$ | $-0.63$ | $1.72$ | $4.60$ | $4.94$ | $5.05$ | $4.85$ | $4.32$
4 | 2 | $\frac{3}{8}\sqrt{\frac{5}{\pi}}\left(x^{2}-y^{2}\right)\left(7z^{2}-1\right)$ | $0.033$ | $0.125$ | $0.184$ | $0.108$ | $0.051$ | $0.001$ | $-0.052$
4 | 4 | $\frac{3}{16}\sqrt{\frac{35}{\pi}}\left(x^{4}-6x^{2}y^{2}+y^{4}\right)$ | $-0.007$ | $-0.005$ | $-0.018$ | $-0.041$ | $-0.088$ | $-0.187$ | $-0.345$
The absence of certain spherical harmonics in expansion Eq. (9) can be
discussed based on group-theoretical arguments. The function
$E_{\text{b}}(\bm{\sigma})$ should be invariant under symmetry
transformations, $g$, of the point-contact,
$E_{\text{b}}(\bm{\sigma})=E_{\text{b}}(g\bm{\sigma})$, including the symmetry
of both the lattice and the given (cycloidal) spin-configuration. Regarding
that the spin-vectors transform as axial vectors, the only allowed
transformation is the reflection onto the (001) plane:
$(x,y,z)\rightarrow(-x,-y,z)$. Thus we conclude that only those function can
enter the expansion of $E_{\text{b}}(\bm{\sigma})$ that contain even powers of
the variables $x$ and $y$. As seen from Table 2, this is fully confirmed by
our calculations. Apparently, the expansion Eq. (9) shows a satisfactory
convergence as the coefficients rapidly decrease with increasing $\ell$. An
obvious exception can, however, be seen for $K_{4}^{0}$ that for $x\geq 0.95$
overweights $K_{3}^{0}$. Noticeably, among the terms with a given $\ell$, the
one associated with the $z$ component of the magnetization ($m=0$), i.e.,
excluding in-plane anisotropy, has the largest weight.
In order to connect the above results to the rotational energy of the domain
wall discussed in Sec. III.3, we relate expansion Eq. (9) to a classical spin-
model. According to a Heisenberg model extended by relativistic corrections
Udvardi _et al._ (2003); Szunyogh _et al._ (2011) the energy in Eq. (9) can
be expressed as
$E(\bm{\sigma})=E_{\text{anis}}(\bm{\sigma})+\bm{\sigma}\sum_{j}\mathbf{J}_{\text{c}j}\bm{\sigma}_{j}\,,$
(10)
where $\mathbf{J}_{\text{c}j}$ denote the exchange coupling tensor between the
central site and the other sites of the contact with classical spin-vectors
$\bm{\sigma}_{j}$ and $E_{\text{anis}}(\bm{\sigma})$ stands for the on-site
anisotropy energy that, due to the tetragonal ($D_{4h}$) point-group symmetry
of the point-contact, can be expanded up to $\ell=4$ as
$E_{\text{anis}}(\bm{\sigma})=K_{2}^{0}R_{2}^{0}(\bm{\sigma})+K_{4}^{0}R_{4}^{0}(\bm{\sigma})+K_{4}^{4}R_{4}^{4}(\bm{\sigma})\,.$
(11)
It is clear that the $(\ell,m)=(1,0)$ term in Eq. (9) is uniquely related to
the exchange coupling and, due to the presence of a cycloidal wall, it
represents a strong Weiss field that orients the magnetic moment at the
central site along the $z$ direction. Because of the increasing distances
between the central site and the other sites of the contact, it is also easy
to understand why this term significantly decreases with increasing stretching
ratio. On the other hand, there is no $(\ell,m)=(1,0)$ term in the rotational
energy of the domain wall, Eq. (8), since in that case the relative
orientation of the spins are unchanged. With other words, repeating the
expansion Eq. (9) in the presence of a helical wall, the leading term
correspond to the spherical harmonics $\propto x$, with practically the same
coefficients as listed in Table 2 for $(\ell,m)=(1,0)$.
In relation to Eq. (11), the terms proportional to $R_{2}^{0}$, $R_{4}^{0}$
and $R_{4}^{4}$ in Eq. (9) can mainly be attributed to on-site anisotropy
contributions to the spin-Hamiltonian, however, the effect of higher order
spin-interactions can not be ruled out. The second-order uniaxial anisotropy
coefficients, $K_{2}^{0}$, are negative in the whole range of stretching,
favoring thus a normal-to-plane direction. Remarkably, the magnitude of
$K_{2}^{0}$ is around 30 meV, with a maximum of
$\left|K_{2}^{0}\right|=33.2\,\mathrm{meV}$ at $x=0.95$. This value should be
compared to some results communicated in the literature: Etz et al. Etz _et
al._ (2008) and Bornemann et al. Bornemann _et al._ (2007) calculated 5.3 meV
and 4.76 meV, respectively, for the MAE of a Co ad-atom on Pt(111) surface,
while, including orbital polarization, Gambardella et al. Gambardella _et
al._ (2003) obtained 18.45 meV for the same system. In a similar geometrical
confinement of an atomic scale junction, W and Ir turned out to be magnetic
with a magnetic anisotropy energy comparable to our values.Thiess _et al._
(2010)
Figure 9: Diamonds: Calculated energy differences between the helical and
cycloidal domain walls, $E_{\text{HW}}-E_{\text{CW}}$, circles: on-site
uniaxial magnetic anisotropy energy of the central atom (see text) as a
function of the stretching parameter, $x$. Thin lines serve as a guide for the
eye.
From Fig. 8 and Table 1 we inferred that the rotational energy of the domain
wall is dominated by the uniaxial magnetic anisotropy term proportional to
$\cos^{2}\theta=z^{2}$. In Fig. 9 the energy differences obtained between the
helical wall configuration and the ground state cycloidal wall configuration
is plotted as a function of the stretching factor, together with that provided
by the uniaxial anisotropy of the central atom,
$\frac{3}{4}\sqrt{\frac{5}{\pi}}K_{2}^{0}$. The values of $\Delta E$ from the
two calculations agree well for $x\geq 0.95$, while for more squeezed contacts
the uniaxial anisotropy of the central atom overestimates the energy
difference between the different types of domain walls. Nevertheless, we can
in general conclude that the main driving force of the formation of a
cycloidal domain wall is a giant uniaxial on-site magnetic anisotropy at the
central atom: in the cycloidal wall the magnetic moment of the central atom is
parallel to the easy axis, while in the helical wall configuration it lies
within the hard plane.
Finally, we briefly comment on the terms corresponding to $(\ell,m)=(2,2)$,
$(3,0)$, $(3,2)$ and $(4,2)$ in Table 2. Since these terms are not invariant
under transformations of the $D_{4h}$ point-group, they can not be accounted
for the on-site anisotropy terms. In terms of a spin-model, these terms
should, therefore, be related to higher order spin-interactions. The
$(\ell,m)=(2,2)$ term can, e.g., be identified as the consequence of
biquadratic interactions,Deak _et al._ (2011)
$\sum_{i}B_{\text{c}i}(\bm{\sigma}\bm{\sigma_{i}})^{2}$, while the $\ell=3$
terms to triquadratic interactions,Boča (2012)
$\sum_{i}T_{\text{c}i}(\bm{\sigma}\bm{\sigma_{i}})^{3}$. Four-spin
interactions have been explicitely calculated and proved to give significant
contributions to a spin-Hamiltonian of Cr trimers deposited on Au(111) surface
by Antal et al.,Antal _et al._ (2008) but recently their presence was
highlighted even in bulk magnets. Lounis and Dederichs (2010)
## IV Summary
In case of deposited magnetic nanostructures the point-group symmetry of the
system might considerably be reduced, therefore, complex magnetic states occur
naturally. Detecting and investigating such magnetic states pose a challenge
for ab initio calculations. We have developed a computational technique based
on a self-consistent embedded cluster Korringa-Kohn-Rostoker method suitable
to find non-collinear ground-states of finite magnetic clusters. The method is
applied to determine the structure of a domain wall formed through an atomic
scale nanocontact between two antiparallelly magnetized cobalt leads. The
obtained ground state is a cycloidal domain wall which remains stable against
squeezing or stretching the contact along the normal-to-plane direction. A
huge enhancement, as well as, anisotropy of the orbital moment are found at
the central site of the contact. The energy of the domain walls was explored
in terms of the magnetic force theorem. Our main observation is that the
formation of the cycloidal wall against a helical wall is primarily driven by
the uniaxial on-site anisotropy at the central site. We also found effects of
higher order spin-interactions as terms in the expansion of the band energy
not complying with the point-group symmetry of the point contact.
###### Acknowledgements.
Financial support is acknowledged to the New Széchenyi Plan of Hungary Project
ID. TÁMOP-4.2.2.B-10/1–2010-0009, the Hungarian Scientific Research Fund
(contracts OTKA PD83353, K77771, K84078) and the Bolyai Research Grant of the
Hungarian Academy of Sciences. UN and LS acknowledge financial support from
the DFG through SFB 767.
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|
arxiv-papers
| 2012-05-21T12:17:37 |
2024-09-04T02:49:31.139526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L\\'aszl\\'o Balogh, Kriszti\\'an Palot\\'as, L\\'aszl\\'o Udvardi,\n L\\'aszl\\'o Szunyogh, Ulrich Nowak",
"submitter": "L\\'aszl\\'o Udvardi",
"url": "https://arxiv.org/abs/1205.4579"
}
|
1205.4618
|
# Supersymmetrization of Quaternion Dirac Equation for Generalized Fields of
Dyons
A. S. Rawat(1), Seema Rawat(2) , Tianjun Li(3) and O. P. S. Negi(3,4) Address
for Correspondence during Feb. 22-April 19, 2012: Institute of Theoretical
Physics,Chinese Academy of Sciences, Zhong Guan Cun East Street 55, P. O. Box
2735, Beijing 100190, P. R. China
###### Abstract
The quaternion Dirac equation in presence of generalized electromagnetic field
has been discussed in terms of two gauge potentials of dyons. Accordingly, the
supersymmetry has been established consistently and thereafter the one, two
and component Dirac Spinors of generalized quaternion Dirac equation of dyons
for various energy and spin values are obtained for different cases in order
to understand the duality invariance between the electric and magnetic
constituents of dyons.
Key words: Supersymmetry, quaternion, Dirac equation, dyons
PACS No.: 11.30.Pb, 14.80.Ly, 03.65.Ge
1\. Department of Physics, H. N. B. Garhwal University, Pauri Campus, Pauri
(Garhwal)-246001, Uttarakhand, India.
2\. Department of Physics, Zakir Husain College, Delhi University, Jawaharlal
Nehru Marg, New Delhi-110002, India.
3\. Institute of Theoretical Physics,Chinese Academy of Sciences, Zhong Guan
Cun East Street 55, P. O. Box 2735, Beijing -100190, P. R. China.
4\. Department of Physics, Kumaun University, S. S. J. Campus, Almora- 263601,
Uttarakhand, India
email: 1. drarunsinghrawat@gmail.com; 2. rawatseema1@rediffmail.com; 3\.
tli@itp.ac.cn; 4. ops_negi@yahoo.co.in
## 1 Introduction:
Symmetries are one of the most powerful tools in the theoretical physics.
Relativistic quantum mechanics is the theory of quantum mechanics that is
consistent with the Einstein’s theory of relativity. Dirac[1] was the first
who attempted in this field followed by Feshback and Villars[2]. Since
relativistic quantum mechanics in 3+1 space-time dimension becomes difficult
because of different dimensionality of time and space. Nevertheless, the use
of quaternions has become essential because quaternion algebra[3] has certain
advantages. It provides 4-dimensional structure to relativistic quantum
mechanics and also provide consistent representation in terms compact
notations. Quaternions have direct link with Pauli spin matrices where the
spin [4, 5] plays an important role in order to make connection between bosons
and fermions. Pioneer work in the field of relativistic quaternionic quantum
mechanics was done by Adler[4] while Rotelli[6] and Leo et al[7, 8] discussed
the quaternionic wave equation. Gürsey[9] and Hestens[10] reformulated the
Dirac equation from quaternion valued terms showing that the algebraic
equivalent of Dirac has been forced to break the automorphism group of
quaternions. Supersymmetric formulation of quaternionic quantum mechanics [4]
has been discussed by Davies [11] into study supersymmetric quantum mechanics.
More over, a lot of literature has been cited [12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23] to describe the supersymmetry (SUSY) as the natural symmetry
of spin \- particles. Nicolai [24] has also introduced the SUSY for spin
system in statistical mechanics. Consequently, supersymmetric method in
quaternionic Dirac equation provides [11] the exact solutions of various
problems. Keeping in view the advantages of SUSY and the applications of
quaternionic algebra, we [25, 26] have also analyzed the supersymmertization
of quaternion quantum mechanics and quaternion Dirac equation for different
masses. Extending our results [26] , in this paper, we have discussed the
quaternion Dirac equation in electromagnetic field where the partial
derivative has been replaced by the quaternion covariant derivative. The
quaternion Dirac equation in electromagnetic field consists of two gauge
fields subjected by two unitary gauge transformations in terms of two gauge
potentials. These two gauge potentials are identified as the gauge potentials
respectively associated with the simultaneous existence of electric and
magnetic charge ( particles named as dyons [27, 28]). Accordingly, we have
obtained the one and two components solutions of generalized quaternion Dirac
equation of dyons for its different cases associated with its electric and
magnetic constituents. Furthermore, we have analyzed, the supersymmertization
of generalized quaternion Dirac equation of dyons for considering different
cases of electric and magnetic fields interacting with electric and magnetic
charges as the consequence of electromagnetic duality of dyons.
## 2 Quaternion Preliminaries:
The algebra $\mathbb{H}$ of quaternion is a four-dimensional algebra over the
field of real numbers $\mathbb{R}$ and a quaternion $\phi$ is expressed in
terms of its four base elements as
$\displaystyle\phi=\phi_{\mu}e_{\mu}=$
$\displaystyle\phi_{0}+e_{1}\phi_{1}+e_{2}\phi_{2}+e_{3}\phi_{3}(\forall\mu=0,1,2,3)$
(1)
where $\phi_{0}$,$\phi_{1}$,$\phi_{2}$,$\phi_{3}$ are the real quartets of a
quaternion and $e_{0},e_{1},e_{2},e_{3}$ are called quaternion units and
satisfies the following relations,
$\displaystyle e_{0}^{2}$ $\displaystyle=e_{0}=1,;\,\,\,\,e_{j}^{2}=-e_{0};$
$\displaystyle e_{0}e_{i}=e_{i}e_{0}$ $\displaystyle=e_{i}(i=1,2,3);$
$\displaystyle e_{i}e_{j}$
$\displaystyle=-\delta_{ij}+\varepsilon_{ijk}e_{k}(\forall\,i,j,k=1,2,3)$ (2)
where $\delta_{ij}$ is the delta symbol and $\varepsilon_{ijk}$ is the Levi
Civita three index symbol having value $(\varepsilon_{ijk}=+1)$ for cyclic
permutation, $(\varepsilon_{ijk}=-1)$ for anti cyclic permutation and
$(\varepsilon_{ijk}=0)$ for any two repeated indices. Addition and
multiplication are defined by the usual distribution law
$(e_{j}e_{k})e_{l}=e_{j}(e_{k}e_{l})$ along with the multiplication rules
given by equation (2). $\mathbb{H}$ is an associative but non commutative
algebra. If $\phi_{0},\phi_{1},\phi_{2},\phi_{3}$ are taken as complex
quantities, the quaternion is said to be a bi- quaternion. Alternatively, a
quaternion is defined as a two dimensional algebra over the field of complex
numbers $\mathbb{C}$. We thus have
$\phi=\upsilon+e_{2}\omega(\upsilon,\omega\in\mathbb{C})$ and
$\upsilon=\phi_{0}+e_{1}\phi_{1}$ , $\omega=\phi_{2}-e_{1}\phi_{3}$ with the
basic multiplication law changes to $\upsilon e_{2}=-e_{2}\bar{\upsilon}$.The
quaternion conjugate $\overline{\phi}$ is defined as
$\displaystyle\overline{\phi}=\phi_{\mu}\bar{e_{\mu}}=$
$\displaystyle\phi_{0}-e_{1}\phi_{1}-e_{2}\phi_{2}-e_{3}\phi_{3}.$ (3)
In practice $\phi$ is often represented as a $2\times 2$ matrix
$\phi=\phi_{0}-i\,\vec{\sigma}\cdot\vec{\phi}$ where
$e_{0}=I,e_{j}=-i\,\sigma_{j}(j=1,2,3)$ and $\sigma_{j}$are the usual Pauli
spin matrices. Then $\overline{\phi}=\sigma_{2}\phi^{T}\sigma_{2}$ with
$\phi^{T}$ is the transpose of $\phi$. The real part of the quaternion
$\phi_{0}$ is also defined as
$\displaystyle Re\,\phi$ $\displaystyle=\frac{1}{2}(\overline{\phi}+\phi)$ (4)
where $Re$ denotes the real part and if $Re\,\phi=0$ then we have
$\phi=-\overline{\phi}$ and imaginary $\phi$ is known as pure quaternion
written as
$\displaystyle\phi=$ $\displaystyle
e_{1}\phi_{1}+e_{2}\phi_{2}+e_{3}\phi_{3}.$ (5)
The norm of a quaternion is expressed as
$N(\phi)=\phi\overline{\phi}=\overline{\phi}\phi=\sum_{j=0}^{3}\phi_{j}^{2}$which
is non negative i.e.
$\displaystyle N(\phi)=\left|\phi\right|=$
$\displaystyle\phi_{0}^{2}+\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=Det.(\phi)\geq
0.$ (6)
Since there exists the norm of a quaternion, we have a division i.e. every
$\phi$ has an inverse of a quaternion and is described as
$\displaystyle\phi^{-1}=$
$\displaystyle\frac{\overline{\phi}}{\left|\phi\right|}.$ (7)
While the quaternion conjugation satisfies the following property
$\displaystyle\overline{\phi_{1}\phi_{2}}=$
$\displaystyle\overline{\phi_{2}}\,\overline{\phi_{1}}.$ (8)
The norm of the quaternion (1) is positive definite and enjoys the composition
law
$\displaystyle N(\phi_{1}\phi_{2})=$ $\displaystyle N(\phi_{1})N(\phi_{2}).$
(9)
Quaternion (1) is also written as $\phi=(\phi_{0},\vec{\phi})$ where
$\vec{\phi}=e_{1}\phi_{1}+e_{2}\phi_{2}+e_{3}\phi_{3}$ is its vector part and
$\phi_{0}$ is its scalar part. So, the sum and product of two quaternions are
described as
$\displaystyle(\alpha_{0}\vec{,\,\alpha})+(\beta_{0}\vec{,\,\beta})$
$\displaystyle=(\alpha_{0}+\beta_{0},\,\vec{\alpha}+\vec{\beta});$
$\displaystyle(\alpha_{0}\vec{,\,\alpha})\cdot(\beta_{0}\vec{,\,\beta})$
$\displaystyle=(\alpha_{0}\beta_{0}-\overrightarrow{\alpha}\cdot\overrightarrow{\beta}\,,\alpha_{0}\overrightarrow{\beta}+\beta_{0}\overrightarrow{\alpha}+\overrightarrow{\alpha}\times\overrightarrow{\beta}).$
(10)
Quaternion elements are non-Abelian in nature and thus represent a non
commutative division ring.
## 3 Quaternion Dirac Equation For Dyons:
The free particle quaternion Dirac equation is described [6] as,
$\displaystyle(i\,\gamma^{\mu}\partial_{\mu}-$ $\displaystyle m)\Psi(x,t)=0$
(11)
where $\Psi(x,t)=\left(\begin{array}[]{c}\Psi_{a}(x,t)\\\
\Psi_{b}(x,t)\end{array}\right)$ is the two component spinor and
$\displaystyle\Psi_{a}(x,t)=\Psi_{0}+e_{1}\,\Psi_{1};$
$\displaystyle\Psi_{b}(x,t)=\Psi_{2}-e_{1}\,\Psi_{3}\,$ (12)
are the components of spinor quaternion
$\Psi=\Psi_{0}+e_{1}\,\Psi_{1}+e_{2}\,\Psi_{2}+e_{3}\,\Psi_{3}$ and Dirac
$\gamma$ matrices are also expressed in terms of quaternion units i.e.
$\displaystyle\gamma_{0}$ $\displaystyle=\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right];\,\,\,\,\,\,\,\,\,\,\,\,\gamma_{j}=ie_{j}\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right](\forall j=1,2,3).$ (17)
So, the set of pure quaternion field (1) remains invariant under the
transformations
$\displaystyle\phi$
$\displaystyle\rightarrow\phi^{\prime}=U\phi\overline{U},\,\,\,\,\,\,\,\,\,\,U\in
Q,\,\,\,\,U\overline{U}=1.$ (18)
Similarly, the quaternion conjugate $\overline{\phi}$ transforms as
$\displaystyle\overline{\phi^{\prime}}$
$\displaystyle=\overline{U\phi\overline{U}}=U\,\overline{\phi}\overline{U\,}=-U\phi\overline{U}=-\phi^{\prime}$
(19)
Any $U\in Q$ has a decomposition like equation (18) which gives rise to a set
$\\{U\in Q;\,\,\,\,U\overline{U}=1\\}\sim SP(1)\sim SU(2).$ Though it has been
emphasized earlier [4] that the automorphic transformation of $Q-$fields are
local but one is free to select them according to the representations. On the
other hand, a $Q-$field is subjected to more general $SO(4)$ transformations
as
$\displaystyle\phi$
$\displaystyle\rightarrow\phi^{\prime}=U_{1}\phi\overline{U}_{2},\,\,\,\,\,\,\,\,\,\,U_{1},U_{2}\in
Q,\,\,\,\,U_{1}\overline{U_{1}}=U_{2}\overline{U_{2}}=1.$ (20)
So, the covariant derivative may then be described [4] in terms of two $Q-$
gauge fields i.e
$\displaystyle D_{\mu}\phi$
$\displaystyle=\partial_{\mu}\phi+A_{\text{\textmu}}\phi-\phi
B_{\text{\textmu}}$ (21)
which is subjected by two gauges $A_{\text{\textmu}}$ and $B_{\text{\textmu}}$
transforming like
$\displaystyle A_{\text{\textmu}}^{\prime}$
$\displaystyle=U_{1}A_{\text{\textmu}}\overline{U_{1}}+(\partial_{\mu}U_{1})\overline{U_{1}};$
$\displaystyle B_{\text{\textmu}}^{\prime}$
$\displaystyle=U_{2}B_{\text{\textmu}}\overline{U_{2}}+(\partial_{\mu}U_{2})\overline{U_{2}};$
(22)
where $A_{\text{\textmu}}$ and $B_{\text{\textmu}}$ may be identified as the
four potentials associated, respectively, with the electric and magnetic
charges of dyons in terms of $U(1)\times U(1)$ gauge theory [28]. Here, the
gauge transformations are Abelian and global. The quaternion covariant
derivative given by equation (21) thus supports the idea of two four
potentials of dyons. Accordingly, we way write the Dirac equation (11) for
dyons on replacing the partial derivative $\partial_{\mu}$ by covariant
derivative $D_{\mu}$as
$\displaystyle(i\,\gamma^{\mu}D_{\mu}-$ $\displaystyle m)\Psi(x,t)=0$ (23)
where the commutator is defined as
$\displaystyle\left[D_{\mu},\,D_{\nu}\right]\Psi$
$\displaystyle=D_{\mu}(D_{\nu}\Psi)-D_{\nu}(D_{\mu}\Psi)=F_{\mu\nu}\Psi-\Psi\widetilde{F_{\mu\nu}}.$
(24)
Here the gauge field strengths $F_{\mu\nu}$ and $\widetilde{F_{\mu\nu}}$ are
described [28] as the generalized anti-symmetric dual invariant
electromagnetic field tensors for dyons and are expressed as
$\displaystyle F_{\mu\nu}=$
$\displaystyle\partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu}-\frac{1}{2}\varepsilon_{\mu\nu\lambda\sigma}(\partial^{\lambda}B^{\sigma}-\partial^{\sigma}B^{\sigma});$
$\displaystyle\widetilde{F_{\mu\nu}}$
$\displaystyle=\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}-\frac{1}{2}\varepsilon_{\mu\nu\lambda\sigma}(\partial^{\lambda}A^{\sigma}-\partial^{\sigma}A^{\sigma});$
(25)
which leads to the following expressions [28] for the generalized
electromagnetic fields of dyons i.e.
$\displaystyle\mathrm{\overrightarrow{\mathrm{E}}}$
$\displaystyle=-\frac{\partial\overrightarrow{A}}{\partial
t}-\overrightarrow{\nabla}\phi-\overrightarrow{\nabla}\times\overrightarrow{B};$
$\displaystyle\overrightarrow{\mathrm{B}}$
$\displaystyle=-\frac{\partial\overrightarrow{B}}{\partial
t}-\overrightarrow{\nabla}\varphi+\overrightarrow{\nabla}\times\overrightarrow{A};$
(26)
where $\left\\{A_{\text{\textmu}}\right\\}=\left\\{\phi,\,-\vec{A}\right\\}$
and $\left\\{B_{\text{\textmu}}\right\\}=\left\\{\varphi,\,-\vec{B}\right\\}$.
Generalized electromagnetic field tensors (25) of dyons satisfy the following
famous covariant form of Generalized Dirac-Maxwell’s (GDM) equations in
presence of magnetic monopoles[1] i.e.
$\displaystyle F_{\mu\nu,\nu}=$ $\displaystyle j_{\mu};$
$\displaystyle\widetilde{F_{\mu\nu,\nu}}=$ $\displaystyle k_{\mu};$ (27)
where
$\left\\{j_{\mu}\right\\}=\left\\{\rho,\,-\overrightarrow{j}\right\\}=\mathbf{e\,\bar{\Psi\gamma_{\mu}\Psi}}$and
$\left\\{k_{\mu}\right\\}=\left\\{\varrho,\,-\overrightarrow{k}\right\\}=\mathbf{g\,\bar{\Psi\gamma_{\mu}\Psi}}$are
described [28] as the four currents respectively associated with the electric
$\mathbf{e}$ and magnetic $\mathbf{g}$ charges of dyons. We may now expend the
four potentials (gauge potentials) in terms of quaternion as
$\displaystyle A_{\text{\textmu}}$
$\displaystyle=A_{\text{\textmu}}^{{}^{0}}e_{0}+A_{\text{\textmu}}^{{}^{1}}e_{1}+A_{\text{\textmu}}^{{}^{2}}e_{2}+A_{\text{\textmu}}^{{}^{3}}e_{3};$
$\displaystyle B_{\mu}=$ $\displaystyle
B_{\text{\textmu}}^{{}^{0}}e_{0}+B_{\text{\textmu}}^{{}^{1}}e_{1}+B_{\text{\textmu}}^{{}^{2}}e_{2}+B_{\text{\textmu}}^{{}^{3}}e_{3}.$
(28)
As such, the Abelian theory of dyons can now be restored by taking the real
part of the quaternion (28) $A_{\text{\textmu}}=\overline{A_{\text{\textmu}}}$
and $B_{\text{\textmu}}=\overline{B_{\text{\textmu}}}$ implying that
$(A_{\text{\textmu}}^{{}^{0}})^{\prime}=(A_{\text{\textmu}}^{{}^{0}})=A_{\mu}$
and
$(B_{\text{\textmu}}^{{}^{0}})^{\prime}=(B_{\text{\textmu}}^{{}^{0}})=B_{\mu}$.
However, if we consider the imaginary quaternion i.e.
$A_{\text{\textmu}}=-\overline{A_{\text{\textmu}}}$ and
$B_{\text{\textmu}}=-\overline{B_{\text{\textmu}}}$ we have the $SU(2)\times
SU(2)$ gauge structure where
$A_{\text{\textmu}}=A_{\text{\textmu}}^{{}^{a}}e_{a}=A_{\text{\textmu}}^{{}^{1}}e_{1}+A_{\text{\textmu}}^{{}^{2}}e_{2}+A_{\text{\textmu}}^{{}^{3}}e_{3}$
and
$B_{\text{\textmu}}=B_{\text{\textmu}}^{{}^{a}}e_{a}=B_{\text{\textmu}}^{{}^{1}}e_{1}+B_{\text{\textmu}}^{{}^{2}}e_{2}+B_{\text{\textmu}}^{{}^{3}}e_{3}$.
Thus, with the implementation of condition
$U_{1}\overline{U_{1}}=U_{2}\overline{U_{2}}=1$ there are only the six gauge
fields $A_{\text{\textmu}}^{{}^{a}}$and $B_{\text{\textmu}}^{{}^{a}}$
associated with the covariant derivative of Dirac equation (23). The
transformation equation (20) is continuous and isomorphic to $SO(4)$ i.e.
$\displaystyle\overline{\phi^{\prime}}\phi^{\prime}$
$\displaystyle=\overline{(U_{1}\phi\overline{U}_{2})}(U_{1}\phi\overline{U}_{2})=U_{2}\overline{\phi}\,\overline{U_{1}}\,U_{1}\phi\overline{U_{2}}=U_{2}\overline{\phi}\phi\overline{U_{2}}=\overline{\phi}\phi.$
(29)
The resulting $Q-$ gauge theory has the correspondence $SO(4)\sim SO(3)\times
SO(3)$ isomorphic to $SU(2)\times SU(2)$. Accordingly, the spinor transforms
as left and right component (electric or magnetic) spinors as
$\displaystyle\Psi_{\mathbf{e}}$
$\displaystyle\mapsto(\Psi_{\mathbf{e}})^{\prime}=U_{1}\Psi_{\mathbf{e}}\,\,\,\,\&\,\,\,\,\,\Psi_{\mathbf{g}}\mapsto(\Psi_{\mathbf{g}})^{\prime}=U_{2}\Psi_{\mathbf{g}}.$
(30)
The following split basis of quaternion units may also be considered as
$\displaystyle u_{0}$
$\displaystyle=\frac{1}{2}(1-i\,e_{3});\,\,\,,\,\,\,\,\,\,u_{0}^{\star}=\frac{1}{2}(1+i\,e_{3});$
$\displaystyle u_{1}$
$\displaystyle=\frac{1}{2}(e_{1}+i\,e_{2});\,\,\,,\,\,\,\,\,\,u_{1}^{\star}=\frac{1}{2}(e_{1}-i\,e_{2});$
(31)
to constitute the $SU(2)$ doublets. As such, we may express the $Q-$classes
into five groups and can expand the theory with these choices. These five
irreducible representations of $SO(4)$ are realized as
$\displaystyle 1.$ $\displaystyle(U_{1},U_{2})\Rightarrow SO(4)\mapsto(2,2)$
$\displaystyle 2.$ $\displaystyle(U_{1},U_{1})\Rightarrow SU(2)\mapsto(3,1)$
$\displaystyle 3.$ $\displaystyle(U_{2},U_{2})\Rightarrow SU(2)\mapsto(1,3)$
$\displaystyle 4.$ $\displaystyle(U_{1},1)\Rightarrow Spinor\mapsto(2,1)$
$\displaystyle 5.$ $\displaystyle(U_{2},1)\Rightarrow Spinor\mapsto(1,2).$
(32)
Accordingly, it is easier to develop a non-Abelian gauge theory of dyons. It
is to be mentioned that the occurrence of two gauge potentials supports the
idea of duality invariance [29] among the electric and magnetic parameters of
dyons.
## 3 Supersymmetrization of Quaternion Dirac Equation for Dyons
Quaternion Dirac equation (11) for dyons may now be written as
$\displaystyle i\gamma_{\mu}D_{\mu}\psi\left(x,t\right)=$ $\displaystyle
m\psi\left(x,t\right)$ (33)
where $\gamma$ matrices satisfy the properties
$\displaystyle\gamma_{0}^{2}=+1;\,\,\,$
$\displaystyle\gamma_{l}^{2}=-1\,(\forall l=1,2,3)$
$\displaystyle\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=$
$\displaystyle-2g_{\mu\nu}\,(g_{\mu\nu}=-1,+1,+1,+1)$ (34)
showing that $\gamma_{0}$ is Hermitian while $\gamma_{l}$ are anti-Hermitian
matrices. Accordingly, the matrix $\gamma_{5}$ may be expressed as
$\displaystyle\gamma_{5}=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}=$
$\displaystyle\left[\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right]$ (37)
which satisfies the relations
$\displaystyle\gamma_{0}\gamma_{5}+\gamma_{5}\gamma_{0}=$ $\displaystyle 0;$
$\displaystyle\gamma_{l}\gamma_{5}+\gamma_{5}\gamma_{l}=$ $\displaystyle
0;\,\,\,\gamma_{5}^{2}=-1.$ (38)
It shows that the matrix $\gamma_{5}$ is pseudo scalar matrix. Furthermore,
the quaternionic Dirac spinor
$\psi=\psi_{0}+e_{1}\psi_{1}+e_{2}\psi_{2}+e_{3}\psi_{3}$ can now be
decomposed as
$\displaystyle\psi=\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)=$ $\displaystyle\left(\begin{array}[]{c}\psi_{0}\\\
\psi_{1}\\\ \psi_{2}\\\ -\psi_{3}\end{array}\right)$ (45)
in terms of two and four components Dirac spinors associated with symplectic
representation of quaternions $\psi=\psi_{a}+e_{2}\psi_{b}$with
$\psi_{a}=\psi_{0}+e_{1}\psi_{1}$ and $\psi_{b}=\psi_{2}-e_{1}\psi_{3}$.
Furthermore, we may also write one component quaternion valued Dirac spinor
which is isomorphic to two component complex spinor and four component real
spinor representation. Substituting the value of $D_{\mu}$ from equation
(2147), we get
$\displaystyle
i\gamma_{\mu}\left(\partial_{\mu}\psi\left(x,t\right)+\mathbf{e}A_{\mu}\psi\left(x,t\right)-\mathbf{g}\psi\left(x,t\right)B_{\mu}\right)=$
$\displaystyle m\psi\left(x,t\right).$ (46)
Splitting $\gamma_{\mu}$ ,$\partial_{\mu}$,$A_{\mu}$and $B_{\mu}$ in terms of
real and quaternionic constituents, we get
$\displaystyle
i\gamma_{0}\left(\partial_{0}\psi+\mathbf{e}A_{0}\psi-\mathbf{g}\psi
B_{0}\right)+i\gamma_{l}\left(\partial_{l}\psi+\mathbf{e}A_{l}\psi-\mathbf{g}\psi
B_{l}\right)=$ $\displaystyle m\psi;$ (47)
which is the general equation of spin-$\frac{1}{2}$ particle (dyon) in
generalized electromagnetic field. Equation (47) may now be reduced as
$\displaystyle i\gamma_{0}\left(-iE\psi+\mathbf{e}A_{0}\psi-\mathbf{g}\psi
B_{0}\right)+i\gamma_{l}\left(ip_{l}\psi+\mathbf{e\,}e_{l}A_{l}\psi-\mathbf{g}\psi
e_{l}B_{l}\right)=$ $\displaystyle m\psi$ (48)
which can also be written explicitly as
$\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]\left(E\psi+i\mathbf{e}A_{0}\psi-i\mathbf{g}\psi
B_{0}\right)$ $\displaystyle+$ (51)
$\displaystyle\left[\begin{array}[]{cc}0&ie_{l}\\\
-ie_{l}&0\end{array}\right]\left(-P_{l}\psi+i\mathbf{e}e_{l}A_{l}\psi-i\mathbf{g}\psi
e_{l}B_{l}\right)-m\psi$ $\displaystyle=0$ (54)
Let us study the above equation for different cases
### 3.1 Case (a) For electric field due to electric charge
Let us discuss the case when we have only pure electric field associated with
electric charge $\mathbf{e}$. In this case we have $A_{0}\neq
0\,,\,A_{l}=0,\,B_{\mu}=0$ so that the equation (54) reduces to
$\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]\left(E+i\,\mathbf{e}\,A_{0}\right)\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)+\left[\begin{array}[]{cc}0&ie_{l}.P_{l}\\\
-ie_{l}.p_{l}&0\end{array}\right]\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)-m\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)=$ $\displaystyle 0$ (65)
which further reproduces two coupled equations
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\psi_{b}=ie_{l}.P_{l}\psi_{b}=\left(E+i\,\mathbf{e\,}A_{0}-m\right)$
$\displaystyle\psi_{a};$
$\displaystyle\mathcal{\widehat{A}}\psi_{a}=ie_{l}.P_{l}\psi_{a}=\left(E+i\,\mathbf{e}\,A_{0}+m\right)$
$\displaystyle\psi_{b};$ (66)
where
$\mathcal{\widehat{A}}=\mathcal{\widehat{A}^{\dagger}}=ie_{l}p_{l}$.These two-
coupled equations (66) can now be decoupled into a single equation leading to
its supersymmetrization as
$\displaystyle P_{l}^{2}\psi_{a,b}=$
$\displaystyle\left\\{\left(E+i\,\mathbf{e\,}A_{0}\right)^{2}-m^{2}\right\\}\psi_{a,b}$
(67)
so that the super partner Hamiltonian may now be written as
$\displaystyle\mathcal{\widehat{H}}_{-}=$
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\mathcal{\widehat{A}}=P_{l}^{2};$
$\displaystyle\mathcal{\widehat{H}}_{+}=$
$\displaystyle\mathcal{\widehat{A}}\mathcal{\widehat{A}^{\dagger}}=P_{l}^{2}.$
(68)
Corresponding Dirac Hamiltonian may be defined in the following manner where
we have used the Pauli-Dirac representation i.e.
$\displaystyle\mathcal{\widehat{H}}_{D}=$
$\displaystyle\left[\begin{array}[]{cc}m&ie_{l}P_{l}\\\
ie_{l}P_{l}&-m\end{array}\right].$ (71)
Let us write equation (71) as compared to the standard Dirac Hamiltonian given
by Thaller [21] as
$\displaystyle\mathcal{\widehat{H}}_{D}=$
$\displaystyle\left[\begin{array}[]{cc}M_{+}&\hat{Q}_{D}^{\dagger}\\\
\hat{Q}_{D}&M_{-}\end{array}\right]$ (74)
which leads to $M_{+}=M_{-}=0$ and $\hat{Q}_{D}=\hat{Q}_{D}^{+}=ie_{l}P_{l}$
along with the following supersymmetric conditions
$\displaystyle\hat{Q}_{D}^{\dagger}M_{-}=$ $\displaystyle
M_{+}\hat{Q}_{D}^{\dagger};$ $\displaystyle\hat{Q}_{D}M_{+}=$ $\displaystyle
M_{-}\hat{Q}_{D}$ (75)
and the following expression for the square of the Dirac Hamiltonian i.e.
$\displaystyle\mathcal{\widehat{H}}_{D}^{2}=$
$\displaystyle\left[\begin{array}[]{cc}\left(P_{l}^{2}+m^{2}\right)&0\\\
0&\left(P_{l}^{2}+m^{2}\right)\end{array}\right].$ (78)
As such, we may write the Schrodinger Hamiltonian $\hat{H}_{s}$ and
Supercharges $\hat{Q}_{s}$ and $\hat{Q}_{s}^{\dagger}$ as
$\displaystyle\hat{H}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}P_{l}^{2}&0\\\
0&P_{l}^{2}\end{array}\right];$ (81) $\displaystyle\hat{Q}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}0&ie_{l}P_{l}\\\
0&0\end{array}\right];$ (84) $\displaystyle\hat{Q}_{s}^{\dagger}=$
$\displaystyle\left[\begin{array}[]{cc}0&0\\\
ie_{l}P_{l}&0\end{array}\right];$ (87)
which satisfy the following well known forms of supersymmetric (SUSY) algebra
i.e.
$\displaystyle\left[\hat{Q}_{s},\hat{H}_{s}\right]=\left[\hat{Q}_{s}^{\dagger},\hat{H}_{s}\right]=$
$\displaystyle 0$
$\displaystyle\left\\{\hat{Q}_{s},\hat{Q}_{s}\right\\}=\left\\{\hat{Q}_{s}^{\dagger},\hat{Q}_{s}^{\dagger}\right\\}=$
$\displaystyle 0$
$\displaystyle\left[\hat{Q}_{s},\hat{Q}_{s}^{\dagger}\right]=$
$\displaystyle\hat{H}_{s}^{+}.$ (88)
We may also obtain the following types of four spinor amplitudes of Dirac
spinors i.e.
* •
One component spinor amplitudes
$\displaystyle\Psi^{1}=$
$\displaystyle(1+e_{2}.\frac{ie_{l}P_{l}}{E_{+}-\mathbf{e}\,A_{0}+m})$
$\displaystyle\,(Energy=+ive,\,spin=\uparrow);$ $\displaystyle\Psi^{2}=$
$\displaystyle(1+e_{2}.\frac{ie_{l}P_{l}}{E_{+}-\mathbf{e}\,A_{0}+m})e_{1}$
$\displaystyle\,(Energy=+ive,\,spin=\downarrow);$ $\displaystyle\Psi^{3}=$
$\displaystyle(e_{2}-\frac{ie_{l}P_{l}}{E_{-}+\mathbf{e}\,A_{0}+m})$
$\displaystyle\,(Energy=-ive,\,spin=\uparrow);$ $\displaystyle\Psi^{4}=$
$\displaystyle(e_{2}-\frac{ie_{l}P_{l}}{E_{-}+\mathbf{e}\,A_{0}+m})e_{1}$
$\displaystyle\,(Energy=-ive,\,spin=\downarrow).$ (89)
* •
Two component spinor amplitudes
$\displaystyle\Psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{ie_{l}P_{l}}{E_{+}-\mathbf{e}\,A_{0}+m}\end{array}\right)\,(Energy=+ive,\,spin=\uparrow);$
(92) $\displaystyle\Psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{ie_{l}P_{l}}{E_{+}-\mathbf{e}\,A_{0}+m}\end{array}\right)e_{1}\,(Energy=+ive,\,spin=\downarrow);$
(95) $\displaystyle\Psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{ie_{l}P_{l}}{E_{-}+\mathbf{e}\,A_{0}+m}\\\
1\end{array}\right)\,(Energy=-ive,\,spin=\uparrow);$ (98)
$\displaystyle\Psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{ie_{l}P_{l}}{E_{-}+\mathbf{e}\,A_{0}+m}\\\
1\end{array}\right)e_{1}\,(Energy=-ive,\,spin=\downarrow).$ (101)
* •
Four component spinor amplitudes may also be obtained by restricting the
direction of propagation along any one axis which we suppose $Z-axis$ i.e
($p_{x}=p_{y}=0)$ and on substituting $e_{l}=-i\sigma_{l}$ and
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\,\,\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\,\,\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ along with the usual definitions of spin up and spin
down amplitudes of spin i.e.
$\displaystyle\psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\frac{\left|\vec{p}\right|}{E_{+}-\mathbf{e}\,A_{0}+m}\\\
0\end{array}\right)(Energy=+ive,\,spin=\uparrow);$ (106)
$\displaystyle\psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
-\frac{\left|\vec{p}\right|}{E_{+}-\mathbf{e}\,A_{0}+m}\end{array}\right)(Energy=+ive,\,spin=\downarrow);$
(111) $\displaystyle\psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{\left|\vec{p}\right|}{E_{-}+\mathbf{e}\,A_{0}+m}\\\
0\\\ 1\\\ 0\end{array}\right)(Energy=-ive,\,spin=\uparrow);$ (116)
$\displaystyle\psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\
\frac{\left|\vec{p}\right|}{E_{-}+\mathbf{e}\,A_{0}+m}\\\ 0\\\
1\end{array}\right)(Energy=-ive,\,spin=\downarrow).$ (121)
As such, we have obtained the solution of quaternion Dirac equation for dyons
in terms of one component quaternion, two component complex and four component
real spinor amplitudes. Equation (121) is same as obtained for the case of
usual Dirac equation in electromagnetic field. Thus we may interpret that the
$N=1$ quaternion spinor amplitude is isomorphic to $N=2$ complex and $N=4$
real spinor amplitude solution of Dirac equation for dyons. We can accordingly
interpret the minimum dimensional representation for Dirac equation is $N=1$
in quaternionic case, $N=2$ in complex case and $N=4$ for real number field.
### 3.2 Case (b): For magnetic field due to electric charge
Let us discuss the case when we have only pure magnetic associated with
electric charge $\mathbf{e}$. In this case we have $A_{0}=0\,,\,A_{l}\neq
0,\,B_{\mu}=0$ so that the equation (54) reduces to
$\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]E\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)+\left[\begin{array}[]{cc}0&ie_{l}\\\
-ie_{l}&0\end{array}\right](-P_{l}+i\,\mathbf{e}e_{l}A_{l})\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)-m\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)=$ $\displaystyle 0$ (132)
which yields two coupled equations i.e.
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\psi_{b}=ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})\psi_{b}=\left(E-m\right)$
$\displaystyle\psi_{a};$
$\displaystyle\mathcal{\widehat{A}}\psi_{a}=ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})\psi_{a}=\left(E+m\right)$
$\displaystyle\psi_{b};$ (133)
where
$\mathcal{\widehat{A}}=\mathcal{\widehat{A}^{\dagger}}=ie_{l}(P_{l}-i\,\mathbf{\,}e_{l}A_{l}).$
These two-coupled equations can be decoupled into a single coupled equation
showing supersymmetry in the following manner
$\displaystyle[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}\psi_{a,b}=$
$\displaystyle\left\\{E-m^{2}\right\\}\psi_{a,b}$ (134)
so that the super partner Hamiltonian may now be written as
$\displaystyle\mathcal{\widehat{H}}_{-}=$
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\mathcal{\widehat{A}}=\mathcal{\widehat{H}}_{+}=\mathcal{\widehat{A}}\mathcal{\widehat{A}^{\dagger}}=[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}.$
(135)
Thus the corresponding Dirac Hamiltonian may be defined in the following
manner
$\displaystyle\mathcal{\widehat{H}}_{D}=$
$\displaystyle\left[\begin{array}[]{cc}m&ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})\\\
ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})&-m\end{array}\right].$ (138)
Like wise, the previous case of electric field, here in case of magnetic field
we may also obtain $M_{+}=M_{-}=m$ and
$\hat{Q}_{D}=\hat{Q}_{D}^{+}=ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})$ along with
the supersymmetric condition (75) and the following expression for the square
of the Dirac Hamiltonian as
$\displaystyle\mathcal{\widehat{H}}_{D}^{2}=$
$\displaystyle\left[\begin{array}[]{cc}[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}+m^{2}&0\\\
0&[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}+m^{2}\end{array}\right].$ (141)
Accordingly we may write the Schrodinger Hamiltonian $\hat{H}_{s}$ and
Supercharges $\hat{Q}_{s}$ and $\hat{Q}_{s}^{\dagger}$ as
$\displaystyle\hat{H}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}&0\\\
0&[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]^{2}\end{array}\right];$ (144)
$\displaystyle\hat{Q}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}0&[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]\\\
0&0\end{array}\right];$ (147) $\displaystyle\hat{Q}_{s}^{\dagger}=$
$\displaystyle\left[\begin{array}[]{cc}0&0\\\
{}[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]&0\end{array}\right].$ (150)
Here, also $\hat{H}_{s}$, $\hat{Q}_{s}$ and $\hat{Q}_{s}^{\dagger}$ satisfy
the well known supersymmetric (SUSY) algebra given by equation (88).
Consequently, we may also obtain the following types of four spinor amplitudes
of Dirac spinors in presence of pure magnetic field as i.e.
* •
One component spinor amplitudes
$\displaystyle\Psi^{1}=$
$\displaystyle(1+e_{2}.\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{+}+m})$
$\displaystyle\,(Energy=+ive,\,spin=\uparrow);$ $\displaystyle\Psi^{2}=$
$\displaystyle(1+e_{2}.\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{+}+m})e_{1}$
$\displaystyle\,(Energy=+ive,\,spin=\downarrow);$ $\displaystyle\Psi^{3}=$
$\displaystyle(e_{2}-\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{-}+m})$
$\displaystyle\,(Energy=-ive,\,spin=\uparrow);$ $\displaystyle\Psi^{4}=$
$\displaystyle(e_{2}-\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{-}+m})e_{1}$
$\displaystyle\,(Energy=-ive,\,spin=\downarrow).$ (151)
* •
Two component spinor amplitudes
$\displaystyle\Psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{+}+m}\end{array}\right)\,(Energy=+ive,\,spin=\uparrow);$
(154) $\displaystyle\Psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{+}+m}\end{array}\right)e_{1}\,(Energy=+ive,\,spin=\downarrow);$
(157) $\displaystyle\Psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{-}+m}\\\
1\end{array}\right)\,(Energy=-ive,\,spin=\uparrow);$ (160)
$\displaystyle\Psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{-}+m}\\\
1\end{array}\right)e_{1}\,(Energy=-ive,\,spin=\downarrow).$ (163)
* •
Four component spinor amplitudes may also be obtained by restricting the
direction of propagation along any one axis which we suppose $Z-axis$ i.e
($p_{x}=p_{y}=0)$ and $(A_{x}=A_{y}=0\Rightarrow H_{z}=0)$. Accordingly,
substituting $e_{l}=-i\sigma_{l}$ and
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\,\,\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\,\,\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ along with the usual definitions of spin up and spin
down amplitudes of spin , we get
$\displaystyle\psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\frac{\left|\vec{p}\right|}{E_{+}+m}\\\
0\end{array}\right)(Energy=+ive,\,spin=\uparrow);$ (168)
$\displaystyle\psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
-\frac{\left|\vec{p}\right|}{E_{+}+m}\end{array}\right)(Energy=+ive,\,spin=\downarrow);$
(173) $\displaystyle\psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{\left|\vec{p}\right|}{E_{-}+m}\\\
0\\\ 1\\\ 0\end{array}\right)(Energy=-ive,\,spin=\uparrow);$ (178)
$\displaystyle\psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\
\frac{\left|\vec{p}\right|}{E_{-}+m}\\\ 0\\\
1\end{array}\right)(Energy=-ive,\,spin=\downarrow).$ (183)
which are the well known usual spinor amplitudes for a Dirac free Particle .
### 3.3 Case (c): For Electric field due to magnetic monopole
Here, we discuss the case when we have only electric field associated with
magnetic charge (pure magnetic monopole) $\mathbf{g}$ only. So, by virtue of
duality of magnetic charge [27, 28, 29, 30], we take $B_{0}=0\,,\,B_{l}\neq
0,\,A_{\mu}=0$. Thus, the equation (54) reduces to
$\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]E\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)+\left[\begin{array}[]{cc}0&ie_{l}\\\
-ie_{l}&0\end{array}\right](-P_{l}+i\,\mathbf{g\,}e_{l}B_{l})\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)-m\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)=$ $\displaystyle 0$ (194)
which yields two coupled equations i.e.
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\psi_{b}=ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})\psi_{b}=\left(E-m\right)$
$\displaystyle\psi_{a};$
$\displaystyle\mathcal{\widehat{A}}\psi_{a}=ie_{l}(P_{l}-i\,\mathbf{g}e_{l}B_{l})\psi_{a}=\left(E+m\right)$
$\displaystyle\psi_{b};$ (195)
where
$\mathcal{\widehat{A}}=\mathcal{\widehat{A}^{\dagger}}=ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l}).$
These two-coupled equations can be decoupled into a single coupled equation
showing supersymmetry in the following manner
$\displaystyle[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]^{2}\psi_{a,b}=$
$\displaystyle\left\\{E-m^{2}\right\\}\psi_{a,b}$ (196)
so that the super partner Hamiltonian may now be written as
$\displaystyle\mathcal{\widehat{H}}_{-}=$
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\mathcal{\widehat{A}}=\mathcal{\widehat{H}}_{+}=\mathcal{\widehat{A}}\mathcal{\widehat{A}^{\dagger}}=[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]^{2}.$
(197)
Thus the corresponding Dirac Hamiltonian may be defined in the following
manner
$\displaystyle\mathcal{\widehat{H}}_{D}=$
$\displaystyle\left[\begin{array}[]{cc}m&ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})\\\
ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})&-m\end{array}\right].$ (200)
Like wise, the previous case of electric field, here in case of magnetic field
we may also obtain $M_{+}=M_{-}=m$ and
$\hat{Q}_{D}=\hat{Q}_{D}^{+}=ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})$ along
with the supersymmetric condition (75) and the following expression for the
square of the Dirac Hamiltonian as
$\displaystyle\mathcal{\widehat{H}}_{D}^{2}=$
$\displaystyle\left[\begin{array}[]{cc}[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]^{2}+m^{2}&0\\\
0&[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})+m^{2}\end{array}\right].$ (203)
Accordingly we may write the Schrodinger Hamiltonian $\hat{H}_{s}$ and
Supercharges $\hat{Q}_{s}$ and $\hat{Q}_{s}^{\dagger}$ as
$\displaystyle\hat{H}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]^{2}&0\\\
0&[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]^{2}\end{array}\right];$ (206)
$\displaystyle\hat{Q}_{s}=$
$\displaystyle\left[\begin{array}[]{cc}0&[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]\\\
0&0\end{array}\right];$ (209) $\displaystyle\hat{Q}_{s}^{\dagger}=$
$\displaystyle\left[\begin{array}[]{cc}0&0\\\
{}[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]&0\end{array}\right].$ (212)
Here, also $\hat{H}_{s}$, $\hat{Q}_{s}$ and $\hat{Q}_{s}^{\dagger}$ satisfy
the well known supersymmetric (SUSY) algebra given by equation (88).
Consequently, we may also obtain the following types of four spinor amplitudes
of Dirac spinors in presence of pure magnetic field as i.e.
* •
One component spinor amplitudes
$\displaystyle\Psi^{1}=$
$\displaystyle(1+e_{2}.\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]}{E_{+}+m})$
$\displaystyle\,(Energy=+ive,\,spin=\uparrow);$ $\displaystyle\Psi^{2}=$
$\displaystyle(1+e_{2}.\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]}{E_{+}+m})e_{1}$
$\displaystyle\,(Energy=+ive,\,spin=\downarrow);$ $\displaystyle\Psi^{3}=$
$\displaystyle(e_{2}-\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]}{E_{-}+m})$
$\displaystyle\,(Energy=-ive,\,spin=\uparrow);$ $\displaystyle\Psi^{4}=$
$\displaystyle(e_{2}-\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]}{E_{-}+m})e_{1}$
$\displaystyle\,(Energy=-ive,\,spin=\downarrow).$ (213)
* •
Two component spinor amplitudes
$\displaystyle\Psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l})]}{E_{+}+m}\end{array}\right)\,(Energy=+ive,\,spin=\uparrow);$
(216) $\displaystyle\Psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l}))]}{E_{+}+m}\end{array}\right)e_{1}\,(Energy=+ive,\,spin=\downarrow);$
(219) $\displaystyle\Psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{[ie_{l}(P_{l}-i\,\mathbf{e}e_{l}A_{l})]}{E_{-}+m}\\\
1\end{array}\right)\,(Energy=-ive,\,spin=\uparrow);$ (222)
$\displaystyle\Psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{[ie_{l}(P_{l}-i\,\mathbf{g\,}e_{l}B_{l}))]}{E_{-}+m}\\\
1\end{array}\right)e_{1}\,(Energy=-ive,\,spin=\downarrow).$ (225)
* •
Four component spinor amplitudes may also be obtained by restricting the
direction of propagation along any one axis which we suppose $Z-axis$ i.e
($p_{x}=p_{y}=0)$ and $(B_{x}=B_{y}=0\Rightarrow E_{z}=0)$. Accordingly,
substituting $e_{l}=-i\sigma_{l}$ and
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\,\,\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\,\,\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ along with the usual definitions of spin up and spin
down amplitudes of spin , we get the four component Dirac spinors same as the
equation (183).
### 3.4 Case (d): For Magnetic field due to magnetic monopole
Let us discuss the case when we have only pure magnetic field associated with
magnetic charge (monopole) $\mathbf{g}$. In this case we have $B_{0}\neq
0\,,\,B_{l}=0,\,A_{\mu}=0$ so that the equation (54) reduces to
$\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]\left(E+i\,\mathbf{g}\,B_{0}\right)\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)+\left[\begin{array}[]{cc}0&ie_{l}.P_{l}\\\
-ie_{l}.p_{l}&0\end{array}\right]\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)-m\left(\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right)=$ $\displaystyle 0$ (236)
which further reduces to two coupled equations
$\displaystyle\mathcal{\widehat{A}^{\dagger}}\psi_{b}=ie_{l}.P_{l}\psi_{b}=\left(E+i\,\mathbf{g\,}B_{0}-m\right)$
$\displaystyle\psi_{a};$
$\displaystyle\mathcal{\widehat{A}}\psi_{a}=ie_{l}.P_{l}\psi_{a}=\left(E+i\,\mathbf{g\,}B_{0}-m\right)$
$\displaystyle\psi_{b};$ (237)
where
$\mathcal{\widehat{A}}=\mathcal{\widehat{A}^{\dagger}}=ie_{l}p_{l}$.These two-
coupled equations (237) can now be decoupled into a single equation leading to
its supersymmetrization as
$\displaystyle P_{l}^{2}\psi_{a,b}=$
$\displaystyle\left\\{\left(E+i\,\mathbf{g\,}B_{0}\right)^{2}-m^{2}\right\\}\psi_{a,b};$
(238)
so that the super partner Hamiltonian may now be written as equation (68).
Corresponding Dirac Hamiltonian then may be defined as equation (71) which can
also be written as (74) after its comparison with the standard Dirac
Hamiltonian given by Thaller [21] and thus, leads to $M_{+}=M_{-}=0$ and
$\hat{Q}_{D}=\hat{Q}_{D}^{+}=ie_{l}P_{l}$ along with the following
supersymmetric conditions given by equation (75) along with the Dirac
Hamiltonian given by
$\displaystyle\mathcal{\widehat{H}}_{D}^{2}=$
$\displaystyle\left[\begin{array}[]{cc}\left(P_{l}^{2}+m^{2}\right)&0\\\
0&\left(P_{l}^{2}+m^{2}\right)\end{array}\right]=\widehat{H}_{s}^{2}+m^{2}\widehat{I}$
(241)
where $\hat{I}$ is unit matrix of order $4$. Consequently, we may write the
Schrodinger Hamiltonian $\hat{H}_{s}$ and Supercharges ( $\hat{Q}_{s}$ and
$\hat{Q}_{s}^{\dagger}$) as given by equation (87 ) leading to well known
supersymmetric (SUSY) algebra relations given by equation (88). Furthermore,
the following types of four spinor amplitudes of Dirac spinors may also be
obtained as
* •
One component spinor amplitudes
$\displaystyle\Psi^{1}=$
$\displaystyle(1+e_{2}.\frac{ie_{l}P_{l}}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)})$
$\displaystyle\,(Energy=+ive,\,spin=\uparrow);$ $\displaystyle\Psi^{2}=$
$\displaystyle(1+e_{2}.\frac{ie_{l}P_{l}}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)})e_{1}$
$\displaystyle\,(Energy=+ive,\,spin=\downarrow);$ $\displaystyle\Psi^{3}=$
$\displaystyle(e_{2}-\frac{ie_{l}P_{l}}{\left(E_{-}+\mathbf{g\,}B_{0}+m\right)})$
$\displaystyle\,(Energy=-ive,\,spin=\uparrow);$ $\displaystyle\Psi^{4}=$
$\displaystyle(e_{2}-\frac{ie_{l}P_{l}}{\left(E_{-}+i\,\mathbf{g\,}B_{0}+m\right)})e_{1}$
$\displaystyle\,(Energy=-ive,\,spin=\downarrow).$ (242)
* •
Two component spinor amplitudes
$\displaystyle\Psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{ie_{l}P_{l}}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)}\end{array}\right)\,(Energy=+ive,\,spin=\uparrow);$
(245) $\displaystyle\Psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
\frac{ie_{l}P_{l}}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)}\end{array}\right)e_{1}\,(Energy=+ive,\,spin=\downarrow);$
(248) $\displaystyle\Psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{ie_{l}P_{l}}{\left(E_{-}+\mathbf{g\,}B_{0}+m\right)}\\\
1\end{array}\right)\,(Energy=-ive,\,spin=\uparrow);$ (251)
$\displaystyle\Psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{ie_{l}P_{l}}{\left(E_{-}+\mathbf{g\,}B_{0}+m\right)}\\\
1\end{array}\right)e_{1}\,(Energy=-ive,\,spin=\downarrow).$ (254)
* •
Four component spinor amplitudes may also be obtained by restricting the
direction of propagation along any one axis which we suppose $Z-axis$ i.e
($p_{x}=p_{y}=0)$ and on substituting $e_{l}=-i\sigma_{l}$ and
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\,\,\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\,\,\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ along with the usual definitions of spin up and spin
down amplitudes of spin i.e.
$\displaystyle\psi^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\frac{\left|\vec{p}\right|}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)}\\\
0\end{array}\right)(Energy=+ive,\,spin=\uparrow);$ (259)
$\displaystyle\psi^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
-\frac{\left|\vec{p}\right|}{\left(E_{+}-\mathbf{g\,}B_{0}+m\right)}\end{array}\right)(Energy=+ive,\,spin=\downarrow);$
(264) $\displaystyle\psi^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\frac{\left|\vec{p}\right|}{\left(E_{-}+g\,B_{0}+m\right)}\\\
0\\\ 1\\\ 0\end{array}\right)(Energy=-ive,\,spin=\uparrow);$ (269)
$\displaystyle\psi^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\
\frac{\left|\vec{p}\right|}{\left(E_{-}+\mathbf{g\,}B_{0}+m\right)}\\\ 0\\\
1\end{array}\right)(Energy=-ive,\,spin=\downarrow).$ (274)
## 4 Discussion and Conclusion
We have discussed the quaternion Dirac equation in electromagnetic field where
the partial derivative has been replaced by the quaternion covariant
derivative in terms of two gauge potentials. These two gauge potentials are
identified as the gauge potentials associated with a particle which contains
the simultaneous existence of electric and magnetic charge (monopole). Such
type of particles are named as dyons. Thereafter, we have established
consistently the SUSY for different cases of quaternion Dirac equation for
dyons. Case (a) deals with the study of supersymmetrization of Dirac equation
when it interacts with electric field produced by electric charge only.
Thereby, we have obtained a single decoupled equation, super partner
Hamiltonians, total (Schrodinger) Hamiltonians and Schrodinger supercharges
consistently followed by the Dirac Hamiltonian and Dirac supercharges. It is
shown that the supercharges and Hamiltonian satisfy the SUSY algebra
performing SUSY transformations. Moreover, in this case, we have also obtained
the solutions of the Dirac equation for one component, two components and four
component Dirac spinors with various energy and spins. Case (b) is described
for a dyon consisting electric charge but moves in only magnetic field.
Likewise, we have followed the same procedure and obtained consistently the
particle Hamiltonian and supercharges to satisfy the SUSY algebra.
Furthermore, we have obtained the consistently one component, two components
and four component Dirac spinors with various energy and spins. Same procedure
has also been extended for Case (c) and Case (d) respectively associated with
the electric and magnetic fields due to the presence of magnetic monopole in
order to establish the consistent formulation of SUSY and Dirac spinors of
various energy and spins. It is concluded that the Case (a) and Case (d) and
likewise, Case (b) and Case (c) are dual invariant. These cases may also be
analyzed by applying the duality transformations between electric and magnetic
constituents of dyons. It may also be concluded that minimal representation
for quaternion Dirac equation is described as $N=1$ quaternionic, $N=2$complex
and $N=4$ real representation. In fact, the one-component spinor amplitudes
are isomorphic to two component complex spinor amplitudes and four component
real spinor amplitudes. As such, the higher dimensional supersymmetric Dirac
equation in generalized electromagnetic fields of dyons may be tackled well in
terms of quaternions splitting into$N=1$ quaternionic, $N=2$complex and $N=4$
real representations of Supersymmetric quantum mechanics.
ACKNOWLEDGMENT: One of us (OPSN) acknowledges the financial support for
UNESCO-TWAS Associateship from Third World Academy of Sciences, Trieste
(Italy) and Chinese Academy of Sciences, Beijing. He is also thankful to
ProfessorYue-Liang Wu, Director ITP for his hospitality and research
facilities at ITP and KITP.
## References
* [1] P. A. M. Dirac, “The principles of quantum mechanics”, (4th ed) Oxford University Press London (1958).
* [2] H. Feshback and F. Villars, Rev. Mod. Phys., 30 (1958), 24.
* [3] J. Souček, J. Phys. A: Math.Gen., 14 (1981), 1629.
* [4] S. L. Adler; “Quaternionic Quantum mechanics”, Oxford University Press, Oxford (1995).
* [5] A. Das, S. Okubo and S.A. Pernice, Modern Physics Letters, A12 (1997), 581.
* [6] P. Rotelli, Modern Phys letters, 4 (1989) 1763.
* [7] S. De Leo and P.Rotelli, Prog.Theor.Phys., 92 (1994), 917.
* [8] S. De Leo and P.Rotelli, Mod.Phys.Lett., A11 (1996), 357.
* [9] F. Gürsey, Rev. fac. Sci., Univ. Istribul (Turkey), A21 (1956), 33.
* [10] D. Hestens, J. Math. Phys., 8 (1967), 778.
* [11] A. J. Davies, Phys. Rev. A49 (1994), 714.
* [12] F. Cooper, A. Khare and U. Sukhatme, Phys. Rep., 251 (1995) 267.
* [13] F. Cooper, A. Khare and U. Sukhatme, Ann. Phys (NY), 187 (1988)1.
* [14] C.V. Sukumar, J. Phys., A 18 (1985), 2917 & 2937\.
* [15] L. P. Singh and B. Ram, Pramana-Journal of Physics; 58 (2002), 591
* [16] S. V.ketov and Ya S. Prager, Acta Phys. Pol., B21 (1990) , 463.
* [17] T. E. Clark and S.T. Love, Nucl.Phys., B231 (1984), 91.
* [18] M. de Crombrugghe and V. Rittenberg, Annals of Physics, 151 (1983), 99.
* [19] B. Thaller; J. Math. Phys ., 29(1988), 247\.
* [20] B. Thaller; “Dirac particle in magnetic fields”, in A.Boulet de Monrel, P. Dita, G. Nenciu and R.Purice Eds, ’Recent Development in quantum mechanics; Mathematical Physics Studies’ Nr.12 , Kluwer Acad. Publ. Dordrechel,(1991), pp.351-366.
* [21] B. Thaller; “The Dirac equation”, Springer Verlag, Berlin, (1992).
* [22] A. A. Andrianov, F. Cannata, J. P. Dedonder and M.V.Ioffe, Int. J. Mod. Phys., A10 (1995), 2683
* [23] E. Witten, Nucl.Phys., B188 (1981), 513.
* [24] H. Nicolai; J.Phys., A 9 (1976), 1497.
* [25] Seema Rawat and O.P.S. Negi, Int. J. Theor. Phys., 48 (2009), 305.
* [26] Seema Rawat and O.P.S. Negi, Int. J. Theor. Phys., 48 (2009), 2222
* [27] P. S. Bisht and O. P. S. Negi, Int. J. Theor. Phys., 47 (2008), 1497.
* [28] O. P. S. Negi and H. Dehnen, Int. J. Theor. Phys., 50 (2011), 2446.
* [29] P. S. Bisht and O. P. S. Negi, Int. J. Theor. Phys., 47 (2008), 3108.
* [30] Y. M. Shnir, “Magnetic Monopoles”, Springer-Verlag Berlin-Heidelberg (2005).
|
arxiv-papers
| 2012-03-07T08:15:48 |
2024-09-04T02:49:31.148898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. S. Rawat, Seema Rawat, Tianjun Li and O. P. S. Negi",
"submitter": "Om Prakash Singh Negi",
"url": "https://arxiv.org/abs/1205.4618"
}
|
1205.4666
|
# A Generalization of the Goldberg-Sachs Theorem and its Consequences
Carlos Batista carlosbatistas@df.ufpe.br Departamento de Física, Universidade
Federal de Pernambuco, 50670-901 Recife - PE, Brazil
###### Abstract
The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds
endowed with torsion-free connection compatible with the metric, the treatment
includes all signatures as well as complex manifolds. It is shown that when
the Weyl tensor is algebraically special severe geometric restrictions are
imposed. In particular it is demonstrated that the simple self-dual
eigenbivectors of the Weyl tensor generate integrable isotropic planes.
Another result obtained here is that if the self-dual part of the Weyl tensor
vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be
Calabi-Yau or symplectic and admits a solution for the source-free Einstein-
Maxwell equations.
Goldberg-Sachs theorem, Weyl tensor, Integrable distributions, Petrov
classification, General relativity.
## I Introduction
The Petrov classification Petrov ; Stephani is a form to classify the Weyl
tensor in four-dimensional space-times that promoted much progress in general
relativity. Besides other contributions it has helped in the search of new
exact solutions to Einstein’s equation, Kerr metric being the most important
example Kerr . Particularly important in this process was the Goldberg-Sachs
theorem Goldberg-Sachs , which associates algebraic constraints in the Weyl
tensor to some integrability properties in the space-time. This theorem states
that in a Ricci-flat (vacuum) four-dimensional Lorentzian manifold there
exists a shear-free null geodesic congruence if, and only if, the Weyl tensor
is algebraically special, where such congruences are generated by the so
called repeated principal null directions.
Recently the Petrov classification was extended to four-dimensional manifolds
of all signatures, as well as complexified manifolds111In this paper the term
”complexified manifold” means a manifold in which the metric can be complex,
so that the Weyl tensor is also generally complex. The here called ”real
manifolds” are the ones with real metric and, consequently, real Weyl tensor.
In general the tangent bundle of the real manifolds will be assumed to be
complexified. Finally, the term ”complex manifold” will mean a manifold that
can be covered by complex charts with analytic transition functions, these
manifolds are sometimes called Hermitian., in an unified treatment based on
the action of the Weyl tensor in the bivector bundle art1 . The intent of the
present article is continue this path and explore the generalized version of
the Goldberg-Sachs(GS) theorem valid in all signatures Plebanski2 using the
bivector approach and in an unified way, so that the results in real manifolds
of any signature follow from the general complex case, by conveniently
choosing a real slice. This different form to attack the problem will prove to
be valuable because it is full of geometric content. For example, the null
eigenbivectors of the Weyl tensor will be shown to generate integrable planes
when the Ricci tensor vanishes. The generalized version of the GS theorem in
complexified manifolds was investigated before in reference Plebanski2 , while
the Euclidean case was treated in Broda , but in both works spinor techniques
were used rather than the bivector approach. Some important steps in the
direction of this work were also taken in Robinson Manifolds ; Nurwoski2 .
Here some progress is made is this subject, in particular it is proved that
when the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold,
there is a covariantly constant rank two tensor other than the metric and a
solution for the source-free Einstein-Maxwell equations. It is also proved in
this article that the Weyl tensor is algebraically special if, and only if, it
admits a simple self-dual eigenbivector. It is relevant pointing out that in
recent years there has been an active research in order to find a suitable
generalization of the GS theorem valid in all dimensions Dur-Reall ; Ortaggio5
; HigherGSisotropic1 ; HigherGSisotropic2 , hopefully the present work can
give new insights on these attempts.
Section II provides a short review of the unified algebraic classification
scheme for the Weyl tensor in four-dimensional manifolds, there the Weyl
tensor is viewed as an operator on the bivector bundle. Next section shows
that in Lorentzian signature the repeated principal null directions (PNDs) are
in a one to one correspondence with the simple self-dual eigenbivectors of the
Weyl tensor. This result serves as the motivation to conjecture that the
generalization of the repeated PNDs to the non-Lorentzian manifolds are the
simple self-dual eigenbivectors of the Weyl tensor. In section IV the
extension of the Goldberg-Sachs theorem to complexified manifolds and real
manifolds of all signatures is enunciated. This theorem implies that the
simple self-dual eigenbivectors of the Weyl tensor span integrable planes just
as the repeated PNDs are related to the integrability of shear-free null
geodesic congruences, supporting the conjecture of preceding section. Next
section provides the interpretations and consequences of the generalized GS
theorem. In particular it is proved that when the self-dual part of the Weyl
tensor vanishes in complex and Euclidean manifolds the manifold is Calabi-Yau,
while in (2,2) signature it can also be symplectic and admits a non-trivial
covariantly constant tensor of rank two. Finally, section VI briefly discuss
the physical applicability of achieved results and of the mathematical
structure behind them.
## II Weyl Tensor Classification by Bivectors in All Signatures
This section will be a quick sum up of the results obtained in art1 . Let
$(M,g_{\mu\nu})$ be a four-dimensional differential manifold endowed with
metric $g_{\mu\nu}$. Sometimes $(M,g_{\mu\nu})$ will also denote a
complexified Riemannian manifold of complex dimension four, as will be clear
in the context. A skew-symmetric rank two tensor field,
$B_{\mu\nu}=-B_{\nu\mu}$, is called a bivector. Denoting the volume form by
$\epsilon_{\mu\nu\rho\sigma}$, the dual of a bivector is defined by:
$\widetilde{B}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}B^{\rho\sigma}\,.$
(1)
It is easy to see that given any two bivectors, $B_{\mu\nu}$ and $F_{\mu\nu}$,
we have
$\widetilde{B}_{\mu\nu}F^{\mu\nu}=B_{\mu\nu}\widetilde{F}^{\mu\nu}.$ (2)
By means of the contraction properties of the volume form, it is obtained that
the double dual of a bivector is a multiple of it:
$\widetilde{\widetilde{B}_{\mu\nu}}=\frac{1}{4}\epsilon_{\mu\nu\rho\sigma}\epsilon^{\rho\sigma\alpha\beta}B_{\alpha\beta}=\varepsilon^{2}B_{\mu\nu}\,.$
(3)
Where $\varepsilon=1$ in complexified manifolds and in real manifolds of
Euclidean or (2,2) signature, while $\varepsilon=i$ in real Lorentzian
manifolds. Equation (3) enables to split the bivector bundle, $\mathfrak{B}$,
into a direct sum of two spaces of the same dimension:
$\mathfrak{B}=\mathfrak{D}^{+}\oplus\,\mathfrak{D}^{-},$ (4)
$\mathfrak{D}^{+}=\\{Z^{+}_{\mu\nu}\in\mathfrak{B}|\widetilde{Z^{+}}_{\mu\nu}=\varepsilon
Z^{+}_{\mu\nu}\\}\;;\;\mathfrak{D}^{-}=\\{Z^{-}_{\mu\nu}\in\mathfrak{B}|\widetilde{Z^{-}}_{\mu\nu}=-\varepsilon
Z^{-}_{\mu\nu}\\}.$
Where $\mathfrak{D}^{+}$ is called the bundle of self-dual bivectors, while
$\mathfrak{D}^{-}$ is the bundle of anti-self-dual bivectors.
A bivector $B_{\mu\nu}$ is called simple if it is possible to find two vector
fields, $X$ and $Y$, such that $B_{\mu\nu}=X_{[\mu}Y_{\nu]}$. This kind of
bivector is naturally associated with planes, if $B_{\mu\nu}=X_{[\mu}Y_{\nu]}$
the bivector $B_{\mu\nu}$ is said to generate the planes spanned by the vector
fields $X$ and $Y$. In four dimensions a bivector is simple if, and only if,
$B_{\mu\nu}\widetilde{B}^{\mu\nu}=0$.
Because of the skew-symmetry in the first and second pairs of Weyl tensor
indices it is natural to define
$C_{\mu\nu\widetilde{\rho\sigma}}=\frac{1}{2}\epsilon_{\rho\sigma\alpha\beta}C_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\;;\;C_{\widetilde{\mu\nu}\rho\sigma}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}C^{\alpha\beta}_{\phantom{\alpha\beta}\rho\sigma}.$
(5)
It can be proved the following important properties of the Weyl tensor art1 :
$C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\rho\sigma}\mu\nu}\;\;;\;\;C_{\mu\nu\widetilde{\rho\sigma}}=C_{\widetilde{\mu\nu}\rho\sigma}.$
(6)
Skew-symmetry in the first and second pairs of the Weyl tensor indices also
enables to view this tensor as an operator in $\mathfrak{B}$ Petrov ; Law1 ;
Hervik-Coley ,
$C:\mathfrak{B}\rightarrow\mathfrak{B}\;\;;\;\;B\mapsto
C(B)=T\,,\;\textrm{where}\;\,T_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}.$
This operator will be called the Weyl operator. It has the important property
of sending self-dual bivectors into self-dual bivectors and anti-self-dual
bivectors into anti-self-dual bivectors. To see this let $Z^{\pm}_{\mu\nu}$
pertain to $\mathfrak{D}^{\pm}$ and $F^{\pm}\equiv C(Z^{\pm})$, then using (6)
and (2) we have:
$\widetilde{F^{\pm}}_{\mu\nu}=C_{\widetilde{\mu\nu}\rho\sigma}Z^{\pm\,\rho\sigma}=C_{\mu\nu\widetilde{\rho\sigma}}Z^{\pm\,\rho\sigma}=C_{\mu\nu\rho\sigma}\widetilde{Z}^{\pm\,\rho\sigma}=\pm\varepsilon
C_{\mu\nu\rho\sigma}Z^{\pm\,\rho\sigma}\equiv\pm\varepsilon
F^{\pm}_{\phantom{\pm}\mu\nu}.$ (7)
So the Weyl operator can be split into a direct sum of two operators,
$C=C^{+}\oplus C^{-}$, where $C^{+}$ is called the self-dual part of Weyl
operator, and $C^{-}$ is the anti-self-dual part. $C^{+}$ sends elements of
$\mathfrak{D}^{+}$ into elements of $\mathfrak{D}^{+}$ and gives zero when
operates in $\mathfrak{D}^{-}$, while $C^{-}$ sends elements of
$\mathfrak{D}^{-}$ into elements of $\mathfrak{D}^{-}$ and has trivial action
in $\mathfrak{D}^{+}$. Restricting the action of $C^{\pm}$ to
$\mathfrak{D}^{\pm}$ and using the fact that both operators have vanishing
trace we find that $C^{\pm}$ can have the following algebraic types:
$\left[\begin{array}[]{ll}\textbf{Type O}^{\pm}\rightarrow\;C^{\pm}=0\\\
\textbf{Type I}^{\pm}\;\,\rightarrow\;C^{\pm}$ allows 3 distinct eigenvalues
$\\\ \textbf{Type D}^{\pm}\rightarrow C^{\pm}$ is diagonalizable with a
repeated non-zero eigenvalue $\\\ \textbf{Type II}^{\pm}\rightarrow C^{\pm}$
is non-diagonalizable and has a repeated non-zero eigenvalue $\\\ \textbf{Type
III}^{\pm}\rightarrow\,(C^{\pm})^{3}=0\,$and$\,(C^{+})^{2}\neq 0$\,(all
eigenvalues are zero)$\\\ \textbf{Type
N}^{\pm}\rightarrow\,(C^{\pm})^{2}=0\,$and$\,C^{+}\neq 0$ \,(all eigenvalues
are zero)$\end{array}\right.$
If operator $C^{\pm}$ is type I± it is called algebraically general, otherwise
it is algebraically special. An algebraic classification for the full Weyl
tensor is made by the composition of the possible types of operators
$C^{\pm}$. For example, Weyl tensor is said to be of type (I,N) if $C^{+}$ is
type I+ and $C^{-}$ is type N-. Since the bundles $\mathfrak{D}^{+}$ and
$\mathfrak{D}^{-}$ are interchanged by a simple change of sign in the volume
form it follows that the type (I,N) is intrinsically equivalent to the type
(N,I) and so on. At the end there are 21 distinct algebraic types for the Weyl
tensor. If the manifold $(M,g_{\mu\nu})$ is complexified or real with (2,2)
signature it follows that the 21 types are allowed. But if $(M,g_{\mu\nu})$ is
a real manifold with Lorentzian or Euclidean signature the reality condition
implies that not all classifications are realizable. In Lorentzian signature
the allowable types are (O,O), (I,I), (D,D), (II,II), (III,III) and (N,N),
which are respectively the well known Petrov types O, I, D, II, III and N,
while in Euclidean case the possible algebraic types for the Weyl tensor are
(O,O), (O,I), (O,D),(I,I), (I,D) and (D,D).
This classification to Weyl tensor in complexified manifolds was first
obtained in Plebanski75 , Euclidean case was treated in Hacyan while (2,2)
signature appeared in Law2 , in all these references spinor techniques are
used. In Law1 the operator method was used to classify the curvature of (2,2)
signature Einstein manifolds, in such reference fewer types are defined
because the vanishing eigenvalues are not distinguished from the non-zero
ones. Some general aspects of the operator method to classify tensors in
higher-dimensional manifolds were addressed in Hervik-Coley . An attempt to
classify Weyl tensor in pseudo-Riemannian manifolds in dimensions grater than
four was described in ColeyPSEUD . More references about the Weyl tensor
classification can be found in art1 .
## III Finding Principal Null Directions from Eigenbivectors
In this section it will be assumed that $(M,g_{\mu\nu})$ is a Lorentzian
manifold of dimension four with non-vanishing Weyl tensor. In this kind of
manifold a real null vector $k^{\mu}\neq 0$, $k^{\mu}k_{\mu}=0$, is said to
point into a principal null direction(PND) if
$k_{[\alpha}C_{\mu]\nu\rho[\sigma}k_{\beta]}k^{\nu}k^{\rho}=0.$ (8)
In general a space-time admits four PNDs, but if the Petrov type of the Weyl
tensor is special, not type I, then some of these principal null directions
coincide and we have less than four independent solutions to equation (8). The
Petrov classification can be done entirely in terms of the degeneracy of these
directions, which is most easily seen using the spinorial approach due to
Penrose Penrose . For example, in this approach the Weyl tensor is type II if
it admits just three independent PNDs, one of which is doubly degenerate
Stephani . The null vector $k^{\mu}$ is said to point into a degenerate
principal null direction if
$C_{\mu\nu\rho[\sigma}k_{\beta]}k^{\nu}k^{\rho}=0.$ (9)
The intent of this section is to show that the repeated PNDs are deeply
related to the eigenbivectors of the Weyl operator, a result that was implicit
in Bel’s article Bel when he defines the Petrov types, but was not explicitly
enunciated and proved. More precisely in this section the following theorem
will be proved.
Theorem: _If $Z_{\mu\nu}\neq 0$ is a self-dual eigenbivector of the Weyl
operator, $C_{\mu\nu\rho\sigma}Z^{\rho\sigma}\propto Z_{\mu\nu}$, and there
exists a real vector, $k^{\mu}\neq 0$, such that $Z_{\mu\nu}k^{\nu}=0$, then
$k^{\mu}$ points into a repeated PND. Conversely, if $k^{\mu}$ is a repeated
PND then the Weyl operator admits a self-dual eigenbivector $Z_{\mu\nu}$ such
that $Z_{\mu\nu}k^{\nu}=0$._
Evidently this theorem continues to be valid if instead of self-dual
eigenbivectors it is used anti-self-dual eigenbivectors, but to avoid double
work only the self-dual case will be treated. To begin let us see that if
$C_{\mu\nu\rho\sigma}Z^{\rho\sigma}\propto Z_{\mu\nu}$ and
$Z_{\mu\nu}k^{\nu}=0$ then $k^{\mu}$ points in a repeated PND. First note that
$k^{\mu}$ is a null vector:
$0=iZ_{\mu\nu}k^{\nu}=\widetilde{Z}_{\mu\nu}k^{\nu}=\frac{1}{2}k^{\nu}\epsilon_{\mu\nu}^{\phantom{\mu\nu}\rho\sigma}Z_{\rho\sigma}=\frac{1}{2}\epsilon_{\mu}^{\phantom{\mu}\nu\rho\sigma}k_{[\nu}Z_{\rho\sigma]}\,\Rightarrow\,k_{[\nu}Z_{\rho\sigma]}=0\,,$
(10)
contracting this equation with $k^{\nu}$ and using $Z_{\mu\nu}k^{\nu}=0$ we
easily get $k^{\nu}k_{\nu}=0$. Now let us see that $Z_{\mu\nu}$ must be a
simple bivector. Let $e^{\mu}$ be a vector such that $e^{\mu}k_{\mu}=1$, then
contracting the right side of equation (10) with $e^{\nu}$ we obtain
$Z_{\rho\sigma}=v_{\rho}k_{\sigma}-k_{\rho}v_{\sigma}\;\;\textrm{with}\;v_{\sigma}\equiv
Z_{\sigma\nu}e^{\nu}.$ (11)
If we contract the above equation with $k^{\sigma}$ we get that
$v_{\sigma}k^{\sigma}=0$. This relation together with (11) produces
$v^{\rho}Z_{\rho\sigma}=(v^{\rho}v_{\rho})k_{\sigma}$. Now using the self-
duality of $Z_{\mu\nu}$ and the definition of $v_{\mu}$ we have,
$i(v^{\rho}v_{\rho})k_{\sigma}=v^{\rho}\widetilde{Z}_{\rho\sigma}=\frac{1}{2}e_{\mu}Z^{\rho\mu}\epsilon_{\rho\sigma\alpha\beta}Z^{\alpha\beta}=\frac{1}{2}e_{\mu}\epsilon_{\rho\sigma\alpha\beta}Z^{[\alpha\beta}Z^{\rho]\mu}$
But since equation (11) shows that $Z_{\mu\nu}$ is a simple bivector it
follows that $Z^{[\alpha\beta}Z^{\rho]\mu}=0$. Thus summarizing we got
$k^{\mu}v_{\mu}=0$ and $v^{\mu}v_{\mu}=0$. Note that $v^{\mu}$ must be
complex, otherwise we would have $v^{\mu}\propto k^{\mu}$ which implies
$Z_{\mu\nu}=0$. Since $k^{\mu}$ is real and $v^{\mu}$ complex it is possible
to define a null tetrad frame for the tangent bundle,
$\\{l,m,\overline{m},n\\}$, such that $l^{\mu}=k^{\mu}$, $m^{\mu}=v^{\mu}$ and
the only non-vanishing contractions between basis vectors are
$l^{\mu}n_{\mu}=1=-m^{\mu}\overline{m}_{\mu}$. Using the hypothesis that
$Z_{\mu\nu}$ is an eigenbivector of the Weyl operator together with equation
(11) we find:
$0=l^{\nu}Z_{\mu\nu}\propto
l^{\nu}C_{\mu\nu\rho\sigma}Z^{\rho\sigma}=2l^{\nu}C_{\mu\nu\rho\sigma}m^{\rho}l^{\sigma}\;\;\;\Rightarrow\;C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}m^{\sigma}=0$
(12)
Using the above equation and its complex conjugate we see that
$2m^{(\alpha}\overline{m}^{\sigma)}C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}=0$ .
But we can reexpress this tensor equation using the expansion of metric in a
null tetrad frame,
$g^{\alpha\sigma}=2l^{(\alpha}n^{\sigma)}-2m^{(\alpha}\overline{m}^{\sigma)}$:
$0=C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}[(l^{\alpha}n^{\sigma}+n^{\alpha}l^{\sigma})-g^{\alpha\sigma}]\;\Rightarrow\;l_{\alpha}C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}n^{\sigma}=C_{\mu\nu\rho\alpha}l^{\nu}l^{\rho}\,.$
From which it follows that $k^{\mu}=l^{\mu}$ satisfies equation (9). This
means that $k^{\mu}$ points in a repeated PND, proving the first part of the
theorem.
Now suppose that $k^{\mu}\neq 0$ points in a repeated PND, which means that it
obeys (9). Let us set $l^{\mu}=k^{\mu}$ and complete the null tetrad frame
$\\{l^{\mu},m^{\mu},\overline{m}^{\mu},n^{\mu}\\}$. Equation (9) says that
$l_{\alpha}C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}=l_{\mu}C_{\alpha\nu\rho\sigma}l^{\nu}l^{\rho}$.
Contracting this with $m^{\mu}n^{\alpha}$ we get
$m^{\mu}C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}=0$. Which implies that
$T_{\rho\sigma}=C_{\rho\sigma\mu\nu}l^{\mu}m^{\nu}$ is a bivector such that
$T_{\rho\sigma}l^{\sigma}=0$. Now since
$\\{l^{\mu},m^{\mu},\overline{m}^{\mu},n^{\mu}\\}$ is a basis for the tangent
bundle the following expansion is always valid:
$T_{\mu\nu}=A\,l_{[\mu}m_{\nu]}+B\,l_{[\mu}\overline{m}_{\nu]}+C\,l_{[\mu}n_{\nu]}+D\,m_{[\mu}\overline{m}_{\nu]}+E\,m_{[\mu}\
n_{\nu]}+F\,\overline{m}_{[\mu}\ n_{\nu]}.$
Where $A,B,C,D,E$ and $F$ are complex numbers. Using
$T_{\rho\sigma}l^{\sigma}=0$ it follows that $C=E=F=0$. Also since the Weyl
operator maps self-dual bivectors into self-dual bivectors and
$l_{[\mu}m_{\nu]}$ is self-dual then $T_{\mu\nu}$ must also be self-dual. This
fact implies that $B=D=0$, so that $T_{\mu\nu}=Al_{[\mu}m_{\nu]}$, i.e.,
$l_{[\mu}m_{\nu]}$ is a self-dual eigenbivector of the Weyl operator such that
$T_{\mu\nu}l^{\nu}=0$, finishing the proof of theorem.
This section dealt only with Lorentzian signature, thus natural questions that
can be raised are: (1) What should be the analogues of repeated PNDs in the
other signatures and in the case of complexified manifolds? (2) Are these
analogues of repeated PNDs related to eigenbivectors of Weyl operator? The
above theorem and its proof give some hint toward the answer of these
questions, since it was shown that to every repeated principal null direction,
$l^{\mu}$, it is associated a simple self-dual eigenbivector of the Weyl
operator, $Z_{\mu\nu}=l_{\mu}m_{\nu}-m_{\mu}l_{\nu}$, and vice versa. So it is
natural to guess that the objects that will replace the repeated PNDs in a
general four-dimensional manifold are the simple self-dual eigenbivectors of
Weyl operator. Indeed, we will see in forthcoming sections that this
conjecture is correct.
## IV Generalization of the Goldberg-Sachs Theorem
The most important theorem related to the Weyl tensor classification in
Lorentzian manifolds is the Goldberg-Sachs theorem Goldberg-Sachs . It states
that a vacuum space-time has an algebraically special Weyl tensor if, and only
if, the repeated principal null direction generates a null congruence that is
geodesic and shear-free. Thirteen years after the proof of this theorem J.
Pleblański created a classification to the Weyl tensor in complexified four-
dimensional manifolds Plebanski75 and proved together with S. Hacyan the
analogue of Goldberg-Sachs theorem in these manifolds Plebanski2 . In this
section this important theorem will be trivially generalized to four-
dimensional manifolds of all signatures, this generalization is an important
ingredient to answer the two questions raised at the end of last section as
well as to investigate the geometric consequences of an algebraically special
Weyl tensor.
Let $(M,g_{\mu\nu})$ be a four-dimensional manifold (complexified or real of
any signature) with vanishing Ricci tensor (vacuum)222Throughout this and the
next section the Ricci tensor will always be assumed to vanish. Also the
tangent bundle is assumed to be endowed with a torsion-free connection
compatible with the metric (Levi-Civita), only this kind connection is
considered in this article., so that the Riemann tensor is equal to the Weyl
tensor. Let $\\{e_{1},e_{2},e_{3},e_{4}\\}$ be a null tetrad frame for the
tangent bundle, defined to be such that the only non-zero contractions are
$e_{1}^{\mu}e_{3\,\mu}=1=-e_{2}^{\mu}e_{4\,\mu}$. The components of the metric
in this basis are denoted by $g_{ab}=g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}$. The
dual frame of 1-forms is denoted by $\\{e^{1},e^{2},e^{3},e^{4}\\}$,
$e^{a}(e_{b})=\delta^{a}_{b}$ 333Note that
$e^{1}_{\phantom{1}\mu}=e_{3\,\mu}$, $e^{2}_{\phantom{2}\mu}=-e_{4\,\mu}$,
$e^{3}_{\phantom{3}\mu}=e_{1\,\mu}$ and $e^{4}_{\phantom{4}\mu}=-e_{2\,\mu}$..
Let us denote a set of ten components of the Weyl tensor by:
$\displaystyle\Psi^{+}_{0}\equiv C_{1212}\;;\;\Psi^{+}_{1}\equiv
C_{1312}\;;\;\Psi^{+}_{2}\equiv C_{1243}\;;\;\Psi^{+}_{3}\equiv
C_{1343}\;;\;\Psi^{+}_{4}\equiv C_{3434}$ $\displaystyle\Psi^{-}_{0}\equiv
C_{1414}\;;\;\Psi^{-}_{1}\equiv C_{1314}\;;\;\Psi^{-}_{2}\equiv
C_{1423}\;;\;\Psi^{-}_{3}\equiv C_{1323}\;;\;\Psi^{-}_{4}\equiv C_{3232}.$
(13)
Where, for example, $C_{1312}\equiv
C_{\mu\nu\rho\sigma}e_{1}^{\phantom{1}\mu}e_{3}^{\phantom{1}\nu}e_{1}^{\phantom{1}\rho}e_{2}^{\phantom{1}\sigma}$
and the scalars $\Psi_{A}^{\pm}$ are called the Weyl scalars. The self-dual
part of the Weyl tensor, $C^{+}$, depends only on the scalars $\Psi_{A}^{+}$,
while $C^{-}$ depends only on $\Psi_{A}^{-}$. The vanishing of the Ricci
tensor and the first Bianchi identity satisfied by the Riemann tensor can be
summarized by the following equations:
$\displaystyle C_{2123}$ $\displaystyle=$ $\displaystyle
C_{4143}=C_{1214}=C_{3234}=0\;;$ $\displaystyle C_{2124}$ $\displaystyle=$
$\displaystyle\Psi^{+}_{1}\;;\;C_{4142}=\Psi_{1}^{-}\;;\;C_{2324}=\Psi_{3}^{-}\;;\;C_{4342}=\Psi^{+}_{3}\;;$
(14) $\displaystyle C_{2424}$ $\displaystyle=$ $\displaystyle
C_{1313}=\Psi^{+}_{2}+\Psi_{2}^{-}\;;\;C_{1324}=\Psi_{2}^{-}-\Psi^{+}_{2}.$
The various algebraic types of the Weyl tensor can be characterized by the
possibility of annihilating some of the Weyl scalars by a suitable choice of
null tetrad art1 . For example, when $C^{+}$ is type N+ it is possible to find
a null frame in which $\Psi^{+}_{0}=\Psi^{+}_{1}=\Psi^{+}_{2}=\Psi^{+}_{3}=0$
and $\Psi^{+}_{4}\neq 0$. The table below shows which Weyl scalars can be set
to zero by conveniently choosing the null tetrad in each of the possible types
of $C^{+}$.
Weyl Scalars that Can be Made to Vanish by a Suitable Choice of Basis
Type O+ \- All $\Psi^{+}_{A}$ | Type I+ \- $\Psi^{+}_{0},\Psi^{+}_{4}$ | Type D+ \- $\Psi^{+}_{0},\Psi^{+}_{1},\Psi^{+}_{3},\Psi^{+}_{4}$
---|---|---
Type II+ -$\Psi^{+}_{0},\Psi^{+}_{1},\Psi^{+}_{4}$ | Type III+ \- $\Psi^{+}_{0},\Psi^{+}_{1},\Psi^{+}_{2},\Psi^{+}_{4}\>$ | Type N+ \- $\Psi^{+}_{0},\Psi^{+}_{1},\Psi^{+}_{2},\Psi^{+}_{3}$
Obviously a similar table can be constructed for the types of $C^{-}$ in terms
of $\Psi_{A}^{-}$. Particularly note that when $C^{+}$ is algebraically
special, i.e. not type I+, it is always possible to find a null frame in which
$\Psi^{+}_{0}=\Psi^{+}_{1}=0$. Conversely, when $\Psi^{+}_{0}=\Psi^{+}_{1}=0$
in some null tetrad then $C^{+}$ is algebraically special.
Now choosing conveniently the sign of the volume form it follows that the
below bivectors form a basis for the space of self-dual and anti-self-dual
bivectors respectively:
$\displaystyle Z^{1+}=e^{4}\wedge e^{3}\;;\;Z^{2+}=e^{1}\wedge
e^{2}\;;\;Z^{3+}=\frac{1}{\sqrt{2}}[e^{1}\wedge e^{3}-e^{2}\wedge e^{4}]$
$\displaystyle Z^{1-}=e^{2}\wedge e^{3}\;;\;Z^{2-}=e^{1}\wedge
e^{4}\;;\;Z^{3-}=\frac{1}{\sqrt{2}}[e^{1}\wedge e^{3}-e^{4}\wedge e^{2}]$ (15)
Note that the bivectors $Z^{3\pm}$ are not simple, since
$\widetilde{Z^{3\pm}}^{\mu\nu}Z^{3\pm}_{\mu\nu}=\pm\varepsilon
Z^{3\pm\,\mu\nu}Z^{3\pm}_{\mu\nu}=\mp 2\varepsilon\neq 0$, while $Z^{1\pm}$
and $Z^{2\pm}$ are obviously simple. Now let $Z_{\mu\nu}$ be a simple and
self-dual bivector, then it is not difficult to see that it is possible to
find a null tetrad frame in which
$Z_{\mu\nu}=Z^{1+}_{\mu\nu}=2e_{1[\mu}e_{2\nu]}$. Using this frame we have:
$C_{\mu\nu\rho\sigma}e_{1}^{\phantom{1}\rho}e_{2}^{\phantom{2}\sigma}=\lambda
e_{1[\mu}e_{2\nu]}\;\Rightarrow\;\Psi^{+}_{0}=\lambda
e_{1\mu}e_{2\nu}e_{1}^{\phantom{1}[\mu}e_{2}^{\phantom{2}\nu]}=0\;\;\textrm{and}\;\;\Psi^{+}_{1}=\lambda
e_{1\mu}e_{2\nu}e_{1}^{\phantom{1}[\mu}e_{3}^{\phantom{2}\nu]}=0.$
This means that if the Weyl operator admits a simple self-dual eigenbivector
then it is possible to find a null tetrad frame where
$\Psi^{+}_{0}=\Psi^{+}_{1}=0$. Conversely, after some algebra it can be proved
that when Weyl operator acts in $Z^{1+}$ we get
$C(Z^{1+})=2\Psi^{+}_{2}Z^{1+}+2\Psi^{+}_{0}Z^{2+}+2\sqrt{2}\Psi^{+}_{1}Z^{3+}$,
thus if $\Psi^{+}_{0}=\Psi^{+}_{1}=0$ then $Z^{1+}$ is a simple self-dual
eigenbivector of the Weyl operator. Analogously $\Psi^{+}_{3}=\Psi^{+}_{4}=0$
if, and only if, $Z^{2+}$ is an eigenbivector of the Weyl operator. From
preceding results can be stated:
_Weyl operator $C^{+}$ is algebraically special if, and only if, it admits a
simple self-dual eigenbivector. Analogously, $C^{-}$ is algebraically special
if, and only if, it admits a simple anti-self-dual eigenbivector._
Previous section established that in Lorentzian signature to every repeated
PND it is associated a simple self-dual eigenbivector of the Weyl operator.
Since the existence of a repeated principal null direction is equivalent to
the Weyl tensor being algebraically special, the above result endorse the
conjecture that the natural extension of repeated PNDs to other signatures are
the simple self-dual eigenbivectors of the Weyl operator. The Goldberg-Sachs
theorem says that in vacuum space-times the repeated PNDs are tangent to
shear-free null geodesic congruences, which is an integrability property
related to these directions. So to the conjecture be correct it is important
to relate these simple self-dual eigenbivectors of the Weyl operator to some
integrability property. This will be accomplished in what follows.
Denoting the Levi-Civita connection by $\nabla_{\mu}$ then the connection
1-forms, $\omega^{a}_{\phantom{a}b}$, and its components in the null frame are
defined by the following equations:
$\nabla_{X}e_{b}\equiv
X^{\mu}\nabla_{\mu}e_{b}=\omega^{a}_{\phantom{a}b}(X)e_{a}\;\;;\;\;\omega_{ab}^{\phantom{ab}c}\equiv\omega^{c}_{\phantom{c}b}(e_{a})\,,$
(16)
where $X$ is an arbitrary vector field. The indices of the connection 1-forms
can be lowered by means of the metric,
$\omega_{ab}=g_{ac}\omega^{c}_{\phantom{c}b}$,
$\omega_{abc}=g_{cd}\omega_{ab}^{\phantom{ab}d}$. Since the components of the
metric in this frame are constant it follows that $\omega_{ab}=-\omega_{ba}$
and $\omega_{abc}=-\omega_{acb}$. The Cartan structure equations when the
Ricci tensor vanishes are given by:
$de^{a}+\omega^{a}_{\phantom{a}b}\wedge
e^{b}=0\;\,;\,\;\frac{1}{2}C^{a}_{\phantom{a}bcd}e^{c}\wedge
e^{d}=d\omega^{a}_{\phantom{a}b}+\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}.$
(17)
By means of equations (IV), (IV) and (IV) it follows that the self-dual part
of the second structure equation can be written in the below form.
$\displaystyle\Psi_{2}^{+}\,Z^{1+}+\Psi_{0}^{+}\,Z^{2+}+\sqrt{2}\Psi_{1}^{+}\,Z^{3+}$
$\displaystyle=$ $\displaystyle
d\omega_{12}+\omega_{12}\wedge(\omega_{24}-\omega_{13})$ (18)
$\displaystyle\Psi_{4}^{+}\,Z^{1+}+\Psi_{2}^{+}\,Z^{2+}+\sqrt{2}\Psi_{3}^{+}\,Z^{3+}$
$\displaystyle=$ $\displaystyle
d\omega_{43}-\omega_{43}\wedge(\omega_{24}-\omega_{13})$ (19)
$\displaystyle-2\Psi_{3}^{+}\,Z^{1+}-2\Psi_{1}^{+}\,Z^{2+}-2\sqrt{2}\Psi_{2}^{+}\,Z^{3+}$
$\displaystyle=$ $\displaystyle
d(\omega_{24}-\omega_{13})+2\omega_{12}\wedge\omega_{43}$ (20)
Making the changes $Z^{i+}\rightarrow Z^{i-}$,
$\Psi_{A}^{+}\rightarrow\Psi_{A}^{-}$, $\omega_{12}\rightarrow\omega_{14}$,
$\omega_{43}\rightarrow\omega_{23}$ and $\omega_{24}\rightarrow\omega_{42}$ in
(18), (19) and (20) we get the other three missing components of the second
structure equation.
When the self-dual part of the Weyl tensor vanishes, $\Psi_{A}^{+}=0$ for all
$A$, it is seen that a possible solution to equations (18), (19) and (20) is
$\omega_{12}=\omega_{34}=0$ and $\omega_{24}=\omega_{13}$. Conversely if
$\omega_{12}=\omega_{34}=0$ then equations (18) and (19) implies that
$\Psi_{A}^{+}=0$ for all $A$. Since this result will be important in the next
section let us stress it:
_When $C^{+}$ vanishes there exists some null frame in which
$\omega_{12}=\omega_{34}=0$ and $\omega_{24}=\omega_{13}$. Conversely, if
$\omega_{12}=\omega_{34}=0$ in some null frame then the self-dual part of the
Weyl tensor is zero. _
Now the generalization of the Goldberg-Sachs(GS) theorem to all four-
dimensional manifolds will be enunciated. In what follows this theorem will be
dubbed the GSHP theorem, since Plebański and Hacyan were responsible for the
extension of GS theorem to the case of complexified manifolds of complex
dimension four. Here the theorem will be extended in a trivial way to real
four-dimensional manifolds of all signatures but the proof will be omitted
because it is basically the same of the complexified case Plebanski2 .
GSHP Theorem: Let $(M,g_{\mu\nu})$ be a four-dimensional manifold
(complexified or real with any signature) with vanishing Ricci tensor, then
the Weyl scalars $\Psi_{0}^{+}$ and $\Psi_{1}^{+}$ vanish if, and only if,
there exists a null tetrad frame in which the connection components
$\omega_{112}$ and $\omega_{221}$ are both zero.
Next section will be devoted to explore the consequences of this theorem in
all signatures as well as in complexified manifolds. Before this, it is worth
mentioning that the GSHP theorem obviously has an analogous version related to
the anti-self-dual part of the Weyl tensor. More precisely can be stated that
the Weyl scalars $\Psi_{0}^{-}$ and $\Psi_{1}^{-}$ vanish if, and only if,
there exists a null tetrad frame in which the connection components
$\omega_{114}$ and $\omega_{441}$ are zero. Also making the changes
$e_{1}\leftrightarrow e_{3}$ and $e_{2}\leftrightarrow e_{4}$ we get that
$\Psi_{4}^{+}$ and $\Psi_{3}^{+}$ vanish if, and only if, there is a null
tetrad frame in which the connection components $\omega_{334}$ and
$\omega_{443}$ vanish, similarly $\Psi_{4}^{-}$ and $\Psi_{3}^{-}$ vanish if,
and only if, there is a null tetrad frame in which $\omega_{332}$ and
$\omega_{223}$ are both zero.
As a last comment it shall be mentioned that the GSHP theorem is also valid if
instead of vanishing Ricci tensor it is assumed that the Ricci tensor is
proportional to the metric, an Einstein manifold Plebanski2 . Recently it was
investigated in Nurwoski2 whether a less restrictive condition can be imposed
to the Ricci tensor while keeping the GSHP theorem valid. In Lorentzian
signature a conformally invariant version of the GS theorem was proved in
reference GS-CottonYork .
## V The Consequences of GSHP Theorem
Let us calculate the Lie bracket of vectors $e_{a}$ and $e_{b}$:
$[e_{a},e_{b}]=e_{a}^{\phantom{a}\mu}\nabla_{\mu}e_{b}-e_{b}^{\phantom{b}\mu}\nabla_{\mu}e_{a}=\nabla_{a}e_{b}-\nabla_{b}e_{a}=(\omega_{ab}^{\phantom{ab}c}-\omega_{ba}^{\phantom{ab}c})e_{c}\,.$
So, using the identity $\omega_{abc}=-\omega_{acb}$, the above relation
implies that
$[e_{1},e_{2}]=(\omega_{12}^{\phantom{12}c}-\omega_{21}^{\phantom{12}c})e_{c}=(\omega_{123}-\omega_{213})e_{1}-(\omega_{124}-\omega_{214})e_{2}+\omega_{121}e_{3}+\omega_{212}e_{4}.$
(21)
Since $[e_{1},e_{1}]$ and $[e_{2},e_{2}]$ are trivially zero, it follows from
(21) that the distribution generated by the vector fields $\\{e_{1},e_{2}\\}$
is integrable if, and only if, $\omega_{112}=\omega_{221}=0$. Thus what the
GSHP theorem says is that the integrability of the planes generated by
$\\{e_{1},e_{2}\\}$ is equivalent to the vanishing of the Weyl scalars
$\Psi^{+}_{0}$ and $\Psi^{+}_{1}$. Since
$e_{1}^{\phantom{1}\mu}e_{1\,\mu}=e_{2}^{\phantom{2}\mu}e_{2\,\mu}=e_{1}^{\phantom{1}\mu}e_{2\,\mu}=0$,
then all vectors tangent to the planes generated by $\\{e_{1},e_{2}\\}$ are
null, this kind of distribution is called isotropic or totally null. More
about isotropic subspaces can be found in Simple Spinors . Thus, in other
words, the theorem proved in the last section states that in a Ricci-flat
four-dimensional manifold the Weyl tensor is algebraically special if, and
only if, the manifold admits an integrable foliation of isotropic planes. In
complexified manifolds this result was obtained in Plebanski2 , where it was
also proved that this two-dimensional foliation is extremal, in the sense that
it can be obtained from the extremization of some functional.
Simple bivectors that generate isotropic distributions are called null
bivectors. For example, the bivector $Z^{1+}_{\mu\nu}=2e_{1[\mu}e_{2\nu]}$ is
null. In four dimensions a bivector is null if, and only if, it is simple and
self-dual or simple and anti-self-dual. In the last section it was
demonstrated that the Weyl tensor admits a null eigenbivector if, and only if,
there exists some null frame in which $\Psi^{+}_{0}=\Psi^{+}_{1}=0$. In this
frame $Z^{1+}$ is an eigenbivector of the Weyl tensor. But from the above
results the distribution generated by $Z^{1+}$, $\\{e_{1},e_{2}\\}$, is
integrable. Thus arriving at the following important result:
_In a Ricci-flat manifold the Weyl tensor admits a null eigenbivector if, and
only if, the isotropic distribution generated by this bivector is integrable._
The author is not aware of any previous literature that arrived at this
statement. Such result shows that the bivector approach to the classification
of the Weyl tensor is useful and fruitful to analyze the integrability
properties in an unified way in all signatures. So the bivector method used by
A. Z. Petrov Petrov and for long time abandoned is, after all, suitable and
convenient for some types of studies. Before proceeding it will be introduced
some definitions and notation that will be important in what follows.
Given a manifold $(M,g_{\mu\nu})$, an almost complex structure on this
manifold is an endomorphism of the tangent bundle, $J:TM\rightarrow TM$, such
that its square is minus the identity map, $J(J(V))=-V$ for all $V\in TM$. Let
us define the following almost complex structure:
$J\equiv i(e_{1}\otimes e^{1}+e_{2}\otimes e^{2})-i(e_{3}\otimes
e^{3}+e_{4}\otimes e^{4})\,.$ (22)
It is easy to see that this almost complex structure has the important
property of leaving the metric invariant, $g(X,Y)=g(J(X),J(Y))$ for all
$X,Y\in TM$. Because of this the metric is said to be Hermitian with respect
to $J$. The operator $J$ naturally splits the tangent bundle into a direct sum
of two bundles of the same dimension,
$TM=TM^{+}\oplus TM^{-}\;\;\;\textrm{with}\;\;\;TM^{\pm}\equiv\\{V\in
TM\,|\,J(V)=\pm iV\\}.$
When both bundles $TM^{+}$ and $TM^{-}$ are integrable the almost complex
structure $J$ is said to be integrable. For the $J$ defined on equation (22)
we have $TM^{+}=\textrm{Span}\\{e_{1},e_{2}\\}$ and
$TM^{-}=\textrm{Span}\\{e_{3},e_{4}\\}$, so that from the previous results we
conclude that $J$ is integrable if, and only if, $\omega_{112}$,
$\omega_{221}$, $\omega_{334}$ and $\omega_{443}$ all vanish. It is easy to
prove that the integrability of $J$ is equivalent to the vanishing of the
Nijenhuis tensor, $N$, defined by the below equation Nakahara .
$N:TM\times TM\rightarrow
TM\;\;\;;\;\;\;N(X,Y)=[X,Y]-[J(X),J(Y)]+J([J(X),Y])+J([X,J(Y)]).$
The Kähler form, $\Omega$, is the 2-form constructed from $J$ and $g$ whose
action on $TM\times TM$ is defined by $\Omega(X,Y)=g(J(X),Y)$. For the $J$
defined on equation (22) we have:
$\Omega\,=\,i(e^{1}\wedge e^{3}+e^{4}\wedge e^{2})\,=\,i\sqrt{2}\,Z^{3+}.$
(23)
For a complex manifold444Where by complex manifold it is meant a manifold
which over the complex field can be covered by charts with analytic transition
functions. if the metric is Hermitian with respect to an integrable $J$ and
$\Omega$ is a closed form, $d\Omega=0$, then the manifold is called a Kähler
manifold. If besides this the curvature is Ricci-flat, as will be assumed in
this section, the manifold is said to be a Calabi-Yau manifold555Actually a
Calabi-Yau manifold is defined to be a Kähler manifold with vanishing first
Chern class. When the Ricci tensor is zero the first Chern class vanishes
trivially. Conversely, it can be proved that a Kähler manifold with vanishing
first Chern class admits a Ricci-flat metric.. For later convenience let us
calculate the exterior derivative of the Kähler form:
$d\Omega=i[de^{1}\wedge e^{3}-e^{1}\wedge de^{3}+de^{4}\wedge
e^{2}-e^{4}\wedge de^{2}]=i[-\omega^{1}_{\phantom{1}a}\wedge e^{a}\wedge
e^{3}+e^{1}\wedge\omega^{3}_{\phantom{3}a}\wedge
e^{a}-\omega^{4}_{\phantom{4}a}\wedge e^{a}\wedge
e^{2}+e^{4}\wedge\omega^{2}_{\phantom{2}a}\wedge e^{a}]=$
$=-2i\omega_{12}\wedge e^{1}\wedge e^{2}+2i\omega_{34}\wedge e^{3}\wedge
e^{4}$ (24)
Since $\omega_{ab}=\omega_{cba}e^{c}$ then the above equation implies that the
closeness of $\Omega$ together with the integrability of $J$ (Kähler
condition) is equivalent to the vanishing of $\omega_{12}$ and $\omega_{34}$.
Using the relations $\nabla_{a}e_{b}=\omega_{ab}^{\phantom{ab}c}e_{c}$ and
$\nabla_{a}e^{b}=\omega_{a\phantom{b}c}^{\phantom{a}b}e^{c}$ it is
straightforward to calculate $\nabla_{a}J$. For example,
$\nabla_{1}J=2\omega_{134}(e_{1}\otimes e^{4}+e_{2}\otimes
e^{3})+2\omega_{121}(e_{3}\otimes e^{2}+e_{4}\otimes e^{1})$. Computing the
other terms we get that $J$ is covariantly constant if, and only if, the
connection 1-forms $\omega_{12}$ and $\omega_{34}$ vanish. Then can be stated:
$J\;\textrm{integrable
and}\;\;d\Omega=0\;\;\;\Leftrightarrow\;\;\;\nabla_{X}J=0\;\;\forall\;X\in
TM\;\;\;\Leftrightarrow\;\;\;\omega_{12}\,=\,\omega_{34}\,=\,0\,.$ (25)
Now the consequences of the above results will be investigated in complexified
manifolds as well as in real manifolds with all types of signature. Important
attempts on the lines presented below can be found in Robinson Manifolds ;
Nurwoski2 , where the integrable isotropic structures are investigated in
detail. In what follows new results are presented, in particular it is proved
that when $C^{+}$ vanishes the manifold admits a covariantly constant rank two
tensor and a solution for the source-free Einstein-Maxwell equations.
### V.1 Complexified Manifolds
In the case of $(M,g_{\mu\nu})$ being a complexified Riemannian manifold of
complex dimension four, the ten complex Weyl scalars, $\Psi_{A}^{\pm}$, are
independent of each other and all the 21 types of classification for the Weyl
tensor are realizable. By means of the canonical forms of each type, described
in reference art1 , we can see that if the type of the Weyl tensor is (I,I)
then the Weyl operator does not have any null eigenbivector, so that the
manifold does not admit integrable isotropic planes. For types (II,I),(III,I)
and (N,I) the Weyl tensor admits just one independent null eigenbivector, thus
just one foliation of isotropic planes. In types (II,II), (II,III), (II,N),
(III,III), (III,N), (N,N) and (D,I) there are two independent distributions of
integrable isotropic planes, while types (D,II), (D,III) and (D,N) allow three
such distributions. Type (D,D) admits four independent integrable isotropic
planes. For types (O,I), (O,II), (O,D), (O,III), (O,N) and (O,O), when $C^{+}$
or $C^{-}$ vanishes, there are infinitely many integrable isotropic planes.
The most interesting results appear when the Weyl tensor is type (D,something)
or type (O,something). In these cases there is some null tetrad frame in which
$\Psi_{0}^{+}=\Psi_{1}^{+}=\Psi_{3}^{+}=\Psi_{4}^{+}=0$ art1 . As mentioned at
the beginning of section IV this implies that
$Z^{1+}_{\mu\nu}=2e_{1[\mu}e_{2\nu]}$ and
$Z^{2+}_{\mu\nu}=2e_{4[\mu}e_{3\nu]}$ are simple self-dual eigenbivectors of
the Weyl tensor, so that, as seen above, the planes generated by
$\\{e_{1},e_{2}\\}$ and $\\{e_{4},e_{3}\\}$ are integrable. But these planes
are the ones that generate $TM^{+}$ and $TM^{-}$ respectively, thus these
bundles are integrable. Conversely suppose that the Ricci-flat manifold
$(M,g)$ has an integrable almost complex structure, $J$. This means that
$TM^{+}$ and $TM^{-}$ are integrable. If $g$ is a Hermitian metric with
respect to $J$ we have that these bundles are isotropic. For example, if
$X^{+},Y^{+}\in TM^{+}$ then
$g(X^{+},Y^{+})=g(J(X^{+}),J(Y^{+}))=g(iX^{+},iY^{+})=-g(X^{+},Y^{+})$, so
$g(X^{+},Y^{+})=0$. Now if $\\{e_{1},e_{2}\\}$ is some basis of $TM^{+}$ then,
since the metric is non-degenerate, it is always possible to find a basis
$\\{E_{3},E_{4}\\}$ for $TM^{-}$ such that $g(e_{1},E_{3})=-g(e_{2},E_{4})=1$,
$g(e_{1},E_{4})=a$, $g(e_{2},E_{3})=b$. If $a=b=0$ put $e_{3}=E_{3}$ and
$e_{4}=E_{4}$, if $a\neq 0$ and $b=0$ put $e_{3}=E_{3}$ and
$e_{4}=E_{4}-aE_{3}$, and if $a\neq 0\neq b$ and $(1+ba)\neq 0$ put
$e_{3}=\frac{1}{1+ba}(E_{3}+bE_{4})$ and $e_{4}=\frac{1}{1+ba}(E_{4}-aE_{3})$.
The case $a\neq 0\neq b$ and $(1+ba)=0$ is not possible, since in this case
there would be an isotropic space of dimension three. Then the vectors
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ form a null tetrad frame and the planes
generated by $\\{e_{1},e_{2}\\}$ and $\\{e_{3},e_{4}\\}$ are integrable, this
implies that $\Psi_{0}^{+}=\Psi_{1}^{+}=0$ and $\Psi_{3}^{+}=\Psi_{4}^{+}=0$,
so Weyl tensor is type (D,something) or type (O,something). This paragraph can
be summarized by the following words:
_In a Ricci-flat complexified manifold the self-dual part of the Weyl
operator, $C^{+}$, is type D+ or type O+ if, and only if, there is some null
tetrad frame in which the almost complex structure defined in (22) is
integrable. In other words, $C^{+}$ is type D+ or type O+ if, and only if, the
Ricci-flat complexified manifold admits an integrable almost complex structure
such that the metric is Hermitian with respect to it. _
In the particular case of Weyl tensor being type (O,something),
$\Psi_{A}^{+}=0$ for all $A\in(0,1,2,3,4)$, equations (18), (19) and (20)
implies that there is some null tetrad frame where the connection 1-forms
$\omega_{12}$ and $\omega_{43}$ vanish while $\omega_{24}=\omega_{13}$. This
together with equation (24) implies that $J$ is integrable and the exterior
derivative of the Kähler form is zero666Note that if $\omega_{12}=0$,
$\omega_{43}=0$ and $\omega_{24}=\omega_{13}$ then all the isotropic
distributions $\\{ae_{1}+be_{4},ae_{2}+be_{3}\\}$ for $a,b$ constants are
integrable. Thus anti-self-dual manifolds admit infinitely many integrable
self-dual isotropic distributions.. The Euclidean version of this result was
previously obtained in Broda . Now according to (25) this is equivalent to $J$
being a constant tensor. Conversely, if $J$ is an integrable almost complex
structure and $d\Omega=0$, then equation (24) together with the reasoning of
the last paragraph implies that there exists a null tetrad frame such that
$\\{e_{1},e_{2}\\}$ generates $TM^{+}$ and $\\{e_{3},e_{4}\\}$ generates
$TM^{-}$ and such that $\omega_{12}=\omega_{34}=0$. Inserting this last
equality into equations (18) and (19) we have
$\Psi_{0}^{+}=\Psi_{1}^{+}=\Psi_{2}^{+}=\Psi_{3}^{+}=\Psi_{4}^{+}=0$. Thus we
can state:
_In a Ricci-flat complexified manifold the self-dual (or the anti-self-dual)
part of the Weyl operator, $C^{+}$ ($C^{-}$), vanishes if, and only if, there
is some null tetrad frame in which the almost complex structure defined in
(22) is covariantly constant. In the case of a complex manifold this means
that $C^{+}$ (or $C^{-}$) vanishes if, and only if, the manifold is Calabi-
Yau._
### V.2 Euclidean Signature
In a real four-dimensional Euclidean Ricci-flat manifold $(M,g)$, given an
orthonormal frame, $\\{E_{1},E_{2},E_{3},E_{4}\\}$ with
$g(E_{a},E_{b})=\delta_{ab}$, it is possible to construct the following null
tetrad frame in the complexified tangent bundle, $\mathbb{C}\otimes TM$:
$e_{1}=\frac{1}{\sqrt{2}}(E_{1}+iE_{2})\,,\,e_{3}=\frac{1}{\sqrt{2}}(E_{1}-iE_{2})\,,\,e_{2}=\frac{1}{\sqrt{2}}(E_{3}+iE_{4})\;\textrm{and}\;e_{4}=\frac{-1}{\sqrt{2}}(E_{3}-iE_{4})\,$.
Note that since $\\{E_{a}\\}$ are real vector fields it follows that
$\overline{e_{1}}=e_{3}$ and $\overline{e_{2}}=-e_{4}$. For the purposes of
this section it can always be assumed that the basis vectors obey to these
reality conditions. Therefore the almost complex structure defined in (22) and
the Kähler form of equation (23) are real tensors, $\overline{J}=J$ and
$\overline{\Omega}=\Omega$, while the Weyl scalars are such that
$\overline{\Psi^{\pm}_{0}}=\Psi^{\pm}_{4}$ ,
$\overline{\Psi^{\pm}_{1}}=-\Psi^{\pm}_{3}$ and
$\overline{\Psi^{\pm}_{2}}=\Psi^{\pm}_{2}$. This implies that the only allowed
types for the Weyl tensor are (O,O), (O,I), (O,D), (I,I), (I,D) and (D,D) art1
. In particular note that if the Weyl tensor is algebraically special, not
type (I,I), then conveniently choosing the sign of the volume form we can
guarantee that $C^{+}$ is type D+ or type O+.
Results of subsection V.1 implies that if $C^{+}$ is type D+ or O+ there is
some null frame such that the almost complex structure defined in (22) is
integrable. If $C^{+}$ is strictly type O+ the Kähler form is closed and the
almost complex structure $J$ is covariantly constant. Since the real 2-form
$\Omega$ is non-degenerate it follows that when the self-dual part of the Weyl
tensor vanishes, $C^{+}$ is type O+, the real manifold is symplectic, with
symplectic form $\Omega$.
An important theorem in complex differential geometry Newlander , the
Newlander-Nirenberg theorem, states that a manifold admits an integrable and
real almost complex structure if, and only if, it is a complex
manifold777Meaning that the manifold over the complex field can be covered by
charts with analytic transition functions.. Since in Euclidean case $J$ is
real it follows that when $C^{+}$ is type D+ or type O+ the manifold over the
complex field is a complex manifold. In the particular case of $C^{+}$ being
type O+ the manifold is a Kähler manifold, more precisely a Calabi-Yau, since
Ricci tensor is assumed to vanish. Conversely, as seen in last subsection, if
the manifold is Calabi-Yau then the self-dual part of the Weyl tensor must
vanish. However it must be noted that a manifold can be Calabi-Yau but
constructed from the complexification of a non-Euclidean real manifold. The
above results are summarized by the following stressed results, one of which
is here dubbed the Euclidean version of the GS theorem:
_When the Weyl tensor in a Ricci-flat Euclidean manifold, $M$, is not type
(I,I) the manifold over the complex field is a complex manifold. Particularly,
if the self-dual part of the Weyl tensor vanishes then the complexification of
$M$ is a Calabi-Yau manifold and the real tensor $J$ is covariantly constant.
Conversely, when the manifold is Calabi-Yau the self-dual part of the Weyl
tensor must vanish, although the manifold may not be the complexification of a
real Euclidean manifold. _
Euclidean version of GS theorem: _In vacuum the Weyl tensor is algebraically
special if, and only if, the tangent bundle admits a real integrable almost
complex structure._
### V.3 Lorentzian Signature
Lorentzian four-dimensional manifolds are characterized by the existence of a
real frame {$e_{t},e_{x},e_{y},e_{z}$} such that the only non-zero
contractions are $e^{\mu}_{t}e_{t\mu}=1$ and
$e^{\mu}_{x}e_{x\mu}=e^{\mu}_{y}e_{y\mu}=e^{\mu}_{z}e_{z\mu}=-1$. Null tetrad
frames can be constructed in the complexification of the tangent bundle, one
example being $e_{1}=l=\frac{1}{\sqrt{2}}(e_{t}+e_{z})$,
$e_{2}=m=\frac{1}{\sqrt{2}}(e_{x}+ie_{y})$,
$e_{3}=n=\frac{1}{\sqrt{2}}(e_{t}-e_{z})$ and
$e_{4}=\overline{m}=\frac{1}{\sqrt{2}}(e_{x}-ie_{y})$. Note that vector fields
$e_{1}$ and $e_{3}$ are real while $e_{2}$ and $e_{4}$ are complex and
conjugates to each other, therefore the Weyl scalars $\Psi_{A}^{-}$ are the
complex conjugates of $\Psi_{A}^{+}$ and only types (O,O), (I,I), (D,D),
(II,II), (III,III) and (N,N) are realizable in this signature art1 .
When the space-time is algebraically special there is some null tetrad frame
in which $\Psi_{0}^{+}=\Psi_{1}^{+}=\Psi_{0}^{-}=\Psi_{1}^{-}=0$. From the
GSHP theorem it follows that for a vacuum space-time in this frame we have
$\omega_{112}=\omega_{221}=\omega_{114}=\omega_{441}=0$, so that:
$\nabla_{1}e_{1}=\omega_{11}^{\phantom{11}a}\,e_{a}=\omega_{113}\,e_{1}-\omega_{114}\,e_{2}+\omega_{111}\,e_{3}-\omega_{112}\,e_{4}=\omega_{113}\,e_{1}.$
This means that $e_{1}$ is a null geodesic vector field. Now let us calculate
the optical scalars of the null congruence generated by $e_{1}$. To accomplish
this we must compute the projection of the tensor
$B_{\mu}^{\phantom{\mu}\nu}=(\nabla_{\mu}e_{1})^{\nu}$ into the space-like
plane generated by $e_{x}=\frac{1}{\sqrt{2}}(e_{2}+e_{4})$ and
$e_{y}=\frac{1}{i\sqrt{2}}(e_{2}-e_{4})$ Wald .
$\nabla_{x}e_{1}=\frac{1}{\sqrt{2}}(\nabla_{2}e_{1}+\nabla_{4}e_{1})\sim-\frac{1}{\sqrt{2}}[(\omega_{214}+\omega_{414})e_{2}+(\omega_{212}+\omega_{412})e_{4}]=-\frac{1}{2}[(\omega_{214}+\omega_{412})e_{x}+i(\omega_{214}-\omega_{412})e_{y}]\\\
\nabla_{y}e_{1}=\frac{1}{i\sqrt{2}}(\nabla_{2}e_{1}-\nabla_{4}e_{1})\sim-\frac{1}{i\sqrt{2}}[(\omega_{214}-\omega_{414})e_{2}+(\omega_{212}-\omega_{412})e_{4}]=-\frac{1}{2i}[(\omega_{214}-\omega_{412})e_{x}+i(\omega_{214}+\omega_{412})e_{y}]$
Where the symbol $\sim$ means equal except for terms proportional to $e_{1}$.
From this equation we conclude that the projection of
$B_{\mu}^{\phantom{\mu}\nu}$ into the plane $\\{e_{x},e_{y}\\}$,
$\widehat{B}_{\mu}^{\phantom{\mu}\nu}$, is:
$\widehat{B}_{\mu}^{\phantom{\mu}\nu}=\left[\begin{array}[]{cc}B_{x}^{\phantom{x}x}&B_{x}^{\phantom{x}y}\\\
B_{y}^{\phantom{y}x}&B_{y}^{\phantom{y}y}\\\
\end{array}\right]=-\frac{1}{2}\left[\begin{array}[]{cc}(\omega_{214}+\omega_{412})&i(\omega_{214}-\omega_{412})\\\
-i(\omega_{214}-\omega_{412})&(\omega_{214}+\omega_{412})\\\
\end{array}\right]$
Since the trace-less symmetric part of the above matrix is zero we conclude
that the congruence generated by $e_{1}$ is shear-free888In the Newman-Penrose
formalism the shear parameter is given by $\sigma=\omega_{212}$, which is zero
in the considered case.. The expansion of the congruence is the trace of
$\hat{B}$, while the rotation is the skew-symmetric part of this matrix. From
section III we know that $e_{1}$ is a repeated PND when
$\Psi_{0}^{+}=\Psi_{1}^{+}=0$, so we arrived at the important result that
algebraically special space-times in vacuum allow a shear-free null geodesic
congruence generated by the repeated principal null direction. From above
results it is easy to see that the converse is also true, which proves the
usual version of the Goldberg-Sachs theorem. In particular, when Weyl tensor
is type D there are two independent repeated PNDs, so two independent shear-
free null geodesic congruences, this was the key property that enabled to find
all type D vacuum solutions of Einstein’s equation typeD .
### V.4 (2,2) Signature
Let $(M,g)$ be a Ricci-flat real manifold of (2,2) signature, then it is
possible to find a real frame $\\{E_{1},E_{2},E_{3},E_{4}\\}$ such that the
only non-zero inner products between the basis vectors are
$E_{1}^{\mu}E_{1\mu}=E_{2}^{\mu}E_{2\mu}=1$ and
$E_{3}^{\mu}E_{3\mu}=E_{4}^{\mu}E_{4\mu}=-1$. From this we can construct the
following null tetrad basis in the complexified tangent bundle:
$e_{1}=\frac{1}{\sqrt{2}}(E_{1}+iE_{2})\,,\,e_{3}=\frac{1}{\sqrt{2}}(E_{1}-iE_{2})\,,\,e_{2}=\frac{1}{\sqrt{2}}(E_{3}+iE_{4})\,,\,e_{4}=\frac{1}{\sqrt{2}}(E_{3}-iE_{4})\,$.
In this complex frame note that $\overline{e_{1}}=e_{3}$ and
$\overline{e_{2}}=e_{4}$. But in this signature it is also possible to form a
real null frame, one example being:
$\check{e}_{1}=\frac{1}{\sqrt{2}}(E_{1}+E_{3})$,
$\check{e}_{3}=\frac{1}{\sqrt{2}}(E_{1}-E_{3})$,
$\check{e}_{2}=\frac{1}{\sqrt{2}}(E_{2}+E_{4})$ and
$\check{e}_{4}=\frac{-1}{\sqrt{2}}(E_{2}-E_{4})$. Using a real null frame it
is easy to see that all the 10 Weyl scalars are real and independent of each
other, so that in (2,2) signature spaces all the 21 algebraic types for the
Weyl tensor are realizable art1 .
When $C^{+}$ is algebraically special there exists some null tetrad frame in
which $\Psi_{0}^{+}=\Psi_{1}^{+}=0$, but the frame can be complex, as
$\\{e_{a}\\}$, or real, as $\\{\check{e}_{a}\\}$. Thus the GSHP theorem
guarantees that if $C^{+}$ is not type I+ then there is some null frame,
complex or real, in which $\omega_{112}=\omega_{221}=0$. This implies that the
planes generated by $\\{e_{1},e_{2}\\}$ or $\\{\check{e}_{1},\check{e}_{2}\\}$
are integrable, in the former case since
$\overline{\omega}_{112}=\omega_{334}$ and
$\overline{\omega}_{221}=\omega_{443}$, then $\omega_{334}$ and $\omega_{443}$
are also zero, so that the planes generated by $\\{e_{3},e_{4}\\}$ are
integrable too. Thus a generalization of Goldberg-Sachs theorem in (2,2)
signature can be as follows.
(2,2) signature version of GS theorem: _In vacuum the Weyl tensor is
algebraically special if, and only if, there is some integrable isotropic
distribution of planes in the manifold. The planes can be complex or real, in
the former case it follows that the complex conjugate planes are also
isotropic and integrable._
In (2,2) signature when dealing with the real null tetrad frames it is useful
to introduce the following analogues of $J$ and $\Omega$:
$\check{J}\equiv(\check{e}_{1}\otimes\check{e}^{1}+\check{e}_{2}\otimes\check{e}^{2})-(\check{e}_{3}\otimes\check{e}^{3}+\check{e}_{4}\otimes\check{e}^{4})\;\;\;;\;\;\;\check{\Omega}=(\check{e}^{1}\wedge\check{e}^{3}+\check{e}^{4}\wedge\check{e}^{2}).$
(26)
By equations (24) and (25) we get the relations
$d\check{\Omega}=-2\omega_{12}\wedge\check{e}^{1}\wedge\check{e}^{2}+2\omega_{34}\wedge\check{e}^{3}\wedge\check{e}^{4}\;\;\;;\;\check{J}\;\textrm{integrable
and}\;\;d\check{\Omega}=0\;\;\;\Leftrightarrow\;\;\;\nabla_{X}\check{J}=0\;\;\forall\;X\in
TM\,.$ (27)
Where $\check{J}$ is called integrable when its invariant subspaces are
integrable. Note that if $X,Y\in TM$ then $\check{J}(\check{J}(X))=X$ and
$\check{\Omega}(X,Y)=g(\check{J}(X),Y)$. The advantage of $\check{J}$ and
$\check{\Omega}$ is that they are real, since the basis $\\{\check{e}_{a}\\}$
is real, just as $J$ and $\Omega$ are real in a complex frame such that
$\overline{e_{1}}=e_{3}$ and $\overline{e_{2}}=e_{4}$. But it is worth noting
that the metric is not invariant under $\check{J}$, instead now we have
$g(\check{J}(X),\check{J}(Y))=-g(X,Y)$. The tensor $\check{J}$ is called a
paracomplex structure, more about this kind of object in the context of GSHP
theorem can be found in parastruct .
Now if $C^{+}$ is type D+ there are two possible cases: (1) The null frame in
which $\Psi_{0}^{+}=\Psi_{1}^{+}=\Psi_{3}^{+}=\Psi_{4}^{+}=0$ is complex, like
$\\{e_{a}\\}$, in this case the complex planes $\\{e_{1},e_{2}\\}$ and
$\\{e_{3},e_{4}\\}$ are integrable, so that the almost complex structure $J$
is integrable. Since the tensor $J$ is real the Newlander-Nirenberg theorem
guarantees that the manifold over the complex field is a complex manifold; (2)
The null frame in which
$\Psi_{0}^{+}=\Psi_{1}^{+}=\Psi_{3}^{+}=\Psi_{4}^{+}=0$ is real, like
$\\{\check{e}_{a}\\}$, in this case the real planes
$\\{\check{e}_{1},\check{e}_{2}\\}$ and $\\{\check{e}_{3},\check{e}_{4}\\}$
are integrable, so that $\check{J}$ is integrable.
When $C^{+}$ is type O+ there is some null tetrad frame where the connection
1-forms are such that $\omega_{12}=0=\omega_{34}$ and
$\omega_{24}=\omega_{13}$. This null frame can be complex or real, so that we
again have two cases: (1) If this null frame is complex we have that $J$ is
integrable, covariantly constant and $d\Omega=0$. The manifold over the real
field is symplectic, since $d\Omega=0$ and $\Omega$ is real and non-
degenerate, and over the complex field the manifold is Calabi-Yau; (2) If this
null frame is real it follows that $\check{J}$ is integrable and covariantly
constant and $d\check{\Omega}=0$, which implies that the manifold is
symplectic, since $\check{\Omega}$ is real and non-degenerate. The above
results are summarized by the following words:
_Let $(M,g)$ be a Ricci-flat real manifold of (2,2) signature, then if the
Weyl tensor is type (D,something) or type (O,something) there are two distinct
families of isotropic planes which are integrable. When these planes are
complex it follows that the manifold over the complex field is a complex
manifold. If the Weyl tensor is strictly type (O,something) there is a
covariantly constant real tensor of rank two and the manifold is symplectic
and if the integrable isotropic planes are complex it follows that over the
complex field the manifold is Calabi-Yau._
Conversely, it is easy to see that if a real four-dimensional manifold of
(2,2) signature admits a real integrable almost complex structure such that
the metric is Hermitian, then $C^{+}$ is type D+ or type O+. If this almost
complex structure is covariantly constant it follows that $C^{+}$ vanishes.
Analogously, when a real four-dimensional manifold of (2,2) signature admits
an integrable real paracomplex structure, $\check{J}$, then $C^{+}$ is type D+
or type O+, if the tensor $\check{J}$ is covariantly constant it follows that
$C^{+}$ vanishes.
### V.5 A Solution for Einstein-Maxwell Equations
The source-free Maxwell’s equations can be put in the form
$dF=0=d\widetilde{F}$, where $F$ is a 2-form and $\widetilde{F}$ its Hodge
dual. In the presence of the electromagnetic field $F$ Einstein’s equation
becomes $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T_{\mu\nu}$, where
$T_{\mu\nu}=F_{\mu\alpha}F_{\nu}^{\phantom{\nu}\alpha}-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}g_{\mu\nu}$
is the energy-momentum tensor of the electromagnetic field.
Note that $\Omega$ defined on equation (23) is self-dual, so if $d\Omega=0$
then it also follows that $d\widetilde{\Omega}=0$. It has been proven in the
preceding subsections that when the self-dual part of the Weyl tensor
vanishes, $C^{+}=0$, in Ricci-flat manifolds, there is some null frame in
which the Kähler form is closed, $d\Omega=0$, which also implies
$d\widetilde{\Omega}=0$. Now putting $F=\Omega$ we have that $F$ obeys to the
source-free Maxwell’s equations. Furthermore computing the energy-momentum
tensor associated to $F$ it is easily found that it vanishes, $T_{\mu\nu}=0$.
Since the Ricci tensor is assumed to vanish it follows that Einstein’s
equation in the presence of this field is satisfied. Thus can be stated:
_When $C^{+}$ vanishes in a Ricci-flat manifold there exists some null tetrad
frame such that the Kähler form defined in (23) is a solution for the source-
free Einstein-Maxwell equations. _
## VI Physical Relevance
Although rather mathematical, the issues treated in this article have multiple
physical applicability. In this section it will be briefly mentioned some
physical areas where the subjects treated on this work are of importance.
The path followed here was to consider complexified manifolds and when
convenient choose some reality condition to obtain a real manifold of the
desired signature. This kind of approach to obtain results on real four-
dimensional Lorentzian manifolds was taken before, for example, in a series of
papers of C. McIntosh and M. Hickman McIntosh , where it was advocated that
the use of complexified manifolds are profitable to obtain real solutions to
Einstein’s equation. In particular this approach is important to understand
better what happens when a Wick rotation is made.
Besides the ubiquitous Lorentzian case, other signatures are also of relevance
in several physical problems. Euclidean manifolds, called gravitational
instantons, are useful when time is Wick rotated, with the intent of
calculating partition functions or in the computation of tunneling
probabilities. Whereas (2,2) signature is physically important in classical
mechanics, since phase spaces are manifolds of split signature. The (2,2)
signature is also of interest for the theory of integrable systems Mason .
From the results of this article one with direct physical applicability is the
existence of covariantly constant real tensors of rank two when $C^{+}$
vanish, these constant tensors furnish conserved quantities, useful for the
incorporation of symmetries in physical problems and for the integrability of
equations of motion.
The physical intuition that causal structure is the key concept in general
relativity lead Roger Penrose to introduce the complex null tetrad in general
relativity, giving rise to the so called Newman-Penrose formalism. This
formalism was used in the solution of many important problems in general
relativity, one example being the finding of all type D vacuum solutions typeD
. A message that can be extracted from this is that null directions are
connected to many physically important properties and even when they are
complex physical content may be encapsulated. Here the isotropic planes were
of central importance, since they are generated by null directions Penrose’s
intuition is enforced by the results of the present paper.
Higher-dimensional space-times have been intensively investigated for long
time. One of the lines of research in this topic is the generalization of
Petrov classification and correlated results to dimensions greater than four.
A successful generalization of Weyl tensor classification in Lorentzian
higher-dimensional manifolds was described in CMPP , the so called CMPP
classification. Since it is well known that Goldberg-Sachs theorem cannot be
trivially generalized to higher dimensions999For instance, reference
FrolovMyers5D proved that the repeated PNDs of 5-dimensional Myers-Perry
black hole are not shear-free, a partial generalization of GS theorem is being
looked for and some important results have been already obtained. In Dur-Reall
it was proved that every Einstein space-time that admits a multiple Weyl
aligned null direction(WAND)101010In CMPP classification the WANDs are natural
higher-dimensional analogues of the four-dimensional principal null directions
CMPP . also admits a multiple WAND that is tangent to a geodesic congruence.
Further in Ortaggio5 it was worked out the restrictions on the optical matrix
of null congruences tangent to geodesic multiple WAND in five-dimensional
Einstein space-times. The present article stressed the importance of null
structures (isotropic planes) in four dimensions. Such line of thinking can be
used for a higher-dimensional generalization of Petrov classification and GS
theorem. This path was followed in HigherGSisotropic1 ; HigherGSisotropic2 ,
where the integrability condition of maximally isotropic hyper-planes is
related to algebraic conditions on the Weyl and Cotton-York tensors, which can
be seen as the generalization of half of the GS theorem. As a last comment it
is worth remembering that maximally isotropic subspaces are associated with
the so called pure spinors, a mathematical object with increasing relevance in
physical theories, string theory being an example Nathan .
## VII Conclusion
It is well known that in vacuum Lorentzian manifolds the repeated principal
null directions of the Weyl tensor are related to the integrability of shear-
free null geodesic congruences. It then follows a connection between algebraic
constraints on the Weyl tensor and geometrical properties of space-time. Here
it has been shown that the same kind of connection happens in Ricci-flat four-
dimensional complexified manifolds as well as in real manifolds with any
signature. The main results presented in this article were:
* •
The analogues of repeated PNDs in non-Lorentzian manifolds are the null
eigenbivectors of the Weyl operator.
* •
When the Ricci tensor vanishes these eigenbivectors generate integrable
isotropic planes, this is the generalization of the Goldberg-Sachs theorem to
non-Lorentzian manifolds.
* •
In Ricci-flat complex manifolds the self-dual part of the Weyl tensor vanishes
if, and only if, the manifold is Calabi-Yau.
* •
In Ricci-flat Euclidean manifolds the Weyl tensor is algebraically special if,
and only if, the manifold has an integrable almost complex structure and the
metric is Hermitian with respect to it. When the self-dual part of the Weyl
tensor vanishes there is a real covariantly constant rank two tensor and the
manifold over the complex field is Calabi-Yau.
* •
In a Ricci-flat (2,2) signature manifold if the self-dual part of the Weyl
tensor vanishes then the manifold is symplectic and has a real covariantly
constant rank two tensor.
* •
In all Ricci-flat manifolds such that the self-dual part of the Weyl tensor
vanishes the Kähler form is a solution to the source-free Einstein-Maxwell
equations.
## Acknowledgments
I want to thank Bruno G. Carneiro da Cunha for the encouragement and for the
manuscript revision. This research was supported by CNPq(Conselho Nacional de
Desenvolvimento Científico e Tecnológico). The final publication is available
at link.springer.com (DOI:10.1007/s10714-013-1539-4).
## References
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* (2) H. Stephani et al., Exact solutions of Einstein’s field equations, Cambridge University Press (2009).
* (3) R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Physical Review Letters 11 (1963), 237.
* (4) J. Goldberg and R. Sachs, A theorem on Petrov types, General Relativity and Gravitation 41 (2009), 433. This is a republication of original 1962 paper.
* (5) C. Batista, Weyl tensor classifcation in four-dimensional manifolds of all signatures, General Relativity and Gravitation 45 (2013), 785. Available at arXiv:1204.5133
* (6) J. F. Plebański and S. Hacyan, Null geodesic surfaces and Goldberg-Sachs theorem in complex Riemannian spaces, Journal of Mathematical Physics 16 (1975), 2403.
* (7) M. Przanowski and B. Broda, Locally Kähler gravitational instantons, Acta Physica Polonica B14 (1983), 637.
* (8) P. Nurowski and A. Trautman, Robinson manifolds as the Lorentzian analogs of Hermite Manifolds, Differential Geometry and its Applications 17(2002), 175.
* (9) A. Gover, C. Hill and P. Nurowski, Sharp version of the Goldberg-Sachs theorem, Annali di Matematica Pura ed Applicata 190 Number 2 (2011), 295. Available at arXiv:0911.3364.
* (10) M. Durkee and H. S. Reall, A higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem, Classical and Quantum Gravity 26 (2009), 245005. Available at arXiv:0908.2771
* (11) M. Ortaggio et al., On a five-dimensional version of the Goldberg-Sachs theorem, Available at arXiv:1205.1119
* (12) A. Taghavi-Chabert, Optical structures, algebraically special spacetimes and the Goldberg-Sachs theorem in five dimensions, Classical and Quantum Gravity 28 (2011), 145010. Available at arXiv:1011.6168
* (13) A. Taghavi-Chabert, The complex Goldberg-Sachs theorem in higher dimensions, Journal of Geometry and Physics 62 (2012), 981. Available at arXiv:1107.2283
* (14) P. R. Law, Neutral Einstein metrics in four dimensions, Journal of Mathematical Physics 32 (1991), 3039.
* (15) A. Coley and S. Hervik, Higher dimensional bivectors and classification of the Weyl operator, Classical and Quantum Gravity 27 (2010), 015002. Available at arXiv:0909.1160
S. Hervik and A. Coley, Curvature operators and scalar curvature invariants,
Classical and Quantum Gravity 27 (2010), 095014. Available at arXiv:1002.0505
* (16) J. Plebański, Some solutions of complex Einstein equations, Journal of Mathematical Physics 16 (1975), 2395.
* (17) S. Hacyan, Gravitational instantons in H-spaces, Physics Letters 75A (1979), 23
* (18) P. R. Law, Classification of the Weyl curvature spinors of neutral metrics in four dimensions, Journal of Geometry and Physics 56 (2006), 2093
* (19) S. Hervik and A. Coley, On the algebraic classification of pseudo-Riemannian spaces, International Journal of Geometric Methods in Modern Physics 8 (2011), 1679. Available at arXiv:1008.3021
* (20) R. Penrose and W. Rindler, Spinors and space-time vol.1 and 2, Cambridge University Press (1984 and 1986).
* (21) L. Bel, Radiation states and the problem of energy in general relativity 32, 2047 (2000)Reprint of a 1962 paper.
* (22) I. Robinson and A. Schild, Generalization of a theorem by Goldberg and Sachs, Journal of Mathematical Physics 4 (1963), 484.
* (23) W. Kopczynski and A. Trautman, Simple spinors and real structures, Journal of Mathematical Physics 33 (1992), 550.
* (24) M. Nakahara, Geometry, Topology and Physics, Taylor$\&$Francis (2003).
* (25) A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds Annals of Mathematics, 65 (1957), 391.
* (26) R. M. Wald, General Relativity, The University of Chicago Press (1984).
* (27) W. Kinnersley, Type D vacuum Metrics, Journal of Mathematical Physics 10 (1969), 1195\.
* (28) S. Ivanov and S. Zamkovoy, Parahermitian and paraquaternionic manifolds, Differential Geometry and its Applications 23 (2005), 205. Available at arXiv:math/0310415
* (29) C. McIntosh and M. Hickman, Complex relativity and real solutions. I: Introduction, General Relativity and Gravitation 17 (1985), 111.
* (30) L. Mason and N. Woodhouse, Integrability, self-duality and twistor theory , Oxford University Press (1996).
* (31) A. Coley, R. Milson, V.Pravda and A. Pravdová, Classification of the Weyl Tensor in Higher Dimensions, Classical and Quantum Gravity 21 (2004), L-35. Available at arXiv:gr-qc/0401008
* (32) V. Frolov and D. Stojković, Particle and light motion in a space-time of a five-dimensional black hole, Physical Review D 68 (2003), 064011. Available at arXiv:gr-qc/0301016
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|
arxiv-papers
| 2012-05-21T18:08:23 |
2024-09-04T02:49:31.157083
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Batista",
"submitter": "Carlos A. Batista da S. Filho",
"url": "https://arxiv.org/abs/1205.4666"
}
|
1205.4703
|
# Disease Persistence in Epidemiological Models:
The Interplay between Vaccination and Migration
Jackson Burton Lora Billings billingsl@mail.montclair.edu Derek A. T.
Cummings Ira B. Schwartz Montclair State University, Department of
Mathematical Sciences, Montclair, NJ 07043 Johns Hopkins Bloomberg School of
Public Health, Department of International Health, Baltimore, MD, 21205 US
Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section,
Plasma Physics Division, Washington, DC 20375
###### Abstract
We consider the interplay of vaccination and migration rates on disease
persistence in epidemiological systems. We show that short-term and long-term
migration can inhibit disease persistence. As a result, we show how migration
changes how vaccination rates should be chosen to maintain herd immunity. In a
system of coupled SIR models, we analyze how disease eradication depends
explicitly on vaccine distribution and migration connectivity. The analysis
suggests potentially novel vaccination policies that underscore the importance
of optimal placement of finite resources.
###### keywords:
epidemics, migration, vaccination, herd immunity
††journal: Mathematical Biosciences
## 1 Introduction
Countries are increasingly connected by travel and economics. Due to economic
disparities and political turmoil, extreme heterogeneity exists in childhood
vaccination coverage across the two sides of multiple national boundaries. It
has been suggested that the immunization coverage of neighboring countries or
those countries well connected by travel can or should be used when crafting
national level immunization policy. In the case of hepatitis B, Gay and
Edmunds [1] argue that it would be four times more cost effective for the
United Kingdom to sponsor a vaccination program in Bangladesh than to
introduce its own universal program. When indigenous wild poliovirus was
eradicated in all but four endemic countries in 2005: India, Nigeria, Pakistan
and Afghanistan, it was exported from northern Nigeria and northern India and
subsequently caused $>50$ outbreaks and paralyzed $>1500$ children in
previously polio-free countries across Asia and Africa [2]. And in 2007, the
WHO estimated that there were 197,000 measles deaths, despite the 82%
worldwide vaccination coverage. In countries where measles has been largely
eliminated, cases imported from other countries remain an important source of
infection [3]. It is clear that a country needs to be concerned with the
vaccination rate of a neighboring country as well as its own.
On another scale, vaccination policies must also take into consideration the
subpopulation dynamics within a country. Wilson, et al. [4] models linked
urban and rural epidemics of HIV and discusses how to optimize a limited
treatment supply to minimize new infections. Cummings et al. [5] uses data to
identify a distinct pattern in the periodicity of measles outbreaks in
Cameroon before the widespread vaccination efforts of the Measles Initiative.
The southern part of Cameroon experienced a significant measles epidemic
approximately every three years. In contrast, the three northern provinces
contend with annual measles epidemics. In 2000 and 2001, these cyclic
outbreaks coincided, exacerbating the situation and causing a much more severe
epidemic [5].
Noting that a small contribution of infections from one population to another
could drive a new type of epidemic that would not normally occur, we study how
migration between populations could change dynamics and respective herd
immunity levels in metapopulation models. We analyze a model of a disease
imported between subpopulations of a region by short-term and long-term
migration with limited vaccination coverage. Our initial study is based on the
analysis of a system of canonical SIR compartmental models. The system allows
the rigorous proof of the qualitative affects migration has on herd immunity.
The model can be enhanced to include more compartments or seasonal forcing,
but most of these systems will require numerical exploration of trends in
spatial synchrony and bifurcation analysis, which will be explored in future
papers. In this article, we revisit the fundamental ways migration is modeled
in metapopulation models and how it fundamentally affects herd immunity.
Migration is often treated as a phenomenological input to maintain incidence
in a population that might experience local fade-out [6]. Long-term migration
has been analyzed by Liebovitch and Schwartz [7], with a thorough derivation
of the linear flux term coupling the patches. This approach also agrees with
the classes of models proposed by Sattenspiel and Dietz [8] and Lloyd and
Jansen [9]. Keeling and Rohani [10] investigated the spatial coupling of
dynamics exhibited in models using multiple formulations of migration
including mass-action coupling and linear flux terms. However, they did not
explore the impact of coupling in the presence of vaccination needed to
maintain disease free states. Additional analysis of mixed long-term and
short-term migration in transport-related disease spread can be found in [11,
12, 13]. These papers derive the global asymptotic stability of the disease
free state for a new disease. Because there is no vaccination, the papers
conclude that it is essential to strengthen restrictions of passenger travel
as soon as the infectious diseases appear.
Our paper considers how migration directly affects the vaccination levels
needed for herd immunity against a known disease and how that would impact
optimum usage of limited vaccination supplies. We investigate the dynamics of
models that include mass-action coupling, an assumption that assumes mixing
occurs at fast time scales, and linear migration, which is more consistent
with mixing occurring at long time scales. The organization of this paper is
as follows: We introduce a coupled compartmental model in Section 2 and
perform stability analysis of the disease free state as a function of the
migration and vaccination rates. We also consider normal forms of the
bifurcations created by the short-term and long-term migration dynamics.
Section 3 describes how vaccination rates should be adjusted with respect to
short-term and long-term migration levels to preserve herd immunity. Section 4
has a summary of our observations and conclusions.
## 2 The Model
We start with the classic Susceptible, Infected, Recovered (SIR) model. Let
$S$, $I$, and $R$ denote the number of people in each of the disease classes
for a population of size $N$. Let the parameters $\beta>0$ denote the contact
rate, $\mu>0$ denote the birth/death rate, and $\kappa>0$ denote the recovery
rate. The vaccination rate, $0\leq v\leq 1$, represents the removal of a
percentage of the incoming newborn population to recovered. The standard form
for this system is
$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle(1-v)\,\mu
N-\frac{\beta SI}{N}-\mu S,$ $\displaystyle\frac{dI}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\kappa I-\mu I,$ (1)
$\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle v\mu N+\kappa
I-\mu R.$
The death rates in the classes balance the births so that the population size
$N>0$ is constant. For a detailed analysis of the single patch formulation of
this system, see Hethcote [14].
We now consider two coupled subpopulations where the disease dynamics of each
population are described by the SIR model. Let $S_{k}$, $I_{k}$, and $R_{k}$
denote the number of people in each of the disease classes, $\mu_{k}>0$ denote
the birth/death rates, and $v_{k}$ denote the vaccination rates of
subpopulations $N_{k}$ for $k=1,2$. To model long-term movement (linear
mixing), let $c_{1}\geq 0$ denote the rate of migration from population two to
population one and vice versa for the rate $c_{2}\geq 0$. To model short-term
movement (mass action mixing), let $0\leq c_{3}\leq 1$ be a scaling of the
number of infectives from one population who move into the other population
for a short time and mix with the susceptibles to produce additional
infections. Because $\beta$ is proportional to the average number of contacts
a person can make per unit time, we distribute the contacts for the
susceptibles between the infected people by mass action within and outside the
population by using the prefactors $c_{3}$ and $(1-c_{3})$ respectively as in
Keeling and Rohani [10]. The coupled two population model is as follows:
$\displaystyle\frac{dS_{1}}{dt}$ $\displaystyle=$
$\displaystyle(1-v_{1})\,\mu_{1}N_{1}-\frac{(1-c_{3})\,\beta\,S_{1}I_{1}}{N_{1}}-\frac{c_{3}\,\beta\,S_{1}I_{2}}{N_{1}}-\mu_{1}S_{1}+c_{1}S_{2}-c_{2}S_{1},$
$\displaystyle\frac{dI_{1}}{dt}$ $\displaystyle=$
$\displaystyle\frac{(1-c_{3})\,\beta\,S_{1}I_{1}}{N_{1}}+\frac{c_{3}\,\beta\,S_{1}I_{2}}{N_{1}}-\kappa
I_{1}-\mu_{1}I_{1}+c_{1}I_{2}-c_{2}I_{1},$ $\displaystyle\frac{dR_{1}}{dt}$
$\displaystyle=$ $\displaystyle v_{1}\mu_{1}N_{1}+\kappa
I_{1}-\mu_{1}R_{1}+c_{1}R_{2}-c_{2}R_{1},$ (2)
$\displaystyle\frac{dS_{2}}{dt}$ $\displaystyle=$
$\displaystyle(1-v_{2})\,\mu_{2}N_{2}-\frac{(1-c_{3})\,\beta\,S_{2}I_{2}}{N_{2}}-\frac{c_{3}\,\beta\,S_{2}I_{1}}{N_{2}}-\mu_{2}S_{2}+c_{2}S_{1}-c_{1}S_{2},$
$\displaystyle\frac{dI_{2}}{dt}$ $\displaystyle=$
$\displaystyle\frac{(1-c_{3})\,\beta\,S_{2}I_{2}}{N_{2}}+\frac{c_{3}\,\beta\,S_{2}I_{1}}{N_{2}}-\kappa
I_{2}-\mu_{2}I_{2}+c_{2}I_{1}-c_{1}I_{2},$ $\displaystyle\frac{dR_{2}}{dt}$
$\displaystyle=$ $\displaystyle v_{2}\mu_{2}N_{2}+\kappa
I_{2}-\mu_{2}R_{2}+c_{2}R_{1}-c_{1}R_{2}.$
We keep the number of people in the subpopulations constant by letting
$\rho=N_{2}/N_{1}$ and setting the constraint $c_{2}=c_{1}\rho$. This system
is overdetermined by the subpopulation constraints, $S_{k}+I_{k}+R_{k}=N_{k}$
for $k=1,2$, and therefore the analysis omits the variables $R_{k}$ for
$k=1,2$.
Motivated by the distinct subpopulation dynamics of Cameroon described in
Cummings et al. [5], numerical simulations will use parameters based on
Cameroon demographics. The values are listed in Table 1. The subpopulation
sizes are totals for the northern and southern regions based on data in [5].
The birth/death rates are averages over the northern and southern regions
based on data in [5]. The recovery rate is a parameter that is derived from
the biological characteristics of measles. The contact rate was estimated
using the average age of incident measles cases over the period 1998-2006 from
passive surveillance data [5]. The specific results here are fairly
insensitive to changes to $\beta$. An SIR model is used here without the
exposed class but we expect the inclusion of an exposed class would not
substantively change our qualitative results.
Table 1: Parameter Values for Model Based on Cameroon Data Parameter | Value | Unit | Description
---|---|---|---
$N_{1}$ | 4,451,000 | people | Northern subpopulation size
$N_{2}$ | 10,212,000 | people | Southern subpopulation size
$\rho$ | $2.2943$ | none | Ratio of $N_{2}/N_{1}$
$\beta$ | $700$ | year-1 | Contact rate
$\kappa$ | $100$ | year-1 | Measles recovery rate
$\mu_{1}$ | $.0428$ | year-1 | Birth and death rate for $N_{1}$
$\mu_{2}$ | $.0329$ | year-1 | Birth and death rate for $N_{2}$
### 2.1 General system analysis
We start with a general analysis of the system to determine the conditions
necessary for the populations to be disease free.
###### Proposition 1.
System (2) has a disease free equilibrium (DFE) and is given by
$(S_{1},I_{1},S_{2},I_{2})=(N_{1}\hat{S}_{1},0,N_{2}\hat{S}_{2},0),$ (3)
for
$\displaystyle\hat{S}_{1}=\frac{(1-v_{1})(\mu_{1}c_{1}+\mu_{1}\mu_{2})+(1-v_{2})\mu_{2}c_{1}\rho}{\mu_{1}c_{1}+\mu_{1}\mu_{2}+\mu_{2}c_{1}\rho},$
(4)
$\displaystyle\hat{S}_{2}=\frac{(1-v_{1})\mu_{1}c_{1}+(1-v_{2})(\mu_{1}\mu_{2}+\mu_{2}c_{1}\rho)}{\mu_{1}c_{1}+\mu_{1}\mu_{2}+\mu_{2}c_{1}\rho}.$
(5)
Note that the DFE does not depend on the short-term migration parameter
$c_{3}$. Without long-term migration ($c_{1}=0$), the DFE simplifies to
$(S_{1},I_{1},S_{2},I_{2})=\left(N_{1}(1-v_{1}),0,N_{2}(1-v_{2}),0\right)$,
which is the steady state for the uncoupled system. Also, there is no steady
state for which the disease dies out in only one of the two subpopulations if
$c_{1}>0$.
The local stability of the DFE can be determined by the eigenvalues of the
Jacobian of the system evaluated at the DFE. The resulting eigenvalues are
$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(c_{1}+c_{1}\rho+\mu_{1}+\mu_{2}+\sqrt{(c_{1}+c_{1}\rho+\mu_{1}-\mu_{2})^{2}-4c_{1}(\mu_{1}-\mu_{2})}\right),$
(6) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(c_{1}+c_{1}\rho+\mu_{1}+\mu_{2}-\sqrt{(c_{1}+c_{1}\rho+\mu_{1}-\mu_{2})^{2}-4c_{1}(\mu_{1}-\mu_{2})}\right),$
(7) $\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(c_{1}+c_{1}\rho+\mu_{1}+\mu_{2}+2\kappa-(1-c_{3})\,\beta\,(\hat{S}_{1}+\hat{S}_{2})\,+\sqrt{W}\right),$
(8) $\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(c_{1}+c_{1}\rho+\mu_{1}+\mu_{2}+2\kappa-(1-c_{3})\,\beta\,(\hat{S}_{1}+\hat{S}_{2})\,-\sqrt{W}\right),$
(9)
for
$W=4\,\beta\,c_{3}\,c_{1}\left(\hat{S}_{2}+\rho\,\hat{S}_{1}\right)+4\left(\beta^{2}c_{3}^{2}\hat{S}_{1}\hat{S}_{2}+c_{1}^{2}\rho\right)+\left((1-c_{3})\,\beta\left(\hat{S}_{1}-\hat{S}_{2}\right)+c_{1}-c_{1}\rho-\mu_{1}+\mu_{2}\right)^{2}.$
(10)
The DFE is locally stable if the maximum value of the real parts of this set
of eigenvalues is negative.
###### Proposition 2.
The eigenvalue $\lambda_{4}$ determines the local stability of the DFE.
###### Proof.
We can show $\lambda_{1}$ and $\lambda_{2}$ are always negative. First, let
$\begin{array}[]{rl}\theta_{1}&=c_{1}+c_{1}\rho+\mu_{1}+\mu_{2},\\\\[7.22743pt]
\theta_{2}&=(c_{1}+c_{1}\rho+\mu_{1}-\mu_{2})^{2}-4c_{1}(\mu_{1}-\mu_{2}).\\\
\end{array}$ (11)
If we consider $\theta_{2}$ as a quadratic expression in $c_{1}$ with leading
coefficient $(\rho+1)^{2}$, it attains an absolute minimum at
$d\theta_{2}/dc_{1}=0$. Solving this equation gives
$c_{1}=(\mu_{1}-\mu_{2})/(\rho+1)^{2}$. Substituting this expression into
$\theta_{2}$ to find the absolute minimum gives
$4\rho(\mu_{1}-\mu_{2})^{2}/(\rho+1)^{2}>0$. Therefore $\theta_{2}>0$ for all
$c_{1}$, which implies $\lambda_{1}$ and $\lambda_{2}$ are real valued.
Upon inspection we see that $\theta_{1}>0$, $\theta_{1}+\sqrt{\theta_{2}}>0$,
and therefore $\lambda_{1}<0$. For $\lambda_{2}<0$, we require
$\theta_{1}>\sqrt{\theta_{2}}$. This is equivalent to
$c_{1}>-\mu_{1}\mu_{2}/(\mu_{1}+\mu_{2}\rho)$. Since we assume $c_{1}>0$, this
is always true. Therefore $\theta_{1}-\sqrt{\theta_{2}}>0$, which implies
$\lambda_{2}<0$.
For our parameter assumptions, we see that $W>0$ by inspection. This implies
$\lambda_{3}$ and $\lambda_{4}$ are real valued and $\lambda_{4}>\lambda_{3}$.
The only way for the DFE to be unstable is for $\lambda_{3}>0$ or
$\lambda_{4}>0$. Because of the ordering, a sign change would have to happen
for $\lambda_{4}$ first. Therefore, $\lambda_{4}$ determines the stability of
the DFE. ∎
Figure 1: Contour plot of $\lambda_{4}$ values as a function of $c_{1}$ and
$c_{3}$. Parameters are given by the values in Table 1, with $v_{1}=0.82$ and
$v_{2}=0.90$. The DFE is unstable for $\lambda_{4}>0$, which occurs for
smaller values of $c_{1}$ and $c_{3}$.
To quantify how each migration type effects the stability of the DFE, we can
monitor the sign of $\lambda_{4}$ as we vary $c_{1}$ and $c_{3}$. As an
example, we show a contour plot of $\lambda_{4}$ in Fig. 1 using the
parameters in Table 1, with $v_{1}=0.82$ and $v_{2}=0.90$. When
$\lambda_{4}>0$, the DFE is unstable. From the figure, you can see that as
$c_{1}$ and $c_{3}$ decrease, $\lambda_{4}$ increases and the DFE becomes
unstable. We now explore the underlying conditions necessary in each
subpopulation for which migration can cause the die out or invasion of a
disease.
In the absence of migration, we recover the basic reproductive numbers scaled
by vaccination for each subpopulation for the uncoupled system [14].
Specifically when $c_{1}=c_{3}=0$,
$\hat{R}_{1}(v_{1})=\frac{\beta(1-v_{1})}{\kappa+\mu_{1}}~{}~{}\mbox{and}~{}~{}\hat{R}_{2}(v_{2})=\frac{\beta(1-v_{2})}{\kappa+\mu_{2}}$
(12)
We omit the arguments for $\hat{R_{k}}$ for $k=1,2$ from here on, unless
otherwise specified. The basic reproductive number is the quantity that
defines the threshold between disease absence and persistence, and for the
canonical SIR model without vaccination $R_{0}=\frac{\beta}{\kappa+\mu}$. We
write these expressions as a function of the vaccination rate in the
subpopulation noting that the inequalities $\hat{R}_{k}<1$ for $k=1,2$ implies
that the DFE in each subpopulation is locally stable. At $\hat{R}_{k}=1$ for
$k=1,2$, these two expressions also represent transcritical bifurcations for
the uncoupled system. As an example, for the parameters used in Fig. 1,
$\hat{R}_{1}(0.82)>1$ and the disease would be endemic in $N_{1}$. Conversely,
$\hat{R}_{2}(0.90)<1$ and the disease would die out in $N_{2}$. This motivates
us to examine the effect migration has on a simple system with a transcritical
bifurcation in each component.
### 2.2 Normal Form for Linear Mixing
To understand how long-term migration directly affects the stability of the
DFE, we rewrite the system as the normal form of a transcritical bifurcation
with linear coupling. Since the SIS model has the same topology near the DFE
as the SIR model, we consider the standard SIS model with births and deaths
[15]
$\displaystyle\frac{ds}{dt}$ $\displaystyle=\mu-\beta si+\kappa i-\mu s,$
(13a) $\displaystyle\frac{di}{dt}$ $\displaystyle=\beta si-\kappa i-\mu i,$
(13b)
with nondimensional variables representing percentages of the population.
Since $s+i=1$, the system is overdetermined and we need only to solve $di/dt$.
In the first subpopulation, let $x=i$ and $1-x=s$. Substitute these variables
into Eq. (13b) and rescale time by $\beta$. We repeat the process using the
variable $y$ to represent the second population. By adding the linear
migration terms to these equations, we find
$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle r_{1}x-x^{2}-\alpha
x+\alpha y,$ (14) $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle
r_{2}y-y^{2}-\alpha y+\alpha x.$
Here, the bifurcation parameter $r_{k}=(\hat{R}_{k}-1)/\hat{R}_{k}$ for
$k=1,2$ from Eq. (12). In addition, we rescaled the long-term migration rate
as the parameter $\alpha=c_{1}/\beta$.
The steady state $(x,y)=(0,0)$ is equivalent to the DFE in the full system in
Eq. (2). In the absence of coupling ($\alpha=0$), a transcritical bifurcation
will occur in the $x$ system, transferring the stability from $x=0$ to
$x=r_{1}$ at $r_{1}=0$. The dynamics are similar for $y$, respectively.
Linearizing about the steady state $(x,y)=(0,0)$ yields two eigenvalues,
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(r_{1}+r_{2}-2\alpha+\sqrt{4\alpha^{2}+(r_{1}-r_{2})^{2}}\right),$
(15) $\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(r_{1}+r_{2}-2\alpha-\sqrt{4\alpha^{2}+(r_{1}-r_{2})^{2}}\right).$
The following analysis uses this linearization approach to conclude when long-
term migration can change the stability of this steady state.
We start by considering the case of two isolated endemic populations, which
have basic reproduction numbers greater than one. We ask if it is possible to
stabilize the die out state through the coupling parameter, $\alpha$.
###### Proposition 3.
If $r_{1},r_{2}>0$, then the fixed point $(x,y)=(0,0)$ is unstable for all
$\alpha\in[0,\infty)$.
###### Proof.
Upon inspection, $\Lambda_{1}$ is the dominant eigenvalue. Assuming
$r_{1},r_{2}>0$, then $\Lambda_{1}>0$ implies
$(r_{1}+r_{2})+\left(\sqrt{4\alpha^{2}+(r_{1}-r_{2})^{2}}\right)>2\alpha.$
(16)
Squaring both sides and simplifying, we find
$r_{1}^{2}+r_{2}^{2}+(r_{1}+r_{2})\sqrt{4\alpha^{2}+(r_{1}-r_{2})^{2}}>0,$
(17)
which is always true. Therefore, $(x,y)=(0,0)$ is unstable for all
$\alpha\in[0,\infty)$. ∎
We can interpret this abstract result in the original system by concluding
that for two isolated endemic populations, the amount of long-term migration
is irrelevant to the persistence of the disease. The stability of the DFE
cannot be changed by migration and intervention by vaccination is necessary
for disease die out.
Next, consider the case where we have sufficient vaccination so that one of
the basic reproductive numbers is less than one, while the other is not.
Again, we ask under what conditions the coupling parameter can stabilize the
die out state.
###### Proposition 4.
Without loss of generality, we assume $r_{1}>0$ and $r_{2}<0$. Case 1: If
$-r_{2}<r_{1}$, then the fixed point $(x,y)=(0,0)$ is unstable for all
$\alpha\in[0,\infty)$. Case 2: If $-r_{2}>r_{1}$, then there exists some
$\alpha^{*}\in[0,\infty)$ such that the fixed point $(0,0)$ is stable for all
$\alpha>\alpha^{*}$.
###### Proof.
In both cases, assume $r_{1}>0$ and $r_{2}<0$.
Case 1: For $-r_{2}<r_{1}$, $\Lambda_{1}>0$ reduces to the relationship in Eq.
(17). This is always true for $0<r_{1}+r_{2}$ and we conclude $(x,y)=(0,0)$ is
unstable for all $\alpha\in[0,\infty)$.
Case 2: For $-r_{2}>r_{1}$, $\Lambda_{1}<0$ reduces to the relationship
$\left(\sqrt{4\alpha^{2}+(r_{1}-r_{2})^{2}}\right)<2\alpha-(r_{1}+r_{2}).$
(18)
Because both sides are positive, we can square both sides to find
$\alpha>\frac{r_{1}r_{2}}{r_{1}+r_{2}}>0.$ (19)
There exists an $\alpha^{*}=r_{1}r_{2}/(r_{1}+r_{2})$ for
$\alpha^{*}\in[0,\infty)$. Therefore, for $\alpha>\alpha^{*}$, it follows that
$\Lambda_{1}<0$ and $(x,y)=(0,0)$ is stable. ∎
This result implies that in a system with one population supporting an endemic
state, there is a minimum amount of migration necessary for the system to
achieve stability of the DFE. In fact, we can interpret the requirement of
$-r_{2}>r_{1}$ as $y=0$ in the uncoupled system is more stable than $x=0$.
Therefore, $y$ is sharing its extra stability with $x$.
For completeness, we can also show that long-term migration cannot change the
stability of a stable die out state for two isolated populations that have
basic reproduction numbers less than one.
###### Proposition 5.
If $r_{1}$, $r_{2}<0$, then the fixed point $(x,y)=(0,0)$ is stable for all
$\alpha\in[0,\infty)$.
###### Proof.
For $r_{1}$, $r_{2}<0$, $\Lambda_{1}<0$ reduces to Eq. (18). Because both
sides are positive, we can square both sides to find
$\alpha>\frac{r_{1}r_{2}}{r_{1}+r_{2}}.$ (20)
Since $\frac{r_{1}r_{2}}{r_{1}+r_{2}}<0$, $\Lambda_{1}<0$ for all
$\alpha\in[0,\infty)$ and $(x,y)=(0,0)$ is stable. ∎
We conclude that long-term migration has a positive effect on the stability of
the DFE. The mixing in all classes diffuses the force of infection, making it
harder for the disease to persist. In applications where migration is common,
this effect might be significant.
### 2.3 Normal Form for Mass Action Mixing
To capture the effect of short-term migration for each subpopulation in Eq.
(2), we follow the construction of model for linear mixing by substituting $x$
and $y$ for $i$ in Eq. (13b). In this system, we use mass action coupling with
the parameter $\sigma=c_{3}$ controlling the mass action mixing strength.
Specifically, the term $\sigma(1-x)y$ represents the infectious person from
$y$ coming into contact with a susceptible from $x$. The system take the form
$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle\left(\frac{r_{1}-\sigma}{1-\sigma}\right)x-x^{2}-\frac{\sigma}{1-\sigma}(1-x)y,$
(21) $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle\left(\frac{r_{2}-\sigma}{1-\sigma}\right)y-y^{2}-\frac{\sigma}{1-\sigma}(1-y)x.$
Again, the bifurcation parameter $r_{k}=(\hat{R}_{k}-1)/\hat{R}_{k}$ for
$k=1,2$ from Eq. (12) and time has been rescaled by $\beta(1-\sigma)$.
Performing the linearization about the steady state $(x,y)=(0,0)$ yields two
eigenvalues,
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2(1-\sigma)}\,\left(r_{1}+r_{2}-2\sigma+\sqrt{4\,{\sigma}^{2}+(r_{1}-r_{2})^{2}}\right),$
(22) $\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2(1-\sigma)}\,\left(r_{1}+r_{2}-2\sigma-\sqrt{4\,{\sigma}^{2}+(r_{1}-r_{2})^{2}}\right).$
(23)
The eigenvalues for this system are of a similar form as those for linear
migration in Eq. (15), but multiplied by $1/(1-\sigma)$. Therefore, the mass
action mixing can change the stability of the DFE as a function of basic
reproduction numbers in the same settings as linear mixing.
## 3 Vaccination responses
This section directly considers how the migration rates change the vaccination
levels necessary to keep the DFE stable, which implies the occurrence of herd
immunity. It explores whether neglecting the short- and long-term migration
rates overestimates or underestimates the minimum vaccination rates necessary
for disease fade-out.
We first restrict our attention to long-term migration only; i.e., letting
$c_{3}=0$ in Eq. (2). For $c_{1}>0$, this is equivalent to identifying the
bifurcation points in $(v_{1},v_{2})$ when $\lambda_{4}=0$ in Eq. (9). Notice
that if $c_{1}=0$, the subpopulations are isolated. The vaccination levels
needed in each for the disease to die out is equivalent to solving for $v_{1}$
and $v_{2}$ in $\hat{R}_{1,2}\leq 1$ from Eq. (12). The constant solutions for
$\hat{R}_{1,2}=1$ using parameter values in Table 1 are shown in Figure 2 as
solid black lines, and the disease will die out in both populations in the top
right quadrant.
Figure 2: Boundary of the region of stability for the DFE as we vary $c_{1}$.
The curves represent $\lambda_{4}=0$ using the parameter values in Table 1 and
$c_{3}=0$.The vertical and horizontal black lines represent the vaccination
rates necessary for a stable DFE in isolated populations ($c_{1}=0$). The
dashed line represents the limiting curve as we increase $c_{1}$.
As we increase $c_{1}$, the boundary for the region of stability for the DFE
spreads away from the $c_{1}=0$ case, increasing the die out region. The limit
is a line, shown by the dotted black line in Figure 2.
###### Proposition 6.
As $c_{1}\rightarrow\infty$, the bifurcation curve bounding the stable region
approaches the line
$v_{2}=-\left(\frac{\mu_{1}}{\mu_{2}\rho}\right)v_{1}+\frac{(\mu_{1}+\mu_{2}\rho)(\beta-\kappa)}{\beta\mu_{2}\rho}-\frac{(\mu_{1}+\mu_{2}\rho)^{2}}{\beta\mu_{2}\rho(1+\rho)}.$
(24)
This line is decreasing in $v_{1}$, with the slope depending on a ratio of the
birth rates and subpopulation sizes. Note that if we use the basic
reproductive numbers for the isolated subpopulations, Eq. (24) is equivalent
to
$v_{2}=-\left(\frac{\mu_{1}}{\mu_{2}\rho}\right)v_{1}+\frac{\left(1+\frac{\mu_{1}}{\mu_{2}\rho}\right)}{(1+\rho)}\left(\left(\frac{\hat{R}_{1}(0)-1}{\hat{R}_{1}(0)}\right)+\left(\frac{\hat{R}_{2}(0)-1}{\hat{R}_{2}(0)}\right)\right).$
(25)
As $\hat{R}_{1}(0)$ and/or $\hat{R}_{2}(0)$ increase, the $v_{2}$ intercept
increases and shifts the line up vertically. Therefore, the attainable stable
DFE region in $(v_{1},v_{2})$ space decreases, as expected.
Next, we consider only short-term migration, i.e. letting $c_{1}=0$.
Similarly, for $0\leq c_{3}\leq 1$, this is equivalent to identifying the
bifurcation points in $(v_{1},v_{2})$ when $\lambda_{4}=0$. We fix the
parameters to the values in Table 1 and vary $c_{3}$ to see the changes to the
boundary of the stable region. The solution for $c_{3}=0$ is shown as solid
black lines in Figure 3. Again the disease will die out in both populations in
the top right quadrant. As we increase $c_{3}$, the boundary for the region of
stability for the DFE smoothly pulls away from top right quadrant, increasing
the die out region.
###### Proposition 7.
The limit as we increase $c_{3}\rightarrow 1$ is
$v_{2}=1-{\frac{\left(\kappa+\mu_{{2}}\right)\left(\kappa+\mu_{{1}}\right)}{\left(1-v_{{1}}\right){\beta}^{2}}}.$
Therefore, in both cases, underestimating the migration between populations
causes an overestimation of the vaccination levels needed for herd immunity.
Figure 3: Boundary of the region of stability for the DFE as we vary $c_{3}$.
The curves represent $\lambda_{4}=0$ using the parameter values in Table 1 and
$c_{1}=0$. The vertical and horizontal black lines represent the vaccination
rates necessary for a stable DFE in isolated populations ($c_{3}=0$).
## 4 Conclusions
In this paper, we consider the effects of short- and long-term migration in
coupled population models in the presence of vaccination. We study the
interplay between the independent vaccination and migration rates across
different populations. We conclude that neglecting migration effects
overestimates the vaccination levels necessary to achieve herd immunity.
We have proven that if two isolated populations support an endemic state
simultaneously, migration cannot change the stability of those endemic states.
Analogously, this is also true for two populations with stable disease free
equilibria. In contrast, migration can lead to disease die out in the mixed
case. If a single population has a vaccination rate sufficient for herd
immunity in isolation, low levels of migration from a population that is
endemic will not necessarily make the disease endemic in both. In fact,
increased levels of migration can lead to disease die out in both populations.
However, migration rates are only physically realistic when they are small.
Our results suggest more efficient vaccination strategies may be identified
for groups of countries with significant migration between them. For example,
instead of increasing the vaccination levels in a population that has already
achieved herd immunity, sending vaccine to the less vaccinated neighboring
country could have a greater impact on outbreak levels. The most efficient
control algorithm would be to target the stable die out region as shown in
Figure 2 or Figure 3.
Conversely, populations for which vaccine delivery is difficult may benefit to
a degree by vaccination of neighboring countries. More specifically, consider
decreasing the migration rates to a country with a lower vaccination rate. We
show in Figure 4 a policy where $N_{2}$, which has a vaccination rate
$v_{2}=0.9$, decreases the long-term migration rate $c_{1}$ with $N_{1}$,
which has a vaccination rate $v_{1}=0.7$. The short term migration rate is
held constant at $c_{3}=0.1$. The decrease in number of new infections for
$N_{2}$ is a small percentage of the increase in infections in $N_{1}$, and we
conclude that the policy meant to help $N_{2}$ has unintended negative
consequences for $N_{1}$.
Figure 4: Time series of infectives in both populations using the parameter
values in Table 1, with $v_{1}=0.7$, $v_{2}=0.9$, and $c_{3}=0.1$. The value
of $c_{1}$ is decreased to the constant noted in each window.
In future directions, the model can be extended to include the effects of
seasonality. A similar analysis of stable periodic behavior can reveal the
sensitivity of synchronization to short-term and long-term migration. For
example, the work of Schwartz [16] predicts new periodic orbits that can be
excited by the mass action coupling in models with seasonal forcing.
Specifically, these orbits exhibit long period outbreaks in small populations
due to mass action coupling. When applying time dependent vaccination
schedules, other parameters must be considered in addition to the average
vaccination rates, such as pulse frequency and phase with respect to periodic
application. Other techniques can be extended to migration models with vaccine
control, such as prediction of future outbreaks as reported in Schwartz, et
al. [17].
## Acknowledgments
We gratefully acknowledge support from the Office of Naval Research. The
authors were also supported by the National Institute of General Medical
Sciences (Award No. R01GM090204). DATC holds a Career Award at the Scientific
Interface from the Burroughs Wellcome Fund and received funding from the Bill
and Melinda Gates Foundation Vaccine Modeling Initiative. The content is
solely the responsibility of the authors and does not necessarily represent
the official views of the National Institute of General Medical Sciences or
the National Institutes of Health. We would also like to thank Leah Shaw and
Luis Mier-Y-Teran for their useful discussions.
## References
* [1] N. J. Gay, W. J. Edmunds, Developed countries could pay for hepatitis B vaccination in developing countries, British Medical Journal 316 (1998) 7142.
* [2] B. Aylward, R. Tangermann, The global polio eradication initiative: Lessons learned and prospects for success, Vaccine Suppl 4 (2011) D80–5.
* [3] World Health Organization (WHO), Measles Fact Sheet No. 286, December 2009, http://www.who.int/mediacentre/factsheets/fs286/en//.
* [4] D. P. Wilson, J. Kahn, S. M. Blower, Predicting the epidemiological impact of antiretroviral allocation strategies in KwaZulu-Natal: The effect of the urban-rural divide, PNAS 103 (2006) 14228–14233.
* [5] D. A. T. Cummings, W. J. Moss, K. Long, C. S. Wiysonge, T. J. Muluh, B. Kollo, E. Nomo, N. D. Wolfe, D. S. Burke, Improved measles surveillance in Cameroon reveals two major dynamic patterns of incidence, International Journal of Infectious Diseases 10 (2) (2006) 148–155.
* [6] N. M. Ferguson, C. A. Donnelly, R. M. Anderson, Transmission dynamics and epidemiology of dengue: Insights from age-stratified sero-prevalence surveys, Phil. Trans. R. Soc. London, Ser. B 354 (1999) 757–768.
* [7] L. S. Liebovitch, I. B. Schwartz, Migration induced epidemics: Dynamics of flux-based multipatch models, Physics Letters A 332 (2004) 256–267.
* [8] L. Sattenspiel, K. Dietz, A structured epidemic model incorporating geographic mobility among regions, Mathematical Biosciences 128 (1995) 71–91.
* [9] A. L. Lloyd, V. A. A. Jansen, Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models, Mathematical Biosciences 188 (2004) 1–16.
* [10] M. J. Keeling, P. Rohani, Estimating spatial coupling in epidemiological systems: A mechanistic approach, Ecology Letters 5 (1) (2002) 20–29.
* [11] J. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, Journal of Theoretical Biology 239 (3) (2006) 376 – 390.
* [12] Y. Takeuchi, X. Liu, J. Cui, Global dynamics of SIS models with transport-related infection, Journal of Mathematical Analysis and Applications 329 (2) (2007) 1460 – 1471.
* [13] J. Liu, Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection, Chaos, Solitons & Fractals 40 (1) (2009) 145 – 158\.
* [14] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000) 599–653.
* [15] J. D. Murray, Mathematical Biology, Springer, Berlin, 1989.
* [16] I. B. Schwartz, Small amplitude, long period outbreaks in seasonally driven epidemics, J. Math. Biology 30 (1992) 473–491.
* [17] I. B. Schwartz, L. Billings, E. M. Bollt, Dynamical epidemic suppression using stochastic prediction and control, Physical Review E 70 (2004) 046220.
|
arxiv-papers
| 2012-05-21T19:26:15 |
2024-09-04T02:49:31.167874
|
{
"license": "Public Domain",
"authors": "Jackson Burton, Lora Billings, Derek A. T. Cummings and Ira B.\n Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1205.4703"
}
|
1205.4762
|
# Very High Resolution Solar X-ray Imaging Using Diffractive Optics
B. R. Dennis1G. K. Skinner2,3M. J. Li4A. Y. Shih1 1 Code 671, NASA Goddard
Space Flight Center email: brian.r.dennis@nasa.gov
2 Code 661 email: gerald.k.skinner@nasa.gov
3 CRESST and Univ of Maryland, College Park
4 Code 553 email: %**** ̵DiffractiveXrayOptics˙21May2012.tex ̵Line ̵100
̵****mary.j.li@nasa.gov
###### Abstract
This paper describes the development of X-ray diffractive optics for imaging
solar flares with better than 0.1 arcsec angular resolution. X-ray images with
this resolution of the $\geq 10$ MK plasma in solar active regions and solar
flares would allow the cross-sectional area of magnetic loops to be resolved
and the coronal flare energy release region itself to be probed. The objective
of this work is to obtain X-ray images in the iron-line complex at 6.7 keV
observed during solar flares with an angular resolution as fine as 0.1 arcsec
- over an order of magnitude finer than is now possible. This line emission is
from highly ionized iron atoms, primarily Fe xxv, in the hottest flare plasma
at temperatures in excess of $\approx$10 MK. It provides information on the
flare morphology, the iron abundance, and the distribution of the hot plasma.
Studying how this plasma is heated to such high temperatures in such short
times during solar flares is of critical importance in understanding these
powerful transient events, one of the major objectives of solar physics. We
describe the design, fabrication, and testing of phase zone plate X-ray lenses
with focal lengths of $\approx$100 m at these energies that would be capable
of achieving these objectives. We show how such lenses could be included on a
two-spacecraft formation-flying mission with the lenses on the spacecraft
closest to the Sun and an X-ray imaging array on the second spacecraft in the
focal plane $\approx$100 m away. High resolution X-ray images could be
obtained when the two spacecraft are aligned with the region of interest on
the Sun. Requirements and constraints for the control of the two spacecraft
are discussed together with the overall feasibility of such a formation-flying
mission.
###### keywords:
Solar flares, X-rays, lenses, imaging
## 1 Introduction
Sect:Introduction
High resolution X-ray imaging has been discussed extensively in astrophysics
over many years. One of the most exciting prospects with the 0.1 to 1 micro-
arcsecond resolution that should be possible with diffractive optics is to be
able to probe the space-time at the event horizon of super-massive black
holes, e.g. Skinner (2001, 2009). However, achieving this resolution requires
lenses with focal lengths of hundreds or thousands of km. Their use would
require flying X-ray detectors on one spacecraft and lenses on a second
spacecraft separated from the first by the focal length in the direction of
the source of interest. Formation flying with such large spacecraft
separations and the stabilization and alignment knowledge necessary to
generate such high-resolution images would be a major advance on existing
capabilities.
We show here that a less demanding form of the same technology can contribute
in heliophysics to the frontier area of solar flare research. The high
resolution X-ray imaging of the Sun possible with diffractive optics on a
scale more modest than that considered for astrophysical applications offers
the possibility of obtaining significantly finer angular resolution than is
possible with the conventional diffraction-limited reflective optics used at
longer wavelengths. Even a 1-m diameter mirror is diffraction limited at
optical wavelengths at $\approx 0.1$ arcsec. The Advanced Technology Solar
Telescope (ATST) in Hawaii will operate at the diffraction limit of a 4-m
diameter mirror in the near infra-red wavelength range of 900 - 2500 nm to
achieve 0.03 arcsec resolution at 500 nm and 0.08 arcsec at 1.6 $\mu$m over a
field of view of 2 to 3 arcmin (Keil et al., 2011). Because of the much
shorter wavelengths of X-rays (1 Å = 0.1 nm), diffraction-limited X-ray optics
of only 1 cm in diameter could equal or even improve on this angular
resolution.
High resolution imaging of solar X-rays would be invaluable in both soft
X-rays ($\lesssim$10 keV) from hot plasma and hard X-rays ($\gtrsim$ 10 keV))
from nonthermal distributions of energetic electrons. Soft X-ray observations
with 0.1 arcsec resolution of the $\gtrapprox$10 MK plasma in solar active
regions and solar flares would allow the cross-sectional area of magnetic
loops to be resolved and could allow the coronal flare energy release region
itself to be probed on physically meaningful spatial scales. At higher
energies, high resolution imaging would be possible of hard X-rays produced by
beams of nonthermal electrons as they stream down from the acceleration site
in the corona into the higher density regions of the lower corona and
chromosphere. The structure of the compact footpoint X-ray sources at the 0.1
arcsec level will provide information on the details of the coronal
acceleration process itself. The ability to detect the relatively weak coronal
emission from the electron beams in the presence of the bright chromospheric
footpoint sources (dynamic range requirement of $\gtrapprox$100:1) will also
allow the acceleration site to be studied in unprecedented detail.
Although there is great interest in X-ray imaging over a broad energy range
extending into the nonthermal domain above 10 keV, technical issues limit the
technique using diffractive X-ray optics to a very narrow energy range.
Consequently, for this initial demonstration of high resolution solar X-ray
imaging we have chosen to concentrate on the soft X-ray range, specifically
the group of Fe emission lines between 6.6 and 6.7 keV. The strongest line is
typically the ‘w’ Fe xxv resonance line at 6.699 keV (1.851Å) as seen in
Figure fig:BCS_spectra. By using detectors with relatively modest energy
resolution or by absorbing lower energy photons, it is possible to arrange
that the diffractive optics image of a flare is dominated by emission in this
line complex. In this way we are able to make images almost free from blurring
due to photons at other energies that come to a focus at different distances
due to the chromatic aberration intrinsic to diffractive optics. We propose to
use this line complex to facilitate the first demonstration of high resolution
imaging using diffractive X-ray optics and readily available detectors. It
will be possible to extend this technique to higher energies in the future but
because the solar flare X-ray emission above 10 keV is all bremsstrahlung
continuum with no narrow lines, detectors with high energy resolution will be
required to isolate the energies in the narrow range that come to a focus at
the detector.
---
Figure 1.: High resolution X-ray spectra of a solar flare showing the complex
of lines between 6.5 and 6.75 keV. The lines are from highly ionized iron
atoms in a plasma at a temperature of $\approx$20 MK, with the most prominent
line from Fe xxv at 6.699 keV that is labeled with a ’w’ in the lower plot
using the naming convention of Gabriel (1972). The spectrum was recorded with
the Bent Crystal Spectrometer (BCS) on the Solar Maximum Mission during a
flare in 1980 June 29. fig:BCS˙spectra
### 1.1 Solar Flare X-ray Emission
Sect:Science
X-rays from solar flares in the energy range between $\approx$1 and 10 keV are
emitted from hot plasma with temperatures from $\approx$$10^{6}$ K (1 MK) to
sometimes in excess of 50 MK. Detailed studies of this radiation, both
spectroscopically and by imaging, provide crucial information on the nature of
the heated plasma, and can provide clues to help us understand how such a
large volume of plasma can be heated to such high temperatures so rapidly
during a flare.
The solar spectrum in this energy range is a combination of both line and
continuum emission. The many narrow lines are emitted by transitions of atoms
of the different elements of the plasma in the solar atmosphere in various
stages of ionization; the continuum emission is from both free-free and free-
bound interactions of electrons with atomic nuclei that produce bremsstrahlung
and recombination radiation, respectively. The most detailed catalog of lines
in the 3-10 keV range is given by Phillips (2004, 2008). Both line and
continuum spectra can be synthesized using the CHIANTI database and software
(Dere et al., 1997, 2009) for a range of possible temperatures, plasma
abundances, and excitation and ionization conditions. The line of greatest
interest here is the Fe xxv resonance line at 6.699 keV (1.851 Å). It is the
strongest and most prominent line of the complex of iron lines between 6.5 and
6.7 keV (Figure fig:BCS_spectra). This line was named the ‘w’ line by Gabriel
(1972) and arises from the $1s^{2~{}1}S_{0}-1s2p^{1}P_{1}$ transition of
helium-like iron nuclei, i.e., with just two electrons remaining of the
original 26 of an unionized iron atom. It is emitted from plasma at
temperatures above $\approx$10 MK, and hence provides information on the
hottest plasma generated in a solar flare. During the flare impulsive phase,
the line shows a broadening that is commonly attributed to turbulence in the
plasma resulting from the impact of nonthermal electrons. The broadening may
amount to 0.03 Å (0.01 keV or 10 eV), equivalent to a few hundred km s-1.
Nevertheless, the line is always sufficiently narrow to allow imaging with
better than 0.1 arcsec resolution using diffractive optics.
It is clear from images taken with existing X-ray instruments and those in
different wavelength bands that there are features that are unresolved at the
currently resolvable angular scale of $\approx$1 arcsec. The ubiquitous
magnetic loops observed before, during, and after flares appear to be
significantly narrower than 1 arcsec. The magnetic reconnection process that
is believed to lead to the energy release from the coronal magnetic field that
heats the plasma takes place on scales much smaller than 1 arcsec. The total
energy and density of the thermal plasma can be estimated from the emission
measure and source volume as revealed by the X-ray images but the values so
obtained are subject to a large, order-of-magnitude, uncertainty because of
the unknown “filling factor,” the ratio of the apparent volume of the plasma
to the actual volume. Only by making images with higher angular resolution can
this uncertainty be diminished and ultimately eliminated once the finest
plasma structures are resolved. These unknowns then set the goal of the
current effort to image in soft X-rays with better than 0.1 arcsec resolution,
fully an order of magnitude better than what has been achieved to date.
### 1.2 Current Capabilities and Limitations
It is clear that the finest possible angular resolution is required to fully
understand the heating of the flare plasma. To date, the highest angular
resolution for solar observations in this energy range is $\approx$1 arcsec
(full width at half maximum, FWHM) achieved with the X-ray Telescope (XRT) on
Hinode (Golub et al., 2007; Weber et al., 2007) using a single monolithic
grazing-incidence mirror with two reflections. With grazing incidence optics,
angular resolution is limited by surface figure and, where nested optics are
employed, by alignment considerations. As pointed out by Davila (2011), for
example, conventional high energy optics techniques cannot obtain the required
$<$0.1 arcsec resolution.
The Ramaty High Energy Solar Spectroscopic Imager (RHESSI) has achieved
$\approx$2 arcsec angular resolution using Fourier-transform imaging at
energies extending from $\approx$3 keV up to 100 keV, and with progressively
poorer resolution up to 17 MeV (Lin et al., 2002). Bi-grid rotating modulation
collimators are used with 1.5 m separation between grids and a finest slit
pitch of 34 $\mu$m. Higher resolution would be possible with finer slits
and/or greater grid separation but at the expense of field of view. In
addition, the achievable dynamic range in any one image using this technique
is limited to less than $\approx$50:1 by side lobes in the response function.
This makes it difficult, for example, to image the relatively weak and
extended coronal X-ray sources in the presence of intense compact footpoint
sources.
## 2 Diffractive X-ray Optics
Sect:Diff_Opt The angular resolution possible with reflecting optics is
critically dependent on the accuracy of the polishing of the mirror surfaces
and on their alignment. In contrast, the tolerances necessary to manufacture
diffractive lenses capable of ultra-high angular resolution are much more
relaxed. This is because diffractive focusing depends on modulation of the
phase and/or amplitude of radiation by an optical element working in
transmission at normal incidence. This means that the lenses can be relatively
thin, and alignment precision is not a serious issue.
Davila (2011) has proposed the use of a photon sieve, a form of Fresnel zone
plate, to achieve high angular resolution in the extreme ultraviolet. We
concentrate here on Fresnel lenses that can offer higher efficiency than zone
plate variants and suffer less from diffraction into spurious orders. The
principles of phase Fresnel lenses and their development for X-ray and gamma-
ray astronomy have been discussed in a series of papers (Skinner, 2001;
Krizmanic et al., 2005; Krizmanic et al., 2007; Skinner, 2009).
### 2.1 Lens Design Considerations
sect:design
The focussing ability of diffractive optics requires that radiation that
passes through different parts of the lens must all arrive at the focal point
with identical phase. This can be achieved if the thickness, $t$, of the lens
with a focal length, $f$, is the following function of the radial distance,
$r$, from its center:
$\displaystyle t(r)$ $\displaystyle=$
$\displaystyle\left(r^{2}/(2f\delta)\pmod{\lambda}\right)+t_{0}$ (1)
$\displaystyle=$
$\displaystyle\left(r^{2}/(2f\lambda)\pmod{1}\right)t_{2\pi}+t_{0},$ (2)
where $\lambda$ is the wavelength and the refractive index is $\mu=1-\delta$
(see Krizmanic et al., 2007). The addition of the $t_{0}$ term reflects the
fact that, apart from absorption, the presence of a constant thickness
substrate does not affect the focussing properties, just the throughput. In
Equation eqn:t2pi, we have written $t_{2\pi}$ for the thickness of material,
$\lambda/\delta$, that changes the phase by $2\pi$. Conveniently, in the soft
X-ray band, this quantity is typically a few $\mu$m leading to structures well
matched to fabrication by Micro-Electro-Mechanical Systems (MEMS) engineering
technology.
The ideal profile corresponding to Equation eqn:t2pi, as illustrated in Figure
fig:profilesa, produces an optic known as a phase Fresnel lens (PFL). It can
be replaced by a multi-step approximation (Figure fig:profilesb), the limiting
case being where there are just two thicknesses - $t_{0}$ (which may be zero)
and $t_{0}+t_{2\pi}/2$ \- as illustrated in Figure fig:profilesc. In a further
approximation, it is possible simply to block the radiation for which the
phase is not within $\pm\pi/2$ of the ideal, leading to the zone plate
structure shown in Figure fig:profilesd. For that reason, a two-level
diffractive lens is often referred to as a phase zone plate or PZP. The ideal
efficiencies, defined as the flux in the first order focus as a fraction of
the total incident flux, are listed in Table table:eff for the different
cases.
---
Figure 2.: (a) The ideal cross-section through a Phase Fresnel Lens (PFL)
with a maximum thickness (t) of $t_{0}$ (which may be zero) $+\hskip
3.61371ptt_{2\pi}$. For X-rays, the refractive index is less than one
($\delta$ is positive) so the configuration shown corresponds to a converging
lens. (b) A four-level approximation to a PFL profile with a thickness of
$t_{0}+0.75\hskip 3.61371ptt_{2\pi}$. (c) A two-level approximation (phase
zone plate, PZP) with a thickness of $t_{0}+0.5\hskip 3.61371ptt_{2\pi}$. (d)
A zone plate, in which alternate zones are opaque to the radiation.
fig:profiles Table 1.: The ideal efficiency of diffraction into the first-
order focus for lenses with profiles such as those illustrated in Figure
fig:profiles. Absorption is assumed to be negligible.
| phase | | | phase |
---|---|---|---|---|---
Approximation | Fresnel lens | $n$-levels | 4-levels | zone plate | zone plate
| (PFL) | | | (PZP) | (ZP)
Figure fig:profiles | (a) | | (b) | (c) | (d)
Efficiency | 1 | $\left[\frac{n}{\pi}\sin(\frac{\pi}{n})\right]^{2}$ | 0.811 | 0.405 | 0.101
table:eff
Irrespective of the profile adopted, at the edge of the lens the structure is
periodic with the minimum period given by
$p_{\mathrm{min}}=2f\lambda/d,\ilabel{eqn:pmin}$ (3)
where $d$ is the lens diameter. The focal length $f$ can be written in terms
of the photon energy, $E$, in the range that is of particular interest here
as:
$f=102\left(\frac{p_{\mathrm{min}}}{1.9\mbox{
$\mu$m}}\right)\left(\frac{E}{6.65\mbox{ keV}}\right)\left(\frac{d}{20\mbox{
mm}}\right)\mbox{ m}\ilabel{eqn:f}.$ (4)
For a diffraction-limited lens, the angular diameter of the Airy disk is :
$\theta_{\mathrm{d}}=2.44\>\lambda/d\>=1.22\>p_{\mathrm{min}}/f.$ (5)
For an ideal lens, the Airy disk contains 84% of the energy in the first order
focus. With typical parameters, its angular size is much smaller than the
resolution targeted here:
$\theta_{\mathrm{d}}=4.7\left(\frac{E}{6.65\mbox{
keV}}\right)^{-1}\left(\frac{d}{20\mbox{ mm}}\right)^{-1}\>{\mbox{milli-
arcsec}}\ilabel{eqn:diff}.$ (6)
For our purposes, a more important consideration than the diffraction limit is
blurring due to chromatic aberration. Equation eqn:f shows that the focal
length is proportional to photon energy so in practice the use of such lenses
is restricted to observations in which a narrow range of energies is dominant
or where the bandwidth is limited in other ways. If the point spread function
(PSF) of the lens is approximated by the best-fit Gaussian and the spectral
line to be imaged is also taken to be Gaussian with a FWHM of $\Delta E$, then
the chromatic aberration contribution to the FWHM angular resolution of the
lens is found to be
$\theta_{\mathrm{c}}=5.2\left(\frac{\Delta
E}{E}\right)\;\left(\frac{d}{20\mbox{ mm}}\right)\;\left(\frac{f}{100\mbox{
m}}\right)^{-1}\>{\mbox{arcsec}}.\ilabel{eqn:chrom}$ (7)
Note that the PSF is strongly cusped and the central peak is much sharper than
this implies. On the other hand, if the criterion adopted is the width that
contains 84% of the energy (the fraction within the central peak of an Airy
disk), then the numerical factor in Equation eqn:chrom is more than doubled.
A typical line width due to thermal broadening of the Fe xxv 6.699 keV line is
about 2 eV, corresponding to a Doppler velocity of $\approx$100 km s-1. There
is often additional broadening due to turbulence, and line shifts due to bulk
motion, both of which can be several times larger than this. Furthermore,
although this spectral line is frequently dominant, it is just one line of
many lines in a complex that can be approximated by a Gaussian with a
characteristic width of about 100 eV. If the detector is energy resolving and
can be used in a photon counting mode, one can attempt to select just those
photons that fall within a narrow energy band. However, for Silicon CCDs and
similar detectors, the attainable FWHM resolution $\Delta E$ at 6.7 keV is
limited to about 150 eV – wider than the line complex. If we take a width of
100 eV and the reference parameters used in Equations eqn:f and eqn:diff, the
limit due to chromatic aberration is 80 milli-arcsec FWHM.
A further limit to the angular resolution potentially arises because of the
spatial resolution of the detector. For a pixel size $\Delta x$ the
corresponding limit is
$\theta_{\mathrm{s}}=\frac{\Delta x}{f}=21\left(\frac{\Delta x}{10{\mbox{
$\mu$m}}}\right)\left(\frac{f}{100\mbox{ m}}\right)^{-1}\>{\mbox{milli-
arcsec}.\ilabel{eqn:detlim}}$ (8)
The work of Young (1972) on aberrations in zone plate imaging can be applied
to Fresnel lens optics. It shows that, for systems of interest for solar
physics or astrophysics, geometric aberrations are very small, and the only
limit on the field of view will be that imposed by the size of a practical
detector.
Thus, none of these limits prevent an angular resolution of the order of 100
milli-arcsec from being obtained in imaging observations in the Fe-line
complex provided that lenses with $p_{\mathrm{min}}$ of 1.5 – 2 $\mu$m can be
made and used in a configuration with a focal length of $\approx$100 m.
Figure 3.: Photograph of a 3-cm diameter silicon phase zone plate fabricated
in the Detector Development Laboratory (DDL) at Goddard Space Flight
Center.fig:PZPphoto
Figure 4.: Scanning electron microscope image of the central area of a
demonstration lens showing the circular slits and the radial support ribs.
fig:Lens˙SEMimage
Figure 5.: Scanning electron microscope image of the outer section of a lens
showing the finest slits and part of a radial support rib. fig:SEMslits
### 2.2 Lenses Fabrication
sect:fab
We have initially concentrated on the fabrication of PZP lenses with the
cross-section shown in Figure fig:profilesc. For a given $t_{2\pi}$, it is
easier to obtain a small $p_{\mathrm{min}}$ with this profile than to
fabricate the tapered PFL profile (Figure fig:profilesa) or a multi-level
approximation to one (Figure fig:profilesb). For a particular focal length, a
smaller $p_{\mathrm{min}}$ allows the lens diameter to be larger (per
Equations (eqn:pmin) and (eqn:f)), and the increased geometric area can
compensate for the lower efficiency. We have fabricated lenses with parameters
given in Table table:params that are similar to the reference values in
Equation (eqn:f) needed to meet the requirements for a flight instrument. A
photograph of such a phase zone plate is shown in Figure fig:PZPphoto with
scanning electron microscope images of the central area in Figure
fig:Lens_SEMimage and the finest slits at the edge in Figure fig:SEMslits.
Table 2.: Comparison of the parameters of the demonstration lenses with a
possible flight design.
Parameter | Demonstration | Possible flight |
---|---|---|---
| | laboratory lenses | design |
Energy | $E$ | 5.411 | 6.65 | keV
Focal length | $f$ | 110.4 | 100 | m
Diameter | $d$ | 30 | 20 | mm
Finest pitch | $p_{\mathrm{min}}$ | 1.7 | 1.9 | $\mu$m
Profile height | $t$ | 8.2 | 8.2 | $\mu$m
table:params
We chose to make the lenses of silicon since MEMS techniques are most advanced
for this material and it is acceptable for lenses working in the X-ray energy
range considered here. At 6.699 keV, the thickness of Si needed to give a
$2\pi$ phase change ($t_{2\pi}$) is 16.8 $\mu$m. This is the peak thickness,
$t$, of the profile for an ideal PFL (Figure fig:profilesa); for a PZP, only
half this thickness is required so $t=8.4$ $\mu$m. In fact we have adopted
$t=8.2$ $\mu$m as by reducing the depth slightly imperfect phase matching is
traded for reduced absorption and a small improvement in overall efficiency is
obtained, while making fabrication marginally easier.
Fabrication of the lenses was conducted in the Detector Development Laboratory
(DDL) at GSFC, a fully equipped semiconductor processing facility center with
Class 10 clean-room capabilities. Phase zone plate lenses (Figure
fig:profilesc) were designed using the DW2000 software tool for mask layout.
Standard photolithography and advanced Deep Reactive Ion Etching (DRIE)
processes were employed with a UV mask aligner (SUSS MicroTec MA-6) and a
high-rate etcher (STS).
Each lens was fabricated from a 4-inch diameter Silicon-On-Insulator (SOI)
wafer. As shown in Figure fig:fab, an SOI wafer consists of three layers - (1)
a thin Si layer on top called the device silicon layer, (2) a thin silicon
dioxide (SiO2) insulating layer, and (3) a thicker Si layer called the handle
silicon layer.
---
Figure 6.: Cross-sections of SOI wafers for illustration of the lens
fabrication process. Top: SOI wafer showing the top device silicon layer, a
thin insulating layer of silicon oxide, and the bottom thicker layer called
the handle silicon layer; middle: lens etched out from the device silicon
layer; bottom: support structure etched from the silicon substrate or handle
silicon layer. fig:fab
The sequence of operations is illustrated in Figure fig:fab. The lens pattern
is first formed on the front (device) side of the SOI wafer by UV exposure of
a photo-resist layer through a chromium-on-glass mask. This is followed by
development and DRIE etching down to the silicon oxide insulating layer. The
SOI wafer is then attached to a glass wafer with wax to protect the front-side
silicon lens features. The procedure is repeated on the backside to produce
the spider-web support structure, again with DRIE etching as far as the oxide
layer.
This DRIE process allows slits with the required high-aspect ratios of up to
20:1 to be etched in silicon. However, the parameters of the etch process had
to be tuned in order to obtain vertical silicon walls with the required width
of remaining silicon between adjacent slits. An anti-reflection coating was
used to make good optical contact between the SOI wafer and the glass mask,
allowing a precise lens pattern with features at the submicron level to be
obtained.
Techniques exist that allow the fabrication of lenses with multiple levels
(Figure fig:profilesb) or even good approximations to the ideal continuous
profile (Figure fig:profilesa) – see for example di Fabrizio et al. (1999) and
Krizmanic et al. (2005). If lenses could be made with the same diameter and
focal length as our two-level lenses but with, say, a 4-level stepped profile,
then the effective area could, in principle, be more than doubled (see Table
table:eff). Fabricating such structures would require etching features finer
by at least a factor of two, and somewhat deeper. In conjunction with the Army
Research Laboratory (ARL) in Adelphi, MD, we are investigating the feasibility
of using e-beam lithography to obtain four-level profiles. Because of the long
writing times involved, initially the objective is to make a sample 1 mm wide
strip across the radius of a 30 mm diameter lens.
### 2.3 Test Results
section:TestResults
Several lenses have been tested at the NASA-GSFC X-ray Interferometry Testbed.
This facility provides a 600-m evacuated path between source and detector
stations with an intermediate station for the focusing optics. Because of the
relatively low melting point of iron and the need for high surface brightness,
it was not possible to use an iron target X-ray tube, which would have
produced the Fe K$\alpha$ line at 6.4 keV, close to the solar Fe xxv $w$ line
at 6.699 keV. Instead, a chromium target X-ray tube was used, producing the Cr
K$\alpha$ line at 5.411 keV. Consequently, although the lenses were designed
to have the finest pitch $p_{\mathrm{min}}$, thickness $t$, and focal length
$f$ similar to those needed for a 20 mm diameter flight lens working at 6.699
keV, the diameter was actually 30 mm. The focal length was chosen to be 110.4
m at the Cr line energy so that, when the lens was positioned at the ‘optics’
station of the testbed 146 m from a 5-$\mu$m wide tungsten slit placed
directly in front of the source, an enlarged image of the slit was formed in
the plane of the detector, 452 m from the lens (see Figure fig:TestSetup).
This magnifying configuration has the advantage that the 13 $\mu$m pixels of
the Roper CCD camera used as a detector were capable of resolving the image of
the slit. The camera was used in a photon-counting mode, accepting events in a
narrow energy range (5.3$-$5.6 keV) containing the Cr K$\alpha$ line while
excluding the K$\beta$ line at 5.947 keV.
Figure 7.: Test setup at Goddard’s 600-m X-ray Interferometry Testbed. A CCD
X-ray camera with 13 $\mu$m pixels was used to record the magnified image of a
5-$\mu$m slit, whose width at a distance of 146 m corresponds to 7 milli-
arcsec. Not to scale. fig:TestSetup
An image produced with this configuration is shown in Figure fig:image and a
cross-section though it in Figure fig:xsect. The core of the response is 66
$\mu$m FWHM which corresponds to 30 milli-arcsec, or 28 milli-arcsec when
allowance is made for broadening by the detector pixelization and the finite
width (5 $\mu$m) of the slit. There is a significant amount of power in the
wings that are rather broader than the core but even measuring the width at
one tenth maximum, the width of 70 milli-arcsec is better than the target of
100 milli-arcsec.
---
Figure 8.: An image of a 5 $\mu$m slit obtained with the test setup in Figure
fig:TestSetup. The pixels correspond to an angular size of 6 milli-arcsec.
fig:image
---
Figure 9.: Intensity as a function of angular offset transverse to the slit
for the image in Figure fig:image. fig:xsect
We have estimated the effective area of the lens by comparing the count rate
in the peak with that when the lens was replaced by a plate with a small hole
of known size. To avoid pulse pile-up effects and to allow any variation of
efficiency across the lens to be investigated, some of the measurements were
made with a 5 mm aperture in front of the lens. The position of the aperture
could be controlled in two axes with stepper-motor drives. We find an average
efficiency of 24% with very little radial variation, leading to an effective
area of just over 2 cm2. This can be compared with the theoretical value for
an ideal phase zone plate of 40.5%, which is expected to be reduced to 26.8%
when absorption, obscuration by the radial support ribs, and the fact that the
profile is optimized for an energy different from the test energy, are taken
into account. A similar calculation leads to an ideal efficiency of 33.7% at
$\approx$6.7 keV.
## 3 Space Mission Concept
Sect:MissionConcept
The major problem associated with flying such phase zone plate lenses in space
and obtaining X-ray images of the Sun with sub-arcsecond resolution is the
required focal length. The lenses we have developed and demonstrated have a
focal length $f$ of $\approx$ 100 m. Extensible booms up to 60 m in length
have been flown in space (Farr et al., 2007) and longer ones proposed (Johnson
et al., 2010, e.g.) but for such distances formation flying of two spacecraft
with lenses on one and detectors on the other becomes an attractive
possibility.
### 3.1 Two-Spacecraft Formation Flying
For our purposes, the formation-flying concept requires control of the
relative positions of the two spacecraft and of their orientations. Table
table:requirements summarizes the technical requirements. Note that the
spacecraft station-keeping and attitude control requirements are relatively
lax. This is because images can be built up from a series of ‘snapshots’ and
milli-arcsec imaging can be achieved providing only that milli-arcsec
knowledge of the lens-detector alignment with respect to the Sun at the time
of each snapshot is available. Aspect control of the individual spacecraft is
required only to a fraction of a degree. The transverse station keeping
requirement is dictated only by the need to keep the region of interest imaged
on the detector. The axial requirement is determined by the necessity of
keeping the out-of-focus blurring to a negligible level. Precise measurement
of the actual alignment at all times then allows the image to be reconstructed
taking into account the actual determined attitude without compromising the
angular resolution. If the detector is a CCD or other device that integrates
between readouts and does not allow accurate time-tagging, then there is a
requirement that the alignment remain stable on the timescale of the
integration – perhaps one or a few second(s).
Table 3.: The principal parameters and station-keeping requirements for a
solar flare X-ray imager using diffractive X-ray optics on a formation-flying
mission. table:requirements Lens - detector requirements
---
Lens - detector separation | $\approx$100 m | Larger separations would allow larger lenses, but would reduce the field of view possible with a reasonable detector size
Pixel size for 0.1” resolution | 25 $\mu$m | X-ray CCD or X-ray pixel detector
Field of View | 3 arcmin | Assumes 10 cm detector diameter to cover a typical active region - could be larger.
Station keeping requirements
Control - transverse | 2 cm | Requirement is to keep image on detector - could move detector or lens independently.
Control - axial | 25 cm | Equivalent to 20 eV energy shift
Stability - transverse | $<$0.1 arcsec = 25 $\mu$m at 100 m over 1 s | Alignment should not change between read-outs
Knowledge - transverse | $<$0.1 arcsec = 25 $\mu$m at 100 m | Alignment of line joining lens and detector must be known relative to the direction to Sun center to an accuracy better than the desired resolution.
Knowledge - axial | 10 cm | To determine the energy of the focussed X-rays to $<$10 eV.
Attitude requirement
Control and knowledge | 10 arcmin | Tilt and tip of lens and detector are not critical
### 3.2 Orbit Considerations
To maintain the configuration of the two spacecraft, a quasi-continuous thrust
will be needed on one, or both, spacecraft to overcome differential
accelerations due to gravity gradients, radiation pressure, and drag. For
spacecraft separations in the range under consideration here of order 100 m,
solar gravity forces are never an important consideration. However, within the
Earth-Moon environment, gravity gradient forces will generally dominate. At a
distance, D, from the center of the Earth, a spacecraft of mass
$M_{\mathrm{sat}}$ will require a thrust, $\Delta F_{\mathrm{g}}$, to keep at
a constant direction and separation $f$ ($f\ll D$) relative to a passive
reference spacecraft. The required thrust is given by –
$\Delta F_{\mathrm{g}}=8\left(\frac{M_{\mathrm{sat}}}{1,000\mbox{
kg}}\right)\left(\frac{D}{10,000\mbox{
km}}\right)^{-3}\left(\frac{f}{100\mbox{ m}}\right)\mbox{
mN.}\ilabel{eqn:diffgrav}$ (9)
The $D^{-3}$ term in Equation (eqn:diffgrav) means that orbits in which the
spacecraft spend most of the time far from Earth are strongly preferred.
Highly eccentric orbits with periods of several days duration are probably a
viable choice, though observations will be interrupted during perigee and the
subsequent time necessary to reconfigure the formation. Krizmanic et al.
(2005) have discussed propulsion solutions for astrophysical missions with
lenses having much longer focal lengths and higher thruster requirements than
implied by Equation (eqn:diffgrav).
Far from the Earth and Moon, the dominant disturbance force is likely to be
differential radiation pressure. At 1 AU, this is only 4.6 to 9.2 $\mu$N per
m2 of difference in effective areas of the two spacecraft normal to the Sun,
depending on the reflection properties of the surfaces. Stationing the
spacecraft pair close to the Sun-Earth L1 Lagrangian point would allow for
almost continuous aligned observations.
We note that ESA’s Proba-3 mission (Vives et al., 2010; Lamy et al., 2010) is
designed to demonstrate many of the capabilities needed for a mission to
perform high angular resolution solar X-ray imaging with diffractive lenses.
Proba-3 will control two spacecraft separated by 25–250 m so that a
coronagraph occulter on the front spacecraft will appear directly in front of
the Sun when viewed by instruments on the other spacecraft. This technology
demonstration mission will be in a 24-h eccentric Earth orbit and so the
required alignment will only be maintained for limited times. The precision of
the specification for the attitude determination falls only a little short of
that needed to take full advantage of the capabilities of the lenses under
discussion here.
A coronagraph could be a natural companion instrument to a diffractive optics
X-ray imager. A lens (or multiple lenses) could be mounted within the
coronagraph occulting disk. Light blocks would ensure that they are opaque to
visible/UV radiation while having negligible X-ray absorption (such light
blocks will, in any case, be desirable for thermal control).
## 4 Simulations of Capabilities
Sect:sims
In order to demonstrate the potential of the X-ray imaging technique
considered here, we have performed Monte Carlo simulations of the operation of
an instrument with a diffractive lens on one spacecraft and a CCD imaging
detector on the other. We used the lens parameters for the possible flight
design given in Table table:params. The assumed lens performance is based on
the lenses that we have developed and tested as described in Sections sect:fab
and section:TestResults. We have modeled a flare of similar intensity and
temperature to the 1980 June 29 event (see Figure fig:BCS_spectra), a class M4
flare, and used the spatial distribution shown in Figure fig:SimImage(top) in
which a loop is assumed to be composed of a large number (30) of fine
intertwined threads, inspired by a model illustrated in Figure 1.18 of
Aschwanden (2005).
For the spectrum of the emission, we have used a CHIANTI simulation of a
plasma with coronal abundances, a temperature of 18.8 MK, and a volume
emission measure of $10^{48.84}~{}\mathrm{cm}^{-3}$ based on the GOES X-ray
data of the flare. This gives approximately the spectral form and
normalization measured with BCS shown in Figure fig:BCS_spectra. In order to
follow changes in the flare loop structures, we would like to be able to
obtain an image in, say, 10 s. We assume that, because of the limitations of a
CCD detector, a 10 s accumulation is composed of 1 s observation periods each
followed by a 1 s readout time, leading to 50% live time.
The simulations demonstrate an important consideration when making
observations of relatively bright, potentially fast-changing objects. The
event rate in all the bright regions of the image precludes operation in the
photon counting mode since there will be multiple photons per pixel in each
1-s frame. However, because the emission is concentrated in a narrow energy
band, the total charge collected provides a good measure of the signal. For
operation in this analog mode, it is important to attenuate the lower energy
photons as much as possible, and for this purpose we assume that a
25-$\mu$m-thick layer of Si and a 10-$\mu$m layer of Fe is placed in the line
of sight, perhaps forming a lens substrate.
Figure fig:SimImage(middle) shows a simulated image derived from the model
image shown in Figure fig:SimImage(top). The image quality is more affected by
photon statistics than by the optics. Figure fig:SimImage(bottom) shows a
corresponding simulation of a 10 times longer observation or a 10 times
stronger flare.
---
Figure 10.: (top) The spatial distribution model used as input to the
simulations shown below. (middle) The image resulting from a 10 s (5 s
effective) observation of the above source distribution. (bottom) The
corresponding image with 10 times the exposure or for a 10 times stronger
flare.fig:SimImage
## 5 Summary and Conclusions
Sect:conclusions
The results presented here show that diffractive X-ray lenses can be made that
are capable of solar imaging in the Fe xxv $w$ emission line and nearby lines.
With such lenses, the capability exists for making X-ray images over narrow
energy ranges with angular resolutions as fine as 0.1 arcsec. This high
angular resolution and the effective area of the demonstration lenses already
fabricated would allow hot plasma generated during solar flares to be studied
on unprecedentedly fine spatial scales and determine the multi-threaded nature
of hot coronal magnetic loops. The detailed structure of the coronal energy
release sites could also be explored in detail during solar flares.
We note that the efficiency of a PZP lens can be improved by using a
multilevel approximation to the ideal Phase Fresnel Lens profile rather than
just the two levels of the lenses we have fabricated. In developments not
reported here, we have demonstrated the capability of using grey-scale
technology to generate a 4-level profile which should double the efficiency of
the lens and give twice the sensitivity. Alternatively, the higher efficiency
could be used to achieve a higher bandwidth with the same sensitivity by
fabricating a smaller lens with a higher focal ratio. The technique proposed
here can be extended to higher energies, though even longer focal distances
would be needed to achieve a useful effective area unless lenses with even
finer slits and higher aspect ratios can be developed. On the other hand, the
lower absorption losses at higher energies may allow the bandwidth to be
increased with achromatic combinations (Skinner, 2010).
Detailed design of a mounting and support structure and demonstration that the
lens can withstand launch and operational environments are planned, as are
further studies of the formation-flying aspects of a possible mission. More
detailed development of the complete instrument concept is underway. These
activities are designed to bring a high resolution X-ray imager concept to the
point where it could be proposed for flight, perhaps as a science demonstrator
on a formation-flying technology mission.
#### Acknowledgements
We thank John Krizmanic and Keith Gendreau for supporting the lens design and
testing effort, Kenneth Phillips for providing the spectra shown in Figure
fig:BCS_spectra and for help with the solar objectives, and Amil Patel, Gang
Hu, and Thitima Suwannasiri for their work fabricating the lenses in Goddard’s
Detector Development Lab. This project was supported with funding from the
Goddard Internal Research and Development (IRAD) program. CHIANTI is a
collaborative project involving George Mason University, the University of
Michigan (USA), and the University of Cambridge (UK).
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|
arxiv-papers
| 2012-05-21T22:31:32 |
2024-09-04T02:49:31.176657
|
{
"license": "Public Domain",
"authors": "B. R. Dennis, G. K. Skinner, M. J. Li, and A. Y. Shih",
"submitter": "Brian Dennis Brian R. Dennis",
"url": "https://arxiv.org/abs/1205.4762"
}
|
1205.4845
|
# Assessing the behavior of modern solar magnetographs and spectropolarimeters
J.C. del Toro Iniesta Instituto de Astrofísica de Andalucía (CSIC), Apdo. de
Correos 3004, E-18080 Granada, Spain, jti@iaa.es V. Martínez Pillet
Instituto de Astrofísica de Canarias, Vía Láctea, s/n, E-28200 La Laguna,
Spain, vmp@iac.es
###### Abstract
The design and later use of modern spectropolarimeters and magnetographs
require a number of tolerance specifications that allow the developers to
build the instrument and then the scientists to interpret the data accuracy.
Such specifications depend both on device-specific features and on the
physical assumptions underlying the particular measurement technique. Here we
discuss general properties of every magnetograph, as the detectability
thresholds for the vector magnetic field and the line-of-sight velocity, as
well as specific properties of a given type of instrument, namely that based
on a pair of nematic liquid crystal variable retarders and a Fabry-Pérot
etalon (or several) for carrying out the light polarization modulation and
spectral analysis, respectively. We derive formulae that give the detection
thresholds in terms of the signal-to-noise ratio of the observations and the
polarimetric efficiencies of the instrument. Relationships are also
established between inaccuracies in the solar physical quantities and
instabilities in the instrument parameters. Such relationships allow, for
example, to translate scientific requirements for the velocity or the magnetic
field into requirements for temperature or voltage stability. We also
demonstrate that this type of magnetograph can theoretically reach the optimum
polarimetric efficiencies of an ideal polarimeter, regardless of the optics in
between the modulator and the analyzer. Such optics induces changes in the
instrument parameters that are calculated too.
###### Subject headings:
Sun: magnetic fields – Sun: photosphere – Sun: polarimetry
††slugcomment: Submitted to ApJ
## 1\. Introduction
Spectropolarimetry and magnetography have become two of the most useful tools
in solar physics because they provide the deepest analysis one can make of
light. Solar information is encoded in the spectrum of the Stokes parameters.
We measure this spectrum and infer solar quantities from it. Recently, less
and less conceptual differences exist between spectropolarimeters and
magnetographs except for the specific devices: the formers usually include
those instruments using a scanning spectrograph; the latter usually employ a
bidimensional filtergraph like a Fabry-Pérot etalon. Some decades ago
magnetographs were only able to sample one or two wavelengths across a
spectral line; nowadays, new technologies provide a better wavelength
sampling, thus enabling the scientists to interpret the data in terms of
sophisticated inversion techniques of the radiative transfer equation, a
procedure similar to the one regularly used with spectropolarimeters. Some of
the instruments mentioned below enter this category. Modern solar
spectropolarimeters and magnetographs are often vectorial because all four
Stokes parameters of the light spectrum are measured. Longitudinal
magnetography (i.e., Stokes $I\pm V$) can be interesting for some specific
applications, but the partial analysis is usually included (if possible) as a
particular case of the more general, full-Stokes polarimetry. Some of these
modern instruments have been recently built or are currently in operation
(e.g., the Tenerife Infrared Polarimeter, TIP, Martínez Pillet et al. 1999,
Collados et al. 2007; the Diffraction-Limited Spectro-Polarimeter, DLSP,
Sankarasubramanian et al. 2003; the air-spaced Fabry-Perot based CRISP
instrument, Scharmer et al. 2008; the spectropolarimeter, SP, Lites et al.
2001, for the Hinode mission, Kosugi et al. 2007; CRISP, Narayan et al. 2008;
the Visible Imaging Polarimeter, VIP, Beck et al. 2010; the Imaging
Magnetograph eXperiment, IMaX, Martínez Pillet et al. 2011, for the Sunrise
mission, Barthol et al. 2011; and the Helioseismic and Magnetic Imager, HMI,
Graham et al. 2003, for the Solar Dynamics Observatory mission, Title 2000).
Some other are being designed and built for near future operation and missions
(e.g., the Polarimetric and Helioseismic Imager, SO/PHI, [formerly called VIM,
Martínez Pillet 2007] for the Solar Orbiter mission, Marsch et al. 2005).
The common interest of users of these instruments is centered in vector
magnetic fields (of components $B$, $\gamma$, and $\phi$) and line-of-sight
(LOS) velocities ($v_{\rm LOS}$). Some spectropolarimeters can provide
information on temperatures as well (and eventually on another thermodynamical
quantity) but that feature is not common to all of them. Therefore, assessing
the magnetograph capabilities in terms of their accuracy for retrieving
magnetographic and tachographic quantities is in order since such an analysis
can diagnose how far reaching is our current knowledge of the solar dynamics
and magnetism. The diagnostics is relevant both for the design of new
instruments in order to maximize their performances and for the analysis of
uncertainties in data coming from currently operating devices. General
considerations can obviously not be made but a few. Specifically, we here
study in Sect. 2 the detection thresholds induced by random noise on the
inferred longitudinal and transverse components of the magnetic field; in the
particular case of photon-induced noise we also find uncertainty formulas.
Both thresholds and relative uncertainties are obtained in terms of the
signal-to-noise ratio of the observations and of the polarimetric efficiencies
of the instrument. Since such efficiencies vary from instrument to instrument,
at that point, the analysis concentrates in a particular type of magnetograph,
namely that consisting of two nematic liquid crystal variable retarders
(LCVRs) as the polarization modulator and a Fabry-Pérot etalon. In Sect. 3, we
demonstrate that these polarimeters can reach the theoretically optimum
efficiencies no matter the optics behind the modulator, including the etalon.
The way for calculating the required retardances for the two LCVRs are
explained along with a number of rules and periodicities in the solutions.
Section 4 analyzes these instruments in terms of the influence of temperature
and voltage instabilities, as well as of thickness inhomogeneities
(roughness), of both the LCVRs and the etalon(s), on the final magnetographic
and tachographic measurements. Finally, Sect. 5 summarizes the results.
## 2\. The thresholding action of random noise
Most astrophysical measurements are nothing but photon counting. Their
accuracy, therefore, depends on photometric accuracy, that is, on a battle
between our ability to detect changes in the solar (stellar) physical
quantities and the noise that hide such changes. The key concept is changes:
we need to discern if a given quantity like the magnetic field strength, $B$,
or the line-of-sight (LOS) velocity, $v_{\rm LOS}$, varies among pixels:
whether or not it is greater or smaller than in the neighbor zones. The only
tool we have to gauge these changes is the observable Stokes parameter changes
that are linked to them through the response functions. Discussing response
functions is out of the scope of this paper as they have been extensively
analyzed elsewhere (e.g., Del Toro Iniesta et al., 2010; Orozco Suárez & Del
Toro Iniesta, 2007; Del Toro Iniesta, 2003; Del Toro Iniesta & Ruiz Cobo,
1996; Ruiz Cobo & Del Toro Iniesta, 1994; Landi Degl’Innocenti & Landi
Degl’Innocenti, 1977; Mein, 1971). As explained by Del Toro Iniesta & Martínez
Pillet (2010), purely phenomenological approaches (e.g., Cabrera Solana et
al., 2005) are also valid to establish a relationship between changes in the
physical quantities and the Stokes parameters. Here we shall concentrate on
noise; on how it establishes the minimum threshold below which no signal
changes can be detected. In spectropolarimetry, the customary estimate for
noise (and thus for the detection threshold) is the standard deviation of the
continuum signal because polarization is assumed to be constant at continuum
wavelengths. Therefore, the noise is calculated either over a continuum window
in a given spatial pixel or over all the spatial pixels of a map in a given
continuum wavelength sample. Both estimates should agree as it is the case in
most observations. If we call $\mathbf{S}\equiv(S_{1},S_{2},S_{3},S_{4})$ the
(pseudo-)vector of Stokes parameters and denote by $\sigma_{i}$, with
$i=1,2,3,4$, the standard deviation of each Stokes parameter, then the signal-
to-noise ratio in each parameter, $(S/N)_{i}$, is defined as the inverse of
this deviation in units of the continuum intensity:
$(S/N)_{i}=\left(\frac{S_{1}}{\sigma_{i}}\right)_{\\!\\!\\!\rm c},$ (1)
where index c refers to continuum. Thus, when we say that our observations
have, for example, a $S/N=1000$, we mean that when signals in a given (non
specified) Stokes parameter are greater than $10^{-3}S_{1,{\rm c}}$ can be
detected and this is certainly valid for that parameter but not necessarily
for the others. As a matter of fact, if noise is random (or uncorrelated with
the signal) and can be represented by a Gaussian distribution (Keller & Snik,
2009), according to Del Toro Iniesta & Collados (2000),
$\sigma_{i}=\frac{\varepsilon_{1}}{\varepsilon_{i}}\sigma_{1},\,\,\,i=1,2,3,4,$
(2)
where $\varepsilon_{i}$ stands for each one of the so-called polarimetric
efficiencies of the instrument. The polarimetric efficiencies depend in a non-
linear way on the modulation matrix elements (cf. Eq. 37) that on their turn
come from the first row of the polarimeter Mueller matrix (Del Toro Iniesta,
2003). Since all the efficiencies are necessarily less than the first (that of
the intensity), Eq. (2) means that the noise is always larger in the
polarization parameters than in the intensity. Then one can easily see that
(see Martínez Pillet et al., 1999)
$(S/N)_{i}=\frac{\varepsilon_{i}}{\varepsilon_{1}}(S/N)_{1},\,\,\,i=2,3,4,$
(3)
that is, that the signal-to-noise ratio for Stokes $S_{2}$, $S_{3}$, and
$S_{4}$ is always less than that for Stokes $S_{1}$. Let us point out here,
however, that other systematic (or instrumental) errors like those introduced
by flat fielding of images may invalidate the above equation. We are
explicitly discarding these other sources of noise from our analysis.
The Stokes parameters cannot be measured with single exposures. Instead, for
vector polarimetry, a number $N_{p}\geq 4$ of single detector shots are
recorded each providing a linear combination of all the four Stokes
parameters. The set of $N_{p}$ individual measurements constitutes a
modulation cycle that is characteristic of an instrument mode of operation.
After demodulation, that is, after solving the set of $N_{p}$ linear equations
made up of the individual exposures, the Stokes vector is measured. To
increase the signal-to-noise ratio of the measurement, many instruments use
$N_{a}$ accumulations, that is, repeat the modulation cycle $N_{a}$ times and
the corresponding polarization images are added together. Since the degree of
polarization of the incoming typical solar beam is fairly small, each single
shot usually has the same light levels and, hence, the same (photon-noise-
dominated) signal-to-noise ratio $s/n$. Thus, the signal-to-noise ratio of
$S_{1}$ is related to the single-shot $s/n$ (see Martínez Pillet, 2007, for an
illustrative description) through
$(S/N)_{1}=(s/n)\,{\varepsilon_{1}}\sqrt{N_{p}N_{a}},$ (4)
because $S_{1}$ is often retrieved from the sum of all the accumulated images
for all the polarization exposures of the given modulation scheme. An
advisable practice for characterizing the signal-to-noise ratio of an
instrument is to always refer to that of intensity and equate $S/N=(S/N)_{1}$.
This is convenient because there is only one intensity while the other
polarization Stokes parameters are three and one would need to specify which
one is meant each time. Let us remark, however, that this convention is not
universal and some authors, always interested in the polarization features,
think of and speak about some of the other three Stokes parameter signal-to-
noise ratios. Such an alternative convention makes sense if one takes into
account the differential character of polarization measurements: demodulation
implies that Stokes $S_{2,3,4}$ are essentially retrieved from image
differences; hence, any systematic error like that produced by flat fielding
is naturally mitigated (or eventually cancels out). On the contrary, the
additive character of Stokes $S_{1}$ implies that intensity noise can be
higher than simple photon noise. We shall hereafter follow the $S/N=(S/N)_{1}$
convention in the paper for the sake of simplicity in the description and in
the equations (thus, explicitly neglecting systematic errors). When people is
more interested in the other three Stokes parameter signal-to-noise ratios,
Eq. (3) provides the obvious help.
As demonstrated by Del Toro Iniesta & Collados (2000), the maximum
efficiencies that an ideal system can have are
${\mathbf{\varepsilon}}=(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, if all the
three last ones are equal. Therefore, the relationship $(S/N)_{i}\leq
1/\sqrt{3}\,(S/N)$, $i=2,3,4$, holds for any random-noise-dominated
polarimetric system.111There may be polarimeters that are designed to measure
not all four Stokes parameters or that aim at better accuracies for given
Stokes parameters. In such cases some (never all) efficiencies can be greater
than $1/\sqrt{3}$. In what follows, however, we assume that our interest is
the same for $S_{2}$, $S_{3}$, and $S_{4}$. Or, in terms of Eq. (2),
necessarily,
$\sigma_{i}\geq\sqrt{3}\,\sigma_{1}.$ (5)
Equation (5) means that the detection threshold is bigger for the polarization
parameters than for the intensity. Detectability is smaller in polarimetry
than in pure photometry.
An important parameter describing the state of any beam of light is its degree
of polarization
$p^{2}\equiv\frac{1}{S_{1}^{2}}\sum_{i=2}^{4}S_{i}^{2}.$ (6)
Now the question naturally arises as to what is the minimum detectable degree
of polarization by a given polarimetric system. If the uncertainties in the
Stokes parameters are uncorrelated (and this should be especially true when
considering a statistics on all the pixels of an image), error propagation in
Eq. (6) gives
$\sigma_{p^{2}}^{2}=\sum_{i=1}^{4}\left(\frac{\partial p^{2}}{\partial
s_{i}}\right)^{2}\sigma_{s_{i}}^{2},$ (7)
where, for convenience we have made $s_{i}\equiv S_{i}^{2}$, $i=1,2,3,4$. Now,
since $\sigma_{s_{i}}=2S_{i}\sigma_{i}$, it is easy to see that
$\sigma_{p^{2}}^{2}=\frac{4}{S_{1}^{4}}\left(p^{4}S_{1}^{2}\sigma_{1}^{2}+\sum_{i=2}^{4}S_{i}^{2}\sigma_{i}^{2}\right)$
(8)
and, according to Eq. (2) and since $\sigma_{p^{2}}=2p\,\sigma_{p}$, one can
write
$\frac{\sigma_{p}^{2}}{p^{2}}=\left(1+\frac{1}{p^{4}}\sum_{i=2}^{4}\frac{S_{i}^{2}}{S_{1}^{2}}\frac{\varepsilon_{1}^{2}}{\varepsilon_{i}^{2}}\right)\frac{\sigma_{1}^{2}}{S_{1}^{2}}.$
(9)
Now, if all the three last efficiencies are the same and certainly less than
their maxima, we finally obtain
$\frac{\sigma_{p}}{p}\geq\frac{\sqrt{1+\frac{3}{p^{2}}}}{S/N},$ (10)
an inequality already published by Del Toro Iniesta & Orozco Suárez (2010),
Martínez Pillet et al. (2011), and Del Toro Iniesta & Martínez Pillet (2010)
without demonstration.
Our instruments are aimed at measuring magnetic fields and velocities.
Therefore, any reasonable design should include lower limits for these
quantities within the overall error budget. Detectability thresholds for the
Stokes parameters imply thresholds for the magnetograph and tachograph signals
as well. The rest of this section is devoted to estimate them. Of course, any
estimation that one can make depends not only on the instrument but also on
the inference technique. Most modern magnetographs use inversion of the
radiative transfer equation to infer values for both the magnetic field vector
and the plasma velocity. These inferences involve all four Stokes parameters
and, hence, should be more accurate than those using just one or two of them.
However, for the sake of clarity in the analytical derivation, we shall
consider errors induced in the magnetographic and tachographic formulas (11),
(12), and (17).
Using classical magnetographic formulas, the longitudinal and transverse
components of the magnetic field are given by
$B_{\rm lon}=k_{\rm lon}\frac{V_{s}}{S_{1,\rm c}}$ (11)
and
$B_{\rm tran}=k_{\rm tran}\sqrt{\frac{L_{s}}{S_{1,\rm c}}},$ (12)
where $k_{\rm lon}$ and $k_{\rm tran}$ are (model-dependent) calibration
coefficients and $V_{s}$ and $L_{s}$ are the circular and linear polarization
signals calculated as
$V_{s}\equiv\frac{1}{n_{\lambda}}\sum_{i=1}^{n_{\lambda}}a_{i}\,S_{4,i},$ (13)
where $a_{i}=1$ or $-1$ depending on whether the sample is to the blue
(including the zero shift) or the red side of the central wavelength of the
line, and
$L_{s}\equiv\frac{1}{n_{\lambda}}\sum_{i=1}^{n_{\lambda}}\sqrt{S_{2,i}^{2}+S_{3,i}^{2}}.$
(14)
In the above equations, $n_{\lambda}$ stands for the number of wavelength
samples. If we now assume that the minimum polarization signals are
$V_{s}=\sigma_{4}$ and $L_{s}=\sigma_{2}=\sigma_{3}$, the minimum detectable
thresholds are
$\delta(B_{\rm lon})=\frac{k_{\rm
lon}}{S/N}\,\frac{\varepsilon_{1}}{\varepsilon_{4}}$ (15)
and
$\delta(B_{\rm tran})=k_{\rm
tran}\sqrt{\frac{\varepsilon_{1}/\varepsilon_{2}}{S/N}}=k_{\rm
tran}\sqrt{\frac{\varepsilon_{1}/\varepsilon_{3}}{S/N}}.$ (16)
Expressions (15) and (16) give the explicit dependence of magnetic
detectability thresholds in terms of the instrument efficiencies and are very
useful in practice. For example, if we use the calibration constants for IMaX
quoted in Martínez Pillet et al. (2011), assuming that the maximum
polarimetric efficiencies have been reached, and typical signal-to-noise
ratios of 1700 ($\sim 1000$ for $S_{2},S_{3}$, and $S_{4}$), the minimum
longitudinal and transverse components of the magnetic field detectable with
that magnetograph are 5 and 80 G, respectively.
As far as the velocity is concerned, we shall assume that the Fourier
tachometer technique (Beckers & Brown, 1978; Brown, 1981; Fernandes, 1992) is
used:
$v_{\rm LOS}=\frac{2{\rm
c}\,\delta\lambda}{\pi\lambda_{0}}\,\arctan\frac{S_{1,-9}+S_{1,-3}-S_{1,+3}-S_{1,+9}}{S_{1,-9}-S_{1,-3}-S_{1,+3}+S_{1,+9}},$
(17)
where c is the speed of light, $\delta\lambda$ is the spectral resolution of
the instrument, and $\lambda_{0}$ is the central wavelength of the line;
$-9,-3,+3,+9$ stand for the sample wavelengths of the intensity, measured in
picometers with respect to $\lambda_{0}$. Let us assume that the minimum
detectable difference between symmetric wavelength samples (such as
$S_{1,-3}-S_{1,+3}$) due to LOS velocity shifts is $\sigma_{1}$. Then, if the
difference between the samples at the same flank of the line is approximated
by $~{}1/2\,\,S_{1,\rm c}$, the minimum detectable LOS velocity can be
approximated by
$\delta(v_{\rm LOS})\simeq\frac{2{\rm
c}\,\delta\lambda}{\pi\lambda_{0}}\,\arctan\frac{2}{S/N}.$ (18)
Likewise Eqs. (10), (15) and (16), this new expression (18) relates the
velocity threshold with the signal-to-noise ratio of the instrument. If we use
again IMaX values ($\delta\lambda=8.5$ pm; $\lambda_{0}=525.02$ nm) and assume
$S/N=1700$, the minimum detectable LOS velocity change is roughly 4 m s-1.
### 2.1. Uncertainties induced by photon noise
Fluctuations in the light levels due to photon statistics necessarily imply
variances in the Stokes parameters that in the end induce uncertainties in the
measured physical quantities, $B_{\rm lon}$, $B_{\rm tran}$, and $v_{\rm
LOS}$. In this section, we are going to establish a relationship between those
variances and uncertainties. Note that we discard for the moment any random
fluctuation in the instrument that will be dealt with in Sect. 4.
Error propagation in Eq. (11) easily yields
$\frac{\sigma^{2}_{B_{\rm lon}}}{B_{\rm
lon}^{2}}=\frac{\sigma_{4}^{2}}{n_{\lambda}V_{s}^{2}}+\left(\frac{1}{S/N}\right)^{2},$
(19)
because $\sigma^{2}_{V_{s}}=\sigma^{2}_{4}/n_{\lambda}$. Using now Eqs. (2),
(11), and (1), one obtains
$\frac{\sigma^{2}_{B_{\rm lon}}}{B_{\rm lon}^{2}}=\left(\frac{k_{\rm
lon}^{2}}{n_{\lambda}B_{\rm
lon}^{2}}\frac{\varepsilon_{1}^{2}}{\varepsilon_{4}^{2}}+1\right)\frac{1}{(S/N)^{2}},$
(20)
that relates the $B_{\rm lon}$ relative error with itself, the signal-to-noise
ratio of the observations, and the polarimetric efficiencies. On its turn,
error propagation in Eq. (12) gives
$\frac{\sigma^{2}_{B_{\rm tran}}}{B_{\rm
tran}^{2}}=\frac{1}{4}\left[\frac{\sigma_{L_{s}}^{2}}{L_{s}^{2}}+\left(\frac{1}{S/N}\right)^{2}\right].$
(21)
Now, since the variances for Stokes $S_{2}$ and Stokes $S_{3}$ should be
approximately the same,
$\sigma_{L_{s}}^{2}\simeq(1/2n_{\lambda})(\sigma_{2}^{2}+\sigma_{3}^{2})$ and,
using Eqs. (2), (12), and (1), Eq. (21) turns out to be
$\frac{\sigma^{2}_{B_{\rm tran}}}{B_{\rm tran}^{2}}=\left[\frac{k_{\rm
tran}^{4}}{8n_{\lambda}B_{\rm
tran}^{4}}\left(\frac{\varepsilon_{1}^{2}}{\varepsilon_{2}^{2}}+\frac{\varepsilon_{1}^{2}}{\varepsilon_{3}^{2}}\right)+\frac{1}{4}\right]\frac{1}{(S/N)^{2}},$
(22)
that again relates the relevant magnetographic quantity relative error with
itself, the photon-induced signal-to-noise ratio of the observations, and the
polarimetric efficiencies of the instrument.
If we use the values for the calibration constants quoted in Martínez Pillet
et al. (2011) for IMaX, $n_{\lambda}=5$ for this instrument, and assume that
the maximum polarimetric efficiencies are reached, then the estimated relative
errors for $B_{\rm lon}$ and $B_{\rm tran}$ induced by a photon noise of
$S/N=1700$ are of 2 and 15%, respectively, for magnitudes in either quantities
of 100 G; for magnitudes of 1000 G, the relative errors drop to 0.2 and 0.1%,
respectively.
After a similar calculation for photon-noise-induced uncertainties in the
tachographic formula (17), one gets
$\sigma_{v_{\rm
LOS}}^{2}=\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{2\sigma_{1}^{2}}{\Delta},$
(23)
where $\Delta=(S_{1,-9}-S_{1,+3})^{2}+(S_{1,+9}-S_{1,-3})^{2}$, and the
variances of the Stokes $S_{1}$ samples are all assumed to be $\sigma_{1}$.
Note that the slight asymmetry between Eq. (23) and Eqs. (20) and (22) is not
such as the ratio $\sigma_{1}^{2}/\Delta$ is a kind of inverse, square signal-
to-noise ratio. A numerical estimate for the IMaX instrument, and using the
FTS spectrum by Brault & Neckel (1987) to evaluate $\Delta$ for its Fe i line
at 525.02 nm, we conclude that the photon-noise-induced uncertainty is 4 m
s-1.
## 3\. An optimum vector plus longitudinal polarimeter
As explained by Martínez Pillet et al. (2004), a versatile polarimeter is
obtained through the combination of two nematic liquid crystal variable
retarders (LCVRs) with their optical axes properly oriented at 0∘and 45∘with
the Stokes $S_{2}$ positive ($X$) direction. This is so because it can provide
optimum modulation schemes for both the vectorial and the longitudinal
$(S_{1}\pm S_{4})$ polarization analyses by simply tuning the voltages that
change their retardances. The theoretical maximum efficiencies mentioned above
can in principle be reached by such an ideal polarimeter. We have assumed
these maximum efficiencies for our instruments so far. However, instrumental
effects may corrupt the measurement so that the final efficiencies are lower.
Let us see in this section what happens if some typical optical elements are
included between the modulator and the analyzer in the analysis.
The corrupting effect of the optical elements of an instrument in the final
polarization analysis is called instrumental polarization. It is well known
that those optical components acting on light after the polarization
modulation do not produce any instrumental polarization. However, nobody has
yet demonstrated whether optimum polarimetric efficiencies can still be
reached no matter the optics in between the modulator and the analyzer. In
this section we are going to show that this is the case with these two-LCVR-
based polarimeters because retardances can be fine tuned by simply changing
the acting voltages. This property certainly makes this type of polarimeters
very versatile and optimum for solar investigations. To understand the result,
let us start by demonstrating that, indeed, a polarimeter made up of two
nematic LCVRs oriented as above plus a linear analyzer can reach the optimum
polarimetric efficiencies.
According to Del Toro Iniesta (2003), the modulation matrix of any
polarimetric system consists of rows that equal the first row of the system
Mueller matrix for each of the measurements. If ${\bf R}(\theta,\delta)$
stands for the Mueller matrix of a general retarder whose fast axis is at an
angle $\theta$ with the $X$ axis and whose retardance is $\delta$, our LCVR
Mueller matrices can be described by ${\bf M}_{1}={\bf R}(0,\rho_{i})$ and
${\bf M}_{2}={\bf R}(\pi/4,\tau_{i})$, where $i=1,2,3,4$ is an index for each
of the four measurements. Hence, in our case, where the analyzer (of Mueller
matrix ${\bf M}_{4}$) is a linear polarizer at 0∘,222Dual-beam polarimeters
use a polarizing beam splitter as a double analyzer. Hence, another analyzer
at 90∘is indeed present simultaneously although the double calculation is not
necessary. such a modulation matrix, disregarding a 1/2 gain factor, is given
by (see Martínez Pillet et al., 2004)
${\bf
O}=\left(\begin{array}[]{llll}1&\cos\tau_{1}&\sin\rho_{1}\sin\tau_{1}&-\cos\rho_{1}\sin\tau_{1}\\\
1&\cos\tau_{2}&\sin\rho_{2}\sin\tau_{2}&-\cos\rho_{2}\sin\tau_{2}\\\
1&\cos\tau_{3}&\sin\rho_{3}\sin\tau_{3}&-\cos\rho_{3}\sin\tau_{3}\\\
1&\cos\tau_{4}&\sin\rho_{4}\sin\tau_{4}&-\cos\rho_{4}\sin\tau_{4}\end{array}\right).$
(24)
As explained in Del Toro Iniesta & Collados (2000), if all the three last
column elements of ${\bf O}$ have a magnitude of $1/\sqrt{3}$ (with their
signs properly altered), then the modulation is optimum and the maximum
efficiencies are reached. $|\cos\tau|=1/\sqrt{3}$ has the four independent
solutions
$\tau=54\hbox{$.\\!\\!^{\circ}$}736,125\hbox{$.\\!\\!^{\circ}$}264,234\hbox{$.\\!\\!^{\circ}$}736,$
and $305\hbox{$.\\!\\!^{\circ}$}264$. With them,
$|\sin\rho\,\sin\tau|=1/\sqrt{3}$ is equivalent to $|\sin\rho|=\sqrt{2}/2$
that has four independent solutions as well:
$\rho=45^{\circ}\/,135^{\circ}\/,225^{\circ}\/$, and $315^{\circ}\/$. The
verification of the above two equations ensures the automatic verification of
that for the third column and, therefore, we have found that several
combination of matrix elements exist that qualify $\bf O$ as the modulation
matrix of an optimum polarimetric scheme, as we aimed at demonstrating.
Real polarimeters, however, have some optics in between the modulator and the
analyzer. Very importantly, modern magnetographs like CRISP, VIP, IMaX, or
SO/PHI have one or several Fabry-Pérot etalons. Such etalons can modify the
Mueller matrix that leads to a modulation matrix like that in Eq. (24) and,
hence, we must check whether or not the resulting modulation matrix, ${\bf
O}^{\prime}$, remains optimum. To do that, let us model the most general
behavior of an etalon as a retarder ${\bf M}_{3}={\bf R}(\theta_{\rm
etalon},\delta_{\rm etalon})$. Then, the final Mueller matrix of the system is
now ${\bf F}={\bf M}_{4}{\bf M}_{3}{\bf M}_{2}{\bf M}_{1}$ and its first row
(again disregarding the gain factor) is given by $F_{11}=1$,
$\begin{array}[]{lll}F_{12}&=&M_{3,22}\cos\tau_{i}+M_{3,24}\sin\tau_{i},\\\
F_{13}&=&M_{3,22}\sin\rho_{i}\,\sin\tau_{i}+M_{3,23}\cos\rho_{i}-M_{3,24}\sin\rho_{i}\,\cos\tau_{i},\\\
F_{14}&=&-M_{3,22}\cos\rho_{i}\,\sin\tau_{i}+M_{3,23}\sin\rho_{i}+M_{3,24}\cos\rho_{i}\,\cos\tau_{i}.\end{array}$
(25)
We do not need any more matrix elements of ${\bf F}$ because the rows of the
new modulation matrix are $O^{\prime}_{ij}=F_{1j}(\tau_{i},\rho_{i})$. Now, we
only need to find out four different combinations of the first and second
retardances that are solutions for Eqs. (25) with $|F_{1k}|=1/\sqrt{3}$, where
$k=2,3,4$. Equations (25) are transcendental and, thus, have to be solved
numerically. However, before proceeding with the numerical exercise we can
realize several features in the solutions. First, the trivial cases, where
$\delta_{\rm etalon}=0$ (that is, no etalon exists or it is not birefringent)
or $\theta_{\rm etalon}=0,\pi/2$, the orthogonal directions of the analyzer
axis, are indeed trivial because the effect of ${\bf M}_{3}$ disappears and
$O^{\prime}_{ij}=O_{ij}$. Second, a number of periodicities can be deduced
from the equations structure:
* •
If $\tau_{0}$ is a solution for the first of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\tau_{0}+(2k+1)\pi$, with $k$ integer, are
solutions for that equation when $F_{12}=-1/\sqrt{3}$ and vice versa.
* •
If $\rho_{0}$ is a solution for the second or the third of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\rho_{0}+(2k+1)\pi$, with $k$ integer, are
solutions for that equation when $F_{12}=-1/\sqrt{3}$ and vice versa.
* •
If $\rho_{0}$ is a solution for the second of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\rho_{0}+(2k+1)\pi/2$, with $k$ even integer, are
solutions for the third equation when $F_{12}=-1/\sqrt{3}$. When $k$ is odd,
then the solution for the third equation is when $F_{12}=1/\sqrt{3}$ as well.
To solve the first of Eqs. (25) let us consider the function
$f(\tau)=F_{12}-M_{3,22}\cos\tau-M_{3,24}\sin\tau,$ (26)
that has extrema where its derivative becomes zero. This occurs at $\tau_{0}$
where either $M_{3,24}=\sin\tau_{0}$ and $M_{3,22}=\cos\tau_{0}$ or
$M_{3,24}=-\sin\tau_{0}$ and $M_{3,22}=-\cos\tau_{0}$. These values imply that
$f(\tau_{\rm max})=F_{12}+1$ and $f(\tau_{\rm min})=F_{12}-1$. That is, the
maximum of the function is positive and the minimum is negative when
$|F_{12}|=1/\sqrt{3}$ (which is required for reaching optimum
efficiencies).333Note that we have selected one out of the infinite solutions
for the derivative of $f$ to be zero, but this is coherent with our neglecting
multiplicative, gain factors in the definition of Mueller matrices. Therefore,
since $f$ is continuous, Bolzano’s theorem ensures that a solution exists in
$(\tau_{\rm min},\tau_{\rm max})$ and this enables us to find that solution,
for instance, through the bisector method. This has to be done just once per
value of $F_{12}$; the other value derives from the first of the above
specified properties.
As a summary, the first of Eqs. (25) has four solutions in $[0,2\pi]$, each
two belonging to one of the signs of $F_{12}$. For each of these four
retardances, $\tau_{i}$, four possibilities are open according to the values
of $F_{13}$ and $F_{14}$. These four solutions for $\rho_{i}$ in the second
and third of Eqs. (25) can be shown to be enclosed in the following single
expression,
$\cos\rho=\pm\frac{(n\mp m)}{\sqrt{3}\,(m^{2}+n^{2})},$ (27)
where $n=M_{3,23}$ and $m=M_{3,22}\sin\tau-M_{3,24}\cos\tau$. A further
property of the solutions thus derives from Eq. (27): if $\rho_{0}$ is a
solution for the second and the third of Eqs. (25), then $2\pi-\rho_{0}$,
$\pi-\rho_{0}$, and $\pi+\rho_{0}$ are solutions as well.
Therefore, the presence of an etalon modeled as a retarder is not a problem
for the two-LCVR-based polarimeter to be optimum. No matter the possible
retardance or orientation the etalon may have, we are always able to find out
more than four combinations of $\rho$ and $\tau$ that ensure theoretical
polarimetric efficiencies for all three Stokes parameters all equal to
$1/\sqrt{3}$. In practice, these new solutions can be achieved by simply
tuning the acting voltages of the two LCVRs.
Figure 1.— Retardances of the first (bottom) and second (top) LCVRs as
functions of the angle $\theta$ of the etalon orientation with respect to the
$S_{2}$ positive direction. Only values between $0$ and $2\pi$ are displayed.
Different line types refer to different values of the etalon retardance (see
text).
Figure 1 displays the LCVR retardances in $[0,2\pi]$ that guarantee optimum
performance as functions of the orientation angle of the etalon. Values for
the second LCVR are in the top panel and those for the first one are in the
bottom panel.444Note that retardances larger than $2\pi$ may be needed for
design convenience. Their values are easily deducible from the above-mentioned
properties. Different colors correspond to different solutions; different line
types correspond to different values of the assumed etalon retardance:
0∘(dotted), 30∘(solid), 45∘(dashed), and 60∘(dashed-dotted). As commented on
before, when $\theta_{\rm etalon}=90^{\circ}\/$, both $\rho$ and $\tau$
recover the same value as if the etalon were absent. Moreover, both
retardances are periodic with $\theta$, with periods $\pi/4$ ($\rho$) and
$\pi/2$ ($\tau$). It is also interesting to note that four out of the eight
solutions for $\rho$ are equal to the other four but phase shifted by $\pi/4$.
Now, it is a little tedious but easy to demonstrate that, regardless of how
many, mirrors can be introduced in the optical path between the modulator and
the analyzer (as in real instruments) without affecting the maximum
polarimetric efficiencies, provided they all are perpendicular to the optical
axis plane. From the Mueller matrix of a single mirror, one can realize
(Collett, 1992) that the matrix of such a mirror train, no matter the angles
between them, keeps always the shape
${\bf E}=\left(\begin{array}[]{cccc}a&b&0&0\\\ b&a&0&0\\\ 0&0&c&d\\\
0&0&-d&e\end{array}\right).$ (28)
The elements of the first row in the new final Mueller matrix of the system
become $F^{\prime}_{1j}=(a+b)F_{1j}$, $j=1,2,3,4$. That is, the final
modulation matrix remains the same as before introducing the mirrors but
scaled by a gain factor that can be disregarded as we have been doing for all
the treatment. Therefore, we can conclude that optimum efficiencies can still
be achieved with as many mirrors as needed. Since a mirror is indeed a
combination of a retarder and a partial polarizer (the $d$ element is zero for
the latter; see e.g. Stenflo, 1994), the same conclusion can be reached for
whatever differential absorption effects for the orthogonal polarization
states that may be located between the modulator and the analyzer. Therefore,
if, for example, the etalon or the diffraction grating of the instrument
display different transmittances for orthogonally polarized beams the
polarimetric efficiencies can still attain maximum values.
## 4\. Instrument-induced inaccuracies
Now that we know that our magnetograph can reach optimum polarimetric
performance, let us study the behavior of this particular instrument against
instabilities in its main optical elements. Photon noise is not the only harm
for magnetographic or tachographic measurements. Instabilities of different
types like those in the temperature or in the tuning voltage of both the LCVRs
and the etalon, or roughness in their final thicknesses, can induce
inaccuracies. For single measurements, the inaccuracies can imply errors in
magnetic field or absolute wavelength calibration. For time series like those
needed in helioseismological studies, such inaccuracies may avoid detection of
some particular oscillatory modes. An assessment on such inaccuracies is
therefore in order for clearly defining design tolerances of these
instrumental quantities. Such tolerances should ensure the fulfillment of the
scientific requirements of the instrument.
### 4.1. Polarimetric inaccuracies
An important example of a scientific requirement is the polarimetric accuracy
of the system. By such we understand any of the inverse signal-to-noise ratios
for $S_{2}$, $S_{3}$, or $S_{4}$ as defined in Eq. (1). Following our general
assumption that these $(S/N)_{i}$ are intended to be the same by design,
aiming at a $S/N$ of 1700 is equivalent to require the system to have a
polarimetric accuracy of $10^{-3}$. Temperature, voltage, and other
instabilities and defects of the LCVRs lead to changes in the retardances
that, on their turn, induce modulation and demodulation changes. Such changes
can be seen as cross-talk between the Stokes parameters that drive to
covariances in the magnetographic measurements (Asensio Ramos & Collados 2008;
see also an interesting discussion on seeing-induced cross-talk in Casini et
al. 2011). Since our modulation matrix (24) is analytical, we attempt an
analytical approach to the study of the effect of these retardance changes
onto the polarimetric accuracy of the system.
Let ${\bf D}$ be the demodulation matrix of the instrument that always exists.
Thus, ${\bf OD}={\bf DO}={1\mskip-7.0mu1}$. The measured Stokes vector is then
given by ${\bf S}={\bf D}{\bf I}_{\rm meas}$, where ${\bf I}_{\rm meas}$
stands for the four intensity measurement vector of each modulation cycle. If
we linearly perturb matrix ${\bf D}$ as a consequence of a small perturbation
in the LCVR retardances ($\sigma_{\rho_{k}},\sigma_{\tau_{k}},\,k=1,2,3,4$),
then the Stokes vector becomes ${\bf S}+{\bf S}^{\prime}=({\bf D}+{\bf
D}^{\prime)}{\bf I}_{\rm meas}$, where the perturbed demodulation matrix ${\bf
D}^{\prime}$ is given by
${\bf D}^{\prime}=\sum_{k=1}^{4}\left(\frac{\partial{\bf
D}}{\partial\rho_{k}}\sigma_{\rho_{k}}+\frac{\partial{\bf
D}}{\partial\tau_{k}}\sigma_{\tau_{k}}\right).$ (29)
It is obvious that the polarimetric accuracy requirement directly implies that
none of the four elements of ${\bf S}^{\prime}={\bf D}^{\prime}{\bf I}_{\rm
meas}$ can be greater than $10^{-3}$. To fulfill that requirement, let us then
study how the perturbation in the retardances can be produced and which are
the tolerances for the instrument quantities whose fluctuations produce them.
If we call $\delta_{L}$ anyone of the retardances, we have by definition that
$\delta_{L}=\frac{\beta t}{\lambda_{0}},$ (30)
where $\beta=n_{\rm e}-n_{\rm o}$ is the birefringence, i.e., the difference
between the extraordinary and the ordinary refractive indices of the liquid
crystal, and $t$ stands for its geometrical thickness. Therefore, it is
evident that
$\frac{\sigma^{2}_{\delta_{L}}}{\delta^{2}_{L}}=\frac{\sigma^{2}_{\beta}}{\beta^{2}}+\frac{\sigma^{2}_{t}}{t^{2}},$
(31)
that is, the relative inaccuracy in the LCVR retardance is the square root of
the sum of the square relative inaccuracies in the birefringence and in the
geometrical thickness at the operating wavelength. Let us ascribe for
convenience any possible local fabrication defect in the LC like an air bubble
to the thickness inaccuracy, so that birefringence can be considered spatially
constant for the whole device.
Figure 2.— Retardance (in degree) of a specific nematic LCVR as a function of
the acting voltage (in volts; solid line) and its derivative with inverted
sign (dashed line).
Variations in the birefringence can be produced by either variations in the
LCVR temperature, the acting voltage, or both. Hence, one can write
$\sigma^{2}_{\beta}=q_{T}^{2}\sigma_{T}^{2}+q_{V}^{2}\sigma_{V}^{2},$ (32)
where $q_{T}$ and $q_{V}$ are values of the (partial) derivatives of $\beta$
with respect to $T$ and $V$ at the given values of voltage and temperature,
respectively. Since we indeed have calibrations of the $\delta_{L}$
dependences rather than those of $\beta$, we better rewrite Eq. (31) as
$\sigma^{2}_{\delta_{L}}=m_{T}^{2}\,\sigma^{2}_{T}+m_{V}^{2}\,\sigma^{2}_{V}+\frac{\delta_{L}^{2}}{t^{2}}\,\sigma^{2}_{t},$
(33)
where $m_{T}^{2}$ and $m_{V}^{2}$ have a clear meaning and can be deduced from
calibrations. According to Martínez Pillet et al. (2011), based on data by
Heredero et al. (2007),
$m_{T}=-1.16+0.305V-0.02V^{2},$ (34)
for $V<8$ volt and $0$ otherwise, and $m_{V}$ can be obtained from data like
those displayed in Fig. 2 where $\delta_{L}$ (solid line) and its derivative
(dashed line; inverted sign) are plotted as functions of the acting voltage
for a particular LCVR.
To get a numerical estimation of the value of real tolerances, that is, of the
maximum $\sigma_{T}$, $\sigma_{V}$, and $\sigma_{t}$ affordable in real
instruments in order not to have the ${\bf S}^{\prime}$-elements greater than
$10^{-3}$, we shall use IMaX parameters555The combination for IMaX retardances
was $\rho=[315,315,225,225]$ and $\tau=[305.264,54.736,125.264,234.736]$, in
degrees. Their corresponding voltages were $V(\rho)=[2.535,2.535,3.112,3.112]$
and $V(\tau)=[2.4,9.0,4.3,2.9]$, in volts. plus the analytic expressions for
${\bf D}$ and its partial derivatives.666An IDL program that evaluates such
analytic expressions is available upon request. Direct (by hand) evaluation is
so tedious that only with the help of software applications like Mathematica
such expressions can be obtained. The last term in Eq. (33) may vary spatially
and is, thus, responsible for the pixel-to-pixel variations of the retardance
but can easily be calibrated if needed. Indeed, roughness in the device
thickness is a fabrication specification and can be checked upon delivery from
the manufacturer. Thickness inaccuracies may produce locally significant
effects (e.g., Alvarez-Herrero et al., 2010). Our own estimation, using Eqs.
(29, 33) indicate that relative errors in the thickness larger than 6 % induce
perturbations of the Stokes vector that are larger than the polarimetric
accuracy. Therefore, should this be the only instrumental instability,
specific pixel-to-pixel calibration of ${\bf D}$ would be needed if the
relative roughness is larger than 6 %. Note that, since the typical
thicknesses of LCVRs are of the order of micrometers, a roughness less than 6%
may mean a stringent requirement for the manufacturer of the order of tens of
nanometers.
Using again Eqs. (29, 33), we find out that instabilities larger than
$\sigma_{T}=600$ mK or $\sigma_{V}=1.5$ mV deteriorate the polarimetric
accuracy below the required $10^{-3}$. These two tolerances and that for the
roughness have been calculated after assuming that each instability is acting
individually. According to Eq. (33), the final uncertainty in the retardance
stems from the three sources simultaneously. Hence, a safety reduction factor
of $\sqrt{3}$ (assuming all the three contribute the same) is advisable.
Therefore, in the end, the final tolerance specification for our instrument to
reach a polarimetric accuracy of $10^{-3}$ is 300 mK for temperature, 1 mV for
voltage, and a 4 % for LCVR roughness.
### 4.2. Magnetographic inaccuracies
The retardance perturbations of Eq. (33) induce changes in the maximum
polarimetric efficiencies of the instrument. Such changes necessarily imply
modifications in the magnetographic measurements. The modifications might
jeopardize the quality of the results. Imagine, for instance, that a
requirement on the repeatability of $B_{\rm lon}$ and $B_{\rm tran}$ applies
because we are interested on a measurement time series: a calculation of
tolerances in the instrument parameters ($T$, $V$, roughness, etc.) that
ensure the fulfillment of the magnetographic repeatability is in order.
Since no explicit dependence of $B_{\rm lon}$ and $B_{\rm tran}$ on
$\varepsilon_{i}$ exists, we cannot analytically gauge the induced
inaccuracies in any magnetographic measurement. Nevertheless, we can obtain a
hint on the global by studying the specific variations of $\delta(B_{\rm
lon})$ and $\delta(B_{\rm tran})$, the minimum detectable values of such
magnetograph quantities.
Error propagation in Eqs. (15) and (16) readily gives
$\sigma^{2}_{{\delta(B_{\rm lon})}}=\frac{{\delta^{2}(B_{\rm
lon})}}{4\varepsilon_{4}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{4}}$ (35)
and
$\sigma^{2}_{{\delta(B_{\rm tran})}}=\frac{{\delta^{2}(B_{\rm
tran})}}{16\varepsilon_{2}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{2}}=\frac{{\delta^{2}(B_{\rm
tran})}}{16\varepsilon_{3}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{3}}.$ (36)
According to Del Toro Iniesta & Collados (2000), the maximum polarimetric
efficiencies that can be reached by any system are
$\varepsilon^{2}_{{\rm max},i}=\frac{\sum_{j=1}^{4}O^{2}_{ji}}{N_{p}},$ (37)
where $O_{ji}$ are the matrix elements of ${\bf O}^{\rm T}$, the transpose of
$\bf O$. For a system with a modulation matrix like that in Eq. (24), it is
easy to see that the efficiency inaccuracies ensuing the thermal instabilities
are
$\sigma^{2}_{\varepsilon^{2}_{{\rm max},1}}=0,$ (38)
as a consequence of using normalized Mueller matrices, and
$\sigma^{2}_{\varepsilon^{2}_{{\rm
max},2}}=\frac{1}{16}\sum_{j=1}^{4}\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}},$
(39) $\displaystyle\sigma^{2}_{\varepsilon^{2}_{{\rm max},3}}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left\\{\sum_{j=1}^{4}\sin^{2}2\rho_{j}\,\,\sin^{4}\tau_{j}\,\,\sigma^{2}_{\rho_{j}}\right.+$
(40)
$\displaystyle\left.\sum_{j=1}^{4}\sin^{4}\rho_{j}\,\,\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}}\right\\},$
$\displaystyle\sigma^{2}_{\varepsilon^{2}_{{\rm max},4}}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left\\{\sum_{j=1}^{4}\sin^{2}2\rho_{j}\,\,\sin^{4}\tau_{j}\,\,\sigma^{2}_{\rho_{j}}\right.+$
(41)
$\displaystyle\left.\sum_{j=1}^{4}\cos^{4}\rho_{j}\,\,\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}}\right\\},$
where $\sigma^{2}_{\tau_{j}}$ and $\sigma^{2}_{\rho_{j}}$ are the variances of
the LCVR retardances and where we have neglected the influence of possible
time-exposure differences or instabilities like those caused by a rolling-
shutter detector or a non-ideal repeatability of a mechanical shutter. Such
influences can be estimated separately for the specific instrument and
directly scale the effective exposure time (either per pixel or per frame).
Using the values for IMaX, and assuming maximum efficiencies, instabilities of
0.3 K or of 1.1 mV produce a 5 % repeatability error in the threshold for
$B_{\rm lon}$. A comparison of Eqs. (35) and (36) readily tells that the
effect is 2 times smaller on the relative repeatability error in the threshold
for $B_{\rm tran}$ but since the threshold itself is 16 times larger, the
absolute error is in the end 8 times larger as well.
### 4.3. Velocity inaccuracies
Error propagation in Eq. (17) yields
$\sigma_{v_{\rm LOS}}^{2}=\left(\frac{\partial v_{\rm
LOS}}{\partial\delta\lambda}\right)^{2}\sigma_{\delta\lambda}^{2}+\left(\frac{\partial
v_{\rm
LOS}}{\partial\lambda_{0}}\right)^{2}\sigma_{\lambda_{0}}^{2}+\sum_{k}\left(\frac{\partial
v_{\rm LOS}}{\partial S_{1,k}}\right)^{2}\sigma_{{1,k}}^{2}$ (42)
with $k=-9,-3,+3,+9$. Uncertainties in the spectral resolution come from
uncertainties in the etalon spacing and related fabrication details that
produce etalon roughness. Errors in the central wavelength come from the
etalon tuning that mostly depends on the ambient temperature, $T$, and on the
tuning voltage, $V$:
$\sigma_{\lambda_{0}}^{2}=k_{T}^{2}\sigma_{T}^{2}+k_{V}^{2}\sigma_{V}^{2},$
(43)
where $k_{T}$ and $k_{V}$ are constants that give the (linear) dependence of
$\lambda_{0}$ on $T$ and $V$. Finally, uncertainties in the Stokes $S_{1}$
samples come both from pure photon noise, as in any photometric measurement,
and from etalon tuning uncertainties. Since the (inexplicit) dependence of
$S_{1,k}$ on $\lambda_{0}$ is non linear, let us linearize it (that is,
introduce a small perturbation and take the first approximation) and write
$\sigma_{1,k}^{2}=\sigma_{1}^{2}+s_{1,k}^{2}\sigma_{\lambda_{0}}^{2},$ (44)
where we have assumed that the photometric contribution is equal for all the
samples and indeed equal to the photon noise as calculated in the continuum;
$s_{1,k}$ stand for the derivatives of the Stokes $S_{1}$ profile at the
corresponding wavelengths.
After a tedious but straightforward algebra, Eq. (42) can be recast as
$\sigma_{v_{\rm LOS}}^{2}=\frac{v_{\rm
LOS}^{2}}{(\delta\lambda)^{2}}\sigma_{\delta\lambda}^{2}+\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{2\sigma_{1}^{2}}{\Delta}+$
$\left[\frac{v_{\rm
LOS}^{2}}{\lambda_{0}^{2}}+\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{d_{1}+d_{2}}{\Delta^{2}}\right](k_{T}^{2}\sigma_{T}^{2}+k_{V}^{2}\sigma_{V}^{2}),$
(45)
where $\Delta$ is defined in Sect. 2.1,
$d_{1}=(S_{1,+9}-S_{1,-3})^{2}(s_{1,-9}^{2}+s_{1,+3}^{2})$ and
$d_{2}=(S_{1,-9}-S_{1,+3})^{2}(s_{1,+9}^{2}+s_{1,-3}^{2})$. Hence, clear
contributions to the final LOS velocity uncertainty can be discerned from the
etalon roughness, the photon noise, the etalon temperature instability, and
the etalon voltage instability.
Quantitative estimates of the various terms in Eq. (45) can be made by using
the FTS spectrum by Brault & Neckel (1987) to evaluate $\Delta$, $d_{1}$, and
$d_{2}$ for a given spectral line and a given instrument. Let assume the HMI
and SO/PHI Fe i line at $\lambda_{0}=617.3$ nm and a spectral resolution of
the etalon of $\delta\lambda=10$ pm. The first term has a clear impact on the
tachographic results: the etalon relative roughness is directly translated
into the same $v_{\rm LOS}$ relative uncertainty. In other words, we cannot
expect better accuracy in the line-of-sight velocity (when measured with the
Fourier tachometer formula) than that limited by the etalon relative
roughness; this means that a mere 0.1 pm, rms resolution uncertainty induces
10 m s-1 errors for speeds of 1 km s-1. The second term coincides with the
right-hand side of Eq. (23) and has been discussed already in Sect. 2.1. Since
the ratio between the second and the first terms within brackets is of the
order of 4$\cdot$109 for velocities of up to 5 km s-1, it is the second one
what really matters in the estimation; this means that the dependence of the
profile shape on the central wavelength of the line is really important. If we
use the same IMaX values of $k_{T}=2.52$ pm/K and $k_{V}=3.35\cdot$10-2 pm/V
for the SO/PHI etalon, Eq. (45) gives instabilities of the order of 3.3 mK or
0.25 V induce the same LOS velocity uncertainty of 4 km s-1 quoted above for
pure photon noise. Another way of seeing the same effect can be explained by
saying that a 100 m s-1 uncertainty is produced by either a 45 mK or a 3.4 V
instabilities. These uncertainties are really important when stability during
given periods of time of the instrument is required as for helioseismic
measurements. For single shots, uncertainties in temperature or voltage imply
thresholds for accurate absolute wavelength (velocity) calibration. The
importance of having included the measurement technique in this error budget
analysis is clear: should one have simply used the $k_{V}$ and $k_{T}$
calibration constants above and the Doppler formula an uncertainty of just 55
m s-1 would have been obtained. Hence, the uncertainty would have been
underestimated by a factor almost 2.
## 5\. Summary and conclusions
An assessment study on the salient features and properties of solar
magnetographs has been presented. An error budget procedure has been followed.
Special care has been devoted in including photon-induced and instrument-
induced noise as well as specific measurement technique contributions to the
final variances. We have first discussed the effect of random noise in the
measurements and deduced useful formulae –general for every device– that
provide some minimum detectable parameters like the degree of polarization of
light, the longitudinal and transverse components of the magnetic field, and
the line-of-sight velocity. The detection thresholds are given as functions of
the polarimetric efficiencies of the instrument and of the signal-to-noise
ratio of the observations. (As a proposal, we have suggested as well to use
the $S/N$ for the Stokes intensity as the signal-to-noise ratio for the
instrument.) When the random noise is photon-induced, we have calculated as
well the relative uncertainty in the magnetographic and tachographic
quantities. Secondly, an analysis is presented for those instruments based on
two nematic liquid crystal variable retarders as a polarization modulator and
a Fabry-Pérot etalon as the spectrum analyzer. Although specific for these
magnetographs, the methodology can easily be followed by others in order to
characterize their capabilities and accuracies. We have demonstrated that this
type of instrument can indeed reach theoretical maximum polarimetric
efficiencies because solutions always exist for the retardances of the two
LCVRs that ensure such efficiencies, hence optimizing the detection thresholds
and the relative uncertainties. Very remarkably, the existence of such
solutions is independent of the optics that is in between the polarization
modulator and the analyzer. Neither retarders nor partial polarizers or
mirrors (the most commonly used devices) alter that property. The LCVR optimum
retardances do depend in such pass-through optics but can be fine-tuned
according to the polarizing properties of the optics. A number of rules and
periodicity properties of the required retardances have also been deduced.
These polarimeters have modulation and demodulation matrices that are
explicitly calculated through an IDL procedure that is available upon request.
Thirdly, error propagation has yielded equations relating the variances of the
measured Stokes vector and the solar physical quantities and instrument
parameters, hence providing a bridge between scientific requirements and
instrument design specifications. The analytic character of this particular
type of instrument has also allowed the quantitative estimation of the
mentioned uncertainties. Hopefully, the discussion presented in this paper
excites (and helps) further diagnostics of other instruments.
This work has been partially funded by the Spanish Ministerio de Ciencia e
Innovación, through Projects No. AYA2009-14105-C06 and AYA2011-29833-C06, and
Junta de Andalucía, through Project P07-TEP-2687, including a percentage from
European FEDER funds.
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|
arxiv-papers
| 2012-05-22T09:00:13 |
2024-09-04T02:49:31.186959
|
{
"license": "Public Domain",
"authors": "J. C. del Toro Iniesta and V. Mart\\'inez Pillet",
"submitter": "Jose Carlos del Toro Iniesta",
"url": "https://arxiv.org/abs/1205.4845"
}
|
1205.4883
|
11institutetext: Computer Science and Engineering Department, Sichuan
University Jinjiang College, 620860 Pengshan, China
greenhat1016@gmail.com,mr.l172586418@gmail.com,cys19900611@gmail.com
# Hybrid Parallel Bidirectional Sieve based on SMP Cluster
Gang Liao Lian Luo Lei Liu
###### Abstract
In this article, hybrid parallel bidirectional sieve is implemented by SMP
Cluster, the individual computational units joined together by the
communication network, are usually shared-memory systems with one or more
multicore processor. To high-efficiency optimization, we propose average
divide data into nodes, generating double-ended queues (deque) for sieve that
are able to exploit dual-cores simultaneously start sifting out primes from
the head and tail.And each node create a FIFO queue as dynamic data buffer to
ache temporary data from another nodes send to. The approach obtains huge
speedup and efficiency on SMP Cluster. ybrid parallel, HPC, SMP Cluster, sieve
###### Keywords:
h
## 1 Introduction
Research into questions involving primes continues today, partly driven by the
importance of primes in modern cryptography. As our computational power
increases, researcher often pays more attention to Data analysis, Climate
modeling, Protein folding, Drug discovery etc. We can also exploit multicores
to efficiency solve some problem in the field of number theory.
M.Aigner and G.M.Ziegler [1] presented six quite different proofs of the
infinitude of primes. Mills[2] has shown that there is a constant $\Theta$
such that the function $f(n)=[{\Theta^{3}}^{n}]$ generates only primes. The
sieve of Eratosthenes-Legendre [3] [4] is an ancient algorithm for finding all
prime numbers up to any given limit. In number theory, tests distinguishing
between primes and composite integers will be crucial. The most basic
primality test is trial division, which tells us that integer $n$ is prime if
and only if it is not divisible by any prime not exceeding $\sqrt{n}$.
The computational complexity of algorithms for determining whether an integer
is prime is measured in terms of the number of binary digits in the integer.
The algorithm using trial divisions to determine whether an integer $n$ is
prime is exponential in terms of the number of binary digits of $n$, or in
terms of $\log_{2}n$ ,because $\sqrt{n}={2}^{log_{2}{n/2}}$.
As n gets large, an algorithm with exponential complexity quickly becomes
impractical. Leonard Adleman, Carl Pomerance, and Robert Rumely [5] [6]
developed an algorithm that can prove an integer is prime using $(\log
n)^{clogloglogn}$ nit operations, where c is a constant. In 2002, M. Agrawal,
N. Kayal, and N. Saxena [7], announced that they had found an algorithm PRIMES
is in P that can produce a certificate of primality for an integer n using
$O((logn)^{12})$ bit operations.
Karl Friedrich Gauss conjectured that $\pi(x)$ increases at the same rate as
the functions $\frac{x}{logx}$ and $Li(x)=\int_{2}^{x}\frac{dt}{logt}$. And
the Prime Number Theorem that the ratio of $\pi(x)$ to $\frac{x}{logx}$
approaches 1 as $x$ grows without bound. One way [11] to evaluate $\pi(x)$
only $O({x}^{\frac{3}{5}+e})$ bit operations without finding all the primes
less than $x$ is to use a counting argument based on the sieve of
Eratosthenes.
In this paper, Hybrid parallel bidirectional sieve based on SMP Cluster is
proposed to improve efficient and speedup. The result is proved to be
effective by MPI and OpenMP [8] [9] [10]. With Hybrid parallel, it has far-
reaching significance in cryptography.
## 2 Communication and Optimization
ILP and TLP provide parallelism at a very low level, they are typically
controlled by the processor and the operating system, and isn’t directly
controlled by the programmer. Parallel hardware is often classified using
Flynn’s taxonomy, which distinguished between the number of instruction
streams and the number of data streams a system can handle. A von Neumann
system is classified as SISD. Vector processors and graphics processing units
(GPU) are often classified as SIMD. MIMD execute multiple independent
instruction streams, each of which can have its own data stream. Shared-memory
or distributed-memory is typically MIMD. And most of the lager MIMD systems
are hybrid systems (Fig.1) in which a number of relatively small share-memory
are connected by an interconnection network. In such systems, the individual
shared-memory systems are sometimes called nodes.
Figure 1: SMP Cluster Architecture.
### 2.1 Interconnection networks
Currently the two most widely used interconnects on shared-memory systems are
buses and crossbars [15]. The key characteristic of a bus is that the
communication wires are shared by the devices that are connected to it. Buses
have the virtue of low cost and flexibility. Crossbars (Fig.2) allow
simultaneous communication among different devices, so they are much faster
than uses. But the cost of the switches and links is relatively high.
Distributed-memory interconnects are often divided into two groups: direct
interconnects and indirect interconnects. One measure of ”number of
simultaneous communications” or ”connectivity” is bisection width. To
understand this measure, imagine that the parallel system is divided into two
halves, and each half contains half of the processors or nodes. An alternate
way of computing the bisection width is to remove the minimum number of links
needed to split the set of nodes into two equal halves.
Figure 2: Shared-memory system simultaneous memory access
The hypercube (Fig.3) is a highly connected direct interconnect that has been
used in actual system. A hypercube of dimension d has $p={2}^{d}$ nodes, and a
switch in a d-dimensional hypercube is directly connected to a processor and d
switches. The bisection width of a hypercube is $\frac{p}{2}$.The switches
support $1+d=1+\log_{2}p$ wires. The hypercube is more powerful and expensive
to construct.
Figure 3: (a) two-dimensional hypercube (b) three-dimensional hypercube (c)
four-dimensional hypercube
The crossbar and the omega network are relatively simple examples of indirect
networks. The omega network (Fig.4) is less expensive than crossbar. The omega
network uses $\frac{1}{2}plog_{2}(p)$ of the 2 x 2 crossbar switches, so it
uses a total of ${2}plog_{2}(p)$ switches, while the crossbar users $p^{2}$.
### 2.2 Hybrid Parallelism
We define the speedup of a parallel program to be
$S=\frac{T_{serial}}{T_{parallel}}$ . Then linear speedup has $S=Pcores$, this
value, $\frac{S}{P}$, is sometimes called the efficiency of the parallel
program as follows:
$E=\frac{S}{P}=\frac{\frac{T_{serial}}{T_{parallel}}}{P}$ (1)
Back in the 1960s, Gene Amdahl [13] that’s become as Amdahl’s Law:
$S_{overall}=\frac{1}{(1-f)+\frac{f}{s}}$ (2)
It means that unless virtually all of a serial program is parallelized, the
possible speedup is going to be very limited-regardless of the number of cores
available. A more mathematical version of this statement is known as
Gustafson’s Law [14].
Unfortunately, there are several mismatch problem between the (hybrid)
programming schemes and the hybrid hardware architecture. Often, one can see
in publications, that applications may or may not benefit from hybrid
programming depending on some application parameters, e.g., in [16][17][18]
[19].
Polf Rabenseifner analyses strategies to overcome typical drawbacks of this
easily usable programming scheme on systems with weaker inter-connects [20].
Best performance can be achieved with overlapping communication and
computation, but this scheme is lacking in ease of use. Often, hybrid MPI $+$
OpenMP programming denotes a programming style with OpenMP shared memory
parallelization inside the MPI processes (i.e., each MPI process itself has
several OpenMP threads) and communication with MPI between the MPI processes,
but only outside of parallel regions.
This hybrid programming scheme will be named materonly in the following
classification, which is based on the question, when and by which thread(s)
the messages are sent between the MPI processes:
* .
Pure MPI
* .
Hybrid MPI $+$ OpenMP
* .
Overlapping communication and computation
* .
Pure OpenMP
Overlapping of communication and computation is a chance for an optimal usage
of the application itself, in the OpenMP parallelization and in the load
balancing. It requires a coarse-grained and thread-rank-based OpenMP
parallelization, the separation of halo-based computation from the computation
that can be overlapped with communication, and the threads with different
tasks must be load balanced. Advantages of the overlapping scheme are:
* .
the problem that one CPU may not achieve the inter-node bandwidth is no longer
relevant as long as there is enough computational work that can be overlapped
with the communication
* .
the saturation problem is solved as long as not more CPUs communicate in
parallel than necessary to achieve the inter-node bandwidth
* .
the sleeping threads problem is solved as long as all computation and
communication is load balanced among the threads.
Figure 4: (a) Crossbar (b) omega network
## 3 Bidirectional Sieve Model
Foster’s methodology [12] provides an outline of steps include
* .
Partitioning.
* .
Communication.
* .
Agglomeration or aggregation
* .
Mapping for parallel programming
### 3.1 Algorithm Design
The sieve of Eratosthenes does so by iteratively marking as composite the
multiples of each prime, starting with the multiples of 2 [4]. We can exploit
and improve the sieve of Eratosthenes based on SMP Cluster (Fig. 5). Assume
that there are some disorder integers which the scale of $n$, and when each
node sieve the integers in the block that the scale of $k$, it could achieve
high-efficiency optimization. We conjectured that the SMP Cluster requires at
least N nodes.The formula as follows:
$N=\frac{n}{k}+(n\bmod k)\And{1}$ (3)
And each node generate one deque and do with dual-cores. One core is located
in the head of the deque. On the contrary, the other one is located in the
tail of the deque. It’s easy to deduction the formula about the amount of
cores($C_{cores}$) and deques($D_{deques}$):
$C_{cores}=D_{deques}={2}N$ (4)
There is another point that’s worth considering. In most cases, the scale of
node $N$ is not exactly equal $k$. We can deal with the state as follows
Alg.1:
Algorithm 1 the scale of node $N^{th}$
0: $K$ denote that the currency scale of node $N^{th}$
0: $k$ denote that the general scale of node
if ${0}\leq K\leq\frac{k}{2}$ then
Node N assign single core to right or left sieve
else
Node N assign dual-cores to simultaneous bidirectional sieve
end if
Figure 5: Construct Bidirectional Sieve
And its flow diagram is shown in Fig.6.
Figure 6: High-level flow diagram of hybrid parallel bidirectional Sieve
### 3.2 Primality Testing : Non-deterministic
Primality testing of a number is perhaps the most common problem concerning
number theory.The problem of detecting whether a given number is a prime
number has been studied extensively but nonetheless,it turns out that all the
deterministic algorithms for this problem are too slow to be used in real life
situations and the better ones amongst them are tedious to code.But,there are
some probabilistic methods which are very fast and very easy to
code.Moreover,the probability of getting a wrong result with these algorithms
is so slow that it can be neglected in normal situations.
All the algorithms which we are going to discuss will require you to
efficiently compute $(a^{b})\bmod c$ (where a,b,c are non-negative integers).
A straightforward algorithm to do the task can be to iteratively multiply the
result with $a$ and take the remainder with $c$ at each step,this algorithm
takes $O(b)$ time and is not very useful in practice. We can do it
$O(\log{b})$ by using what is called as exponentiation by squaring as follows:
$f(n)=$ $\begin{cases}(a^{2})^{\frac{b}{2}},&\mbox{if }b\mbox{ is even and b
$>$ 0}\\\ a(a^{2})^{\frac{b-1}{2}},&\mbox{if }b\mbox{ is odd}\\\ $1$,&\mbox{if
}b\mbox{ $=0$}\end{cases}\\\ $
Algorithm 2 modulo(a,b,c) : Exponentiating by squaring to $(a^{b})\bmod c$
0: $x=1,y=a$
0: $(a^{b})\bmod c$
while $b>0$ do
if $b\And 1$ then
$x=(x*y)\bmod c$
end if
$y=(y*y)\bmod c$
$b>>=1$
end while
return $x\bmod c$
Pierre de Fermat first stated the Fermat’s Little Theorem in a letter dated
October 18, 1640, to his friend and confidant Fr$\acute{e}$nicle de Bessy as
the following [7]:
$a^{p}=a\pmod{p}$ (5)
or alternatively:
$a^{p-1}=1\pmod{p}$ (6)
According to Fermat’s Little Theorem[7], if $p$ is a prime number and a is
positive integer less than $p$ ($a<p$),and then calculate $a^{p-1}\bmod p$. If
the result is not 1, then by Fermat’s Little Theorem p cannot be prime.The
more iterations we do, the higher is the probability that our result is
correct.
Algorithm 3 Fermat(p,iterations) : Fermat′s primality test
if $p=1$ then
return $false$
end if
for $i:=1$ to $iterations$ do
$a=rand()\bmod(p-1)+1$
if modulo(a,p-1,p)!=1 (Alg.2) then
return $false$
end if
end for
return $true$
Though Fermat is highly accurate in practice there are certain composite
numbers $p$ known as Carmichael numbers for which all values of $a<p$ for
which $gcd(a,p)=1$,$(a^{p-1})\bmod p=1$.And in that case,the Fermat’s test
will return wrong result with very high probability.Out of the Carmichael
numbers less than ${10}^{16}$,about $95\%$ of them are divisible by primes
$<1000$.However,there are other improved primality tests which don’t have this
flaw as Fermat’s(e.g.Rabin-Miller test[21][22],Solovay-Strassen test [23]).
## 4 Performance Analysis
Figure 7: statistics and analysis hybrid parallel bidirectional sieve with
general method
Different programming schemes on clusters of SMPs show different performance
benefits or penalties in this paper. Fig.7 summarizes the result of hybrid
parallel bidirectional sieve .It’s obvious that nodes communication would
waste most of time when data scale is tiny.Even its slower than general
method.However, if there are hyper-data scale,hybrid parallel show huge
efficiency and optimization.Indeed,sometimes the waste of communication could
be neglected.In that case,multicores parallelism is effective approach to
solve some problem in number theory.
To achieve an optimal usage of the hardware,one can also try to use the idling
CPU’s for other applications,especially low-priority single-threaded or multi-
threaded non-MPI application if the parallel high-priority hybrid application
does not use the total memory of the SMP nodes.
## 5 Conclusion
In this study we haven shown that hybrid parallel on SMP cluster is an
applicable method to implement bidirectional sieve . The analysis demonstrated
that even hybrid parallel bidirectional sieve is efficiency and optimization
solution.
As our computational power increases,Most HPC system are clusters of shared
memory nodes.Parallel programming must combine the distributed memory
parallelization on the node inter-connect with shared memory parallelization
inside of each node.And Each parallel programming schema on hybrid
architecture has one or more significant drawbacks(e.g. sleeping-thread and
saturation problem). However,Hybrid parallel also has far-reaching
significance in many fields(e.g.Cryptography,Data analysis, Climate modeling,
Protein folding, Drug discovery).
We believe that hybrid parallel bidirectional sieve can be properly modeled
using techniques form number theory and this article is just an early trial of
using hybrid parallelism to improve speedup and efficiency.
## References
* [1] M. Aigner , G. M. Ziegler: Proofs from THE BOOK,3rd ed.,Springer-Verlag,Berlin,(2003)
* [2] W.H. Mills: A prime-representing function, Bulletin of the American Mathematical Society, Volume 53,604,(1947)
* [3] H.Halberstam , H.-E.Richert: Sieve Methods, Academic Press, London,(1974)
* [4] Horsley, Rev. Samuel, F. R. S.: The Sieve of Eratosthenes. Being an Account of His Method of Finding All the Prime Numbers, Philosophical Transactions (1683 C1775), Vol. 62., pp. 327 C347,(1772)
* [5] L.M. Adleman, C. Pomerance, R.S. Rumly, On distinguishing prime numbers from composite numbers, Annals of Mathematics, Volume 117 (1983).
* [6] R. Rumely,:Recent advances in primality testing, Notices of American Mathmatical Society, Volume 30 , 475-477,(1983)
* [7] M.A Agrawal, N. Kayal, N. Saxena : PRIMES is in P, Department of Computer Science & Engineering, Indian Institute of Technology, Kanpur, India,(2002)
* [8] R. Chandra, et al.: Parallel Programming in OpenMP, Morgan Kaufmann, San Francisco,(2001)
* [9] P. Pacheco: Parallel Programming with MPI, Morgan Kaufmann, San Francisco,(1997)
* [10] M.Quinn: Parallel Programming in C with MPI and OpenMP, McGraw-Hill Higher Education, Boston,(2004).
* [11] J.C. Lagarias , A.M. Odlyzko: New algorithm for computing PI(x), Bell Laboratories Technical Memorandum TM-82-11218-57.
* [12] I. Foster: Designing and Building Parallel Programs, Addison-Wesley, Reading, MA, 1995. Also available from http://www.mcs.anl.gov/ itf/dbpp/ (accessed 21.09.10)
* [13] G.M. Amdahl: Validity of the single processor approach to achieving large scale computing capabilities, in: Proceedings of the American Federation of Information Processing Societies Conference, vol. 30, issue 2, Atlantic City, NJ, 1967, pp. 483-485
* [14] J.L. Gustafson: Reevaluating Amdahl’s law, Commun. ACM 31 (5) (1988) 532-533
* [15] Peter S. Pacheco: An introduction to PARALLEL PROGRAMMING, Elsevier (Singapore) Pte Ltd,(2011)
* [16] Georg Hager, Frank Deserno, Gerhaed Wellein: Pseudo-Vectorization and RISC Optimization Techniques for the Hitachi SR8000 Architecture, in High Performance Computing in Science and Engineering in Munich ’02, Springer-Verlag Berlin Heidelberg, (2003)
* [17] D. S. Henty: Performance of hybrid message-passingand shared-memory parallelism for discrete element modeling, in Proc. Supercomputing’00, Dallas, TX, (2000).
* [18] Richard D. Loft, Stephen J. Thomas, John M. Dennis: Terascale spectral element dynamical core for atmospheric general circulation models, in proceedings, SC 2001, NOW. 2001, Nov. 2001, Denver, USA. www.sc2001.org/papers/pap.pap189.pdf
* [19] Gerhard Wellein, GeorgHager, Achim Basermann, Holger Fehske: Fast sparse matrix-vector multiplication for TeraFlop/s computers, in proceedings of VECPAR’2002, 5th Int’l Conference on High Performance Computing and Computational Science, Porto, Portugal, June 26-28, 2002, part I, pp 57-70. http://vecpar.fe.up.pt/
* [20] Rolf Rabenseifner: Hybrid Parallel Programming: Performance Problems and Chances, in proceeding of the 45th CUG Conference 2003, Columbus, Ohio, USA, May 12-16,2003, www.cug.org
* [21] Miller, Gary L.:Riemann’s Hypothesis and Tests for Primality, Journal of Computer and System Sciences 13 (3): 300 C317, doi:10.1145/800116.803773,(1976)
* [22] Rabin, Michael O.:Probabilistic algorithm for testing primality, Journal of Number Theory 12 (1): 128 C138, doi:10.1016/0022-314X(80)90084-0,(1980)
* [23] Solovay, Robert M.Strassen, Volker.:A fast Monte-Carlo test for primality”. SIAM Journal on Computing 6 (1): 84 C85. doi:10.1137/0206006,(1977)
|
arxiv-papers
| 2012-05-22T11:27:10 |
2024-09-04T02:49:31.197732
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gang Liao, Lian Luo, Lei Liu",
"submitter": "Gang Liao",
"url": "https://arxiv.org/abs/1205.4883"
}
|
1205.4896
|
# Vector exchanges in production of light meson pairs and elementary atoms.
S. R. Gevorkyan, E. A. Kuraev, M. K. Volkov Joint Institute for Nuclear
Research, 141980, Dubna, Russia
###### Abstract
The production of pseudoscalar and scalar mesons pairs and bound states
(positronium or pionium atoms) in high energy $\gamma\gamma$ collisions at
high energies provided by photon or vector meson exchanges are considered. The
vector exchanges lead to nondecreasing with energy cross section of binary
process $\gamma+\gamma\to h_{a}+h_{b}$ with $h_{a},h_{b}$ states in the
fragmentation regions of initial particles. The production of light mesons
pairs $\pi\pi,\eta\eta,\eta^{\prime}\eta^{\prime},\sigma\sigma$ as well as a
pairs of positronium $Ps$ and pionium $A_{\pi}$ atoms in peripheral kinematics
are discussed. Unlike the photon exchange the vector meson exchange needs a
reggeization, leading to fall with energy. Nevertheless due to peripheral
kinematics out of very forward production angles the vector meson exchanges
dominated.
The proposed approach allows to express the matrix elements of the considered
processes through impact factors, which can be calculated in perturbation
models like Chiral Perturbation Theory (ChPT) or Nambu-Jona-Lasinio (NJL)
model or determined from $\gamma\gamma$ sub-processes or vector mesons
radiative decay widths.
We obtain the cross sections for pionium atom production in collisions of high
energy pions and electrons with protons. The possibility to measure these
processes in experiment are discussed.
## 1 Introduction
The next large project after LHC should be likely a linear $e^{+}e^{-}$
accelerator at energy $\sqrt{s}=0.5-1TeV$, giving exciting challenge to study
$\gamma\gamma$ interactions at energies of hundreds GeV. The technology of
obtaining the beams of high energy photons is based on the backward Compton
scattering of laser light on high energy electrons [1], idea known for many
years [2, 3].
Exclusive processes with hadronic final states test various model calculations
and hadron production mechanisms. So far the meson pairs production in two
photon collisions are measured [4, 5, 6] at $\gamma\gamma$ center of mass
energy $W\leq 4$ GeV and scattering angle $|cos{\theta}|<0.8$.
In this work we investigate the production of light mesons pairs and
elementary atoms (positronium Ps and pionium $A_{\pi}$ atoms) in high energy
$\gamma\gamma$ collisions in peripheral kinematics:
$\displaystyle\gamma(k_{1})+\gamma(k_{2})\to
h_{a}(p_{1})+h_{b}(p_{2});~{}~{}~{}h_{a},h_{b}=\pi,\eta,\eta\prime,\sigma,Ps,A_{\pi}$
(1)
Due to peripheral kinematics (
$s=(k_{1}+k_{2})^{2},t=q^{2}=(p_{1}-k_{1})^{2};~{}~{}~{}s>>|q^{2}|$ ) the
created objects $h_{a},h_{b}$ have energies approximately equal to the
energies of colliding photons and move along the directions of initial
particles motion (center of mass of initial particles implied).
The dominant contribution to peripheral processes comes from large orbital
momenta in scattering amplitude expansion. The background from low orbital
momentum in peripheral kinematics is strongly suppressed unlike the processes
allowing production at any angles. A typical example is the Born-term
amplitude ( $\pi$ exchange in the t channel) of the process
$\gamma\gamma\to\pi^{+}\pi^{-}$ whose differential cross section at small
angles has additional suppression due to wide phase volume of the final state.
The another remarkable property of the relevant cross sections-they become
independent from center of mass energy s of colliding particles starting from
some threshold energy $\sqrt{s}\sim 2-3GeV$. The nondecreasing feature of
pairs yield is a result of vector nature of the interaction (photon or vector
meson exchanges in the t-channel (Fig.1)).
In peripheral kinematics one can use the perturbation models of hadrons like
Chiral Perturbation Theory [7, 8] (ChPT) or Nambu-Jona-Lasinio [9] (NJL) model
to describe the sub-processes at the relevant vertexes. One can expresses the
matrix element of reaction (1) through so called impact factors, which are
nothing else than the matrix elements of sub-processes (Fig.1a):
$\gamma(k_{1})+\gamma^{*}(q)\to h_{a}$ and $\gamma(k_{2})+\gamma^{*}(q)\to
h_{b}$ or (Fig. 1b): $\gamma(k_{1})+V(q)\to h_{a}$ and $\gamma(k_{2})+V(q)\to
h_{b}$ .
Figure 1: (a) The photon exchange in the process $\gamma+\gamma\to
h_{a}+h_{b}$ ; (b) Exchange by vector meson.
Let us briefly discuss the connection of matrix element of reaction (1) with
relevant impact factors $M^{a},M^{b}$ (the details can be found in [10]).
According to the general rules the matrix element of the process (1) reads :
$\displaystyle
M=\frac{J^{a}_{\rho}J^{b}_{\sigma}}{q^{2}-m_{V}^{2}}g^{\rho\sigma},$ (2)
$J^{a,b}$ are currents associated with blocks $a,b$ of relevant Feynman
diagram.
Making use the infinite momentum frame parametrization of the transferred
momentum:
$\displaystyle q=\alpha k_{1}+\beta
k_{2}+q_{\bot};~{}~{}~{}q_{\bot}k_{1}=q_{\bot}k_{2}=0;~{}q_{\bot}^{2}=-\vec{q}^{2}<0;~{}k_{1}^{2}=k_{2}^{2}=0.$
(3)
and written metric tensor in the Gribov’s form:
$\displaystyle
g^{\rho\sigma}=g_{\bot}^{\rho\sigma}+\frac{2}{s}(k_{1}^{\rho}k_{2}^{\sigma}+k_{2}^{\rho}k_{1}^{\sigma}).$
(4)
one obtains the connection of matrix element of process (1) with impact
factors (with power accuracy):
$\displaystyle
M=\frac{2s}{q^{2}-m_{V}^{2}}M^{a}M^{b};~{}~{}~{}M^{a}=\frac{J^{a}_{\mu}k_{2}^{\mu}}{s};~{}~{}~{}M^{b}=\frac{J^{b}_{\nu}k_{1}^{\nu}}{s}.$
(5)
The impact factors $M^{a},M^{b}$ don’t decrease with energy and can be
described in terms of perturbation strong interaction models like Nambu-Jona-
Lasinio model or Chiral Perturbation Theory. The cross section of the
processes (1) is connected with matrix element (5) in the standard way:
$\displaystyle d\sigma^{ab\to h_{a}h_{b}}=\frac{1}{8s}\sum{|M|^{2}}d\Gamma$
(6)
Expressing the phase volume of the two final particles $d\Gamma$ through the
Sudakov parameters (3) one can rewritten the two particles phase space volume:
$\displaystyle
d\Gamma=(2\pi)^{4}\delta(k_{1}+k_{2}-p_{1}-p_{2})\frac{d^{3}p_{1}}{2(2\pi)^{3}E_{1}}\frac{d^{3}p_{2}}{2(2\pi)^{3}E_{2}}$
(7)
in the following form [10]:
$\displaystyle d\Gamma=\frac{d^{2}q}{2(2\pi)^{2}s}$ (8)
As a result the differential cross section of the processes (1) reads:
$\displaystyle d\sigma^{ab\to
h_{a}h_{b}}=\frac{d^{2}q}{(4\pi)^{2}(\vec{q}^{2}+m_{V}^{2})^{2}}\sum_{spins}|M^{a}|^{2}\sum_{spins}|M^{b}|^{2}$
(9)
Thus the knowledge of relevant impact factors allows one to calculate the
cross sections of processes (1).
## 2 Mesons production. Photon exchange.
We start with the production of $\pi^{0}\pi^{0}$ pair in $\gamma\gamma$
collisions with photon exchange in the t-channel (Fig. 1a). The current
algebra gives for the matrix element of neutral pion decay to two photons
$\pi^{0}(p)\to\gamma(k_{1},e_{1})+\gamma(k_{2},e_{2})$:
$\displaystyle M(\pi^{0}\to\gamma\gamma)=\frac{\alpha}{\pi
f_{\pi}}(k_{1}e_{1}k_{2}e_{2}),$ (10)
where
$(abcd)=a^{\alpha}b^{\beta}c^{\gamma}d^{\delta}\epsilon_{\alpha\beta\gamma\delta}$
and $k_{i},e_{i}(k_{i})$ are the momenta and polarization vectors of real
photons, $\alpha=\frac{e^{2}}{4\pi}=1/137$ is the fine structure constant and
$f_{\pi}=92.2MeV$ is the pion decay constant measured in the
$\pi^{+}\to\mu^{+}\nu_{\mu}$ decay rate.
The pion radiative decay width is given by the textbook formula:
$\displaystyle\Gamma(\pi^{0}\to
2\gamma)=(\frac{m_{\pi}}{4\pi})^{3}(\frac{\alpha}{f_{\pi}})^{2}=7.76eV$ (11)
The decay amplitude (10) can be used as impact factor in $\pi^{0}\pi^{0}$
production.
More elaborated impact factors considering the photon virtuality can be
obtained if one calculates the triangle fermion loop with the light u and d
quarks as a fermions [9]. Quarks charges and number of colors result in a
factor $3((2/3)^{2}-(1/3)^{2})=1$. After standard procedure of denominators
joining, calculating the relevant trace in the fermions spin indices and
integration over the loop momenta we obtain:
$\displaystyle M(\pi^{0}\to\gamma\gamma^{*})=\frac{\alpha}{2\pi
f_{\pi}}|\left[\vec{e},\vec{q}\right]|F_{\pi}(z),~{}~{}~{}z=\frac{\vec{q}^{2}}{m_{q}^{2}}$
$\displaystyle
F_{\pi}(z)=N_{\pi}\int\limits_{0}^{1}dx\int\limits_{0}^{1}\frac{ydy}{1-\rho_{\pi}^{2}y^{2}x(1-x)+zy(1-y)x},~{}~{}~{}\rho_{\pi}=\frac{m_{\pi}}{m_{q}}.$
(12)
Here $m_{q}$ is the constituent quark mass which we put
$m_{q}=m_{u}=m_{d}\approx 280MeV$, whereas $N_{\pi}$ is the normalization
constant $F_{\pi}(0)=1$.
The similar expression for the sub-process of the scalar meson decay
$\sigma\to\gamma\gamma^{\ast}$ reads:
$\displaystyle M(\sigma\to\gamma\gamma^{*})=\frac{5\alpha}{6\pi
f_{\sigma}}|\left(\vec{e},\vec{q}\right)|F_{\sigma}(z);~{}~{}~{}F_{\sigma}(0)=1,~{}~{}~{}f_{\sigma}\approx
f_{\pi}$ $\displaystyle
F_{\sigma}(z)=N_{\sigma}\int\limits_{0}^{1}dx\int\limits_{0}^{1}\frac{y(1-4y^{2}x(1-x))dy}{1-\rho_{\sigma}^{2}y^{2}x(1-x)+zy(1-y)x},~{}~{}~{}\rho_{\sigma}=\frac{m_{\sigma}}{m_{q}}.$
(13)
The combination of quark charges and color factor give a coefficient
$3((2/3)^{2}+(1/3)^{2})=5/3$. The nontrivial difference in numerators of (12)
and (13) is a result of scalar nature of $\sigma$ meson. As to the decay
$\eta(\eta^{\prime})\to\gamma\gamma^{*}$ it is enough to do the relevant
replacements of masses in equation (12).
The amplitudes
$M(\pi^{0}\to\gamma\gamma^{*}),~{}~{}~{}M(\sigma\to\gamma\gamma^{*})$ are
nothing else than impact factors, one needs to calculate the cross sections of
neutral mesons pairs production . Now we are in position to estimate the
influence of the photon virtuality on the cross section of the reaction
$\gamma\gamma\to\pi^{0}\pi^{0}$ from (1). Making use the relation:
$\displaystyle\int\limits_{0}^{2\pi}\frac{d\phi}{2\pi}[\vec{q}\vec{e}_{1}]^{2}[\vec{q}\vec{e}_{2}]^{2}=\frac{(\vec{q}^{2})^{2}}{8}(1+2\cos^{2}\phi_{0}),$
(14)
with $\phi_{0}$ the azimuthal angle between the initial photons polarization
vectors. Substituting expression (12) for $\pi^{0}$ impact factor in (9) we
get:
$\displaystyle\frac{d\sigma}{dz}=\frac{m_{q}^{2}}{8\pi}(\frac{\alpha}{4\pi
f_{\pi}})^{4}\frac{(zF_{\pi}(z))^{4}}{(1+z^{2})^{2}}(1+2cos^{2}{\phi_{0}}).$
(15)
In the case of pions production one can safely neglect the small term
$y^{2}x(1-x)m_{\pi}^{2}/m_{q}^{2}<0.05$ in the denominator of (12) with the
result:
$\displaystyle
F_{\pi}(z)=2\int\limits_{0}^{1}dx\int\limits_{0}^{1}\frac{ydy}{1+zy(1-y)x}=\frac{4}{z}\ln^{2}\left(\sqrt{1+\frac{z}{4}}+\sqrt{\frac{z}{4}}\right).$
(16)
The total cross section of the two neutral pions production:
$\displaystyle\sigma^{\gamma\gamma\to\pi_{0}\pi_{0}}=\sigma_{0}(1+2\cos^{2}\phi_{0})I,~{}~{}~{}\sigma_{0}=\frac{\alpha^{4}m_{q}^{2}}{2^{7}\pi^{5}f_{\pi}^{4}}\approx
2,6\times 10^{-2}pb,$ $\displaystyle
I=\frac{1}{4}\int\limits_{0}^{\infty}\frac{dz}{z^{4}}\ln^{8}(\sqrt{1+z}+\sqrt{z})=0.3557.$
(17)
Thus the expressions (9), (12), (13) allow to calculate the yields of any
combination of light meson pairs produced in $\gamma\gamma$ collisions.
## 3 Bound states production
The considered approach is especially efficient in investigation of bound
states formation in collisions of high energy particles. As a typical examples
we examine the production of simplest atoms being the bound state of two
charged pions (pionium atom $A_{\pi}$) and atom constructed from two fermions
(positronium atom Ps).
To determine the pionium impact factor $\gamma\gamma^{\ast}\to A_{\pi}$ we
take advantage of well known QED amplitude [11] for the process
$\gamma(k_{1},e_{1})+\gamma^{\ast}(q)\to\pi^{-}(q_{-})+\pi^{+}(q_{+})$:
$\displaystyle
M^{\gamma\gamma^{\ast}\to\pi\pi}=\frac{4\pi\alpha}{s}[\frac{(2q_{-}e_{1})((-2q_{+}+q)k_{2})}{2q_{-}k_{1}}+\frac{(-2q_{+}e_{1})((2q_{-}-q)k_{2})}{-2q_{+}k_{1}}-2(e_{1}k_{2})]$
Account on that in atom pions have almost the same velocity $q_{+}=q_{-}=p/2$;
$2(pk_{1})=4m_{\pi}^{2}+\vec{q}^{2}$ and expressing $p,e$ through the Sudakov
variables:
$\displaystyle
p=\alpha_{p}k_{2}+\beta_{p}k_{1}+q_{\bot};~{}~{}~{}e=\beta_{e}k_{1}+e_{\bot},$
(19)
Making use the relation:
$\displaystyle(pe_{1})((p-q)k_{2}-(pk_{1})(ek_{2})=-2s(\vec{q}\vec{e}_{1})$
(20)
the amplitude of two pions production with the same velocities takes the form:
$\displaystyle
M^{\gamma\gamma^{\ast}\to\pi\pi}=\frac{8\pi\alpha(\vec{e}\vec{q})}{\vec{q}^{2}+4m_{\pi}^{2}}$
(21)
In order to obtain the amplitude for pionium production we use the relation
[12] allowing to connect the amplitude of two free scalar mesons production
with the production amplitude of their bound state $A_{\pi}$ 111The square of
pionium ground state wave function at origin has the form:
$|\Psi(0)|^{2}=\frac{\alpha^{3}m^{3}}{8\pi}$
$\displaystyle M^{\gamma\gamma^{\ast}\to
A_{\pi}}=M^{\gamma\gamma^{\ast}\to\pi\pi}\frac{i\Psi(\vec{r}=0)}{\sqrt{m_{\pi}}}$
(22)
Finally for the amplitude of pionium production in $\gamma\gamma^{\ast}$
collisions we get:
$\displaystyle M^{\gamma\gamma^{\ast}\to
A_{\pi}}=\frac{8i\pi\alpha(\vec{e}\vec{q})}{4m_{\pi}^{2}+\vec{q}^{2}}\frac{\Psi(0)}{\sqrt{m_{\pi}}},$
(23)
To obtain the impact factor for para-positronium creation
$\gamma(k_{1},e_{1})+\gamma^{\ast}(q)\to Ps(p)$ we take advantage of the
receipt [13, 14] of passage from free $e^{+}e^{-}$ pair to their bound state
and textbook expression [11] for $e^{+}e^{-}$ pair creation in $\gamma\gamma$
collisions. As a result the matrix element for bound state creation takes the
form:
$\displaystyle M^{\gamma\gamma\ast\to Ps}$ $\displaystyle=$ $\displaystyle
i\frac{4\pi\alpha}{s}\frac{m_{e}\sqrt{\alpha^{3}}}{\sqrt{4\pi}}\frac{1}{4}Tr[\hat{e}_{1}\frac{\hat{q}_{-}-\hat{k}_{1}+m_{e}}{(q_{-}-k_{1})^{2}-m_{e}^{2}}\hat{k}_{2}$
(24) $\displaystyle+$
$\displaystyle\hat{k}_{2}\frac{-\hat{q}_{+}+\hat{k}_{1}+m_{e}}{(-q_{+}+k_{1})^{2}-m_{e}^{2}}\hat{e}_{2}](\hat{p}+m_{Ps})\gamma_{5}.$
Making use the relations $q+k_{1}=p=q_{+}+q_{-}$ and $q_{+}=q_{-}=p/2$ one
obtains:
$\displaystyle M^{\gamma\gamma^{\ast}\to
Ps}=\frac{4im_{e}\sqrt{\pi\alpha^{5}}}{4m_{e}^{2}+\vec{q}^{2}}|[\vec{e}_{1},\vec{q}]|.$
(25)
With the help of equation (9) and impact factors (23), (25) one can calculates
the differential cross section of elementary atoms creation in the processes:
$\displaystyle\gamma+\gamma\to
S_{1}+S_{2};~{}~{}~{}~{}S_{1},S_{2}=A_{\pi},Ps.$ (26)
For reader convenience and rough estimates of the order of total cross
sections of bound state production by photon exchange mechanism (Fig.1a) we
cite the expressions for the total cross sections relevant to reactions (26):
$\displaystyle\sigma^{\gamma\gamma\to
PsPs}=\frac{\pi\alpha^{8}}{96}r_{e}^{2}(1+2\cos^{2}\phi_{0});~{}~{}~{}~{}\sigma^{\gamma\gamma\to
A_{\pi}A_{\pi}}=(\frac{r_{e}}{4r_{\pi}})^{2}\sigma^{\gamma\gamma\to PsPs};$
$\displaystyle\sigma^{\gamma\gamma\to
PsA_{\pi}}=\frac{\pi\alpha^{8}}{64}r_{\pi}^{2}(3-2\cos^{2}\phi_{0});~{}~{}~{}\sigma^{\gamma\gamma\to\pi^{0}Ps}=\frac{\alpha^{7}}{32\pi^{2}f_{\pi}^{2}}(1+2\cos^{2}\phi_{0});$
$\displaystyle r_{e}=\frac{\alpha}{m_{e}},r_{\pi}=\frac{\alpha}{m_{\pi}}.$
(27)
Rough estimates of these cross sections show that they are really very small
quantity of order $10^{-8}nb$.
## 4 Vector meson exchange in pairs production
Up to now we considered production processes provided by photon exchanges
(Fig.1a). From the other hand the vector meson (Fig. 1b) exchanges also give
nondecreasing with energy contribution to the processes (1). The problem with
such type exchanges is connected with the fact that the Born approximation
depicted on Fig. 1b badly violated for strong interactions.
To take into account the higher order contributions of strong interaction one
would replaces the exchanged vector meson propagator in (9) by its reggeized
analog [15]
$\displaystyle\frac{1}{t-m_{V}^{2}}\to\alpha^{\prime}\frac{1-e^{-i\pi\alpha(t)}}{2}\Gamma{(1-\alpha(t))}(\frac{s}{s_{0}})^{\alpha(t)},$
(28)
where $\alpha(t)$ is the Regge trajectory of vector meson
$\displaystyle\alpha(t)=\alpha(0)+\alpha^{\prime}t$ (29)
The $\Gamma$ function contains the pole propagator $sin^{-1}(\pi\alpha(t))$
and in the limit $t\to m_{V}^{2}$ the expression (28) reduced to the standard
pole propagator. The detailed characteristics of Regge trajectories of
different vector mesons can be found in work [16] and references therein.
Later on for estimation of vector mesons contribution to the relevant cross
sections we use the simplified suppression factor:
$\displaystyle
R(s,t)=(\frac{s}{s_{0}})^{2(\alpha(t)-1)}\approx\frac{s_{0}}{s};~{}~{}s_{0}\approx
1GeV^{2}.$ (30)
The impact factors corresponding to the vector mesons exchanges depend on the
considered process and would be obtained as it has been done above for photon
exchanges.
As an example let us consider the process of two charged pions production
$\gamma\gamma\to\pi^{+}\pi^{-}$ for which the photon exchange is absent. The
main contribution to this reaction at high energies gives the $\rho$
exchange.222The pion exchange [17] dominates only at small transfer momenta
$t\leq 4m_{\pi}^{2}\leq 0.1GeV^{2}$ and fall off with energy much stronger
than vector exchanges. The matrix element of radiative decay of charged meson
$\rho^{+}(p,e_{1})\to\pi^{+}(p_{\pi})+\gamma(k,e_{2})$ reads
$M=g_{+}(pe_{1}ke_{2})$, where the constant $g_{+}$ can be determined from the
relevant decay width :
$\displaystyle\Gamma^{\rho^{+}\to\pi^{+}\gamma}=\frac{g_{+}^{2}}{96\pi}(\frac{m_{\rho}^{2}-m_{\pi}^{2}}{m_{\rho}})^{3}.$
(31)
Comparing this relation with the experimental value of the
$\rho^{+}\to\pi^{+}\gamma$ branching ratio [18] $B=4.5\times 10^{-4}$,
$\Gamma=67keV$ one gets $g_{+}\approx 0.21GeV^{-1}$.
In the case when one of the photons is virtual it is enough to do the simple
replacement $g_{+}\to g_{+}F(z)$ with
$\displaystyle
F(z)=\frac{4}{z}\ln^{2}(\sqrt{1+\frac{z}{4}}+\sqrt{\frac{z}{4}});~{}~{}~{}z=\frac{\vec{q}^{2}}{m_{q}^{2}}.$
(32)
The differential cross section of the process
$\gamma\gamma\ast\to\pi^{+}\pi^{-}$ in peripheral kinematic takes the form
$\displaystyle d\sigma$ $\displaystyle=$
$\displaystyle\frac{d\vec{q}^{2}d\phi}{32\pi^{2}}\frac{|M^{(1)}|^{2}|M^{(2)}|^{2}}{(\vec{q}^{2}+m_{\rho}^{2})^{2}},$
$\displaystyle M^{(1)}$ $\displaystyle=$
$\displaystyle\frac{g_{+}}{2}[\vec{q}\vec{e}_{1}]F(z);~{}~{}~{}M^{(2)}=\frac{g_{+}}{2}[\vec{q}\vec{e}_{2}]F(z).$
(33)
Averaging over azimuthal angle according to equation (14) for the total cross
section we obtain:
$\displaystyle\sigma(\gamma\gamma\to\pi^{+}\pi^{-})=\frac{g_{+}^{4}m_{q}^{2}}{32\pi}(1+2\cos^{2}\phi_{0})I;$
$\displaystyle
I=\int\limits_{0}^{\infty}\frac{dz}{z^{2}(z+(\frac{m_{\rho}}{2m_{q}})^{2})^{2}}\ln^{8}(\sqrt{1+\frac{z}{4}}+\sqrt{\frac{z}{4}})\approx
0.372$ $\displaystyle\sigma^{\gamma\gamma\to\pi^{+}\pi^{-}}\approx
60(1+2\cos^{2}\phi_{0})(\frac{s_{0}}{s})nb.$ (34)
In the same way one can estimates the contribution from $\rho,\omega$
exchanges to the process of two neutral pions production
$\gamma\gamma\to\pi^{0}\pi^{0}$, determining the constants
$g_{\rho},g_{\omega}$ from experimental data on the relevant decay rates [18]
$\displaystyle\Gamma(\rho^{0}\to\pi^{0}\gamma)$ $\displaystyle=$
$\displaystyle 8.9\times 10^{-5}GeV;~{}~{}~{}g_{\rho}=0.25GeV^{-1}$
$\displaystyle\Gamma(\omega\to\pi^{0}\gamma)$ $\displaystyle=$ $\displaystyle
70\times 10^{-5}GeV;~{}~{}~{}g_{\omega}=0.71GeV^{-1}.$ (35)
For the total cross section of the process $\gamma\gamma\to\pi^{0}\pi^{0}$
provided by vector meson exchanges we obtain:
$\displaystyle\sigma^{\gamma\gamma\to\pi^{0}\pi^{0}}=3(1+2\cos^{2}\phi_{0})(\frac{s_{0}}{s})\mu
b.$ (36)
## 5 Pionium atom production in $ep$ and $\pi p$ collisions.
In recent years there has been a significant effort to extract the $\pi\pi$
s-wave scattering lengths $a_{I}$ with total isospin I=0, 2 from experimental
data on pionium atom $A_{\pi}$ creation. The scattering lengths determination
with high precision allows to check the predictions of low-energy hadron
theories such as Chiral Perturbation Theory (CHPT) or Nambu-Jona-Lasinio model
(NJL) which give it value with unprecedented for strong interaction accuracy
$\sim 2\%$ [19].
The main goal of experiment DIRAC [20] at PS CERN has been the determination
of pions scattering lengths difference $a_{0}-a_{2}$ from the measurement of
pionium atom lifetime, which is connected with this difference by the relation
[21]:
$\displaystyle\Gamma=\frac{1}{\tau}=\frac{2}{9}\sqrt{\frac{2(m_{\pi^{+}}-m_{\pi^{0}})}{m_{\pi}}}(a_{0}^{0}-a_{0}^{2})^{2}m_{\pi}^{3}\alpha^{3}.$
(37)
At present due to experiment Dirac and experiments on kaons decays [22, 23]
the scattering lengths determined from experimental data with precision
comparable with theoretical predictions.
Below we will consider the peripheral mechanism of creation of two charged
pions in collision of high energy electron with the proton and similar one
with the initial high energy negatively charged $\pi$-meson instead electron
$\displaystyle e(p_{1})+p(p_{2})\to
e(p_{1}^{\prime})+A_{\pi}(p)+p(p_{2}^{\prime})$ (38)
$\displaystyle\pi(p_{1})+p(p_{2})\to\pi(p_{1}^{\prime})+A_{\pi}(p)+p(p_{2}^{\prime})$
(39)
$s=(p_{1}+p_{2})^{2}>>m_{p}^{2}$ with $m_{p}$ a proton mass.
For the case of electron-proton collision the pion pair is created in the
collision of virtual photon emitted by electron and virtual $\rho$ ($\omega$)
meson emitted by proton (Fig. 2a). In the case of $\pi$-meson proton
interaction the pion pair is produced by two virtual $\rho$ mesons (Fig. 2b).
Figure 2: a) The pionium electroproduction in the process $e+p\to
e+p+A_{\pi}$ ; b) Pionium production by pions $\pi+p\to\pi+p+A_{\pi}$
The matrix element corresponding to these processes has the form:
$\displaystyle
M=\frac{G}{(q_{1}^{2}-m_{1}^{2})(q_{2}^{2}-m_{2}^{2})}J_{1}(p_{1})_{\mu_{1}}T_{\mu\nu}J_{p}(p_{2})_{\nu_{1}}G^{\mu\mu_{1}}G^{\nu\nu_{1}},$
(40)
with $G$ is the product of the relevant coupling constants, $m_{1,2}$-masses
of the exchanged vector particles; $J_{1},J_{p}$ are the currents connected
with the colliding particles; tensor $T_{\mu\nu}$ describes the conversion of
two vector mesons to pion pair.
The main contribution in peripheral kinematics (non-vanishing in limit
$s\to\infty$) arises from relevant Green functions:
$\displaystyle
G^{\mu\mu_{1}}=\frac{2}{s}p_{2}^{\mu}p_{1}^{\mu_{1}};G^{\nu\nu_{1}}=\frac{2}{s}p_{2}^{\nu}p_{1}^{\nu_{1}}.$
(41)
Matrix element of the sub-process of creation of pion pair with equal
4-momenta by two virtual vector particles
$\displaystyle V_{\mu}(q_{1})+V_{\nu}(q_{2})\to\pi^{+}(q)+\pi^{-}(q)$ (42)
is described by the tensor:
$\displaystyle
T_{\mu\nu}=\frac{2}{D}[q_{2\mu}q_{1\nu}+Dg_{\mu\nu}],D=-\frac{1}{2}[4m^{2}+\vec{q}_{1}^{2}+\vec{q}_{2}^{2}],$
(43)
Combining these expressions one gets for the matrix element of the process
$e+p\to e^{\prime}+p^{\prime}+A_{\pi}$:
$\displaystyle M^{ep\to
epA_{\pi}}=\frac{4s}{\vec{q}_{1}^{2}+m_{e}^{2}\beta_{1}^{2}}\frac{G_{e}}{\vec{q}_{2}^{2}+m_{V}^{2}}\Phi_{e}\Phi_{A}\Phi_{p}\Psi(0),$
(44)
where $G_{e}=4\pi\alpha g_{\pi}g_{p}$, ($g_{\pi},g_{p}$-coupling constants of
$\rho$-meson with pion and proton, which we put $g_{\pi}=g_{\rho}=3$)
$\displaystyle\Phi_{e}=\frac{1}{s}\bar{u}(p_{1}^{\prime})\hat{p}_{2}u(p_{1});~{}~{}\Phi_{A}=\frac{1}{s}p_{1}^{\mu}p_{2}^{\nu}T_{\mu\nu}=-2\frac{\vec{q}_{1}\vec{q}_{2}}{D};$
$\displaystyle\Phi_{p}=\frac{1}{s}\bar{u}(p_{2}^{\prime})\Gamma_{\mu}u(p_{1})p_{2}^{\mu},\Gamma_{\mu}=\gamma_{\mu}F_{1}+\frac{1}{4M_{p}}(\hat{q}_{2}\gamma_{\mu}-\gamma_{\mu}\hat{q}_{2})F_{2},$
(45)
Here $F_{1}=F_{1}(q_{2}^{2}),F_{2}=F_{2}(q_{2}^{2})$ are Dirac and Pauli form-
factors of proton.
The phase volume of the three particles in the final state:
$\displaystyle
d\Gamma=\frac{(2\pi)^{4}}{(2\pi)^{9}}\frac{d^{3}p_{1}^{\prime}}{2E_{1}^{\prime}}\frac{d^{3}p_{2}^{\prime}}{2E_{2}^{\prime}}\frac{d^{3}p_{A}}{2E_{A}}\delta^{4}(p_{1}+p_{2}-p_{1}^{\prime}-p_{2}^{\prime}-p_{A}),$
(46)
can be reduced using the Sudakov variables to the following form:
$\displaystyle
d\Gamma=\frac{1}{(2\pi)^{5}}\frac{1}{4s}\frac{d\beta_{1}}{\beta_{1}}d^{2}\vec{q}_{1}d\vec{q}_{2}.$
(47)
Making use the summed over spin states of the squares of matrix elements of
the relevant sub-processes:
$\displaystyle\sum|\Phi_{e}|^{2}=2;~{}~{}\sum|\Phi_{p}|^{2}=2[F_{1}^{2}+\frac{\vec{q}_{2}^{2}}{4m_{p}^{2}}F_{2}^{2}];$
$\displaystyle|\Phi_{A}|^{2}=\frac{4(\vec{q}_{1}\vec{q}_{2})^{2}}{(4m^{2}+\vec{q}_{1}^{2}+\vec{q}_{2}^{2})^{2}},$
(48)
where m is the pion mass. The cross section of the process $e+p\to
e+p+A_{\pi}$ takes the form:
$\displaystyle d\sigma^{ep\to epA_{\pi}}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{5}g_{\pi}^{2}g_{p}^{2}}{2\pi^{2}}\frac{m^{2}\vec{q}_{1}^{2}d\vec{q}_{1}^{2}\vec{q}_{2}^{2}d\vec{q}_{2}^{2}}{(4m^{2}+\vec{q}_{1}^{2}+\vec{q}_{2}^{2})^{2}(\vec{q}_{1}^{2}+m_{e}^{2}\beta_{1}^{2})^{2}(\vec{q}_{2}^{2}+m_{\rho}^{2})^{2}}$
(49) $\displaystyle\times$
$\displaystyle[F_{1}^{2}+\frac{\vec{q}_{2}^{2}}{4m_{p}^{2}}F_{2}^{2}]\frac{d\beta_{1}(1-\beta_{1})}{\beta_{1}};~{}~{}~{}\frac{4m^{2}}{s}<\beta_{1}<1.$
Similar expression for the cross section with initial $\pi$ meson instead of
the electron:
$\displaystyle d\sigma^{\pi p\to\pi pA_{\pi}}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{3}g_{\pi}^{6}g_{p}^{2}}{64\pi^{4}}\frac{m^{2}\vec{q}_{1}^{2}d\vec{q}_{1}^{2}\vec{q}_{2}^{2}d\vec{q}_{2}^{2}}{(4m^{2}+\vec{q}_{1}^{2}+\vec{q}_{2}^{2})^{2}(\vec{q}_{1}^{2}+m_{\rho}^{2})^{2}(\vec{q}_{2}^{2}+m_{\rho}^{2})^{2}}$
(50) $\displaystyle\times$
$\displaystyle[F_{1}^{2}+\frac{\vec{q}_{2}^{2}}{4m_{p}^{2}}F_{2}^{2}]\frac{d\beta_{1}(1-\beta_{1})}{\beta_{1}}.$
Integrating these expressions over phase volume one obtains the total yield of
pionium atom. In the case of the electroproduction:
$\displaystyle\sigma(ep\to
epA_{\pi})=\sigma_{e}D_{e},~{}~{}\sigma_{e}=\frac{\alpha^{5}g_{\pi}^{2}g_{p}^{2}m^{2}}{2\pi^{2}m_{\rho}^{4}}\approx
0.3pb;$ $\displaystyle
D_{e}=J_{N}[l_{m}^{2}+l_{\pi}(l_{m}-1)-2],J_{N}=\int\limits_{0}^{\infty}\frac{xN^{2}dx}{(x+4)^{2}(x+N)^{2}}\approx
0.845;$ $\displaystyle
l_{m}=\ln\frac{s}{4m^{2}},~{}~{}~{}l_{\pi}=\ln\frac{m^{2}}{m_{e}^{2}}.$ (51)
For $s=100GeV^{2}$ the cross section $\sigma(ep\to epA_{\pi})\approx 30pb$ is
too small to be measured at present accelerators.
As to the pionium production by pions we obtain:
$\displaystyle\sigma(\pi p\to\pi
pA_{\pi})=\sigma_{\pi}D_{\pi},~{}~{}~{}\sigma_{\pi}=\frac{\alpha^{3}g^{8}m^{2}}{64\pi^{2}m_{\rho}^{4}}\approx
217nb;$ $\displaystyle
D_{\pi}=(l_{m}-1)I,I=\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\frac{x_{1}x_{2}dx_{1}dx_{2}}{(x_{1}+x_{2})^{2}(x_{1}+1)^{2}(x_{2}+1)^{2}}\approx
0.133.$ (52)
The total cross section turns out to be of the order $\sigma(\pi p\to\pi
pA_{\pi})\approx 178nb$ for s=80 $GeV^{2}$ (IHEP, Protvino) and thus can be
measured at modern facilities.
In conclusion we note that the contribution from the channels with exchange of
two photons is of order
$\displaystyle\sigma_{e}^{\gamma\gamma}=\sigma_{0}D_{e}^{\gamma\gamma};~{}~{}\sigma_{\pi}^{\gamma\gamma}=\sigma_{0}D_{\pi}^{\gamma\gamma},\sigma_{0}=\frac{8\alpha^{7}}{m^{2}}\approx
1,8\times 10^{-3}pb.$ (53)
In spite of a rather large enhancement factors $D_{e}^{\gamma\gamma}\sim
10D_{\pi}^{\gamma\gamma}\sim 10^{2}$ the relevant contributions can be safely
neglected.
## 6 The vector meson exchange reggeization
As was mentioned above the consideration of hadronic processes in peripheral
kinematics in Born approximation is non-adequate. The effect of converting the
ordinary vector mesons to the relevant Regge poles must be taken into account.
It results in an additional suppression factor to the total cross sections of
processes (38), (39):
$\displaystyle R=\left(\frac{s_{1}s_{2}}{s_{0}^{2}}\right)^{2(\alpha(0)-1)},$
(54)
Keeping in mind the kinematical relation $s_{1}s_{2}\approx 4m^{2}s$ and
puting $\alpha(0)\approx 0.5$ :
$\displaystyle R\approx\frac{s_{0}^{2}}{4sm^{2}}.$ (55)
For instance at $s=80GeV^{2}$ it results in the suppression factor
$\displaystyle R\approx 0.16.$ (56)
So the realistic cross section for this energies is about $\sigma^{\pi}\approx
28nb$.
Let us note that in the double pomeron exchanges in the process (39) (or
pionium photoproduction off pomeron in the case of reaction (38)) such
suppression factor is absent and at enough high energies the pomeron exchanges
dominated. It is useful to estimate the energies from which the photon
exchange becomes comparable with vector mesons one. For instance to obtain the
matrix element for pionium electroproduction by two photon exchanges from the
matrix element with one vector meson exchange (fig.2a) it is enough to do a
simple replacement:
$\displaystyle g_{\pi}g_{p}\frac{s_{0}}{2m\sqrt{s}}\to 4\pi\alpha$ (57)
Thus only from energies $s\sim 10^{5}GeV^{2}$ the contribution with two photon
exchanges in pionium electroproduction becomes larger than the one with vector
meson exchange.
## 7 Acknowledgements
Authors are grateful to A. Ahmadov, N. Kochelev and R. Togoo for discussions.
The work of E. Kuraev was partially supported by RFBR-01201164165 and
Belorussian grants.
## References
## References
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* [2] F. R. Arutjunian , V. A. Tumanyan, ZETP, 44, 2100 (1963)
* [3] V. G. Serbo, Nucl. Instr. Meth. , 472, 260 (2010)
* [4] ALEPH Collaboration, Phys. Lett.B569 140 (2003)
* [5] L3 Collaboration, Phys. Lett.B615 19 (2005)
* [6] Belle Collaboration, hep-ex/0711.1926
* [7] S. Weinberg, Physica, A96, 327 (1979)
* [8] J. Gasser, H. Leutwyler, Nucl. Phys., B250, 465 (1985)
* [9] M. K. Volkov, A. E. Radzhabov, arXiv: hep-ph/0508263
* [10] A. B. Arbuzov et al., Particles and Nuclei, 41, 1113 (2010).
* [11] A. I. Akhiezer, V. B. Berestetskij, Quantum Electrodynamics, Moscow, 1959.
* [12] R. Staffin, Phys. Rev.D16, 726 (1977)
* [13] V. A. Novikov et al., Phys. Rep. 41C (1978)
* [14] S. R. Gevorkyan et al. , Phys. Rev. A 58, 4556 (1998)
* [15] M. Guidal, J. M. Laget, M. Vanderhaeghen, Nucl. Phys. A627 645 (1997)
* [16] A. V. Titov, B. Kampfer arXiv: hep-ph/0807.1822
* [17] N. Schmitz, Nucl. Phys. B 36,145 (1994)
* [18] C. Amsler et al. (PDG), Phys. Lett. B667, 1 (2008)
* [19] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B603, 125 (2001)
* [20] B. Adeva et al., Phys. Lett. B704, 24 (2011)
* [21] J. Uretsky, J.Palfrey, Phys. Rev. 121, 1798 (1961)
* [22] J. R. Batley et al., Eur. Phys. J. C64, 589 (2009)
* [23] S. R. Gevorkyan, A.V. Tarasov, O. O. Voskresenskaya, Phys.Lett. B649, 159 (2007)
|
arxiv-papers
| 2012-05-22T12:38:18 |
2024-09-04T02:49:31.203674
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. R. Gevorkyan, E. A. Kuraev, M. K. Volkov",
"submitter": "Sergey Gevorkyan",
"url": "https://arxiv.org/abs/1205.4896"
}
|
1205.5081
|
# A comparative study of dark matter in the MSSM and its singlet extensions: a
mini review
Wenyu Wang Institute of Theoretical Physics, College of Applied Science,
Beijing University of Technology, Beijing 100124, China
###### Abstract
In this note we briefly review the recent studies of dark matter in the MSSM
and its singlet extensions: the NMSSM, the nMSSM, and the general singlet
extension. Under the new detection results of CDMS II, XENON, CoGeNT and
PAMELA, we find that (i) the latest detection results can exclude a large part
of the parameter space which allowed by current collider constraints in these
models. The future SuperCDMS and XENON can cover most of the allowed parameter
space; (ii) the singlet sector will decouple from the MSSM-like sector in the
NMSSM, however, singlet sector makes the nMSSM quite different from the MSSM;
(iii) the NMSSM can allow light dark matter at several GeV exists. Light CP-
even or CP-odd Higgs boson must be present so as to satisfy the measured dark
matter relic density. In case of the presence of a light CP-even Higgs boson,
the light neutralino dark matter can explain the CoGeNT and DAMA/LIBRA
results; (iv) the general singlet extension of the MSSM gives a perfect
explanation for both the relic density and the PAMELA result through the
Sommerfeld-enhanced annihilation. Higgs decays in different scenario are also
studied.
###### pacs:
14.80.Ly,11.30.Pb,95.35.+d
## I Introduction
Although there are many theoretical or aesthetical arguments for the necessity
of TeV-scale new physics, the most convincing evidence is from the WMAP
(Wilkinson Microwave Anisotropy Probe) observation of the cosmic cold dark
matter, which naturally indicates the existence of WIMPs (Weakly Interacting
Massive Particle) beyond the prediction of the Standard Model (SM). By
contrast, the neutrino oscillations may rather imply trivial new physics
(plainly adding right-handed neutrinos to the SM) or new physics at some very
high see-saw scale unaccessible to any foreseeable colliders. Therefore, the
TeV-scale new physics to be unraveled at the Large Hadron Collider (LHC) is
the most likely related to the WIMP dark matter.
If WIMP dark matter is chosen by nature, it will give a strong argument for
low-energy supersymmetry (SUSY) with R-parity which can give a good candidate.
Nevertheless, SUSY is motivated for solving the hierarchy problem elegantly.
It can also solve other puzzles of the SM, such as the $3\sigma$ deviation of
the muon anomalous magnetic moment from the SM prediction. In the framework of
SUSY, the most intensively studied model is the minimal supersymmetric
standard model (MSSM) mssm , which is the most economical realization of SUSY.
However, this model suffers from the $\mu$-problem. The $\mu$-parameter is the
only dimensional parameter in the SUSY conserving sector. From a top down
view, one would expect the $\mu$ to be either zero or at the Planck scale. But
in the MSSM, the relation of the electro-weak (EW) scale soft parameters
($\tilde{m}_{d}^{2},~{}\tilde{m}_{u}^{2}$) sugrawgc
$\frac{1}{2}\,M_{Z}^{2}={\tilde{m}_{d}^{2}-\tilde{m}_{u}^{2}\tan^{2}\beta\over\tan^{2}\beta-1}-\mu^{2},$
(1)
makes that $\mu$ must be at the EW scale, while LEP constraints on the
chargino mass require $\mu$ to be non-zero lepsusy . A simple solution is to
promote $\mu$ to a dynamical field in extensions of the MSSM that contain an
additional singlet superfield $\hat{S}$ which does not interact with the MSSM
fields other than the two Higgs doublets. An effective $\mu$ can be reasonably
got at EW scale when $\hat{S}$ denotes the vacuum expectation value (VEV) of
the singlet field. Among these extension models the next-to-minimal
supersymmetric model (NMSSM) NMSSM and the nearly minimal supersymmetric
model (nMSSM) xnMSSM1 ; xnMSSM2 caused much attention recently. Note that the
little hierarchy problem which is also a trouble of the MSSM is relaxed
greatly in the NMSSM.
If introduce a singlet superfield to the MSSM, the Higgs sector will have one
more CP even component and one more CP odd component, and the neutralino
sector will have one more singlino component. These singlet multiplets compose
a “singlet sector” of the MSSM. It can make the phenomenologies of SUSY dark
matter and Higgs different from the MSSM. More and more precision results of
dark matter detection give us an opportunity to test if this singlet sector
really exists. For example, experiments for the underground direct detection
of cold dark matter $\tilde{\chi}$ have recently made significant progress.
While the null observation of $\tilde{\chi}$ in the CDMS and XENON100
experiments has set rather tight upper limits on the spin-independent (SI)
cross section of $\tilde{\chi}$-nucleon scattering CDMSII ; XENON100 . The
CoGeNT experiment CoGeNT reported an excess which cannot be explained by any
known background sources but seems to be consistent with the signal of a light
$\tilde{\chi}$ with mass around 10 GeV and scattering rate $\hbox{(1--
2)}\times 10^{-40}$ cm2. Intriguingly, this range of mass and scattering rate
are compatible with dark matter explanation for both the DAMA/LIBRA data and
the preliminary CRESST data Hooper . Though CoGeNT result is not consistent
with the CDMS or XENON results, it implies that the mass of dark matter can
range a very long scope at EW scale, that is from a few GeV to several TeV.
The indirect detection PAMELA also observed an excess of the cosmic ray
positron in the energy range 10-100 GeV pamela which may be explained by dark
matter.
In this paper, We will give a short review on the difference between the MSSM
and the MSSM with a singlet sector under the constraints of new dark matter
detection results. As the Higgs hunting on colliders has delicate relation
with dark matter detections, the implication on Higgs searching is also
reviewed. The content is based on our previous work Wang:2009rj ; Cao:2010fi ;
Cao:2011re . the paper is organized as following, in sec. II, we will give a
short review on the structures of the MSSM, the NMSSM and the nMSSM. In sec.
III we will give a comparison on the models under the constraints of CDMS,
XENON, and CoGeNT. In sec. IV, a general singlet extension of the MSSM is
discussed, and a summary is given in sec. V.
## II the MSSM and its Singlet Extensions
As the economical realization of supersymmetry, the MSSM has the minimal
content of particles, while the NMSSM and the nMSSM extend the MSSM by only
adding one singlet Higgs superfield $\hat{S}$. The difference between these
models is reflected in their superpotential:
$\displaystyle W_{\rm MSSM}$ $\displaystyle=$ $\displaystyle
W_{F}+\mu\hat{H}_{u}\cdot\hat{H}_{d},$ (2) $\displaystyle W_{\rm NMSSM}$
$\displaystyle=$ $\displaystyle
W_{F}+\lambda\hat{H}_{u}\cdot\hat{H}_{d}\hat{S}+\frac{1}{3}\kappa\hat{S}^{3},$
(3) $\displaystyle W_{\rm nMSSM}$ $\displaystyle=$ $\displaystyle
W_{F}+\lambda\hat{H}_{u}\cdot\hat{H}_{d}\hat{S}+\xi_{F}M_{n}^{2}\hat{S},$ (4)
where
$W_{F}=Y_{u}\hat{Q}\cdot\hat{H}_{u}\hat{U}-Y_{d}\hat{Q}\cdot\hat{H}_{d}\hat{D}-Y_{e}\hat{L}\cdot\hat{H}_{d}\hat{E}$
with $\hat{Q}$, $\hat{U}$ and $\hat{D}$ being the squark superfields, and
$\hat{L}$ and $\hat{E}$ being the slepton superfields. $\hat{H}_{u}$ and
$\hat{H}_{d}$ are the Higgs doublet superfields, $\lambda$, $\kappa$ and
$\xi_{F}$ are dimensionless coefficients, and $\mu$ and $M_{n}$ are parameters
with mass dimension. Note that there is no explicit $\mu$-term in the NMSSM or
the nMSSM, and an effective $\mu$-parameter (denoted as $\mu_{\rm eff}$) can
be generated when the scalar component ($S$) of $\hat{S}$ develops a VEV. Also
note that the nMSSM differs from the NMSSM in the last term with the trilinear
singlet term $\kappa\hat{S}^{3}$ of the NMSSM replaced by the tadpole term
$\xi_{F}M_{n}^{2}\hat{S}$. As pointed out in Ref. xnMSSM1 , such a tadpole
term can be generated at a high loop level and naturally be of the SUSY
breaking scale. The advantage of such replacement is the nMSSM has no discrete
symmetry thus free of the domain wall problem which the NMSSM suffers from.
Corresponding to the superpotential, the Higgs soft terms in the scalar
potentials are also different between the three models (the soft terms for
gauginos and sfermions are the same thus not listed here)
$\displaystyle V_{\rm soft}^{\rm MSSM}$ $\displaystyle=$
$\displaystyle\tilde{m}_{d}^{2}|H_{d}|^{2}+\tilde{m}_{u}^{2}|H_{u}|^{2}+\left(B\mu
H_{u}\cdot H_{d}+\mbox{h.c.}\right)$ (5) $\displaystyle V_{\rm soft}^{\rm
NMSSM}$ $\displaystyle=$
$\displaystyle\tilde{m}_{d}^{2}|H_{d}|^{2}+\tilde{m}_{u}^{2}|H_{u}|^{2}+\tilde{m}_{s}^{2}|S|^{2}+\left(A_{\lambda}\lambda
SH_{d}\cdot H_{u}+\frac{\kappa}{3}A_{\kappa}S^{3}+\mbox{h.c.}\right),$ (6)
$\displaystyle V_{\rm soft}^{\rm nMSSM}$ $\displaystyle=$
$\displaystyle\tilde{m}_{d}^{2}|H_{d}|^{2}+\tilde{m}_{u}^{2}|H_{u}|^{2}+\tilde{m}_{s}^{2}|S|^{2}+\left(A_{\lambda}\lambda
SH_{d}\cdot H_{u}+\xi_{S}M_{n}^{3}S+\mbox{h.c.}\right).$ (7)
After the scalar fields $H_{u}$,$H_{d}$ and $S$ develop their VEVs $v_{u}$,
$v_{d}$ and $s$ respectively, they can be expanded as
$\displaystyle
H_{d}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}\left(v_{d}+\phi_{d}+i\varphi_{d}\right)\\\
H_{d}^{-}\end{array}\right)\,,H_{u}=\left(\begin{array}[]{c}H_{u}^{+}\\\
\frac{1}{\sqrt{2}}\left(v_{u}+\phi_{u}+i\varphi_{u}\right)\end{array}\right)\,,S=\frac{1}{\sqrt{2}}\left(s+\sigma+i\xi\right)\,.$
(12)
The mass eigenstates can be obtained by unitary rotations
$\displaystyle\left(\begin{array}[]{c}h_{1}\\\ h_{2}\\\
h_{3}\end{array}\right)=U^{H}\left(\begin{array}[]{c}\phi_{d}\\\ \phi_{u}\\\
\sigma\end{array}\right),~{}\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\
G_{0}\end{array}\right)=U^{A}\left(\begin{array}[]{c}\varphi_{d}\\\
\varphi_{u}\\\ \xi\end{array}\right),~{}\left(\begin{array}[]{c}G^{+}\\\
H^{+}\end{array}\right)=U^{H^{+}}\left(\begin{array}[]{c}H_{d}^{+}\\\
H_{u}^{+}\end{array}\right),$ (29)
where $h_{1,2,3}$ and $a_{1,2}$ are respectively the CP-even and CP-odd
neutral Higgs bosons, $G^{0}$ and $G^{+}$ are Goldstone bosons, and $H^{+}$ is
the charged Higgs boson. Including the scalar part of the singlet sector in
the NMSSM and the nMSSM leads to a pair of charged Higgs bosons, three CP-even
and two CP-odd neutral Higgs bosons. In the MSSM, we only have two CP-even and
one CP-odd neutral Higgs bosons in addition to a pair of charged Higgs bosons.
The MSSM predicts four neutralinos $\chi^{0}_{i}$ ($i=1,2,3,4$), i.e. the
mixture of neutral gauginos (bino $\lambda^{\prime}$ and neutral wino
$\lambda^{3}$) and neutral higgsinos ($\psi_{H_{u}}^{0},\psi_{H_{d}}^{0}$),
while the NMSSM and the nMSSM predict one more neutralino corresponding to the
singlino $\psi_{S}$ from the fermion part of singlet sector. In the basis
$(-i\lambda^{\prime},-i\lambda^{3}2,\psi_{H_{u}}^{0},\psi_{H_{d}}^{0},\psi_{S})$
(for the MSSM $\psi_{S}$ is absent) the neutralino mass matrix is given by
$\displaystyle\left(\begin{array}[]{ccccc}M_{1}&0&m_{Z}s_{W}s_{b}&-m_{Z}s_{W}c_{b}&0\\\
0&M_{2}&-m_{Z}c_{W}s_{b}&m_{Z}c_{W}c_{b}&0\\\
m_{Z}s_{W}s_{b}&-m_{Z}s_{W}s_{b}&0&-\mu&-\lambda vc_{b}\\\
-m_{Z}s_{W}c_{b}&-m_{Z}c_{W}c_{b}&-\mu&0&-\lambda vs_{b}\\\ 0&0&-\lambda
vc_{b}&-\lambda vs_{b}&\scriptstyle\left\\{\begin{array}[]{c}\scriptstyle
2\frac{\kappa}{\lambda}\mu~{}~{}{\rm{~{}~{}for~{}the~{}NMSSM}}\\\ \scriptstyle
0{\rm~{}~{}~{}~{}~{}~{}for~{}the~{}nMSSM}\end{array}\right.\end{array}\right),$
(37)
where $M_{1}$ and $M_{2}$ are respectively $U(1)$ and $SU(2)$ soft gaugino
mass parameters, $s_{W}=\sin\theta_{W}$, $c_{W}=\cos\theta_{W}$,
$s_{b}=\sin\beta$ and $c_{b}=\cos\beta$ with $\tan\beta\equiv v_{u}/v_{d}$.
The lightest neutralino $\tilde{\chi}^{0}_{1}$ is assumed to be the lightest
supersymmetric particle (LSP), serving as the SUSY dark matter particle. It is
composed by
$\displaystyle\tilde{\chi}^{0}_{1}=N_{11}(-i\lambda^{\prime})+N_{12}(-i\lambda^{3})+N_{13}\psi_{H_{u}}^{0}+N_{14}\psi_{H_{d}}^{0}+N_{15}\psi_{S},$
(38)
where $N$ is the unitary matrix ($N_{15}$ is zero for the MSSM) to diagonalize
the mass matrix in Eq. (37). For the mass matrices above we should note that
the following two points
1. 1.
For a moderate value of $\kappa$, the neutralino sector of the NMSSM can go
back to the MSSM when $\lambda$ approaches to zero. This is because in such
case the singlino component will become super heavy and decouple from EW
scale. The singlet scalar will not mix with the two Higgs doublet, then the
NMSSM will be almost the same as the MSSM at EW scale.
2. 2.
Since the $\psi_{S}\psi_{S}$ element of Eq. (37) is zero in the nMSSM, the
singlino will not decouple when $\lambda$ approaches to zero. In fact, in the
nMSSM the mass of the LSP can be written as
$\displaystyle m_{\chi_{1}^{0}}\simeq\frac{2\mu_{\rm
eff}\lambda^{2}(v_{u}^{2}+v_{d}^{2})}{2\mu_{\rm
eff}^{2}+\lambda^{2}(v_{u}^{2}+v_{d}^{2})}\frac{\tan\beta}{\tan^{2}\beta+1}.$
(39)
This formula shows that to get a heavy $\tilde{\chi}^{0}_{1}$, we need a large
$\lambda$, a small $\tan\beta$ as well as a moderate $\mu_{\rm eff}$.
The chargino sector of these three models is the same except that in the
NMSSM/nMSSM the parameter $\mu$ is replaced by $\mu_{\rm eff}$. The charginos
$\tilde{\chi}^{\pm}_{1,2}$ ($m_{\chi^{\pm}_{1}}\leq m_{\chi^{\pm}_{2}}$) are
the mixture of charged Higgsinos $\psi_{H_{u,d}}^{\pm}$ and winos
$\lambda^{\pm}=(\lambda^{1}\pm\lambda^{2})/\sqrt{2}$, whose mass matrix in the
basis of $(-i\lambda^{\pm},\psi_{H_{u,d}}^{\pm})$ is given by
$\displaystyle\left(\begin{array}[]{cc}M_{2}&\sqrt{2}m_{W}s_{b}\\\
\sqrt{2}m_{W}c_{b}&\mu_{\rm eff}\end{array}\right).$ (42)
So the chargino $\tilde{\chi}^{\pm}_{1}$ can be wino-dominant (when $M_{2}$ is
much smaller than $\mu$) or higgsino-dominant (when $\mu$ is much smaller than
$M_{2}$). Since the composing property (wino-like, bino-like, higgsino-like or
singlino-like) of the LSP and the chargino $\tilde{\chi}^{\pm}_{1}$ is very
important in SUSY phenomenologies, we will show such a property in our
following study.
## III Comparison with the MSSM and the MSSM with a Singlet sector
### III.1 In light of CDMS II and XENON
First let’s see the MSSM, the NMSSM and the nMSSM under the constraints of
results of CDMS II and XENON100. As both current and future limits of
$\tilde{\chi}$-nucleon of CDMS and XENON are similar to each other, we will
show only one of them. Nevertheless, as a good substitute of the SM, SUSY
model must satisfy all the results of current collider and detector
measurements. In our study we consider the following experimental constraints:
Nakamura:2010zzi (1) we require $\tilde{\chi}^{0}_{1}$ to account for dark
matter relic density $0.105<\Omega h^{2}<0.119$; (2) we require the SUSY
contribution to explain the deviation of the muon $a_{\mu}$, i.e.,
$a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(25.5\pm 8.0)\times 10^{-10}$, at
$2\sigma$ level; (3) the LEP-I bound on the invisible $Z$-decay,
$\Gamma(Z\to\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1})<1.76$ MeV, and the LEP-
II upper bound on
$\sigma(e^{+}e^{-}\to\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{i})$, which is
$5\times 10^{-2}~{}{\rm pb}$ for $i>1$, as well as the lower mass bounds on
sparticles from direct searches at LEP and the Tevatron; (4) the constraints
from the direct search for Higgs bosons at LEP-II, including the decay modes
$h\to h_{1}h_{1},a_{1}a_{1}\to 4f$, which limit all possible channels for the
production of the Higgs bosons; (5) the constraints from $B$ physics
observable such as $B\to X_{s}\gamma$, $B_{s}\to\mu^{+}\mu^{-}$,
$B^{+}\to\tau^{+}\nu$, $\Upsilon\to\gamma a_{1}$, the $a_{1}$–$\eta_{b}$
mixing and the mass difference $\Delta M_{d}$ and $\Delta M_{s}$; (6) the
constraints from the precision EW observable such as $\rho_{\rm lept}$,
$\sin^{2}\theta_{\rm eff}^{\rm lept}$, $m_{W}$ and $R_{b}$; (7) the
constraints from the decay $\Upsilon\to\gamma h_{1}$, and the Tevatron search
for a light Higgs boson via $4\mu$ and $2\mu 2\tau$ signals Dark-Higgs . The
constraints (1–5) have been encoded in the package NMSSMTools NMSSMTools . We
use this package in our calculation and extend it by adding the constraints
(6, 7). As pointed out in Ref. Dark-Higgs , the constraints (7) are important
for a light Higgs boson. In addition to the above experimental limits, we also
consider the constraint from the stability of the Higgs potential, which
requires that the physical vacuum of the Higgs potential with non-vanishing
VEVs of Higgs scalars should be lower than any local minima.
For the calculation of cross section of $\tilde{\chi}$-nucleon scattering, we
use the formulas in Ref. Drees ; susy-dm-review for the MSSM and extend them
to the NMSSM/nMSSM. It is sufficient to consider only the SI interactions
between $\tilde{\chi}^{0}_{1}$ and nucleon (denoted by $f_{p}$ for proton and
$f_{n}$ for neutron susy-dm-review ) in the calculation. The leading order of
these interactions are induced by exchanging the SM-like Higgs boson at tree
level. For moderately light Higgs bosons, $f_{p}$ is approximated by susy-dm-
review (similarly for$f_{n}$)
$\begin{split}f_{p}\simeq\sum_{q=u,d,s}\frac{f_{q}^{H}}{m_{q}}m_{p}f_{T_{q}}^{(p)}+\frac{2}{27}f_{T_{G}}\sum_{q=c,b,t}\frac{f_{q}^{H}}{m_{q}}m_{p}~{},\end{split}$
(43)
where $f_{Tq}^{(p)}$ denotes the fraction of $m_{p}$ (proton mass) from the
light quark $q$ while $f_{T_{G}}=1-\sum_{u,d,s}f_{T_{q}}^{(p)}$ is the heavy
quark contribution through gluon exchange. $f_{q}^{H}$ is the coefficient of
the effective scalar operator. The $\tilde{\chi}^{0}$-nucleus scattering rate
is then given by susy-dm-review
$\sigma^{SI}=\frac{4}{\pi}\left(\frac{m_{\tilde{\chi}^{0}}m_{T}}{m_{\tilde{\chi}^{0}}+m_{T}}\right)^{2}\times\bigl{(}n_{p}f_{p}+n_{n}f_{n}\bigr{)}^{2},$
(44)
where $m_{T}$ is the mass of target nucleus and $n_{p}(n_{n})$ is the number
of proton (neutron) in the target nucleus. In our numerical calculations we
take $f_{T_{u}}^{(n)}=0.023$, $f_{T_{d}}^{(n)}=0.034$,
$f_{T_{u}}^{(p)}=0.019$, $f_{T_{d}}^{(p)}=0.041$ and
$f_{T_{s}}^{(p)}=f_{T_{s}}^{(n)}=0.38$. Note that the scattering rate is very
sensitive to the value of $f_{T_{s}}$ Ellis:2008hf . Recent lattice simulation
lattice gave a much smaller value of $f_{T_{s}}$ (0.020), it reduces the
scattering rate significantly which can be seen in Ref. Cao1 .
Considering all the constraints listed above, we scan over the parameters in
the following ranges
$\displaystyle
100{\rm~{}GeV}\leq\left(M_{\tilde{q}},M_{\tilde{\ell}},~{}m_{A},~{}\mu\right)\leq
1{\rm~{}TeV},$ $\displaystyle 50{\rm~{}GeV}\leq M_{1}\leq
1{\rm~{}TeV},~{}~{}1\leq\tan\beta\leq 40,$
$\displaystyle\left(|\lambda|,|\kappa|\right)\leq 0.7,~{}~{}|A_{\kappa}|\leq
1{\rm~{}TeV},$ (45)
where $M_{\tilde{q}}$ and $M_{\tilde{\ell}}$ are the universal soft mass
parameters of the first two generations of squarks and the three generations
of sleptons respectively. To reduce the number of the relevant soft
parameters, we worked in the so-called $m_{h}^{max}$ scenario with following
choice of the soft masses for the third generation squarks:
$M_{\tilde{Q}_{3}}=M_{\tilde{U}_{3}}=M_{\tilde{D}_{3}}=800$ GeV, and
$X_{t}=A_{t}-\mu\cot\beta=-1600$ GeV. The advantage of such a choice is that
other SUSY parameters more easily survive the constraints (so that the bounds
we obtain are conservative). Moreover, we assume the grand unification
relation for the gaugino masses:
$M_{1}:M_{2}:M_{3}\simeq 1:1.83:5.26~{}.$ (46)
This relation is often assumed in studies of SUSY at the TeV scale for it can
be easily generated in the mSUGRA model Nilles:1983ge . Note that relaxing
this relation will give a large effect on the light neutralino scenario
Feldman:2010ke .
Figure 1: The scatter plots (taken for Ref. Cao:2010fi ) for the spin-
independent elastic cross section of $\tilde{\chi}$-nucleon scattering. The
‘$+$’ points (red) are excluded by CDMS limits (solid line), the ‘$\times$’
(blue) would be further excluded by SuperCDMS 25kg supercdms in case of non-
observation (dash-dotted line), and the ‘$\circ$’ (green) are beyond the
SuperCDMS sensitivity.
The surviving points for the three model are displayed in Fig. 1 for the spin-
independent elastic cross section of $\tilde{\chi}$-nucleon scattering. We see
that for each model the CDMS II limits can exclude a large part of the
parameter space allowed by current collider constraints and the future
SuperCDMS (25 kg) limits can cover most of the allowed parameter space. For
the MSSM and the NMSSM dark matter mass is roughly in range of 50-400 GeV,
while for the nMSSM dark matter mass is constrained below 40 GeV by current
experiments and further constrained below 20GeV by SuperCDMS in case of non-
observation.
Figure 2: Same as Fig. 1, but projected on the plane of $|N_{11}|^{2}$ and
$|N_{15}|^{2}$ versus dark matter mass. (taken for Ref. Cao:2010fi ) Figure 3:
Same as Fig. 1, but showing the chargino mass $m_{\chi^{+}_{1}}$ versus the
LSP mass. The dashed lines indicate $m_{\chi^{+}_{1}}=m_{\chi^{0}_{1}}$.
(taken for Ref. Cao:2010fi )
From Fig. 1, we can see that the $\tilde{\chi}$-nucleon scattering plot of the
MSSM and the NMSSM are very similar to each other, but very different from
nMSSM. This implies that under the experiment constraints, the singlet sector
will decouple from the MSSM-like sector in the NMSSM, then the NMSSM will
perform almost the same as the MSSM, However, the singlet components change EW
scale phenomenology greatly in the nMSSM. This can also be seen in Fig. 2 and
Fig. 3. We can see that for both the MSSM and the NMSSM $\tilde{\chi}_{1}^{0}$
is bino-dominant, while for the nMSSM $\tilde{\chi}_{1}^{0}$ is singlino-
dominant, and the region allowed by CDMS limits (and SuperCDMS limits in case
of non-observation) favors a more bino-like $\tilde{\chi}_{1}^{0}$ for the
MSSM/NMSSM and a more singlino-like $\tilde{\chi}_{1}^{0}$ for the nMSSM. For
the MSSM/NMSSM the LSP lower bound around 50 GeV is from the chargino lower
bound of 103.5 GeV plus the assumed GUT relation $M_{1}\simeq 0.5M_{2}$; while
the upper bound around 400 GeV is from the bino nature of the LSP ($M_{1}$
cannot be too large, must be much smaller than other relevant parameters) plus
the experimental constraints like the muon g-2 and B-physics. If we do not
assume the GUT relation $M_{1}\simeq 0.5M_{2}$, then $M_{1}$ can be as small
as 40 GeV and the LSP lower bound in the MSSM/NMSSM will not be sharply at 50
GeV. (We talk about it in the following section.) For both the MSSM and the
NMSSM, the CDMS limits tend to favor a heavier chargino and ultimately the
SuperCDMS limits tend to favor a wino-dominant chargino with mass about
$2m_{\chi^{0}_{1}}$. Note that, there still can be a singlino dominant LSP in
some parameter space of the NMSSM Belanger:2005kh , but in the scan range Eq.
(45) listed above, getting such singlino dominant LSP needs some fine-tuning,
thus we do not focus on it.
Figure 4: Same as Fig. 1, but projected on the plane of $|\lambda|$ versus the
charged Higgs mass in the NMSSM and the nMSSM. (taken for Ref. Cao:2010fi )
In Fig. 4 we show the value of $|\lambda|$ versus the charged Higgs mass in
the NMSSM and the nMSSM. This figure indicates that $\lambda$ larger than 0.4
is disfavored by the NMSSM. The underlying reason is that
$h_{1}\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$ depends on $\lambda$
explicitly and large $\lambda$ can enhance $\tilde{\chi}$-nucleon scattering
rate. By contrast, although CDMS has excluded some points with large $\lambda$
in the nMSSM, there are still many surviving points with $\lambda$ as large as
0.7. We have talked the reason above: to get a heavy $\chi^{0}_{1}$, one need
a large $\lambda$, a small $\tan\beta$ as well as a moderate $\mu_{\rm eff}$.
Figure 5: Same as Fig. 1, but projected for the decay branching ratio of
$h_{\rm SM}\to\chi^{0}_{1}\chi^{0}_{1}$ versus the mass of the Higgs boson
$h_{\rm SM}$. (taken for Ref. Cao:2010fi )
From the survived parameter space for all the model above, we should know that
the Higgs decay will be similar for the MSSM and the NMSSM, but quite
different from the nMSSM. This can be seen in Fig. 5 which shows decay
branching ratio of $h_{1}\to\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$ versus
the mass of the SM-like Higgs boson $h_{\rm SM}$ ( which is $h_{1}$ here, and
it is Higgs doublet $\hat{H}_{u}$ and $\hat{H}_{d}$ dominant ). Such a decay
is strongly correlated to the $\tilde{\chi}$-nucleon scattering because the
coupling $h_{1}\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$ is involved in both
processes. We see that in the MSSM and the NMSSM this decay mode can open only
in a very narrow parameter space since $\tilde{\chi}_{1}^{0}$ cannot be so
light, and in the allowed region this decay has a very small branching ratio
(below $10\%$). However, in the nMSSM this decay can open in a large part of
the parameter space since the LSP can be very light, and its branching ratio
can be quite large (over $80\%$ or $90\%$).
### III.2 light dark matter in the NMSSM
As talked in the introduction, the data of CoGeNT experiment favors a light
dark matter around 10 GeV. However, we scan the parameter space in the MSSM
and find that it is very difficult to find a neutralino $\tilde{\chi}_{1}^{0}$
lighter than about 28 GeV, unless when it is associated with a light stau as
the next to the lightest supersymmetric particle (NLSP), but such scenario
always needs a fine-tuning in the parameter space Dreiner:2009ic . The main
reason for the absence of a lighter $\tilde{\chi}_{1}^{0}$ is that the
dominant annihilation channel for $\tilde{\chi}_{1}^{0}$ in the early universe
is $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to b\bar{b}$ through $s$-channel
exchange of the pseudoscalar Higgs boson ($A$) and the measured dark matter
relic density requires $m_{A}\sim(90\hbox{--}100)$ GeV and $\tan\beta\sim 50$,
this is in conflict with the constraints from the LEP experiment and $B$
physics MSSM-light ; NMSSM-light ; Belanger . The LHC data gives an even more
stronger constraint on the light pseudoscalar scenario Dermisek:2009fd such
that light dark matter seems impossible in the MSSM. Though in the nMSSM the
neutralino $\tilde{\chi}_{1}^{0}$ can be as light as 10 GeV (shown in Fig. 1),
the scattering rate is much lower under the CoGeNT-favored region. In the
NMSSM, however, with the participation of singlet sector one can get very
light NMSSM Higgs. This feature is particularly useful for light
$\tilde{\chi}_{1}^{0}$ scenario since it opens up new important annihilation
channels for $\tilde{\chi}_{1}^{0}$, i.e., either into a pair of $h_{1}$ (or
$a_{1}$) or into a pair of fermions via $s$-channel exchange of $h_{1}$ (or
$a_{1}$) Belanger ; light-anni ; Cao-nMSSM . For the former case,
$\tilde{\chi}_{1}^{0}$ must be heavier than $h_{1}$ ($a_{1}$); while for the
latter case, due to the very weak couplings of $h_{1}$ ($a_{1}$) with
$\tilde{\chi}_{1}^{0}$ and with the SM fermions, a resonance enhancement (i.e.
$m_{h_{1}}$ or $m_{a_{1}}$ must be close to $2m_{\tilde{\chi}_{1}^{0}}$) is
needed to accelerate the annihilation. So a light $\tilde{\chi}_{1}^{0}$ may
be necessarily accompanied by a light $h_{1}$ or $a_{1}$ to provide the
required dark matter relic density. From the discussion in the upper section,
light $\tilde{\chi}_{1}^{0}$ can be obtained by releasing the GUT relation Eq.
(46), thus LSP in the NMSSM may explain the detection of CoGeNT. Note that, as
the LSP in the nMSSM is singlino dominant, relaxing the GUT relation will not
the change the phenomenology of dark matter and Higgs too much.
Figure 6: The scatter plots (taken for Ref. Cao:2011re ) of the parameter
samples which survive all constraints, with ‘$\times$’ (red) and
‘$\blacktriangle$’ (green) corresponding to a light $h_{1}$ and a light
$a_{1}$, respectively. The left frame is projected on the $\sigma^{\rm
SI}$-$m_{\chi}$ plane, while the right frame is projected on the $\sigma^{\rm
SI}$-$m_{h_{1}}$ plane (denoted by ‘$\times$’) and the $\sigma^{\rm
SI}$-$m_{a_{1}}$ plane (denoted by ‘$\blacktriangle$’). The curves are the
limits from CoGeNT CoGeNT , CDMS CDMSII , while the contour is the CoGeNT-
favored region CoGeNT .
Now we discuss how to get a light $h_{1}$ or $a_{1}$ in the NMSSM. A light
$a_{1}$ can be easily obtained when the theory is close to the U(1)R or U(1)PQ
symmetry limit, which can be realized by setting the product $\kappa
A_{\kappa}$ to be negatively small NMSSM . In contrast, a light $h_{1}$ can
not be obtained easily. However, as shown below, it can still be achieved by
somewhat subtle cancelation via tuning the value of $A_{\kappa}$. We note that
for any theory with multiple Higgs fields, the existence of a massless Higgs
boson implies the vanishing of the determinant of its squared mass matrix and
vice versa. For the NMSSM, at tree level the parameter $A_{\kappa}$ only
enters the mass term of the singlet Higgs bosons, so the determinant
($\mathop{\rm Det}{\cal{M}}^{2}$) of the mass matrix of the CP-even Higgs
bosons depends on $A_{\kappa}$ linearly NMSSM . When other relevant parameters
are fixed, one can then obtain a light $h_{1}$ by varying $A_{\kappa}$ around
the value $\tilde{A}_{\kappa}$ which is the solution to the equation
$\mathop{\rm Det}{\cal{M}}^{2}=0$. In practice, one must include the important
radiative corrections to the Higgs mass matrix, which will complicate the
dependence of ${\cal{M}}^{2}$ on $A_{\kappa}$. However, we checked that the
linear dependence is approximately maintained by choosing the other relevant
parameters at the SUSY scale, and one can solve the equation iteratively to
get the solution $\tilde{A}_{\kappa}$.
In Fig. 6 we display the surviving parameter samples, showing the
$\tilde{\chi}$-nucleon scattering cross section versus the neutralino dark
matter mass (left frame) and versus the mass of $h_{1}$ or $a_{1}$ (right
frame). It shows that the scattering rate of the light dark matter can reach
the sensitivity of CDMS and, consequently, a sizable parameter space is
excluded by the CDMS data supercdms . The future CDMS experiment can further
explore (but cannot completely cover) the remained parameter space. Note that
in the light-$h_{1}$ case the scattering rate can be large enough to reach the
sensitivity of CoGeNT and to cover the CoGeNT-favored region. The underlying
reason is that the $\tilde{\chi}$-nucleon scattering can proceed through the
$t$-channel exchange of the CP-even Higgs bosons, which can be enhanced by a
factor $1/m_{h_{1}}^{4}$ for a light $h_{1}$ light-anni ; while a light
$a_{1}$ can not give such an enhancement because the CP-odd Higgs bosons do
not contribute to the scattering in this way. We noticed that the studies in
NMSSM-light ; Das-light claimed that the NMSSM is unable to explain the
CoGeNT data because they did not consider the light-$h_{1}$ case.
Figure 7: Same as Fig. 6, but showing the decay branching ratios of the SM-
like Higgs boson $h_{\rm SM}$. Here $Br(h_{\rm
SM}\to\tilde{\chi}_{i}^{0}\tilde{\chi}_{j}^{0})$ denotes the total rates for
all possible $h_{\rm SM}\to\tilde{\chi}_{i}^{0}\tilde{\chi}_{j}^{0}$ decays.
(taken for Ref. Cao:2011re )
In the light $\tilde{\chi}_{1}^{0}$ scenario, $h_{\rm SM}$ may decay
exotically into $\tilde{\chi}_{i}^{0}\tilde{\chi}_{j}^{0}$, $h_{1}h_{1}$ or
$a_{1}a_{1}$, and consequently the conventional decays are reduced. This
feature is illustrated in Fig. 7, which shows that the sum of the exotic decay
branching ratios may exceed $50\%$ and the traditional decays $h_{\rm SM}\to
b\bar{b},\tau\bar{\tau},WW^{\ast},\gamma\gamma$ can be severely suppressed.
Numerically, we find that the branching ratio of $h_{\rm SM}\to b\bar{b}$ is
suppressed to be below $30\%$ for all the surviving samples in the
light-$h_{1}$ ($h_{2}$ is $h_{\rm SM}$) case and for about $96\%$ of the
surviving samples in the light-$a_{1}$ ($h_{1}$ is $h_{\rm SM}$) case (for the
remaining $4\%$ of the surviving samples in the light-$a_{1}$ case, the decay
$h_{\rm SM}\to a_{1}a_{1}$ is usually kinematically forbidden so that the
ratio of $h_{\rm SM}\to b\bar{b}$ may exceed $60\%$). Another interesting
feature shown in Fig. 7 is that, due to the open-up of the exotic decays,
$h_{\rm SM}$ may be significantly lighter than the LEP bound. This situation
is favored by the fit of the precision electro-weak data and is of great
theoretical interest Gunion .
Figure 8: Same as Fig. 6, but showing the diphoton production rate of the SM-
like Higgs boson at the LHC.
Since the conventional decay modes of $h_{\rm SM}$ may be greatly suppressed,
especially in the light-$h_{1}$ case which can give a rather large
$\tilde{\chi}$-nucleon scattering rate, the LHC search for $h_{\rm SM}$ via
the traditional channels may become difficult. Now the LHC observed a new
particle in the mass region around 125-126 GeV which is the most probable the
long sought Higgs boson cern . In this mass range, the most important
discovering channel of $h_{\rm SM}$ at the LHC is the di-photon signal. In
Fig. 8 we give the ratio of the di-photon production rate to the SM at the LHC
with $\sqrt{s}=7$ TeV. In calculating the rate, we used the narrow width
approximation and only considered the leading contributions to $pp\to h_{\rm
SM}$ from top quark, bottom quark and the squark loops.
Fig. 8 indicates that, compared with the SM prediction, the ratio in the NMSSM
in the light $\tilde{\chi}_{1}^{0}$ scenario is suppressed to be less than 0.4
for the light-$h_{1}$ case. For the light-$a_{1}$ case, most samples (about
$96\%$) predict the same conclusion. Since in the light-$h_{1}$ case the
$\tilde{\chi}$-nucleon scattering rate can reach the CoGeNT sensitivity, this
means that in the framework of the NMSSM the CoGeNT search for the light dark
matter will be correlated with the LHC search for the Higgs boson via the di-
photon channel. We checked that, once the future XENON experiment fails in
observing dark matter, less than $1\%$ of the surviving samples in light
$a_{1}$ case predict the ratio of di-photon signal larger than 0.4.
## IV General extension for the explanation to PAMELA
To explain the PAMELA excess by dark matter annihilation, there are some
challenges. First, dark matter must annihilate dominantly into leptons since
PAMELA has observed no excess of anti-protons pamela (However, as pointed in
Ref. kane , this statement may be not so solid due to the significant
astrophysical uncertainties associated with their propagation). Second, the
explanation of PAMELA excess requires an annihilation rate which is too large
to explain the relic abundance if dark matter is produced thermally in the
early universe. To tackle these difficulties, a new theory of dark matter was
proposed in Ref. sommerfeld2 . In this new theory the Sommerfeld effect of a
new force in the dark sector can greatly enhance the annihilation rate when
the velocity of dark matter is much smaller than the velocity at freeze-out in
the early universe, and dark matter annihilates into light particles which are
kinematically allowed to decay to muons or electrons.
The above fancy idea is hard to realize in the MSSM, because there is not a
new force in the neutralino dark matter sector to induce the Sommerfeld
enhancement and neutralino dark matter annihilates largely to final states
consisting of heavy quarks or gauge and/or Higgs bosons susy-dm-review ; neu .
However, as discussed in Ref. Hooper:2009gm , in a general extension of the
MSSM by introducing a singlet Higgs superfield, the idea in Ref. sommerfeld2
can be realized by the singlino-like neutralino dark matter:
* (i)
The singlino dark matter annihilates to the light singlet Higgs bosons and the
relic density can be naturally obtained from the interaction between singlino
and singlet Higgs bosons.
* (ii)
The singlet Higgs bosons, not related to electro-weak symmetry breaking, can
be light enough to be kinematically allowed to decay dominantly into muons or
electrons through the tiny mixing with the Higgs doublets.
* (iii)
The Sommerfeld enhancement needed in dark matter annihilation for the
explanation of PAMELA result can be induced by the light singlet Higgs boson.
In the following section, we will show how does this happen, the Higgs decay
are also investigated.
### IV.1 Higgs and neutralinos spectrum
If introduce a singlet Higgs to the MSSM in general, the renormalizable
holomorphic superpotential of Higgs is given by Ref. Hooper:2009gm
$\displaystyle
W=\mu\widehat{H}_{u}\cdot\widehat{H}_{d}+\lambda\widehat{S}\widehat{H}_{u}\cdot\widehat{H}_{d}+\eta\widehat{S}+\frac{1}{2}\,\mu_{s}\widehat{S}^{2}+\frac{1}{3}\kappa\widehat{S}^{3}\
,$ (47)
which include linear term, quadratic term, cubic term of singlet superfield
(like Wess-Zumino model wzmodel ). Note that in such case, we do not require
the singlet to solve the $\mu$ problem. The soft SUSY-breaking terms are given
by
$\displaystyle V_{\rm soft}$ $\displaystyle=$
$\displaystyle\tilde{m}_{u}^{2}|H_{u}|^{2}+\tilde{m}_{d}^{2}|H_{d}|^{2}+\tilde{m}_{s}^{2}|S|^{2}$
(48) $\displaystyle+(B\mu H_{u}\cdot H_{d}+\lambda A_{\lambda}\ H_{u}\cdot
H_{d}S+C\eta S+\frac{1}{2}\,B_{s}\mu_{s}S^{2}+\frac{1}{3}\,\kappa A_{\kappa}\
S^{3}+\mathrm{h.c.})\,.$
After the Higgs fields develop the VEVs $v_{u}$, $v_{d}$ and $s$, i.e., we get
the similar Higgs spectrum as the NMSSM and the nMSSM which is
* (1)
The CP-even Higgs mass matrix in the basis $(\phi_{u},\phi_{d},\sigma)$ is
given by
$\displaystyle{\cal M}_{h,11}$ $\displaystyle=$ $\displaystyle
g^{2}v_{u}^{2}+\cot\beta\left[\lambda s(A_{\lambda}+\kappa
s+\mu_{s})+B\mu\right],$ (49) $\displaystyle{\cal M}_{h,22}$ $\displaystyle=$
$\displaystyle g^{2}v_{d}^{2}+\tan\beta\left[\lambda s(A_{\lambda}+\kappa
s+\mu_{s})+B\mu\right],$ (50) $\displaystyle{\cal M}_{h,33}$ $\displaystyle=$
$\displaystyle\lambda(A_{\lambda}+\mu_{s})\frac{v_{u}v_{d}}{s}\,-\lambda\frac{\mu}{s}(v_{u}^{2}+v_{d}^{2})+\kappa
s(A_{\kappa}+4\kappa s+3\mu_{s})-\frac{C\eta}{s},$ (51) $\displaystyle{\cal
M}_{h,12}$ $\displaystyle=$
$\displaystyle(2\lambda^{2}-g^{2})v_{u}v_{d}-\lambda s(A_{\lambda}+\kappa
s+\mu_{s})-B\mu,$ (52) $\displaystyle{\cal M}_{h,13}$ $\displaystyle=$
$\displaystyle 2\lambda(\mu+\lambda s)v_{u}-\lambda v_{d}(A_{\lambda}+2\kappa
s+\mu_{s}),$ (53) $\displaystyle{\cal M}_{h,23}$ $\displaystyle=$
$\displaystyle 2\lambda(\mu+\lambda s)v_{d}-\lambda v_{u}(A_{\lambda}+2\kappa
s+\mu_{s}),$ (54)
where $g^{2}=(g_{1}^{2}+g_{2}^{2})/2$ with $g_{1}$ and $g_{2}$ being
respectively the coupling constant of SU(2) and U(1) in the SM.
* (2)
The CP-odd Higgs mass matrix ${\cal M}_{a}$ is given by
$\displaystyle{\cal M}_{a,11}$ $\displaystyle=$
$\displaystyle(\tan\beta+\cot\beta)[\lambda s(A_{\lambda}+\kappa
s+\mu_{s})+B\mu],$ (55) $\displaystyle{\cal M}_{a,22}$ $\displaystyle=$
$\displaystyle 4\lambda\kappa
v_{u}v_{d}+\lambda(A_{\lambda}+\mu_{s})\frac{v_{u}v_{d}}{s}-\lambda\frac{\mu}{s}(v_{u}^{2}+v_{d}^{2})$
(56) $\displaystyle-\kappa
s(3A_{\kappa}+\mu_{s})-\frac{C\eta}{s}-2B_{s}\mu_{s},$ $\displaystyle{\cal
M}_{a,12}$ $\displaystyle=$
$\displaystyle\lambda\sqrt{v_{u}^{2}+v_{d}^{2}}\,(A_{\lambda}-2\kappa
s-\mu_{s}).$ (57)
Note that here we have dropped the Goldstone mode, thus there left a $2\times
2$ mass matrix in the basis ($\tilde{A},\xi$). and it can be diagonalized by
an orthogonal $2\times 2$ matrix $P^{\prime}$ and the physical CP-odd states
$a_{i}$ are given by (ordered as $m_{a_{1}}<m_{a_{2}}$)
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
P_{11}^{\prime}\tilde{A}+P_{12}^{\prime}S_{I}=P_{11}^{\prime}(\cos\beta\varphi_{u}+\sin\beta\varphi_{d})+P_{12}^{\prime}\xi,$
(58) $\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle
P_{21}^{\prime}\tilde{A}+P_{22}^{\prime}S_{I}=P_{21}^{\prime}(\cos\beta\varphi_{u}+\sin\beta\varphi_{d})+P_{22}^{\prime}\xi.$
(59)
* (3)
The charged Higgs mass matrix ${\cal M}_{\pm}$ in the basis
$\left(H_{u}^{+},H^{+}_{d}\right)$ is given by
${\cal M}_{\pm}=\left(\lambda s(A_{\lambda}+\kappa
s+\mu_{s})+B\mu+h_{u}h_{d}(\frac{g_{2}^{2}}{2}-\lambda^{2})\right)\left(\begin{array}[]{cc}\cot\beta&1\\\
1&\tan\beta\end{array}\right),$ (60)
* (4)
The neutralino mass matrix is :
${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&m_{Z}s_{W}s_{b}&-m_{Z}s_{W}c_{b}&0\\\
0&M_{2}&-m_{Z}c_{W}s_{b}&m_{Z}c_{W}c_{b}&0\\\
m_{Z}s_{W}s_{b}&-m_{Z}s_{W}s_{b}&0&-\mu&-\lambda vc_{b}\\\
-m_{Z}s_{W}c_{b}&-m_{Z}c_{W}c_{b}&-\mu&0&-\lambda vs_{b}\\\ 0&0&-\lambda
vc_{b}&-\lambda vs_{b}&2\kappa s+\mu_{s}\end{array}\right).$ (61)
### IV.2 Explanation of PAMELA and implication on Higgs decays
To explain the observation of PAMELA, $a_{1}$ is singlet-dominant, while
$h_{1}$ is singlet-dominant and the next-to-lightest $h_{2}$ is doublet-
dominant ($h_{\rm SM}$). We use the notation:
$a\equiv a_{1},~{}~{}~{}~{}h\equiv h_{1},~{}~{}~{}~{}h_{\rm SM}\equiv h_{2}.$
(62)
As discussed in Ref. Hooper:2009gm , when the lightest neutralino
$\tilde{\chi}^{0}_{1}$ in Eq. (38) is singlino-dominant, it can be a perfect
candidate for dark matter. As shown in Fig. 9, such singlino dark matter
annihilates to a pair of light singlet Higgs bosons followed by the decay
$h\to aa$ ($h$ has very small mixing with the Higgs doublets and thus has very
small couplings to the SM fermions). In order to decay dominantly into muons,
$a$ must be light enough. Further, in order to induce the Sommerfeld
enhancement, $h$ must also be light enough. From the superpotential term
$\kappa\hat{S}^{3}$ we know that the couplings
$h\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$ and
$a\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$ are proportional to $\kappa$. To
obtain the relic density of dark matter, $\kappa$ should be ${\cal O}(1)$.
$h,a$ are singlet-dominant and $\tilde{\chi}^{0}_{1}$ is singlino-dominant,
this implies small mixing between singlet and doublet Higgs fields. From the
superpotential in Eq.(47) we see that this means the mixing parameter
$\lambda$ must be small enough. On the other hand, from Eq. (51) and Eq. (56)
lightness of $h_{1}$ and $a_{1}$ also require $\lambda$ and other term
approaching to zero. Therefore, in our scan we require parameters $A_{\kappa}$
and $B_{s}$ has the relation:
$\displaystyle A_{\kappa}$ $\displaystyle\sim$ $\displaystyle\left(-4\kappa
s-3\mu_{s}+\frac{C\eta}{\kappa s^{2}}\right),$ (63) $\displaystyle
2B_{s}\mu_{s}$ $\displaystyle\sim$ $\displaystyle\left(-3A_{\kappa}\kappa
s-\mu_{s}\kappa s-\frac{C\eta}{s}\right),$ (64)
to realize light $h_{1}$ and $a_{1}$.
Figure 9: Feynman diagrams for singlino dark matter annihilation where
Sommerfeld enhancement is induced by exchanging $h$. (taken from Ref.
Wang:2009rj ) Figure 10: The scatter plots showing the decay branching ratios
$a\to\mu^{+}\mu^{-}$ (muon), $a\to gg$ (gluon) and $a\to s\bar{s}$ ($s$-quark)
versus $m_{a}$ for $\lambda=10^{-3}$. (taken from Ref. Wang:2009rj ) Figure
11: Same as Fig. 10, but showing the Sommerfeld enhancement factor induced by
$h$. (taken from Ref. Wang:2009rj )
The numerical results of this model are displayed in different planes in
Figs.10-12. We see from Fig. 10 that in the range $2m_{\mu}<m_{a}<2m_{\pi}$,
$a$ decays dominantly into muons. It is clear that $h$ can be as light as a
few GeV, which is light enough to induce the necessary Sommerfeld enhancement
as shown in Fig. 11.
Figure 12: Same as Fig. 10, but showing branching ratio of $h_{\rm SM}\to
aa,hh$ versus $m_{h_{\rm sm}}$ and $|A_{\kappa}|$ versus the branching ratio
of $h_{\rm SM}\to aa,hh$. (taken from Ref. Wang:2009rj )
In left plot of Fig. 12, we show the branching ratios of $h_{\rm SM}$ decays.
We see that in the allowed parameter space $h_{\rm SM}$ tends to decay into
$aa$ or $hh$ instead of $b\bar{b}$. This can be understood as following, the
MSSM parameter space is stringently constrained by the LEP experiments if
$h_{\rm SM}$ is relatively light and decays dominantly to $b\bar{b}$, and to
escape such stringent constraints $h_{\rm SM}$ tends to have exotic decays
into $aa$ or $hh$. As a result, the allowed parameter space tends to favor a
large $A_{\kappa}$, as shown in right plot of Fig. 12, which greatly enhances
the couplings $h_{\rm SM}aa$ and $h_{\rm SM}hh$ through the soft term $\kappa
A_{\kappa}S^{3}$ although $S$ has a small mixing with the doublet Higgs
bosons. Such an enhancement can be easily seen. Take the coupling $h_{\rm
SM}hh$ as an example, the soft term $\kappa A_{\kappa}S^{3}$ gives a term
$\kappa A_{\kappa}\sigma^{3}$ which then gives the interaction $\kappa
A_{\kappa}~{}(U_{13}^{H})^{2}U^{H}_{23}~{}h_{\rm SM}hh$ because
$\sigma=U^{H}_{13}h_{1}+U^{H}_{23}h_{2}+U^{H}_{33}h_{3}$ with $h_{1}\equiv h$
and $h_{2}\equiv h_{\rm SM}$ (see Eqs. (29) and (62)). Although the mixing
$(U_{13}^{H})^{2}U^{H}_{23}$ is small for a small $\lambda$, a large
$A_{\kappa}$ can enhance the coupling $h_{\rm SM}hh$. Note that as the mass of
the observed Higgs boson at the LHC is around 125 GeV, thus in the MSSM, the
dominant decay mode of $h_{\rm SM}$ is $b\bar{b}$. In this general singlet
extension of the MSSM, its dominant decay mode may be changed to $aa$ or $hh$,
as shown in our above results.
Finally, we note that for the specified singlet extensions like the nMSSM and
the NMSSM, the explanation of PAMELA and relic density through Sommerfeld
enhancement is not possible. The reason is that the parameter space of such
models is stringently constrained by various experiments and dark matter relic
density as shown in the above section, and, as a result, the neutralino dark
matter may explain either the relic density or PAMELA, but impossible to
explain both via Sommerfeld enhancement. For example, in the nMSSM various
experiments and dark matter relic density constrain the neutralino dark matter
particle in a narrow mass range dm-nmssm , which is too light to explain
PAMELA.
## V Summary
At last we summarize here, the SUSY dark matter and Higgs physics will be
changed if introducing a singlet to the MSSM. Under the latest results of dark
matter detection, we have:
1. 1.
In the MSSM, the NMSSM and the nMSSM, the latest detection result can exclude
a large part of the parameter space allowed by current collider constraints
and the future SuperCDMS and XENON can cover most of the allowed parameter
space.
2. 2.
Under the new dark matter constraints, the singlet sector will decouple from
the MSSM-like sector in the NMSSM, thus the phenomenologies of dark matter and
Higgs are similar to the MSSM. The singlet sector make the nMSSM quite
different from the MSSM, the LSP in the nMSSM are singlet dominant, and the
SM-like Higgs will mainly decay into the singlet sector. Future precision
measurements will give us an opportunity to determine whether the new scalar
is from standard model or from SUSY. Perhaps the nMSSM will be the first model
be excluded for its much larger branching ratio of invisible Higgs decay.
3. 3.
The NMSSM can allow light dark matter at several GeV exists. Light CP-even or
CP-odd Higgs boson must be present so as to satisfy the measured dark matter
relic density. In case of the presence of a light CP-even Higgs boson, the
light neutralino dark matter can explain the CoGeNT and DAMA/LIBRA results.
Further, we find that in such a scenario the SM-like Higgs boson will decay
predominantly into a pair of light Higgs bosons or a pair of neutralinos and
the conventional decay modes will be greatly suppressed.
4. 4.
The general singlet extension of the MSSM gives a perfect explanation for both
the relic density and the PAMELA result through the Sommerfeld enhanced
annihilation into singlet Higgs bosons ($a$ or $h$ followed by $h\to aa$) with
$a$ being light enough to decay dominantly to muons or electrons. Although the
light singlet Higgs bosons have small mixing with the Higgs doublets in the
allowed parameter space, their couplings with the SM-like Higgs boson $h_{SM}$
can be enhanced by the soft parameter $A_{\kappa}$. In order to meet the
stringent LEP constraints, the $h_{SM}$ tends to decay into the singlet Higgs
pairs $aa$ or $hh$ instead of $b\bar{b}$.
## Acknowledgment
This work was supported in part by the NSFC No. 11005006, No. 11172008 and
Doctor Foundation of BJUT No. X0006015201102.
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|
arxiv-papers
| 2012-05-23T01:15:48 |
2024-09-04T02:49:31.216561
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenyu Wang",
"submitter": "Wenyu Wang",
"url": "https://arxiv.org/abs/1205.5081"
}
|
1205.5168
|
# A Determination of the Intergalactic Redshift Dependent UV-Optical-NIR
Photon Density Using Deep Galaxy Survey Data and the Gamma-ray Opacity of the
Universe
Floyd W. Stecker Astrophysics Science Division, NASA/Goddard Space Flight
Center Greenbelt, MD 20771; Floyd.W.Stecker@nasa.gov Department of Physics
and Astronomy, University of California, Los Angeles Los Angeles,
CA90095-1547 Matthew A. Malkan Department of Physics and Astronomy,
University of California, Los Angeles Los Angeles, CA90095-1547;
malkan@astro.ucla.edu Sean T. Scully Department of Physics, James Madison
University Harrisonburg, VA 22807; scullyst@jmu.edu
###### Abstract
We calculate the intensity and photon spectrum of the intergalactic background
light (IBL) as a function of redshift using an approach based on observational
data obtained in many different wavelength bands from local to deep galaxy
surveys. This allows us to obtain an empirical determination of the IBL and to
quantify its observationally based uncertainties. Using our results on the
IBL, we then place 68% confidence upper and lower limits on the opacity of the
universe to $\gamma$-rays, free of the theoretical assumptions that were
needed for past calculations. We compare our results with measurements of the
extragalactic background light and upper limits obtained from observations
made by the Fermi Gamma-ray Space Telescope.
diffuse radiation – galaxies:observations – gamma rays:theory
## 1 Introduction
### 1.1 Empirical Approach to Determining the Intergalactic Background
Radiation
The purpose of this paper is to present the results of a new, fully empirical
approach to calculating the intergalactic background light (IBL) as well as
the $\gamma$-ray opacity of the Universe. This methodology, hitherto
unavailable, is now enabled by very recent data from deep galaxy surveys
spanning the electromagnetic spectrum from millimeter to UV wavelengths and
using galaxy luminosity functions for redshifts $0\leq z\leq 8$ in the UV and
for redshifts up to 2 or 3 in other wavelength ranges. We stress that this
approach is both capable of delineating empirically based uncertainties on the
determination of the IBL, and the $\gamma$-ray opacity of the Universe.
In this paper (Paper I) we specifically consider the frequency range from the
far ultraviolet (FUV) to the near infrared $I$ band (NIR), as this range is of
particular relevance to the $\gamma$-ray opacity studies in the $\sim$0.1-200
GeV energy range being made by the Fermi $\gamma$-ray space telescope. A
follow-up paper (Paper II) will address the frequency range from the NIR to
the far-IR (FIR). That range has particular relevance for opacity studies by
ground-based air Čerenkov telescopes.
Previous calculations the IBL at different redshifts have been based on
various theoretical models and assumptions. These include backward evolution
models (Malkan & Stecker 1998, 2001; Stecker, Malkan & Scully 2006;
Franceschini et al. 2008), semi-analytical forward evolution models (e.g.,
Gilmore et al. 2009; Somerville et al. 2011) and other models based on the
evolution of galaxy parameters such as star formation rate and stellar
population synthesis models (Salamon & Stecker 1998 (hereafter SS98); Kneiske
et al. 2004). Kneiske & Dole (2010) have recently used a forward evolution
model to derive lower limits on the EBL. Finke, Razzaque & Dermer (2010)
employed a triple blackbody approximation to extimate the EBL. Domínguez et
al. (2011) used an approach based on the redshift evolution of the $K$-band
galaxy luminosity functions (LFs) derived by Cirasuolo et al. (2010), together
with model templates based on Spitzer-based $0.2\leq z\leq 1$ infrared galaxy
SEDs and AEGIS data. To obtain $K$-band LFs for $1<z<4$, Cirasuolo et al.
(2010) used 8 $\mu$m Spitzer/IRAC (Infrared Array Camera) channels combined
with population synthesis models of Bruzual & Charlot (2003), including a
correction for dust obscuration. Most recently, a semi-analytic model of the
EBL has been published by Gilmore et el. (2012). The earlier exploration of
the EBL using direct measurements, galaxy counts, and indirect constraints was
reviewed some time ago by Hauser & Dwek (2001).
We note that previous studies had to adopt at least some assumptions about how
galaxy LFs evolves with cosmic time, starting either at the present (well-
measured epoch) and going back in time, or starting with the simulations of
the galaxy formation epoch using semi-analytic models (see above) or modeled
galaxy SEDs. However, the latest observations have become sufficiently
extensive and accurate to allow direct integration of observational data on
galaxy LFs from the deep galaxy surveys at many wavelengths, where we can
interpolate between observationally determined LFs at many wavelengths from
the far UV to near infrared and the redshift range extending in the UV from
$z=0$ to $z\geq 8$. Thus, the first goal of our paper is to determine the IBL
based on empirical data from deep survey galaxy observations. This avoids the
complications entailed by theoretical calculations that have need of making
various assumptions for stellar population synthesis models, stellar initial
mass functions, unknown amounts of dust extinction, and poorly known stellar
metallicity-age modeling for different evolving galaxy types (e.g., Wilkins et
al. 2012). This is because the observational data are the direct result of all
of the physical processes involved in producing galactic emission. Thus our
treatment only involves uncertainties inherent in the analyses discussed in
the observational survey papers that we used.
### 1.2 Gamma Ray Opacity and the IBL
The second goal of our paper is to use our results on the IBL to determine the
$\gamma$-ray opacity of the universe as a function of energy and redshift. It
was first suggested by Stecker, De Jager & Salamon (1992) that $\gamma$-ray
observations from high redshift sources such as blazars (and later
$\gamma$-ray bursts) could be used to probe the IBL. Such studies make use of
the opacity caused by the annihilation of $\gamma$-rays owing to interactions
with low energy photons that produce $e^{+}e^{-}$ pairs. The Fermi Gamma-Ray
Space Telescope (Fermi) is now being used to probe the high redshift IBL at
optical and UV wavelengths by constraining the opacity of the universe to
multi-GeV $\gamma$-rays (Abdo et al. 2010). This is accomplished by measuring
the energy of the highest energy photons observed by Fermi that have been
emitted by GRBs and blazars at known redshifts.
Observations of TeV $\gamma$-ray emitting blazars utilizing modern air
Čerenkov telescope arrays also probe, or at least constrain, the nearby
(redshift $z\sim 0-0.5$) intergalactic infrared background radiation. Attempts
to constrain the IBL have been made by various authors (Stecker & de Jager
1993; Aharonian et al. 2006 (but see Stecker, Baring & Summerlin 2009); Mazin
& Raue 2007; Georganopoulos, Fincke & Reyes 2010; Abdo et al. 2010; Orr,
Krennrich & Dwek 2011, but see Stecker, Baring & Summerlin 2009).
Our methodology will also be used to define secure upper and lower limits on
the opacity of the universe to high energy $\gamma$-rays based on the
observational uncertainties in the deep survey data. We then compare the
opacity range defined by these limits with the upper limits derived using the
Fermi observations of multi-GeV $\gamma$-rays from high redshift sources Abdo,
et al (2010).
## 2 Intergalactic Photon Energy Densities and Emissivities from Galaxies
The co-moving radiation energy density $u_{\nu}(z)$ is derived from the co-
moving specific emissivity ${\cal E}_{\nu}(z)$, which, in turn is derived from
the galaxy luminosity function (LF). The galaxy luminosity function,
$\Phi_{\nu}(L)$, is defined as the distribution function of galaxy
luminosities at a specific frequency or wavelength. The specific emissivity at
frequency $\nu$ and redshift $z$ (also referred to in the literature as the
luminosity density, $\rho_{L_{\nu}}$) , is the integral over the luminosity
function
${\cal E}_{\nu}(z)=\int_{L_{min}}^{L_{max}}dL_{\nu}\,L_{\nu}\Phi(L_{\nu};z)$
(1)
There are many references in the literature where the LF is given and fit to
Schechter parameters, but where $\rho_{L_{\nu}}$ is not given. In those cases,
we could not determine the covariance of the errors in the Schechter
parameters used to determine the dominant statistical errors in their
analyses. Thus, we could not ourselves accurately determine the error on the
emissivity from equation (1). We therefore chose to use only the papers that
gave values for $\rho_{L_{\nu}}(z)={\cal E}_{\nu}(z)$ with errors. We did not
consider cosmic variance, but this uncertainly should be minimized since we
used data from many surveys.
In compiling the observational data on ${\cal E}_{\nu}(z)$, we scaled all of
the results to a value of h = 0.7. Thus results using h = 0.5 were scaled by a
factor of (7/5)111Using the most recent and accurate value of 0.74 (Riess et
al. 2011) would increase all of our results by $\sim 6\%$.
The co-moving radiation energy density $u_{\nu}(z)$ is the time integral of
the co-moving specific emissivity ${\cal E}_{\nu}(z)$,
$u_{\nu}(z)=\int_{z}^{z_{\rm max}}dz^{\prime}\,{\cal
E}_{\nu^{\prime}}(z^{\prime})\frac{dt}{dz}(z^{\prime})e^{-\tau_{\rm
eff}(\nu,z,z^{\prime})},$ (2)
where $\nu^{\prime}=\nu(1+z^{\prime})/(1+z)$ and $z_{\rm max}$ is the redshift
corresponding to initial galaxy formation (Salamon & Stecker 1998, hereafter
SS98), and
$\frac{dt}{dz}{(z)}={[H_{0}(1+z)\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+z)^{3}}}]^{-1},$
(3)
with $\Omega_{\Lambda}=0.72$ and $\Omega_{m}=0.28$.
The opacity factor for frequencies below the Lyman limit is dominated by dust
extinction. In the model of SS98, which relied on the population synthesis
studies of Bruzual & Charlot (1993), dust absorption was not included. Our
earlier paper (Stecker, Malkan & Scully 2006) used a rough approximation of
the results obtained by Salamon & Stecker (1998) (SS98) and therefore, also
did not take dust absorption into account. However, since we are here using
actual observations of galaxies rather than models, dust absorption is
implicitly included. The remaining opacity $\tau_{\nu}$ refers to the
extinction of ionizing photons with frequencies above the rest frame Lyman
limit of $\nu_{LyL}\equiv 3.29\times 10^{15}$ Hz by interstellar and
intergalactic hydrogen and helium. It has been shown that this opacity is very
high, corresponding to the expectation of very small fraction of ionizing
radiation in intergalactic space compared with radiation below the Lyman limit
(Lytherer et al.1995; SS98). In fact, the Lyman limit cutoff is used as a
tool; when galaxies disappear when using a filter at a given waveband (e.g.,
”$U$-dropouts”, ”$V$-dropouts”) it is an indication of the redshift of the
Lyman limit. We thus replace equation (2) with the following expression
$u_{\nu}(z)=\int_{z}^{z_{\rm max}}dz^{\prime}\,{\cal
E}_{\nu^{\prime}}(z^{\prime})\frac{dt}{dz}(z^{\prime}){{\cal
H}(\nu(z^{\prime})-\nu^{\prime}_{LyL})},$ (4)
where ${\cal H}(x)$ is the Heavyside step function.
### 2.1 Empirical Specific Emissivities
#### 2.1.1 Luminosity Densities
We have used the results of many galaxy surveys to compile a set of luminosity
densities, $\rho_{L_{\nu}}(z)={\cal E}_{\nu}(z)$ (LDs), at all observed
redshifts, and at rest-frame wavelengths from the far-ultraviolet, FUV = 150
nm to the $I$ band, $I$ = 800 nm. Figure 1 shows the redshift evolution of the
luminosity ${\cal E}_{\nu}(z)$ for the various wavebands based on those
published in the literature.222Table 1 references used to construct Figure 1
are as follows: Bouwens et al. (2007)(BO07), Bouwens et al. (2010)(BO10),
Budavári et al.(2005)(BU05), Burgarella et al. (2007)(BU07), Chen et al.(2003)
(CH03), Cucciati et al. (2012)(CU12), Dahlen et al. (2007)(DA07), Faber et al.
(2007)(FA07) and references therein, Iwata et al. (2007)(IW07), Ly et al.
(2009)(LY09), Reddy & Steidel (2009)(RE09), Marchesini et al. (2007)(MA07),
Marchesini & Van Dokkum 2007 (MAV07), Marchesini et al. (2012)(MA12), Oesch et
al. (2010)(OE10), Paltani et al. (2007)(PA07), Reddy et al. (2008)(RE08),
Sawicki & Thompson (2006)(SA06), Schiminovich et al. (2005)(SC05), Steidel et
al. (1999)(ST99), Tresse et al. (2007) (TR07), Wolf et al.(2003) (WO03), Wyder
et al. (2005)(WY05), Yoshida et al. (2006)(YO06). The lower right panel shows
all of the observational determinations of galaxy LDs from the references in
footnote 2. The specific waveband and mean redshift identifications for these
data are listed in Table 1 using the key abbreviations indicated in footnote
2. This table reflects the fact that direct determinations of galaxy LDs are
only available out to an observed wavelength of about 2.2 $\mu$m (rest
wavelength $2.2/(1+z)~{}\mu$m). This is because any attempt to survey large
areas of the sky with ground-based telescopes in wavebands longer than 2$\mu$m
is prevented by the sudden increases in background noise.333This 2$\mu$m
barrier is only circumvented by using space-based mid-infrared (3 to 8$\mu$m)
telescopes such as AKARI (with its Infrared Camera, IRC), and Spitzer (with
its Infrared Array Camera, IRAC). These telescopes have only conducted multi-
band imaging and redshift surveys with the necessary sensitivity to measure
the high-redshift ($z\geq 2$) galaxy population in a few, relatively small
deep fields.
Thus, at redshifts above 1.6, the longest rest-wavelengths under consideration
no longer have well measured LDs. At these longer wavelengths, we are obliged
to fall back on a secondary method for estimating galaxy luminosities: we use
the closest available LDs, and extrapolate them using the average observed
color of galaxies from measurements at that redshift. This ’minimal
extrapolation’ should be reliable because the average galaxy colors,
especially at long wavelengths, change only gradually with redshift. For
example, the galaxies that are included in the rest-frame $R$ band LD at
$z=2.2$ by Marchesini et al. are very similar to those of the galaxies that
would have been included in an $I$-band LD at that redshift. Since we are only
extrapolating by a small step in wavelength ($\Delta\lambda/\lambda\sim
0.15$), it is quite reasonable to shift the $R$-band LD using the average
$R-I$ colors observed at that redshift. The incremental color shifts we apply
become large only at $z\geq 4$, where, as we show in Section 4, the overall
contributions to the IBL $\gamma$-ray opacity are not very substantial. Our
color relations, which are also used to interpolate between the closely spaced
wavebands, are given the next subsection. They are given as a function of
redshift, $z$, since galaxies tend to be bluer on average at higher redshifts.
#### 2.1.2 Average Colors
It is hardly surprising that there are often large apparent jumps, or changes,
in the shape and the normalization of the LDs going from one waveband to an
immediately adjacent one. We therefore applied an independent test of the
consistency of these LDs, by comparing the integrated ratios of LDs at
adjacent wavebands to the published average colors measured by observers. This
test has the great advantage of not requiring accurate estimates of volume
incompleteness or even very accurate redshifts. Broadband colors (i.e., local
continuum slopes) are easier to measure than LDs. The main problem is that all
galaxy samples at all redshifts show a wide observed range of broadband
colors. The typical $1\sigma$ scatter we found in published color
distributions was $\pm~{}0.5$ mag. A few rest-frame colors that are very
sensitive to stellar population, such as $U-B$, often show even larger
variation.
In order to determine the redshift evolution of the LD in each of the bands
out to a redshift of $\sim$ 8, we utilized color relations to transform data
from other bands. We have chosen to include all data possible in excess of
$z=1.5$ to fill in the gaps for various wavebands mostly at higher
redshifts.444The most comprehensive observations of galaxies in the best
observed Deep Fields include extremely sensitive Spitzer/IRAC photometry. The
IRAC data are most complete in its Band 1 (3.6 $\mu$m observed) wavelength,
and gradually become less sensitive out to the reddest IRAC band at 8 $\mu$m
observed wavelength corresponding to a rest wavelength of $8/(1+z)~{}\mu$m..
This also provides both an overlap to existing data and multiple sources of
data as a check for consistency of our color relations.
Published estimates of average colors from galaxy surveys at various wavebands
and redshifts tend to be bluer at shorter wavelengths, and redder at longer
wavelengths. This is due to the composite nature of stellar populations in
galaxies, with hot young stars making a stronger contribution in the UV
portion of the spectrum while red giants dominate the long wavelengths. Thus,
the galaxies that are included in a UV LF and not all the same galaxies as
those included in an LF in the $R$ band.
There is a clear trend with redshift over all wavelengths, which is well
known. Redder galaxies (e.g., local E and S0 galaxies) are more and more
outnumbered by blue, actively star-forming galaxies, at higher redshifts. The
average characteristic age of stellar populations decreases with redshift. Our
color relations agree with this trend. At the highest redshifts most known
galaxies are dominated by young starburst populations of O and B stars. This
tends to produce very blue overall spectral energy distribution without very
much sensitivity to the exact details of the star formation. These factors are
automatically taken into account when one uses the actual observational data
on the LDs at various wavelengths and redshifts.
Defining the average wavelengths of the various bands in $nm$ as follows:
FUV = 150, NUV = 280, $U$ = 365, $B$ = 445, $V$ = 551, $R$ = 658, $I$ = 806 nm
We then use the commonly measured astronomical parameter $\beta$, which is
defined by the relation between the differential flux and wavelength of a
galaxy, $f_{\lambda}\propto\lambda^{\beta}$. We have adopted the following
relations (colors) for $\beta_{\Delta\lambda}(z)$:
$\beta(FUV-NUV)=-1.0-1.25log(1+z),\ log(1+z)\leq 0.8$
derived from Bouwens, et al. (2009); Budavári et al.(2005); Castellano et al.
(2012); Cucciati, et al. (2012); Dunlop et al. (2012); Willott, et al. (2012);
Wyder et al.(2005),
$\beta(B-V)=+0.3-1.6log(1+z),\ log(1+z)\leq 0.6$
derived from Arnouts et al.(2007); Brammer (2011),
$\beta(NUV-U)=+0.5-1.2log(1+z),\ log(1+z)\leq 0.6$
derived from Tresse et al. (2007),
$\beta(NUV-R)=+2.5-6.0log(1+z),\ log(1+z)\leq 0.6$
$\beta(U-V)=+1.3-3.0log(1+z),\ log(1+z)\leq 0.6$
derived from Arnouts, et al. (2007); Brammer (2011): Ly et al. (2009),
$\beta(U-B)=+3.0-5.0log(1+z),\ log(1+z)\leq 0.6$
derived from Marchesini et al. (2007); González et al. (2011),
For the FUV-NUV relation we set $\beta[log(1+z)>0.8]=\beta[log(0.8)].$ For all
of the other relations we set $\beta[log(1+z)>0.6]=\beta[log(0.6)].$
We used the above redshift-dependent relations where appropriate in our
analysis. We stress that in the redshift ranges where they overlap, the
colored (observational) data points shown for the various wavelength bands in
Figure 1 agree quite well, within the uncertainties, with the black data
points that were extrapolated from the shorter wavelength bands using our
color relations. Also, where there is no overlap at the higher redshifts, the
uncertainty bands in photon density (see next section) show no
discontinuities.
### 2.2 Photon Density Calculations
The observationally determined LDs, combined with the color relations, extend
our coverage of galaxy photon production from the FUV to the NIR in the galaxy
rest frame. We have at least one or two determinations at each wavelength
across the most crucial redshift range $0\leq z\leq 2.5$. However, to
calculate the opacity for photons at energies higher than $\sim 250/(1+z)$ GeV
(see next section), requires the determination of galaxy LDs at longer rest
wavelengths and higher redshifts. These regimes are less well constrained by
observations, since they require measurement of very faint galaxies at long
wavelengths (mid-IR observed frame.) We will address this topic further in
Paper II. We have assumed a constant color at high redshift at the longer
wavelengths as stated above. However, we stress that our final results are not
very sensitive to errors in our average color relations because the
interpolations that we make cover very small fractional wavelength intervals,
$\Delta\lambda(z)$. We have directly tested this by numerical trial.
The second goal of our paper is to place upper and lower limits (within a 68%
confidence band) on the opacity of the universe to $\gamma$-rays . These
limits are a direct result of the 68% confidence band upper and lower limits
of the IBL determined from the observational data on $\rho_{L_{\nu}}$ . In
order to determine these limits, we make no assumptions about the luminosity
density evolution. We derive a luminosity confidence band in each waveband by
using a robust rational fitting function characterized by
$\rho_{L_{\nu}}={\cal E}_{\nu}(z)={{ax+b}\over{cx^{2}+dx+e}}$ (5)
where $x=\log(1+z)$ and $a$,$b$,$c$,$d$,and $e$ are free parameters.
The 68% confidence band is then computed from Monte Carlo simulation by
finding 100,000 realizations of the data and then fitting the rational
function. In order to best represent the tolerated confidence band,
particularly at the highest redshifts, we have chosen to equally weight all
FUV points in excess of a redshift of 2. Our goal is not to find the best fit
to the data but rather the limits tolerated by the current observational data.
In order to perform the Monte Carlo of the fitting function, a likelihood is
determined at each redshift containing data. The shape of the function is
taken to be Gaussian (or the sum of Gaussians where multiple points exist) for
symmetric errors quoted in the literature. Where symmetric errors are not
quoted it is impossible to know what the actual shape of the likelihood
functions is. We have chosen to utilize a skew normal distribution to model
asymmetric errors. This assumption has very little impact on the determination
of the confidence bands. The resulting bands are shown along with the
luminosity density data in Figure 1.
With the confidence bands established, we take the upper and lower limits of
the bands to be our high and low IBL respectively. We then interpolate each of
these cases separately between the various wavebands to find the upper and
lower limit rest frame luminosity densities. The calculation is extended to
the Lyman limit using the slope derived from our color relationship between
the near and far UV bands.
The specific emissivity is then the derived high and low IBL luminosity
densities ${\cal E}_{\nu}(z)=\rho_{L_{\nu}}(z)$. The co-moving radiation
energy density is determined from equation 4. Figure 2 shows the resulting
photon density determined by dividing the energy density by the energy in each
frequency for high and low IBL. This result is used as input for the
determination of the optical depth of the universe to $\gamma$-rays .
The photon densities
$\epsilon n(\epsilon,z)=u(\epsilon,z)/\epsilon\ \ ,$ (6)
with $\epsilon=h\nu$, as calculated using equation (2), are shown in Figure 2.
## 3 Comparison of z = 0 IBL with Data and Constraints
As a byproduct of our determination of the IBL as a function of redshift using
LDs from galaxy surveys, we have also determined the local ($z=0$) IBL, also
known as the extragalactic background light (EBL). Determining the EBL
directly has been the object of intense observational effort, although the
various estimates and limits in the published literature are far from
consistent with each other. Nonetheless, since these observations provide a
potential consistency check on our calculations, we consider them here.
Using equation (2), together with our empirically based determinations given
the confidence band derived for our specific emissivities, ${\cal
E}_{\nu}(z)$, we have evaluated the EBL within the 68% confidence band upper
and lower limits within the wavelength range of our calculations. This band is
indicated by the gray zone in Figure 3. We also show recent measurements using
the Hubble Wide-field Planetary Camera 2 (Bernstein 2007), the dark field from
Pioneer 10/11 (Matsuoka et al. 2011) and the preliminary analysis of Mattila
et al. (2011) using differential measurements using the ESO VLT (very large
telescope array). Figure 3 also shows the various lower limits from galaxy
counts obtained by Gardner et al. (2000) from the ST Imaging Spectrograph
data, by Madau & Pozzetti (2000) using Hubble Deep Field South data, and by Xu
et al. (2005) from GALEX (Galaxy Evolution Explorer) data, all indicated by
upward-pointing arrows.
## 4 The Optical Depth from Interactions with Intergalactic Low Energy
Photons
The cross section for photon-photon scattering to electron-positron pairs can
be calculated using quantum electrodynamics (Breit & Wheeler 1934). The
threshold for this interaction is determined from the frame invariance of the
square of the four-momentum vector that reduces to the square of the threshold
energy, $s$, required to produce twice the electron rest mass in the c.m.s.:
$s=2\epsilon E_{\gamma}(1-\cos\theta)=4m_{e}^{2}$ (7)
This invariance is known to hold to within one part in $10^{15}$ (Stecker &
Glashow 2001; Jacobson, Liberati, Mattingly & Stecker 2004).
With the co-moving energy density $u_{\nu}(z)$ evaluated, the optical depth
for $\gamma$-rays owing to electron-positron pair production interactions with
photons of the stellar radiation background can be determined from the
expression (Stecker, De Jager, & Salamon 1992)
$\tau(E_{0},z_{e})=c\int_{0}^{z_{e}}dz\,\frac{dt}{dz}\int_{0}^{2}dx\,\frac{x}{2}\int_{0}^{\infty}d\nu\,(1+z)^{3}\left[\frac{u_{\nu}(z)}{h\nu}\right]\sigma_{\gamma\gamma}[s=2E_{0}h\nu
x(1+z)],$ (8)
In equations (7) and (8), $E_{0}$ is the observed $\gamma$-ray energy at
redshift zero, $\nu$ is the frequency at redshift $z$, $z_{e}$ is the redshift
of the $\gamma$-ray source at emission, $x=(1-\cos\theta)$,
$\theta$ being the angle between the $\gamma$-ray and the soft background
photon, $h$ is Planck’s constant, and the pair production cross section
$\sigma_{\gamma\gamma}$ is zero for center-of-mass energy
$\sqrt{s}<2m_{e}c^{2}$, $m_{e}$ being the electron mass. Above this threshold,
the pair production cross section is given by
$\sigma_{\gamma\gamma}(s)=\frac{3}{16}\sigma_{\rm
T}(1-\beta^{2})\left[2\beta(\beta^{2}-2)+(3-\beta^{4})\ln\left(\frac{1+\beta}{1-\beta}\right)\right],$
(9)
where $\sigma_{T}$ is the Thompson scattering cross section and
$\beta=(1-4m_{e}^{2}c^{4}/s)^{1/2}$ (Jauch & Rohrlich 1955).
It follows from equation (7) that the pair-production cross section energy has
a threshold at $\lambda=4.75\ \mu{\rm m}\cdot E_{\gamma}({\rm TeV})$. Since
the maximum $\lambda$ that we consider here is in the rest frame I band at 800
nm at redshift $z$, and we observe $E_{\gamma}$ at redshift 0, so that its
energy at interaction in the rest frame is $(1+z)E_{\gamma}$, we then get a
conservative upper limit on $E_{\gamma}$ of $\sim 200(1+z)^{-1}$ GeV as the
maximum $\gamma$-ray energy affected by the photon range considered here.
Allowing for a small error, our opacities are good to $\sim 250(1+z)^{-1}$
GeV. The 68% opacity ranges for $z=0.1,0.5,1,3~{}$and $5$, calculated using
equation (8) are plotted in Figure 4.
The widths of the grey uncertainty ranges in the LDs shown in Figure 1
increase towards higher redshifts, especially at the longest rest wavelengths.
This reflects the decreasing amount of long-wavelength data and the
corresponding increase in uncertainties about the galaxies in those regimes.
However, these uncertainties do not greatly influence the opacity
calculations. Because of the short time interval of the emission from galaxies
at high redshifts their photons do not contribute greatly to the opacity at
lower redshifts. Indeed, Figure 4 shows that the opacities determined for
redshifts of 3 and 5 overlap within the uncertainties.
## 5 Results and Implications
We have determined the IBL using local and deep galaxy survey data, together
with observationally produced uncertainties, for wavelengths from 150 nm to
800 nm and redshifts out to $z>5$. We have presented our results in terms of
68% confidence band upper and lower limits. As expected, our $z=0$ (EBL) 68%
lower limits are higher than those obtained by galaxy counts alone, since the
EBL from galaxies is not completely resolved. Our results are also above the
theoretical lower limits given recently by Kneiske and Dole (2010). In Figure
3, we compare our $z=0$ result with both published and preliminary
measurements and limits.
Figure 5 shows our 68% confidence band for $\tau=1$ on an energy-redshift plot
compared with the Fermi data on the highest energy photons from extragalactic
sources at various redshifts as given by Abdo et al. (2010). It can be seen
that none of the photons from these sources would be expected to be
significantly annihilated by pair production interactions with the IBL. This
point is brought out further in Figure 6. This figure compares the 68%
confidence band of our opacity results with the 95% confidence upper limits on
the opacity derived for specific blazars by Abdo et al. (2010).
For purposes of discussion, we mention some points of comparison with previous
work. Our EBL results for $z=0$, while lower than the fast evolution model of
our previous work, are generally higher than those modeled more recently. As
an example, at a wavelength of 200 nm in the FUV range our uncertainty range
is a factor of 1.8 - 4.2 higher than the recent fiducial semi-analytic model
of Gilmore et al. (2012) and similarly higher than the previous model result
of Dominguez et al. (2011). Our opacity results at $z\simeq 1$ are comparable
to, or lower than, the models of Kneiske et al. (2004). They are also
consistent with the results of the non-metallicity corrected model of SS98.
However, they are higher than the models of Franceschini et al. (2008),
Gilmore et al. (2009), and Finke et al. (2010), as indicated by comparing
Figure 3 of Abdo et al. (2010) with our Figure 5. We stress that these
comparisons are for illustrative purposes only. Because our new methodology is
based on the direct use of luminosity densities derived directly from
observations, we take the position that they stand by themselves and should be
compared primarily with the observational data as shown in our Figures 3, 5
and 6. In that regard, we find full consistency within our observationally
determined uncertainties.555While we were preparing our revised manuscript for
publication a similar empirically based calculation by Helgason & Kashlinsky
(2012) appeared on the arXiv. These authors calculated the EBL and
$\gamma$-ray opacity based on galaxy luminosity functions compiled by
Helgason, Ricotti & Kashlinsky (2012) extrapolated to $z\geq 2$ using an
exponential cutoff in $z$. Their opacity results are generally consistent with
the results presented here.
Our result bears on questions regarding the possible modification of the pair-
production opacity effect on the $\gamma$-ray flux from distant extragalactic
sources, either by line-of-sight photon-axion oscillations during propagation
(e.g., De Angelis et al. 2009) or by the addition of a component of secondary
$\gamma$-rays from interactions of blazar-produced cosmic-rays with photons
along the line-of-sight to the blazar (e.g., Essey et al. 2010; Essey &
Kusenko 2012). Future theoretical studies and future $\gamma$-ray observations
of extragalactic sources with Fermi and the Čerenkov Telescope Array, which
will be sensitive to extragalactic sources at energies above 10 GeV (Gernot
2011), should help to clarify these important aspects of high energy
astrophysics.
## 6 Our Results Online
Our results in numerical form are available at the following link:
http://csma31.csm.jmu.edu/physics/scully/opacities.html
## Acknowledgments
We would like to thank Luis Reyes and Anita Reimer for supplying us with the
Fermi results shown in Figure 5. We thank Richard Henry for a helpful
discussion of the UV background data. We also thank Tonia Venters for helpful
discussions. This research was partially supported by a NASA Astrophysics
Theory Grant and a NASA Fermi Guest Investigator Grant.
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## Table 1. Identification of References for Fig. 1 Data by Waveband and
Redshift
z | FUV | NUV | U | B | V | R | I
---|---|---|---|---|---|---|---
.05 | SC05, WY05 | WY05 | | | | |
.1 | BU05,CU12 | BU05,CU12 | | | | |
.15 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
.20 | BU05 | BU05 | | | | |
.25 | WO03 | WO03 | | | | WO03 |
.3 | SC05,CU12,SC05,TR07 | TR07,CU12 | TR07,DA05 | TR07,DA05,FA07 | TR07 | TR07 | TR07
.35 | | DA07, WO03 | | WO03 | | WO03 |
.45 | | WO03 | DA05 | DA05, WO03 | | WO03 |
.5 | SC05, CU12, TR07 | TR07 | TR07 | TR07, FA07 | TR07 | TR07 | TR07
.55 | | DA07, WO03 | | WO03 | | WO03 |
.6 | | | DA05 | DA05 | | CH03 |
.65 | | WO03 | | WO03 | MA12 | WO03 |
.7 | | TR07,CU12 | TR07 | TR07, FA07 | TR07 | TR07 | TR07
.75 | | WO03 | | WO03 | | WO03 |
.85 | | WO03 | | WO03 | | WO03 |
.9 | TR07,CU12 | TR07,CU12 | TR07, DA05 | TR07, DA05, FA07 | TR07 | TR07 | TR07
.95 | | WO03 | DA05 | WO03, DA05 | MA12 | WO03, DA05 |
1.0 | SC05 | WO03 | | WO03 | | WO03 |
1.1 | CU12, TR07, DA07, BU07 | DA07,TR07,CU12, WO03 | TR07 | TR07, FA07, WO03 | TR07 | TR07, WO03 | TR07
1.2 | | | DA05 | DA05 | | CH03, DA05 |
1.3 | CU12, TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
1.4 | CU12 | CU12 | | | | |
1.5 | | | DA05 | DA05 | | DA05 |
1.6 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
1.7 | | | DA05 | DA05 | | DA05 |
1.8 | DA07 | DA07 | | | MA12 | |
1.9 | | | DA05 | DA05 | | DA05 |
2.0 | SC05 | | | | | |
2.1 | CU12 | CU12 | | | | |
2.2 | RE08, SA06 | | | MA07 | MA07 | MA07 |
2.3 | LY09 | | | | | |
2.4 | | | | | MA12 | |
2.9 | SC05 | | | | | |
3.0 | CU12 | CU12 | | MA07 | MA07, MA12 | |
3.5 | PA07 | | | | | |
3.8 | BO07 | | | | MA12 | |
4.0 | YO06,CU12 | | | | | |
4.1 | SA06 | | | | | |
4.8 | IW07 | | | | | |
5.0 | BO07 | | | | | |
5.9 | BO07 | | | | | |
6.8 | BO11 | | | | | |
7.0 | OE10 | | | | | |
8.2 | BO10 | | | | | |
Figure Captions
Figure 1: The observed specific emissivities in our fiducial wavebands. The
lower right panel shows all of the observational data from the references in
footnote 1. In the other panels, non-band data have been shifted using the
color relations given in the text in order to fully determine the specific
emissivities in each waveband. The symbol designations are FUV: black filled
circles, NUV: magenta open circles, $U$: green filled squares, $B$: blue open
squares, $V$: brown filled triangles, $R$: orange open triangles, $I$: yellow
open diamonds. Grey shading: 68% confidence bands (see text).
Figure 2: The photon densities $\epsilon n(\epsilon)$ shown as a continuous
function of photon energy and redshift for both the high (upper panel) and low
(lower panel) IBL.
Figure 3: Our empirically-based determination of the EBL together with lower
limits and data as described in the text. The legend is as follows: Madau &
Pozzetti(2000):Black Cicles, Xu et al.(2005):Crosses, Gardner et
al.(2000):Open Squares, Matsuoka et al.(2011):Open Circles, Mattilla et
al.(2011)(preliminary):Black Squares, Bernstein(2007):Black Diamonds. The
upper limit from Mattilla et al.(2011) is thickened for clarity.
Figure 4: Our empirically determined opacities for redshifts of 0.1, 0.5, 1,
3, 5. The dashed lines are for $\tau=1$ and $\tau=3$.
Figure 5: A $\tau=1$ energy-redshift plot (Fazio & Stecker 1970) showing our
uncertainty band results compared with the Fermi plot of their highest energy
photons from FSRQs (red), BL Lacs (black) and and GRBs (blue) vs. redshift
(from Abdo et al. 2010).
Figure 6: Our opacity results for the redshifts of the blazars compared with
95% confidence opacity upper limits (red arrows) and 99% confidence limits
(blue arrows) as given by the Fermi analysis of Abdo, et al. (2010).
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
|
arxiv-papers
| 2012-05-23T13:36:34 |
2024-09-04T02:49:31.227476
|
{
"license": "Public Domain",
"authors": "Floyd W. Stecker (NASA/GSFC and UCLA), Matthew A. Malkan (UCLA) and\n Sean T. Scully (JMU)",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/1205.5168"
}
|
1205.5266
|
# Dark Energy in F(R,T) Gravity
Ratbay Myrzakulov111Email: rmyrzakulov@gmail.com; rmyrzakulov@csufresno.edu
Eurasian International Center for Theoretical Physics and Department of
General
$\&$ Theoretical Physics, Eurasian National University, Astana 010008,
Kazakhstan
###### Abstract
Since the discovery of cosmic acceleration, modified gravity theories play an
important role in the modern cosmology. In particular, the well-known
F(R)-theories reached great popularity motivated by the easier formalism and
by the prospect to find a final theories of gravity for the dark scenarios. In
the present work, we study some generalizations of F(R) and F(T) gravity
theories. At the beginning, we briefly review the formalism of such theories.
Then, we will consider one of their generalizations, the so-called
F(R,T)-theory. The point-like Lagrangian is explicitly presented. Based on
this Lagrangian, the field equations of F(R,T)-gravity are found. For the
specific model $F(R,T)=\mu R+\nu T,$ the corresponding exact solutions are
derived. Furthermore, we will consider the physical quantities associated to
such solutions and we will find how for some values of the parameters the
expansion of our universe can be accelerated without introducing any dark
component.
## 1 Introduction
Recent observational data imply -against any previous belief- that the current
expansion of the universe is accelerating [1]. Since this discovery, the so-
called Dark Energy issue has probably become the most ambitious and
tantalizing field of research because of its implications in fundamental
physics. There exist several descriptions of the cosmic acceleration. Among
them, the simplest one is the introduction of small positive Cosmological
Constant in the framework of General Relativity (GR), the so-called
$\Lambda$CDM Model, but it is well accepted the idea according to which this
is not the ultimate theory of gravity, but an extremely good approximation
valid in the present day range of detection. A generalization of this simple
modification of GR consists in considering modified gravitational theories [1,
2]. In the last years the interest in modified gravity theories like $F(R)$
and $F(G)$-gravity as alternatives to the $\Lambda$CDM Model grew up.
Recently, a new modified gravity theory, namely the $F(T)$-theory, has been
proposed. This is a generalized version of the teleparallel gravity originally
proposed by Einstein [3]-[16]. It also may describe the current cosmic
acceleration without invoking dark energy. Unlike the framework of GR, where
the Levi-Civita connection is used, in teleparallel gravity (TG) the used
connection is the Weitzenböck’one. In principle, modification of gravity may
contain a huge list of invariants and there is not any reason to restrict the
gravitational theory to GR, TG, $F(R)$ gravity and/or $F(T)$ gravity. Indeed,
several generalizations of these theories have been proposed (see e.g. the
quite recent review [17]). In this paper, we study some other generalizations
of $F(R)$ and $F(T)$ gravity theories. At the beginning, we briefly review the
formalism of $F(R)$ gravity and $F(T)$ gravity in Friedmann-Robertson-Walker
(FRW) universe. The flat FRW space-time is described by the metric
$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (1.1)
where $a=a(t)$ is the scale factor. The orthonormal tetrad components
$e_{i}(x^{\mu})$ are related to the metric through
$g_{\mu\nu}=\eta_{ij}e_{\mu}^{i}e_{\nu}^{j}\,,$ (1.2)
where the Latin indices $i$, $j$ run over 0…3 for the tangent space of the
manifold, while the Greek letters $\mu$, $\nu$ are the coordinate indices on
the manifold, also running over 0…3.
$F(R)$ and $F(T)$ modified theories of gravity have been extensively explored
and the possibility to construct viable models in their frameworks has been
carefully analyzed in several papers (see [17] for a recent review). For such
theories, the physical motivations are principally related to the possibility
to reach a more realistic representation of the gravitational fields near
curvature singularities and to create some first order approximation for the
quantum theory of gravitational fields. Recently, it has been registred a
renaissance of $F(R)$ and $F(T)$ gravity theories in the attempt to explain
the late-time accelerated expansion of the Universe [17]. In the modern
cosmology, in order to construct (generalized) gravity theories, three
quantities – the curvature scalar, the Gauss –Bonnet scalar and the torsion
scalar – are usually used (about our notations see below):
$\displaystyle R_{s}$ $\displaystyle=$ $\displaystyle g^{\mu\nu}R_{\mu\nu},$
(1.3) $\displaystyle G_{s}$ $\displaystyle=$ $\displaystyle
R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma},$ (1.4)
$\displaystyle T_{s}$ $\displaystyle=$
$\displaystyle{S_{\rho}}^{\mu\nu}\,{T^{\rho}}_{\mu\nu}.$ (1.5)
In this paper, our aim is to replace these quantities with the other three
variables in the form
$\displaystyle R$ $\displaystyle=$ $\displaystyle u+g^{\mu\nu}R_{\mu\nu},$
(1.6) $\displaystyle G$ $\displaystyle=$ $\displaystyle
w+R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma},$ (1.7)
$\displaystyle T$ $\displaystyle=$ $\displaystyle
v+{S_{\rho}}^{\mu\nu}\,{T^{\rho}}_{\mu\nu},$ (1.8)
where $u=u(x_{i};g_{ij},\dot{g_{ij}},\ddot{g_{ij}},...;f_{j})$,
$v=v(x_{i};g_{ij},\dot{g_{ij}},\ddot{g_{ij}},...;g_{j})$ and
$w=w(x_{i};g_{ij},\dot{g_{ij}},\ddot{g_{ij}},...;h_{j})$ are some functions to
be defined. As a result, we obtain some generalizations of the known modified
gravity theories. With the FRW metric ansatz the three variables (1.3)-(1.5)
become
$\displaystyle R_{s}$ $\displaystyle=$ $\displaystyle 6(\dot{H}+2H^{2}),$
(1.9) $\displaystyle G_{s}$ $\displaystyle=$ $\displaystyle
24H^{2}(\dot{H}+H^{2}),$ (1.10) $\displaystyle T_{s}$ $\displaystyle=$
$\displaystyle-6H^{2},$ (1.11)
where $H=(\ln a)_{t}$. In the contrast, in this paper we will use the
following three variables
$\displaystyle R$ $\displaystyle=$ $\displaystyle u+6(\dot{H}+2H^{2}),$ (1.12)
$\displaystyle G$ $\displaystyle=$ $\displaystyle w+24H^{2}(\dot{H}+H^{2}),$
(1.13) $\displaystyle T$ $\displaystyle=$ $\displaystyle v-6H^{2}.$ (1.14)
This paper is organized as follows. In Sec. 2, we briefly review the formalism
of $F(R)$ and $F(T)$-gravity for FRW metric. In particular, the corresponding
Lagrangians are explicitly presented. In Sec. 3, we consider $F(R,T)$ theory,
where $R$ and $T$ will be generalized with respect to the usual notions of
curvature scalar and torsion scalar. Some reductions of $F(R,T)$ gravity are
presented in Sec. 4. In Sec. 5, the specific model $F(R,T)=\mu R+\nu T$ is
analized and in Sec. 6 the exact power-law solution is found; some
cosmological implications of the model will be here discussed. The Bianchi
type I version of $F(R,T)$ gravity is considered in Sec. 7. Sec. 8 is devoted
to some generalizations of some modified gravity theories. Final conclusions
and remarks are provided in Sec. 9.
## 2 Preliminaries of $F(R)$, $F(G)$ and $F(T)$ gravities
At the beginning, we present the basic equations of $F(R)$, $F(T)$ and $F(G)$
modified gravity theories. For simplicity we mainly work in the FRW spacetime.
### 2.1 $F(R)$ gravity
The action of $F(R)$ theory is given by
${\cal S}_{R}=\int d^{4}xe[F(R)+L_{m}],$ (2.1)
where $R$ is the curvature scalar. We work with the FRW metric (1.1). In this
case $R$ assumes the form
$R=R_{s}=6(\dot{H}+2H^{2}).$ (2.2)
The action we rewrite as
${\cal S}_{R}=\int dtL_{R},$ (2.3)
where the Lagrangian is given by
$L_{R}=a^{3}(F-RF_{R})-6F_{R}a\dot{a}^{2}-6F_{RR}\dot{R}a^{2}\dot{a}-a^{3}L_{m}.$
(2.4)
The corresponding field equations of $F(R)$ gravity read
$\displaystyle 6\dot{R}HF_{RR}-(R-6H^{2})F_{R}+F$ $\displaystyle=$
$\displaystyle\rho,$ (2.5)
$\displaystyle-2\dot{R}^{2}F_{RRR}+[-4\dot{R}H-2\ddot{R}]F_{RR}+[-2H^{2}-4a^{-1}\ddot{a}+R]F_{R}-F$
$\displaystyle=$ $\displaystyle p,$ (2.6) $\displaystyle\dot{\rho}+3H(\rho+p)$
$\displaystyle=$ $\displaystyle 0.$ (2.7)
### 2.2 $F(T)$ gravity
In the modified teleparallel gravity, the gravitational action is
${\cal S}_{T}=\int d^{4}xe[F(T)+L_{m}],$ (2.8)
where $e={\rm det}\,(e_{\mu}^{i})=\sqrt{-g}\,$, and for convenience we use the
units $16\pi G=\hbar=c=1$ throughout. The torsion scalar $T$ is defined as
$T\equiv{S_{\rho}}^{\mu\nu}\,{T^{\rho}}_{\mu\nu}\,,$ (2.9)
where
$\displaystyle{T^{\rho}}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-e^{\rho}_{i}\left(\partial_{\mu}e^{i}_{\nu}-\partial_{\nu}e^{i}_{\mu}\right)\,,$
(2.10) $\displaystyle{K^{\mu\nu}}_{\rho}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}\left({T^{\mu\nu}}_{\rho}-{T^{\nu\mu}}_{\rho}-{T_{\rho}}^{\mu\nu}\right)\,,$
(2.11) $\displaystyle{S_{\rho}}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left({K^{\mu\nu}}_{\rho}+\delta^{\mu}_{\rho}{T^{\theta\nu}}_{\theta}-\delta^{\nu}_{\rho}{T^{\theta\mu}}_{\theta}\right)\,.$
(2.12)
For a spatially flat FRW metric (1.1), as a consequence of equations (2.9) and
(1.1), we have that the torsion scalar assumes the form
$T=T_{s}=-6H^{2}.$ (2.13)
The action (2.8) can be written as
${\cal S}_{T}=\int dtL_{T},$ (2.14)
where the point-like Lagrangian reads
$L_{T}=a^{3}\left(F-F_{T}T\right)-6F_{T}a\dot{a}^{2}-a^{3}L_{m}.$ (2.15)
The equations of F(T) gravity look like
$\displaystyle 12H^{2}F_{T}+F$ $\displaystyle=$ $\displaystyle\rho,$ (2.16)
$\displaystyle 48H^{2}F_{TT}\dot{H}-F_{T}\left(12H^{2}+4\dot{H}\right)-F$
$\displaystyle=$ $\displaystyle p,$ (2.17)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(2.18)
### 2.3 $F(G)$ gravity
The action of $F(G)$ theory is given by
${\cal S}_{G}=\int d^{4}xe[F(G)+L_{m}],$ (2.19)
where the Gauss – Bonnet scalar $G$ for the FRW metric is
$G=G_{s}=24H^{2}(\dot{H}+H^{2}).$ (2.20)
## 3 A naive model of $F(R,T)$ gravity
Our aim in this section is to present a naive version of $F(R,T)$ gravity. We
assume that the relevant action of $F(R,T)$ theory is given by [14]
${\cal S}_{37}=\int d^{4}xe[F(R,T)+L_{m}],$ (3.1)
where $R=u+R_{s}$ and $T=v+T_{s}$ are some dynamical geometrical variables to
be defined, and $R_{s}$ and $T_{s}$ are the usual curvature scalar and the
torsion scalar for the FRW spacetime. It is the so-called M37 \- model [14].
In this paper we will restrict ourselves to the simple case where for FRW
spacetime $R$ and $T$ are given by
$\displaystyle R$ $\displaystyle=$ $\displaystyle
u+6(\dot{H}+2H^{2})=u+R_{s},$ (3.2) $\displaystyle T$ $\displaystyle=$
$\displaystyle v-6H^{2}=v+T_{s}.$ (3.3)
As we can see these two variables ($R,T$) are some analogies (generalizations)
of the usual curvature scalar $(R_{s}$) and torsion scalar ($T_{s}$) and for
obvious reasons we will still continue to call them as the ”curvature” scalar”
and the ”torsion” scalar. We note that, in general,
$u=u(t,a,\dot{a},\ddot{a},\dddot{a},...;f_{i})$ and
$v=v(t,a,\dot{a},\ddot{a},\dddot{a},...;g_{i})$ are some real functions,
$H=(\ln a)_{t}$, while $f_{i}$ and $g_{i}$ are some unknown functions related
with the geometry of the spacetime. Finally we can write the M37 \- model for
the FRW spacetime as
$\displaystyle S_{37}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (3.4) $\displaystyle R$ $\displaystyle=$
$\displaystyle u+6(\dot{H}+2H^{2}),$ (3.5) $\displaystyle T$ $\displaystyle=$
$\displaystyle v-6H^{2}.$ (3.6)
In this paper we restrict ourselves to the case $u=u(a,\dot{a})$ and
$v=v(a,\dot{a})$. The scale factor $a(t)$, the curvature scalar $R$ and the
torsion scalar $T(t)$ are taken as independent dynamical variables. Then,
after some algebra the action (3.4) becomes
${\cal S}_{37}=\int dtL,$ (3.7)
where the point-like Lagrangian is given by
$L_{37}=a^{3}(F-TF_{T}-RF_{R}+vF_{T}+uF_{R})-6(F_{R}+F_{T})a\dot{a}^{2}-6(F_{RR}\dot{R}+F_{RT}\dot{T})a^{2}\dot{a}-a^{3}L_{m}.$
(3.8)
The corresponding equations of the M37 \- model assume the form [14]
$\displaystyle D_{2}F_{RR}+D_{1}F_{R}+JF_{RT}+E_{1}F_{T}+KF$ $\displaystyle=$
$\displaystyle-2a^{3}\rho,\,\,$ $\displaystyle
U+B_{2}F_{TT}+B_{1}F_{T}+C_{2}F_{RRT}+C_{1}F_{RTT}+C_{0}F_{RT}+MF$
$\displaystyle=$ $\displaystyle 6a^{2}p,$ (3.9)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
Here
$\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle-6\dot{R}a^{2}\dot{a},$
(3.10) $\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle-6a\dot{a}^{2}+a^{3}u_{\dot{a}}\dot{a}-a^{3}(u-R),$ (3.11)
$\displaystyle J$ $\displaystyle=$ $\displaystyle-6a^{2}\dot{a}\dot{T},$
(3.12) $\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle-6a\dot{a}^{2}+a^{3}v_{\dot{a}}\dot{a}-a^{3}(v-T),$ (3.13)
$\displaystyle K$ $\displaystyle=$ $\displaystyle-a^{3}$ (3.14)
and
$\displaystyle U$ $\displaystyle=$ $\displaystyle
A_{3}F_{RRR}+A_{2}F_{RR}+A_{1}F_{R},$ $\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle-6\dot{R}^{2}a^{2},$ (3.15) $\displaystyle A_{2}$
$\displaystyle=$
$\displaystyle-6\ddot{R}a^{2}-12\dot{R}a\dot{a}+a^{3}\dot{R}u_{\dot{a}},$
(3.16) $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle-6\dot{a}^{2}-12a\ddot{a}+3a^{2}\dot{a}u_{\dot{a}}+a^{3}\dot{u}_{\dot{a}}-3a^{2}(u-R)-a^{3}u_{a},$
(3.17) $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-12\dot{T}a\dot{a}+a^{3}\dot{T}v_{\dot{a}},$ (3.18)
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-6\dot{a}^{2}-12a\ddot{a}+3a^{2}\dot{a}v_{\dot{a}}+a^{3}\dot{v}_{\dot{a}}-3a^{2}(v-T)-a^{3}v_{a},$
(3.19) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-12a^{2}\dot{R}\dot{T},$ (3.20) $\displaystyle C_{1}$
$\displaystyle=$ $\displaystyle-6a^{2}\dot{T}^{2},$ (3.21) $\displaystyle
C_{0}$ $\displaystyle=$
$\displaystyle-12\dot{R}a\dot{a}-12\dot{T}a\dot{a}-6a^{2}\ddot{T}+a^{3}\dot{R}v_{\dot{a}}+a^{3}\dot{T}u_{\dot{a}},$
(3.22) $\displaystyle M$ $\displaystyle=$ $\displaystyle-3a^{2}.$ (3.23)
We can rewrite the system (3.9) in terms of $H$ as
$\displaystyle DF_{RR}+D_{1}F_{R}+JF_{RT}+E_{1}F_{T}+KF$ $\displaystyle=$
$\displaystyle-2a^{3}\rho,$ $\displaystyle
U+B_{2}F_{TT}+B_{1}F_{T}+C_{2}F_{RRT}+C_{1}F_{RTT}+C_{0}F_{RT}+MF$
$\displaystyle=$ $\displaystyle 6a^{2}p,$ (3.24)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0,$
where
$\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle-6\dot{R}a^{2}\dot{a}=-6a^{3}H\dot{R},$ (3.25) $\displaystyle
D_{1}$ $\displaystyle=$
$\displaystyle-6a^{3}H^{2}+a^{3}u_{\dot{a}}\dot{a}+6a^{3}(\dot{H}+2H^{2})=a^{3}u_{\dot{a}}\dot{a}+6a^{3}(\dot{H}+H^{2}),$
(3.26) $\displaystyle J$ $\displaystyle=$ $\displaystyle-6a^{3}H\dot{T},$
(3.27) $\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle-6a^{3}H^{2}+a^{3}v_{\dot{a}}\dot{a}-6a^{3}H^{2}=-12a^{3}H^{2}+a^{3}v_{\dot{a}}\dot{a},$
(3.28) $\displaystyle K$ $\displaystyle=$ $\displaystyle-a^{3}.$ (3.29)
and
$\displaystyle U$ $\displaystyle=$ $\displaystyle
A_{3}F_{RRR}+A_{2}F_{RR}+A_{1}F_{R},$ $\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle-6\dot{R}^{2}a^{2},$ (3.30) $\displaystyle A_{2}$
$\displaystyle=$
$\displaystyle-6\ddot{R}a^{2}-12\dot{R}a\dot{a}+a^{3}\dot{R}u_{\dot{a}}=-6\ddot{R}a^{2}-12\dot{R}a\dot{a}+a^{3}\dot{R}u_{\dot{a}},$
(3.31) $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle-6\dot{a}^{2}-12a\ddot{a}+3a^{2}\dot{a}u_{\dot{a}}+a^{3}\dot{u}_{\dot{a}}-3a^{2}(u-R)-a^{3}u_{a},$
(3.32) $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-12\dot{T}a\dot{a}+a^{3}\dot{T}v_{\dot{a}},$ (3.33)
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-6\dot{a}^{2}-12a\ddot{a}+3a^{2}\dot{a}v_{\dot{a}}+a^{3}\dot{v}_{\dot{a}}-3a^{2}(v-T)-a^{3}v_{a},$
(3.34) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-12a^{2}\dot{R}\dot{T},$ (3.35) $\displaystyle C_{1}$
$\displaystyle=$ $\displaystyle-6a^{2}\dot{T}^{2},$ (3.36) $\displaystyle
C_{0}$ $\displaystyle=$
$\displaystyle-12\dot{R}a\dot{a}-12\dot{T}a\dot{a}-6a^{2}\ddot{T}+a^{3}\dot{R}v_{\dot{a}}+a^{3}\dot{T}u_{\dot{a}},$
(3.37) $\displaystyle M$ $\displaystyle=$ $\displaystyle-3a^{2}.$ (3.38)
## 4 Reductions. Preliminary classification
Note that the system (3.9) or (3.24) admits some important reductions. Let us
now present these particular cases.
### 4.1 Case: $F=R$
Now we consider the particular case $F=R$. Thus, the system (3.24) becomes
$\displaystyle D_{1}+KR$ $\displaystyle=$ $\displaystyle-2a^{3}\rho,$
$\displaystyle A_{1}+MR$ $\displaystyle=$ $\displaystyle 6a^{2}p,$ (4.1)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0$
or
$\displaystyle 3H^{2}+0.5(u-\dot{a}u_{\dot{a}})$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle
2\dot{H}+3H^{2}-0.5(\dot{a}u_{\dot{a}}+\frac{1}{3}a\dot{u}_{\dot{a}}-u)$
$\displaystyle=$ $\displaystyle-p,$ (4.2) $\displaystyle\dot{\rho}+3H(\rho+p)$
$\displaystyle=$ $\displaystyle 0.$
Let us rewrite this system as
$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle\rho+\rho_{c},$
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle-(p+p_{c}),$
(4.3) $\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0,$
where
$\displaystyle\rho_{c}$ $\displaystyle=$
$\displaystyle-0.5(u-\dot{a}u_{\dot{a}}),$ (4.4) $\displaystyle p_{c}$
$\displaystyle=$
$\displaystyle-0.5(\dot{a}u_{\dot{a}}+3^{-1}a\dot{u}_{\dot{a}}-u)$ (4.5)
are the corrections to the energy denisty and pressure. Note that if $u=0$ we
obtain the standard equations of GR,
$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle\rho,$ $\displaystyle
2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle-p,$ (4.6)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0$
### 4.2 Case: $F=T$
Let us now to consider $F=T$. Then the system (3.24) leads to
$\displaystyle E_{1}+KT$ $\displaystyle=$ $\displaystyle-2a^{3}\rho,$
$\displaystyle B_{1}+MT$ $\displaystyle=$ $\displaystyle 6a^{2}p,$ (4.7)
$\displaystyle\dot{\rho}-3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0,$
or
$\displaystyle 3H^{2}+0.5(v-\dot{a}v_{\dot{a}})$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle
2\dot{H}+3H^{2}-0.5(\dot{a}v_{\dot{a}}+\frac{1}{3}a\dot{v}_{\dot{a}}-v)$
$\displaystyle=$ $\displaystyle-p,$ (4.8) $\displaystyle\dot{\rho}+3H(\rho+p)$
$\displaystyle=$ $\displaystyle 0.$
The above system can be rewritten as
$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle\rho+\rho_{c},$
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle-(p+p_{c}),$
(4.9) $\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0,$
where
$\displaystyle\rho_{c}$ $\displaystyle=$
$\displaystyle-0.5(v-\dot{a}v_{\dot{a}}),$ (4.10) $\displaystyle p_{c}$
$\displaystyle=$
$\displaystyle-0.5(\dot{a}v_{\dot{a}}+3^{-1}a\dot{v}_{\dot{a}}-v)$ (4.11)
are the corrections to the energy density and pressure. Obviously, if $v=0$ we
obtain the standard equations of GR (4.6).
### 4.3 Case: $F=F(T),\quad u=v=0$
Let us take $F=F(T),\quad u=v=0$. Then, the system (3.24) becomes
$\displaystyle E_{1}F_{T}+KF$ $\displaystyle=$ $\displaystyle-2a^{3}\rho,$
(4.12) $\displaystyle B_{2}F_{TT}+B_{1}F_{T}+MF$ $\displaystyle=$
$\displaystyle 6a^{2}p,$ (4.13) $\displaystyle\dot{\rho}+3H(\rho+p)$
$\displaystyle=$ $\displaystyle 0$ (4.14)
or
$\displaystyle-12a\dot{a}^{2}F_{T}-a^{3}F$ $\displaystyle=$
$\displaystyle-2a^{3}\rho,$ (4.15)
$\displaystyle-12\dot{T}a\dot{a}F_{TT}-(36\dot{a}^{2}+12a\ddot{a})F_{T}-3a^{2}F$
$\displaystyle=$ $\displaystyle 6a^{2}p,$ (4.16)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(4.17)
This system can be rewritten as
$\displaystyle-2TF_{T}+F$ $\displaystyle=$ $\displaystyle 2\rho,$ (4.18)
$\displaystyle-8\dot{H}TF_{TT}+2(T-2\dot{H})F_{T}-F$ $\displaystyle=$
$\displaystyle 2p,$ (4.19) $\displaystyle\dot{\rho}-3H(\rho+p)$
$\displaystyle=$ $\displaystyle 0$ (4.20)
that is the same as (2.16)-(2.18) of $F(T)$ gravity.
### 4.4 Case: $F=F(R),\quad u=v=0$
We get the second reduction if we consider the case where $F=F(R),\quad
u=v=0$. Then the system (3.9) leads to
$\displaystyle D_{2}F_{RR}+D_{1}F_{R}+KF$ $\displaystyle=$
$\displaystyle-2a^{3}\rho,$ (4.21) $\displaystyle
A_{3}F_{RRR}+A_{2}F_{RR}+A_{1}F_{R}+MF$ $\displaystyle=$ $\displaystyle
6a^{2}p,$ (4.22) $\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$
$\displaystyle 0,$ (4.23)
where
$\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle-6\dot{R}^{2}a^{2},$
(4.24) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle-6\ddot{R}a^{2}-12\dot{R}a\dot{a},$ (4.25) $\displaystyle A_{1}$
$\displaystyle=$ $\displaystyle-6\dot{a}^{2}-12a\ddot{a}+3a^{2}R,$ (4.26)
$\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle-6\dot{R}a^{2}\dot{a},$
(4.27) $\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle-6a\dot{a}^{2}+a^{3}R,$ (4.28) $\displaystyle K$
$\displaystyle=$ $\displaystyle-a^{3}.$ (4.29)
This system can be written as
$\displaystyle-6\dot{R}a^{2}\dot{a}F_{RR}+[-6a\dot{a}^{2}+a^{3}R]F_{R}-a^{3}F$
$\displaystyle=$ $\displaystyle-2a^{3}\rho,$ (4.30)
$\displaystyle-6\dot{R}^{2}a^{2}F_{RRR}+[-12\dot{R}a\dot{a}-6\ddot{R}a^{2}]F_{RR}+[-6\dot{a}^{2}-12a\ddot{a}+3a^{2}R]F_{R}-3a^{2}F$
$\displaystyle=$ $\displaystyle 6a^{2}p,$ (4.31)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(4.32)
As a consequence,
$\displaystyle 6\dot{R}HF_{RR}-(R-6H^{2})F_{R}+F$ $\displaystyle=$
$\displaystyle 2\rho,$ (4.33)
$\displaystyle-2\dot{R}^{2}F_{RRR}+[-4\dot{R}H-2\ddot{R}]F_{RR}+[-2H^{2}-4a^{-1}\ddot{a}+R]F_{R}-F$
$\displaystyle=$ $\displaystyle 2p,$ (4.34)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(4.35)
This system corresponds to the one in equations (2.5)-(2.7). We have shown
that our model contents $F(R)$ and $F(T)$ gravity models as particular cases.
In this sense it is the generalizations of these two known modified gravity
theories.
### 4.5 The M37A \- model
For the M37A \- model we have $u\neq 0,\quad v=0$ so that
$\displaystyle S_{37A}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.36) $\displaystyle R$ $\displaystyle=$
$\displaystyle u+6(\dot{H}+2H^{2}),$ (4.37) $\displaystyle T$ $\displaystyle=$
$\displaystyle-6H^{2}.$ (4.38)
### 4.6 The M37B \- model
If we consider the case $u=0,\quad v\neq 0$, then we get the M37B \- model
with
$\displaystyle S_{37B}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.39) $\displaystyle R$ $\displaystyle=$
$\displaystyle 6(\dot{H}+2H^{2}),$ (4.40) $\displaystyle T$ $\displaystyle=$
$\displaystyle v-6H^{2}.$ (4.41)
### 4.7 The M37C \- model
Now we consider the case $v=\zeta(u)$. We get the M37C \- model with
$\displaystyle S_{37B}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.42) $\displaystyle R$ $\displaystyle=$
$\displaystyle u+6(\dot{H}+2H^{2}),$ (4.43) $\displaystyle T$ $\displaystyle=$
$\displaystyle\zeta(u)-6H^{2},$ (4.44)
where in general $\zeta$ is a function to be defined e.g.
$\zeta=\zeta(t;a,\dot{a},\ddot{a},\dddot{a},...;\varsigma;u)$ and $\varsigma$
is an unknown function.
### 4.8 The M37D \- model
Now we consider the particular case of $u=\xi(v)$ and we get the M37D \- model
with
$\displaystyle S_{37B}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.45) $\displaystyle R$ $\displaystyle=$
$\displaystyle\xi(v)+6(\dot{H}+2H^{2}),$ (4.46) $\displaystyle T$
$\displaystyle=$ $\displaystyle v-6H^{2},$ (4.47)
where in general $\xi$ is a function to be defined e.g.
$\xi=\xi(t;a,\dot{a},\ddot{a},\dddot{a},...;\varsigma;v)$ and $\varsigma$ is
an unknown function.
### 4.9 The M37E \- model
Finally we consider the case $u=v=0$ and we get the M37E \- model with
$\displaystyle S_{37E}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.48) $\displaystyle R$ $\displaystyle=$
$\displaystyle 6(\dot{H}+2H^{2}),$ (4.49) $\displaystyle T$ $\displaystyle=$
$\displaystyle-6H^{2}.$ (4.50)
About this model we have some doubt related with the equation
$\dot{T}=-2(R+3T)\sqrt{-\frac{T}{6}}$ (4.51)
which follows from (4.49)-(4.50) by avoiding the variable $H$. This equation
tell us that we have only one independent dynamical variable $R$ or $T$. It
turns out that the model (4.48)-(4.50) is not of the type of $F(R,T)$ gravity,
but is equivalent to $F(R)$ or $F(T)$ gravity only. This is why in this paper
we introduced some new functions like $u,v$ and $w$ with the (temporally?)
unknown geometrical nature.
### 4.10 The M37F \- model
The M37F \- model corresponds to the case
$R=0,\quad T\neq 0$ (4.52)
that is
$u=-6(\dot{H}+2H^{2})$ (4.53)
As a consequence the M37F \- model reads
$\displaystyle S_{37J}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.54) $\displaystyle R$ $\displaystyle=$
$\displaystyle 0,$ (4.55) $\displaystyle T$ $\displaystyle=$ $\displaystyle
v-6H^{2}.$ (4.56)
We can see that the M37F \- model is in fact a generalization of $F(T)$
gravity.
### 4.11 The M37G \- model
We obtain the M37G \- model by assuming
$R\neq 0,\quad T=0$ (4.57)
that is
$v=6H^{2}.$ (4.58)
In this way we write the M37G \- model as
$\displaystyle S_{37J}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (4.59) $\displaystyle R$ $\displaystyle=$
$\displaystyle u+6(\dot{H}+2H^{2}),$ (4.60) $\displaystyle T$ $\displaystyle=$
$\displaystyle 0.$ (4.61)
This model is in fact a generalization of $F(R)$ gravity.
## 5 The particular model: $F(R,T)=\mu R+\nu T$
The equations of $F(R,T)$ gravity are much more complicated with respect to
the ones of GR even for FRW metric. For this reason let us consider the
following simplest particular model
$F(R,T)=\nu T+\mu R,$ (5.1)
where $\mu$ and $\nu$ are some real constants. The equations system of
$F(R,T)$ gravity becomes
$\displaystyle\mu D_{1}+\nu E_{1}+K(\nu T+\mu R)$ $\displaystyle=$
$\displaystyle-2a^{3}\rho,$ (5.2) $\displaystyle\mu A_{1}+\nu B_{1}+M(\nu
T+\mu R)$ $\displaystyle=$ $\displaystyle 6a^{2}p,$ (5.3)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0,$ (5.4)
where
$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle-6a\dot{a}^{2}+a^{3}u_{\dot{a}}\dot{a}-a^{3}(u-R)=6a^{2}\ddot{a}+a^{3}\dot{a}u_{\dot{a}}=a^{3}(6\frac{\ddot{a}}{a}+\dot{a}u_{\dot{a}}),$
(5.5) $\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle-6a\dot{a}^{2}+a^{3}v_{\dot{a}}\dot{a}-a^{3}(v-T)=-12a\dot{a}^{2}+a^{3}\dot{a}v_{\dot{a}}=a^{3}(-12\frac{\dot{a}^{2}}{a^{2}}+\dot{a}v_{\dot{a}}),$
(5.6) $\displaystyle K$ $\displaystyle=$ $\displaystyle-a^{3},$ (5.7)
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
12\dot{a}^{2}+6a\ddot{a}+3a^{2}\dot{a}u_{\dot{a}}+a^{3}\dot{u}_{\dot{a}}-a^{3}u_{a},$
(5.8) $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-24\dot{a}^{2}-12a\ddot{a}+3a^{2}\dot{a}v_{\dot{a}}+a^{3}\dot{v}_{\dot{a}}-a^{3}v_{a},$
(5.9) $\displaystyle M$ $\displaystyle=$ $\displaystyle-3a^{2},$ (5.10)
$\displaystyle R$ $\displaystyle=$ $\displaystyle
u+6\frac{\ddot{a}}{a}+6\frac{\dot{a}^{2}}{a^{2}}=u+6(\dot{H}+2H^{2}),$ (5.11)
$\displaystyle T$ $\displaystyle=$ $\displaystyle
v-6\frac{\dot{a}^{2}}{a^{2}}=v-6H^{2}.$ (5.12)
We get
$\displaystyle-6(\mu+\nu)\frac{\dot{a}^{2}}{a^{2}}+\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu v$ $\displaystyle=$ $\displaystyle-2\rho,$ (5.13)
$\displaystyle-2(\mu+\nu)(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a})+\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu
v+\frac{\mu}{3}a(\dot{u}_{\dot{a}}-u_{a})+\frac{\nu}{3}a(\dot{v}_{\dot{a}}-v_{a})$
$\displaystyle=$ $\displaystyle 2p,$ (5.14)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(5.15)
May rewrite it as
$\displaystyle
3(\mu+\nu)\frac{\dot{a}^{2}}{a^{2}}-0.5(\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu v)$ $\displaystyle=$ $\displaystyle\rho,$ (5.16)
$\displaystyle(\mu+\nu)(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a})-0.5(\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu
v)-\frac{\mu}{6}a(\dot{u}_{\dot{a}}-u_{a})-\frac{\nu}{6}a(\dot{v}_{\dot{a}}-v_{a})$
$\displaystyle=$ $\displaystyle-p,$ (5.17)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(5.18)
or
$\displaystyle
3(\mu+\nu)H^{2}-0.5(\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu u-\nu v)$
$\displaystyle=$ $\displaystyle\rho,$ (5.19)
$\displaystyle(\mu+\nu)(2\dot{H}+3H^{2})-0.5(\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu
v)-\frac{\mu}{6}a(\dot{u}_{\dot{a}}-u_{a})-\frac{\nu}{6}a(\dot{v}_{\dot{a}}-v_{a})$
$\displaystyle=$ $\displaystyle-p,$ (5.20)
$\displaystyle\dot{\rho}-3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(5.21)
This system contents 2 equations and 5 unknown functions ($a,\rho,p,u,v$).
Note that the EoS parameter is given by
$\omega=\frac{p}{\rho}=-1-\frac{2(\mu+\nu)\dot{H}-\frac{\mu}{6}a(\dot{u}_{\dot{a}}-u_{a})-\frac{\nu}{6}a(\dot{v}_{\dot{a}}-v_{a})}{3(\mu+\nu)H^{2}-0.5(\mu\dot{a}u_{\dot{a}}+\nu\dot{a}v_{\dot{a}}-\mu
u-\nu v)}.$ (5.22)
Now we assume
$u=\alpha a^{n},\quad v=\beta a^{m},$ (5.23)
where $n,m,\alpha,\beta$ are some real constants so that we have
$u=\alpha\left(\frac{v}{\beta}\right)^{\frac{n}{m}},\quad
v=\beta\left(\frac{u}{\alpha}\right)^{\frac{m}{n}},$ (5.24)
Then, the previous system (5.16)-(5.18) leads to
$\displaystyle 3(\mu+\nu)\frac{\dot{a}^{2}}{a^{2}}+0.5(\mu\alpha
a^{n}+\nu\beta a^{m})$ $\displaystyle=$ $\displaystyle\rho,$ (5.25)
$\displaystyle(\mu+\nu)(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a})+\frac{\mu\alpha(n+3)}{6}a^{n}+\frac{\nu\beta(m+3)}{6}a^{m}$
$\displaystyle=$ $\displaystyle-p,$ (5.26)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0$ (5.27)
or
$\displaystyle 3(\mu+\nu)H^{2}+0.5(\mu\alpha a^{n}+\nu\beta a^{m})$
$\displaystyle=$ $\displaystyle\rho,$ (5.28)
$\displaystyle(\mu+\nu)(2\dot{H}+3H^{2})+\frac{\mu\alpha(n+3)}{6}a^{n}+\frac{\nu\beta(m+3)}{6}a^{m}$
$\displaystyle=$ $\displaystyle-p,$ (5.29)
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0.$
(5.30)
## 6 Cosmological implications. Dark energy
Here we are interested in the cosmological implications of the model relating
to the dark energy problem. In order to satisfy our interest, let us consider
the power-law solution in the form
$a=a_{0}t^{\eta},$ (6.1)
where $a_{0}$ and $\eta$ are contants. Thus,
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle
3(\mu+\nu)\eta^{2}t^{-2}+0.5(\mu\alpha a_{0}^{n}t^{\eta n}+\nu\beta
a_{0}^{m}t^{\eta m}),$ (6.2) $\displaystyle p$ $\displaystyle=$
$\displaystyle-[(\mu+\nu)(-2\eta+3\eta^{2})t^{-2}+\frac{\mu\alpha(n+3)}{6}a_{0}^{n}t^{\eta
n}+\frac{\nu\beta(m+3)}{6}a_{0}^{m}t^{\eta m}].$ (6.3)
Figure 1: The evolution of the EoS parameter $\omega(t)$ with respect of the
cosmic time $t$ for Eq. (125)
The EoS parameter reads
$\omega=\frac{p}{\rho}=-1-\frac{-2\eta(\mu+\nu)+\frac{\mu\alpha
n}{6}a_{0}^{n}t^{\eta n}+\frac{\nu\beta m}{6}a_{0}^{m}t^{\eta
m}}{3(\mu+\nu)\eta^{2}t^{-2}+0.5(\mu\alpha a_{0}^{n}t^{\eta n}+\nu\beta
a_{0}^{m}t^{\eta m})}.$ (6.4)
These expressions still content some unknown constant parameters. We assume
that these parameters have the following values, namely
$\mu=\nu=1=m=n=\alpha=\beta=a_{0}$, $\eta=2/3$. TIn this case one has
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{8}{3}t^{-2}+t^{2/3},$
(6.5) $\displaystyle p$ $\displaystyle=$ $\displaystyle-\frac{4}{3}t^{2/3},$
(6.6)
so that the EoS takes the form
$\rho=\frac{512}{81p^{3}}+\frac{3p}{4}.$ (6.7)
Furthermore, the EoS parameter becomes
$\omega(t)=\frac{p}{\rho}=-\frac{4}{3+8t^{-8/3}}=-\frac{4t^{8/3}}{3t^{8/3}+8}.$
(6.8)
Hence, we see that $\omega(0)=0$, $\omega(1)=-4/11=\approx-0.36$ and
$\omega(\infty)=-4/3\approx-1,33$, so that our particular case admits the
phantom crossing for $\omega=-1$ as $t_{0}=8^{3/8}$. In Fig.1 we plot the
evolution of the EoS parameter with respect to the cosmic time $t$. It is
interesting to compare this result with the torsionless case with
$\nu=\alpha=\beta=0$, by taking the same values for all the other parameters,
namely $\mu=1$ and $\eta=2/3$, which is the case of GR. As a consequence $p=0$
and $\rho=\frac{8}{3t^{2}}$, which describe the dust matter.
## 7 $F(R,T)$ gravity: Bianchi type I model
The results of the section 3 can be extendent to the other metric. As an
example, let us consider the M37 \- model for the Bianchi type spacetime. The
corresponding metric is given by
$ds^{2}=-d\tau^{2}+A^{2}dx_{1}^{2}+B^{2}dx_{2}^{2}+C^{2}dx_{3}^{2},$ (7.1)
In this case the M37 \- model reads as
$\displaystyle S_{39}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(R,T)+L_{m}],$ (7.2) $\displaystyle R$ $\displaystyle=$
$\displaystyle
u+2\left(\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{A}\dot{C}}{AC}+\frac{\dot{B}\dot{C}}{BC}\right),$
(7.3) $\displaystyle T$ $\displaystyle=$ $\displaystyle
v-2\left(\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{A}\dot{C}}{AC}+\frac{\dot{B}\dot{C}}{BC}\right).$
(7.4)
Here
$u=u(t,A,B,C,\dot{A},\dot{B}.\dot{C},\ddot{A},\ddot{B},\ddot{C},,...;f_{i})$
and
$v=v(t,A,B,C,\dot{A},\dot{B}.\dot{C},\ddot{A},\ddot{B},\ddot{C},,...;g_{i}).$
## 8 Other generalizations of some generalized gravity models
### 8.1 The $F(G)$ with $w$ field
Now we consider the M39 \- model which looks like
$\displaystyle S_{39}$ $\displaystyle=$ $\displaystyle\int
d^{4}xe[F(G)+L_{m}],$ (8.1) $\displaystyle G$ $\displaystyle=$ $\displaystyle
w+24H^{2}(\dot{H}+H^{2}),$ (8.2) $\displaystyle w$ $\displaystyle=$
$\displaystyle w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i}),$ (8.3)
where, again, $w=w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i})$ is a real
function and $h_{i}$ is an unknown function related to the geometry of the
spacetime. If $w=0$ the M39 \- model reduces to the usual $F(G)$ gravity with
$G=G_{s}=24H^{2}(\dot{H}+H^{2})$.
### 8.2 The M40 \- model
Now we consider the M40 \- model which reads
$S_{40}=\int d^{4}xe[F(R,G)+L_{m}],$ (8.4)
where
$\displaystyle R$ $\displaystyle=$ $\displaystyle u+6(\dot{H}+2H^{2}),$ (8.5)
$\displaystyle G$ $\displaystyle=$ $\displaystyle w+24H^{2}(\dot{H}+H^{2}),$
(8.6) $\displaystyle u$ $\displaystyle=$ $\displaystyle
u(t,a,\dot{a},\ddot{a},\dddot{a},...;f_{i}),$ (8.7) $\displaystyle w$
$\displaystyle=$ $\displaystyle w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i}).$
(8.8)
Here, $u=u(t,a,\dot{a},\ddot{a},\dddot{a},...;f_{i})$ and
$w=w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i})$ are some real functions and
$f_{i},h_{i},g_{i}$ are some unknown functions relatedto the geometry of the
spacetime. Note that if we put $u=w=0$, the M40 \- model reduces to the usual
$F(R,G)$ gravity.
### 8.3 The M38 \- model
Let us consider the following action of the M38 \- model
${\cal S}_{38}=\int d^{4}xe[F(G,T)+L_{m}],$ (8.9)
where
$\displaystyle G$ $\displaystyle=$ $\displaystyle w+24H^{2}(\dot{H}+H^{2}),$
(8.10) $\displaystyle T$ $\displaystyle=$ $\displaystyle v-6H^{2},$ (8.11)
$\displaystyle w$ $\displaystyle=$ $\displaystyle
w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i}),$ (8.12) $\displaystyle v$
$\displaystyle=$ $\displaystyle v(t,a,\dot{a},\ddot{a},\dddot{a},...;g_{i}).$
(8.13)
Here in general $w=w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i})$ and
$v=v(t,a,\dot{a},\ddot{a},\dddot{a},...;g_{i})$ are some real functions and
$h_{i}$ and $g_{i}$ are some unknown functions related with the geometry of
the spacetime.
### 8.4 The M41 \- model
Now we consider the M41 \- model with the following action
${\cal S}_{41}=\int d^{4}xe[F(R,G,T)+L_{m}],$ (8.14)
where
$\displaystyle R$ $\displaystyle=$ $\displaystyle u+6(\dot{H}+2H^{2}),$ (8.15)
$\displaystyle G$ $\displaystyle=$ $\displaystyle w+24H^{2}(\dot{H}+H^{2}),$
(8.16) $\displaystyle T$ $\displaystyle=$ $\displaystyle v-6H^{2},$ (8.17)
$\displaystyle u$ $\displaystyle=$ $\displaystyle
u(t,a,\dot{a},\ddot{a},\dddot{a},...;f_{i}),$ (8.18) $\displaystyle w$
$\displaystyle=$ $\displaystyle w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i}),$
(8.19) $\displaystyle v$ $\displaystyle=$ $\displaystyle
v(t,a,\dot{a},\ddot{a},\dddot{a},...;g_{i}).$ (8.20)
Here, again, $u=u(t,a,\dot{a},\ddot{a},\dddot{a},...;f_{i})$,
$w=w(t,a,\dot{a},\ddot{a},\dddot{a},...;h_{i})$ and
$v=v(t,a,\dot{a},\ddot{a},\dddot{a},...;g_{i})$ are some real functions and
$f_{i},h_{i},g_{i}$ are some unknown functions related to the geometry of the
spacetime.
### 8.5 The M42 \- model
Let us consider the M42 \- model with the action
${\cal S}_{42}=\int d^{4}xe[F(R,T)+L_{m}],$ (8.21)
where
$\displaystyle R$ $\displaystyle=$ $\displaystyle T\phi+6(\dot{H}+2H^{2}),$
(8.22) $\displaystyle T$ $\displaystyle=$ $\displaystyle R\varphi-6H^{2}$
(8.23)
Here $u=T\phi,\quad v=R\varphi$, where
$\phi=\phi(t,a,\dot{a},\ddot{a},\dddot{a},...;\phi_{i})$ and
$\varphi=\varphi(t,a,\dot{a},\ddot{a},\dddot{a},...;\varphi_{i})$ are some
unknown functions. This model admits (at least) two important particular
cases.
a) The M42A – model. Let us take $R=0$. Then $F(R,T)=F(T)$, $T=-6H^{2}$ and
$\phi=\phi_{0}=2+H^{-2}\dot{H}$, so that we get purely $F(T)$ gravity.
b) The M42B – model. Let us take now $T=0$. Then, $F(R,T)=F(R)$,
$R=6(\dot{H}+2H^{2})$ and $\varphi=\varphi_{0}=H^{2}(\dot{H}+2H^{2})^{-1}$.
This case corresponds to the purely $F(R)$ gravity.
## 9 Conclusion
As it is well known, modified gravity theories play an important role in
modern cosmology. In particular, the well-known $F(R)$ and $F(T)$ theories are
useful tools to study dark energy phenomena motivated at a fundamental level.
In the present work, we have considered the more general theory, namely the
$F(R,T)$\- models.
At first, we have written the equations of the model and we have found their
several reductions. In particular, the Lagrangian has been explicitly
constructed. The corresponding exact solutions are found for the specific
model $F(R,T)=\mu R+\nu T$ theory, for which the universe expands as
$a(t)=a_{0}t^{\eta}$. Furthermore, we have considered the physical quantities
corresponding to the exact solution, and we have found that it can describe
the expansion of our universe in an accelerated way without introducing the
dark energy.
Some remarks are in order. Of course many aspects of $F(R,T)$ theory are
actually unexplored. For example, we do not have any realistic model which
fits the cosmological data, unlike $F(R)$ or $F(T)$ theory. We do not know
viability conditions of the models, , what forms of $F(R,T)$ can be derived
from fundamental theories and so on (it may be extremely important to
reconstract a $F(R,T)$-theory by starting from some basical principles). On
the other hand, we have here shown that the $F(R,T)$ models can be serious
candidates as modified gravity models for the dark energy. Also we note that
the behaviour and the results of $F(R,T)$-gravity may be extremely different
with respect to the ones of GR, $F(R)$ and $F(T)$ gravity theories, so that
only the observation of our universe may discriminate between the various
gravity theories. We not want here discuss merits and demerits of the models
above, since we think that it requires some more accurate investigations
related to cosmological applications.
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|
arxiv-papers
| 2012-05-23T18:53:30 |
2024-09-04T02:49:31.237443
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ratbay Myrzakulov",
"submitter": "Ratbay Myrzakulov",
"url": "https://arxiv.org/abs/1205.5266"
}
|
1205.5357
|
# On the Hilbert function of one-dimensional local complete intersections 111
2010 Mathematics Subject Classification. Primary 13H10; Secondary 13H15
Key words and Phrases: one dimensional local rings, Hilbert functions,
complete intersections.
J. Elias Partially supported by MTM2010-20279-C02-01 M. E. Rossi G. Valla
###### Abstract
The Hilbert function of standard graded algebras are well understood by
Macaulay’s theorem and very little is known in the local case, even if we
assume that the local ring is a complete intersection. An extension to the
power series ring $R$ of the theory of Gröbner bases (w.r.t. local degree
orderings) enable us to characterize the Hilbert function of one dimensional
quadratic complete intersections $A=R/I$, and we give a structure theorem of
the minimal system of generators of $I$ in terms of the Hilbert function. We
find several restrictions for the Hilbert function of $A$ in the case that $I$
is a complete intersection of type $(2,b).$ Conditions for the Cohen-
Macaulyness of the associated graded ring of $A$ are given.
## 1 Introduction and preliminaries
Let $G$ be a standard graded K-algebra; by this we mean $G=P/I$ where $P={{\rm
K}[x_{1},\dots x_{n}]}$ is a polynomial ring over the field K and $I$ an
homogeneous ideal. It is clear that for every $t\geq 0$ the set $I_{t}$ of the
forms of degree $t$ in $P$ is a K-vector space of finite dimension. For every
positive integer $t$ the Hilbert function of $G$ is defined as follows:
$HF_{G}(t)=dim_{\rm K}G_{t}=dim_{\rm K}P_{t}-dim_{\rm
K}I_{t}=\binom{n+t-1}{t}-\ dim_{\rm K}I_{t}.$
Its generating function
$HS_{G}(\theta)=\sum_{t\in\mathbb{N}}HF_{G}(t)\theta^{t}$ is the Hilbert
Series of $G.$
The relevance of this notion comes from the fact that in the case $I$ is the
defining ideal of a projective variety $V,$ the dimension, the degree and the
arithmetic genus of $V$ can be immediately computed from the Hilbert Series of
$P/I.$
A fundamental theorem by Macaulay describes exactly those numerical functions
which occur as the Hilbert functions of a standard graded K-algebra.
Macaulay’s Theorem says that for each $t$ there is an upper bound for
$HF_{G}(t+1)$ in terms of $HF_{G}(t)$, and this bound is sharp in the sense
that any numerical function satisfying it can be realized as the Hilbert
function of a suitable homogeneous standard K-algebra. These numerical
functions are called “admissible” and will be described in the next section.
It is not surprising that additional properties yield further constraints on
the Hilbert function. Thus, for example, the Hilbert function of a Cohen-
Macaulay standard graded algebra is completely described by another theorem of
Macaulay which says that the Hilbert series admissible for a Cohen-Macaulay
standard graded algebra of dimension d, are of the type
$\frac{1+h_{1}\theta+\dots h_{s}\theta^{s}}{(1-z)^{d}}$
where $1+h_{1}\theta+\dots h_{s}\theta^{s}$ is admissible.
The Hilbert function of a local ring $A$ with maximal ideal $\mathfrak{m}$ and
residue field K is defined as follows: for every $t\geq 0$
$HF_{A}(t)=dim_{\rm
K}\left(\frac{\mathfrak{m}^{t}}{\mathfrak{m}^{t+1}}\right).$
It is clear that $HF_{A}(t)$ is equal to the minimal number of generators of
the ideal $\mathfrak{m}^{t}$ and we can see that the Hilbert function of the
local ring $A$ is the Hilbert function of the following standard graded
algebra
$gr_{\mathfrak{m}}(A)=\oplus_{t\geq 0}\ \mathfrak{m}^{t}/\mathfrak{m}^{t+1}.$
This algebra is called the associated graded ring of the local ring
$(A,\mathfrak{m})$ and corresponds to a relevant geometric construction in the
case $A$ is the localization at the origin O of the coordinate ring of an
affine variety $V$ passing through O. It turns out that $gr_{\mathfrak{m}}(A)$
is the coordinate ring of the Tangent Cone of $V$ at O, which is the cone
composed of all lines that are the limiting positions of secant lines to $V$
in O.
Despite the fact that the Hilbert function of a standard graded K-algebra $G$
is so well understood in the case $G$ is Cohen-Macaulay, very little is known
in the local case. This mainly because, passing from the local ring $A$ to its
associated graded ring, many of the properties can be lost. This is the reason
why we are very far from a description of the admissible Hilbert functions for
a Cohen-Macaulay local ring when $gr_{\mathfrak{m}}(A)$ is not Cohen-Macaulay.
We only have some small knowledge of the behavior of these numerical
functions.
An example by Herzog and Waldi (see [10]) shows that the Hilbert function of a
one dimensional Cohen-Macaulay local ring can be decreasing, even the number
of generators of the square of the maximal ideal can be less than the number
of generators of the maximal ideal itself. Further, without restrictions on
the embedding dimension, the Hilbert function of a one dimensional Cohen-
Macaulay local ring can present arbitrarily many ”valleys” (see [5]).
Even if we restrict ourselves to the case of a complete intersection, very
little is known. In [16] it has been proved that the Hilbert function of a
positive dimensional codimension two complete intersection $R/(f,g)$ is non
decreasing, but we have no answer to the question asked by Rossi (see [17])
whether the same is true for every one dimensional Gorenstein local ring.
In the case that the embedding dimension of the local ring is at most three,
the first author gave a positive answer to a question stated by J. Sally, by
proving that the Hilbert function of a one dimensional Cohen-Macaulay local
ring is increasing (see [4]). But examples show that this is not true anymore
if the embedding dimension is bigger than three.
All this amount of results shows that, without strong assumptions, the Hilbert
function of a one-dimensional Cohen-Macaulay local ring could be very wild.
This is the reason why, in this paper, we restrict ourselves to the case
$A={{\rm K}[\\![x,y,z]\\!]}/I,$ where the ideal $I\subseteq(x,y,z)^{2}$ is
generated by a regular sequence $\\{f,g\\}$ of elements of $R$. We will see
that, even with all these strong assumptions, the problem of determining the
admissible Hilbert functions is not so easy, possibly because it is strictly
related to the study of curve singularities in ${\mathbb{A}}^{3}.$
If we consider the corresponding Artinian problem, then we deal with a pair of
plane curves. Several papers have been written in which the Hilbert function
of an Artinian complete intersection ring $A={{\rm K}[\\![x,y]\\!]}/(f,g)$ has
been studied in terms of the invariants of the curves $f=g=0$ (see Iarrobino
[12], Goto, Heinzer, Kim [8], Kothari [13], ….). It is an early result due to
Macaulay that the Hilbert function of such a ring $A$ verifies for every
positive integer $n$ the following inequality:
$|HF_{A}(n+1)-HF_{A}(n)|\leq 1.$
It has been proved that given such a numerical function, there exists a
complete intersection $I=(f,g)\subseteq{{\rm K}[\\![x,y]\\!]}$ with that
Hilbert function, [1], [8]. Hence the problem is solved in the Artinian case
and, more in general, when $gr_{\mathfrak{m}}(A)$ is Cohen-Macaulay.
Conditions on the Cohen-Macaulayness of $gr_{\mathfrak{m}}(A)$ have been
studied by Goto, Heinzer and Kim in [6], [7].
Classical results concerning Cohen-Macaulay local rings of dimension one will
be useful in this paper. For example it is well known, see [14],[4], [18],
that there exists an integer $e\geq 1$, the multiplicity of $A,$ such that
1. (i)
$HF_{A}(n)\leq e$ for all $n$,
2. (ii)
If $HF_{A}(j)=e$ for some $j$, then $HF_{A}(n)=e$ for all $n\geq j$,
3. (iii)
For every $j\geq 0$ we have $HF_{A}(j)\geq\min\\{j+1,e\\}.$ In particular
$HF_{A}(e-1)=e.$
The least integer $r$ such that $HF(r)=e$ coincides with the reduction number
of $\mathfrak{m},$ which is the least integer $r$ such that
$\mathfrak{m}^{r+1}=x\mathfrak{m}^{r}$ for some (hence any) superficial
element $x\in\mathfrak{m}.$ We say that the Hilbert function of $A$ is
increasing (resp. strictly increasing ) if $HF(n)\leq HF_{A}(n+1)$ (resp.
$HF(n)<HF_{A}(n+1)$) for all $n=0,\cdots,r-1.$
Throughout the whole paper ${\rm K}$ denotes an algebraically closed field of
characteristic zero. Let $R={{\rm K}[\\![x_{1},\dots x_{n}]\\!]}$ be the ring
of formal power series in the indeterminates $\\{x_{1},\cdots,x_{n}\\}$ with
coefficients in ${\rm K}$ and maximal ideal
$\mathcal{M}=(x_{1},\cdots,x_{n}).$ We denote by $\mathbb{U}(R)$ the group of
units of $R$. Let $I$ be an ideal of $R$ and consider the local ring $A=R/I$
whose maximal ideal is $\mathfrak{m}:={\mathcal{M}}/I.$
We have seen that the Hilbert function of a local ring $A$ is the same as that
of the associated graded ring $gr_{\mathfrak{m}}(A).$ Hence it will be useful
to recall the presentation of this standard graded algebra. For every power
series $f\in R\setminus\\{0\\}$ we can write $f=f_{v}+f_{v+1}+\cdots$, where
$f_{v}$ is not zero and $f_{j}$ is an homogeneous polynomial of degree $j$ in
$P$ for every $j\geq v.$ We say that $v$ is the order of $f$, denote $f_{v}$
by $f^{*}$ and call it the initial form of $f.$ If $f=0$ we agree that its
order is $\infty.$ It is well known that $gr_{\mathfrak{m}}(A)=P/I^{*}$, where
$I^{*}$ is the homogeneous ideal of the polynomial ring $P$ generated by the
initial forms of the elements of $I.$ A set of power series
$f_{1},\cdots,f_{r}\in I$ is a standard basis of $I$ if
$I^{*}=(f_{1}^{*},\cdots,f_{r}^{*})$, (see [11]). It is clear that every ideal
$I$ has a standard basis and that every standard basis is a basis. However not
every basis is a standard basis. To determine a standard basis of a given
ideal of $R$ is a classical hard problem, even in the very special case we are
involved with in this paper.
In order to determine the Hilbert function of such local complete
intersections it seems to be hopeless to use only the theory of tangent cones.
Instead we found crucial to consider the extension to the power series ring of
the theory of Gröbner bases introduced by Buchberger for ideals in the
polynomial ring. We can say that a mixture of the theory of enhanced standard
basis with that of the ideals of initial forms has been the winning strategy
for us. The use of the theory of enhanced standard bases for studying of the
Hilbert function of a local ring seems to be unusual, while there are several
papers in the Theory of Singularities where this topic is essential.
We recall that the notion of Gröbner basis is defined by considering a term
ordering on the terms of $P$ (i.e. a monomial ordering where all the terms are
bigger than $1$). Instead, we need here to consider the so called local degree
ordering, see [9], Chapter 6, a monomial ordering on the terms of $P$ which
is not a term ordering.
We denote by ${\mathbb{T}^{n}}$ the set of terms or monomials of $P$; let
$\tau$ be a term ordering in ${\mathbb{T}^{n}}$, and we assume that
$x_{1}>\cdots>x_{n}$. We define a new total order ${\overline{\tau}}$ on
${\mathbb{T}^{n}}$ in the following way: given
$m_{1},m_{2}\in{\mathbb{T}^{n}}$ we let $m_{1}>_{{\overline{\tau}}}m_{2}$ if
and only if ${\rm{deg}}(m_{1})<{\rm{deg}}(m_{2})$ or
${\rm{deg}}(m_{1})={\rm{deg}}(m_{2})$ and $m_{1}>_{\tau}m_{2}$. Given $f\in R$
we denote by ${\rm{Supp}}(f)$ the support of $f$, i.e. if
$f=\sum_{{\underline{i}}\in\mathbb{N}^{n}}a_{{\underline{i}}}x^{{\underline{i}}}$
then ${\rm{Supp}}(f)$ is the set of terms $x^{\underline{i}}$ such that
$a_{{\underline{i}}}\neq 0$. We remark that, given $f$ in $R$, there is a
monomial which is the biggest of the monomials in ${\rm{Supp}}(f)$ with
respect to ${\overline{\tau}}$: namely, since the support of $f^{*}$ is a
finite set, we can take the maximum with respect to $\tau$ of the elements of
this set. This monomial is called the leading monomial of $f$ with respect to
${\overline{\tau}}$ and is denoted by $Lt_{{\overline{\tau}}}(f).$ By
definition we have
$Lt_{{\overline{\tau}}}(f)=Lt_{\tau}(f^{*}).$
As usual we define the leading term ideal associated to an ideal $I\subset R$
as the monomial ideal ${\rm{Lt}}_{{\overline{\tau}}}(I)$ generated in $R$ by
${\rm{Lt}}_{{\overline{\tau}}}(f)$ with $f$ running in $I$.
In [1] a set $\\{f_{1},\dots,f_{r}\\}$ of elements of $I$ is called an
enhanced standard basis of $I$ if the corresponding leading terms generate
${\rm{Lt}}_{{\overline{\tau}}}(I).$ Every enhanced standard basis is also a
standard basis, but the converse is not true. In [9] an enhanced standard
basis of $I$ is simply called a standard basis. We have
${\rm{Lt}}_{{\overline{\tau}}}(I)P=Lt_{\tau}(I^{*})$ (see [1] Proposition
1.5.) so that
$HF_{R/I}=HF_{P/I^{*}}=HF_{R/{\rm{Lt}}_{{\overline{\tau}}}(I)}.$
In the Theory of enhanced standard basis a crucial result is the Grauert’s
Division theorem, [9, Theorem 6.4.1]. It claims the following. Given a set of
formal power series $f,f_{1},\cdots,f_{m}\in R$ there exist power series
$q_{1},\dots,q_{m},r\in R$ such that $f=\sum_{j=1}^{m}q_{j}f_{j}+r$ and, for
all $j=1,\dots,m$,
1. (1)
No monomial of $r$ is divisible by $Lt_{{\overline{\tau}}}(f_{j})$,
2. (2)
$Lt_{{\overline{\tau}}}(q_{j}f_{j})\leq Lt_{{\overline{\tau}}}(f)$ if
$q_{j}\neq 0.$
With the above result we can define
$NF(f|\\{f_{1},\dots,f_{m}\\}):=r$
and obtain in this way a reduced normal form of any power series $f$ with
respect to a given finite subset of $R$. The existence of a reduced normal
form is the basis to obtain, in the formal power series ring, all the
properties of Gröbner basis already proved in the classical case. In
particular Buchberger’s criterion holds for the power series ring ${{\rm
K}[\\![x_{1},\dots x_{n}]\\!]}$, see [9, Theorem 1.7.3]. A similar approach
was introduced by Mora in 1982 in the localization of $P,$ (see [15]).
We come now to describe the content of the paper. The main result is the
description of all the numerical functions which are the Hilbert functions of
what we call a quadratic complete intersection of codimension two in ${{\rm
K}[\\![x,y,z]\\!]}.$ By this we mean local rings of type ${{\rm
K}[\\![x,y,z]\\!]}/(f,g)$ where $f$ and $g$ are power series of order two
which form a regular sequence in ${{\rm K}[\\![x,y,z]\\!]}$ with the property
that $g^{*}\notin(f^{*}).$
We first prove in Proposition 2.2 that for the Hilbert function of such local
rings with a given multiplicity $e,$ there are only two possibilities:
1. (1)
either is increasing by one up to reach the multiplicity, say
$\\{1,3,4,5,6,7,...,e-1,e,e,e....\\},$
2. (2)
or it is increasing by one with a flat in position $n$ which is unique, by
which we mean that for some $n\leq e-3$ we have the sequence
0 | 1 | 2 | 3 | 4 | … | n-1 | n | n+1 | n+2 | … | e-2 | e-1 | e | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 3 | 4 | 5 | 6 | … | n+1 | n+2 | n+2 | n+3 | … | e-1 | e | e | …
It turns out that if the Hilbert function is increasing by one, case (1),
there is no constriction on the multiplicity. Instead, if the Hilbert function
has a flat, case (2), the multiplicity $e$ cannot be too big, namely we must
have $e\leq 2n.$ This unexpected result is proved in Theorem 3.6 which is the
main result of this paper. Examples 2.3 and 3.7 show that the above Hilbert
functions are realizable.
We present also two more results on the Cohen-Macaulayness of the tangent cone
of such complete intersections. First, in Proposition 2.5, we prove that a
quadratic complete intersection of codimension two in ${{\rm
K}[\\![x,y,z]\\!]}$ with Hilbert function increasing by one has an associated
graded ring which is Cohen-Macaulay. Finally, as a second application of the
methods we used in the proof of the main theorem, we are able to prove in
Proposition 3.8 that for a quadratic complete intersection $A={{\rm
K}[\\![x,y,z]\\!]}/I,$ the tangent cone is Cohen-Macaulay in the case the
vector space $I^{*}_{\ 2}$ does not contain a square of a linear form.
Section four is devoted to give a structure theorem, modulo analytic
isomorphisms, of the minimal system of generators of quadratic complete
intersection ideals $I$ of codimension two in ${{\rm K}[\\![x,y,z]\\!]}$,
Theorem 4.1 and Theorem 4.2. These results are obtained by taking into account
the two possible Hilbert functions that can occur for such an ideal.
In the last section of the paper we give several examples to illustrate our
results, as well as possible extensions.
## 2 Ideals of type $(2,b)$
From now on we assume that $A={{\rm K}[\\![x,y,z]\\!]}/I$ where $I$ is a
codimension two complete intersection ideal of $R={{\rm K}[\\![x,y,z]\\!]}$.
Given the integers $b\geq a\geq 2,$ we say that $A$ is of type $(a,b)$, or $I$
is of type $(a,b),$ if $I$ can be generated by a regular sequence $\\{f,g\\}$
such that order(f)=a, order(g)=b and $g^{*}\not\in(f^{*}).$ In the language of
[11, Chapter III, Section 1] we write $\nu^{*}(I)=(a,b)$ with the meaning that
$I$ is of type $(a,b).$
In this paper we will be mainly concerned with local rings of type $(2,2)$;
however in this section properties of local rings of type $(2,b)$ will be
considered.
In the following Proposition we prove that the Hilbert function of a local
ring of type $(2,b)$ verifies for every $n\geq 1$ the inequalities
$0\leq HF_{A}(n+1)-HF_{A}(n)\leq 1.$ (1)
The question now is whether every numerical function $H$ such that $H(0)=1$,
$H(1)=3$ and verifies (1) is the Hilbert function of some local ring of type
$(2,b).$ This is not the case because, for example, the numerical function
$\\{1,3,4,5,5,6,7,7,....\\}$ verifies (1) but we will see later that it cannot
be the Hilbert function of a local ring of type $(2,b).$
A local ring $A$ of type $(2,b)$ is Cohen-Macaulay of embedding dimension
three so that we know that the Hilbert function is not decreasing. We say that
$HF_{A}$ admits a flat in position $n$ if
$HF_{A}(n)=HF_{A}(n+1)<e.$
The first basic properties of the Hilbert function of a local ring of type
$(2,b)$ are collected in the following proposition which is an easy
consequence of the classical Macaulay Theorem.
We recall that given two positive integer $n$ and $c$, the $n$-binomial
expansion of c is
$c=\binom{c_{n}}{n}+\binom{c_{n-1}}{n-1}+\cdots\binom{c_{j}}{j}$
where $c_{n}>c_{n-1}>\cdots c_{j}\geq j\geq 1.$ We let
$c^{<n>}=\binom{c_{n}+1}{n+1}+\binom{c_{n-1}+1}{n}+\cdots\binom{c_{j}+1}{j+1}.$
The Theorem of Macaulay states that a numerical function
$\\{h_{0},h_{1},\cdots,h_{i},\cdots,\\}$ is the Hilbert function of a standard
graded algebra if and only if $h_{0}=1$ and $h_{i+1}\leq h_{i}^{<i>}$ for
every $i\geq 1.$ We remark that if $n+1\leq c\leq 2n$ then the $n$-binomial
expansion of c is
$c=\binom{n+1}{n}+\binom{n-1}{n-1}+\cdots\binom{2n-c+1}{2n-c+1},$
so that $c^{<n>}=c+1.$
Further, if $f_{1},\dots,f_{r}$ are elements of order $d_{1},\dots,d_{r}$ in
the regular local ring $(R,\mathcal{M})$ and $J$ the ideal they generate, it
is known that
$J^{*}_{n}=(J\cap\mathcal{M}^{n}+\mathcal{M}^{n+1})/\mathcal{M}^{n+1}$
and
$(f_{1}^{*},\dots,f^{*}_{r})_{n}=(\sum_{i=1}^{r}\mathcal{M}^{n-d_{i}}f_{i}+\mathcal{M}^{n+1})/\mathcal{M}^{n+1}$
for every non negative integer $n.$ With this notation we have the following
basic lemma.
###### Lemma 2.1.
Let $I=(f,g)$ be an ideal of $(R,\mathcal{M})$ with $order(f)=2\leq
order(g)=b.$ Then
1. (i)
$I^{*}_{\ j}=(f^{*})_{j}$ for every integer $2\leq j<b.$
2. (ii)
$I^{*}_{\ b}=(f^{*},g^{*})_{b}.$
3. (iii)
If $g^{*}\notin(f^{*})$ then $I^{*}_{\ b+1}=(f^{*},g^{*})_{b+1}.$
###### Proof.
Since $j+1\leq b$ we have
$g\in\mathcal{M}^{b}\subseteq\mathcal{M}^{j+1}\subseteq\mathcal{M}^{j}$, hence
$(f,g)\cap\mathcal{M}^{j}+\mathcal{M}^{j+1}=(g)+(f)\cap\mathcal{M}^{j}+\mathcal{M}^{j+1}=f\mathcal{M}^{j-2}+\mathcal{M}^{j+1}.$
The first assertion follows. We prove now (ii). We have:
$(f,g)\cap\mathcal{M}^{b}=(g)+(f)\cap\mathcal{M}^{b}=(g)+f\mathcal{M}^{b-2}.$
As for (iii) we need to prove that if $g^{*}\notin(f^{*})$ then
$(f,g)\cap\mathcal{M}^{b+1}=f\mathcal{M}^{b-1}+g\mathcal{M}.$ The inclusion
$\supseteq$ is clear, so let $\alpha=cf+dg\in\mathcal{M}^{b+1}.$ If
$d\in\mathcal{M}$ then $cf\in\mathcal{M}^{b+1}$ and this implies
$c\in\mathcal{M}^{b-1}$ as required. If $d\notin\mathcal{M}$ then
$g\in((f)+\mathcal{M}^{b+1})\cap\mathcal{M}^{b}=\mathcal{M}^{b+1}+f\mathcal{M}^{b-2}$
which implies $g^{*}\in(f^{*}),$ a contradiction. ∎
###### Proposition 2.2.
Let $A=R/I$ be a local ring of type $(2,b)$ and $I=(f,g)$ with
${\rm{order}}(f)=2$, ${\rm{order}}(g)=b$ and $g^{*}\not\in(f^{*}).$ Then the
following properties hold.
1. (i)
$HF_{A}(j)=2j+1$ if $j<b.$
2. (ii)
$HF_{A}(b)=2b.$
3. (iii)
$HF_{A}(j-1)\leq HF_{A}(j)\leq HF_{A}(j-1)+1$ if $j\geq b.$
4. (iv)
$HF_{A}$ admits at most $b-1$ flats.
###### Proof.
By (i) of the above Lemma we have for every $j<b$
$HF_{A}(j)=HF_{P/I^{*}}(j)=HF_{P/(f^{*})}(j)=2j+1.$
We prove now the second assertion. By (ii) of the above Lemma we have
$HF_{A}(b)=HF_{P/I^{*}}(b)=HF_{P/(f^{*},g^{*})}(b).$
Since $g^{*}\not\in(f^{*})$ we get $HF_{A}(b)=HF_{P/(f^{*})}(b)-1=2b+1-1=2b$
as required.
As for (iii) we need only to prove that $HF_{A}(j)\leq HF_{A}(j-1)+1$ if
$j\geq b.$ We have $HF_{A}(b)=2b,$ $HF_{A}(b-1)=2b-1,$ hence we can argue by
induction on $j.$ Let $j\geq b$ and assuming $HF_{A}(j)\leq HF_{A}(j-1)+1$ we
need to prove that $HF_{A}(j+1)\leq HF_{A}(j)+1.$
We have $j+1\leq HF_{A}(j)<HF_{P/(f^{*})}(j)=2j+1,$ hence, by the remark
before the Lemma, we get
$HF_{A}(j+1)\leq HF_{A}(j)^{<j>}=HF_{A}(j)+1$
as wanted.
Finally we prove (iv). We have $HF_{A}(b)=2b$ and at each step $HF_{A}$ goes
up at most by one. Hence, if there are p flats between b and j, we have
$HF_{A}(j)=2b+j-b-p.$ But $HF_{A}(j)\geq j+1,$ so that $p\leq b-1.$ ∎
From the above proposition it follows that the Hilbert function of a local
ring of type $(2,b)$ either is strictly increasing or it has one or more flats
(no more than $b-1$); if the first is the case, it has the following shape
$HF_{A}(j)=\begin{cases}2j+1&\ \ \text{$j=0,\dots,b-1$},\\\ j+b&\ \
\text{$b\leq j\leq e-b,$}\\\ e&\ \ \text{$j\geq e-b+1.$}\\\ \end{cases}$ (2)
where $e$ and $b$ are integers, $b\geq 2$ and $e\geq 2b.$
We show with the following example that given a numerical function $H$ as in
(2) we can find a local ring of type $(2,b)$ with multiplicity $e$ whose
Hilbert function is $H.$
###### Example 2.3.
Let $b\geq 2$ and $e\geq 2b.$ We claim that the above numerical function is
the Hilbert function of the following local ring of type $(2,b)$ and
multiplicity $e.$
Let $I=(x^{2}+y^{e-2b+2},xy^{b-1})$ and $A={{\rm K}[\\![x,y,z]\\!]}/I.$ We fix
an ordering on the monomials of $P$ with the property that $x>y.$ We let
$f:=x^{2}+y^{e-2b+2},\ g:=xy^{b-1}$ and claim that
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy^{b-1},y^{e-b+1}).$
Since $e\geq 2b$ and $x>y$ it is clear that
${\rm{Lt}}_{{\overline{\tau}}}(f)=x^{2}.$ We have
$S(f,g)=y^{b-1}f-xg=y^{b-1}(x^{2}+y^{e-2b+2})-xxy^{b-1}=y^{e-b+1}.$
Let $h:=S(f,g)=y^{e-b+1},$ then
$S(f,h)=y^{e-b+1}f-x^{2}h=y^{e-b+1}(x^{2}+y^{e-2b+2})-x^{2}y^{e-b+1}=y^{2e-3b+3}=y^{e-2b+2}h$
and $S(g,h)=0.$ It follows that
$\displaystyle{\rm NF}(S(f,g)\ |\\{h\\})$ $\displaystyle=$ $\displaystyle{\rm
NF}(h\ |\\{h\\})\;=\;0,$ $\displaystyle{\rm NF}(S(f,h)\ |\\{h\\})$
$\displaystyle=$ $\displaystyle{\rm NF}(y^{e-2b+2}h\ |\\{h\\})\;=\;0,$
$\displaystyle{\rm NF}(S(g,h)\ |\\{h\\})$ $\displaystyle=$ $\displaystyle{\rm
NF}(0\ |\\{h\\})\;=\;0.$
By Buchberger criterion we get that
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy^{b-1},y^{e-b+1})$ as claimed. With
the aim of a simple computation we can prove that
$K[x,y,z]/(x^{2},xy^{b-1},y^{e-b+1})$ has the above Hilbert function; clearly
the same is true for the local ring ${{\rm
K}[\\![x,y,z]\\!]}/(x^{2}+y^{e-2b+2},xy^{b-1}).$
We end this section by proving that for a local ring of type $(2,b)$ the
condition that the Hilbert function is strictly increasing is equivalent to
the Cohen-Macaulayness of the tangent cone. First we need to prove that the
property of having type $(a,b)$ can be carried on the quotient modulo a
suitable superficial element. We recall that an element $\ell\in{\mathcal{M}}$
is superficial for ${\mathcal{M}}/I$ if $\ell$ does not belong to any of the
associated primes of $I^{*}$ different from the homogeneous maximal ideal.
Since the residual field if infinite the existence of superficial elements is
guaranteed. Moreover, it is easy to prove:
###### Proposition 2.4.
Let $I$ be an ideal of $R$ of type $(a,b)$ with $2\leq a\leq b.$ There exists
$\ell\in{\mathcal{M}}\setminus{\mathcal{M}}^{2}$ such that
1. (i)
the coset of $\ell$ in $R/I$ is superficial for ${\mathcal{M}}/I$,
2. (ii)
$\bar{I}=I+(\ell)/(\ell)$ is an ideal of $R/(\ell)$ of type $(a,b).$
###### Proof.
It is well known that $\ell$ verifies $(i)$ if $\ell^{*}$ does not belong to
any of the associated prime ideals of $I^{*}$ (different from the homogeneous
maximal ideal). Let $I=(f,g)$ be with $order(f)=a\leq order(g)=b.$ Then it is
easy to see that $\bar{I}$ satisfies $(ii)$ provided:
a) $\ell^{*}$ does not divide $f^{*}$
b) $g^{*}\not\in(f^{*},\ell^{*})$.
In fact $\bar{I}=(\bar{f},\bar{g})$ in $R/\ell$ and condition a) assures
$order(\bar{f})=a$ and condition b) gives
$\bar{g}~{}^{*}\not\in(\bar{f}~{}^{*}).$ Since
$\operatorname{depth}P/(f^{*},g^{*})\geq 1$ ($P={\rm K}[x,y,z]$), it is easy
to see that for having a) and b) it is enough to choose
$\ell\in{\mathcal{M}}\setminus{\mathcal{M}}^{2}$ such that $\ell^{*}$ is
regular in $P/(f^{*},g^{*}).$ Clearly, if this is the case, $\ell^{*}$ does
not divide $f^{*}$ and if $g^{*}\in(f^{*},\ell^{*}),$ then $g^{*}=\alpha
f^{*}+\beta\ell^{*}$ with $\alpha,\beta\in P.$ Since $\ell^{*}$ is
$P/(f^{*},g^{*})$-regular, then $\beta\in(f^{*},g^{*}).$ Hence $g^{*}=\alpha
f^{*}+\ell^{*}(\beta_{1}g^{*}+\beta_{2}\ell^{*}),$ so
$g^{*}(1-\ell\beta_{1})\in(f^{*}),$ a contradiction because
$g^{*}\not\in(f^{*}).$ Since the residue field is infinite, an element
$\ell\in{\mathcal{M}}\setminus{\mathcal{M}}^{2}$ verifying the conditions of
the proposition can be selected by avoiding the associated prime ideals to
$I^{*}$ and to $(f^{*},g^{*}).$ ∎
It is well known that if the associated graded ring $gr_{\mathfrak{m}}(A)$ is
Cohen-Macaulay, then the Hilbert function of $A$ is strictly increasing.
However the converse is in general very rare. In the following result we will
show a special case where this implication holds true.
###### Proposition 2.5.
Let $A=R/I$ be a local ring of type $(2,b).$ Then $gr_{\mathfrak{m}}(A)$ is
Cohen-Macaulay if and only if $HF_{A}$ is strictly increasing.
###### Proof.
Let $I=(f,g)$ with ${\rm{order}}(f)=2,$ ${\rm{order}}(g)=b$ and
$g^{*}\not\in(f^{*}).$ If the associated graded ring is Cohen-Macaulay, then
its Hilbert function is strictly increasing and thus the Hilbert function of
$A$ is strictly increasing as well. Assume that $HF_{A}$ is strictly
increasing. From Proposition 2.2, a simple computation gives
$\Delta HF_{A}(n):=HF_{A}(n+1)-HF_{A}(n)=\begin{cases}1&\ \ \text{$n=0$},\\\
2&\ \ \text{$n=1,\dots,b-1$},\\\ 1&\ \ \text{$n=b,\dots,r-1$},\\\ 0&\ \
\text{$n\geq r$}\\\ \end{cases}$
with $r=e-b+1$.
From Proposition 2.4 there exists a superficial element $x\in A$ such that
$HF_{A/xA}(n)=\begin{cases}1&\ \ \text{$n=0$},\\\ 2&\ \
\text{$n=1,\dots,b-1$},\\\ 1&\ \ \text{$n=b$}.\\\ \end{cases}$
From Macaulay’s characterization of Hilbert functions and the fact that
$e(A/xA)=e(A),$ we get $\Delta HF_{A}=HF_{A/xA}$. Hence $gr_{\mathfrak{m}}(A)$
is Cohen-Macaulay, [19]. ∎
Notice that the above proposition cannot be extended to local rings of type
$(a,b)$ with $a>2,$ as the following example shows. Consider the local ring
$A=R/I$ where $I=(x^{4},x^{2}y+z^{4})\subseteq R={{\rm K}[\\![x,y,z]\\!]};$
$A$ is a one-dimensional Gorenstein local ring and
$HF_{A}=\\{1,3,6,9,11,13,14,15,16,16,\dots,\\}$
is strictly increasing. Now it is clear that $x^{4},x^{2}y\in I^{*}$ and since
$x^{2}(x^{2}y+z^{4})-yx^{4}\in I,$ also $x^{2}z^{4}\in I^{*}.$ This implies
that $x^{3}z^{3}(x,y,z)\subseteq I^{*};$ since $x^{3}z^{3}\notin I^{*}$
$gr_{\mathfrak{m}}(A)$ is not Cohen-Macaulay.
A natural and general problem would be to characterize the Hilbert functions
of all the ideals $I$ of type $(2,b).$ If the Hilbert function has one or more
flat, the behavior is difficult to control. However if we denote by $p$ the
number of flats, by Proposition 2.2 we know that $p\leq b-1.$ With the aid of
huge computations made with CoCoa, we ask the following question.
###### Question 2.6.
Let $A=R/I$ be a local ring of type $(2,b)$ with $b\geq 2$ and multiplicity
$e.$ Let $n:=\min\\{j:HF_{R/I}(j)=HF_{R/I}(j+1)<e\\}$ and let $p$ be the
number of flats. Then
$e\leq(p+1)n\ (\leq bn).$
The main result of the paper answer the question in the case $a=b=2.$
## 3 The main result
In this section we present a complete characterization of the numerical
functions which are the Hilbert functions of local rings of type $(2,2).$ In
particular we prove that certain monomial ideals cannot be the initial ideals
of a complete intersection, a relevant task even in the graded setting (see
for example [3]).
By the definition we gave in the above section, a local ring $A={{\rm
K}[\\![x,y,z]\\!]}/I$ of type $(2,2)$ is of dimension one, $HF_{A}(1)=3$ and
$HF_{A}(2)=4.$ In particular $I$ can be generated by a regular sequence, say
$I=(f,g),$ where $f$ and $g$ are power series of order two such that $f^{*}$
and $g^{*}$ are linearly independent in the vector space $K[x,y,z]_{2}.$ We
recall that by Proposition 2.2 we have $I^{*}_{\ 2}=<f^{*},g^{*}>$ and
$I^{*}_{\ 3}=(f^{*},g^{*})_{3}=<f^{*}x,f^{*}y,f^{*}z,g^{*}x,g^{*}y,g^{*}z>.$
Since $HF_{A}(2)=4$ we know that $4\leq HF_{A}(3)\leq HF_{A}(2)^{<2>}=5.$ If
$HF_{A}(3)=4,$ then $6=dim(I^{*}_{\ 3})=\operatorname{dim}_{\rm
K}<f^{*}x,f^{*}y,f^{*}z,g^{*}x,g^{*}y,g^{*}z>.$ This easily implies that
$f^{*}$ and $g^{*}$ form a regular sequence in $K[x,y,z].$ As a consequence
$I^{*}=(f^{*},g^{*})$ and the Hilbert function of $A$ is
$\\{1,3,4,4,4,....\\}$ which is as in (2) with $b=2,$ $e=4.$
We want to study the remaining case, when $HF_{A}(3)=5.$ We first remark that
in this case $f^{*}$ and $g^{*}$ share a common factor, say $L,$ which must be
a linear form because $f^{*}$ and $g^{*}$ are linearly independent. Hence we
can write
$f^{*}=LM,\ \ \ \ g^{*}=LN$
where $L,M,N$ are linear forms in $K[x,y,z]$ such that $M$ and $N$ are
linearly independent. In particular $I^{*}_{\ 2}=<LM,LN>.$
We have two possibilities, either $L,M,N$ are linearly independent or are
linearly dependent. We remark that this property depends on the ideal $I$ and
not on the generators of $I$. Namely, if we say that $I^{*}_{\ 2}$ is square
free with the meaning that it does not contain a square of a linear form, we
can prove the following easy result:
###### Lemma 3.1.
With the above assumption, the vectors $L,M,N$ are linearly independent if and
only if $I^{*}_{\ 2}$ is square-free.
###### Proof.
Let us first assume that $L,M,N$ are linearly dependent. Since $M,N$ are
linearly independent we have $L=\alpha M+\beta N$ so that $L^{2}=\alpha
LM+\beta LN\in<I^{*}_{\ 2}>.$ Hence $I^{*}_{\ 2}$ is not square-free.
We prove now that if $L,M,N$ are linearly independent then $I^{*}_{\ 2}$ is
square-free. Let $P$ be a linear form such that $P^{2}\in I^{*}_{\
2}=<LM,LN>$; then $P\in(L)$ so that $P=\lambda L.$ We have
$\lambda^{2}L^{2}=\alpha LM+\beta LN$ hence $\lambda^{2}L=\alpha M+\beta N;$
since $L,M,N$ are linearly independent this implies $\lambda=0$ and finally
$P=0.$ ∎
For completeness, we need now to recall the notion of k-algebra isomorphism.
Given a set of minimal generators $\underline{y}=\\{y_{1},y_{2},...,y_{n}\\}$
of the maximal ideal of $R={{\rm K}[\\![x_{1},\dots x_{n}]\\!]}$, we let
$\phi_{\underline{y}}$ be the automorphism of $R$ which is the result of
substituting $y_{i}$ for $x_{i}$ in a power series
$f(x_{1},x_{2},...,x_{n})\in R.$ Given two ideals $I$ and $J$ in $R$ it is
well known that there exist a $K$-algebra isomorphism $\alpha:R/I\to R/J$ if
and only if for some generators $y_{1},y_{2},...,y_{n}$ of the maximal ideal
of $R$, we have $I=\phi_{\underline{y}}(J).$
We start now by deforming, up to isomorphism, the generators $f$ and $g$ of
the given ideal $I.$
###### Lemma 3.2.
Let $A=R/I$ be a local ring of type $(2,2)$ such that $HF_{A}(3)=5.$ $(i)$ If
$I^{*}_{\ 2}$ is not square-free we may assume, up to isomorphism, that
$I=(f,g)$ with $f^{*}=x^{2}$ and $g^{*}=xy$.
$(ii)$ If $I^{*}_{\ 2}$ is square-free we may assume, up to isomorphism, that
$I=(f,g)$ with $f^{*}=xy$ and $g^{*}=xz$,
###### Proof.
Let us first assume that $I^{*}_{2}$ is not square-free; then $f^{*}=LM,$
$g^{*}=LN$ with $L,M,N$ linearly dependent; since $M$ and $N$ are linearly
independent, we must have $L=\lambda M+\rho N$ for suitable $\lambda$ and
$\rho$ in $K$ with $(\lambda,\rho)\neq(0,0).$ By symmetry we may assume
$\lambda\neq 0.$ Then it is easy to see that $L$ and $N$ are linearly
independent so that we can consider an automorphism $\phi$ sending $x\to
L,y\to N.$ We have $f=LM+a$ and $g=LN+b$ for suitable $a,b\in\mathcal{M}^{3},$
and further
$L^{2}=\lambda LM+\rho LN=\lambda f+\rho g-\lambda a-\rho b.$
We get
$I=(f,g)=(\lambda f,g)=(L^{2}-\rho g+\lambda a+\rho b,g)=(L^{2}+\lambda a+\rho
b,g)=$ $=(L^{2}+\lambda a+\rho b,LN+b)=\phi((x^{2}+\phi^{-1}(\rho b+\lambda
a),xy+\phi^{-1}(b)).$
The conclusion follows.
Now we assume that $I^{*}_{\ 2}$ is square-free. Then $f^{*}=LM$ and
$g^{*}=LN$ where $L,M,N$ are linear forms in $K[x,y,z]$ which are linearly
independent. As before we have $f=LM+a$ and $g=LN+b$ for suitable
$a,b\in\mathcal{M}^{3},$
Let us consider the automorphism $\phi$ sending $x\to L,y\to M,z\to N.$ We
have
$I=(f,g)=(LM+a,LN+b)=\phi((xy+\phi^{-1}(a),xz+\phi^{-1}(b)))$
and the conclusion follows. ∎
Using Grauert division theorem, we can prove a first useful preparation result
in the case $x^{2}={\rm{Lt}}_{{\overline{\tau}}}(f)$ and
$xy={\rm{Lt}}_{{\overline{\tau}}}(g).$
###### Lemma 3.3.
Let $A=R/I$ be a local ring of type $(2,2)$ such that $I=(f,g),$
${\rm{Lt}}_{{\overline{\tau}}}(f)=x^{2},$
${\rm{Lt}}_{{\overline{\tau}}}(g)=xy.$ Then we can write
$I=(x^{2}+axz^{p}+F(y,z),xy+bxz^{q}+G(y,z))$
where $p,q\geq 1$, $a=0$ or $a\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $b=0$ or
$b\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $F,G\in{{\rm K}[\\![y,z]\\!]}_{\geq
2}.$
###### Proof.
By the assumption we have $f=x^{2}+F$ with
${\rm{Lt}}_{{\overline{\tau}}}(F)<_{{\overline{\tau}}}x^{2}$ and $g=xy+G$ with
${\rm{Lt}}_{{\overline{\tau}}}(G)<_{{\overline{\tau}}}xy.$ Applying Grauert’s
division theorem to the power series $F,f,g$ we get
$F=\alpha f+\beta g+r$
where $\alpha,\beta,r\in R$, no monomial of ${\rm{Supp}}(r)$ is divisible by
$x^{2}$ or $xy$, and
${\rm{Lt}}_{{\overline{\tau}}}(\alpha f),{\rm{Lt}}_{{\overline{\tau}}}(\beta
g)\leq_{{\overline{\tau}}}{\rm{Lt}}_{{\overline{\tau}}}(F)<_{{\overline{\tau}}}{\rm{Lt}}_{{\overline{\tau}}}(f)=x^{2}.$
We can write $\alpha=\sum_{i\geq 0}\alpha_{i},$ where, for every $i,$
$\alpha_{i}$ is a degree $i$ form in ${{\rm K}[\\![x,y,z]\\!]}.$ It is clear
that the initial form of $\alpha f=\alpha(x^{2}+F)$ is
$\alpha_{0}x^{2}+\alpha_{0}F_{2}$ so that $\alpha_{0}=0,$ otherwise
${\rm{Lt}}_{{\overline{\tau}}}(\alpha f)=x^{2}.$ In particular $1-\alpha$ is a
unit. Since
$(1-\alpha)f=f-\alpha f=x^{2}+F-\alpha f=x^{2}+r+\beta g$
we get
$I=(f,g)=(x^{2}+r,g).$
We apply now Grauert’s Division Theorem to the power series $G,x^{2}+r,f$
where $G=g-xy$ and ${\rm{Lt}}_{{\overline{\tau}}}(G)<_{{\overline{\tau}}}xy.$
We get
$g-xy=G=t(x^{2}+r)+sg+r^{\prime}$
where no monomial of ${\rm{Supp}}(r^{\prime})$ is divisible by
${\rm{Lt}}_{{\overline{\tau}}}(x^{2}+r)={\rm{Lt}}_{{\overline{\tau}}}(x^{2}+F-\alpha
f-\beta g)=x^{2}$ or by ${\rm{Lt}}_{{\overline{\tau}}}(g)=xy.$
Since $g=xy+t(x^{2}+r)+sg+r^{\prime},$ we get
$g(1-s)=t(x^{2}+r)+r^{\prime}+xy$
and we claim that $1-s$ is a unit. Namely,
${\rm{Lt}}_{{\overline{\tau}}}(sg)\leq{\rm{Lt}}_{{\overline{\tau}}}(G)<xy$
and, as before,
$sg=s(xy+G)=s_{0}(xy+G)+s_{1}(xy+G)+....$
This implies $s_{0}=0,$ otherwise ${\rm{Lt}}_{{\overline{\tau}}}(sg)=xy.$ This
proves the claim.
Now we have
$I=(x^{2}+r,g)=(x^{2}+r,(1-s)g)=(x^{2}+r,t(x^{2}+r)+r^{\prime}+xy)=(x^{2}+r,xy+r^{\prime}),$
where no monomial of ${\rm{Supp}}(r)$ and ${\rm{Supp}}(r^{\prime})$ is
divisible by $x^{2}$ or $xy$.
It is easy to see that this implies
$\displaystyle r$ $\displaystyle=$ $\displaystyle axz^{p}+F(y,z)$
$\displaystyle r^{\prime}$ $\displaystyle=$ $\displaystyle bxz^{q}+G(y,z)$
with $p,q\geq 1$, $F,G\in{{\rm K}[\\![y,z]\\!]}_{\geq 2}$, $a=0$ or
$a\in\mathbb{U}({{\rm K}[\\![z]\\!]})$, and $b=0$ or $b\in\mathbb{U}({{\rm
K}[\\![z]\\!]})$. ∎
We can prove now the main preparation result.
###### Theorem 3.4.
Let $A=R/I$ be a local ring of type $(2,2)$ such that $HF_{A}(3)=5.$ a) If
$I^{*}_{2}$ is not square-free then, up to isomorphism, we can write
$I=(x^{2}+axz^{p}+F(y,z),xy+G(y,z))$
where $p\geq 2$, $a\in\\{0,1\\}$, and $F,G\in{{\rm K}[\\![y,z]\\!]}_{\geq 3}.$
b) If $I^{*}_{2}$ is square-free then, up to isomorphism, we can write
$I=(x^{2}+xz+F(y,z),xy+dyz+\alpha y^{r}+\beta z^{s})$
where $F\in{{\rm K}[\\![y,z]\\!]}_{\geq 3},$ $d\in{{\rm K}[\\![y,z]\\!]},$
$d(0,0)=1$, $r,s\geq 3$, $\alpha=0$ or $\alpha\in\mathbb{U}({{\rm
K}[\\![y]\\!]})$, $\beta=0$ or $\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$.
###### Proof.
By Lemma 3.2, up to isomorphism, we can find generators $f$ and $g$ of $I$
such that either $f^{*}=x^{2}$ and $g^{*}=xy$ or $f^{*}=xy$ and $g^{*}=xz.$
Let us first assume that $f^{*}=x^{2}$ and $g^{*}=xy$; then
${\rm{Lt}}_{{\overline{\tau}}}(f)=Lt_{\tau}(f^{*})=Lt_{\tau}(x^{2})=x^{2}$
${\rm{Lt}}_{{\overline{\tau}}}(g)=Lt_{\tau}(g^{*})=Lt_{\tau}(xy)=xy,$
so that, as remarked at the end of the proof of Lemma 3.3, we have
$I=(x^{2}+r,xy+r^{\prime})$ where no monomial of ${\rm{Supp}}(r)$ and
${\rm{Supp}}(r^{\prime})$ is divisible by $x^{2}$ or $xy.$
Since $f^{*}=x^{2}$ and $g^{*}=xy,$ we also have $f=x^{2}+h,$ $g=xy+s$ where
${\rm{order}}(h),{\rm{order}}(s)\geq 3.$ This implies
$I=(x^{2}+r,xy+r^{\prime})=(x^{2}+h,xy+s)$ and $I^{*}_{2}=<x^{2},xy>$, the
vector space spanned by $x^{2}$ and $xy.$. Since the degree 2 component of $r$
is a linear combination of the monomials $xz,y^{2},yz,z^{2},$ it must be zero,
otherwise the leading form of $x^{2}+r$ cannot be in $I^{*}_{2}=<x^{2},xy>.$
This proves that the order of $r$ is at least 3. Exactly in the same way we
can prove that this holds true also for $r^{\prime}$.
It is easy to see that this implies
$r=axz^{p}+D(y,z),\ \ \ \ \ r^{\prime}=bxz^{q}+E(y,z)$
where $p,q\geq 2$, $a=0$ or $a\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $b=0$ or
$b\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ and $D,E\in{{\rm
K}[\\![y,z]\\!]}_{\geq 3}.$
Now let $\phi$ be the automorphism of ${{\rm K}[\\![x,y,z]\\!]}$ defined by
$x\rightarrow x,\ \ y\rightarrow y-bz^{q},\ \ z\rightarrow z$
and let $S:=\phi(D)$ and $T:=\phi(E).$ Then $S,T\in{{\rm
K}[\\![y,z]\\!]}_{\geq 3}$ and we have
$\phi(f)=\phi(x^{2}+r)=\phi(x^{2}+axz^{p}+D(y,z))=x^{2}+axz^{p}+S(y,z))$
and
$\phi(g)=\phi(xy+r^{\prime})=\phi(xy+bxz^{q}+E(y,z))=x(y-bz^{q})+bxz^{q}+\phi(E(y,z))=xy+T(y,z)).$
This implies that, up to isomorphism, we may assume
$I=(x^{2}+axz^{p}+S(y,z),xy+T(y,z))$
with $p\geq 2$, $a=0$ or $a\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $b=0$ or
$b\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ and $S,T\in{{\rm
K}[\\![y,z]\\!]}_{\geq 3}.$
Now if $a=0$ we are done, otherwise let $a\neq 0.$ Since the ground field
${\rm K}$ is algebraically closed and $a$ is invertible in $K[[z]]$, a
straightforward application of Hensel Lemma enables us to find an element
$c\in R$ such that $c^{p}=a.$
Let us consider the automorphism $\phi:{{\rm K}[\\![x,y,z]\\!]}\to{{\rm
K}[\\![x,y,z]\\!]}$ defined by
$x\rightarrow x,\ \ y\rightarrow y,\ \ z\rightarrow cz.$
If $F$ and $G$ are power series in $K[[z]]$ such $\phi(F)=S$ and $\phi(G)=T,$
then
$\phi(x^{2}+xz^{p}+F)=x^{2}+xc^{p}z^{p}+S=x^{2}+axz^{p}+S,\ \ \ \ \ \ \
\phi(xy+G)=xy+T.$
The conclusion easily follows.
We need now to consider the other case when $f^{*}=xy,$ $g^{*}=xz.$ As before
we choose a monomial order $\tau$ such that $x>_{\tau}z$ and let $\phi$ be the
automorphism of ${{\rm K}[\\![x,y,z]\\!]}$ defined by
$x\rightarrow x+z,\ \ y\rightarrow x,\ \ z\rightarrow y.$
We have $f=xy+d,$ $g=xz+e$ where $d$ and $e$ have order at least 3. Hence
$\phi(f)=(x+z)x+\phi(d)=x^{2}+xz+h,\ \ \ \ \phi(g)=(x+z)y+\phi(e)=xy+yz+s$
where $h:=\phi(d)$ and $s:=\phi(e)$ have order $\geq 3.$ Thus, up to
isomorphism, we may assume that $I$ is generated by the power series
$x^{2}+xz+h$ and $xy+yz+s;$ this implies that $I^{*}_{2}=<x^{2}+xz,xy+yz>.$
Since $x^{2}>_{\tau}xz$ and $xy>_{\tau}yz,$ we get
${\rm{Lt}}_{{\overline{\tau}}}(x^{2}+xz+h)=Lt_{\tau}((x^{2}+xz+h)^{*})=Lt_{\tau}(x^{2}+xz)=x^{2}$
${\rm{Lt}}_{{\overline{\tau}}}(xy+yz+s)=Lt_{\tau}((xy+yz+s)^{*})=Lt_{\tau}(xy+yz)=xy$
and we may use Lemma 3.3 to get
$I=(x^{2}+axz^{p}+S(y,z),xy+bxz^{q}+M(y,z))$
where $p,q\geq 1$, $a=0$ or $a\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $b=0$ or
$b\in\mathbb{U}({{\rm K}[\\![z]\\!]}),$ $S,M\in{{\rm K}[\\![y,z]\\!]}_{\geq
2}.$
Now let $\alpha:=x^{2}+axz^{p}+S(y,z);$ if $p\geq 2$ then
$\alpha^{*}=x^{2}+S(y,z)_{2}$ is an element of the vector space
$I^{*}_{2}=<x^{2}+xz,xy+yz>,$ a contradiction. Hence $p=1$ and thus we get
$\alpha^{*}=x^{2}+a_{0}xz+S(y,z)_{2}\in<x^{2}+xz,xy+yz>.$ This clearly implies
$a_{0}=1$ and $S(y,z)_{2}=0$, so that the order of $S(y,z)$ is at least 3.
Now let $\beta:=xy+bxz^{q}+M(y,z);$ if $b\neq 0$ and $q=1$ then $b_{0}\neq 0$
and we have
$\beta^{*}=xy+b_{0}xz+M(y,z)_{2}\in I^{*}_{2}=<x^{2}+xz,xy+yz>,$
a contradiction. Hence it must be either $b=0$ or $q\geq 2;$ in both cases we
have
$\beta^{*}=xy+M(y,z)_{2}\in<x^{2}+xz,xy+yz>$
which implies $M(y,z)=yz+H(y,z)$ where $H(y,z)$ is a power series in
$K[[y,z]]$ with order at least 3.
At this point we have $I=(x^{2}+axz+S(y,z),xy+bxz^{q}+yz+H(y,z))$ with
$a_{0}=1,$ S and H $\in K[[y,z]]_{\geq 3}$ and either $b=0$ or $q\geq 2$ .
Let us consider the automorphism $\phi$ given by
$x\to x,\ \ y\to y-bz^{q},\ \ z\to z.$
We get
$\phi(x^{2}+axz+S(y,z))=x^{2}+axz+B(y,z),$
and
$\phi(xy+bxz^{q}+yz+H(y,z))=x(y-bz^{q})+bxz^{q}+(y-bz^{q})z+\phi(H)=xy+yz+L(y,z)$
where $B(y,z):=\phi(S)$ and $L(y,z):=-bz^{q+1}+\phi(H)\in K[[y,z]]_{\geq 3}.$
Hence, up to isomorphism, we may assume $I=(x^{2}+axz+B(y,z),xy+yz+L(y,z))$
with $a_{0}=1$ and $B,L\in K[[y,z]]_{\geq 3}.$ Now it is clear that since
$L(y,z)$ has order at least 3, we can write $L(y,z)=cyz+\alpha y^{r}+\beta
z^{s}$ with $c\in K[y,z]_{\geq 1},$ $r,s\geq 3$, $\alpha=0$ or
$\alpha\in\mathbb{U}({{\rm K}[\\![y]\\!]})$, and $\beta=0$ or
$\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$. Hence we get
$I=(x^{2}+axz+B(y,z),xy+yz+cyz+\alpha y^{r}+\beta z^{s}).$ We let $d:=1+c$ so
that $d\in K[[y,z]],$ $d(0,0)=1+c(0,0)=1$ and
$I=(x^{2}+axz+B(y,z),xy+dyz+\alpha y^{r}+\beta z^{s}).$
Finally let us consider the automorphism $\phi$ given by
$x\to x,\ \ y\to y,\ \ z\to az.$
Let $F(y,z):=\phi^{-1}(B(y,z))$ and
$f:=x^{2}+xz+F(y,z)\ \ \ \ g:=xy+\phi^{-1}(d/a)yz+\alpha
y^{r}+\phi^{-1}(\beta/a^{s})z^{s}.$
Then we get
$\phi(f)=x^{2}+axz+B(y,z)$ $\phi(g)=xy+(d/a)yaz+\alpha
y^{r}+(\beta/a^{s})(a^{s}z^{s})=xy+dyz+\alpha y^{r}+\beta z^{s}.$
We remark that the constant term of the power series $d/a$ is 1 and the power
series $\beta/a^{s}$ is invertible if not zero. Hence the same holds for
$\phi^{-1}(d/a)$ and $\phi^{-1}(\beta/a^{s}).$ The conclusion follows. ∎
We recall that in this section we are assuming that $A={{\rm
K}[\\![x,y,z]\\!]}/I$ is a local ring of type (2,2) such that $HF_{A}(1)=3,$
$HF_{A}(2)=4$ and $HF_{A}(3)=5.$ This implies that if we let $n$ to be the
least integer such that $HF_{A}(n)=HF_{A}(n+1),$ then $n\geq 3.$ Also it is
easy to see that $n\leq r,$ the reduction number of $A$. The integer $n$ plays
a relevant work in the paper. With the aid of this integer $n$ and as a
consequence of Proposition 2.2, the Hilbert function of a local ring $A$ of
type (2,2) and multiplicity $e$ has the following shape:
$HF_{A}(t)=\begin{cases}1&\ \ \text{$t=0$},\\\ t+2&\ \
\text{$t=1,\cdots,n$},\\\ t+1&\ \ \text{$t=n+1,\cdots,e-1$},\\\ e&\ \
\text{$t\geq e$}.\\\ \end{cases}$ (3)
for some integer $n\leq e-2.$ We say that $HF_{A}$ has a flat in position $n$.
It is clear that we have two possibilities, either $e=n+2$ or $e\geq n+3.$ In
the first case the Hilbert function is increasing by one up to reach the
multiplicity, while in the second case the Hilbert function has a flat in
position $n$ and is increasing by one in all the other positions, before
reaching the multiplicity.
We are ready to prove the main result of this paper. It says that, quite
unexpectedly, if the Hilbert function of a local ring of type $(2,2)$ has a
flat in position $n,$ then the multiplicity cannot be too big, namely it
cannot overcome $2n.$
First we need this easy Lemma.
###### Lemma 3.5.
Let $J\subset P=k[x,y,z]$ be a monomial ideal such that $x^{2},xy\in J.$ If
for some $n\geq 2$ we have $\ HF_{P/J}(n+1)=n+2\ $ and
$HF_{P/J}(n~{}+~{}2)=~{}n+3,$ then $xz^{n}$ is the unique monomial of degree
$n+1$ which is in $J$ and not in $(x^{2},xy).$
If we have also $\ HF_{P/J}(n)=n+2,$ then $J_{d}=(x^{2},xy)_{d}$ for all
$2\leq d\leq n$.
###### Proof.
Since $HF_{P/J}(n+1)=n+2<HF_{P/(x^{2},xy)}(n+1)=n+3$ there is a monomial $m$
of degree n+1 which is in $J$ and not in $(x^{2},xy).$ If $m\neq xz^{n}$ it
should be $m=y^{n+1-j}z^{j}$ for some $j=0,...,n+1.$ But then the monomials of
the vector space $(x^{2},xy)_{n+2}$ and $my,mz$ would be linearly independent.
This implies that
$n+3=HF_{P/J}(n+2)\leq
HF_{P/(x^{2},xy,my,mz)}(n+2)=HF_{P/(x^{2},xy)}(n+2)-2=n+2,$
a contradiction. Hence $m=xz^{n}.$
Let us assume that also $\ HF_{P/J}(n)=n+2;$ if for some $t\leq n-1$ we have
$HF_{P/J}(t)\leq t+1,$ then
$HF_{P/J}(t+1)\leq HF_{P/J}(t)^{<t>}\leq(t+1)^{<t>}=t+2$
and going on in this way we would have $HF_{P/J}(n)\leq n+1,$ a contradiction.
It follows that for all $2\leq d\leq n$ we have
$HF_{P/J}(d)=HF_{P/(x^{2},xy)}(d)$ and, as a consequence,
$J_{d}=(x^{2},xy)_{d}$ for the same $d$’s. ∎
###### Theorem 3.6.
Let $A$ be a local ring of type (2,2) and multiplicity $e$. If the Hilbert
function of $A$ has a flat in position $n,$ then $e\leq 2n.$
###### Proof.
As usual, we consider a monomial ordering $\tau$ on the terms of $K[x,y,z]$
such that $x>_{\tau}z.$ In order to cover both case a) and b) in Theorem 3.4,
we may assume $I=(f,g)$ where
$f:=x^{2}+axz^{p}+F(y,z)\ \ \ \ g:=xy+G(y,z)$
are power series such that $p\geq 1$ and $a=0$ or $a\in\mathbb{U}({{\rm
K}[\\![z]\\!]}).$
Since it is clear that $(x^{2},xy)\nsubseteq{\rm{Lt}}_{{\overline{\tau}}}(I)$,
the elements $f$ and $g$ are not a standard basis for $I;$ thus, by
Buchberger’s criterion, we should have
$h:={\rm NF}(S(f,g),\\{f,g\\})\neq 0.$
It is clear that $h\in I$ and if we let
$m:={\rm{Lt}}_{{\overline{\tau}}}(h)=\rm{Lt}_{\tau}(h^{*})$, then
$m\in{\rm{Lt}}_{{\overline{\tau}}}(I)$ and by 1.6.4 in [19]
$m\notin(x^{2},xy).$ We claim that $m$ is a monomial of degree $n+1.$
Namely, by the second statement of Lemma 3.5 applied to the monomial ideal
${\rm{Lt}}_{{\overline{\tau}}}(I),$ it is clear that $m$ has degree at least
$n+1.$ Let us assume that $m$ has degree $\geq n+2,$ so that
${\rm{order}}(h)\geq n+2.$ Since for every $s$ and $G$ one can easily prove
that ${\rm{order}}(s)\leq{\rm{order}}({\rm NF}(s\ |G)),$ we get
${\rm{order}}({\rm NF}(S(f,h)|\\{f,g,h\\}))\geq{\rm{order}}(S(f,h))\geq
max\\{{\rm{order}}(f),{\rm{order}}(h)\\}\geq n+2.$
In the same way we can also prove that ${\rm{order}}({\rm
NF}(S(g,h)|\\{f,g,h\\}))\geq n+2.$ Now recall that, accordingly to the
Buchberger algorithm, in order to determine a standard basis of
${\rm{Lt}}_{{\overline{\tau}}}(I)$, one has to compute ${\rm
NF}(S(f,h)|\\{f,g,h\\}),$ ${\rm NF}(S(g,h)|\\{f,g,h\\}),$ to add those of them
which are not zero to the list and go on in this way up to the end. At each
step of this procedure the order of the elements can only increase; hence if
$m$ has degree $\geq n+2$ then ${\rm NF}(S(f,h)|\\{f,g,h\\})$ and ${\rm
NF}(S(g,h)|\\{f,g,h\\})$ have degree at least $n+2$ and we cannot obtain, as
Lemma 3.5 requires, the monomial $xz^{n}$ which has degree $n+1.$ This proves
the claim. By Lemma 3.5 the claim implies that
$m={\rm{Lt}}_{{\overline{\tau}}}(h)={\rm{Lt}}_{\tau}(h^{*})=xz^{n}.$
We want now to compute ${\rm NF}(S(f,g)\ |\ \\{f,g\\})$. First it is clear
that we can write $G(y,z)=yH(y,z)+\alpha z^{c}$ with $\alpha=0$ or invertible
in $K[[z]]$ and $c\geq 0.$ Hence $I=(f,g)$ where
$f=x^{2}+axz^{p}+F(y,z),\ \ \ g=xy+yH(y,z)+\alpha z^{c}$
with $c\geq 0,$ $p\geq 1$ and $a$ and $\alpha$ either zero or invertible in
$K[[z]].$ We have
$S(f,g)=yf-xg=axyz^{p}+yF(y,z)-xyH(y,z)-\alpha xz^{c}=g(az^{p}-H(y,z))-\alpha
xz^{c}+M(y,z)$
where $M(y,z)=yF(y,z)-(yH(y,z)+\alpha z^{c})(az^{p}-H(y,z))\in K[[y,z]].$
We claim that ${\rm NF}(S(f,g)\ |\ \\{f,g\\})=M(y,z)-\alpha xz^{c}.$ Namely we
have
$S(f,g)=0\cdot f+(az^{p}-H(y,z))g+M(y,z)-\alpha xz^{c}$
and we need to prove:
a) no monomial in the support of $M(y,z)-\alpha xz^{c}$ is divisible by
$x^{2}$ or $xy$
b)
${\rm{Lt}}_{{\overline{\tau}}}(g(az^{p}-H(y,z)))\leq{\rm{Lt}}_{{\overline{\tau}}}(S(f,g)).$
Now a) is true because $\alpha$ is zero or invertible in $K[[z]].$ As for b)
it is clear that we have
${\rm{Lt}}_{{\overline{\tau}}}(g(az^{p}-H(y,z)))=xy\cdot{\rm{Lt}}_{{\overline{\tau}}}(az^{p}-H(y,z)).$
This monomial is not in the support of $M(y,z)-\alpha xz^{c},$ hence it is in
the support of $S(f,g).$ This implies b) and the claim
$h={\rm NF}(S(f,g)\ |\ \\{f,g\\})=M(y,z)-\alpha xz^{c}$ (4)
is proved.
Since ${\rm{Lt}}_{{\overline{\tau}}}(h)=xz^{n}$ it follows
$\alpha\in\mathbb{U}({{\rm K}[\\![z]\\!]})$, $c=n$, and ${\rm{order}}(M)\geq
n+1$. In particular we deduce
$g=xy+\alpha z^{n}+yH(y,z).$ (5)
Let $J:=I+(y)=(x^{2}+axz^{p}+F(0,z),z^{n},y);$ it is clear that
${\rm{Lt}}_{{\overline{\tau}}}(J)\supseteq(x^{2},z^{n},y).$ Since $R/J$ is
Artinian, $\overline{y}$ is a parameter in $A=R/I;$ hence
$e=e(R/I)\leq length(R/J)=length(R/{\rm{Lt}}_{{\overline{\tau}}}(J)\leq
length(R/(x^{2},z^{n},y))=2n.$
The conclusion follows. ∎
In example 2.3, we have seen that however we fix an integer $e\geq 4,$ there
is a local ring of type $(2,2)$ with multiplicity $e$ and strictly increasing
Hilbert function. For each pair of integers $(n,e)$ such that $n\geq 3$ and
$n+3\leq e\leq 2n,$ we exhibit now local rings of type $(2,2)$ and
multiplicity $e$ whose Hilbert function has a flat in position $n.$
###### Example 3.7.
Given the integers $n$ and $e$ such that $n\geq 3$, $n+3\leq e\leq 2n,$ the
ideal
$I=(x^{2}-y^{e-2},xy-z^{n})$
is a complete intersection ideal of $R={{\rm K}[\\![x,y,z]\\!]}$ of type
$(2,2)$ with multiplicity $e,$ whose Hilbert function has a flat in position
$n.$
###### Proof.
Let us consider a monomial ordering $\tau$ such that $x>y>z;$ we are going to
prove that
$\\{f=x^{2}-y^{e-2},\ \ g=xy-z^{n},\ \ h=-y^{e-1}+xz^{n},\ \
k=y^{e}-z^{2n}\\}$
is a standard basis for $I$. Namely, if this is the case, we get
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy,xz^{n},y^{e})$ and from this an
easy computation shows that the local ring
$K[x,y,z]]/(x^{2}-y^{e-2},xy-z^{n})$has multiplicity $e$ and Hilbert function
with a flat in position $n.$
We have:
$S(f,g)=yf-xg=y(x^{2}-y^{e-2})-x(xy-z^{n})=xz^{n}-y^{e-1}$
and since $e\geq n+3$ implies $e-1\geq n+2>n+1$, we get
${\rm{Lt}}_{{\overline{\tau}}}(S(f,g))=xz^{n}.$
We let
$h:=S(f,g)=xz^{n}-y^{e-1}.$
Now
$S(f,h)=z^{n}f-xh=z^{n}(x^{2}-y^{e-2})-x(xz^{n}-y^{e-1})=xy^{e-1}-z^{n}y^{e-2}=y^{e-2}g$
so that
${\rm{Lt}}_{{\overline{\tau}}}(S(f,h))=y^{e-2}{\rm{Lt}}_{{\overline{\tau}}}(g)=xy^{e-1}.$
Further
$S(g,h)=z^{n}g-yh=z^{n}(xy-z^{n})-y(xz^{n}-y^{e-1})=y^{e}-z^{2n}$
and since $e\leq 2n$ and $y>z,$ we have
${\rm{Lt}}_{{\overline{\tau}}}(S(g,h))=y^{e}.$
We let
$k:=S(g,h)=y^{e}-z^{2n}$
with
${\rm{Lt}}_{{\overline{\tau}}}(S(g,h))={\rm{Lt}}_{{\overline{\tau}}}(k)=y^{e}.$
Now
$S(f,k)=y^{e}f-x^{2}k=y^{e}(x^{2}-y^{e-2})-x^{2}(y^{2}-z^{2n})=x^{2}z^{2n}-y^{2e-2}=z^{2n}f-y^{e-2}k$
and since $2e-2\geq 2(n+3)-2=2n+4>2n+2,$ we have
${\rm{Lt}}_{{\overline{\tau}}}(S(f,k))=x^{2}z^{2n}.$
Also
$S(g,k)=y^{e-1}g-xk=y^{e-1}(xy-z^{n})-x(y^{2}-z^{2n})=xz^{2n}-y^{e-1}z^{n}=z^{n}h$
so that
${\rm{Lt}}_{{\overline{\tau}}}(S(g,k))=z^{n}{\rm{Lt}}_{{\overline{\tau}}}(h)=xz^{2n}.$
Finally
$S(h,k)=y^{2}h-xz^{n}k=y^{e}(xz^{n}-y^{e-1})-xz^{n}(y^{e}-z^{2n})=xz^{3n}-y^{2e-1}=z^{2n}h-y^{e-1}k.$
Here we can only remark that
${\rm{Lt}}_{{\overline{\tau}}}(S(h,k))=\operatorname{max}\\{xz^{3n},y^{2e-1}\\}.$
From these computations we get
${\rm NF}(S(f,g)\ |\ \\{h\\})={\rm NF}(h\ |\ \\{h\\})=0$ ${\rm NF}(S(f,h)\ |\
\\{g\\})={\rm NF}(y^{e-2}g\ |\ \\{g\\})=0$ ${\rm NF}(S(g,h)\ |\ \\{k\\})={\rm
NF}(k\ |\ \\{k\\})=0$ ${\rm NF}(S(f,k)\ |\ \\{f,k\\})={\rm
NF}(z^{2n}f-y^{e-2}k\ |\ \\{f,k\\})=0$
because
${\rm{Lt}}_{{\overline{\tau}}}(z^{2n}f-y^{e-2}k)=x^{2}z^{2n}\geq{\rm{Lt}}_{{\overline{\tau}}}(z^{2n}f)=x^{2}z^{2n},{\rm{Lt}}_{{\overline{\tau}}}(y^{e-2}k)=y^{2e-2}.$
${\rm NF}(S(g,k)\ |\ \\{h\\})={\rm NF}(z^{n}h\ |\ \\{h\\})=0$ ${\rm
NF}(S(h,k)\ |\ \\{h,k\\})={\rm NF}(z^{2n}h-y^{e-1}k\ |\ \\{h,k\\})=0$
because
${\rm{Lt}}_{{\overline{\tau}}}(z^{2n}h-y^{e-1}k)=\operatorname{max}\\{xz^{3n},y^{2e-1}\\}\geq{\rm{Lt}}_{{\overline{\tau}}}(z^{2n}h)=xz^{3n},{\rm{Lt}}_{{\overline{\tau}}}(y^{e-1}k)=y^{2e-1}.$
By Buchberger’s criterion the conclusion follows.∎
We prove now that if $I^{*}_{2}$ is square-free then the Hilbert function is
strictly increasing, so that the associated graded ring is Cohen-Macaulay.
###### Proposition 3.8.
Let $A=R/I$ be a local ring of type $(2,2).$ If $I^{*}_{\ 2}$ is square-free,
then the Hilbert function of $A$ is strictly increasing and thus the
associated graded ring $gr_{\mathfrak{m}}(A)$ is Cohen-Macaulay.
###### Proof.
From Proposition 3.4 $(ii)$ we may assume, up to isomorphism of $R$, that
$I=(x^{2}+xz+F(y,z),xy+byz+\alpha y^{r}+\beta z^{s})$
where $F\in{{\rm K}[\\![y,z]\\!]}_{\geq 3},$ $b\in{{\rm K}[\\![y,z]\\!]}$ with
$b(0,0)=1$, $r,s\geq 3$, $\alpha=0$ or $\alpha\in\mathbb{U}({{\rm
K}[\\![y]\\!]})$, and $\beta=0$ or $\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$.
If $HF_{R/I}(n+2)=n+2,$ then $e=n+2$ and we conclude by Proposition 2.5.
Assume that $HF_{R/I}(n+2)=n+3$, then by Lemma 3.5 we have that
$xz^{n}\in{\rm{Lt}}_{{\overline{\tau}}}(I)$. By Burchberger’s criterion we
should have $xz^{n}={\rm{Lt}}_{{\overline{\tau}}}({\rm
NF}(S(f,g),\\{f,g\\}))$. The $S$-polynomial of the pair $f,g$ is
$h:=S(f,g)=z(b-1)A+\alpha y^{r-1}A+yF-\beta xz^{s},$
$A=byz+\alpha y^{r}+\beta z^{s}$. We write $L=z(b-1)A+\alpha y^{r-1}A+yF$;
notice that $L\in{{\rm K}[\\![y,z]\\!]}$ and $\beta\in{{\rm K}[\\![z]\\!]}$ so
$h={\rm NF}(h,\\{f,g\\})$. Since ${\rm{Lt}}_{{\overline{\tau}}}(h)=xz^{n}$ we
deduce $\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$, $s=n$, and
${\rm{order}}(L)\geq n+1$.
Let now consider
$\displaystyle S(h,g)=W$ $\displaystyle=$ $\displaystyle\beta z^{n}g+yh$
$\displaystyle=$ $\displaystyle\beta byz^{n+1}+\alpha\beta
y^{r}z^{n}+\beta^{2}z^{2n}+yL.$
Notice that since $b,\beta\neq 0$
${\rm{order}}(\alpha\beta y^{r}z^{n}),{\rm{order}}(\beta^{2}z^{2n})\geq
n+3>n+2={\rm{order}}(\beta byz^{n+1})$
and ${\rm{Lt}}_{{\overline{\tau}}}(\beta byz^{n+1})=yz^{n+1}$.
Recall that ${\rm{order}}(L)\geq n+1$, so in order to prove that
${\rm{Lt}}_{{\overline{\tau}}}(W)=yz^{n+1}$ we should prove that in
${\rm{Supp}}(yL)$ there is not the monomial $yz^{n+1}$. This is equivalent to
prove that in ${\rm{Supp}}(L)$ there is not the monomial $z^{n+1}.$ At this
end we set $y=0$ in $L$ and we get
$L(0,z)=(b(0,z)-1)\beta z^{n+1}.$
recall that $b(0,0)=1$ so ${\rm{order}}(L(0,z))\geq n+2$. Hence we have that
${\rm{Lt}}_{{\overline{\tau}}}(k)=yz^{n+1}$.
Let us consider now the monomial ideal
$J=(x^{2},xy,xz^{n},yz^{n+1})\subset{\rm{Lt}}_{{\overline{\tau}}}(I)$. We have
$HF_{R/I}(n+2)\leq HF_{R/J}(n+2)=n+2,$
a contradiction. ∎
Notice that if $I^{*}_{2}$ is square-free then by Lemma 3.2 $(ii)$ we may
assume, up to isomorphisms, that $f^{*}=xy,g^{*}=xz$. Hence Proposition 3.8
recover [7, Corollary 4.6] in the case $(2,2)$.
The following example shows that the converse of the above theorem does not
hold. Let $I=(x^{2}-y^{2}z,xy-y^{3})\subseteq{{\rm K}[\\![x,y,z]\\!]}$. It is
clear that $x^{2}\in I^{*}_{2}$ so that $I^{*}_{2}$ is not square-free. It is
easy to see that the Hilbert function of $A={{\rm K}[\\![x,y,z]\\!]}/I$ is
strictly increasing, namely is $\\{1,3,4,5,5,5,5,.......\\}.$ By Proposition
2.5 the associated graded ring of $A$ is Cohen-Macaulay.
We close this section by describing the possible minimal free resolutions of
the associated graded ring of a local ring of type $(2,2).$
We have seen in (3) that the Hilbert function of a local ring $A$ of type
$(2,2)$ has the following shape
$HF_{A}(t)=\begin{cases}1&\ \ \text{$t=0$},\\\ t+2&\ \
\text{$t=1,\cdots,n$},\\\ t+1&\ \ \text{$t=n+1,\cdots,e-1$},\\\ e&\ \
\text{$t\geq e$}.\\\ \end{cases}$ (6)
where $n$ is the least integer such that $HF_{A}(n)=HF_{A}(n+1).$ We have
$3\leq n\leq e-2$ and it is easy to see that the lex-segment ideal with the
above Hilbert function is the following ideal $L:=(x^{2},xy,xz^{n},y^{e}).$ We
can compute the minimal free resolution of $P/L$ by using the well known
formula of Eliaouh and Kervaire. We get
$0\to P(-n-3)\to P(-3)\oplus P(-n-2)^{2}\oplus P(-e-1)\to$ $\to
P(-2)^{2}\oplus P(-n-1)\oplus P(-e)\to P\to P/L\to 0.$
It is clear that in the case $e\geq n+3$ there is no possible cancelation.
Hence every homogeneous ideal $J$ with Hilbert function as in (3) and with
$e\geq n+3$ has the same resolution of the corresponding lex-segment ideal.
In the other case, when $e=n+2,$ we can either have the above resolution or
one of the following obtained by cancelation:
$0\to P(-n-3)\to P(-3)\oplus P(-n-2)\oplus P(-n-3)\to P(-2)^{2}\oplus
P(-n-1)\to P\to P/J\to 0$
$0\to P(-3)\oplus P(-n-2)\to P(-2)^{2}\oplus P(-n-1)\to P\to P/J\to 0.$
It is clear that if $P/J$ is Cohen-Macaulay only the last shorter resolution
is available.
We apply this to the associated graded ring of a local ring of type (2,2) and
we get the following result.
###### Proposition 3.9.
Let $A$ be a local ring of type (2,2), $e$ the multiplicity of $A$ and let $n$
be an integer such that $n\leq e-2.$ If $e=n+2,$ then $gr_{\mathfrak{m}}(A)$
is Cohen-Macaulay with minimal free resolution:
$0\to P(-3)\oplus P(-n-2)\to P(-2)^{2}\oplus P(-n-1)\to P\to
gr_{\mathfrak{m}}(A)\to 0.$
If $e\geq n+3,$ then $e\leq 2n$ and $gr_{\mathfrak{m}}(A)$ is not Cohen-
Macaulay with minimal free resolution
$0\to P(-n-3)\to P(-3)\oplus P(-n-2)^{2}\oplus P(-e-1)\to$ $\to
P(-2)^{2}\oplus P(-n-1)\oplus P(-e)\to P\to gr_{\mathfrak{m}}(A)\to 0.$
###### Proof.
It is enough to remark that by Proposition 2.5 the associated graded ring of a
local ring of type (2,2) is Cohen-Macaulay when $e=n+2.$ ∎
## 4 A structure’s theorem for quadratic complete intersections of
codimension two
The aim of this section is to give a structure, up analytic isomorphism, of
the minimal system of generators of ideals $I$ of type $(2,2)$ such that
$A={{\rm K}[\\![x,y,z]\\!]}/I$ is of multiplicity $e$. This a first step
towards the difficult problem of the analytic classification of the ideals of
type $(2,2)$. In this direction we show in Example 5.8 two ideals of type
$(2,2)$ with same Hilbert function that are not analytic isomorphic.
Accordingly with Proposition 2.2, Example 2.3, and Example 3.7 the Hilbert
function of $A$ take exactly the following shapes
$H(e)(t):=\begin{cases}1&\ \ \text{$t=0$},\\\ t+2&\ \
\text{$t=1,\cdots,e-3$},\\\ e&\ \ \text{$t\geq e-2$}\end{cases}$ (7)
or
$H(n,e)(t):=\begin{cases}1&\ \ \text{$t=0$},\\\ t+2&\ \
\text{$t=1,\cdots,n$},\\\ t+1&\ \ \text{$t=n+1,\cdots,e-2$},\\\ e&\ \
\text{$t\geq e-1$}.\\\ \end{cases}$ (8)
###### Theorem 4.1.
Let $A$ be a local ring of type $(2,2)$ and multiplicity $e.$ The following
conditions are equivalent:
$(i)$ $HF_{A}=H(n,e)$ for some integer $n\geq 3.$
$(ii)$ Up analytic isomorphism, $I$ is generated in $R={{\rm
K}[\\![x,y,z]\\!]}$ by:
$\displaystyle f$ $\displaystyle=$ $\displaystyle x^{2}+az^{p}(x+H)-H^{2}+L$
$\displaystyle g$ $\displaystyle=$ $\displaystyle xy+\alpha z^{n}+yH$
where
* •
$a\in\\{0,1\\}$, $p\geq 2$, $\alpha\in\mathbb{U}({{\rm K}[\\![z]\\!]})$,
* •
$H,L\in{{\rm K}[\\![y,z]\\!]}$ with ${\rm{order}}(L)\geq n+1$ ,
${\rm{order}}(H)\geq 2$,
* •
$n+3\leq e\leq 2n,$
* •
${\rm{order}}(2\alpha z^{n}H-a\alpha z^{n+p}+yL)\geq e-1$ and the equality
holds whenever $e<2n.$
###### Proof.
Taking advantage of Proposition 3.4, the proof is based on the computation of
${\rm{Lt}}_{{\overline{\tau}}}(I)$ accordingly with Buchberger’s criterion. As
usual assume $x>y,x>z.$
First we prove $(i)$ implies $(ii)$. Since $HF_{A}=H(n,e),$ then by
Proposition 2.5 $gr_{\mathfrak{m}}(A)$ is not Cohen-Macaulay and, by Theorem
3.6, $n+3\leq e\leq 2n.$ By Proposition 3.8, $I^{*}_{2}$ contains a square of
a linear form. Hence we may assume that $f^{*}=x^{2}$ and $g^{*}=xy$. Notice
that $x^{2},xy\in{\rm{Lt}}_{{\overline{\tau}}}(I),$ hence because $(i),$ by
Lemma 3.5, ${\rm{Lt}}_{{\overline{\tau}}}(I)\supseteq(x^{2},xy,xz^{n}).$ From
the particular shape of the Hilbert function it is easy to see that
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy,xz^{n},m)$ where $m$ is a monomial
in $K[y,z]_{e}.$ From Lemma 3.4 we may also assume
$\displaystyle f$ $\displaystyle=$ $\displaystyle x^{2}+axz^{p}+F(y,z),$
$\displaystyle g$ $\displaystyle=$ $\displaystyle xy+G(y,z),$
where $a\in\\{0,1\\}$, $p\geq 2$, $F,G\in{{\rm K}[\\![y,z]\\!]}$ with
${\rm{order}}(F),{\rm{order}}(G)\geq 3$. Moreover, from the equation (5) of
the proof of Theorem 3.6, (5), we get
$G(y,z)=yH(y,z)+\alpha z^{n}$
where $H\in{{\rm K}[\\![y,z]\\!]}_{\geq 2}$ and $\alpha\in\mathbb{U}({{\rm
K}[\\![z]\\!]}).$ We recall that $S(f,g)=yf-xg=axyz^{p}+yF-\alpha xz^{n}-xyH.$
In particular a standard computation gives
$h:={\rm NF}(S(f,g),\\{f,g\\})=-\alpha xz^{n}+yL+\alpha z^{n}(H-az^{p})$
where $L=F-az^{p}H+H^{2}.$ Notice that ${\rm{order}}(\alpha
z^{n}(H-az^{p}))\geq n+2.$ Notice that
$xz^{n}={\rm{Lt}}_{{\overline{\tau}}}(h)$.
A simple calculation shows that ${\rm NF}(S(h,f),\\{h,f,g\\})=0$. On the other
hand
$S(h,g)={\rm NF}(S(h,g),\\{h,f,g\\})=\alpha^{2}z^{2n}+y(2\alpha z^{n}H-a\alpha
z^{n+p}+yL)\neq 0$
because $\alpha\neq 0$ and $z^{2n}$ does not appear in the support of the
remaining part. As a consequence $m={\rm{Lt}}_{{\overline{\tau}}}(S(h,g)),$
and hence ${\rm{order}}(S(h,g))=e.$ It follows ${\rm{order}}(2\alpha
z^{n}H-a\alpha z^{n+p}+yL)\geq e-1$. In particular ${\rm{order}}(L)\geq n+1$,
and ${\rm{order}}(2\alpha z^{n}H-a\alpha z^{n+p}+yL)=e-1$ if $e<2n$,.
Conversely, assuming $(ii),$ it is enough to apply Buchberger’s criterion for
computing ${\rm{Lt}}_{{\overline{\tau}}}(I).$ By following the previous
computations we get ${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy,xz^{n},m)$
where $m={\rm{Lt}}_{{\overline{\tau}}}(\alpha^{2}z^{2n}+y(2\alpha
z^{n}H-a\alpha z^{n+p}+yL))$ and $(i)$ follows. ∎
If the Hilbert function is increasing, i.e. of type $H(e)$, we present a
structure’s theorem under the assumption that $I^{*}$ does not contain the
square of a linear form.
###### Theorem 4.2.
Let $A$ be a local ring of type $(2,2)$ and multiplicity $e.$ The following
conditions are equivalent:
$(i)$ $HF_{A}=H(e)$ and $I^{*}$ does not contain the square of a linear form
$(ii)$ Up analytic isomorphism, $I$ is generated in $R={{\rm
K}[\\![x,y,z]\\!]}$ by:
$\displaystyle f$ $\displaystyle=$ $\displaystyle x^{2}+xz+F$ $\displaystyle
g$ $\displaystyle=$ $\displaystyle xy+dyz+\alpha y^{r}+\beta z^{s}$
where
* •
$r\geq 3$,
* •
$F\in{{\rm K}[\\![y,z]\\!]}$ and ${\rm{order}}(F)\geq 3$,
* •
$d\in\mathbb{U}({{\rm K}[\\![y,z]\\!]})$, with $d(0,0)=1$,
* •
$\alpha=0$ or $\alpha\in\mathbb{U}({{\rm K}[\\![y]\\!]})$, $\beta=0$ or
$\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$ and $s\geq e-1\geq 3,$
* •
${\rm{order}}(F+d(d-1)z^{2}+\alpha(2d-1)zy^{r-1}+\alpha^{2}y^{2(r-1)})=e-2.$
###### Proof.
As usual, consider a monomial ordering $\tau$ with $x>y,x>z.$ We prove $(i)$
implies $(ii).$ By Theorem 3.4 $(ii)$, we may assume that
$I=(x^{2}+xz+F(y,z),xy+dyz+\alpha y^{r}+\beta z^{s})$
$F\in{{\rm K}[\\![y,z]\\!]}_{\geq 3},$ $d\in{{\rm K}[\\![y,z]\\!]}$ with
$d(0,0)=1$, $r,s\geq 3$, $\alpha=0$ or $\alpha\in\mathbb{U}({{\rm
K}[\\![y]\\!]})$, and $\beta=0$ or $\beta\in\mathbb{U}({{\rm K}[\\![z]\\!]})$.
Since the Hilbert function is increasing up to $n=e-2\geq 2$ and
$HF_{R/I}(t)=e$ for all $t\geq e-2,$ then
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy,m)$ where $m\in K[y,z]$ is a
monomial of degree $e-1$.
By Buchberger’s criterion necessarily $m={\rm{Lt}}_{{\overline{\tau}}}({\rm
NF}(S(f,g),\\{f,g\\}))$. Now
$S(f,g)=-\beta xz^{s}+yF(y,z)+xy[(1-d)z-\alpha y^{r-1}]$
After a computation we get
${\rm NF}(S(f,g),\\{f,g\\})=-\beta xz^{s}+yW+\alpha\beta
y^{r-1}z^{s}+\beta(d-1)z^{s+1}$
where $W=F+d(d-1)z^{2}+\alpha(2d-1)zy^{r-1}+\alpha^{2}y^{2(r-1)}.$ Since
$yW\in{{\rm K}[\\![y,z]\\!]}$, $r\geq 3$ and $1-d\in(y,z){{\rm
K}[\\![y,z]\\!]}$ we get that if $\beta\neq 0,$ then $xz^{s}$ appears in the
support of ${\rm NF}(S(f,g),\\{f,g\\}).$ Since
${\rm{Lt}}_{{\overline{\tau}}}({\rm NF}(S(f,g),\\{f,g\\}))\in K[y,z]_{e-1},$
it follows ${\rm{order}}(W)=e-2$ and, if $\beta\neq 0,$ then $s\geq e-1.$
Conversely if we assume $(ii),$ then it is easy to see that $I^{*}$ does not
contain the square of a linear form because $I^{*}_{2}=(x^{2}+xz,xy+yz)$ which
is reduced. Moreover by repeating Buchberger’s algorithm, looking at the
previous computation on $S(f,g),$ we get
${\rm{Lt}}_{{\overline{\tau}}}(I)=(x^{2},xy,y{\rm{Lt}}_{{\overline{\tau}}}(W)),$
hence $HF_{A}=H(e).$ ∎
## 5 Examples
The aim of this section is to present examples supporting the results of the
previous sections or detecting the possible extensions to the non quadratic
case. All computations are performed by using CoCoA system ([2]). Here
$HS_{A}(\theta)$ denotes the Hilbert series of $A,$ that is
$HS_{A}(\theta)=\sum_{t\geq 0}HF_{A}(t)\theta^{t}.$
We have seen in Proposition 3.9 that the minimal free resolution of the
tangent cone of a local ring of type $(2,2)$ has no possible cancelation, both
in the case the Hilbert function is strictly increasing and in the case of a
flat. One can ask if this is the case also for local rings of type $(a,b)$
with $3\leq a\leq b.$
The first two examples that we propose show that the answer is negative.
###### Example 5.1.
Let $A=R/I$ where $I=(x^{3},z^{5}+xz^{3}+x^{2}y).$ The local ring $A$ has type
$(3,3)$ and $I^{*}=(x^{3},x^{2}y,x^{2}z^{3},-xyz^{5}+xz^{6},-xz^{7},z^{10}).$
The resolution of $P/I^{*}$ is the following
$0\to P(-7)\oplus P(-10)\to P(-4)\oplus P^{2}(-6)\oplus P(-8)\oplus
P^{2}(-9)\oplus P(-11)\to$ $\to P^{2}(-3)\oplus P(-5)\oplus P(-7)\oplus
P(-8)\oplus P(-10)\to P\to P/I^{*}\to 0.$
It is clear that we have a possible cancelation and the Hilbert function
$\\{1,3,6,8,10,11,13,14,14,15,15,.......\\}$
has a flat in position 7.
###### Example 5.2.
Let $A=R/I$ where $I=(x^{4},z^{4}+x^{2}y).$ The local ring $A$ has type
$(3,4)$ and $I^{*}=(x^{2}y,x^{4},x^{2}z^{4},z^{8}).$ The resolution of
$P/I^{*}$ is the following
$0\to P(-9)\to P(-5)\oplus P(-7)\oplus P(-8)\oplus P(-10)\to$ $\to P(-3)\oplus
P(-4)\oplus P(-6)\oplus P(-8)\to P\to P/I^{*}\to 0.$
It is clear that we have a possible cancelation and the Hilbert function
$\\{1,3,6,9,11,13,14,15,16,16,16,........\\}$
is strictly increasing.
The following example shows that Proposition 2.5 cannot be extended to local
rings of type $(a,b)$ with $a>2.$
###### Example 5.3.
Let us consider the ideal $I=(x^{4},x^{2}y+z^{4})\subseteq R=k[[x,y,z]].$ Then
$A=R/I$ has strictly increasing Hilbert function, in fact the Hilbert series
is:
$HS_{A}(\theta)=(1+2\theta+3\theta^{2}+3\theta^{3}+2\theta^{4}+2\theta^{5}+\theta^{6}+\theta^{7}+\theta^{8})/(1-\theta).$
Nevertheless $I^{*}=(x^{2}y,x^{4},x^{2}z^{4},z^{8}),$ hence
$gr_{\mathfrak{m}}(A)$ is not Cohen-Macaulay.
The following example, due to T. Shibuta, shows that the Hilbert function of a
one-dimensional local domain of type $(2,b)$ can have $b-1$ flats, the maximum
number accordingly to Proposition 2.
###### Example 5.4.
(see [7], example 5.5) Let $b\geq 2$ be an integer. Consider the family of
semigroup rings
$A=k[[t^{3b},t^{3b+1},t^{6b+3}]].$
It is easy to see that $A=k[[x,y,z]]/I_{b}$ where
$I_{b}=(xz-y^{3},z^{b}-x^{2b+1}).$ Thus $A$ is a one-dimensional local domain
of type $(2,b).$ For every $b\geq 2$ the Hilbert function of $A$ has $b-1$
flats. Namely
$HF_{A}(t)=\begin{cases}1&\ \ \text{$t=0$},\\\ 2t+2&\ \
\text{$t=1,\cdots,b-1$},\\\ 2b&\ \ \text{$t=b$},\\\ 2b+1&\ \
\text{$t=b+1$},\\\ 2b+k&\ \ \text{$t=b+2k,\ \ k=1,\cdots,b-1$},\\\ 2b+k+1&\ \
\text{$t=b+2k+1,\ \ k=1,\cdots,b-1$},\\\ 3b&\ \ \text{$t\geq 3b-1$}.\\\
\end{cases}$ (9)
In the above example the Hilbert function of the local ring of type $(2,b)$
presents $b-1$ flats which are not consecutive. The following example shows
that we can also have $b-1$ consecutive flats, that is a strip like this:
$HF(n)=HF(n+1)=\dots=HF(n+b-1)<e.$
###### Example 5.5.
Let us consider the ideal $I=(x^{2},xy^{2}+z^{5}+xy^{3}z^{2})\subseteq
R=k[[x,y,z]].$ Then $A=R/I$ is of type $(2,3)$ and its Hilbert function
presents two (=b-1) flats which are consecutive: namely we have
$HF(5)=HF(6)=HF(7)=8<e=10.$ In particular the Hilbert series is:
$HS_{A}(\theta)=(1+2\theta+2\theta^{2}+\theta^{3}+\theta^{4}+\theta^{5}+\theta^{8}+\theta^{9})/(1-\theta).$
The Hilbert function of a local ring of type $(a,b),$ with $3\leq a\leq b,$ is
at the moment far from our understanding. In order to show how the problem is
difficult when $a$ and $b$ are increasing, we present two more examples, the
first of type $(3,3)$ with one very large platform consisting of 13
consecutive flats, the second of type $(4,4)$ with nine flats and three
platforms.
###### Example 5.6.
Let $I=(x^{3}-zy^{14},x^{2}y+xz^{7})\subseteq R=k[[x,y,z]].$ The local ring
$A=R/I$ is of type $(3,3)$ and
$HF_{A}(15)=HF_{A}(16)=\dots\dots=HF_{A}(29)=31<e=32.$
In particular the Hilbert series is:
$HS_{A}(\theta)=(1+2\theta+3\theta^{2}+2\theta^{3}+2\theta^{4}+2\theta^{5}+2\theta^{6}+2\theta^{7}+2\theta^{8}+\theta^{9}+2x\theta^{10}+2\theta^{11}+2\theta^{12}+$
$+2\theta^{13}+2\theta^{14}+\theta^{15}+\theta^{16}+\theta^{30}+\theta^{31})/(1-\theta)$
and $I^{*}=(x^{3},x^{2}y,x^{2}z^{7},xz^{14},xy^{15}z,y^{31}z).$
###### Example 5.7.
Let $I=(x^{4},xy^{3}-z^{6})\subseteq R=k[[x,y,z]].$ The local ring $A=R/I$ is
of type $(4,4)$ and
$HF_{A}(8)=HF_{A}(9)=HF_{A}(10)=HF_{A}(11)=18;$
$HF_{A}(13)=HF_{A}(14)=HF_{A}(15)=HF_{A}(16)=20;$
$HF_{A}(18)=HF_{A}(19)=HF_{A}(20)=HF_{A}(21)=22<e=24;$
In particular the Hilbert series is:
$HS_{A}(\theta)=(1+2\theta+3\theta^{2}+4\theta^{3}+3\theta^{4}+2\theta^{5}+\theta^{6}+\theta^{7}+\theta^{8}+\theta^{12}+\theta^{13}+\theta^{17}+\theta^{18}+\theta^{22}+\theta^{23})/(1-\theta)$
and $I^{*}=(xy^{3},x^{4},x^{3}z^{6},x^{2}z^{12},xz^{18},z^{24}).$
It would be very interesting to describe the isomorphism classes of local
rings of type $(2,2)$ which have the same given Hilbert function. But this is
a difficult task, as the following examples show.
First we are given the Hilbert function $\\{1,3,4,5,5,6,6,...\\}$ which has a
flat in position 3 and multiplicity 6. The two ideals which we are going to
prove that are not isomorphic are obtained one from the other with very little
modifications, namely by adding a monomial to one of the two generators.
###### Example 5.8.
Let us consider the ideals
$I=(x^{2}-y^{4},xy+z^{3}),\ \ J=(x^{2}+xz^{2}-y^{4},xy+z^{3})$
in $R={{\rm K}[\\![x,y,z]\\!]}.$
They are of type $(2,2)$, they have the same Hilbert function
$\\{1,3,4,5,5,6,6,...\\}$ and the same leading ideal
${\rm{Lt}}_{{\overline{\tau}}}(I)={\rm{Lt}}_{{\overline{\tau}}}(J)=(x^{2},xy,xz^{3},y^{6}).$
On the other hand the ideals of initial forms differ in degree $6$:
$I^{*}=(x^{2},xy,xz^{3},y^{6}-z^{6}),\ \ \
J^{*}=(x^{2},xy,xz^{3},y^{6}+yz^{5}-z^{6}).$
We prove that ${{\rm K}[\\![x,y,z]\\!]}/I$ and ${{\rm K}[\\![x,y,z]\\!]}/J$
are not isomorphic.
If there exists an analytic isomorphism $\phi$ such that $\phi(I)=J$ then we
can find power series $f,g,h$ of order $1$ such that $\mathcal{M}=(f,g,h)$ and
$\phi$ is the result of substituting $f$ for $x$, $g$ for $y$ and $h$ for $z$
in any power series of $R.$ We have $f=L_{1}+F\ \ \ \ g=L_{1}+G\ \ \ \
h=L_{3}+H$ where $L_{1},L_{2},L_{3}$ are linearly independent linear forms in
$K[x,y,z]$ and $F,G,H$ are power series of order $\geq 2.$
We let for $i=1,2,3$
$L_{i}=\lambda_{i1}x+\lambda_{i2}y+\lambda_{i3}z$
with $\lambda_{ij}\in{\rm K}.$ Since $x^{2}-y^{4}\in I$ we have
$\phi(x^{2}-y^{4})=f^{2}-g^{4}\in J$, hence $L_{1}^{2}\in J^{*}.$ Since
$I^{*}_{\ 2}$ is the ${\rm K}$-vector space $I^{*}_{\ 2}=<x^{2},xy>,$ we have
$(\lambda_{11}x+\lambda_{12}y+\lambda_{13}z)^{2}=px^{2}+qxy$
with $p,q\in{\rm K};$ this clearly implies $\lambda_{12}=\lambda_{13}=0.$
In the same way, since $xy+z^{3}\in I,$ we have $\phi(xy+z^{3})=fg+h^{3}\in
J$, hence $L_{1}L_{2}\in J^{*}.$ Thus we get
$(\lambda_{11}x)(\lambda_{21}x+\lambda_{22}y+\lambda_{23}z)=rx^{2}+sxy$
with $r,s\in{\rm K}.$ This implies $\lambda_{23}=0$ because $\lambda_{11}\neq
0.$
Finally we have
$y^{6}-z^{6}=-y^{2}(x^{2}-y^{4})+(xy+z^{3})(xy-z^{3})\in I$
so that $\phi(y^{6}-z^{6})=g^{6}-h^{6}\in J,$ and, as before,
$L_{2}^{6}-L_{3}^{6}\in J^{*}.$ Looking at the generators of the vector space
$J^{*}_{\ 6}$ we get as a consequence
$(\lambda_{21}x+\lambda_{22}y)^{6}-(\lambda_{31}x+\lambda_{32}y+\lambda_{33}z)^{6}=Ax^{2}+Bxy+Cxz^{3}+D(y^{6}+yz^{5}-z^{6})$
where $A,B,C,D$ are forms of degree $4,4,2,0$ respectively in the polynomial
ring ${\rm K}[x,y,z].$
Since $L_{1},L_{2},L_{3}$ are linearly independent, we must have
$\lambda_{33}\neq 0.$ Hence, looking at the coefficient of the monomial
$y^{5}z$ in the above formula, we get $\lambda_{32}=0.$ But then, looking at
the coefficient of the monomial $yz^{5},$ we certainly get $D=0$ and finally,
looking at the coefficient of the monomial $z^{6},$ we get $\lambda_{33}=0.$
This is a contradiction, so that the algebras $R/I$ and $R/J$ are not in the
same isomorphism class.
The case when the Hilbert function is strictly increasing is not more easy to
handle. Here we consider the Hilbert function $\\{1,3,4,5,6,6,6,....\\}$ which
is strictly increasing and we look at the possible isomorphism classes of
local rings with that Hilbert function.
###### Example 5.9.
Let us consider the two ideals
$I:=(x^{2}+y^{4},xy),\ \ \ \ \ J:=(x^{2}+y^{4}+z^{4},xy).$
They have the same Hilbert function $\\{1,3,4,5,6,6,6,....\\}$ and different
tangent cones, namely
$I^{*}=(x^{2},xy,y^{5})\ \ \ \ \ \ J^{*}=(x^{2},xy,y^{5}+yz^{4}).$
A calculation as before shows that ${{\rm K}[\\![x,y,z]\\!]}/I$ and ${{\rm
K}[\\![x,y,z]\\!]}/J$ are not isomorphic.
## References
* [1] V. Bertella, _Hilbert function of local Artinian level rings in codimension two_ , J. Algebra 321 (2009), no. 5, 1429–1442.
* [2] CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
* [3] A. Conca and J. Sidman, _Generic initial ideals of points and curves_ , J. Symbolic Comput. 40 (2005), no. 3, 1023–1038.
* [4] J. Elias, _The conjecture of Sally on the Hilbert function for curve singularities_ , J. Algebra 160 (1993), no. 1, 42–49.
* [5] , _Roller coaster curve singularities_ , J. Algebra 168 (1994), no. 3, 864–867.
* [6] S. Goto, W. Heinzer, and M.-K. Kim, _The leading ideal of a complete intersection of height two_ , J. Algebra 298 (2006), no. 1, 238–247.
* [7] S. Goto, W. Heinzer, and M.-K. Kim, _The leading ideal of a complete intersection of height two, Part II_ , J. Algebra 312 (2007), no. 2, 709–732.
* [8] S. Goto, W. Heinzer, and M.-K. Kim, _The leading ideal of a complete intersection of height two in a 2-dimensional regular local ring_ , Comm. Algebra 36 (2008), no. 5, 1901–1910.
* [9] G.-M. Greuel and G. Pfister, _A singular introduction to commutative algebra_, second extended ed., Springer, 2008, with contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann.
* [10] J. Herzog and R. Waldi, _A note on the Hilbert function of a one-dimensional Cohen-Macaulay ring_ , Man. Math. 16 (1975), 251–260.
* [11] H. Hironaka, _Resolution of singularities of an algebraic variety over a field of characteristic zero_ , Annals of Math. 79 (1964), no. 1, 2, 109–326.
* [12] A. Iarrobino, _Associated graded algebra of a Gorenstein Artin algebra_ , Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115.
* [13] S. C. Kothari, _The local Hilbert function of a pair of plane curves_ , Proc. Amer. Math. Soc. 72 (1978), no. 3, 439–442.
* [14] E. Matlis, _1-dimensional Cohen-Macaulay rings_ , L.N.M. Springer Verlag, 327 (1977).
* [15] F. Mora, _A constructive characterization of standard bases_ , Boll. Un. Mat. Ital. D (6) 2 (1983), no. 1, 41–50.
* [16] T. Puthenpuracal, _The Hilbert function of a maximal Cohen-Macaulay module_ , Math. Z. 251 (2005), 551–573.
* [17] M. E. Rossi, _Hilbert functions of Cohen-Macaulay local rings_ , Proceedings PASI (A. Corso and C. Polini, eds.), Contemporary Mathematics, vol. 555, A.M.S., 2011.
* [18] M. E. Rossi and G. Valla, _Hilbert functions of filtered modules_ , Lecture Notes of the Unione Matematica Italiana, vol. 9, Springer-Verlag, 2010.
* [19] B. Singh, _Effect of a permisible blowing-up on the local Hilbert function_ , Inv. Math. 26 (1974), 201–212.
* [20] P. Valabrega and G. Valla, _Form rings and regular sequences_ , Nagoya Math. J. 72 (1978), 93–101.
Juan Elias
Departament d’Àlgebra i Geometria
Universitat de Barcelona
Gran Via 585, 08007 Barcelona, Spain
e-mail: elias@ub.edu
Maria Evelina Rossi
Dipartimento di Matematica
Università di Genova
Via Dodecaneso 35, 16146 Genova, Italy
e-mail: rossim@dima.unige.it
Giuseppe Valla
Dipartimento di Matematica
Università di Genova
Via Dodecaneso 35, 16146 Genova, Italy
e-mail: valla@dima.unige.it
|
arxiv-papers
| 2012-05-24T07:53:57 |
2024-09-04T02:49:31.246617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Elias, M. E. Rossi, and G. Valla",
"submitter": "Juan Elias",
"url": "https://arxiv.org/abs/1205.5357"
}
|
1205.6020
|
# Non-Markovian dynamics for an open two-level system without rotating wave
approximation: Indivisibility versus backflow of information
Hao-Sheng Zeng111Corresponding author email: hszeng@hunnu.edu.cn, Ning Tang,
Yan-Ping Zheng and Tian-Tian Xu Key Laboratory of Low-Dimensional Quantum
Structures and Quantum Control of Ministry of Education, and Department of
Physics, Hunan Normal University, Changsha 410081, China
###### Abstract
By use of the two measures presented recently, the indivisibility and the
backflow of information, we study the non-Markovianity of the dynamics for a
two-level system interacting with a zero-temperature structured environment
without using rotating wave approximation (RWA). In the limit of weak coupling
between the system and the reservoir, and by expanding the time-
convolutionless (TCL) generator to the forth order with respect to the
coupling strength, the time-local non-Markovian master equation for the
reduced state of the system is derived. Under the secular approximation, the
exact analytic solution is obtained and the sufficient and necessary
conditions for the indivisibility and the backflow of information for the
system dynamics are presented. In the more general case, we investigate
numerically the properties of the two measures for the case of Lorentzian
reservoir. Our results show the importance of the counter-rotating terms to
the short-time-scale non-Markovian behavior of the system dynamics, further
expose the relations between the two measures and their rationality as non-
Markovian measures. Finally, the complete positivity of the dynamics of the
considered system is discussed.
PACS numbers: 03.65.Ta, 03.65.Yz, 42.50.Lc
## I Introduction
Realistic quantum systems cannot avoid interactions with their environments,
thus the study of open quantum systems is very important. It is not only
relevant for better understanding of quantum theory, but also fundamental for
various modern applications of quantum mechanics, especially for quantum
communication, cryptography and computation Nielsen . The early study of
dynamics of open quantum systems usually consists in the application of an
appropriate Born-Markov approximation, that is, neglects all the memory
effects, leading to a master equation which can be cast in the so-called
Lindblad form Lindblad ; Gorini . Master equations in Lindblad form can be
characterized by the fact that the dynamics of the system satisfies both the
semigroup property and the complete positivity, thus ensuring the preservation
of positivity of the density matrix during the time evolution. We usually
attribute the dynamical processes with these evolutional properties to the
well-known Markovian ones.
However, people recently found that Many relevant physical systems, such as
the quantum optical system Breuer3 , quantum dot Kubota , superconductor
system Yinghua , could not be described simply by Markovian dynamics.
Similarly, quantum chemistry Shao and the excitation transfer of a biological
system Chin also need to be treated as non-Markovian processes. Quantum non-
Markovian processes can lead to distinctly different effects on decoherence
and disentanglement Dijkstra ; Anastopoulos of open systems compared to
Markovian processes. These non-markovian effects can on the one hand enrich
the basic theory of quantum mechanics, on the other hand benefit the quantum
information processing. Because of these distinctive properties and extensive
applications, more and more attention and interest have been devoted to the
study of non-Markovian processes of open systems, including the measures of
non-Markovianity Breuer ; Laine ; Rivas ; Wolf ; Usha ; Lu ; Hou ; Xu ; He ,
the positivity Breuer1 ; Shabani ; Breuer2 , and some other dynamical
properties Haikka ; Chang ; Krovi ; Chru ; Haikka1 and approaches Jing ; Koch
; Wu of non-Markovian processes. Experimentally, the simulation Xu1 ; Xu2 of
non-Markovian environment has been realized.
The measure of non-Markovianity of quantum evolution is a fundamental problem
which aims to detect whether a quantum process is non-Markovian and how much
degrees it deviates from a Markovian one. Based on the distinguishability of
quantum states, Breuer, Laine and Piilo (BLP) Breuer proposed a measure to
detect the non-Markovianity of quantum processes which is linked to the flow
of information between the system and environment. Alternatively, Rivas,
Huelga and Plenio Rivas (RHP) also presented a measure of non-Markovianity by
employing the dynamical divisibility of a trace-preserving completely positive
map. It is clear that the BLP measure is based on the physical features of the
system-reservoir interactions, while the RHP definition is based on the
mathematical property of the dynamical maps. It has been shown that the two
measures agree for several important and commonly-used models Zeng , but do
not agree in general Dariusz . In this paper, we will use both the two
measures to describe the non-Markovianity of the dynamics of the considered
system, so as to more clearly see their relation, as well as the rationality
as the measure of non-Markovianity.
The study of the dynamics of non-Markovian open quantum systems is typically
very involved and often requires some approximations. Almost all the previous
treatments are based on the RWA, that is, neglect the counter-rotating terms
in the microscopic system-reservoir interaction Hamiltonian. However, the
counter-rotating terms which are responsible for the virtual exchanges of
energy between the system and the environment not always can be neglected. For
example, for the wide-frequency-spectra reservoir or when the frequency
distribution of the structured environment is detuned large enough from the
transition of the system, the RWA is invalid. Another motivation of this paper
is thus to study the effect of the counter-rotating terms on the non-Markovian
dynamics of the considered open quantum system.
The article is organized as follows. In Sec. II we introduce the microscopic
Hamiltonian model between the system and its environment, and derive the non-
Markovian time-local master equation for a two-level system weakly coupled to
a vacuum reservoir, by using the TCL approach to the forth-order but without
employing RWA in the interaction Hamiltonian. In Sec. III, we investigate the
non-Markovianity of the system dynamics in terms of both the RHP and BLP
measures. Through the analytical solution in the secular approximation, we
obtain the sufficient and necessary conditions for the dynamical
indivisibility and the backflow of information, showing the effect of the
counter-rotating terms on the non-Markovian dynamics of the system, and
exposing the relations between the BLP and RHP measures. In sec. IV, by
choosing the Lorentzian spectra reservoir as an exemplary example, we further
demonstrate the effect of the counter-rotating terms on the dynamical
indivisibility and the backflow of information, and clarify the rationality of
the two non-Markovian measures. Finally in Sec.V, we discuss simply the
complete positivity of the system dynamics. And the conclusion is arranged in
Sec.VI.
## II The microscopic model
Consider a two-level atom with Bohr frequency $\omega_{0}$ interacting with a
zero-temperature bosonic reservoir modeled by an infinite chain of quantum
harmonic oscillators. The total Hamiltonian for this system in the Schrödinger
picture is given by
$H=\frac{1}{2}\omega_{0}\sigma_{z}+\sum_{k}\omega_{k}b^{+}_{k}b_{k}+\sum_{k}g_{k}(\sigma_{+}+\sigma_{-})(b_{k}+b^{+}_{k}),$
(1)
where $\sigma_{z}$ and $\sigma_{\pm}$ are the Pauli and inversion operators of
the atom, $\omega_{k}$, $b_{k}$ and $b_{k}^{+}$ are respectively the
frequency, annihilation and creation operators for the $k$-th harmonic
oscillator of the reservoir. The coupling strength $g_{k}$ is assumed to be
real for simplicity. The distinct feature of this Hamiltonian is the
reservation of the counter-rotating terms, $\sigma_{+}b_{k}^{+}$ and
$\sigma_{-}b_{k}$, which is the so-called without RWA we call in this paper.
Note however that our starting point is the dipole interaction Hamiltonian
between the atom and its environment, whose derivation starting with the
canonical Hamiltonian involves the discarding of a term which is quadratic
with respect to the radiation field. The discarding is not based on the RWA,
but the fact that for low-intensity radiation, the quadratic term is much
small compared to the dipole interaction one Claude .
The time-convolutionless projection operator technique is most effective in
dealing with the dynamics of open quantum systems. In the limit of weak
coupling between the system and the environment, by expanding the TCL
generator to the forth order with respect to coupling strength, the non-
Markovian master equation describing the evolution of the reduced system, in
the interaction picture, can be written as [For the main clue of its
derivation, see appendix A.]
$\frac{d\rho(t)}{dt}=-i[H_{LS}(t),\rho(t)]+D[\rho(t)]+D^{\prime}[\rho(t)],$
(2)
where
$H_{LS}(t)=S_{+}(t)\sigma_{+}\sigma_{-}+S_{-}(t)\sigma_{-}\sigma_{+},$ (3)
is the Lamb shift Hamiltonian which describes a small shift in the energy of
the eigenstates of the two-level atom. In many theoretical researches Haikka ,
this term was neglected usually. But in this paper, we will take it into the
consideration. The Lamb shift includes the second and forth order
contributions,
$S_{\pm}(t)=S_{\pm}^{II}(t)+S_{\pm}^{IV}(t),$ (4)
which respectively come from the second and forth order perturbative expansion
of the TCL generator. The second order Lamb shift is
$S_{\pm}^{II}(t)=\pm\int^{t}_{0}d\tau\int d\omega
J(\omega)\sin[(\omega_{0}\mp\omega)\tau],$ (5)
with $J(\omega)=\sum_{k}|g_{k}|^{2}\delta(\omega-\omega_{k})$ the spectral
distribution of the environment. The expression for the forth order Lamb shift
$S_{\pm}^{IV}(t)$ is cumbersome which is presented in the appendix A.
The dissipator $D[\rho(t)]$ that describes the secular motion of the system
has the form
$\displaystyle D[\rho(t)]$ $\displaystyle=$
$\displaystyle\Gamma_{-}(t)\textbf{\\{}\sigma_{-}\rho(t)\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho(t)\\}\textbf{\\}}$
$\displaystyle+$
$\displaystyle\Gamma_{+}(t)\textbf{\\{}\sigma_{+}\rho(t)\sigma_{-}-\frac{1}{2}\\{\sigma_{-}\sigma_{+},\rho(t)\\}\textbf{\\}}$
$\displaystyle+$
$\displaystyle\Gamma_{0}(t)\textbf{\\{}\sigma_{+}\sigma_{-}\rho(t)\sigma_{+}\sigma_{-}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho(t)\\}\textbf{\\}},$
where the first line describes the dissipation of the atom to the vacuum
environment with time-dependent decay rate $\Gamma_{-}(t)$, and the second
line denotes the heating of the atom in the vacuum environment with time-
dependent heating rate $\Gamma_{+}(t)$. This heating is related to the
dissipation, for a ground-state atom in a zero-temperature environment, there
is no heating effect. Dissipation and heating are usually accompanied by
decoherence. The last line in eq.(6) describes the pure decoherence with time-
dependent decoherence rate $\Gamma_{0}(t)$. The time-dependent transition
rates $\Gamma_{\pm}(t)$ also include the second and forth order perturbative
contributions of the TCL generator,
$\Gamma_{\pm}(t)=\Gamma_{\pm}^{II}(t)+\Gamma_{\pm}^{IV}(t),$ (7)
with the second order contribution as
$\Gamma_{\pm}^{II}(t)=2\int_{0}^{t}d\tau\int d\omega
J(\omega)\cos[(\omega_{0}\pm\omega)\tau].$ (8)
While $\Gamma_{0}(t)$ completely comes from the forth-order perturbative
contribution. All the forth-order contributions are presented in the appendix
A. Eq.(6) indicates that the dissipative model of eq.(1), except for inducing
the energy exchange between the system and its environment, also makes
decoherence of the system. But the rate of decoherence is much less than that
of energy dissipation, because $\Gamma_{0}(t)$ is only a forth-order
contribution term of TCL perturbative expansion.
The dissipator $D^{\prime}[\rho(t)]$ represents the contribution of the so-
called nonsecular terms, that is, terms oscillating rapidly with Bohr
frequency $\omega_{0}$,
$D^{\prime}[\rho(t)]=[\alpha(t)+i\beta(t)]\sigma_{+}\rho(t)\sigma_{+}+h.c.,$
(9)
here $h.c.$ denotes the Hermitian conjugation. These nonsecular terms
sometimes may also be neglected under the so-called secular approximation
Maniscalco . The time-dependent coefficients $\alpha(t)$ and $\beta(t)$ also
include the second and forth order contributions,
$\alpha(t)=\alpha^{II}(t)+\alpha^{IV}(t),$ (10)
$\beta(t)=\beta^{II}(t)+\beta^{IV}(t),$ (11)
with
$\alpha^{II}(t)=2\int_{0}^{t}d\tau\int d\omega
J(\omega)\cos[\omega(t-\tau)]\cos[\omega_{0}(t+\tau)],$ (12)
and
$\beta^{II}(t)=2\int_{0}^{t}d\tau\int d\omega
J(\omega)\cos[\omega(t-\tau)]\sin[\omega_{0}(t+\tau)].$ (13)
The forth-order contributions are listed in the appendix A.
Note that the dynamics for a two-level system embedded in a zero-temperature
structured environment, under RWA, can be solved exactly, where the
corresponding master equation has the Lindblad-like form Breuer3 ,
$\frac{d}{dt}\rho(t)=-\frac{i}{2}S(t)[\sigma_{+}\sigma_{-},\rho(t)]+\gamma(t)\textbf{\\{}\sigma_{-}\rho(t)\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho(t)\\}\textbf{\\}},$
(14)
where the time-dependent decay rate $\gamma(t)$ and Lamb shift $S(t)$ are
related to the correlation function of the reservoir. Comparing this equation
with eq.(2), we see that the last two terms in the dissipator $D[\rho(t)]$,
that is, the heating and the pure decoherence terms, as well as the nonsecular
dissipator $D^{\prime}[\rho(t)]$ and the Lamb shift $S_{-}(t)$, are completely
from the contribution of the counter-rotating terms presented in the
interaction Hamiltonian. While the decay rate $\Gamma_{-}(t)$ and the Lamb
shift $S_{+}(t)$ include the contributions of both rotating and counter-
rotating terms, but the main contributions [i.e., the second-order terms
$\Gamma^{II}_{-}(t)$ and $S^{II}_{+}(t)$] come from the rotating terms. In
fact, by expanding the decay rate $\gamma(t)$ and the Lamb shift $S(t)$ to the
second order with respect to coupling strength, one obtain
$\Gamma^{II}_{-}(t)$ and $S^{II}_{+}(t)$ Breuer3 . In the following, we will
show that the contributions that come from the counter-rotating terms are
important, in particular to the short-time-scale non-Markovian behaviors.
## III Measures of Non-Markovianity
Recently, people have been interested in the study of non-Markovianity of open
quantum systems. Several definitions or measures Breuer ; Rivas ; Wolf ; Usha
; Lu of non-Markovian dynamics have been presented. In this section, we will
employ two of the measures, i.e., the RHP Rivas and BLP Breuer measures, to
investigate the non-Markovian dynamics of the considered system so as to see
the effect of the counter-rotating terms on non-Markovianity and the relation
between the two measures.
### III.1 Divisible and indivisible dynamics
A trace-preserving completely positive map $\varepsilon(t_{2},0)$ that
describes the evolution from times zero to $t_{2}$ is divisible if it
satisfies composition law,
$\varepsilon(t_{2},0)=\varepsilon(t_{2},t_{1})\varepsilon(t_{1},0),$ (15)
with $\varepsilon(t_{2},t_{1})$ being completely positive for any $t_{2}\geq
t_{1}\geq 0$. Due to the continuity of time, eq.(15) is always fulfilled in
form. The key point for divisibility is actually the complete positivity of
$\varepsilon(t_{2},t_{1})$ for any $t_{2}\geq t_{1}\geq 0$. If there exist
times $t_{1}$ and $t_{2}$ such that the map $\varepsilon(t_{2},t_{1})$ is not
completely positive, then the dynamical map $\varepsilon(t_{2},0)$ is
indivisible. RHP Rivas defined all the divisible maps to be Markovian.
Therefore, the indivisibility of a map advocates its dynamical non-
Markovianity. It was shown that all the evolutions governed by Lindblad-type
master equation with positive transition rates are divisible Alicki , thus
Markovian.
It was proved Rivas that the indivisibility of map $\varepsilon(t,0)$ is
equivalent to the complete positivity of the quantity,
$g(t)=\lim_{\epsilon\rightarrow
0^{+}}\frac{\|[\varepsilon(t+\epsilon,t)\otimes
I]|\Phi\rangle\langle\Phi|\|-1}{\epsilon}.$ (16)
Only for divisible map, $g(t)=0$. Where $|\Phi\rangle$ is a maximally
entangled state between the system of interest and an ancillary particle, and
the map $\varepsilon$ performs only on the state of the system. Using the
time-local master equation $\frac{d\rho}{dt}=\mathcal{L}_{t}(\rho)$, this
expression may be equivalently written as Rivas
$g(t)=\lim_{\epsilon\rightarrow 0^{+}}\frac{\|[I+(\mathcal{L}_{t}\otimes
I)\epsilon]|\Phi\rangle\langle\Phi|\|-1}{\epsilon}.$ (17)
The function $g(t)$ is the so-called RHP non-Markovian measure. If and only if
$g(t)=0$ for every time $t\in\\{0,t_{2}\\}$, the map $\varepsilon(t_{2},0)$ is
Markovian. Otherwise it is non-Markovian. The distinctive advantage of RHP
non-Markovian measure is that its calculation can be processed only by the use
of time-local master equation, not requiring the exact form of the dynamical
map $\varepsilon(t,0)$. In the following, we call the time interval that
satisfies $g(t)>0$ the indivisible dynamical interval (IDI). For a non-
Markovian process, there must exist one or several IDIs.
For the open two-level system considered in this paper, suppose that
$|\Phi\rangle=\frac{1}{\sqrt{2}}[|01\rangle+|10\rangle]$, a straightforward
deduction using equations (2) and (17) gives
$\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{1}{4}|\Gamma_{-}+\Gamma_{+}+\sqrt{(\Gamma_{-}-\Gamma_{+})^{2}+4(\alpha^{2}+\beta^{2})}|$
$\displaystyle+$
$\displaystyle\frac{1}{4}|\Gamma_{-}+\Gamma_{+}-\sqrt{(\Gamma_{-}-\Gamma_{+})^{2}+4(\alpha^{2}+\beta^{2})}|$
$\displaystyle+$
$\displaystyle\frac{1}{4}[|\Gamma_{0}|-\Gamma_{0}-2\Gamma_{-}-2\Gamma_{+}],$
where for compactness we omit the argument of all the time-dependent
coefficients. Obviously, the Lamb shift $H_{LS}(t)$ has no effect on the
indivisibility of the system dynamics.
### III.2 Backflow of information
The second measure of non-Markovianity for quantum processes of open systems
we employ is proposed by BLP Breuer which is based on the consideration in
purely physics. Note that Markovian processes always tend to continuously
reduce the trace distance between any two states of a quantum system, thus an
increase of the trace distance during any time interval implies the emergence
of non-Markovianity. BLP further linked the change of the trace distance to
the flow of information between the system and its environment, and concluded
that the back flow of information from environment to the system is the key
feature of a non-Markovian dynamics. In quantum information science, the trace
distance for quantum states $\rho_{1}$ and $\rho_{2}$ is defined as Nielsen
$D(\rho_{1},\rho_{2})=\frac{1}{2}tr|\rho_{1}-\rho_{2}|,$ (19)
with $|A|=\sqrt{A^{+}A}$. For a given pair of initial states $\rho_{1,2}(0)$
of the system, the change of the dynamical trace-distance can be described by
its time derivative
$\sigma\textbf{(}t,\rho_{1,2}(0)\textbf{)}=\frac{d}{dt}D\textbf{(}\rho_{1}(t),\rho_{2}(t)\textbf{)},$
(20)
where $\rho_{1,2}(t)$ are the dynamical states of the system with the initial
states $\rho_{1,2}(0)$. For Markovian processes, the monotonically reduction
of the trace distance implies $\sigma\textbf{(}t,\rho_{1,2}(0)\textbf{)}\leq
0$ for any initial states $\rho_{1,2}(0)$ and at any time $t$. If there exists
a pair of initial states of the system such that for some evolutional time
$t$, $\sigma\textbf{(}t,\rho_{1,2}(0)\textbf{)}>0$, then the information takes
backflow from environment to the system, and the process is non-Markovian.
In order to calculate the BLP measure, we must solve the dynamics of the
system. For this purpose, we write the alternative Bloch equation of eq.(2) as
[see appendix B for their derivation],
$\displaystyle\dot{b}_{x}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}-2\alpha)b_{x}+(S_{-}-S_{+}-\beta)b_{y},$
(21) $\displaystyle\dot{b}_{y}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}+2\alpha)b_{y}-(S_{-}-S_{+}+\beta)b_{x},$
(22) $\displaystyle\dot{b}_{z}$ $\displaystyle=$
$\displaystyle-(\Gamma_{-}+\Gamma_{+})b_{z}+\Gamma_{+}-\Gamma_{-},$ (23)
where the three components of the Bloch vector are defined as
$b_{j}(t)=\texttt{Tr}[\rho(t)\sigma_{j}]$ with $j=x,y,z$ and $\sigma_{j}$ the
Pauli operators. In terms of Bloch vector, the trace distance of eq.(19) may
be expressed as
$D(t)=\frac{1}{2}\sqrt{(\Delta b_{x})^{2}+(\Delta b_{y})^{2}+(\Delta
b_{z})^{2}}$ (24)
where $\Delta b_{j}=b_{1j}(t)-b_{2j}(t)$ are the differences between the two
Bloch components at evolutional time $t$. Correspondingly, the derivative of
this trace distance becomes
$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-\frac{1}{4}[(\Delta
b_{x})^{2}+(\Delta b_{y})^{2}+(\Delta
b_{z})^{2}]^{-1/2}\textbf{\\{}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}-2\alpha)(\Delta
b_{x})^{2}$ $\displaystyle+$
$\displaystyle(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}+2\alpha)(\Delta
b_{y})^{2}+4\beta(\Delta b_{x})(\Delta b_{y})+2(\Gamma_{-}+\Gamma_{+})(\Delta
b_{z})^{2}\textbf{\\}},$
where we have used the Bloch eqs.(21)-(23) in the deduction process. According
BLP’s criterion, $\sigma>0$ indicates the backflow of information from
environment to the system. In the following, we call the time intervals in
which $\sigma(t)>0$ the information-backflow intervals (IBIs).
### III.3 Secular approximation
In order to see the effect of counter-rotating terms and make a distinct
comparison between the BLP and RHP measures in the current system, we now
consider the case where the nonsecular term $D^{\prime}[\rho(t)]$ can be
neglected, i.e., performing the so-called secular approximation. Here for the
sake of discrimination, we call as in many literatures _the rotating-wave
approximation that used after tracing over the bath degrees of freedom_ the
secular approximation. In other words, the secular approximation and the RWA
have the same mathematical approaches–throwing away the rapidly oscillating
terms in time, merely the times the approximations taking place are different.
Just as pointed out in reference Maniscalco , this kind of secular
approximation though also is an average over rapidly oscillating terms, it
does not wash out the effect of the counter-rotating terms present in the
coupling Hamiltonian. Under the secular approximation, the master equation (2)
has the Lindblad-like form with time-dependent transition rates,
$\Gamma_{\pm}(t)$, $\Gamma_{0}(t)$ and Lamb shift $H_{LS}(t)$. Employing the
method proposed in Michael , the corresponding Bloch eqs.(21)-(23) in this
case can be solved exactly which gives
$\displaystyle b_{x}(t)$ $\displaystyle=$ $\displaystyle
e^{-\Theta(t)}[b_{x}(0)\cos\delta(t)-b_{y}(0)\sin\delta(t)],$ (26)
$\displaystyle b_{y}(t)$ $\displaystyle=$ $\displaystyle
e^{-\Theta(t)}[b_{x}(0)\sin\delta(t)+b_{y}(0)\cos\delta(t)],$ (27)
$\displaystyle b_{z}(t)$ $\displaystyle=$ $\displaystyle
e^{-\Lambda(t)}\left\\{b_{z}(0)+\int_{0}^{t}dse^{\Lambda(s)}[\Gamma_{+}(s)-\Gamma_{-}(s)]\right\\},$
(28)
with
$\Theta(t)=\frac{1}{2}\int_{0}^{t}ds[\Gamma_{-}(s)+\Gamma_{+}(s)+\Gamma_{0}(s)],$
(29) $\Lambda(t)=\int_{0}^{t}ds[\Gamma_{-}(s)+\Gamma_{+}(s)],$ (30)
and
$\delta(t)=\int_{0}^{t}ds[S_{+}(s)-S_{-}(s)].$ (31)
Inserting these solutions into eq.(25), we get
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle-\frac{1}{4}I(t)\textbf{\\{}e^{-2\Theta(t)}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0})[\textbf{(}\Delta
b_{x}(0)\textbf{)}^{2}+\textbf{(}\Delta b_{y}(0)\textbf{)}^{2}]$
$\displaystyle+$ $\displaystyle
2e^{-2\Lambda(t)}(\Gamma_{-}+\Gamma_{+})\textbf{(}\Delta
b_{z}(0)\textbf{)}^{2}\textbf{\\}},$
where $I(t)=\\{e^{-2\Theta(t)}[\textbf{(}\Delta
b_{x}(0)\textbf{)}^{2}+\textbf{(}\Delta
b_{y}(0)\textbf{)}^{2}]+e^{-2\Lambda(t)}\textbf{(}\Delta
b_{z}(0)\textbf{)}^{2}\\}^{-1/2}$ is a positive function and $\Delta
b_{j}(0)=b_{1j}(0)-b_{2j}(0)$ is the difference between the two initial Bloch
components. This expression shows that the sufficient and necessary conditions
for the backflow of information from environment to the system are
$\Gamma_{-}(t)+\Gamma_{+}(t)+\Gamma_{0}(t)<0,$ (33)
or
$\Gamma_{-}(t)+\Gamma_{+}(t)<0.$ (34)
Because if at some time $t$, one of this conditions is satisfied, then we can
always find a pair of initial states such that $\sigma(t)>0$. For example, if
eq.(33) fulfils, it suffices to choose the initial states satisfying $\Delta
b_{z}(0)=0$. Conversely, if $\sigma(t)>0$ at some time $t$, then at least one
of the two conditions must be satisfied.
On the other hand, under secular approximation, eq.(18) is simplified as
$g=\frac{1}{4}\\{2|\Gamma_{-}|+2|\Gamma_{+}|+|\Gamma_{0}|-2\Gamma_{-}-2\Gamma_{+}-\Gamma_{0}\\}.$
(35)
Obviously, when one of the three rate functions, $\Gamma_{-}(t)$,
$\Gamma_{+}(t)$ or $\Gamma_{0}(t)$, is negative, then $g>0$, vice versa. Thus
the sufficient and necessary conditions for the indivisibility of the dynamics
are
$\Gamma_{-}(t)<0,or\hskip 5.69046pt\Gamma_{+}(t)<0,or\hskip
5.69046pt\Gamma_{0}(t)<0.$ (36)
Eqs. (33), (34) and (36) demonstrate two important results. One the one hand,
the counter-rotating terms [which induce $\Gamma_{+}(t)$, $\Gamma_{0}(t)$ and
a part of $\Gamma_{-}^{IV}(t)$] may have important effect to the non-Markovian
dynamics of the system, according to RHP and BLP measures. On the other hand,
they show that the conditions for the backflow of information are much more
rigorous than that of indivisibility. The later only requires one of the
transition rates to be negative, while the former further requires the sum of
the two or the total transition rates to be negative. This conditionality once
again validates the previous results: The backflow of information must lead
the indivisibility of the dynamics, but the reverse is not true Dariusz .
However, for Lindblad-like master equation with only single transition rate,
the sufficient and necessary conditions for the two measures become clearly
the same, denoting the consistency of the two measures in this case Zeng .
## IV Non-Markovian dynamics for Lorentzian spectrum
In order to further demonstrate quantitatively the effect of the counter-
rotating terms, as well as the rationality of the two non-Markovian measures,
we specify our study to a particular reservoir spectra, Lorentzian spectra,
$J(\omega)=\frac{\gamma_{0}\lambda^{2}}{2\pi[(\omega_{0}-\omega-\Delta)^{2}+\lambda^{2}]},$
(37)
which describes the interaction of an atom with an imperfect cavity and is
widely used in literatures. Where $\omega_{0}$ denotes the transition
frequency of the atom, $\Delta=\omega_{0}-\omega_{c}$ is the frequency
detuning between the atom and the cavity mode. $\lambda$ is the width of
Lorentzian distribution, which is connected to the reservoir correlation time
$\tau_{R}=\lambda^{-1}$. The parameter $\gamma_{0}$ can be regarded as the
decay rate for the excited atom in the Markovian limit of flat spectrum which
is related to the relaxation time $\tau_{S}=\gamma_{0}^{-1}$. For the
Lorentzian spectra, all the time-dependent coefficients including
$S_{\pm}(t)$, $\Gamma_{\pm}(t)$, $\Gamma_{0}(t)$, $\alpha(t)$ and $\beta(t)$
can be calculated analytically, but the expressions are too complicated. We
thus study them only numerically.
In Fig.1, we show the time evolution of these coefficients. For our purpose,
we intentionally choose three special sets of parameters. It shows that for
narrow spectrum and small detuning [In Fig.1 (a) and (d),
$\lambda/\omega_{0}=0.2\%$, $\Delta/\omega_{0}=2\%$], $\Gamma_{-}(t)$ plays
the dominant role, while $\Gamma_{+}(t)$ and $\Gamma_{0}(t)$ are almost zero.
The nonsecular coefficients $\alpha(t)$ and $\beta(t)$ in this case behave
fast oscillations [Fig.1 (d)], so that on average in time the effect can also
be neglected. These results imply that for this set of parameters, the
counter-rotating terms in Hamiltonian (1) play little effect actually to the
system dynamics and the commonly-used RWA is valid. However, for wider
spectrum or/and larger detuning, the results are different [see Fig.1 (b) and
(c)], where though $\Gamma_{0}$ is still near zero Note , $\Gamma_{+}$ clearly
can not be neglected. Thus the counter-rotating terms in these cases are
important and the RWA is invalid. Of course, for very wide spectrum, one may
expect that the dynamics tends to be Markovian. The positivity of the
$\Gamma_{\pm}(t)$ and $\Gamma_{0}(t)$ in Fig.1 (c) confirms this point. In
addition, when $\lambda$ is small, the correlation time of the environment is
longer, thus $\Gamma_{-}$ in Fig.1 (a) oscillates to emerge negative values in
a relatively longer time. With the increasing of $\lambda$, the correlation
time becomes small and small, and the times for $\Gamma_{\pm}$ to be negative
shorten or even vanish [Fig.1 (b) and (c)]. Note that the observable negative
values of $\Gamma_{+}$ in Fig.1 (b) demonstrate the contribution of the
counter-rotating terms to the non-Markovianity of the system dynamics.
Figure 1: Evolution of the time-dependent coefficients. The dot-dash line,
dot line and solid line in (a),(b) and (c) correspond to respectively the
evolutions of $\Gamma_{-}$, $\Gamma_{+}$ and $\Gamma_{0}$, while the solid and
dot lines in (d) refer to the evolutions of $\alpha$ and $\beta$. Where we
choose $\omega_{0}=100\gamma_{0}$, and $\lambda=0.2\gamma_{0}$,
$\Delta=2\gamma_{0}$ for (a); $\lambda=5\gamma_{0}$, $\Delta=50\gamma_{0}$ for
(b); $\lambda=400\gamma_{0}$, $\Delta=10\gamma_{0}$ for (c). The parameters
for (d) are the same as that of (a).
Note that in the RWA, the corresponding master equation (14) may be solved
exactly. For the Lorentzian spectrum, the RHP and BLP measures may be
expressed as Zeng ,
$g(t)=\left\\{\begin{array}[]{lll}0&\mathrm{for}&\gamma(t)\geq 0\\\
-\gamma(t)&\mathrm{for}&\gamma(t)<0\\\ \end{array}\right.$ (38)
and
$\sigma(t)=-\gamma(t)F(t).$ (39)
where
$\gamma(t)=\texttt{Re}\left[\frac{2\gamma_{0}\lambda\sinh(dt/2)}{d\cosh(dt/2)+(\lambda-i\Delta)\sinh(dt/2)}\right],$
(40)
with $d=\sqrt{(\lambda-i\Delta)^{2}-2\gamma_{0}\lambda}$. The positive real
function $F(t)$ is defined as,
$F(t)=\frac{a^{2}e^{-\frac{3}{2}\Gamma(t)}+|b|^{2}e^{-\frac{1}{2}\Gamma(t)}}{\sqrt{a^{2}e^{-\Gamma(t)}+|b|^{2}}},$
(41)
with $\Gamma(t)=\int_{0}^{t}dt^{\prime}\gamma(t^{\prime})$, and $a=\langle
1|\rho_{1}(0)|1\rangle-\langle 1|\rho_{2}(0)|1\rangle$, $b=\langle
1|\rho_{1}(0)|0\rangle-\langle 1|\rho_{2}(0)|0\rangle$ being the differences
of the population and of coherence respectively for the two given initial
states. Eqs.(38) and (39) show that under the RWA, the distributions of IDIs
and IBIs are exactly the same, which are determined by $\gamma(t)<0$. In the
following, we study numerically the evolution of the measures $\sigma$ and
$g$, under the condition without using RWA, so as to further highlight the
non-Markovian effect of the counter-rotating terms, as well as the rationality
of BLP and RHP measures.
In Fig.2, we show the time evolution of the measure $\sigma$ in the same
parameters as in Fig.1, where the solid lines are plotted according to eq.(25)
and the dot lines according to eq.(39). We choose the pair of initial states
to be $\rho_{1}(0)=|1\rangle\langle 1|$ and $\rho_{2}(0)=|0\rangle\langle 0|$,
which can maximize the BLP measure Breuer . For evidence, we only give the
time intervals of $\sigma>0$, i.e., the IBIs. We can see clearly the
corrections of the counter-rotating terms on the BLP measure. In Fig.2 (a),
both the distributions of the IBIs and the shapes of the two curves are
similar, responding that the counter-rotating terms make lesser effect to the
backflow of information in this case which is in line with the idea of RWA.
The dips on each peaks of the solid-line in Fig.2 (b) are due to the
negativity of $\Gamma_{+}(t)$ at that times [see Fig.1 (b)], implying that
$\Gamma_{+}$ has the offset on the backflow of information. There is no IBI in
Fig.2 (c), denoting that under the choice of this set of parameters, there is
no backflow of information, or equivalently the dynamics is Markovian
according to BLP measure, which is in line with the non-negativity of
$\Gamma_{\pm}(t)$ and $\Gamma_{0}(t)$. In addition, the time scale for the
backflow of information is consistent with the reservoir correlation time
$\lambda^{-1}$ [Fig.2 (a), (b)]. All these results show that on the one hand
the counter-rotating terms can affect the backflow of information, and on the
other hand the correction of the counter-rotating terms on the backflow of
information is reasonable.
Figure 2: Time evolution of $\sigma(t)$, with the solid and dot lines
corresponding to respectively eqs.(25) and (39). The parameters in (a),(b) and
(c) are set to be in accordance with that in Fig.1.
In Fig.3, we plot the time evolution of the measure $g$ in the same parameters
as in Fig.1. We see that when the counter-rotating terms are omitted, the
distribution of the IDIs agrees with that of IBIs [see the dot lines in Figs.
2 and 3]. The non-Markovian time scale predicted by measure $g$ is also in
accordance with the reservoir correlation time $\lambda^{-1}$. The horizontal
dot line in Fig.3(c) denotes that under the choice of those parameters, the
dynamics is actually Markovian. All these results show that with no counter-
rotating terms, the RHP and BLP measures agree. Both of them can depict
rightly the non-Markovianity of the underlying dynamics. However, when the
counter-rotating terms are considered, the case is distinctly different: The
IDIs now become $(0,\infty)$ [see the solid lines in Fig.3], which are clearly
inconsistent with the practice. Because first of all, the non-Markovian time
scales in the underlying conditions are never infinite. Next, in Fig.3(a), the
choice of the parameters is consistent with the RWA, the result after
considering counter-rotating terms should has some tiny, not distinct
amendments, over the result under RWA. For the parameters in Fig.3(c), the
reservoir correlation time $\lambda^{-1}$ is very short and the system
dynamics is actually Markovian, should not appearing long-time non-
Markovianity. These egregious results denote that the RHP measure in these
cases is invalid. Note that the reason for resulting in these unpractical
phenomena is mainly due to the nonsecular coefficients $\alpha(t)$ and
$\beta(t)$. When these nonsecular coefficients are neglected, eq.(35) is not
seen to deviate obviously from the practice.
Figure 3: Time evolution of $g(t)$, where the solid and dot lines are plotted
according to eqs.(18) and (38) respectively. The parameters are set to be the
same as in Fig.1.
## V Complete positivity
The evolution of a real physical state should be not only positive but also
complete positive. In practical theoretical study, however, due to the
application of some assumptions and approximations, the positivity or the
complete positivity may not always be satisfied. Here we present a study of
the complete positivity for our considered model, i.e., the master equation of
(2). As the damping matrix has the block diagonal form (see appendix B), thus
we can directly use the conditions for complete positivity presented by Hall
Michael . The necessary condition of the complete positivity, for the master
equation (2), may be given by two inequalities:
$\Lambda(t)\geq 0,$ (42) $2\Theta(t)\geq\Lambda(t),$ (43)
with $\Theta(t)$ and $\Lambda(t)$ given by eqs.(29)-(30). The sufficient
condition is also given by two inequalities. The first one coincides with
eq.(42) and the second one may be expressed as
$\chi(t)\cosh\theta(t)\leq 1+A^{2}(t)-\kappa^{2}(t)-2|A(t)-\chi(t)|,$ (44)
where $\chi(t)=e^{-2\Theta(t)}$, $A(t)=e^{-\Lambda(t)}$,
$\kappa(t)=A(t)\int_{0}^{t}ds[\Gamma_{+}(s)-\Gamma_{-}(s)]A^{-1}(s)$, and
$\theta(t)=2\int_{0}^{t}ds\sqrt{\alpha^{2}(s)+\beta^{2}(s)}$ with $\alpha(t)$,
$\beta(t)$ given by eqs.(10)-(11). Using inequality (43) to release the
modulus in the right-hand side, we get
$\chi(t)\cosh\theta(t)\leq[1-A(t)]^{2}+2\chi(t)-\kappa^{2}(t),$ (45)
Note that the left-hand side of eq.(45) is relevant to the nonsecular motion,
but the right-hand side only depends on the secular motion. As $\theta(t)$
increases with time $t$, eq.(45) is not satisfied for long times. But in short
non-Markovian time scales we are interested in, it may be fulfilled. In order
to see this, we plot the time evolution of function
$G(t)\equiv[1-A(t)]^{2}+2\chi(t)-\kappa^{2}(t)-\chi(t)\cosh\theta(t)$ as in
Fig.4, for the same parameters as in Fig.1 and under the Lorentzian spectra.
Obviously, in the scale of the correlation times $\lambda^{-1}$, $G(t)>0$. The
condition of eqs.(42)-(43) is satisfied for all times in this case. Thus in
the short non-Markovian time scales, the evolution of the system is physical.
In the secular regime, the sufficient condition eq.(45) can be relaxed to
$[1-A(t)]^{2}+\chi(t)-\kappa^{2}(t)\geq 0,$ (46)
which can be satisfied for much more longer times for the Lorentzian
reservoir.
Figure 4: Time evolution of $G(t)$, where the solid, dash and dot lines
correspond to respectively the parameters in Fig.1 (a), (b) and (c).
## VI Conclusion
In conclusion, we have studied the non-Markovianity of the dynamics for a two-
level system interacting with a zero-temperature structured environment
without using RWA. In the limit of weak coupling between the system and its
reservoir, by expanding the TCL generator to the forth order with respect to
the coupling strength, we have derived the time-local non-Markovian master
equation for the reduced state of the system. Under the secular approximation,
the TCL master equation has the Lindblad-like form with time-dependent
transition rates. We have obtained the exact analytic solution. The sufficient
and necessary conditions for the indivisibility and the backflow of
information for the system dynamics were presented, which showed two important
results: First, the counter-rotating terms may play important roles to the
indivisibility and the backflow of information for the system dynamics.
Second, it showed explicitly that the BLP and RHP measures generally do not
coincide. It demonstrated more clearly the previous result: The backflow of
information must lead to the indivisibility of dynamics, but the reserve is
not true.
When the nonsecular terms are included, we have investigated numerically the
non-Markovian properties of the system dynamics by assuming that the
environment spectrum is Lorentzian. By compared with the result under RWA, we
found that the BLP measure is corrected appropriately, but the RHP measure is
inconsistent with practice, showing that the RHP measure has finite applicable
range.
Finally, we have discussed the complete positivity of the underlying dynamics.
We have presented the sufficient and necessary conditions of the complete
positivity. Numerical simulation showed that these conditions can be satisfied
in the short non-Markovian time scale.
The measure of non-Markovianity is a fundamental problem in the study of open
quantum system dynamics. Although several measures of non-Markovianity have
been presented already, it is noted that these measures are not completely
equivalent to each other. Therefore, the problem for measuring the non-
Markovianity of quantum processes still remains elusive and, in some sense,
controversial. At present stage, it is meaningful and necessary to expose the
characteristics of various measures and their relations in some concrete
systems.
The investigation of a two-level system interacting with a bath of harmonic
oscillators, i.e., the spin-boson model, is of particular interest in the
theory of open quantum system. In the context of quantum computation, it
represents a qubit coupled to an environment, which can produce dissipation
and decoherence. Though in the numerical simulations we have only considered
the Lorentzian environment, our analytic results adapt to other structured
environments, such as the Ohmic reservoir, the photonic band-gap material
Woldeyohannes , etc. By properly engineering the structure of the environment,
one can control the non-Markovian dynamics of the open quantum system, so as
to effectively control the evolution of some interesting physical quantities,
such as the quantum coherence, quantum entanglement and discord, etc.
Therefore, our work will be helpful for the quantum information processing.
Of course, our model is not fully general. First of all, we have considered
only a two-level system weakly coupled a zero-temperature environment. Next,
our starting point is based on the dipole interaction Hamiltonian between the
atom and its environment, not on the canonical Hamiltonian. Finally, we have
used the TCL perturbation expansion for the derivation of master equation
eq.(2). Thus our results are still conditional and further investigations may
be necessary.
###### Acknowledgements.
This work is supported by the National Natural Science Foundation of China
(Grant No.11075050), the National Fundamental Research Program of China (Grant
No.2007CB925204), the Program for Changjiang Scholars and Innovative Research
Team in University under Grant No.IRT0964, and the Construct Program of the
National Key Discipline.
## Appendix A Derivation of the master equation and the time-dependent forth-
order coefficients
In our study, the derivation of the forth-order TCL master equation (2) is
very cumbersome. Here we can present only the main clue about the deduction.
Our calculation is based on the description of reference Breuer3 about the
TCL projection operator technique. By assuming a factoring initial condition
$\rho(0)=\rho_{S}(0)\otimes\rho_{B}$ for the system and environment, one
obtains a homogeneous TCL master equation [see (9.33) of Breuer3 ]
$\frac{\partial}{\partial
t}\mathcal{P}\rho(t)=\mathcal{K}(t)\mathcal{P}\rho(t).$ (47)
Due to the assumption of vacuum reference state $\rho_{B}=|0\rangle\langle 0|$
for the environment, the TCL generator $\mathcal{K}(t)$ only has even-order
terms in its perturbation expansion. The second- and forth-order TCL
generators may be calculated directly via eqs.(9.61)-(9.62) of reference
Breuer3 , where the related operators $F_{k}$ and $Q_{k}$ in the interaction
picture are given by,
$\displaystyle F_{k}(t)$ $\displaystyle=$
$\displaystyle\sigma_{+}e^{i\omega_{0}t}+\sigma_{-}e^{-i\omega_{0}t},$ (48)
$\displaystyle Q_{k}(t)$ $\displaystyle=$ $\displaystyle
g_{k}(b_{k}e^{-i\omega_{k}t}+b_{k}^{+}e^{i\omega_{k}t}).$ (49)
Calculating the second- and forth-order TCL generators and sorting them in
operators, then eq.(A1) reduces to the required master equation.
In the master equation (2), each of the time-dependent coefficients consists
of in principle two parts–the second and the forth order parts. The second-
order parts have relatively simple expressions, but the expressions of the
forth-order parts are very complex. In terms of abbreviation
$t_{ij}=t_{i}-t_{j}$ with $t_{0}\equiv t$, $C(t)=\int d\omega
J(\omega)\cos\omega t$, $S(t)=\int d\omega J(\omega)\sin\omega t$ and
$\texttt{T}\int=\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}$,
the forth-order coefficients may be written in the following,
$\displaystyle S_{+}^{IV}(t)$ $\displaystyle=$ $\displaystyle
2\texttt{T}\int\textbf{\\{}[S(t_{02})\sin(\omega_{0}t_{03})-3C(t_{02})\cos(\omega_{0}t_{03})]C(t_{13})\sin(\omega_{0}t_{12})$
$\displaystyle+$
$\displaystyle[C(t_{02})\sin(\omega_{0}t_{03})-S(t_{02})\cos(\omega_{0}t_{03})]S(t_{13})\sin(\omega_{0}t_{12})$
$\displaystyle+$
$\displaystyle[S(t_{03})\sin(\omega_{0}t_{02})-3C(t_{03})\cos(\omega_{0}t_{02})]C(t_{12})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[C(t_{03})\sin(\omega_{0}t_{02})-S(t_{03})\cos(\omega_{0}t_{02})]S(t_{12})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[-S(t_{03})\sin(\omega_{0}t_{01})-C(t_{03})\cos(\omega_{0}t_{01})]C(t_{12})\sin(\omega_{0}t_{23})$
$\displaystyle+$
$\displaystyle[-C(t_{03})\sin(\omega_{0}t_{01})+S(t_{03})\cos(\omega_{0}t_{01})]S(t_{12})\sin(\omega_{0}t_{23})\textbf{\\}},$
$\displaystyle S_{-}^{IV}(t)$ $\displaystyle=$ $\displaystyle
2\texttt{T}\int\textbf{\\{}[S(t_{02})\sin(\omega_{0}t_{03})+C(t_{02})\cos(\omega_{0}t_{03})]C(t_{13})\sin(\omega_{0}t_{12})$
$\displaystyle+$
$\displaystyle[C(t_{02})\sin(\omega_{0}t_{03})-S(t_{02})\cos(\omega_{0}t_{03})]S(t_{13})\sin(\omega_{0}t_{12})$
$\displaystyle+$
$\displaystyle[S(t_{03})\sin(\omega_{0}t_{02})+C(t_{03})\cos(\omega_{0}t_{02})]C(t_{12})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[C(t_{03})\sin(\omega_{0}t_{02})-S(t_{03})\cos(\omega_{0}t_{02})]S(t_{12})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[-S(t_{03})\sin(\omega_{0}t_{01})-C(t_{03})\cos(\omega_{0}t_{01})]C(t_{12})\sin(\omega_{0}t_{23})$
$\displaystyle+$
$\displaystyle[-C(t_{03})\sin(\omega_{0}t_{01})+S(t_{03})\cos(\omega_{0}t_{01})]S(t_{12})\sin(\omega_{0}t_{23})\textbf{\\}},$
$\displaystyle\Gamma_{\pm}^{IV}(t)$ $\displaystyle=$
$\displaystyle-8\texttt{T}\int\textbf{\\{}[C(t_{13})\sin(\omega_{0}t_{03})\pm
S(t_{13})\cos(\omega_{0}t_{03})]C(t_{02})\sin(\omega_{0}t_{12})$
$\displaystyle+$ $\displaystyle[C(t_{12})\sin(\omega_{0}t_{02})\pm
S(t_{12})\cos(\omega_{0}t_{02})]C(t_{03})\sin(\omega_{0}t_{13})$
$\displaystyle\mp$
$\displaystyle[S(t_{03})C(t_{12})+C(t_{03})S(t_{12})]\sin(\omega_{0}t_{23})\cos(\omega_{0}t_{01})\textbf{\\}},$
$\displaystyle\Gamma_{0}(t)$ $\displaystyle=$ $\displaystyle
16\texttt{T}\int\textbf{\\{}[C(t_{02})C(t_{13})+S(t_{02})S(t_{13})]\sin(\omega_{0}t_{03})\sin(\omega_{0}t_{12})$
$\displaystyle+$
$\displaystyle[C(t_{03})C(t_{12})+S(t_{03})S(t_{12})]\sin(\omega_{0}t_{02})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[C(t_{03})C(t_{12})-S(t_{03})S(t_{12})]\sin(\omega_{0}t_{01})\sin(\omega_{0}t_{23})\textbf{\\}},$
$\displaystyle\alpha^{IV}(t)$ $\displaystyle=$
$\displaystyle-8\texttt{T}\int\textbf{\\{}S(t+t_{2})S(t_{13})\sin\omega_{0}(t+t_{3})\sin(\omega_{0}t_{12})$
$\displaystyle+$ $\displaystyle
S(t+t_{3})S(t_{12})\sin\omega_{0}(t+t_{2})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[C(t_{03})C(t_{12})-S(t_{03})S(t_{12})]\sin\omega_{0}(t+t_{1})\sin(\omega_{0}t_{23})\textbf{\\}},$
$\displaystyle\beta^{IV}(t)$ $\displaystyle=$ $\displaystyle
8\texttt{T}\int\textbf{\\{}S(t_{02})S(t_{13})\cos\omega_{0}(t+t_{3})\sin(\omega_{0}t_{12})$
$\displaystyle+$ $\displaystyle
S(t_{03})S(t_{12})\cos\omega_{0}(t+t_{2})\sin(\omega_{0}t_{13})$
$\displaystyle+$
$\displaystyle[C(t_{03})C(t_{12})-S(t_{03})S(t_{12})]\cos\omega_{0}(t+t_{1})\sin(\omega_{0}t_{23})\textbf{\\}}.$
## Appendix B Derivation of Bloch equation
According to the definition of Bloch vector
$b_{j}(t)=\texttt{Tr}[\rho(t)\sigma_{j}]$, we have
$\dot{b}_{j}(t)=\texttt{Tr}[\dot{\rho}(t)\sigma_{j}]$. By inserting master
equation (2) into it and after some deduction, one can obtain the required
Bloch equation. For example, for the component equation concerning
$\dot{b}_{x}$ we have,
$\dot{b}_{x}(t)=-i\texttt{Tr}\\{[H_{LS}(t),\rho(t)]\sigma_{x}\\}+\texttt{Tr}\\{D[\rho(t)]\sigma_{x}\\}+\texttt{Tr}\\{D^{\prime}[\rho(t)]\sigma_{x}\\}.$
(56)
By use of the circulation property of trace operation and the Pauli algorithm,
one easily get
$-i\texttt{Tr}\\{[H_{LS}(t),\rho(t)]\sigma_{x}\\}=(S_{-}-S_{+})b_{y}$,
$\texttt{Tr}\\{D[\rho(t)]\sigma_{x}\\}=-\frac{1}{2}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0})b_{x}$,
and $\texttt{Tr}\\{D^{\prime}[\rho(t)]\sigma_{x}\\}=\alpha b_{x}-\beta b_{y}$.
Summing up them, we thus obtain eq.(21).
The Bloch eqs.(21)-(23) can also be written as the compact vector form,
$\dot{\textbf{b}}=M\textbf{b}+\textbf{v}$, with the damping matrix $M$ and
drift matrix v given respectively by
$M=\left(\begin{array}[]{ccc}-\frac{1}{2}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}-2\alpha)&S_{-}-S_{+}-\beta&0\\\
-(S_{-}-S_{+}+\beta)&-\frac{1}{2}(\Gamma_{-}+\Gamma_{+}+\Gamma_{0}+2\alpha)&0\\\
0&0&-(\Gamma_{-}+\Gamma_{+})\\\ \end{array}\right),$ (57)
and $\textbf{v}^{T}=\left(\begin{array}[]{ccc}0,&0,&\Gamma_{+}-\Gamma_{-}\\\
\end{array}\right).$ Note that the damping matrix is in block diagonal form.
## References
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* (28) Jing J and Yu T 2010 Phys. Rev. Lett. 105, 240403
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* (36) Cohen-Tannoudji, Dupont-Roc J and Grynberg G 1989 _Photons and Atoms: Introduction to Quantum Electrodynamics_ (John Wiley and Sons, New York)
* (37) Alicki R and Lendi K 2007 _Quantum Dynamical Semigroups and Applications_ (Springer, Berlin Heidelberg)
* (38) The reason for $\Gamma_{0}$ always very much smaller than $\Gamma_{\pm}$ is that the former is the forth-order small quantity of the TCL generator, while the later are the second-order small quantities.
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|
arxiv-papers
| 2012-05-28T02:56:34 |
2024-09-04T02:49:31.277445
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hao-Sheng Zeng, Ning Tang, Yan-Ping Zheng and Tian-Tian Xu",
"submitter": "Hao-Sheng Zeng",
"url": "https://arxiv.org/abs/1205.6020"
}
|
1205.6077
|
# Nonadiabatic quantum chaos in atom optics
S.V. Prants prants@poi.dvo.ru, tel.007-4232-312602, fax 007-4232-312573 [
Laboratory of Nonlinear Dynamical Systems, Pacific Oceanological Institute of
the Russian Academy of Sciences, Baltiiskaya St., 43, 690041 Vladivostok,
Russia
###### Abstract
Coherent dynamics of atomic matter waves in a standing-wave laser field is
studied. In the dressed-state picture, wave packets of ballistic two-level
atoms propagate simultaneously in two optical potentials. The probability to
make a transition from one potential to another one is maximal when centroids
of wave packets cross the field nodes and is given by a simple formula with
the single exponent, the Landau–Zener parameter $\kappa$. If $\kappa\gg 1$,
the motion is essentially adiabatic. If $\kappa\ll 1$, it is (almost) resonant
and periodic. If $\kappa\simeq 1$, atom makes nonadiabatic transitions with a
splitting of its wave packet at each node and strong complexification of the
wave function as compared to the two other cases. This effect is referred as
nonadiabatic quantum chaos. Proliferation of wave packets at $\kappa\simeq 1$
is shown to be connected closely with chaotic center-of-mass motion in the
semiclassical theory of point-like atoms with positive values of the maximal
Lyapunov exponent. The quantum-classical correspondence established is
justified by the fact that the Landau–Zener parameter $\kappa$ specifies the
regime of the semiclassical dynamical chaos in the map simulating chaotic
center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found
in the behavior of the momentum and position probabilities.
###### keywords:
cold atom , matter wave , quantum chaos
url]http://dynalab.poi.dvo.ru
## 1 Introduction
The mechanical action of light upon neutral atoms has been comprehensively
studied since the pioneer works of Lebedev, Gerlah and Stern, Kapitza and
Dirac and Frisch. The light pressure force provides optical cooling and
trapping of atoms [1]. In the last two decades, cold atoms in standing-wave
optical fields have been used to study quantum chaos. The proposal [2] to
study atomic dynamics in a far-detuned modulated standing wave made atomic
optics a testing ground for quantum chaos. A number of impressive experiments
have been carried out in accordance with this proposal [3, 4, 5]. New
possibilities are opened if one works near the atom-field resonance where the
interaction between the internal and external atomic degrees of freedom is
intense [6, 7, 8].
Dynamical chaos in classical mechanics is a special kind of random-like motion
without any noise and/or random parameters. It is characterized by exponential
sensitivity of trajectories in the phase space to small variations in initial
conditions and/or control parameters. Such sensitivity does not exist in
isolated quantum systems because their evolution is unitary, and there is no
well-defined notion of a quantum trajectory. Thus, there is a fundamental
problem of emergence of classical dynamical chaos from more profound quantum
mechanics which is known as quantum chaos problem and the related problem of
quantum-classical correspondence. In a more general context it is a problem of
wave chaos. It is clear now that quantum chaos, microwave, optical, and
acoustic chaos [9, 10, 11, 12] have much in common. The common practice is to
construct an analogue for a given wave object in a semiclassical (ray)
approximation and study its chaotic properties (if any) by well-known methods
of dynamical system theory. Then, it is necessary to solve the corresponding
linear wave equation in order to find manifestations of classical chaos in the
wave-field evolution in the same range of the control parameters. If one
succeeds in that the quantum-classical or the wave-ray correspondence are
announced to be established.
In atom optics [13] one quantizes both the atomic internal and translational
degrees of freedom. The atom is treated as a wave packet which undergoes
deformations in the process of exchange of energy and momentum quanta with a
light wave. Quantization of the translation motion provides an entanglement of
the internal and external degrees of freedom. Any changes in the form of the
wave packet will affect the internal state of the atom and vice versa [14,
15]. The optical field provides a tool to manipulate the atomic matter waves.
In atom optics the Schrödinger equation for the probability amplitudes
constitutes a linear infinite-dimensional dynamical system which is governed
by an external force if the field is treated as a classical wave.
In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle with the Hamilton–Schrödinger equations of
motion constituting a nonlinear dynamical system. A number of nonlinear
Hamiltonian and dissipative dynamical effects have been found with such a
system including chaotic Rabi oscillations, chaotic atomic transport,
dynamical fractals, synchronization, chaotic walking, and Lévy flights [16,
17, 18, 19, 20, 21]. Similar and new effects have been found numerically and
described analytically with two-level atoms in a losseless cavity with a
single quantized mode in the framework of the Jaynes-Cummings model [22, 23,
24, 25] and the Tavis-Cummings model [26]. It has been shown that the coupled
atom-field dynamics in a cavity can be unstable under appropriate conditions
in the absence of any kind of interaction with environment. This kind of
quantum instability manifests itself in fractal chaotic scattering of atoms
[22, 23, 24, 25], in strong variations of reduced quantum purity and entropy
[24, 25, 26], correlating with the respective maximal Lyapunov exponent, and
in exponential sensitivity of fidelity of quantum states to small variations
of the detuning [24, 25].
The main aim of this paper is to establish a kind of the quantum-classical
correspondence in transport properties of point-like atoms and atomic matter
waves moving in a standing-waves field. Is the coherent evolution of the
atomic matter waves really complicated in that range of the control parameters
where the corresponding center-of-mass motion has been shown to be chaotic?
## 2 Wave-packet motion in a standing light wave
The Hamiltonian of a two-level atom, moving along a one-dimensional classical
standing-wave laser field, can be written in the frame rotating with the laser
frequency $\omega_{f}$ as follows:
$\hat{H}=\frac{\hat{P}^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\hbar\Omega\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}},$
(1)
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $\hat{X}$ and $\hat{P}$ are the atomic position and
momentum operators, $\omega_{a}$ and $\Omega$ are the atomic transition and
Rabi frequencies, respectively. We will work in the momentum representation
and expand the state vector as follows:
${|\Psi(t)\closeket}=\int[a(P,t){|2\closeket}+b(P,t){|1\closeket}]{|P\closeket}dP,$
(2)
where $a(P,t)$ and $b(P,t)$ are the probability amplitudes to find atom at
time $t$ with the momentum $P$ in the excited, ${|2\closeket}$, and ground,
${|1\closeket}$, states, respectively. After some algebra one gets the
normalized Schrödinger equation for the probability amplitudes [14]
$\displaystyle
i\dot{a}(p)=\frac{1}{2}(\omega_{r}p^{2}-\Delta)a(p)-\frac{1}{2}[b(p+1)+b(p-1)],$
(3) $\displaystyle
i\dot{b}(p)=\frac{1}{2}(\omega_{r}p^{2}+\Delta)b(p)-\frac{1}{2}[a(p+1)+a(p-1)],$
where the dot denotes differentiation with respect to dimensionless time
$\tau\equiv\Omega t$, $p\equiv P/\hbar k_{f}$, and $x\equiv k_{f}X$. The
normalized recoil frequency $\omega_{r}\equiv\hbar k_{f}^{2}/m_{a}\Omega$ and
the atom-field detuning $\Delta\equiv(\omega_{f}-\omega_{a})/\Omega$ are the
control parameters.
The probability to find an atom with the momentum $p$ at the moment of time
$\tau$ is ${\cal P}(p,\tau)=|a(p,\tau)|^{2}+|b(p,\tau)|^{2}$. The internal
atomic state is described by the following real-valued combinations of the
probability amplitudes: $u_{q}(\tau)\equiv 2\operatorname{Re}\int
dp\left[a(p,\tau)b^{*}(p,\tau)\right]$,
$v_{q}(\tau)\equiv-2\operatorname{Im}\int dp[a(p,\tau)b^{*}(p,\tau)]$,
$z_{q}(\tau)\equiv\int dp[|a(p,\tau)|^{2}-|b(p,\tau)|^{2}]$, which are
expected values of the synchronized (with the laser field) and a quadrature
components of the atomic electric dipole moment ($u_{q}$ and $v_{q}$,
respectively) and the atomic population inversion, $z_{q}$. Varying the value
of the Rabi frequency $\Omega$, we can change the value of the dimensionless
recoil frequency $\omega_{r}$ with the same atom. Working, say, with a cesium
atom ($m_{a}=133$ a.u., $\lambda_{f}=852.1$ nm, and $\nu_{\rm rec}\simeq 2$
KHz), we get $\omega_{r}=10^{-5}$ at $\Omega=100$ MHz.
We will interpret the wave-packet motion in the dressed-state basis [13, 27]
${|+\closeket}_{\Delta}={|2\closeket}\sin{\Theta}+{|1\closeket}\cos{\Theta},\
{|-\closeket}_{\Delta}={|2\closeket}\cos{\Theta}-{|1\closeket}\sin{\Theta},$
(4)
where $\Theta$ is the mixing angle
$\tan{\Theta}\equiv\frac{\Delta}{2\cos{x}}-\sqrt{\left(\frac{\Delta}{2\cos{x}}\right)^{2}+1}.$
(5)
These states are eigenstates of an atom at rest in a laser field with the
eigenvalues of the quasienergy
$E_{\Delta}^{(\pm)}=\pm\sqrt{\frac{\Delta^{2}}{4}+\cos^{2}{x}}.$ (6)
The probability amplitudes to find the atom at point $x$ in those potentials
are, respectively
$C_{+}(x)=a(x)\sin{\Theta}+b(x)\cos{\Theta},C_{-}(x)=a(x)\cos{\Theta}-b(x)\sin{\Theta},$
(7)
where the amplitudes in the bare-state basis $a(x)$ and $b(x)$ may be computed
in the position representation with the help of the Fourier transform
$a(x)={\rm
const}\int_{-\infty}^{\infty}dp^{\prime}e^{ip^{\prime}x}a(p^{\prime}),\
b(x)={\rm
const}\int_{-\infty}^{\infty}dp^{\prime}e^{ip^{\prime}x}b(p^{\prime}).$ (8)
Let us assume that we are able to prepare an atom exactly in one of its
dressed states, ${|+\closeket}_{\Delta}$ or ${|-\closeket}_{\Delta}$. Then the
atom will move in one of the potentials, $E_{\Delta}^{(+)}$ or
$E_{\Delta}^{(-)}$, along a single trajectory. In quantum mechanics, there is
a nonzero probability to make a transition to another potential. To estimate
this probability we write the Hamiltonian of the internal degree of freedom of
a two-level atom in the basis ${|\pm\closeket}_{\Delta}$
$\hat{H}_{\rm int}=\hat{\sigma}_{z}\cos{x}+\frac{\Delta}{2}\hat{\sigma}_{x}.$
(9)
Let us linearize the cosine in the vicinity of a node of the standing wave and
estimate a small distance the atom makes when crossing the node as follows:
$\delta x=\omega_{r}|p_{\rm node}|\tau$ [13]. The quantity $\omega_{r}|p_{\rm
node}|$ is a normalized Doppler shift for an atom moving with the momentum
$|p_{\rm node}|$, i.e., $\omega_{D}\equiv\omega_{r}|p_{\rm node}|\equiv
k_{f}|v_{\rm node}|/\Omega$. The Schrödinger equation for the probability
amplitudes $C_{\pm}(x)$ in the position representation can be written in the
form of the second-order equation
$\ddot{C}_{+}(x)+\left[i\omega_{D}+\frac{\Delta^{2}}{4}+(\omega_{D}\tau)^{2}\right]C_{+}(x)=0.$
(10)
The asymptotic solution of Eq. (10),
$P_{LZ}=\exp(-\kappa),$ (11)
gives the probability to make a nonadiabatic or Landau-Zener transition from
one of the nonresonant potentials to another one specified by the Landau–Zener
parameter
$\kappa\equiv\pi\frac{\Delta^{2}}{\omega_{D}}.$ (12)
There are three regimes of atomic motion.
1. 1.
$\kappa\gg 1$. The probability to make the transition is exponentially small
even when an atom crosses a node. The evolution of the atomic wave packet is
adiabatic in this case.
2. 2.
$\kappa\ll 1$. The distance between the potentials at the nodes is small and
the atom changes the potential each time when crossing any node with the
probability close to unity. In the limit case $\Delta=0$, the atom moves in
the resonant potentials.
3. 3.
$\kappa\simeq 1$. The probability to change the potential or to remain in the
same one, upon crossing a node, are of the same order. In this regime one may
expect a proliferation of components of the atomic wave packet at the nodes
and complexification of the wave function.
## 3 Simulation of ballistic wave-packet propagation
We simulate the evolution of a Gaussian wave packet with the variance in the
momentum space, $\sigma_{p}^{2}=50$, $p_{0}=10^{3}$, $x_{0}=0$ and
$\omega_{r}=10^{-5}$. The initial average kinetic energy,
$\omega_{r}p^{2}/2=5$, is greater than the depth of the potential wells, so
the atom will move ballistically along the positive direction of the standing-
wave axis.
To study all the regimes of the wave-packet motion, we simulate Eqs. (3) at
different values of the Landau–Zener parameter $\kappa$ (12). The normalized
Doppler shift $\omega_{D}$ nearby a node of the standing wave is estimated to
be $\omega_{D}\simeq\omega_{r}p_{0}=0.01$. If we choose, say, $\Delta=0.3$ we
get the first case in our nomenclature, $\kappa\gg 1$, with exponentially
small probability of nonadiabatic transitions. The wave packet, initially
prepared in the ground state which is a superposition of the dressed states
with approximately equal weights, splits from the beginning (Fig. 1a) into two
components, ${|+\closeket}_{\Delta}$ and ${|-\closeket}_{\Delta}$, each of
which moves in its own nonresonant potential, $E_{\Delta}^{(+)}$ or
$E_{\Delta}^{(-)}$. We really do not observe in Fig. 1a any splitting at the
nodes, and the motion of the wave packet at $\Delta=0.3$ is adiabatic and
practically periodic. If $\kappa\ll 1$ (the motion near the resonance), one
expects to observe the periodic motion in the two resonant potentials
simultaneously without any splitting [14].
At $\Delta=0.1$, we get $\kappa\simeq 1$ and expect nonadiabatic transitions
at the nodes of the standing wave in accordance with formula (11). The initial
ground state ${|1\closeket}$ is now a superposition of the dressed states with
practically the same weights. The initial bifurcation is accompanied by
splittings (see Fig. 1b) that can be proved to occur at the nodes of the
standing wave. Let us start to analyze the wave-packet motion with the
${|+\closeket}_{\Delta}$-component (the upper curve in the figure). The first
splitting occurs at $\tau_{1}^{(+)}\simeq 150$. It is easy to prove that it is
the moment of time when the centroid of the ${|+\closeket}_{\Delta}$-component
crosses the first node at $x=\pi/2$:
$\tau^{(+)}_{1}=\pi/2\omega_{r}\overline{p}_{0,1}^{(+)}\simeq 150$, where
$\overline{p}_{0,1}^{(+)}$ is the average momentum of the centroid of the
${|+\closeket}_{\Delta}$ wave packet between $x=0$ and $x=\pi/2$. Thus, the
wave packet, crossing the node, splits into two parts. The first one prolongs
its motion in the potential $E_{\Delta}^{(+)}$ after passing the point
$x=\pi/2$. It is the lower curve in Fig. 1b starting at $\tau_{1}^{(+)}\simeq
150$. The corresponding packet slows down because this component loses its
kinetic energy going up to the top of the potential $E_{\Delta}^{(+)}$. As to
the second trajectory (the upper curve starting at $\tau_{1}^{(+)}$), it
appears due to the nonadiabatic transition to the potential
$E_{\Delta}^{(-)}$. That is why it accelerates from the beginning and reaches
its maximal velocity at $x=\pi$. The ${|-\closeket}_{\Delta}$-component (the
lower curve starting at $\tau=0$) splits at the first node at
$\tau_{1}^{(-)}=\pi/2\omega_{r}\overline{p}_{0,1}^{(-)}\simeq 156$. In course
of time both the components split at every node of the standing wave at the
moments $\tau_{n}^{(\pm)}$ that can be estimated with the simple formula
$\omega_{r}\overline{p}_{n-1,n}^{(\pm)}\tau^{(\pm)}_{n}=(2n-1)\frac{\pi}{2},\
n=2,3,\ldots,$ (13)
where $\overline{p}_{n-1,n}^{(\pm)}$ is an average momentum of the centroid of
the corresponding component between the $(n-1)$-th and $n$-th nodes. Such a
proliferation at the nodes means a complexification of the atomic wave
function both in the momentum and position spaces as compared to the adiabatic
and resonant cases.
Now we go to the position space and compute the probability
$|C(x,\tau)|^{2}=|C_{-}(x,\tau)|^{2}+|C_{+}(x,\tau)|^{2}$ to be at point $x$
at time $\tau$. In Fig. 2 we show the result of simulation in the case of
adiabatic and nonadiabatic motion at $\Delta=0.3$ and $\Delta=0.1$
corresponding, respectively, to Fig. 1a and b in the momentum space. It is a
plot of the position probability in the frame moving with the initial atomic
velocity $\omega_{r}p_{0}=0.01$ where the slope straight lines mark positions
of the nodes of the standing wave in the moving frame. At $\Delta=0.3$, the
evolution is simple without any transitions at the nodes (Fig. 2a). The
splitting of the total probability $|C(x,\tau)|^{2}$ is caused by the initial
bifurcation of the wave packet due to its bipotential motion.
The situation is cardinally different when we work in the regime with
nonadiabatic transitions at the field nodes ($\Delta=0.1$). Splitting at the
nodes in the momentum space (see Fig.1b) manifest itself in the position space
in Fig. 2b. In this case one observes visible changes in the proba6bility
$|C(x)|^{2}$ exactly at the node lines. It is a clear evidence of the
nonadiabatic transitions that occur in the specific range of the control
parameters, $\kappa\simeq 1$. This results in a proliferation of components of
the wave packet at the nodes and, therefore, a complexification of the wave
function both in the momentum and position spaces.
## 4 Quantum-classical correspondence and nonadiabatic quantum chaos
In this section we compare the quantum results, obtained in the preceding
sections, with those obtained for the same problem but in the semiclassical
approximation when the translational motion has been treated as a classical
one [6, 7, 8, 17, 19]. Coherent semiclassical evolution of a point-like two-
level atom is governed by the Hamilton-Schrödinger equations with the same
normalization as in the quantum case
$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\sin x,\quad\dot{u}=\Delta
v,\\\ \dot{v}=-\Delta u+2z\cos x,\quad\dot{z}=-2v\cos x,\end{gathered}$ (14)
where
$u\equiv 2\operatorname{Re}(a_{0}b_{0}^{*}),\
v\equiv-2\operatorname{Im}(a_{0}b_{0}^{*}),\ z\equiv|a_{0}|^{2}-|b_{0}|^{2}$
(15)
are the atomic-dipole components ($u$ and $v$) and population-inversion ($z$),
and $a_{0}$ and $b_{0}$ are the complex-valued probability amplitudes to find
the atom in the excited and ground states, respectively. The system (14) has
two integrals of motion, the total energy
$H\equiv\frac{\omega_{r}}{2}p^{2}-u\cos x-\frac{\Delta}{2}z,$ (16)
and the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$.
Equations (14) constitute a nonlinear Hamiltonian autonomous system with two
and half degrees of freedom and two integrals of motion. It has been shown in
Ref. [19] to have positive values of the maximal Lyapunov exponent $\lambda$
in a wide range of values of the control parameters and initial states. This
fact implies dynamical chaos in the sense of exponential sensitivity to small
changes in initial conditions and/or control parameters. The result of
computation of the maximal Lyapunov exponent in dependence on the detuning
$\Delta$ and the initial Doppler shift $\omega_{D}=\omega_{r}p_{0}$ is shown
in Fig. 3 at $\omega_{r}=10^{-5}$. In white regions of the plot the values of
$\lambda$ are almost zero, and the atomic motion is regular in the
corresponding ranges of $\Delta$ and $\omega_{D}$. In shadowed regions
positive values of $\lambda$ imply unstable motion. At exact resonance, we get
$\lambda=0$ because at $\Delta=0$ the semiclassical equations of motion (14)
become integrable due to an additional integral of motion, $u={\rm const}$. We
stress that the local instability produces chaotic center-of-mass motion in a
rigid standing wave without any modulation of its parameters in difference
from the situation with atoms in a periodically kicked optical lattice [3, 4,
5]. In dependence on the initial conditions and the parameter values, an atom
may oscillate in a well of the lattice or it may have enough kinetic energy to
overcome the potential barrier. In some cases the center-of-mass motion
resembles a random walking. It means that an atom in a deterministic standing-
wave field alternates between flying through the lattice, and being trapped in
its wells. Moreover, it may change the direction of motion in a random-like
way (see Ref. [17, 19] for coherent Hamiltonian dynamics and Refs. [20, 21]
for a dissipative one with spontaneous emission included).
It follows from (14) that the translational motion is described by the
equation for a nonlinear physical pendulum with the frequency modulation
$\ddot{x}+\omega_{r}u(\tau)\sin x=0,$ (17)
where $u$ is a function of all the other dynamical variables. It has been
shown in Ref. [19] that the regime of the center-of-mass motion is specified
by the character of oscillations of the component $u$ of the Bloch vector. In
a chaotic regime sudden “jumps” of the variable $u$ occur when an atom crosses
the field nodes. Figure 4a demonstrates more or less periodic oscillations of
$u$ at the detuning value $\Delta=0.3$ at which the corresponding maximal
Lyapunov exponent is zero (Fig. 3). In the chaotic regime at $\Delta=0.1$ $u$
demonstrates shallow oscillations interrupted by jumps of different amplitudes
upon crossing the nodes (Fig. 4b).
Approximating the variable $u$ between the nodes by constant values, the
following stochastic map has been constructed in Ref. [19]
$u_{m}=\sin(\Theta\sin\varphi_{m}+\arcsin u_{m-1}),$ (18)
where $\Theta\equiv\sqrt{\pi\Delta^{2}/\omega_{r}p_{\rm{node}}}$ is an angular
amplitude of the jump, $u_{m}$ is a value of $u$ just after the $m$-th node
crossing, $\varphi_{m}$ are random phases to be chosen in the range
$[0,2\pi]$, and $p_{\rm{node}}\equiv\sqrt{2H/\omega_{r}}$ is the value of the
atomic momentum at the instant when the atom crosses a node (which is the same
with a given value of the energy $H$ for all the nodes). With given values of
$\Delta$, $\omega_{r}$ and $p_{\rm{node}}$, the map (18) has been shown
numerically to give a satisfactory probabilistic distribution of magnitudes of
changes in the variable $u$ just after crossing the nodes. The stochastic map
(18) is valid under the assumptions of small detunings ($|\Delta|\ll 1$) and
comparatively slow atoms ($|\omega_{r}p|\ll 1$). Furthermore, it is valid only
for those ranges of the control parameters and initial conditions where the
motion of the basic system (14) is unstable. For example, in those ranges
where all the Lyapunov exponents are zero, $u$ becomes a quasi-periodic
function and cannot be approximated by the map (see Fig. 4a).
The key result in the context of the quantum-classical correspondence is that
the squared angular amplitude of the map (18) is exactly the Landau–Zener
parameter (12), i.e., $\Theta^{2}=\kappa$. Rewriting the map (18) for $\arcsin
u_{m}$, one gets
$\arcsin u_{m}=\sqrt{\kappa}\sin\varphi_{m}+\arcsin u_{m-1},$ (19)
where the jump magnitude does not depend on a current value of the variable.
The map (19) visually looks as a random motion of the point along a circle of
unit radius (see Fig. 4 in Ref. [19]). If $\kappa\simeq 1$, then the internal
atomic variable $\arcsin u_{m}$ just after crossing the $m$-th node may take
with the same probability practically any value from the range
$[-\pi/2,\pi/2]$. It means semiclassically that the momentum of a ballistic
atom changes chaotically upon crossing the field nodes. In accordance with the
quantum formula (12), the corresponding atomic wave packet makes nonadiabatic
transitions when crossing the nodes and splits at each node (see Figs. 1b and
2b). As the result, the wave packet of a single atom becomes so complex that
it may be called a chaotic one in the sense that it is much more complicated
than the wave packets propagating adiabatically. Thus, nonadiabatic wave chaos
and semiclassical dynamical chaos occur in the same range of the control
parameters and are specified by the same Landau–Zener parameter $\kappa\simeq
1$. In two limit cases with $\kappa\ll 1$ and $\kappa\gg 1$ both the
semiclassical and quantized translational ballistic motion are regular.
In quantum mechanics there is no well-defined notion of a trajectory in the
phase space and, hence, the Lyapunov exponents can not be computed. In quantum
mechanics there is no exponential sensitivity to small variations in initial
conditions because the time evolution of an isolated quantum system is
unitary, and the overlap of any two different quantum state vectors is a
constant in course of time. Moreover, quantum phase space is discrete due to
the Heisenberg uncertainty principle unlike continuous classical phase space.
Namely the continuity of the classical phase space provides a possibility of
chaotic mixing which exploits more and more fine structures in the classical
phase space in course of time whereas the quantum evolution stops to do that
over a rather short Ehrenfest time.
The semiclassical (14) and quantum (3) equations of motion look very
different. The semiclassical ones constitute a five-dimensional nonlinear
dynamical system of ODEs with two integrals of motion that has been shown to
be chaotic in a certain range of control parameters with exponential
sensitivity to small variations to initial conditions [19]. The quantum ones
constitute an infinite-dimensional set of linear equations. It is not evident
a priori that their solutions might demonstrate a kind of correspondence in
the same range of the control parameters. Nevertheless, a sort of quantum-
classical correspondence both in regular and chaotic regimes of the center-of-
mass motion has been found. It should be stressed that this correspondence
manifests itself in behavior of the quantum Bloch variable $u$ in the
semiclassical equations of motion (14).
However, the quantum-classical correspondence is not and could not be absolute
because the Planck constant is equal to 1 with our normalization. It cannot
tend to be zero in order to achieve a classical limit as it could be done with
an effective Planck constant (see, for example, Refs. [2, 5]) depending on the
system’s parameters. We work in this sense in a deep quantum regime. The
quantum-classical dualism with cold atoms resembles the wave-ray one in a
classical wave motion. In the context of this paper we might compare ray-like
trajectories of atoms with their wave-like motion.
To illustrate correspondence and difference that inavoidably appears when
comparing quantum evolution with the classical one (that is only an
approximation to the quantum one), we compute with Eqs. (14) the evolution of
a Gaussian distribution over classical momentum $p$ and position $x$ with the
same parameter’s values as in simulation of the wave-packet propagation shown
in Figs. 1 and 2. In accordance with the Lyapunov map in Fig. 3, one expects a
regular center-of-mass motion at the detuning $\Delta=0.3$ and a weakly
chaotic one at $\Delta=0.1$. In Figs. 5a and b evolution of classical momenta
is shown for the regular ($\Delta=0.3$) and chaotic ($\Delta=0.1$) regimes of
the center-of-mass motion. Visible spreading in $p$ with chaotically moving
atoms, as compared to regularly moving ones, is one of the signs of classical
dynamical chaos. Figure 1 demonstrates similar spreading of the momentum
probability distribution of a Gaussian wave packet with nonadiabatic
transitions at $\Delta=0.1$ (Fig. 1b) as compared to the adiabatically moving
wave packet at $\Delta=0.3$ (Fig. 1a). The difference between the classical
and quantum evolution is also evident: the semiclassical equations of motion
(14) are not able to simulate the splitting of wave packets due to purely
quantum effect of motion in two optical potentials simultaneously.
In Figs. 6a and b we plot, respectively, regular and chaotic trajectories in
the frame of reference moving with the initial atomic velocity
$\omega_{r}p_{0}=0.01$. The bundle of chaotically moving atoms in Fig. 6b
diverges in a short time significantly as compared to the regular one in Fig.
6a. This property can be used to detect chaotic scattering in a real
experiment with atoms crossing a standing laser wave [15]. As to quantum
motion in the position space, it is evident that the wave packet with
nonadiabatic transitions (Fig. 2b) becomes much broader in course of time due
to splitting at the standing-wave nodes than the adiabatic wave packet in Fig.
2a resembling a broadening of the bundle of chaotic point-like atoms (Fig. 6b)
as compared to the regular one (Fig. 6a). However, we do not observe any
splitting of the classical bundles because a classical trajectory simulates
only the motion of the centroid of a quantum wave packet and cannot simulate
of course its splitting due to purely quantum effect of motion in two optical
potentials simultaneously. There is only one optical potential in the
semiclassical approximation.
## 5 Conclusion
We have studied coherent dynamics of ballistic atomic wave packets in a one-
dimensional standing-wave laser field. The problem has been considered in the
momentum representation and in the dressed-state basis where the motion of a
two-level atom was interpreted as a motion in two optical potentials. The
character of that motion has been shown to depend strongly on the value of the
Landau–Zener parameter $\kappa$ (12). If $\kappa\gg 1$, then the probability
of transitions from one of the potential to another one, which is described by
the Landau–Zener formula (11), is exponentially small. Under such a condition,
atoms move in the adiabatic regime. If $\kappa\ll 1$, the formula (11) gives
almost unity probability to change the potential when crossing the nodes. In
the intermediate case, $\kappa\simeq 1$, the probabilities for an atom to
change or not to change the nonresonant potential, when crossing a node, are
of the same order. The corresponding nonadiabatic transitions manifest
themselves as a splitting of the atomic wave packets in the momentum space
when their centroids cross the nodes. This nonadiabatic quantum chaos occurs
exactly in the same range of the detuning and the Doppler shift where the
semiclassical dynamics has been shown to be chaotic. It is remarkable that the
same Landau–Zener parameter $\kappa$ specifies both semiclassical and quantum
chaos with ballistic atoms in a deterministic optical lattice.
We hope that the results obtained can be used to study manifestations of
quantum chaos with Bose-Einstein condensates in optical lattices [28] with
coupled degrees of freedom. From the theoretical point of view, the dynamics
of condensates of ultracold atoms is described correctly by the Gross-
Pitaevskii equation which is a kind of a nonlinear Schrödinger equation with
possible chaotic solutions. Experimentally, one of the possibilities is to
prepare two Bose-Einstein condensates in different internal states [29].
Another possibility can be realized with a Bose-Einstein condensate in an
optical lattice subject to a static tilted force [30, 31, 32]. Viewing
transitions between the Bloch bands of a condensate in such a tilted optical
lattice as a two-state problem, we get a mesoscopic quantum system with
coupled different degrees of freedom (I am thankful to an anonymous referee
for that comment).
## Acknowledgments
This work was supported by the Russian Foundation for Basic Research (project
no. 09-02-00358), by the Integration grant from the Far-Eastern and Siberian
branches of the Russian Academy of Sciences, and by the Program “Fundamental
Problems of Nonlinear Dynamics” of the Russian Academy of Sciences. I thank
L.E. Konkov and V.O. Vitkovsky for their help in preparing some figures.
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Figure 1: (Color online) Momentum probability distribution ${\cal P}(p,\tau)$
of a Gaussian wave packet vs time with $p_{0}=1000,\sigma_{p}^{2}=50$, and
$\omega_{r}=10^{-5}$ at (a) $\Delta=0.3$, adiabatic motion, and (b)
$\Delta=0.1$, motion with nonadiabatic transitions. The color codes the values
of ${\cal P}(p,\tau)$.
Figure 2: (Color online) The position probability $|C(x)|^{2}$ in the moving
frame of reference with the slope straight lines marking positions of the
nodes. (a) Adiabatic motion in the position space at $\Delta=0.3$. (b) Wave-
packet propagation in the position space with nonadiabatic transitions at the
field nodes. Figure 3: Maximal Lyapunov exponent $\lambda$ vs atom-field
detuning $\Delta$ and the initial Doppler shift $\omega_{D}=\omega_{r}p_{0}$.
Color codes the values of $\lambda$.
Figure 4: Semiclassical evolution of the atomic-dipole component $u$ in (a)
regular ($\Delta=0.3$) and (b) chaotic ($\Delta=0.1$) regimes of the ballistic
motion of a point-like atom.
Figure 5: Atomic trajectories in the momentum space computed with the
classical Gaussian distribution at the same parameter’s values as in
simulation with Gaussian wave packets. (a) Regular center-of-mass motion
($\Delta=0.3$) corresponding to adiabatic quantum motion in Fig. 1a and (b)
weakly chaotic motion ($\Delta=0.1$) corresponding to nonadiabatic quantum
motion in Fig. 1b.
Figure 6: Classical atomic trajectories in the moving frame of reference. (a)
Regular bundle ($\Delta=0.3$) corresponding to adiabatic quantum motion in
Fig. 2a and (b) weakly chaotic bundle ($\Delta=0.1$) corresponding to
nonadiabatic quantum motion in Fig. 2b.
|
arxiv-papers
| 2012-05-28T10:59:16 |
2024-09-04T02:49:31.287101
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1205.6077"
}
|
1205.6087
|
11institutetext: S.V. Prants 22institutetext: Laboratory of Nonlinear
Dynamical Systems, Pacific Oceanological Institute of the Russian Academy of
Sciences, 43 Baltiiskaya st., 690041 Vladivostok, Russia, 22email:
prants@poi.dvo.ru
# Hamiltonian chaos with a cold atom in an optical lattice
S.V. Prants
###### Abstract
We consider a basic model of the lossless interaction between a moving two-
level atom and a standing-wave single-mode laser field. Classical treatment of
the translational atomic motion provides the semiclassical Hamilton-
Schrödinger equations of motion which are a five-dimensional nonlinear
dynamical system with two integrals of motion. The atomic dynamics can be
regular or chaotic (in the sense of exponential sensitivity to small
variations in initial conditions and/or the system’s control parameters) in
dependence on values of the control parameters, the atom-field detuning and
recoil frequency. We develop a semiclassical theory of the chaotic atomic
transport in terms of a random walk of the atomic electric dipole moment $u$
which is one of the components of a Bloch vector. Based on a jump-like
behavior of this variable for atoms crossing nodes of the standing laser wave,
we construct a stochastic map that specifies the center-of-mass motion. We
find the relations between the detuning, recoil frequency and the atomic
energy, under which atoms may move in a rigid optical lattice in a chaotic
way. We obtain the analytical conditions under which deterministic atomic
transport has fractal properties and explain a hierarchical structure of the
dynamical fractals. Quantum treatment of the atomic motion in a standing wave
is studied in the dressed state picture where the atom moves in two optical
potentials simultaneously. If the values of the detuning and a characteristic
atomic frequency are of the same order, than there is a probability of
nonadiabatic transitions of the atom upon crossing nodes of the standing wave.
At the same condition exactly, we observe sudden changes (jumps) in the atomic
dipole moment $u$ when the atom crosses the nodes. Those jumps are accompanied
by splitting of atomic wave packets at the nodes. Such a proliferation of wave
packets at the nodes of a standing wave is a manifestation of classical atomic
chaotic transport. In particular, the effect of simultaneous trapping of an
atom in a well of one of the optical potential and its flight in the other
potential is a quantum analogue of a chaotic classical walking of an atom. At
large values of the detuning, the quantum evolution is shown to be adiabatic
in accordance with a regular character of the classical atomic motion.
## 1 Short historical background
The fundamental model for the interaction of a radiation with matter,
comprising a collection of two-level quantum systems coupled with a single-
mode electromagnetic field, provides the basis for laser physics and describes
a rich variety of nonlinear dynamical effects. The discovery that a single-
mode laser, a symbol of coherence and stability, may exhibit deterministic
instabilities and chaos is especially important since lasers provide nearly
ideal systems to test general ideas in statistical physics. From the stand
point of nonlinear dynamics, laser is an open dissipative system which
transforms an external excitation into a coherent output in the presence of
loss. In 1975 Haken Haken has shown that a single-mode, homogeneously
broadened laser, operating on resonance with the gain center can be described
in the rotating-wave approximation by three real semiclassical Maxwell-Bloch
equations which are isomorphic to the famous Lorenz equations. Some
manifestations of a Lorenz-type strange attractor and dissipative chaos have
been observed with different types of lasers.
In the same time George Zaslavsky with co-workers Zas have studied
interaction of an ensemble of two-level atoms with their own radiation field
in a perfect single-mode cavity without any losses and external excitations,
which is known as the Dicke model Dicke . They were able to demonstrate
analytically and numerically dynamical instabilities and chaos of Hamiltonian
type in a semiclassical version of the Dicke model without rotating-wave
approximation. It was the first paper that opened the door to study
Hamiltonian atomic chaos in the rapidly growing fields of cavity quantum
electrodynamics, quantum and atomic optics. Semiclassical equations of motion
for this system may be reduced to Maxwell-Bloch equations for three real
independent variables which, in difference from the laser theory, do not
include losses and pump. Those equations are, in general, nonintegrable, but
they become integrable immediately after adopting the rotating-wave
approximation Jaynes that implies the existence of an additional integral of
motion, conservation of the so-called number of excitations. Numerical
experiments have shown that prominent chaos arises when the density of atoms
is very large (approximately $10^{20}$ cm3 in the optical range Zas ). The
following progress in this field has been motivated, mainly, by a desire to
find manifestations of Hamiltonian atomic chaos in the models more suitable
for experimental implementations. Twenty years after that pioneer paper,
manifestations of Hamiltonian chaos have been found in experiments with kicked
cold atoms in a modulated laser field. Nowdays, a few groups in the USA,
Australia, New Zealand, Germany, France, England, Italy and in other countries
can perform routine experiments on Hamiltonian chaos with cold atoms in
optical lattices and traps (for a review see Hens03 ).
In this paper we review some results on theory of Hamiltonian chaos with a
single two-level atom in a standing-wave laser field that have been obtained
in our group in Vladivostok. In spite of we published with George only one
paper on this subject PRA02 , our work in this field has been mainly inspired
by his paper Zas written in 1975 in Krasnoyarsk, Siberia.
## 2 Introduction
An atom placed in a laser standing wave is acted upon by two radiation forces,
deterministic dipole and stochastic dissipative ones Kaz . The mechanical
action of light upon neutral atoms is at the heart of laser cooling, trapping,
and Bose-Einstein condensation. Numerous applications of the mechanical action
of light include isotope separation, atomic lithography and epitaxy, atomic-
beam deflection and splitting, manipulating translational and internal atomic
states, measurement of atomic positions, and many others. Atoms and ions in an
optical lattice, formed by a laser standing wave, are perspective objects for
implementation of quantum information processing and quantum computing.
Advances in cooling and trapping of atoms, tailoring optical potentials of a
desired form and dimension (including one-dimensional optical lattices),
controlling the level of dissipation and noise are now enabling the direct
experiments with single atoms to study fundamental principles of quantum
physics, quantum chaos, decoherence, and quantum-classical correspondence (for
recent reviews on cold atoms in optical lattices see Ref. GR01 ; MO06 ).
Experimental study of quantum chaos has been carried out with ultracold atoms
in $\delta$-kicked optical lattices MR94 ; RB95 ; Hens03 . To suppress
spontaneous emission and provide a coherent quantum dynamics atoms in those
experiments were detuned far from the optical resonance. Adiabatic elimination
of the excited state amplitude leads to an effective Hamiltonian for the
center-of-mass motion GSZ92 , whose 3/2 degree-of-freedom classical analogue
has a mixed phase space with regular islands embedded in a chaotic sea. De
Brogile waves of $\delta$-kicked ultracold atoms have been shown to
demonstrate under appropriate conditions the effect of dynamical localization
in momentum distributions which means the quantum suppression of chaotic
diffusion MR94 ; RB95 ; Hens03 . Decoherence due to spontaneous emission or
noise tend to suppress this quantum effect and restore classical-like
dynamics. Another important quantum chaotic phenomenon with cold atoms in far-
detuned optical lattices is a chaos-assisted tunneling. In experiments Steck01
; HH01 ultracold atoms have been demonstrated to oscillate coherently between
two regular regions in mixed phase space even though the classical transport
between these regions is forbidden by a constant of motion (other than
energy).
The transport of cold atoms in optical lattices has been observed to take the
form of ballistic motion, oscillations in wells of the optical potential,
Brownian motion Chu85 , anomalous diffusion and Lévy flights BB02 ; ME96 . The
Lévy flights have been found in the context of subrecoil laser cooling BB02
in the distributions of escape times for ultracold atoms trapped in the
potential wells with momentum states close to the dark state. In those
experiments the variance and the mean time for atoms to leave the trap have
been shown to be infinite.
A new arena of quantum nonlinear dynamics with atoms in optical lattices is
opened if we work near the optical resonance and take the dynamics of internal
atomic states into account. A single atom in a standing-wave laser field may
be semiclassically treated as a nonlinear dynamical system with coupled
internal (electronic) and external (mechanical) degrees of freedom PRA01 ;
JETPL01 ; JETPL02 . In the semiclassical and Hamiltonian limits (when one
treats atoms as point-like particles and neglects spontaneous emission and
other losses of energy), a number of nonlinear dynamical effects have been
analytically and numerically demonstrated with this system: chaotic Rabi
oscillations PRA01 ; JETPL01 ; JETPL02 , Hamiltonian chaotic atomic transport
and dynamical fractals JETP03 ; PLA03 ; PRA07 ; PU06 , Lévy flights and
anomalous diffusion PRA02 ; JETPL02 ; JRLR06 . These effects are caused by
local instability of the CM motion in a laser field. A set of atomic
trajectories under certain conditions becomes exponentially sensitive to small
variations in initial quantum internal and classical external states or/and in
the control parameters, mainly, the atom-laser detuning. Hamiltonian evolution
is a smooth process that is well described in a semiclassical approximation by
the coupled Hamilton-Schrödinger equations. A detailed theory of Hamiltonian
chaotic transport of atoms in a laser standing wave has been developed in our
recent paper PRA07 .
## 3 Semiclassical dynamics
### 3.1 Hamilton-Schrödinger equations of motion
We consider a two-level atom with mass $m_{a}$ and transition frequency
$\omega_{a}$ in a one-dimensional classical standing laser wave with the
frequency $\omega_{f}$ and the wave vector $k_{f}$. In the frame rotating with
the frequency $\omega_{f}$, the Hamiltonian is the following:
$\hat{H}=\frac{\hat{P}^{2}}{2m_{a}}+\frac{1}{2}\hbar(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\hbar\Omega\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}}.$
(1)
Here $\hat{\sigma}_{\pm,z}$ are the Pauli operators which describe the
transitions between lower, ${|1\closeket}$, and upper, ${|2\closeket}$, atomic
states, $\Omega$ is a maximal value of the Rabi frequency. The laser wave is
assumed to be strong enough, so we can treat the field classically. Position
$\hat{X}$ and momentum $\hat{P}$ operators will be considered in section
“Semiclassical dynamics” as $c$-numbers, $X$ and $P$. The simple wavefunction
for the electronic degree of freedom is
${|\Psi(t)\closeket}=a(t){|2\closeket}+b(t){|1\closeket},$ (2)
where $a$ and $b$ are the complex-valued probability amplitudes to find the
atom in the states ${|2\closeket}$ and ${|1\closeket}$, respectively. Using
the Hamiltonian (1), we get the Schrödinger equation
$\displaystyle i\frac{da}{dt}$
$\displaystyle=\frac{\omega_{a}-\omega_{f}}{2}a-\Omega b\cos k_{f}X,$ (3)
$\displaystyle i\frac{db}{dt}$
$\displaystyle=\frac{\omega_{f}-\omega_{a}}{2}b-\Omega a\cos k_{f}X.$
Let us introduce instead of the complex-valued probability amplitudes $a$ and
$b$ the following real-valued variables:
$u\equiv 2\operatorname{Re}\left(ab^{*}\right),\quad
v\equiv-2\operatorname{Im}\left(ab^{*}\right),\quad
z\equiv\left|a\right|^{2}-\left|b\right|^{2},$ (4)
where $u$ and $v$ are a synchronized (with the laser field) and a quadrature
components of the atomic electric dipole moment, respectively, and $z$ is the
atomic population inversion.
In the process of emitting and absorbing photons, atoms not only change their
internal electronic states but their external translational states change as
well due to the photon recoil. In this section we will describe the
translational atomic motion classically. The position and momentum of a point-
like atom satisfy classical Hamilton equations of motion. Full dynamics in the
absence of any losses is now governed by the Hamilton-Schrödinger equations
for the real-valued atomic variables
$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\sin x,\quad\dot{u}=\Delta
v,\\\ \dot{v}=-\Delta u+2z\cos x,\quad\dot{z}=-2v\cos x,\end{gathered}$ (5)
where $x\equiv k_{f}X$ and $p\equiv P/\hbar k_{f}$ are normalized atomic
center-of-mass position and momentum, respectively. Dot denotes
differentiation with respect to the dimensionless time $\tau\equiv\Omega t$.
The normalized recoil frequency, $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega\ll 1$, and the atom-field detuning,
$\Delta\equiv(\omega_{f}-\omega_{a})/\Omega$, are the control parameters. The
system has two integrals of motion, namely the total energy
$H\equiv\frac{\omega_{r}}{2}p^{2}-u\cos x-\frac{\Delta}{2}z,$ (6)
and the Bloch vector $u^{2}+v^{2}+z^{2}=1$. The conservation of the Bloch
vector length follows immediately from Eqs. (4).
Equations (5) constitute a nonlinear Hamiltonian autonomous system with two
and half degrees of freedom which, owing to two integrals of motion, move on a
three-dimensional hypersurface with a given energy value $H$. In general,
motion in a three-dimensional phase space in characterized by a positive
Lyapunov exponent $\lambda$, a negative exponent equal in magnitude to the
positive one, and zero exponent. The maximum Lyapunov exponent characterizes
the mean rate of the exponential divergence of initially close trajectories
and serves as a quantitative measure of dynamical chaos in the system. The
result of computation of the maximum Lyapunov exponent in dependence on the
detuning $\Delta$ and the initial atomic momentum $p_{0}$ is shown in Fig. 1.
Color in the plot codes the value of the maximum Lyapunov exponent $\lambda$.
Figure 1: Maximum Lyapunov exponent $\lambda$ vs atom-field detuning $\Delta$
and initial atomic momentum $p_{0}$: $\omega_{r}=10^{-5}$,
$u_{0}=z_{0}=0.7071$, $v_{0}=0$.
In white regions the values of $\lambda$ are almost zero, and the atomic
motion is regular in the corresponding ranges of $\Delta$ and $p_{0}$. In
shadowed regions positive values of $\lambda$ imply unstable motion.
Figure 1 demonstrates that the center-of-mass motion becomes unstable if the
dimensionless momentum exceeds the value $p_{0}\approx 300$ that corresponds
(with our normalization) to the atomic velocity $v_{a}\approx 3$ m/s for an
atom with $m_{a}\approx 10^{-22}$ g in the field with the wavelength close to
the transition wavelength $\lambda_{a}\simeq 800$ nm. With these estimates for
the atomic and lattice parameters and $\Omega/2\pi=10^{9}$, one gets the
normalized value of the recoil frequency equal to $\omega_{r}=10^{-5}$. The
detuning $\Delta$ will be varied in a wide range, and the Bloch variables are
restricted by the length of the Bloch vector.
### 3.2 Regimes of motion
The case of exact resonance, $\Delta=0$, was considered in detail in Ref.
PRA01 ; JRLR06 . Now we briefly repeat the simple results for the sake of
self-consistency. At zero detuning, the variable $u$ becomes a constant,
$u=u_{0}$, and the fast ($u$, $v$, $z$) and slow ($x$, $p$) variables are
separated allowing one to integrate exactly the reduced equations of motion.
The total energy (6) is equal to $H_{0}=H(u=u_{0},\Delta=0)$, and the atom
moves in a simple cosine potential $u_{0}\cos x$ with three possible types of
trajectories: oscillator-like motion in a potential well if $H_{0}<u_{0}$
(atoms are trapped by the standing-wave field), motion along the separatrix if
$H_{0}=u_{0}$, and ballistic-like motion if $H_{0}>u_{0}$. The exact solution
for the center-of-mass motion is easily found in terms of elliptic functions
(see PRA01 ; JRLR06 ).
As to internal atomic evolution, it depends on the translational degree of
freedom since the strength of the atom-field coupling depends on the position
of atom in a periodic standing wave. At $\Delta=0$, it is easy to find the
exact solutions of Eqs. (5)
$\displaystyle v(\tau)$ $\displaystyle=\pm\sqrt{1-u^{2}}\
\cos\left(2\int\limits_{0}^{\tau}\cos x\,d\tau^{\prime}+\chi_{0}\right),$ (7)
$\displaystyle z(\tau)$ $\displaystyle=\mp\sqrt{1-u^{2}}\
\sin\left(2\int\limits_{0}^{\tau}\cos x\,d\tau^{\prime}+\chi_{0}\right),$
where $u=u_{0}$, and $\cos[x(\tau)]$ is a given function of the translational
variables only which can be found with the help of the exact solution for $x$
PRA01 ; JRLR06 . The sign of $v$ is equal to that for the initial value
$z_{0}$ and $\chi_{0}$ is an integration constant. The internal energy of the
atom, $z$, and its quadrature dipole-moment component $v$ could be considered
as frequency-modulated signals with the instant frequency $2\cos[x(\tau)]$ and
the modulation frequency $\omega_{r}p(\tau)$, but it is correct only if the
maximum value of the first frequency is much greater than the value of the
second one, i. e., for $|\omega_{r}p_{0}|\ll 2$.
The maximum Lyapunov exponent $\lambda$ depends both on the parameters
$\omega_{r}$ and $\Delta$, and on initial conditions of the system (5). It is
naturally to expect that off the resonance atoms with comparatively small
values of the initial momentum $p_{0}$ will be at once trapped in the first
well of the optical potential, whereas those with large values of $p_{0}$ will
fly through. The question is what will happen with atoms, if their initial
kinetic energy will be close to the maximum of the optical potential.
Numerical experiments demonstrate that such atoms will wander in the optical
lattice with alternating trappings in the wells of the optical potential and
flights over its hills. The direction of the center-of-mass motion of
wandering atoms may change in a chaotic way (in the sense of exponential
sensitivity to small variations in initial conditions). A typical chaotically
wandering atomic trajectory is shown in Fig. 2.
Figure 2: Typical atomic trajectory in the regime of chaotic transport:
$x_{0}=0$, $p_{0}=300$, $z_{0}=-1$, $u_{0}=v_{0}=0$, $\omega_{r}=10^{-5}$,
$\Delta=-0.05$.
It follows from (5) that the translational motion of the atom at $\Delta\neq
0$ is described by the equation of a nonlinear physical pendulum with the
frequency modulation
$\ddot{x}+\omega_{r}u(\tau)\sin x=0,$ (8)
where $u$ is a function of all the other dynamical variables.
### 3.3 Stochastic map for chaotic atomic transport
Chaotic atomic transport occurs even if the normalized detuning is very small,
$|\Delta|\ll 1$ (Fig. 1). Under this condition, we will derive in this section
approximate equations for the center-of-mass motion. The atomic energy at
$|\Delta|\ll 1$ is given with a good accuracy by its resonant value $H_{0}$.
Returning to the basic set of the equations of motion (5), we may neglect the
first right-hand term in the fourth equation since it is very small as
compared with the second one there. However, we cannot now exclude the third
equation from the consideration. Using the solution (7) for $v$, we can
transform this equation as
$\dot{u}=\pm\Delta\sqrt{1-u^{2}}\ \cos\chi,\quad\chi\equiv
2\int\limits_{0}^{\tau}\cos x\,d\tau^{\prime}+\chi_{0}.$ (9)
Far from the nodes of the standing wave, Eq. (9) can be approximately
integrated under the additional condition, $|\omega_{r}p|\ll 1$, which is
valid for the ranges of the parameters and the initial atomic momentum where
chaotic transport occurs. Assuming $\cos x$ to be a slowly-varying function in
comparison with the function $\cos\chi$, we obtain far from the nodes the
approximate solution for the $u$-component of the atomic dipole moment
$u\approx\sin\left(\pm\frac{\Delta}{2\cos x}\sin\chi+C\right),$ (10)
where $C$ is an integration constant. Therefore, the amplitude of oscillations
of the quantity $u$ for comparatively slow atoms ($|\omega_{r}p|\ll 1$) is
small and of the order of $|\Delta|$ far from the nodes.
At $|\Delta|=0$, the synchronized component of the atomic dipole moment $u$ is
a constant whereas the other Bloch variables $z$ and $v$ oscillate in
accordance with the solution (7). At $|\Delta|\neq 0$ and far from the nodes,
the variable $u$ performs shallow oscillations for the natural frequency
$|\Delta|$ is small as compared with the Rabi frequency. However, the behavior
of $u$ is expected to be very special when an atom approaches to any node of
the standing wave since near the node the oscillations of the atomic
population inversion $z$ slow down and the corresponding driving frequency
becomes close to the resonance with the natural frequency. As a result, sudden
“jumps” of the variable $u$ are expected to occur near the nodes. This
conjecture is supported by the numerical simulation. In Fig. 3 we show a
typical behavior of the variable $u$ for a comparatively slow and slightly
detuned atom.
Figure 3: Typical evolution of the atomic dipole-moment component $u$ for a
comparatively slow and slightly detuned atom: $x_{0}=0$, $p_{0}=550$,
$v_{0}=0$, $u_{0}=z_{0}=0.7071$, $\omega_{r}=10^{-5}$, $\Delta=-0.01$.
The plot clearly demonstrates sudden “jumps” of $u$ near the nodes of the
standing wave and small oscillations between the nodes.
Approximating the variable $u$ between the nodes by constant values, we can
construct a discrete mapping PRA07
$u_{m}=\sin(\Theta\sin\phi_{m}+\arcsin u_{m-1}),$ (11)
where $\Theta\equiv|\Delta|\sqrt{\pi/\omega_{r}p_{\text{node}}}$ will be
called an angular amplitude of the jump, $u_{m}$ is a value of $u$ just after
the $m$-th node crossing, $\phi_{m}$ are random phases to be chosen in the
range $[0,2\pi]$, and $p_{\text{node}}\equiv\sqrt{2H/\omega_{r}}$ is the value
of the atomic momentum at the instant when the atom crosses a node (which is
the same with a given value of the energy $H$ for all the nodes). With given
values of $\Delta$, $\omega_{r}$, and $p_{\text{node}}$, the map (11) has been
shown numerically to give a satisfactory probabilistic distribution of
magnitudes of changes in the variable $u$ just after crossing the nodes. The
stochastic map (11) is valid under the assumptions of small detunings
($|\Delta|\ll 1$) and comparatively slow atoms ($|\omega_{r}p|\ll 1$).
Furthermore, it is valid only for those ranges of the control parameters and
initial conditions where the motion of the basic system (5) is unstable. For
example, in those ranges where all the Lyapunov exponents are zero, $u$
becomes a quasi-periodic function and cannot be approximated by the map.
### 3.4 Statistical properties of chaotic transport
With given values of the control parameters and the energy $H$, the center-of-
mass motion is determined by the values of $u_{m}$ (see Eq. (8)). One can
obtain from the expression for the energy (6) the conditions under which atoms
continue to move in the same direction after crossing a node or change the
direction of motion not reaching the nearest antinode. Moreover, as in the
resonance case, there exist atomic trajectories along which atoms move to
antinodes with the velocity going asymptotically to zero. It is a kind of
separatrix-like motion with an infinite time of reaching the stationary
points.
The conditions for different regimes of motion depend on whether the crossing
number $m$ is even or odd. Motion in the same direction occurs at
$(-1)^{m+1}u_{m}<H$, separatrix-like motion — at $(-1)^{m+1}u_{m}=H$, and
turns — at $(-1)^{m+1}u_{m}>H$. It is so because even values of $m$ correspond
to $\cos x>0$, whereas odd values — to $\cos x<0$. The quantity $u$ during the
motion changes its values in a random-like manner (see Fig. 3) taking the
values which provide the atom either to prolong the motion in the same
direction or to turn. Therefore, atoms may move chaotically in the optical
lattice. The chaotic transport occurs if the atomic energy is in the range
$0<H<1$. At $H<0$, atoms cannot reach even the nearest node and oscillate in
the first potential well in a regular manner (see Fig. 1). At $H>1$, the
values of $u$ are always satisfy to the flight condition. Since the atomic
energy is positive in the regime of chaotic transport, the corresponding
conditions can be summarized as follows: at $|u|<H$, atom always moves in the
same direction, whereas at $|u|>H$, atom either moves in the same direction,
or turns depending on the sign of $\cos x$ in a given interval of motion. In
particular, if the modulus of $u$ is larger for a long time then the energy
value, then the atom oscillates in a potential well crossing two times each of
two neighbor nodes in the cycle.
The conditions stated above allow to find a direct correspondence between
chaotic atomic transport in the optical lattice and stochastic dynamics of the
Bloch variable $u$. It follows from Eq. (11) that the jump magnitude
$u_{m}-u_{m-1}$ just after crossing the $m$-th node depends nonlinearly on the
previous value $u_{m-1}$. For analyzing statistical properties of the chaotic
atomic transport, it is more convenient to introduce the map for $\arcsin
u_{m}$ PRA07
$\theta_{m}\equiv\arcsin u_{m}=\Theta\sin\phi_{m}+\arcsin u_{m-1},$ (12)
where the jump magnitude does not depend on a current value of the variable.
The map (12) visually looks as a random motion of the point along a circle of
unit radius (Fig. 4). The vertical projection
Figure 4: Graphic representation for the maps of $u_{m}$ and
$\theta_{m}\equiv\arcsin u_{m}$. $H$ is a given value of the atomic energy.
Atoms either oscillate in optical potential wells (trapping) or fly through
the optical lattice (flight).
of this point is $u_{m}$. The value of the energy $H$ specifies four regions,
two of which correspond to atomic oscillations in a well, and two other ones —
to ballistic motion in the optical lattice.
We will call “a flight” such an event when atom passes, at least, two
successive antinodes (and three nodes). The continuous flight length $L>2\pi$
is a distance between two successive turning points at which the atom changes
the sign of its velocity, and the discrete flight length is a number of nodes
$l$ the atom crossed. They are related in a simple way, $L\simeq\pi l$, for
sufficiently long flight.
Center-of-mass oscillations in a well of the optical potential will be called
“a trapping”. At extremely small values of the detuning, the jump magnitudes
are small and the trapping occurs, largely, in the $2\pi$-wide wells, i. e.,
in the space interval of the length $2\pi$. At intermediate values of the
detuning, it occurs, largely, in the $\pi$-wide wells, i. e. in the space
interval of the length $\pi$. Far from the resonance, $|\Delta|\gtrsim 1$,
trapping occurs only in the $\pi$-wide wells. Just like to the case of
flights, the number of nodes $l$, atom crossed being trapped in a well, is a
discrete measure of trapping.
The PDFs for the flight $P_{\text{fl}}(l)$ and trapping $P_{\text{tr}}(l)$
events were analytically derived to be exponential in a case of large jumps
PRA07 . In a case of small jumps, the kind of the statistics depends on
additional conditions imposed on the atomic and lattice parameters, and the
distributions $P_{\text{fl}}(l)$ and $P_{\text{tr}}(l)$ were analytically
shown to be either practically exponential or functions with long power-law
segments with the slope $-1.5$ but exponential “tails”. The comparison of the
PDFs computed with analytical formulas, the stochastic map, and the basic
equations of motion has shown a good agreement in different ranges of the
atomic and lattice parameters PRA07 . We will use the results obtained to find
the analytical conditions, under which the fractal properties of the chaotic
atomic transport can be observed, and to explain the structure of the
corresponding dynamical fractals.
Since the period and amplitude of the optical potential and the atom-field
detuning can be modified in a controlled way, the transport exponents of the
flight and trapping distributions are not fixed but can be varied
continuously, allowing to explore different regimes of the atomic transport.
Our analytical and numerical results with the idealized system have shown that
deterministic atomic transport in an optical lattice cannot be just classified
as normal and anomalous one. We have found that the flight and trapping PDFs
may have long algebraically decaying segments and a short exponential “tail”.
It means that in some ranges of the atomic and lattice parameters numerical
experiments reveal anomalous transport with Lévy flights. The transport
exponent equal to $-1.5$ means that the first, second, and the other
statistical moments are infinite for a reasonably long time. The corresponding
atomic trajectories computed for this time are self-similar and fractal. The
total distance, that the atom travels for the time when the flight PDF decays
algebraically, is dominated by a single flight. However, the asymptotic
behavior is close to normal transport. In other ranges of the atomic and
lattice parameters, the transport is practically normal both for short and
long times.
### 3.5 Dynamical fractals
Various fractal-like structures may arise in chaotic Hamiltonian systems Gas ;
Zas05 . In Ref. PLA03 ; JETP03 ; JRLR06 ; PU06 we have found numerically
fractal properties of chaotic atomic transport in cavities and optical
lattices. In this section we apply the analytical results of the theory of
chaotic transport, developed in the preceding sections, to find the conditions
under which the dynamical fractals may arise.
We place atoms one by one at the point $x_{0}=0$ with a fixed positive value
of the momentum $p_{0}$ and compute the time $T$ when they cross one of the
nodes at $x=-\pi/2$ or $x=3\pi/2$. In these numerical experiments we change
the value of the atom-field detuning $\Delta$ only. All the initial conditions
$p_{0}=200$, $z_{0}=-1$, $u_{0}=v_{0}=0$ and the recoil frequency
$\omega_{r}=10^{-5}$ are fixed.
Figure 5: Fractal-like dependence of the time of exit of atoms $T$ from a
small region in the optical lattice on the detuning $\Delta$: $p_{0}=200$,
$z_{0}=-1$, $u_{0}=v_{0}=0$. Magnifications of the detuning intervals are
shown.
The exit time function $T(\Delta)$ in Fig. 5 demonstrates an intermittency of
smooth curves and complicated structures that cannot be resolved in principle,
no matter how large the magnification factor. The second and third panels in
Fig. 5 demonstrate successive magnifications of the detuning intervals shown
in the upper panel. Further magnifications reveal a self-similar fractal-like
structure that is typical for Hamiltonian systems with chaotic scattering Gas
; BUP04 . The exit time $T$, corresponding to both the smooth and unresolved
$\Delta$ intervals, increases with increasing the magnification factor.
Theoretically, there exist atoms never crossing the border nodes at $x=-\pi/2$
or $x=3\pi/2$ in spite of the fact that they have no obvious energy
restrictions to do that. Tiny interplay between chaotic external and internal
atomic dynamics prevents those atoms from leaving the small space region.
Various kinds of atomic trajectories can be characterized by the number of
times $m$ atom crosses the central node at $x=\pi/2$ between the border nodes.
There are also special separatrix-like trajectories along which atoms
asymptotically reach the points with the maximum of the potential energy,
having no more kinetic energy to overcome it. In difference from the
separatrix motion in the resonant system ($\Delta=0$), a detuned atom can
asymptotically reach one of the stationary points even if it was trapped for a
while in a well. Such an asymptotic motion takes an infinite time, so the atom
will never reach the border nodes.
The smooth $\Delta$ intervals in the first-order structure (Fig. 5, upper
panel) correspond to atoms which never change the direction of motion ($m=1$)
and reach the border node at $x=3\pi/2$. The singular points in the first-
order structure with $T=\infty$, which are located at the border between the
smooth and unresolved $\Delta$ intervals, are generated by the asymptotic
trajectories. Analogously, the smooth $\Delta$ intervals in the second-order
structure (second panel in Fig. 5) correspond to the $2$-nd order ($m=2$)
trajectories, and so on.
The set of all the values of the detunings, generating the separatrix-like
trajectories, was shown to be a countable fractal in Refs. JETP03 ; JRLR06 ,
whereas the set of the values generating dynamically trapped atoms with
$m=\infty$ seems to be uncountable. The exit time $T$ depends in a complicated
way not only on the values of the control parameters but on initial conditions
as well.
In Fig. 6 JRLR06 we presented a two-dimensional image of the time of exit $T$
in the space of the initial atomic momentum $p_{0}$ and the atom-field
detuning $\Delta$. A self-similarity of this function is evident.
Figure 6: The scattering function in the regime of chaotic wandering. The time
of exit $T$ vs the detuning $\Delta$ and the initial momentum $p_{0}$. The
function is shown in a shaded relief regime.
The length of all smooth segments in the $m$-th order structure in Fig. 5 is
proportional to the number of atoms $N(m)$ leaving the space $[-\pi/2,3\pi/2]$
after crossing the central node $m$ times. An exponential scaling
$N(m)\sim\exp(-\gamma m)$ has been found numerically with $\gamma\simeq 1$.
The trapping PDFs, computed with the basic and reduced equations of motion at
the detunings in the range shown in Fig. 5, have been found to have
exponential tails. It is well known Gas that Hamiltonian systems with fully
developed chaos demonstrate, as a rule, exponential decay laws, whereas the
systems with a mixed phase space (containing islands of regular motion)
usually have more slow algebraic decays due to the effect of stickiness of
trajectories to the boundaries of such islands Zas05 . We have not found
visible regular islands in our system at the values of the control parameters
used to compute the fractal in Fig. 5 and we may conclude that the exponential
scaling is a result of completely chaotic wandering of atoms in the space
interval $[-\pi/2,3\pi/2]$ resembling chaotic motion in hyperbolic systems.
The fractal-like structure with smooth and unresolved components may appear if
atoms have an alternative either to turn back or to prolong the motion in the
same direction just after crossing the node at $x=\pi/2$. For the first-order
structure in the upper panel in Fig. 5, it means that the internal variable
$u$ of an atom, just after crossing the node for the first time ($\cos x<0$),
satisfies either to the condition $u_{1}<H$ (atom moves in the same
direction), or to the condition $u_{1}>H$ (atom turns back). If $u_{1}=H$,
then the exit time $T$ is infinite. The jumps of the variable $u$ after
crossing the node are deterministic but sensitively dependent on the values of
the control parameters and initial conditions. We have used this fact when
introducing the stochastic map. Small variations in these values lead to
oscillations of the quantity $\arcsin u_{1}$ around the initial value $\arcsin
u_{0}$ with the angular amplitude $\Theta$. If this amplitude is large enough,
then the sign of the quantity $u_{1}-H$ alternates and we obtain alternating
smooth (atoms reach the border $x=3\pi/2$ without changing their direction of
motion) and unresolved (atoms turns a number of times before exit) components
of the fractal-like structure.
If the values of the parameters admit large jump magnitudes of the variable
$u$, then the dynamical fractal arises in the energy range $0<H<1$, i. e., at
the same condition under which atoms move in the optical lattice in a chaotic
way. In a case of small jump magnitudes, fractals may arise if the initial
value of an atom $u_{0}$ is close enough to the value of the energy $H$, i.
e., the atom has a possibility to overcome the value $u=H$ in a single jump.
Therefore, the condition for appearing in the fractal $T(\Delta)$ the first-
order structure with singularities is the following:
$|\arcsin u_{0}-\arcsin H|<\Theta.$ (13)
The generation of the second-order structure is explained analogously. If an
atom made a turn after crossing the node for the first time, then it will
cross the node for the second time. After that, the atom either will turn or
cross the border node at $x=-\pi/2$. What will happen depend on the value of
$u_{2}$. However, in difference from the case with $m=1$, the condition for
appearing an infinite exit time with $m=2$ is $u_{2}=-H$. Furthermore, the
previous value $u_{1}$ is not fixed (in difference from $u_{0}$) but depends
on the value of the detuning $\Delta$. In any case we have $u_{1}>H$ since the
second-order structure consists of the trajectories of those atoms which
turned after the first node crossing. In order for an atom would be able to
turn after the second node crossing, the magnitude of its variable $u$ should
change sufficiently to be in the range $u_{2}<-H$. The atoms, whose variables
$u$ could not “jump” so far, leave the space $[-\pi/2,3\pi/2]$. The
singularities are absent in the middle segment of the second-order structure
shown in the second panel in Fig. 5 because all the corresponding atoms left
the space after the second node crossing. The variable $u_{2}$ oscillates with
varying $\Delta$ generating oscillations of the exit time. The condition for
appearing singularities in the second-order structure is the following:
$2\arcsin H<\Theta.$ (14)
With the values of the parameters taken in the simulation, we get the energy
$H=0.2+\Delta/2$. It is easy to obtain from the inequality (14) the
approximate value of the detuning $|\Delta|\approx 0.0107$ for which the
second-order singularities may appear. In the lower panel in Fig. 5 one can
see this effect. No additional conditions are required for generating the
structures of the third and the next orders.
Inequality (14) is opposite to the inequality that determines the condition
for appearing power law decays in the flight PDF. Therefore, dynamical fractal
may appear in those ranges of the control parameters where the Lévy flights
are impossible and vice versa. However, the trapping PDF may have a power law
decay. Inequality (14) in difference from (13) is strongly related with the
chosen concrete scheme for computing exit times. It is not required with other
schemes, say, with three antinodes between the border nodes.
## 4 Quantum dynamics
In this section we will treat atomic translational motion quantum
mechanically, i. e., atom is supposed to be not a point particle but a wave
packet. The corresponding Hamiltonian $\hat{H}$ has the form (1) with
$\hat{X}$ and $\hat{P}$ being the position and momentum operators. We will
work in the momentum space with the state vector
${|\Psi(t)\closeket}=\int\left(a(P,t){|2\closeket}+b(P,t){|1\closeket}\right){|P\closeket}dP,$
(15)
which satisfies to the Schrödinger equation
$i\hbar\frac{d{|\Psi\closeket}}{dt}=\hat{H}{|\Psi\closeket}.$ (16)
The normalized equations for the probability amplitudes have the form
$\displaystyle i\dot{a}(p)$
$\displaystyle=\frac{1}{2}(\omega_{r}p^{2}-\Delta)a(p)-\frac{1}{2}[b(p+1)+b(p-1)],$
(17) $\displaystyle i\dot{b}(p)$
$\displaystyle=\frac{1}{2}(\omega_{r}p^{2}+\Delta)b(p)-\frac{1}{2}[a(p+1)+a(p-1)],$
with the same normalization and the control parameters as in the semiclassical
theory. When deriving (17), we used the following property of the momentum
operator $\hat{P}$:
$\cos
k_{f}\hat{X}{|P\closeket}\equiv\frac{1}{2}\left(e^{ik_{f}\hat{X}}+e^{-ik_{f}\hat{X}}\right){|P\closeket}=\frac{1}{2}\left({|P+\hbar
k_{f}\closeket}+{|P-\hbar k_{f}\closeket}\right).$ (18)
Equations (17) are an infinite-dimensional set of ordinary differential
complex-valued equations of the first order with coupled amplitudes $a(p\pm
n)$ and $b(p\pm m)$. To characterize the internal atomic state, let us
introduce the following variables;
$\displaystyle u(\tau)$ $\displaystyle\equiv 2\operatorname{Re}\int
dx\left[a(x,\tau)b^{*}(x,\tau)\right],$ (19) $\displaystyle v(\tau)$
$\displaystyle\equiv-2\operatorname{Im}\int dx[a(x,\tau)b^{*}(x,\tau)],$
$\displaystyle z(\tau)$ $\displaystyle\equiv\int
dx[|a(x,\tau)|^{2}-|b(x,\tau)|^{2}],$
which are quantum versions of the Bloch components (4), and we denote them by
the same letters.
Figure 7: Resonant $E_{0}^{(\pm)}$ and nonresonant $E_{\Delta}^{(\pm)}$
potentials for an atom in a standing wave. The optical Stern-Gerlach effect in
the resonant potential is shown: splitting of an atomic wave packet launched
at the node of the wave ($x_{0}=\pi/2$, $p_{0}=0$). The wave packet, placed
initially at the antinode ($x_{0}=0$, $p_{0}=0$), appears to be simultaneously
at the top of $E_{0}^{(+)}$ and the bottom of $E_{0}^{(-)}$ potentials. Its
${|+\closeket}$-component slides down both the sides of $E_{0}^{(+)}$ and the
${|-\closeket}$-component oscillates at the bottom of $E_{0}^{(-)}$.
## 5 Dressed states picture and nonadiabatic transitions
Interpretation of the atomic wave-packet motion in a standing-wave field is
greatly facilitated in the basis of atomic dressed states which are
eigenstates of a two-level atom in a laser field. The adiabatic dressed states
$\begin{gathered}{|+\closeket}_{\Delta}=\sin{\Theta}{|2\closeket}+\cos{\Theta}{|1\closeket},\quad{|-\closeket}_{\Delta}=\cos{\Theta}{|2\closeket}-\sin{\Theta}{|1\closeket},\\\
\tan{\Theta}\equiv\frac{\Delta}{2\cos{x}}-\sqrt{\left(\frac{\Delta}{2\cos{x}}\right)^{2}+1}\end{gathered}$
(20)
are eigenstates at a nonzero detuning. The corresponding values of the
quasienergy are
$E_{\Delta}^{(\pm)}=\pm\sqrt{\frac{\Delta}{2}^{2}+\cos^{2}{x}}.$ (21)
Figure 7 shows a spatial variation of the quasienergies along the standing-
wave axis. It follows from Eqs.(20) and (21) that, in general case, atom moves
in the two potentials $E_{\Delta}^{(\pm)}$ simultaneously.
At exact resonance, $\Delta=0$, the dressed states have the simple form
${|+\closeket}=\frac{1}{\sqrt{2}}({|1\closeket}+{|2\closeket}),\quad{|-\closeket}=\frac{1}{\sqrt{2}}({|1\closeket}-{|2\closeket})$
(22)
and are called diabatic states. The resonant potentials,
$E_{0}^{(\pm)}=\pm\cos x$, cross each other at the nodes of the standing wave,
$x=\pi/2+\pi m$, $(m=0,\pm 1,\ldots)$. What will happen if we place the
centroid of an atomic wave packet exactly at the node, $x_{0}=\pi/2$, in the
ground state ${|1\closeket}$ and suppose its initial mean momentum to be zero,
$p_{0}=0$? The initial ground state is the superposition of the diabatic
states: ${|1\closeket}=({|+\closeket}+{|-\closeket})/\sqrt{2}$. One part of
the initial wave packet at the top of the potential $E_{0}^{(+)}$ will start
to move to the right under the action of the gradient force
$F^{(+)}=-dE_{0}^{(+)}/dx=\sin x$, and another one — to the left to be forced
by $F^{(-)}=-\sin x$ (see Fig. 7). It is the well-known optical Stern-Gerlach
effect K75 ; Kaz ; Sleator . If the maximal expected value of the atomic
kinetic energy does not exceed the potential one, the atom will be trapped in
the potential well. Two splitted components of the initial wave packet will
oscillate in the well with the period of oscillations
$T\simeq 4\sqrt{\frac{\pi}{\omega_{r}}}.$ (23)
The wave packet, with $p_{0}=0$, placed at the antinode, say, at $x_{0}=0$, is
simultaneously at the top of the potential $E_{0}^{(+)}$ and at the bottom of
$E_{0}^{(-)}$. Therefore, its ${|+\closeket}$-component will slide down the
both sides of the potential curve $E_{0}^{(+)}$, and the
${|-\closeket}$-component will oscillate around $x=0$ (see Fig. 7).
Out off resonance, $\Delta\neq 0$, the atomic wave packet moves in the
bipotential $E_{\Delta}^{(\pm)}$ (21). The distance between the quasienergy
curves is minimal at the nodes of the standing wave and equal to $\Delta$ (see
Fig. 7). The spatial period and the modulation depth of the resonant
potentials $E_{0}^{(\pm)}$ are twice as much as those for the nonresonant
potentials $E_{\Delta}^{(\pm)}$.
The probability of nonadiabatic transitions between the dressed states
${|+\closeket}_{\Delta}$ and ${|-\closeket}_{\Delta}$ can be estimated in a
simple way. The time of flight over a short distance $\delta x$ in
neighbourhood of a node is $\delta x/\omega_{r}p_{\text{node}}$. If the time
of transition between the quasienergy levels, $2/\Delta$, is of the order of
the flight time, the transition probability is close to $1$. It is easy to get
the characteristic frequency of atomic motion from that condition Kaz
$\Delta_{0}=\sqrt{\omega_{r}p_{\text{node}}},$ (24)
where $p_{\text{node}}$ is a value of the momentum in the vicinity of a node.
Depending on the relation between $\Delta$ and $\Delta_{0}$, there are three
typical cases.
1. 1.
If $|\Delta|\ll\Delta_{0}$, the nonadiabatic transition probability between
the states ${|+\closeket}_{\Delta}$ and ${|-\closeket}_{\Delta}$ upon crossing
any node is close to $1$. However, the diabatic states ${|+\closeket}$ and
${|-\closeket}$ are not mixed, and atom moves in one of optical resonant
potentials.
2. 2.
If $|\Delta|\simeq\Delta_{0}$, the atom may or may not undergo a transition
upon crossing any node from one of the nonresonant potentials to another one
with the probabilities of the same order.
3. 3.
If $|\Delta|\gg\Delta_{0}$, the nonadiabatic transition probability is
exponentially small, and atom moves in one of the nonresonant potentials.
### 5.1 Wave packet motion in the momentum space
The atom at $\tau=0$ is supposed to be prepared as a Gaussian wave packet in
the momentum space
$a_{0}(p)=0,\quad b_{0}(p)=\frac{1}{\sqrt{\sqrt{\pi}\Delta
p}}\exp\left[-\frac{(p-p_{0})^{2}}{2(\Delta p)^{2}}-i(p-p_{0})x_{0}\right],$
(25)
with the momentum width $\Delta p=10$ corresponding to the spatial width
$\Delta X=\lambda_{f}/40\pi$ that is much smaller than the optical wavelength
$\lambda_{f}$. We compute the probability to find a two-level atom at the
moment of time $\tau$ with the momentum $p$
$W(p,\tau)=|a(p,\tau)|^{2}+|b(p,\tau)|^{2},$ (26)
by integrating Eqs.(17) with the initial condition (25). The recoil frequency,
$\omega_{r}=10^{-5}$, is fixed and the centroid of the wave packet is placed
at the antinode $x_{0}=0$, in all the numerical experiments.
#### Adiabatic evolution at exact resonance
Figure 8: Time dependence of the momentum probability function $W(p,\tau)$ for
a ballistic atom at resonance prepared initially in the ground state
($\Delta=0$, $\omega_{r}=10^{-5}$, $x_{0}=0$, $p_{0}=800$).
At exact resonance, $\Delta=0$, the wave functions of the diabatic states
${|+\closeket}$ and ${|-\closeket}$ evolve independently, each one evolves in
its own potential $E_{0}^{(+)}$ and $E_{0}^{(-)}$, respectively. The atom,
prepared initially in the ground state
${|1\closeket}=({|+\closeket}+{|-\closeket})/\sqrt{2}$ with the mean initial
momentum $p_{0}=800$, will start to move from the top of $E_{0}^{(+)}$ and the
bottom of $E_{0}^{(-)}$ potentials (see Fig.7). Thus, the initial wave packet
will split into two components ${|+\closeket}$ and ${|-\closeket}$. Time
evolution of the probability function (26) for each of the components is shown
in Fig.8. Pay, please, attention that the values of $p$ on this and similar
plots increase downwards. Color in this figure codes the values of
$W(p,\tau)$. The ${|+\closeket}$-component (the lower trajectory in the
figure) slides down the curve $E_{0}^{(+)}$ and, therefore, moves with an
increasing velocity up to the next antinode at $x=\pi$, and then it slows down
approaching the antinode at $x=2\pi$. The atom moves in the positive direction
of the axis $x$ and the process repeats periodically with the period
$\tau^{(+)}_{0}=2\pi/\omega_{r}\bar{p}^{(+)}_{0,2\pi}\simeq 690$, where
$\bar{p}^{(+)}_{0,2\pi}$ is a mean momentum of the ${|+\closeket}$-component
upon the atomic motion between $0$ and $2\pi$.
The ${|-\closeket}$-component (the upper trajectory in Fig.8) moves upward the
potential curve $E_{0}^{(-)}$ and slows down up to reaching the top of
$E_{0}^{(-)}$ at $x=\pi$. Then it moves with an increasing momentum up to
$x=2\pi$. Since the mean momentum of the ${|-\closeket}$-component is smaller
than that of the ${|+\closeket}$ one, the corresponding period is longer,
$\tau^{(-)}_{0}\simeq 980$.
#### Proliferation of wave packets at the nodes of the standing wave
Figure 9: Proliferation of atomic wave packets at the nodes of the standing
wave at the detuning $\Delta=0.05$. The atom is prepared initially in the
dressed state ${|+\closeket}$. Other conditions are the same as in Fig.8.
New features in propagation of atomic wave packets through the standing wave
appear under the condition $\Delta\simeq\Delta_{0}$. Using the semiclassical
expression for the total atomic energy (6), let us estimate the value of the
atomic momentum at the nodes of the standing wave if the detuning is not
large, $|\Delta|\ll 1$. If the atom is prepared initially in the state
${|+\closeket}$, i.e., $u_{0}=1$, $z_{0}=0$, and $x_{0}=0$ then we have
$H=H_{0}=2.2$ at $p_{0}=800$. Since the total energy is a constant, we get
immediately from Eq. (6)
$p_{\text{node}}\simeq\sqrt{2H/\omega_{r}}\simeq 665.$ (27)
Using the same formula (6), we get the values of the minimal and maximal
momenta if the atom starts to move with the initial mean momentum $p_{0}=800$:
$p_{\text{min}}\simeq\sqrt{2(H_{0}-1)/\omega_{r}}\simeq 490$ and
$p_{\text{max}}\simeq\sqrt{2(H_{0}+1)/\omega_{r}}\simeq 800$.
The formula (24) gives us the value of the characteristic frequency under the
chosen conditions, $\Delta_{0}\simeq 0.08$. We fix $\Delta=0.05$ in this
section, so $\Delta\simeq\Delta_{0}$. The initial state ${|+\closeket}$ is the
following superposition of the adiabatic states:
${|+\closeket}=\frac{1}{\sqrt{2}}[(\cos{\Theta}+\sin{\Theta}){|+\closeket}_{\Delta}+(\cos{\Theta}-\sin{\Theta}){|-\closeket}_{\Delta}].$
(28)
With the help of (21) we can estimate the mixing angle at $\Delta=0.05$ to be
equal to $\theta\simeq-\pi/4$. Then it follows from (28) that
${|+\closeket}\simeq{|-\closeket}_{\Delta}$, i. e., practically all the wave
packet is initially at the bottom of the potential $E_{\Delta}^{(-)}$ (Fig.
7). Figure 9 demonstrates that the wave packet really slows down, and its
centroid intersects the node $x=\pi/2$ at $\tau_{1}^{(-)}\simeq 215$. Under
the condition $\Delta\simeq\Delta_{0}$, the atom has a probability to change
the potential for another one upon crossing a node and a probability to stay
in its present potential. This is exactly what we see in fig. 9: the wave
packet splits at the node $x=\pi/2$ with the ${|-\closeket}$-component
climbing over the potential $E_{\Delta}^{(-)}$ (see the upper trajectory in
this figure) and the ${|+\closeket}$-component sliding down the curve
$E_{\Delta}^{(+)}$ with an increasing momentum (see the lower trajectory).
Just after crossing the node, the most part of the probability density moves
in the potential $E_{\Delta}^{(-)}$ because the corresponding probability is
larger. The ${|+\closeket}$-component increases its velocity upon approaching
the antinode at $x=\pi$ and then slows down up to the second node at
$x=3\pi/2$ where it splits into two components at $\tau_{2}^{(+)}\simeq 640$.
After that, one of the components will move in the potential
$E_{\Delta}^{(+)}$ decreasing the velocity up to the next antinode at
$x=2\pi$, and the other one will move in $E_{\Delta}^{(-)}$ increasing its
velocity in the same space interval. The probability density of this
${|-\closeket}$-component is only a few percents, and we draw a solid curve
along this trajectory in order to visualize the motion.
Figure 10: The same as in Fig.9 but for the atom prepared initially in the
ground state.
The ${|-\closeket}$-component of the packet, splitted after crossing the first
node at $x=\pi/2$, has a smaller mean momentum than the ${|+\closeket}$-one.
Therefore, it reaches the second node later, at $\tau_{2}^{(-)}\simeq 800$,
where it splits into two parts: the upper ${|+\closeket}$-component will move
in the potential $E_{\Delta}^{(+)}$ and the lower ${|-\closeket}$-one — in
$E_{\Delta}^{(-)}$. Such a proliferation of atomic wave packets takes places
upon crossing all the next nodes of the standing wave.
The moment of time $\tau_{n}^{(\pm)}$, when the centroids of the
${|\pm\closeket}$-components cross the $n$-th node, can be estimated by the
simple formula (we suppose that the centroid of the atomic wave packet was at
$x=0$ at $\tau=0$):
$\omega_{r}\overline{p}_{n-1,n}^{(\pm)}\tau_{n}^{(\pm)}=(2n-1)\frac{\pi}{2},\quad
n=2,3,\dots.$ (29)
where $\overline{p}_{n-1,n}^{(\pm)}$ is a mean momentum of the
${|\pm\closeket}$-components upon their movement between $(n-1)$-th and $n$-th
nodes. This quantity for the ${|-\closeket}$-component, moving between $x=0$
and $x=\pi/2$, is $\bar{p}_{0,1}^{(-)}=(p_{0}+p_{\text{node}})/2\simeq 732.5$.
So, the centroid of this wave packet crosses the first node at
$\tau_{1}^{(-)}\simeq 214$. The lower ${|+\closeket}$-component crosses the
second node at $x=3\pi/2$ at $\tau_{2}^{(+)}\simeq 642$. For the upper
${|-\closeket}$-component we get
$\bar{p}_{1,2}^{(-)}=(p_{\text{node}}+p_{\text{min}})/2\simeq 577.5$ and
$\tau_{2}^{(-)}\simeq 815$. All the other moments of time, $\tau_{n}^{(\pm)}$,
can be estimated in the same way. The estimates obtained fit well the
numerical data (see Fig.9). The interference fringes on the upper trajectory
at $\tau\simeq 1000$ and $p\simeq 500$ and on the lower one at $\tau\simeq
900$ and $p\simeq 800$ reflect the fine-scale splitting of the corresponding
wave packets.
Let us now compute the probability map for the atom prepared initially in the
ground state ${|1\closeket}$ which has the following form in the adiabatic
state basis:
${|1\closeket}=\cos{\Theta}{|+\closeket}_{\Delta}-\sin{\Theta}{|-\closeket}_{\Delta},$
(30)
It follows from (21) that (30) is almost a $50\%$–$50\%$ superposition of the
${|+\closeket}_{\Delta}$ and ${|-\closeket}_{\Delta}$ states. All the other
conditions are assumed to be the same as before. The atomic wave packet splits
from the beginning into two components with the ${|+\closeket}$-one sliding
down the curve $E_{\Delta}^{(+)}$ (the lower trajectory in Fig. 10) and the
${|-\closeket}$-one climbing over the potential $E_{\Delta}^{(-)}$ (the upper
trajectory). Each of the components splits at the first node with a small time
difference between the events. The subsequent proliferation of the wave
packets occurs for the upper and lower parts of the probability density
independently on each other in accordance with the same scenario as described
above. In difference from the preceding case, the atom, prepared initially in
the ground state, acquired the values of the momentum that are larger then the
initial momentum $p_{0}=800$.
Figure 11: Time dependence of the dipole moment $u$ and the population
inversion $z$ at the same conditions as in Fig. 9.
The nonadiabatic transitions are accompanied by drastic changes in the
internal state of the atom which is characterized by the values of the
synphased component of the electric dipole moment $u$ and the population
inversion $z$. In Fig. 11 we demonstrate their behavior for the atom prepared
initially in the state ${|+\closeket}$. Both the variables change their values
abruptly in the time intervals with the centers at $\tau\simeq 215$, $640$ and
$815$, i. e., when the centroids of the atomic wave packets cross the first
two nodes.
#### Adiabatic motion at large detunings
For comparison with the results of the preceding section, we demonstrate in
Fig. 12 the evolution of the momentum distribution function $W(p,\tau)$ with
the ground initial state at $\Delta=2$ and the other same conditions as in the
preceding section. The detuning $\Delta=2$ is large as compared to the
characteristic frequency $\Delta_{0}\simeq 0.09$ that is estimated from (24)
at $p_{0}=800$. It follows from (20) and (21) that at $\Delta=2$ the initial
state ${|1\closeket}$ is a superposition of approximately $70\%$ of the state
${|+\closeket}_{\Delta}$ and $\sim 30\%$ of the state
${|-\closeket}_{\Delta}$. So the main part of the initial packet begins to
move in the potential $E_{\Delta}^{(+)}$ increasing the momentum upon
approaching the node at $x=\pi/2$, and the other part moves in
$E_{\Delta}^{(-)}$ decreasing the momentum in the same space interval (see
Fig. 12). Upon crossing the nodes, the probability of transition between the
states ${|\pm\closeket}_{\Delta}$ is small if $|\Delta|\gg\Delta_{0}$, and
each of the component will continue to move in its own potential. The process
is repeated and we see the periodic variations of the mean momentum of each of
the components. The same picture is observed if we take the state
${|+\closeket}=({|1\closeket}+{|2\closeket})/\sqrt{2}$ as the initial one. At
$\Delta=2$, the state ${|+\closeket}$ is a mix of $70\%$ of
${|-\closeket}_{\Delta}$ and $30\%$ of ${|+\closeket}_{\Delta}$, so the main
part of the initial ${|+\closeket}$ wave packet will move in the potential
$E_{\Delta}^{(-)}$. The evolution of the internal atomic variables $z$ and $u$
is shown in Fig. 13. There are no jumps of $z$ and $u$ when the atom crosses
nodes. Instead of that, we see fast oscillations of those variables when the
atom crosses the first antinodes.
Figure 12: Adiabatic evolution of the momentum probability function
$W(p,\tau)$ for a ballistic atom at the large detuning $\Delta=2$.
Figure 13: The same as in Fig.11 but at the large detuning $\Delta=2$.
Thus, at $|\Delta|\gg\Delta_{0}$, there are no nonadiabatic transitions due to
the corresponding small probability and, therefore, no proliferation of wave
packets at the nodes. The evolution of the atomic wave packet is adiabatic.
#### An atom can fly and be trapped simultaneously
An intriguing effect of simultaneous trapping of an atom in a well of the
optical potential and its ballistic flight through the optical lattice is
observed at comparatively small values of the detuning. Let us prepare an atom
in the ground state ${|1\closeket}$ with such a mean initial value of the
momentum $p_{0}$ that its ${|-\closeket}$-component would not be able to
overcome the barrier of the potential $E_{\Delta}^{(-)}$ but its
${|+\closeket}$-component would have a sufficient kinetic energy to overcome
the barrier of the $E_{\Delta}^{(+)}$ potential. Now one could expect periodic
oscillations in the first well of the potential $E_{\Delta}^{(-)}$ and a
simultaneous ballistic flight in the $E_{\Delta}^{(+)}$ potential with a
proliferation of wave packets of the ${|+\closeket}$-component at the nodes of
the standing wave.
Figure 14 demonstrates this effect at $p_{0}=300$, $\Delta=-0.05$ and the same
other conditions as before. We see that the momentum of the
${|-\closeket}$-component (the upper trajectory in this figure) oscillates in
the range ($300$, $-300$), and this component is trapped in the first well
($-\pi/2\leq x\leq\pi/2$). Whereas the ${|+\closeket}$-component moves in the
positive direction splitting at each node. Estimates of the period of
oscillations of the ${|-\closeket}$-component, $T\simeq 2240$, with the help
of (23) and of the time when the centroid of the ${|+\closeket}$-component
crosses the first node, $\tau_{1}^{(+)}\simeq 380$ (formula (29)), fit well
the data in Fig. 14.
Figure 14: Effect of simultaneous trapping of an atom in a well of the optical
potential and its flight through the wave. The ground initial state,
$\Delta=-0.05$, $p_{0}=300$.
## 6 Quantum-classical correspondence and manifestations of dynamical chaos
in wave-packet atomic motion
Dynamical chaos in classical systems is characterized by exponentially fast
divergence of initially close trajectories in a bounded phase space. Such a
behavior is possible because of the continuity of the classical phase space
whose points (therefore, classical system’s states) can be arbitrary close to
each other. The trajectory concept is absent in quantum mechanics whose phase
space is not continuous due to the Heisenberg uncertainty relation. The
evolution of an isolated quantum system is unitary, and there can be no chaos
in the sense of exponential sensitivity of its states to small variations in
initial conditions. What is usually understand under “quantum chaos” is
special features of the unitary evolution of a quantum system in the range of
its parameter values and initial conditions at which its classical analogue is
chaotic.
The question “what happens to classical motion in the quantum world” is a core
of the problem of quantum-classical correspondence. In spite of years of
discussions from the beginning of the quantum era, it is still unclear how
classical features appear from the underlying quantum equations. It is
especially difficult to specify what happens to classical dynamical chaos in
the quantum world BZ78 ; Casati79 ; Z81 ; Gutzwiller ; Reichl ; Haake ;
Shtokman . The interest to the problem of “quantum chaos” is motivated by our
desire to understand the quantum origin of the observed classical chaos.
In this section we establish a correspondence between the quantized motion of
a two-level atom in a standing laser wave and its semiclassical analogue
considered in the third section. Semiclassical equations (5) represent a
nonlinear dynamical system with positive values of the maximal Lyapunov
exponent in a wide range of the initial conditions and control parameters
$\omega_{r}$ and $\Delta$. In other words, trajectories in the five-
dimensional phase space are exponentially sensitive to small variations in
initial conditions and/or parameters in those ranges. That local dynamical
instability is a reason for chaotic Rabi oscillations and chaotic motion of
the atomic center of mass discussed in the third section. In particular, it
has been found that an atom is able to walk chaotically in a strictly periodic
optical lattice without any noise or other random processes (see Fig. 2). The
chaotic behavior is caused by jumps of the electric-dipole moment $u$ at the
nodes of the standing wave (Fig. 3). It follows from Eqs. (5) that this
quantity governs the atomic momentum. A stochastic map for the quantity $u$
(11) allowed to derive analytic expressions for probability density functions
of the atomic trapping and flight events that have been shown to fit well
numerical simulation PRA07 .
It has been shown that sudden changes in the behavior of $u$ take place when
we quantized the atomic motion (see Fig. 11) under the condition
$\Delta\simeq\Delta_{0}$. Those changes are more smooth than the jumps of $u$
in the semiclassical case because a delocalized wave packet crosses a node for
a finite time interval. The quantum analysis provides a clear reason for those
jumps at $\Delta\simeq\Delta_{0}$, namely, it is nonadiabatic transitions
between the quasienergy states ${|+\closeket}_{\Delta}$ and
${|-\closeket}_{\Delta}$ which occur when an atom crosses any node of the
standing wave. Those jumps are accompanied by splitting of wave packets at the
nodes. We may conclude that the proliferation of wave packets at the nodes of
the standing wave is a manifestation of classical chaotic transport of an atom
in an optical lattice that has been shown in Refs. JETP03 ; JRLR06 ; PRA07 to
take place in exactly the same ranges of initial conditions and control
parameters. In particular, the effect of simultaneous trapping of an atom in a
well of the optical potential and its flight in the same potential (Fig. 14)
is a quantum analogue of a chaotic walking of an atom shown in Fig. 2.
In conclusion we would like to discuss briefly the role of dissipation. We did
not take into account any losses in our treatment. Coherent evolution of the
atomic state in a near-resonant standing-wave laser field is interrupted by
spontaneous emission events at random moments of times. The semiclassical
Hamiltonian evolution between these events has been shown to be regular or
chaotic depending on the values of the detuning $\Delta$ and the initial
momentum $p_{0}$. We stress that dynamical chaos may happen without any noise
and any modulation of the lattice parameters. It is a specific kind of
dynamical instability in the fundamental interaction between the matter and
radiation.
Dissipative transport of spontaneously emitting atoms in a 1D standing-wave
laser field has been studied in detail in Ref. PRA08 in the regimes where the
underlying semiclassical Hamiltonian dynamics is regular and chaotic. A Monte
Carlo stochastic wavefunction method was applied to simulate semiclassically
the atomic dynamics with coupled internal and translational degrees of
freedom. It has been shown in numerical experiments and confirmed analytically
that chaotic atomic transport can take the form either of ballistic motion or
a random walking with specific statistical properties. The character of
spatial and momentum diffusion in the ballistic atomic transport was shown to
change abruptly in the atom-laser detuning regime where the Hamiltonian
dynamics is irregular in the sense of dynamical chaos. A clear correlation
between the behavior of the momentum diffusion coefficient and Hamiltonian
chaos probability has been found.
What one could expect if spontaneous emission would be taken into
consideration with our fully quantum equations of motion? Any act of
spontaneous emission interrupts a coherent evolution of an atom at a random
time moment and is accompanied by a momentum recoil and a sudden transition of
the atom into the ground state which is a superposition of the dressed states.
The coherent evolution starts again after that. A collapse of the atomic wave
function and a splitting of atomic wave packets are expected just after any
spontaneous emission event. That additional splitting of wave packets at
random time moments, besides of their proliferation at the nodes of a standing
wave at $\Delta\simeq\Delta_{0}$, can improve the quantum-classical
correspondence in the regime of Hamiltonian chaos.
I thank L. Konkov and M. Uleysky for their help in preparing some figures.
This work was supported by the Russian Foundation for Basic Research (project
no. 09-02-00358) and by the Program “Fundamental Problems of Nonlinear
Dynamics” of the Russian Academy of Sciences.
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|
arxiv-papers
| 2012-05-28T11:56:56 |
2024-09-04T02:49:31.295319
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1205.6087"
}
|
1205.6158
|
18 [2]footnote blue
Running Head: GROUP ANALYSIS OF SELF-ORGANIZING MAPS
Group Analysis of Self-organizing Maps based on Functional MRI using
Restricted Frechet Means
Arnaud P. Fournelab, Emanuelle Reynaudb, Michael J. Brammera,
Andrew Simmonsac, and Cedric E. Ginestetac.
aDepartment of Neuroimaging, Institute of Psychiatry, King’s College London,
UK, bLaboratoire d’Etude des Mécanismes Cognitifs (EMC), EA 3082, Université
Lumière Lyon II, France, cNational Institute of Health Research (NIHR)
Biomedical Research Centre for Mental Health.
### Acknowledgments
This work was supported by a fellowship and core funds from the UK National
Institute for Health Research (NIHR) Biomedical Research Centre for Mental
Health (BRC-MH) at the South London and Maudsley NHS Foundation Trust and
King’s College London. This work has also been funded by the Guy’s and St
Thomas’ Charitable Foundation as well as the South London and Maudsley
Trustees. APF has received financial support from the Region Rhône-Alpes and
the Université Lumière Lyon 2 through an Explora’Doc grant. We would also like
to thank three anonymous reviewers for their valuable inputs.
### Correspondence
Correspondence concerning this article should be sent to Cedric Ginestet at
the Centre for Neuroimaging Sciences, NIHR Biomedical Research Centre,
Institute of Psychiatry, Box P089, King’s College London, De Crespigny Park,
London, SE5 8AF, UK. Email may be sent to cedric.ginestet@kcl.ac.uk
###### Abstract
Studies of functional MRI data are increasingly concerned with the estimation
of differences in spatio-temporal networks across groups of subjects or
experimental conditions. Unsupervised clustering and independent component
analysis (ICA) have been used to identify such spatio-temporal networks. While
these approaches have been useful for estimating these networks at the
subject-level, comparisons over groups or experimental conditions require
further methodological development. In this paper, we tackle this problem by
showing how self-organizing maps (SOMs) can be compared within a Frechean
inferential framework. Here, we summarize the mean SOM in each group as a
Frechet mean with respect to a metric on the space of SOMs. The advantage of
this approach is twofold. Firstly, it allows the visualization of the mean SOM
in each experimental condition. Secondly, this Frechean approach permits one
to draw inference on group differences, using permutation of the group labels.
We consider the use of different distance functions, and introduce two
extensions of the classical sum of minimum distance (SMD) between two SOMs,
which take into account the spatio-temporal pattern of the fMRI data. The
validity of these methods is illustrated on synthetic data. Through these
simulations, we show that the three distance functions of interest behave as
expected, in the sense that the ones capturing temporal, spatial and spatio-
temporal aspects of the SOMs are more likely to reach significance under
simulated scenarios characterized by temporal, spatial and spatio-temporal
differences, respectively. In addition, a re-analysis of a classical
experiment on visually-triggered emotions demonstrates the usefulness of this
methodology. In this study, the multivariate functional patterns typical of
the subjects exposed to pleasant and unpleasant stimuli are found to be more
similar than the ones of the subjects exposed to emotionally neutral stimuli.
In this re-analysis, the group-level SOM output units with the smallest sample
Jaccard indices were compared with standard GLM group-specific $z$-score maps,
and provided considerable levels of agreement. Taken together, these results
indicate that our proposed methods can cast new light on existing data by
adopting a global analytical perspective on functional MRI paradigms.
KEYWORDS: Barycentre, Frechet Mean, fMRI, Group Comparison, Karcher mean,
Multivariate analysis, Self-Organizing Maps, Unsupervised Learning.
## Introduction
Self-organizing Maps (SOMs) were originally introduced by Teuvo Kohonen et al.
(2000). A SOM is an unsupervised artificial neural network that describes a
training data set as a (typically planar) layer of neurons or output units.
Each neuron learns to become a prototype for a number of input units, until
convergence of the algorithm. The resulting SOM therefore represents a
projection of the inputs into a two-dimensional grid. In this sense, SOMs can
be regarded as a dimension-reduction clustering algorithm. One of the main
advantages of this unsupervised method is that the relative position of the
neurons on the grid can be directly interpreted, in the sense that proximity
of two units on the map indicates similarity of the prototypal profiles of
these units.
SOMs have proved to be useful for data-driven analysis and have become popular
tools in the machine learning community (Tarca et al., 2007). In neuroimaging,
these methods have been successfully applied to the detection of fMRI response
patterns related to different cognitive tasks (Liao et al., 2008, Ngan et al.,
2002, Ngan and Hu, 1999, Wismüller et al., 2004). Since SOMs are non-
parametric unsupervised neural networks, they do not require the specification
of temporal signal profiles, such as haemodynamic response function or
anatomical regions of interest in order to generate meaningful summaries of
spatio-temporal patterns of brain activity. As a result, these methods have
also been used to identify variations in low-frequency functional connectivity
(Peltier et al., 2003). Statistically, however, observe that the absence of a
probabilistic model can also be a limitation, as this does not allow for a
formal evaluation of the goodness-of-fit of the method.
When used for clustering, the SOMs have the following main advantages.
Firstly, starting with a sufficient number of neurons, the SOM procedure is
able to identify features in the data even when these features are only
typical of a small number of input vectors. Secondly, the resulting layer of
neurons is arranged according to the similarity of these prototypes in the
original data space. This ‘topology-preserving’ property is generally not
available in other data-reduction techniques, such as independent component
analysis (ICA) or $k$-means clustering. This is one desirable property that
SOMs share with multidimensional scaling. This topological structure
facilitates the merging of nodes in order to form ‘superclusters’, which
provide a way to visualize and compare high-dimensional fMRI data sets.
Fischer and Hennig (1999), for instance, have demonstrated the specific
relevance of these advantages to the analysis of experimental fMRI data.
One of the outstanding questions in the application of SOMs to fMRI data is
whether one can summarize several subject-specific SOMs into a ‘mean map’,
which would pool information over several subjects. In addition, it may be of
interest to draw inference over group differences, by comparing the mean maps
of several groups of subjects. Here, the term group is used interchangeably
with the concept of experimental condition. Hence, two distinct groups need
not be composed of different subjects, but may only represent different sets
of measurements on the same individuals. One may, for instance, be interested
in extracting the SOM that summarizes functional brain activity during a
particular cognitive task; or in comparing the resting-state SOM signature of
schizophrenic patients with that of normal subjects. Few studies, however,
have tackled the problem of formally comparing two or more families of SOMs.
Although several authors have proposed distance functions on spaces of SOMs
(Kaski and Lagus, 1996, Deng, 2007, Kirt et al., 2007), to the best of our
knowledge, none of these researchers have attempted to draw statistical
inference on the basis of such comparisons. This lack of methodology
highlights a pressing need for developing new strategies that would permit the
extension of such multivariate methods from single-subject analysis to
multiple group comparison.
SOM group analysis can naturally be articulated within a Frechean statistical
framework. In 1948, Frechet introduced the concept of Frechet mean, sometimes
referred to as barycentre or Karcher mean in the context of Euclidean and
Riemannian geometry, respectively (Karcher, 1977). The Frechet mean extends
this concept to any metric space. This quantity is a generalization of the
traditional arithmetic mean, applied to abstract-valued random variables,
defined over a metric space. The definition of a generalized notion of the
arithmetic mean therefore solely relies on the specification of a metric on
the data space of interest. Once such a metric has been specified, the Frechet
mean is simply the element that minimizes a convex combination of the squared
distances from all the elements in the space of interest. Hence, we can
construct a metric space of SOMs by choosing a metric on that space, which
permits the comparison of any two given SOMs in that space. Note that, in that
context, the chosen pairwise distance function should be a proper metric in
the sense that it should satisfy the four metric axioms: (i) non-negativity,
(ii) coincidence, (iii) symmetry and (iv) the triangle inequality (see
Searcóid, 2007, for an introduction to metric spaces). In the sequel, we
consider the use of different distance functions on spaces of SOMs, which do
not satisfy the triangle inequality. Nonetheless, we will show that such
distance functions can easily be transformed into proper metrics, using a
straightforward manipulation (Mannila and Eiter, 1997). See appendix A. The
concept of the Frechet mean has proved to be useful in several domains of
applications, including image analysis (Thorstensen et al., 2009, Bigot and
Charlier, 2011), statistical shape analysis (Dryden and Mardia, 1998), and in
the study of phylogenetic trees (Balding et al., 2009).
In this paper, our purpose is to use the concept of the Frechet mean for
drawing statistical inference over several families of subject-specific SOMs.
We thus construct Frechean independent and paired-sample $t$-statistics, by
analogy with the classical treatment of real-valued random variables.
Statistical inference for these different tests are then drawn using
permutation of the group labels. In the paper at hand, these statistics will
be constructed using the restricted Frechet mean, which has been shown to have
desirable asymptotic properties (Sverdrup-Thygeson, 1981), but is more
convenient to use from a computational perspective. The restricted Frechet
mean is defined as the element in the sample space, which minimizes the
squared distances from all the elements in the sample. This formal approach to
group inference on families of subject-specific SOMs has the advantage of
allowing a direct representation of the mean SOM in each group, thereby
pooling together subject-specific information. In addition, the proposed
methods also allow to formally draw inference at the group-level in terms of
the chosen distance function.
The paper is organized as follows. In the next section, we give a general
introduction to SOMs, and how they are computed highlighting the specific
algorithm, which will be used throughout the rest of the paper. In a third
section, we describe our proposed Frechean framework for drawing inference on
several groups of SOMs. This strategy is entirely reliant on the choice of
metric for comparing two given SOMs, and we therefore dedicate a fourth
section to the description of several distance functions on spaces of SOMs,
which appear particularly well-suited for the analysis of fMRI data. These
methods are tested on synthetic data, under a range of different conditions in
section five, and on a real data set in section six. We close the paper by
discussing the potential usefulness of this statistical strategy with an
emphasis on the critical importance of the choice of the distance function.
## Self-Organizing Maps (SOMs)
We assume here that an fMRI data set is available, which consists of several
spatio-temporal volumes $\text{\bf{X}}_{i}$, with $i=1,\ldots,n_{j}$, for
$n_{j}$ subjects in the $j^{\text{th}}$ experimental group. Each
$\text{\bf{X}}_{i}$ is a $V\times T$ matrix, with $V$ voxels and $T$ time
points. In the sequel, it will be of interest to compare several families of
such volumes, such that $j=1,\ldots,J$, for $J$ experimental conditions. When
describing the SOM inference algorithms, however, we will focus on a single
subject-specific data set, X.
### Sequential algorithm
A SOM, denoted M, consists of $K$ output units or neurons arranged in a two-
dimensional rectangular grid of size $K$ where $K=k_{1}\times k_{2}$. For
convenience, we here assume that the grids of interest are square grids, such
that $k_{1}=k_{2}$. Thus, the units of a SOM will be indexed by
$k=1,\ldots,K$, where $k$ ‘reads’ the units in SOM from left to right and top
to bottom. Each entry in M is hence denoted by $\text{\bf{m}}_{k}$, and
corresponds to the coordinates of that unit in M. That is, $\text{\bf{m}}_{k}$
is a two-dimensional vector representing the position of $\text{\bf{m}}_{k}$
in M, such that, for instance, $\text{\bf{m}}_{1}=(1,1)$, and
$\text{\bf{m}}_{2}=(1,2)$, and so forth. Each output unit has an associated
weight vector $\text{\bf{w}}_{k}$, which is, in our case, a time series over
$T$ data points.
The sequential SOM algorithm takes a set of $V$ input units,
$\text{\bf{x}}_{v}$’s, corresponding to the rows of the input data X. The
steps of the procedure will be indexed by $\gamma=0,\ldots,\Gamma$, which
denote the iterations of the algorithm, and $\Gamma$ is the final step at
which a stopping condition is satisfied. In our case, the stopping rule is
simply the number of iterations, but more sophisticated convergence-based
criteria can be used. We firstly initialize the output units in M as random
draws from a uniform distribution on $\mathbb{R}^{T}$. Secondly, an input
vector, denoted $\text{\bf{x}}_{v}$, is randomly chosen amongst the $V$ time
series. All $V$ voxels are selected at each step of the algorithm, and these
input vectors are therefore dependent on $\gamma$. We will thus denote this
dependence on the iterations by $\text{\bf{x}}_{v}(\gamma)$. For each input
vector presented to M, we identify the unit in M, which is the ‘closest’ to
the input $\text{\bf{x}}_{v}(\gamma)$. Here, closeness is generally measured
in terms of Euclidean distance with respect to the values taken by the input
vectors. The unit in M, which is the closest to $\text{\bf{x}}_{v}(\gamma)$ is
referred to as the Best Matching Unit (BMU). The index of that BMU, for a
given input vector $\text{\bf{x}}_{v}$ at iteration $\gamma$, is defined as
follows,
$c(v,\gamma)=\operatornamewithlimits{argmin}_{k\in\\{1,\ldots,K\\}}\|\text{\bf{x}}_{v}(\gamma)-\text{\bf{w}}_{k}\|,$
(1)
with $\|\cdot\|$ denoting the Euclidean norm on $\mathbb{R}^{T}$. Here,
$\text{\bf{x}}_{v}(\gamma)$ and $\text{\bf{w}}_{k}$ are $T$-dimensional time
series. Thirdly, we update the BMU and its neighbors. The new values of these
units are defined as a linear relationship of the input vector
$\text{\bf{x}}_{v}(\gamma)$. For a given $\text{\bf{x}}_{v}(\gamma)$, the
updating rule for the BMU and its neighbors is the following,
$\text{\bf{w}}_{k}(\gamma+1)=\text{\bf{w}}_{k}(\gamma)+\alpha(\gamma)K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})\Big{(}\text{\bf{x}}_{v}(\gamma)-\text{\bf{w}}_{k}(\gamma)\Big{)},$
(2)
for every $k=1,\ldots,K$. After updating their weights, the BMU and its
neighbours are closer to $\text{\bf{x}}_{v}(\gamma)$ in the sense that they
constitute a better representation of that input vector. These steps are
repeated for a fixed number of iterations, $\Gamma$. The updating rule in
equation (2) contains two key parameters: (i) the learning rate, denoted
$\alpha(\gamma)$ and (ii) the kernel function represented by
$K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})$, which grows
smaller as we consider units in M, which are further away from the BMU in the
space of coordinates of M. We describe these two quantities in turn.
The learning rate, $\alpha(\gamma)$ in equation (2), is a decreasing function
of the number of iterations, $\gamma$, which controls the amount of learning
accomplished by the algorithm –that is, the dependence of the values of the
units in M on the inputs. By convention, we have $\alpha(\gamma)\in[0,1]$ for
every $\gamma$. Three common choices for $\alpha(\cdot)$ are a linear
function, a function inversely proportional to the number of iterations and a
power function, such as the following recursive definition,
$\alpha(\gamma+1)=\left(\frac{\alpha(0)}{\alpha(\gamma)}\right)^{\gamma/\Gamma},$
(3)
for every $\gamma=1,\ldots,\Gamma$. A popular initialization for the learning
rate is $\alpha(0)=0.1$ (Peltier et al., 2003, González and Dasgupta, 2003).
Clearly, as the algorithm progresses towards $\Gamma$, the value of
$\alpha(\gamma)$ decreases towards $0$. Note, however, that, in the paper at
hand, we use the batch version of this algorithm, which does not require the
specification of a learning rate.
In equation (2), we have also made use of the neighborhood kernel,
$K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})$. As for the
learning rate, the value taken by this kernel decreases with the number of
iterations, and is therefore dependent on $\gamma$. This dependence on
$\gamma$ has been emphasized through a subscript on $K$. For a given output
unit $c(v,\gamma)$ in the map M, the neighborhood kernel,
$K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})$, quantifies how
‘close’ is $\text{\bf{m}}_{k}$ to the BMU, which has index $c(v,\gamma)$.
Observe that this closeness is expressed in terms of Euclidean distances on
the grid coordinates. As commonly done in this field, we here choose a
standard Gaussian kernel to formalize the dependence of each unit on the
values of its neighbors, such that
$K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})=\exp\left(-\frac{\|\text{\bf{m}}_{k}-\text{\bf{m}}_{c(v,\gamma)}\|^{2}}{2\sigma(\gamma)^{2}}\right),$
(4)
where $\|\cdot\|^{2}$ represents the two-dimensional dot product. Here,
$\sigma(\gamma)$ is a linear function of the number of iterations, which
controls the size of the neighborhood around the BMU. This function is defined
recursively as $\sigma(\gamma+1)=\sigma(0)(1-\gamma/\Gamma)$, where
$\sigma(0)$ is a parameter value that represents the initial neighborhood
radius. This parameter is commonly initialized with respect to the size of the
two-dimensional grid, M, such that $\sigma(0)=k_{1}$, which is the ‘height’ of
the output SOM.
### Batch algorithm
A popular alternative to the sequential SOM algorithm described in the
previous section is the batch SOM algorithm, which has the advantage of being
more computationally efficient than its sequential counterpart (Vesanto and
Alhoniemi, 2000). It has been successfully used in the context of fMRI
analysis (Ngan et al., 2002), in natural language processing (Kohonen et al.,
2000), and in the face recognition literature (Tan et al., 2005). The main
difference between these two approaches is that the entire training set is
considered at once in the batch SOM algorithm, which permits the updating of
the target SOM with the net effect of all the inputs.
This ‘global’ updating is performed by replacing the input vector, denoted
$\text{\bf{x}}_{v}(\gamma)$ in the previous section, with a weighted average
of the input vectors, where the relative weight of each input vector is
proportional to the neighborhood kernel values. At the $\gamma^{\text{th}}$
step of the algorithm, we are therefore conducting the following global
updating,
$\text{\bf{w}}_{k}(\gamma+1)=\frac{\sum_{v=1}^{V}\text{\bf{x}}_{v}K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})}{\sum_{v=1}^{V}K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})},$
(5)
for every $k=1,\ldots,K$. It can easily be seen from equation (5) that
$\text{\bf{w}}_{k}(\gamma+1)$ is a convex linear combination of the input
vectors, $\text{\bf{x}}_{v}$’s, where each of the $V$ inputs is weighted by
$K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})/\sum_{v=1}^{V}K_{\gamma}(\text{\bf{m}}_{k},\text{\bf{m}}_{c(v,\gamma)})$,
and the sum of these weights is equal to 1. Another non-negligible advantage
of the batch SOM algorithm is that it removes the dependence of the outputs on
the learning rate parameter, denoted $\alpha(\gamma)$, as stated in the
previous section.
Throughout the rest of the paper, we will make use of the batch algorithm,
with $\sigma(0)=k_{1}$, and $k_{1}=k_{2}=3$, thereby producing SOMs of
dimensions $3\times 3$. Output units in all SOMs are initialized randomly.
Other groups of researchers in neuroimaging have used square SOMs (Peltier et
al., 2003, Liao et al., 2008). However, we have also investigated using
simulated data, whether the specification of rectangular maps had a
significant impact on our proposed inferential methods (see appendix B).
## SOM Group Frechean Inference
The question of inferring the statistical significance of the difference
between two families of SOMs can be addressed through the use of abstract-
valued random variables as advocated by Fréchet (1948). In this approach,
random variables are solely defined with respect to a probability measure on a
metric space, $(\mathcal{X},d)$ (see Parthasarathy, 1967, chap. 2). Hence, it
suffices to define a metric on the space of interest, in order to obtain a
valid statistical framework. Once such a metric has been chosen, one can
construct the mean element in that space, which is commonly referred to as the
Frechet mean.
In the paper at hand, we are considering a space of SOMs, which we may denote
by $(\mathcal{M},d)$, where $d$ is a metric on that space. A range of
different distance functions for such spaces of SOMs will be described in the
next section. As in most standard fMRI designs, we assume that we have $J$
experimental conditions, with $n_{j}$ subjects in each condition, thereby
allowing for a different number of subjects in each experimental condition. A
full data set will be summarized as an array of SOMs,
$\\{\text{\bf{M}}_{ij}\\}$, with $i=1,\ldots,n_{j}$ and $j=1,\ldots,J$, such
that $\text{\bf{M}}_{ij}$ corresponds to the SOM of the $i^{\text{th}}$
subject in the $j^{\text{th}}$ condition. Given such a sample of SOMs, we can
then define the Frechet mean for the $j^{\text{th}}$ condition as follows,
$\widehat{\text{\bf{M}}}_{j}=\operatornamewithlimits{argmin}_{\text{\bf{M}}^{\prime}\in\mathcal{M}}\frac{1}{(n_{j}-1)}\sum_{i=1}^{n_{j}}d(\text{\bf{M}}_{ij},\text{\bf{M}}^{\prime})^{2},$
(6)
where we have used the Bessel’s correction (i.e. $n_{j}-1$) by analogy with
the real-valued setting. Given the complexity of the underlying space of SOMs,
such a minimization may be unwieldy. As a result, it is computationally more
practical to consider the restricted Frechet mean, as introduced by Sverdrup-
Thygeson (1981). The classical Frechet mean in equation (6) is obtained by
identifying the element in the population of SOMs, which minimizes the average
squared distances from all the elements in the sample. The restricted Frechet
mean, by contrast, is obtained by identifying the element in the sample, which
has this property. Hence, the restricted Frechet mean is computed as follows,
$\overline{\text{\bf{M}}}_{j}=\operatornamewithlimits{argmin}_{\text{\bf{M}}^{\prime}\in\bm{\Lambda}_{j}}\frac{1}{(n_{j}-1)}\sum_{i=1}^{n_{j}}d(\text{\bf{M}}_{ij},\text{\bf{M}}^{\prime})^{2},$
(7)
where $\bm{\Lambda}_{j}$ denotes the sampled $n_{j}$ SOMs in the
$j^{\text{th}}$ condition, such that
$\bm{\Lambda}_{j}=\\{\text{\bf{M}}_{1,j},\ldots,\text{\bf{M}}_{n_{j},j}\\}$.
The restricted Frechet mean has been shown to be consistent, through a
generalization of the strong law of large numbers due to Sverdrup-Thygeson
(1981). Asymptotically, $\overline{\text{\bf{M}}}_{j}$ converges almost surely
to a subset of the theoretical restricted mean, which takes values in the
support of the target population distribution. In the sequel, the theoretical
restricted Frechet mean for the $j^{\text{th}}$ condition will be denoted by
$\mu_{j}$, following standard convention.
Similarly, one can define the condition-specific sample Frechet variances.
These quantities are simply the values taken by the criteria, which are
minimized in equation (7), such that the (restricted) Frechet variance for the
$j^{\text{th}}$ condition is defined with respect to the restricted Frechet
mean in the following manner, for every $j=1,\ldots,J$,
$S_{j}^{2}=\frac{1}{(n_{j}-1)}\sum_{i=1}^{n_{j}}d(\text{\bf{M}}_{ij},\overline{\text{\bf{M}}}_{j})^{2}.$
(8)
Using the restricted Frechet mean and variance, it is now possible to
construct a non-parametric $t$-test on the metric space of SOMs. Here, we
therefore assume that we solely have two experimental conditions, such that
$J=2$. The null hypothesis stating that the (restricted) Frechet means of
these two distributions are $\delta_{0}$-separated, can be formally expressed
as follows, $H_{0}:d(\mu_{1},\mu_{2})=\delta_{0}$. Naturally, our interest
will especially lie in testing the null hypothesis stating that there is no
difference between the theoretical restricted Frechet means, which corresponds
to $H_{0}:d(\mu_{1},\mu_{2})=0$. This can be tested using the following
Frechet $t$-statistic,
$t_{F}=\frac{d(\overline{\text{\bf{M}}}_{1},\overline{\text{\bf{M}}}_{2})-\delta_{0}}{S_{p}\left(1/n_{1}+1/n_{2}\right)^{1/2}},$
(9)
where the denominator, $S_{p}$, is the classical pooled sample variance, which
is defined by analogy with the real-valued setting as
$S^{2}_{p}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S^{2}_{2}}{n_{1}+n_{2}-2}.$
In addition, if one is considering two samples of equal sizes and assuming
equal Frechet variances, then the aforementioned $t_{F}$-statistic for such a
mean difference can be defined as follows
$t_{F}=\frac{d(\overline{\text{\bf{M}}}_{1},\overline{\text{\bf{M}}}_{2})}{S_{p}/\sqrt{N}},$
(10)
where, in this case, the pooled variance is simply the sum of the variances of
the two samples, such that $S^{2}_{p}=S^{2}_{1}+S^{2}_{2}$, and with
$N=n_{1}+n_{2}$. Statistical inference is then conducted using permutation on
the group labels. Although our proposed $t_{F}$-statistic is a real-valued
random variable, its asymptotic distribution is unknown. Indeed, the behavior
of this statistic depends on a large number of other random variables, which
are combined using the non-linear procedure for obtaining group-level SOMs. As
a result, the permutation-based distribution of $t_{F}$ under the null
hypothesis is not expected to follow a standard $t$-distribution. In
particular, the null distribution of $t_{F}$ need not be symmetric. Since we
are here solely considering a generalization of the $t$-test, but will be
applying this statistic to more than two experimental conditions, we will also
make use of the standard Bonferroni correction for multiple testing.
## Choice of Distance Functions
In our proposed approach to group comparison, the choice of the metric on the
space of SOMs is paramount. Different distance functions capture different
aspects of the SOMs under scrutiny. It is therefore of interest to evaluate
group differences with respect to several choices of distance functions. We
here review the main distance functions, which have been previously proposed
in the literature for comparing two given SOMs. In addition, we introduce a
spatio-temporal sum of minimum distances, which is especially relevant for the
study of fMRI-based SOMs.
### Quantization Error and Other Measures
This measure is not a metric, but a popular tool for evaluating the accuracy
of the SOM generated from a given data set. The so-called quantization error
measures the average quantization error of the target SOM (Kohonen, 2001). It
is defined as the sum of the Euclidean distances between each input unit,
$\text{\bf{x}}_{v}$, and its best matching prototype on M –that is, the BMU of
$\text{\bf{x}}_{v}$. The quantization error, denoted $Q_{e}$, is thus formally
defined as follows,
$Q_{e}(\text{\bf{M}},\text{\bf{X}})=\sum_{v=1}^{V}\|\text{\bf{x}}_{v}-\text{\bf{w}}_{c(v)}\|,$
where, as before, $\|\cdot\|$ denotes the $T$-dimensional Euclidean norm and
$c(v)$ is the index of the BMU in M with respect to $\text{\bf{x}}_{v}$ as
described in equation (1). This measure is a good indicator of the convergence
of a SOM, and is often used when assessing the behavior of the algorithms
described in the previous section. In this paper, we will use a variant of the
quantization error in order to identify the output units, which explain the
largest amount of between-subject ‘variance’ in the data. However, the
quantization error does not allow the computation of the distance between two
given SOMs.
Kaski and Lagus (1996) have proposed a measure of dissimilarity between two
SOMs. They proceeded by comparing the shortest path on each SOM after matching
a given pair of input vectors. This dissimilarity measure is computed by
comparing the distances between all pairs of data samples on the feature maps.
This method, however, is not computationally efficient, and would be
especially challenging when considering fMRI data sets, where neuroscientists
are commonly handling about 100,000 input vectors –that is, the voxel-specific
time series– for every subject.
### Sum of Minimum Distances (T-SMD)
The Sum of Minimum Distances (SMD) was originally introduced by Mannila and
Eiter (1997) and has been widely used in image recognition and retrieval
(Kriegel, 2004, Takala et al., 2005, Tungaraza et al., 2009). Moreover, the
SMD function and some of its variants have already been used in order to
tackle the problem of comparing several SOMs (Deng, 2007). Given two SOMs,
denoted $\text{\bf{M}}_{x}$ and $\text{\bf{M}}_{y}$ for input data sets X and
Y, respectively, the SMD can be computed as follows. For every unit,
$\text{\bf{w}}_{x}$ in $\text{\bf{M}}_{x}$, we calculate the Euclidean
distance between $\text{\bf{w}}_{x}$ and every unit $\text{\bf{w}}_{y}$ in
$\text{\bf{M}}_{y}$ in order to retain the unit in $\text{\bf{M}}_{y}$ that
minimizes this distance. These minimal distances are summed and then
normalized by the total number of input vectors, denoted $V$, in our case.
This gives an $\text{\bf{M}}_{x}$-to-$\text{\bf{M}}_{y}$ score. The same
procedure is performed in the opposite direction in order to produce an
$\text{\bf{M}}_{y}$-to-$\text{\bf{M}}_{x}$ score. The average of the
$\text{\bf{M}}_{x}$-to-$\text{\bf{M}}_{y}$ and the
$\text{\bf{M}}_{y}$-to-$\text{\bf{M}}_{x}$ scores is then defined as the
overall SMD between $\text{\bf{M}}_{x}$ and $\text{\bf{M}}_{y}$. Therefore,
this distance function compares SOMs on the basis of the dissimilarity of the
time series underlying each output unit. It follows that this procedure mainly
emphasizes temporal differences between the fMRI volumes of interest. Thus, we
will label this classical SMD as temporal SMD, and denote it by T-SMD. It is
formally defined as
$\operatorname{T-SMD}(\text{\bf{M}}_{x},\text{\bf{M}}_{y})=\frac{1}{2V}\Biggl{(}\sum_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}\min_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}d_{e}(\text{\bf{w}}_{x},\text{\bf{w}}_{y})+\sum_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}\min_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}d_{e}(\text{\bf{w}}_{y},\text{\bf{w}}_{x})\Biggr{)},$
(11)
where
$d_{e}(\text{\bf{w}}_{x},\text{\bf{w}}_{y})=\|\text{\bf{w}}_{x}-\text{\bf{w}}_{y}\|$
is the Euclidean distance between $\text{\bf{w}}_{x}$ and $\text{\bf{w}}_{y}$
on $\mathbb{R}^{T}$, where $\text{\bf{w}}_{x}$ and $\text{\bf{w}}_{y}$
represent $T$-dimensional prototypal time series for maps $\text{\bf{M}}_{x}$
and $\text{\bf{M}}_{y}$, respectively.
It is important to note that the SMD function can be re-written by treating a
map, $\text{\bf{M}}_{x}$, as a set of weight vectors, $\text{\bf{w}}_{x}$. In
this case, we consider the metric space of all weight vectors,
$\text{\bf{w}}_{x}$. This metric space is $(\mathbb{R}^{T},d_{e})$. By a
slight abuse of notation, the SOM, $\text{\bf{M}}_{x}$, will be used to denote
the set of all output vectors, $\text{\bf{w}}_{x}$ associated with the units
in $\text{\bf{M}}_{x}$. Therefore, we have
$\text{\bf{M}}_{x}\subset\mathbb{R}^{T}$. As a result, we can apply the
classical definition of the distance between the subset of a metric space and
an element of that space, $\text{\bf{w}}_{x}\in\mathbb{R}^{T}$, such that
$d(\text{\bf{w}}_{x},\text{\bf{M}}_{x})=\min\\{d(\text{\bf{w}}_{x},\text{\bf{w}}^{\prime}_{x}):\text{\bf{w}}^{\prime}_{x}\in\text{\bf{M}}_{x}\\}$.
Using these conventions, it becomes possible to reformulate the SMD function
in equation (11) in the following manner as stated by Mannila and Eiter
(1997),
$\operatorname{T-SMD}(\text{\bf{M}}_{x},\text{\bf{M}}_{y})=\frac{1}{2V}\Biggl{(}\sum_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}d_{e}(\text{\bf{w}}_{x},\text{\bf{M}}_{y})+\sum_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}d_{e}(\text{\bf{w}}_{y},\text{\bf{M}}_{x})\Biggr{)}.$
In addition, observe that the SMD function is not in general a proper metric,
in the sense that the triangle inequality may fail to be satisfied (see
Mannila and Eiter, 1997, for a counterexample). However, one can easily
produce a proper metric through the identification of the shortest paths
between any two elements in the space of interest, and then define a new
metric with respect to these shortest paths (see appendix A). It can easily be
shown that such a transformation necessarily produces proper metrics, when
considering metrics based on the SMD function (Mannila and Eiter, 1997). This
particular procedure can easily be implemented in our case, because we have
focused our attention on the restricted Frechet mean, where the minimization
required to identify the mean element is solely conducted over the space of
the sampled elements. As a result, there exists a small number of possible
shortest paths between every pair of elements in the sample, which greatly
facilitates the required transformation for producing a proper metric. This
procedure was systematically conducted in the sequel, and therefore all the
variants of the SMD function utilized in this paper are indeed proper metrics.
We will thus assume throughout this paper that all distance functions have
been adequately transformed. We now introduce two novel variants of the SMD
function, which take into account the spatial and spatio-temporal properties
of the fMRI data.
### Spatial SMD
One may also be interested in quantifying the amount of ‘spatial overlap’
between two given SOMs. This question is especially pertinent when analyzing
SOMs based on fMRI data sets. Here, we therefore wish to evaluate whether the
units in two different maps correspond to similar subsets of voxels in the
original images. Such a distance can be quantified through a slight
modification of the aforementioned SMD metric, where the Hamming distance is
used in the place of the Euclidean distance.
$\displaystyle\operatorname{S-SMD}(\text{\bf{M}}_{x},\text{\bf{M}}_{y})=\frac{1}{2V}\Biggl{(}\sum_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}$
$\displaystyle\min_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}\operatorname{Ham}(S(\text{\bf{w}}_{x}),S(\text{\bf{w}}_{y}))$
(12)
$\displaystyle+\sum_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}\min_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}\operatorname{Ham}(S(\text{\bf{w}}_{y}),S(\text{\bf{w}}_{x}))\Biggr{)}.$
with $V$ denoting the number of voxels in the fMRI volumes of interest, and
where $S(\text{\bf{w}}_{x})$ denotes the binarized index vector of the voxels,
which have been assigned to unit $\text{\bf{w}}_{x}$ in $\text{\bf{M}}_{x}$.
That is, if the voxel $v$ has been assigned to $\text{\bf{w}}_{x}$, then
$S_{v}(\text{\bf{w}}_{x})=1$, otherwise, we have $S_{v}(\text{\bf{w}}_{x})=0$.
In addition, we have here made use of the celebrated Hamming distance, which
takes the following form (Hamming, 1950),
$\operatorname{Ham}\Big{(}S(\text{\bf{w}}_{x}),S(\text{\bf{w}}_{y})\Big{)}=\frac{1}{V}\sum_{v=1}^{V}\mathcal{I}\Big{\\{}S_{v}(\text{\bf{w}}_{x})=S_{v}(\text{\bf{w}}_{y})\\}\Big{\\}},$
(13)
where $\mathcal{I}\\{f(x)\\}$ stands for the indicator function taking a value
of $1$ if $f(x)$ is true, and $0$ otherwise. Here, the term spatial refers to
the spatial distribution of the voxels allocated to a particular output unit.
Hence, S-SMD does not emphasize the spatial location of the output units, as
these allocations are arbitrary, but rather the spatial distribution of the
voxels allocated to that output unit. Observe that we have here solely
considered differences in the spatial distributions of the best matched pair
of SOM units, where that matching is done through the minimization reported in
equation (12). This approach, however, omits to take into account the
similarity of these SOM units as prototypal time series. Both the spatial and
the temporal aspects of these maps can nonetheless be combined, as described
in our proposed spatio-temporal SMD.
### Spatio-Temporal SMD
In this novel variant of the classical SMD, we are quantifying the amount of
spatial overlap between any pair of output units in two distinct maps. In
contrast to the S-SMD described in the previous section, however, we are here
comparing the spatial distributions (i.e. the sets of voxel indexes assigned
to a particular unit) of the units that are the closest in terms of time
series profiles, thereby combining the temporal and spatial properties of the
data. This spatio-temporal version of the SMD function is defined as follows,
$\operatorname{ST-
SMD}(\text{\bf{M}}_{x},\text{\bf{M}}_{y})=\frac{1}{2}\Biggl{(}\sum_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}\operatorname{Ham}(S(\text{\bf{w}}_{x}),S(\text{\bf{w}}_{y}^{\ast}))+\sum_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}\operatorname{Ham}(S(\text{\bf{w}}_{y}),S(\text{\bf{w}}^{\ast}_{x}))\Biggr{)},$
(14)
where
$\text{\bf{w}}_{y}^{\ast}=\operatornamewithlimits{argmin}_{\text{\bf{w}}_{y}\in\text{\bf{M}}_{y}}d_{e}(\text{\bf{w}}_{x},\text{\bf{w}}_{y}),\qquad\text{and}\qquad\text{\bf{w}}_{x}^{\ast}=\operatornamewithlimits{argmin}_{\text{\bf{w}}_{x}\in\text{\bf{M}}_{x}}d_{e}(\text{\bf{w}}_{y},\text{\bf{w}}_{x});$
where, again,
$d_{e}(\text{\bf{w}}_{x},\text{\bf{w}}_{y})=\|\text{\bf{w}}_{x}-\text{\bf{w}}_{y}\|$
is the Euclidean distance on $\mathbb{R}^{T}$, and $S(\text{\bf{w}}_{x})$ is
the index set of the voxels in X, whose time series are best represented by
$\text{\bf{w}}_{x}$. Albeit the formulae in equation (14) is somewhat
convoluted, it corresponds to, perhaps, the most intuitive perspective on the
problem of comparing SOMs, when using fMRI data. Indeed, we are quantifying
the amount of spatial overlap between units, which are as similar as possible
in terms of their temporal profiles.
We summarize this section with a concise description of these three different
types of SOM distance functions:
1. i.
Temporal SMD (T-SMD) is based on the sum of the minimum Euclidean distances
between the time series of the output units.
2. ii.
Spatial SMD (S-SMD) is based the sum of the minimum Hamming distances between
the sets of voxels allocated to the output units.
3. iii.
Spatio-temporal SMD (ST-SMD) is based on the sum of Hamming distances between
the sets of voxels allocated to the output units, which are the most similar
in terms of their time series.
## Synthetic Data Simulations
The proposed methods were tested on three different simulated scenarios, with
varying degrees of difficulty. In particular, by isolating different types of
differences between the two groups of interest, each of these scenarios
emphasizes the need for the use of specific metrics, capturing different
aspects of the spatio-temporal process under investigation. In this sense, our
simulations strive to produce realistic differences between the two families
of fMRI volumes, where group differences may be either spatial, temporal or
both spatial and temporal.
Figure 1: Description of the three simulated scenarios ordered by increasing
levels of difficulty. In panels (a-c), we have reported the spatial
distributions of the input vectors for each scenario, where each data set is
composed of $(10\times 10)$-images over $T=50$ time points. In panels (d-f),
we have represented the three types of time series used in these simulations.
Here, SC1, SC2 and SC3 correspond to the three different scenarios, where the
two groups exhibit spatio-temporal differences (SC1), temporal differences
(SC2), and spatial differences (SC3), respectively.
### Simulation Scenarios
For each simulated data set we have constructed two groups of 20 subjects,
where each subject-specific data set is composed of two-dimensional images
with $10\times 10$ voxels, over 50 time points, as represented in panels (a-c)
of figure 1. The time series at each voxel can be of three different types, as
illustrated in panels (d-f) of figure 1, composed of two different signals and
one background time series. The two signals represented in panels (d) and (e)
are sinusoids oscillating over $[-1,1]$, with a frequency of $1/10$Hz and
$1/20$Hz, respectively. We have then added a vector of Gaussian random noise,
z, to these two types of time series, such that $z_{t}\sim N(0,\sigma^{2})$,
for every $t=1,\ldots,50$, and for different choices of $\sigma$. The
background noise time series, in panel (f), is solely composed of the random
noise for a given $\sigma$.
The three scenarios in panels (a-c) of figure 1 are ordered in terms of the
degree of ‘separability’ of the two groups, where the easiest scenario is on
the left and the most difficult one is on the right. The first data set (SC1)
was built with three different time series at three different locations,
corresponding to the purple, blue and gray colors. In this scenario, the
groups differ both in terms of the temporal profiles of some of their voxels
and in terms of the spatial locations of these voxels. The second scenario
(SC2) was constructed with two different types of time series. Here, the two
groups solely differ in terms of the temporal profiles of some of their
voxels. Finally, in the third scenario (SC3), the two groups only differ in
terms of the spatial locations of the voxels, which have been assigned the
second temporal profile.
In addition, we also studied the effect of the signal-to-noise ratio (SNR) on
the performance of our inferential methods. In particular, we varied our
choice of $\sigma$, when generating the different time series displayed in
panels (d-f) of figure 1, in order to produce different SNRs. In these
simulations, the ‘signal’ of interest was defined as the amplitude of the
original sinusoids, which oscillated between -1 and 1, thus giving an
amplitude of $\lambda=2$. Since the noise affecting this signal was specified
to be Gaussian, the SNR was defined, in our case, as
$\operatorname{SNR}=\lambda/2\sigma$. Thus, by setting $\sigma$ to either
$1/2$, $1$ or $2$, we produced three different SNRs of $2$, $1$ and $1/2$,
respectively.
These synthetic data sets were analyzed using our proposed inferential
framework. For each simulated subject-specific volume, a SOM was computed, and
the restricted Frechet mean was identified for each group. In all scenarios,
SOMs were produced by using the batch SOM algorithm. The output grid was of
size $3\times 3$ with $K=9$; the number of iterations was set to $100$ steps;
and we used a decreasing neighborhood kernel of size $k_{1}=3$, as commonly
done in this field (Kohonen et al., 2000). For computational convenience,
statistical inference was drawn in each scenario after 100 permutations. Each
simulated scenario was reproduced 100 times, and constructed for the three
different levels of SNR, thereby totalling $900$ distinct simulations.
Scenarios and Factors | T-SMD | S-SMD | ST-SMD
---|---|---|---
SC1 (Spatio-temporal) | $\operatorname{SNR}=2$ | $0\pm 0$ | $0.012\pm 0.033$ | $0\pm 0$
| $\operatorname{SNR}=1$ | $0\pm 0.001$ | $0.518\pm 0.171$ | $0.003\pm 0.012$
| $\operatorname{SNR}=0.5$ | $0.030\pm 0.066$ | $0.800\pm 0.235$ | $0.049\pm 0.123$
SC2 (Temporal) | $\operatorname{SNR}=2$ | $0\pm 0$ | $0.499\pm 0.303$ | $0\pm 0.006$
| $\operatorname{SNR}=1$ | $0\pm 0$ | $0.499\pm 0.295$ | $0.001\pm 0.005$
| $\operatorname{SNR}=0.5$ | $0.017\pm 0.070$ | $0.484\pm 0.296$ | $0.022\pm 0.030$
SC3 (Spatial) | $\operatorname{SNR}=2$ | $0.472\pm 0.294$ | $0.014\pm 0.057$ | $0.029\pm 0.055$
| $\operatorname{SNR}=1$ | $0.464\pm 0.286$ | $0.525\pm 0.167$ | $0.109\pm 0.122$
| $\operatorname{SNR}=0.5$ | $0.525\pm 0.279$ | $0.783\pm 0.271$ | $0.101\pm 0.141$
Table 1: Significance levels based on synthetic data with 100 simulations in
every cell, with the mean $p$-value and the standard deviation for that
distribution of $p$-values. These results are reported for the three scenarios
described in figure 1, which are denoted by SC1, SC2 and SC3, for three
different levels of SNR, and for the three different distance functions under
scrutiny, denoted by T-SMD, S-SMD and ST-SMD, which stand for the temporal
SMD, spatial SMD, and spatio-temporal SMD, respectively.
### Simulation Results
The summary results of the analysis of these synthetic data sets are reported
in table 1 and figure 2. Overall, the different metrics of interest were found
to successfully capture the aspects of the simulated SOMs that they were
expected to identify. That is, in the first column of table 1, one can see
that T-SMD, which solely takes into account the differences in voxel-specific
temporal profiles, attains its most significant values under the temporal
scenario, SC2. Similarly, in the second column of table 1, the spatial version
of the SMD metric, denoted S-SMD exhibits its best performance under the first
and third scenarios, denoted SC1 and SC3, respectively. Indeed, these two
scenarios are the only ones, where the two groups can be discriminated in
terms of the spatial locations of the different types of time series. Finally,
in the third column of table 1, the spatio-temporal metric, ST-SMD, appears to
be optimal under the first scenario, where group differences can be
characterized through the spatio-temporal properties of the simulated images.
In addition, we have also evaluated the effect of sample size on the capacity
of the metrics to detect group differences. These results are not reported in
this paper, but we have observed, as expected, that the statistical power of
all the studied metrics improved as the number of subjects in each group
increases. In particular, it was noted that for the ST-SMD, we solely needed
$n\geq 15$ in each group to identify significant differences under SNR=1, and
group sizes of $n\geq 20$ under SNR=0.5; for all scenarios. Exemplary null
distributions for $t_{F}$-statistics in the three different scenarios and the
three different levels of SNR with $n=15$ are reported in figure 2.
In sum, one can note that these three distance functions exhibit different
levels of robustness. In particular, S-SMD appeared to be especially sensitive
to noise. Although S-SMD succeeded to capture the spatial differences
simulated in SC3, it only outperformed ST-SMD for high SNRs. Also, in the
spatio-temporal scenario (SC1), T-SMD behaved as well or better than ST-SMD.
This suggests that the T-SMD function is more ‘powerful’ than the ST-SMD, even
when the group differences are characterized by both spatial and temporal
properties. However, the use of ST-SMD remains justified, because it also
succeeds to capture spatial differences, whereas T-SMD fails to do so. In
general, we therefore recommend the joint use of T-SMD and ST-SMD: If only
T-SMD indicates the presence of group differences, then one can conclude that
such differences are mainly of a temporal nature; whereas when the use of ST-
SMD indicates greater group differences, this suggests that such differences
also have a spatial character. Overall, these simulations highlight the
importance of using several types of distance functions, as there may not
exist a single type of metric, which would capture all of the aspects of the
data of interest.
Figure 2: Histograms of the null distributions of $t_{F}$-values obtained
through permutation. These null distributions are given for a single synthetic
data set under the three different simulation scenarios, denoted SC1, SC2 and
SC3, respectively, and for three different metrics on the space of SOMs,
denoted T-SMD, S-SMD and ST-SMD, which stand for sums of minimum distances,
spatial T-SMD and spatio-temporal T-SMD, respectively. The red dashed line
indicates the value of the actual $t_{F}$-statistic for the simulation of
interest. These histograms were constructed using data based on an
$\operatorname{SNR}$ of 1, and for $15$ subjects in each group.
## Experimental Data
We also evaluated our methods with the re-analysis of a classical data set,
originally published by Mourao-Miranda et al. (2006). Since this first
publication, this data set has been re-analyzed several times with different
machine learning algorithms, as conducted by Mourao-Miranda et al. (2007)
using spatio-temporal support vector machine (SVM), and Hardoon et al. (2007)
with unsupervised methods.
### Subjects and Experimental Design
This data set consists of fMRI data from 16 right-handed males with a mean age
of 23 years. All participants had normal eyesight and no history of
neurological or psychiatric disorders, and gave written informed consent to
participate in the study, in accordance with the local ethics committee of the
University of North Carolina (see Mourao-Miranda et al., 2006). Data were
acquired using an experimental block design, composed of three different
conditions: (i) exposure to unpleasant visual stimuli (i.e. photos of
dermatological diseases), (ii) exposure to neutral visual stimuli (i.e. photos
of neutral day-to-day scenes including human actors) and (iii) exposure to
male-relevant pleasant visual stimuli (i.e. scantly dressed women or women in
swimsuits). The entire experimental design consisted of six blocks, where each
block contained seven images, which were each presented to the subjects for
three seconds. Each block was followed by a resting block period where
subjects were solely exposed to a fixation cross. All blocks were of 21$s$ in
length.
Metrics | Tests | $p$-values
---|---|---
T-SMD | Pleasant vs. Neutral | 0.0
| Unpleasant vs. Neutral | 0.0
| Pleasant vs. Unpleasant | 0.258
S-SMD | Pleasant vs. Neutral | 0.412
| Unpleasant vs. Neutral | 0.518
| Pleasant vs. Unpleasant | 0.423
ST-SMD | Pleasant vs. Neutral | 0.021
| Unpleasant vs. Neutral | 0.013
| Pleasant vs. Unpleasant | 0.128
Table 2: Significance results of all pairwise comparisons for the three
conditions of interest, where subjects were exposed to pleasant, unpleasant
and neutral stimuli. These results are reported independently for three
different metrics, denoted T-SMD, and S-SMD and ST-SMD, which stand for
temporal SMD, spatial SMD and spatio-temporal SMD, respectively. Observe that
since three tests were conducted for each pair of conditions and we therefore
necessitate the application of a Bonferroni correction for testing for these
three pairwise comparisons. Hence, only $p$-values below $0.016$ should be
regarded as statistically significant.
### Data Acquisition and Pre-Processing
Blood-oxygenation-level-dependent (BOLD) signal was measured using a 3-Tesla
Allegra head-only MRI System at the Magnetic Resonance Imaging Research Center
in the University of North Carolina. The scanning parameters were specified as
follows. Voxel size was $3\times 3\times 3mm^{3}$, TR was $3s$, TE was $30ms$,
FA was $80$, FOV was $192\times 192mm$ and each MRI volume had dimensions
$64\times 64\times 49$. In each experiment, a total of $254$ functional
volumes were acquired for each participant.
Data were pre-processed using the FSL Software suite (Smith et al., 2004a);
through the use of the Nipype Python Library (Gorgolewski et al., 2011). All
fMRI volumes were first motion-corrected and the skulls were removed, after
tissue segmentation. The voxel time series were detrended and filtered in time
and space: that is, low-frequency (drift) fluctuations were reduced using a
band-pass temporal filter comprised between 0.008Hz and 0.1Hz. The use of a
such a band-pass filter is typical of resting-state connectivity analysis
(Cordes et al., 2001). In addition, spatial smoothing was performed using an
8mm full-width at half-maximum Gaussian kernel.
The first two volumes of each block were discarded, to allow for the between-
block lag in haemodynamic response. The remaining volumes were concatenated to
form three distinct time series representing the three different conditions.
Time series concatenation in the context of functional connectivity has been
introduced by Fair et al. (2007) and has been implemented by several authors
for the study of functional MRI networks (Ginestet and Simmons, 2011, Ginestet
et al., 2011).
| Sample Jaccard Index
---|---
Ranked SOM Units | Neutral | Pleasant | Unpleasant
Units up to 1 | 0.68 | 0.32 | 0.55
Units up to 2 | 0.11 | 0.28 | 0.16
Units up to 3 | 0.08 | 0.04 | 0.03
Units up to 4 | 0.01 | 0.07 | 0.07
Units up to 5 | 0.03 | 0.08 | 0.03
Units up to 6 | 0.01 | 0.05 | 0.04
Units up to 7 | 0.05 | 0.08 | 0.05
Units up to 8 | 0.01 | 0.02 | 0.05
Units up to 9 | 0.02 | 0.06 | 0.02
Table 3: Individual percentage overlaps between the group-level SOM components
and thresholded GLM $z$-score maps, where the SOM components have been ranked
with respect to the sample Jaccard index. In bold, we have highlighted the
three ‘best’ condition-specific SOM output units based on T-SMD, which are
visually described in figures 3 to 5. Each column sums to 1.0, since the
concatenated SOM output units cover all the voxels in the fMRI volumes of
interest.
### Results of Real-data Analysis
The significance results of all the pairwise comparisons are reported in table
2, after applying the Bonferroni correction for multiple testing. Our re-
analysis of this data set has highlighted a substantial degree of difference
between the neutral and pleasant conditions, on one hand, and between the
neutral and unpleasant conditions, on the other hand. These pairwise
differences were found to be highly significant under the T-SMD distance
function. The mean SOM in the unpleasant condition was also found to be
significantly different from the mean SOM in the neutral condition with
respect to the ST-SMD metric ($p<0.016$), albeit to a lesser extent than under
the T-SMD function. The difference between the mean SOMs in the pleasant
condition and the neutral one was found to be just above significance level
under the ST-SMD metric ($p=0.063$). By contrast, none of these differences
reached significance under the S-SMD, indicating that differences in spatial
allocation of the different output units of the SOMs in these experimental
conditions were not sufficient to distinguish between the group-level SOMs.
Importantly, the mean SOMs under the pleasant and unpleasant conditions were
not found to be significantly different under all three metrics.
As noted in the analysis of the synthetic data, the fact that the SOMs are
computed on the basis of the similarities between the voxel-specific time
series is likely to be responsible for the important differences that we are
reporting between the metrics capturing the temporal aspects of the process
and S-SMD, which does not emphasize differences in temporal profiles.
Intriguingly, it should also be noted that the differences between the SOMs in
the pleasant and unpleasant conditions under ST-SMD is lower than under the
T-SMD, thereby perhaps suggesting that the differences between these two
conditions is characterized by an ‘interaction’ between the temporal and
spatial properties of these functional patterns.
Figure 3: Representation of three output units of the restricted Frechet mean
SOM for the unpleasant condition in red, with thresholded ($p\leq 0.05$) GLM
$z$-score maps in blue. The output units have been projected in MNI-normalized
space. These three output units are the ones that explain the largest amount
of sample Jaccard index in that SOM. They are ordered by Jaccard index from
panels (a) to (c), with the unit exhibiting the smallest Jaccard index in (a).
Figure 4: Representation of three output units of the restricted Frechet mean
SOM for the unpleasant condition in red, with thresholded ($p\leq 0.05$) GLM
$z$-score maps in blue. The output units have been projected in MNI-normalized
space. These three output units are the ones that explain the largest amount
of sample Jaccard index in that SOM. They are ordered by Jaccard index from
panels (a) to (c), with the unit exhibiting the smallest Jaccard index in (a).
Figure 5: Representation of three output units of the restricted Frechet mean
SOM for the unpleasant condition in red, with thresholded ($p\leq 0.05$) GLM
$z$-score maps in blue. The output units have been projected in MNI-normalized
space. These three output units are the ones that explain the largest amount
of sample Jaccard index in that SOM. They are ordered by Jaccard index from
panels (a) to (c), with the unit exhibiting the smallest Jaccard index in (a).
### Visualization of Group-level SOMs
In each of the three conditions, we have represented the subject-specific
restricted Frechet mean in order to produce robust visual summaries of the
different output units in each condition. This was conducted by identifying
the output units that explained the largest amount of ‘variance’ in the data,
in terms of sample Jaccard index. This measure quantifies how representative
is a mean output unit in terms of the overlap of this unit with the output
units of all the subjects in that group.
For each experimental condition, we identified the output unit in the subject-
specific SOMs that explained the largest amount of Jaccard index. For each
experimental condition, we have plotted in figures 3, 4 and 5, the three
output units that are associated with the least amount of Jaccard index over
all subjects. That is, for the $j^{\text{th}}$ condition, the sample Jaccard
index of the $k^{\text{th}}$ output unit in the restricted Frechet mean for
that condition was defined as follows,
$J(\text{\bf{m}}_{kj})=\frac{1}{n_{j}}\sum_{i=1}^{n_{j}}\text{\bf{J}}^{k}\left(\text{\bf{M}}_{j},\text{\bf{X}}_{ij}\right),$
Here,
$\text{\bf{J}}^{k}(\text{\bf{M}},\text{\bf{X}})=\sum_{v=1}^{V}\operatorname{Jacc}\\{S(\text{\bf{x}}_{v}),S(\text{\bf{w}}_{k})\\}$
denotes the global Jaccard distance between a mean SOM and a subject-specific
SOM. Moreover, the classical vector-specific Jaccard distance is
$\operatorname{Jacc}\\{\text{\bf{x}},\text{\bf{y}}\\}=\frac{C_{10}+C_{01}}{C_{11}+C_{01}+C_{10}},$
where $C_{01}$ is the number of elements satisfying $x_{h}=1$ and $y_{h}=0$,
for $h$ running from $1$ to the length of these vectors; and $C_{10}$ and
$C_{11}$ are defined similarly. The three output units minimizing the sample
Jaccard index in each of the three experimental conditions have been plotted
in figures 3, 4 and 5, for the neutral, pleasant, and unpleasant conditions,
respectively. Allocation of a voxel to the particular output unit of a SOM is
‘hard’, in the sense, that either a voxel is included into an output unit or
it is included in a different one. Thus, in figures 3, 4 and 5, we have
provided the spatial location of the voxels that have been assigned to
particular output units in each condition.
From these three figures, one can observe that the neutral condition is
characterized by a distinct network of brain regions identified as the third
output unit in the neutral condition, as can be seen from panel (c) of figure
3. The set of regions associated with this particular output unit may be
interpreted as a visual network, since it contains a considerable number of
regions located in the occipital lobe. This particular output unit was not
found to be present in the three units with the least Jaccard index in either
the pleasant or unpleasant condition, as can be noted from figures 4 and 5,
respectively.
### Comparison with Standard GLM Maps
The group-level SOM output units selected using the sample Jaccard index were
compared with standard general linear model (GLM) $z$-score maps. Separate GLM
analyses were conducted for each experimental condition, using the FEAT
function provided in the FSL software suite (Smith et al., 2004b). The
$z$-score maps were thresholded at $p=0.05$, for comparison purposes. These
binarized $z$-score maps were compared with the maps of the three ‘best’
group-level output units, selected on the basis of their sample Jaccard
indices, as described in the previous section. These thresholded GLM maps have
been overlaid in blue over the selected output units in figures 3, 4 and 5.
In each experimental condition, we computed the percentage overlap between the
maps obtained using these two different techniques. That is, in each
condition, we evaluated how many voxels were present in both the thresholded
$z$-score maps and each output unit, normalized by the number of voxels
included in the $z$-score maps. These numerical comparisons are reported in
table 3, where we have described the individual percentage overlap of the nine
different output units, ranked with respect to their sample Jaccard indices.
The combined percentage overlap of the three output units exhibiting the
lowest sample Jaccard indices with the thresholded GLM $z$-score maps was
$87\%$, $64\%$, and $74\%$ for the neutral, pleasant and unpleasant
conditions, respectively. Although our proposed SOM-based methods summarize
such fMRI volumes in a non-linear manner, these numerical comparisons show
that the resulting output units exhibit a considerable degree of agreement
with standard GLM analysis.
## Discussion
### Advantages of Proposed Methods
The main contribution of this paper is the construction of an inferential
framework for the comparison of group-level SOMs. Although some previous
researchers have considered various ways of comparing SOMs (Kaski and Lagus,
1996, Deng, 2007, Kirt et al., 2007), to the best of our knowledge, no authors
have yet treated the problem of evaluating the statistical significance of
such group differences. In addition, observe that our Frechean approach to
statistical inference can be conducted for any choice of metrics. Indeed, the
idea of defining a group distance statistic, such as the Frechean $t$-test in
equation (9) and then evaluating its significance using permutation can be
implemented for any choice of distance functions. Therefore, this allows the
specification of a rich array of different distance functions capturing
different aspects of the SOMs under scrutiny. As illustrated in the main body
of the paper, we have shown that classical distance functions such as the SMD
can be modified in order to emphasize spatial, temporal or spatio-temporal
differences between the groups of interest. Here, the choice of hypothesis is
therefore superseded by the choice of metrics over the space of SOMs. In
particular, this inferential framework allows to test hitherto untestable
group-level hypotheses.
Another substantial advantage of combining a Frechean approach with the
computation of subject-specific SOMs is that this bypasses the problem of
multiple testing correction. In standard mass-univariate analyses of MRI
volumes, one needs to control for the inflation of the number of false
positives introduced by performing a large number of non-independent
statistical tests. By contrast, we are here conducting a single test, which
identifies whether the volumes of interest are different at a multivariate
level, through the comparison of two non-parametric unsupervised
representations of the original data.
Finally, one should also note that the use of the restricted Frechet mean in
our proposed framework is advantageous for several reasons. On the one hand,
the restricted Frechet mean greatly reduces the computational cost of our
overall analytical procedure. This is especially true, because inference was
drawn using permutations of the group labels, and it is not clear whether such
a large number of permutations would have been possible, lest for the use of
the restricted Frechet mean. On the other hand, the restricted Frechet mean
has also the advantage of quasi-automatically transforming any distance
function into a proper metric that satisfies the four metric axioms. This
results in a non-negligible simplification of the probabilistic theory needed
to justify our inferential approach. Indeed, most of the asymptotic results,
which have been previously established relies on the postulate that the
distance function of interest is a proper metric (Ziezold, 1977, Sverdrup-
Thygeson, 1981).
### Limitations of the Frechean Framework
One can identify three substantial limitations to our proposed Frechean
inferential framework for SOMs, which are (i) a lack of contrast maps, (ii) a
reliance on permutation for statistical inference, and (iii) the use of the
restricted Frechet mean in the place of the unrestricted mean element in the
space of all SOMs. We address these three limitations in turn.
Firstly, one of the important limitations of our current method is that it
does not directly permit the production of a ‘group-difference SOM’
representing the difference between two group-specific Frechet means. In
particular, this implies that we cannot represent such differences by plotting
a differential pattern of activation or connectivity, as is commonly done
using standard mass-univariate approaches. See the statistical parametric
network (SPN) approach, advocated by Ginestet and Simmons (2011) for example,
when conducting functional network analyses. From a neuroscientific
perspective, this is a considerable limitation, as it diminishes the
interpretability of the results. We will consider different ways of tackling
this issue and producing group-difference SOMs in future work.
Secondly, our inferential framework has relied on permutation for evaluating
the statistical significance of the test statistics under scrutiny. This was
made computationally feasible, because this choice of inferential method was
used in conjunction with the restricted Frechet mean. That is, for each
permutation of the group labels, the computation of the group-specific Frechet
means was straightforward, because the identification of the restricted
Frechet mean can be conducted by using the margins of the dissimilarity matrix
of the sample points –that is, the dissimilarity matrix of all the subject-
specific SOMs. Hence, the cost of calculating the group means at each
permutation was small, and the full inference could be drawn within a couple
of hours on a standard desktop computer. Such a level of computational
efficiency may not have been achieved if we had attempted to derive the
unrestricted Frechet mean, as described in equation (6), which would have
necessitated to perform a minimization over the space of all possible SOMs.
However, although the use of the restricted Frechet mean was advantageous from
a computational perspective, this particular methodological choice has also
its limitations. Indeed, the use of the restricted Frechet mean in the place
of the unrestricted mean results in a loss of the classical benefits usually
associated with computing an average of real numbers. In decision-theoretic
parlance and when considering real-valued random variables, the arithmetic
mean is the quantity that minimizes the squared error loss (SEL) (see Berger,
1980, for an introduction to decision theory). The restricted version of the
arithmetic mean for real-valued random variables would also minimize the
restricted SEL. However, the restricted arithmetic mean would necessarily
achieve a sample variance greater or equal to the one of the unrestricted
frechet mean. Note, however, that the problems associated with the utilization
of the restricted Frechet mean are also mitigated by the fact that computing
this quantity quasi-automatically makes the space of interest a metric space,
regardless of the particular choice of distance function.
We have here used the sample Jaccard index for selecting the most ‘relevant’
output units in the group-level SOMs. It certainly does not follow from such a
selection criterion that these output units are of greater physiological
relevance. This criterion is entirely statistical, and the resulting
interpretation of these output units should remain statistical. In practice,
it is advisable to visualize the entire set of output units obtained after
this type of SOM analysis, in order to identify relevant physiological
differences based on prior neuroanatomical knowledge.
### Possible Extensions of these Methods
Our proposed Frechean inferential framework could be extended in a range of
different directions. One of the most natural extensions of this method would
be to devise an $F$-test, which would generalize the aforementioned two-sample
$t_{F}$-statistic. A Frechet $F$-statistic may take the following form. Let a
data set of the form $\text{\bf{M}}_{ij}\in(\mathcal{M},d)$, where
$i=1,\ldots,n_{j}$ labels the objects in the $j^{\text{th}}$ group with
$j=1,\ldots,J$. By analogy with the classical real-valued setting, the
$F$-statistic can be defined as the ratio of the between-group to within-group
variances, $F_{F}=\operatorname{SS}_{1}/\operatorname{SS}_{0}$, where these
quantities are here defined with respect to the Frechet moments, such that
$\operatorname{SS}_{1}=(J-1)^{-1}\sum_{j=1}^{J}n_{j}d(\overline{\text{\bf{M}}}_{j},\overline{\overline{\text{\bf{M}}}})^{2}$,
and
$\operatorname{SS}_{0}=(N-J)^{-1}\sum_{j=1}^{J}\sum_{i=1}^{n_{j}}d(\text{\bf{M}}_{ij},\overline{\text{\bf{M}}}_{j})^{2}$,
using standard notation for the Frechet sample group means,
$\overline{\text{\bf{M}}}_{j}$, and grand mean,
$\overline{\overline{\text{\bf{M}}}}$. One can then test for the null
hypothesis that $H_{0}:\sigma^{2}_{1}=\sigma^{2}_{0}$, where $\sigma^{2}_{1}$
and $\sigma^{2}_{0}$ are the theoretical equivalents of
$\operatorname{SS}_{1}$ and $\operatorname{SS}_{0}$, respectively. Statistical
inference can, again, be conducted using permutation of the group labels.
In addition, the analytical strategy that we have here described could also be
improved through the use of different types of SOM algorithm. In the present
paper, we have made use of the batch SOM algorithm. However, several other
alternatives to the traditional sequential SOM algorithm have been proposed in
the literature. In particular, Vesanto and Alhoniemi (2000) have showed that
the SOMs obtained when using the batch SOM algorithm with an initialization of
the maps based on the eigenvectors of the input data can produce more robust
results. Since every SOM is computed independently for each subject, such an
improvement of the existing batch SOM algorithm could easily be incorporated
in our proposed inferential framework.
One of the outstanding questions that is implicitly raised in this paper is
the possibility of separately weighting the individual contributions of the
spatial and temporal properties contributing to the overall SOM difference.
Such a question is likely to be arduous to answer, however, since the temporal
and spatial properties of the fMRI volumes of interest necessarily live in
distinct abstract spaces. On one hand, the temporal differences in T-SMD were
quantified using a Euclidean distance in a $T$-dimensional vector space;
whereas, on the other hand, the spatial differences in S-SMD were quantified
using the Hamming distance on binary vectors of varying sizes. It is unclear
whether the magnitude of the distances in these different metric spaces could
be normalized in order to ensure a modicum of comparability.
## Conclusions
In this paper, we have described a formal framework for drawing group-level
inference between unsupervised multivariate summaries of fMRI data. Our
proposed approach proceeds by computing subject-specific SOMs, and computing
the sample Frechet mean in each group of subjects. Despite the unwieldy nature
of the space of all possible SOMs, this can be done efficiently by identifying
the restricted Frechet mean. Statistical inference on the difference between
the group restricted Frechet means can be conducted using permutation on the
group labels. This framework can be implemented for any choice of metrics
quantifying the difference between pairs of SOMs. As such, the specification
of a particular distance function is equivalent to the choice of a particular
hypothesis test. Different researchers may therefore be interested in
evaluating different metrics, which capture different aspects of the SOMs.
We have hence described and evaluated several types of distance functions for
SOMs based on fMRI data. In particular, we have considered variants of the
classic SMD function, which has previously been used to compare pairs of SOMs.
Our proposed variants distinguish between the temporal, spatial and spatio-
temporal properties of the data under scrutiny. Our inferential framework and
these metrics were tested on both synthetic and real data, Our analysis of the
simulated data showed that the distance functions of interest were indeed
capturing the aspects of the data that they were purported to measure. In
addition, the findings of the re-analysis of an fMRI experiment has
demonstrated the capacity of these methods to extract new information from
existing data sets. In this paradigm, the differences of the restricted mean
SOMs in the pleasant and unpleasant conditions were found to be smaller than
the differences between the mean SOMs in any of these two conditions with
respect to the one in the neutral condition.
Taken together, the analyses of these synthetic and real data sets have
underlined the robustness and potential usefulness of these methods. It is
hoped that this type of global inferential perspective on neuroimaging data
will inspire other neuroscientists to follow this research avenue. One could
imagine a range of other subject-specific abstract-valued random variables
that could be suitably analyzed using this type of Frechet inferential
framework. In fact, the very use of mass-univariate approaches in the context
of neuroimaging could be superseded by a more global perspective, where a
single statistical test is conducted, thereby bypassing the need for exacting
multiple testing penalties.
## Appendices
### A. From Distance Functions to Metrics
Let $d$ be a distance function on a finite space of SOMs,
$\bm{\Lambda}=\\{\text{\bf{M}}_{1},\ldots,\text{\bf{M}}_{n}\\}$, which
satisfies the positivity, coincidence and symmetry axioms. In order to
transform the distance function $d$ into a proper metric $\widetilde{d}$
satisfying the triangle inequality, we need to construct a saturated graph
$G=(V,E)$ representing the topology of $\bm{\Lambda}$. The vertex set of $G$
is defined as $V(G)=\bm{\Lambda}$. Its edge set is composed of all the
possible links between the elements of $\bm{\Lambda}$. That is, $G$ is a
saturated graph, in the sense that it contains the maximal number of edges.
Each of these edges is denoted by $\text{\bf{M}}_{i}\text{\bf{M}}_{j}\in
E(G)$, for any $0\leq i\neq j\leq n$.
A path in $G$ is a non-empty subgraph $P\subseteq G$ of the form
$V(P)=\\{\text{\bf{M}}_{0},\ldots,\text{\bf{M}}_{k}\\}$ and
$E(P)=\\{\text{\bf{M}}_{0}\text{\bf{M}}_{1},\ldots,\text{\bf{M}}_{k-1}\text{\bf{M}}_{k}\\}$,
where the $\text{\bf{M}}_{i}$’s are all distinct. Following an idea proposed
by Mannila and Eiter (1997), it is now possible to construct a new distance
function, denoted $\widetilde{d}$, defined as the set of shortest paths in
$\bm{\Lambda}$, such that for any
$\text{\bf{M}},\text{\bf{M}}^{\prime}\in\bm{\Lambda}$, we have
$\widetilde{d}(\text{\bf{M}},\text{\bf{M}}^{\prime})=\min_{P\in\mathcal{P}(\text{\bf{M}},\text{\bf{M}}^{\prime})}\sum_{\text{\bf{M}}_{i}\text{\bf{M}}_{j}\in
E(P)}d(\text{\bf{M}}_{i},\text{\bf{M}}_{j}),$
where $\mathcal{P}(\text{\bf{M}},\text{\bf{M}}^{\prime})$ is the set of all
paths in $G$ between M and $\text{\bf{M}}^{\prime}$. By construction, it
immediately follows that $\widetilde{d}$ satisfies the triangle inequality.
Therefore, $(\bm{\Lambda},d)$ forms a proper metric space.
### B. Choice of SOM Dimensions
A supplemental set of simulations was conducted in order to investigate the
effect of the choice of SOM dimensions on group-level statistical inference.
We assessed the effect of rectangular SOMs, as well as the effect of
increasing the dimensions of these maps. The synthetic data used for these
simulations followed the design described in the section entitled Synthetic
Data Simulations, based on the three different scenarios, and using the three
types of SMD functions described in this paper, and setting
$\operatorname{SNR}=1$. These results are reported in table 4.
The results of these simulations were consistent with the ones described in
our first analysis of these synthetic data. In particular, for any choice of
SOM dimensions, we obtained strong corroborations of the previous findings.
Under both SC1 and SC2, the T-SMD tended to outperform its counterparts for
any choice of SOM dimensions. As before, S-SMD performed poorly throughout
these simulations, irrespective of the choice of SOM dimensions. Finally, the
ST-SMD function exhibited good performance on all scenarios, and outperformed
the T-SMD under SC3, although ST-SMD did not reach significance level for this
particular scenario.
Scenarios | SOM Dimensions | T-SMD | S-SMD | ST-SMD
---|---|---|---|---
SC1 (Spatio-temporal) | $10\times 10$ | $0\pm 0$ | $0.626\pm 0.236$ | $0.002\pm 0.064$
| $5\times 5$ | $0\pm 0$ | $0.498\pm 0.256$ | $0.001\pm 0.031$
| $4\times 6$ | $0\pm 0$ | $0.516\pm 0.285$ | $0.001\pm 0.012$
| $6\times 8$ | $0\pm 0$ | $0.464\pm 0.279$ | $0.001\pm 0.078$
SC2 (Temporal) | $10\times 10$ | $0\pm 0$ | $0.612\pm 0.350$ | $0\pm 0$
| $5\times 5$ | $0\pm 0$ | $0.557\pm 0.298$ | $0.011\pm 0.014$
| $4\times 6$ | $0\pm 0$ | $0.474\pm 0.288$ | $0.002\pm 0.012$
| $6\times 8$ | $0\pm 0.006$ | $0.487\pm 0.269$ | $0.003\pm 0.097$
SC3 (Spatial) | $10\times 10$ | $0.505\pm 0.296$ | $0.523\pm 0.282$ | $0.180\pm 0.171$
| $5\times 5$ | $0.519\pm 0.206$ | $0.504\pm 0.279$ | $0.149\pm 0.144$
| $4\times 6$ | $0.482\pm 0.272$ | $0.559\pm 0.268$ | $0.108\pm 0.162$
| $6\times 8$ | $0.482\pm 0.300$ | $0.451\pm 0.281$ | $0.149\pm 0.103$
Table 4: Simulation results with varying SOM dimensions summarized as mean
significance levels and standard deviations of these distributions, based on
synthetic data with 100 simulations in every cell and $\operatorname{SNR}=1$.
These results are reported for the three scenarios described in figure 1,
which are denoted by SC1, SC2 and SC3, for three different levels of SNR, and
for the three different distance functions under scrutiny, denoted by T-SMD,
S-SMD and ST-SMD, which stand for the temporal SMD, spatial SMD, and spatio-
temporal SMD, respectively. These results are consistent with the ones of
table 1.
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|
arxiv-papers
| 2012-05-28T16:53:38 |
2024-09-04T02:49:31.308878
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arnaud P. Fournel, Emanuelle Reynaud, Michael J. Brammer, Andrew\n Simmons and Cedric E. Ginestet",
"submitter": "Cedric Ginestet",
"url": "https://arxiv.org/abs/1205.6158"
}
|
1205.6338
|
# Sterile Neutrino Sensitivity with Wrong-Sign Muon Appearance at $\nu$STORM
C.D. Tunnell
IDS-NF-035
Figure 1: A diagram of the proposed $\nu$STORM facility.
Neutrinos from STORed Muons111The facility was previously called the Very Low
Energy Neutrino Factory (VLENF). ($\nu$STORM) is a proposed experiment that
uses 3.8 GeV/c muon decay to produce a well-understood beam of electron and
muon neutrinos that can be used for short baseline physics (Fig. 1). A
magnetized far detector allows for the wrong-sign muon appearance physics of
$\nu_{e}\to\nu_{\mu}$ and provides more sensitivity to sterile neutrinos than
other proposals (See comparisons in [1]) because of the relative ease to which
muon tracks can be identified. Other physics such as $\nu_{e}$ and $\nu_{\mu}$
cross section measurements are possible. For further details, see the Letter
of Intent [16].
An explanation of the $\nu$STORM appearance analysis will follow. This work is
a continuation of the work presented in [20]. For disappearance measurement
work, see [21].
## 1 Short Baseline Oscillations
LEP experiments revealed that there are three light neutrinos that couple to
the $Z$-boson (ie. _active neutrinos_), however, there are theoretical and
experimental motivations [1] for neutrinos without Standard Model interactions
called _sterile_ neutrinos. The (3+1) scenario is the case of three active
neutrinos with an additional heavy sterile neutrino –
$m_{4}>>m_{\text{others}}$ – and only this situation is considered although
the results are generalizable.
The probability $\nu_{e}\to\nu_{\mu}$ depends on the mixing matrix $U$
(Reviewed in [5]). Let $R_{ij}$ be a rotation between the $i$-th and $j$-th
mass eigenstates without a CP violating phase: CP violation cannot be observed
in oscillations with large $\Delta m^{2}$ dominance (See p.g. 273 of [7]). For
$N$ neutrinos, $R_{ij}$ has dimension $N\times N$ and takes the form:
$\displaystyle R_{ij}=\begin{pmatrix}1&\ldots&0&\ldots&0&\ldots&0\\\
\vdots&&\vdots&&\vdots&&\vdots\\\
0&\ldots&\cos\theta_{ij}&\ldots&\sin\theta_{ij}&\ldots&0\\\
\vdots&&\vdots&&\vdots&&\vdots\\\
0&\ldots&-\sin\theta_{ij}&\ldots&\cos\theta_{ij}&\ldots&0\\\
\vdots&&\vdots&&\vdots&&\vdots\\\ 0&\ldots&0&\ldots&0&\ldots&1\\\
\end{pmatrix}.$ (1)
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”) implying
$U_{e4}=\sin(\theta_{14})$ and $U_{\mu 4}=\sin(\theta_{24})\cos(\theta_{14})$.
The oscillation probabilities for appearance and disappearance, respectively,
are:
$\displaystyle\text{P}_{\nu_{e}\to\nu_{\mu}}$ $\displaystyle=$ $\displaystyle
4|U_{e4}|^{2}|U_{\mu 4}|^{2}\sin^{2}\left(\frac{\Delta
m^{2}_{41}L}{4E}\right)$ (2) $\displaystyle=$
$\displaystyle\sin^{2}(2\theta_{e\mu})\sin^{2}\left(\frac{\Delta
m^{2}_{41}L}{4E}\right),$ (3)
$\displaystyle\text{P}_{\nu_{\alpha}\to\nu_{\alpha}}$ $\displaystyle=$
$\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).$ (4)
in this short baseline limit where the definition
$\sin^{2}(2\theta_{e\mu})=4|U_{e4}|^{2}|U_{\mu 4}|^{2}$ has been introduced.
Electron and muon neutrino disappearance measurements will constrain
$|U_{e4}|^{2}$ ([21]) and $|U_{\mu 4}|^{2}$ while the appearance channel
analysis could measure the product $|U_{e4}|^{2}|U_{\mu 4}|^{2}$. Information
about the matrix element $U_{e4}$ also arises from jointly analyzing
$\bar{\nu}_{\mu}$ disappearance and $\nu_{\mu}$ appearance. The remaining
matrix element $U_{\tau 4}$ can be extracted by analyzing NC rates
$|U_{s4}|^{2}=\sum_{e,\mu,\tau}|U_{\alpha 4}|^{2}$, using the other channels
to constrain $|U_{e4}|^{2}$ and $|U_{\mu 4}|^{2}$, and assuming unitarity.
## 2 The Neutrino Flux: $\Phi$
Table 1: Matrix elements for muon decay | $f_{0}(x)$ | $f_{1}(x)$
---|---|---
$\nu_{\mu}$ | $2x^{2}(3-2x)$ | $2x^{2}(1-2x)$
$\nu_{e}$ | $12x^{2}(1-x)$ | $12x^{2}(1-x)$
Muon-decay beams contrast pion-decay beams because the beam characteristics
and production mechanisms are well-known. The neutrino flux arises from the
Electroweak decay of $\mu\to\nu_{\mu}\bar{\nu}_{e}e$ and it is sufficient to
compute matrix elements at tree level. The neutrino spectrum for a
$\mu^{\pm}\to e^{\pm}+\nu_{e}(\bar{\nu}_{e})+\bar{\nu_{\mu}}(\nu_{\mu})$ decay
in the rest frame of the muon follows:
$\displaystyle\frac{\operatorname{d}n}{\operatorname{d}x\operatorname{d}\Omega}=\frac{1}{4\pi}\left[f_{0}(x)\mp\mathcal{P}f_{1}(x)\cos\theta\right]$
(5)
where $x=2E^{\text{c.o.m.}}_{\nu}/m_{\mu}\in[0,1]$ is the scaled neutrino
energy in the rest frame, $\Omega$ is the solid angle in the rest frame,
$f_{0}(x)$ and $f_{1}(x)$ are muon decay parameters, and $\mathcal{P}$ is the
polarization. Electron and neutrino masses are negligible for this process and
ignored, hence the inclusive range for values of $x$. These muon decay
paramters can be computed to leading order with Electroweak theory (See, for
example, chapter 6 of Ref. [18]) and are neutrino flavor dependent (See Table
1).
Figure 2: The flux of $\nu_{e}$ and $\bar{\nu}_{\mu}$ for a 3.8GeV/c muon-
decay without oscillations at 2000 meters. No smearing due to accelerator
effects has been performed.
The polarization $\mathcal{P}$ is set to zero, similar to other studies, and
has been shown to average to zero due to Thomas Precession. Boosting the
neutrino distributions into the lab frame leads to:
$\displaystyle\frac{\operatorname{d}^{2}N_{\mu}}{\operatorname{d}y\operatorname{d}A}$
$\displaystyle=$ $\displaystyle\frac{4n_{\mu}}{\pi
L^{2}m^{6}_{\mu}}E^{4}_{\mu}y^{2}(1-\beta\cos\phi)\left[3m^{2}_{\mu}-4E^{2}_{\mu}y(1-\beta\cos\phi)\right]$
(6)
$\displaystyle\frac{\operatorname{d}^{2}N_{e}}{\operatorname{d}y\operatorname{d}A}$
$\displaystyle=$ $\displaystyle\frac{24n_{\mu}}{\pi
L^{2}m^{6}_{\mu}}E^{4}_{\mu}y^{2}(1-\beta\cos\phi)\left[m^{2}_{\mu}-2E^{2}_{\mu}y(1-\beta\cos\phi)\right]$
(7)
where $y=E_{\nu}/E_{\mu}$ is the scaled neutrino energy in the lab frame,
$\beta=\sqrt{1-m^{2}_{\mu}/E^{2}_{\mu}}$, $A$ is an area, and $n_{\mu}$ is the
number of muons. These neutrino distributions (Fig. 2) are for a point source
so they are not directly applicable to the decay straight of $\nu$STORM.
The number of muons assumed is $1.8\times 10^{18}$ and is based on $10^{21}$
protons on target (POT) at 60 GeV/c. It corresponds to roughly 5 years of
running with a 100 kW target station. The number of useful muon decays is
motivated in [16].
Figure 3: The unoscillated flux of $\nu_{e}$ and $\bar{\nu}_{\mu}$ for a
$(3.8\pm 0.38)\text{ GeV/c}$ muon-decay at 2000 meters. Accelerator effects
are included; see the text for details. Figure 4: The flux at the far detector
for a $(3.8\pm 0.38)\text{ GeV/c}$ muon for initial $\nu_{e}$ states including
integration over the beam envelope and detector volume. Final states include
$\nu_{e}$ without oscillations and both $\nu_{e}$ and $\bar{\nu}_{\mu}$ with
best fit short baseline oscillations. The normalization is $10^{21}$ POT.
When computing the flux for $\nu$STORM, the far detector approximation of a
point-source accelerator and detector no longer is applicable since the size
of the detector and accelerator straight (150 meters) are comparable to the
baseline of 2000 meters. The neutrino fluxes are computed by integrating over
the decay straight, transverse beam phase space, and detector volume. The beam
occupies a 6D phase space ($x$, $y$, $z$, $p_{x}$, $p_{y}$, $p_{z}$) and the
detector has a $5\text{ m}\times 5\text{ m}$ cross section with the depth set
by the desired fiducial mass of 1.3 kt. Both transverse 2D phase spaces are
represented by the Twiss parameters $\alpha=0$ and $\beta=40\text{ m}$ where
the $1\sigma$ Gaussian geometric emittance is assumed to be $2.1\text{ mm}$.
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\in[0,150\text{ m}]$ – accurate to 0.5% – and
$p_{z}\in[3.8\pm 0.38\text{ GeV/c}]$.
The flux is computed by Monte Carlo (MC) integration: random points are chosen
within the beam phase space and within the detector volume to determine the
expected flux. This integration introduces a new computational requirement:
the baseline is a variable that affects both the oscillation probability
($L/E$) and the flux ($L^{-2}$ geometric factor). The GLoBES software (version
3.1.10) [12, 11] that is used for neutrino factory phenomenology treats these
as separable problems and was modified to compute this flux (and later the
event rates and sensitivities). Specifically, GLoBES is modified such that
both the flux and oscillation probability are computed in the _oscillation
probability engine_. The code for the analysis is available [19] under the GPL
license [9].
The resulting flux after the integration (Fig. 3) is corrected for accelerator
effects. The corrections are small for far detector physics (Compare to Fig.
2) but are important for near detector physics where the baseline is smaller
than the decay straight.
## 3 The Oscillation Probability: (Prob.)
Figure 5: The oscillation probability for the “golden channel”
$\nu_{e}\to\nu_{\mu}$ from Eq. 2 using the (3+1) oscillation parameters in
TABLE 2. A baseline of 2000 meters is assumed.
This section will discuss how sterile oscillation phenomenology relates to
conducting the proposed experiment. For instance, for a point-source baseline
of 2000 meters, it is possible to determine the oscillation probability (Fig.
5) using Eq. 2 for any combination of $L$ and $E$.
Table 2: Best-fit oscillation parameters for the (3+1) sterile neutrino scenario using combined MB $\bar{\nu}$ and LSND $\bar{\nu}$ data [8]. Parameter | Value
---|---
$\Delta m^{2}_{41}$ [$\text{eV}^{2}$] | 0.89
$|U_{e4}|^{2}$ | 0.025
$|U_{\mu 4}|^{2}$ | 0.023
Table 3: Values for $3\times 3$ oscillations used. $\sin^{2}\theta_{12}=0.319$
---
$\sin^{2}\theta_{23}=0.462$
$\sin^{2}\theta_{13}=0.010$
$\Delta m^{2}_{21}=7.59\times 10^{-5}\text{ eV}^{2}$
$\Delta m^{2}_{31}=2.46\times 10^{-3}\text{ eV}^{2}$
The best fit parameters for the “short baseline anomaly” and $3\times 3$
mixing (_i.e._ $\sin^{2}(2\theta_{13})$, $\Delta m^{2}_{12}$, etc.) are used
throughout the analysis. The best fit parameters for the LSND anomaly come
from [8] (See TABLE 2) and agree with those published by the LSND
collaboration [2]. For completeness, oscillations between known mass
eigenstates are included despite not influencing the sensitivity: the
correction is order $10^{-5}$. The best fit data from [10] is used to specify
standard $3\times 3$ oscillations. Without loss of generality, normal
hierarchy is assumed and the values of known $3\times 3$ mixing can be seen in
Table 3. Errors associated with these quantities are ignored.
Computationally, the SNU (version 1.1) add-on [13, 14] has been used to extend
computations in GLoBES to $4\times 4$ mixing matrices.
## 4 Cross section: $\sigma$
(a) CC
(b) NC
Figure 6: Neutrino cross sections per nucleon.
Cross sections are required for each neutrino flavor ($\nu_{\mu}$,
$\bar{\nu}_{\mu}$, $\nu_{e}$, $\bar{\nu}_{e}$) and each interaction type (CC
or NC). The nucleon cross sections (Fig. 6) are calculated in [15] and [17]
for the low energy and high energies, respectively. NC cross sections are
flavor independent. The CC cross sections are approximately flavor
independent: _Fermi’s Second Golden Rule_ results in the same matrix elements
and, at these energies, the phase spaces for the final-state electrons and
muons are equal.
The total cross section requires knowing the number of nucleons in addition to
the nucleon cross section. The fiducial mass of 1.3 kt determines the number
of nucleons via Avogadro’s number.
## 5 Interaction rates: $N_{\text{int.}}$
(a) Appearance with stored $\mu^{+}$
(b) Appearance with stored $\mu^{-}$
(c) Disappearance with stored $\mu^{+}$
(d) Disappearance with stored $\mu^{-}$
Figure 7: True channel rate energy distributions assuming the LSND anomaly
best fit values. The transitions $\nu_{e}\to\nu_{\mu}$,
$\bar{\nu}_{e}\to\bar{\nu}_{\mu}$, $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$, and
$\nu_{\mu}\to\nu_{\mu}$ are shown.
The number of neutrino interactions is computed which does not require
assumptions about the detector. The interaction rates can be computed by
$N_{\text{int.}}=\Phi\times\text{(Prob.)}\times\sigma$, for flux $\Phi$,
oscillation probability $(\text{Prob.})$, and cross section $\sigma$, where
all of these quantities have been computed in the previous sections.
Using the LSND anomaly best fit (TABLE 2) as a example for a sterile neutrino
signal, the event rates for $\mu^{+}$ and $\mu^{-}$ decays are shown in TABLE
4. Various deductions can be made about these event rates and their
statistical significance. With either stored $\mu^{+}$s or stored $\mu^{-}$s,
the statistical significance of all channels is greater than $10\sigma$.
Combining the NC channels together results in a statistical significance of
$20\sigma$ and $17\sigma$ for stored $\mu^{+}$ and $\mu^{-}$, respectively.
There are no known physics backgrounds to neither $\nu_{e}\to\nu_{m}u$ CC nor
$\bar{nu}_{e}\to\bar{\nu}_{m}u$ CC interactions except to negligible solar-
term oscillations, so the backgrounds will arise from how well the detector
can differentiate these interactions.
The number of events can also be determined as a function of energy since the
evolution of $\rho$, $\sigma$, and $(\text{Prob.})$ as a function of energy is
known. These distributions are shown in Fig. 7.
There are numerous channels with reach into the sterile neutrino parameter
space. Most other experiments have one channel to explore (See [1] for list of
experiments), whereas in the best case $\nu$STORM allows for 10 signals and in
the worst case 6 (i.e. combine $\nu_{e}\to\nu_{e}$ CC and all NC channels).
Table 4: Truth event rates for $10^{21}$ POT for the no oscillations and short
baseline oscillations described by TABLE 2. The statistical significances are
computed. The combined statistical significance of NC events are 20 and 17 for
stored $\mu^{+}$ and $\mu^{-}$, respectively. There are no physics backgrounds
to $\nu_{e}\to\nu_{m}u$ CC interactions.
Channel | $N_{\textrm{osc.}}$ | $N_{\textrm{null}}$ | Diff. | $(N_{\textrm{osc.}}-N_{\textrm{null}})/\sqrt{N_{\textrm{null}}}$
---|---|---|---|---
$\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ CC | 117 | 0 | $\infty$ | $\infty$
$\bar{\nu}_{e}\to\bar{\nu}_{e}$ NC | 30511 | 32481 | -6.1% | -10.9
$\nu_{\mu}\to\nu_{\mu}$ NC | 66037 | 69420 | -4.9% | -12.8
$\bar{\nu}_{e}\to\bar{\nu}_{e}$ CC | 77600 | 82589 | -6.0% | -17.4
$\nu_{\mu}\to\nu_{\mu}$ CC | 197284 | 207274 | -4.8% | -21.9
(a) Stored $\mu^{-}$.
Channel | $N_{\textrm{osc.}}$ | $N_{\textrm{null}}$ | Diff. | $(N_{\textrm{osc.}}-N_{\textrm{null}})/\sqrt{N_{\textrm{null}}}$
---|---|---|---|---
$\nu_{e}\to\nu_{\mu}$ CC | 332 | 0 | $\infty$ | $\infty$
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ NC | 47679 | 50073 | -4.8% | -10.7
$\nu_{e}\to\nu_{e}$ NC | 73941 | 78805 | -6.2% | -17.3
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ CC | 122322 | 128433 | -4.8% | -17.1
$\nu_{e}\to\nu_{e}$ CC | 216657 | 230766 | -6.1% | -29.4
(b) Stored $\mu^{+}$.
## 6 Event rates after cuts
It must be determined how many of the raw events pass analysis cuts. Similar
analyses have been performed for Neutrino Factories exploring CP violation at
energies ranging from 25 GeV [4] to 5 GeV [6], but never at 3.8 GeV/c.
Preexisting experience and knowledge exists as to fractional background levels
and analysis difficulties; work had to be performed in order to tune the
analysis for this energy range.
The detector performance can be represented by _migration matrices_ (also
known as response matrices or energy smearing matrices) that describe both the
energy resolution and detection efficiency. If events are binned in terms of
true neutrino energy then the migration matrix is needed to transform the
distribution into the space of measured neutrino energies. For example, take
the histogram:
$\vec{h}^{\text{true}}=(N^{\text{true}}_{\text{0.0 - 0.1
\text{GeV}}},N^{\text{true}}_{\text{0.1 - 0.2
\text{GeV}}},\ldots,N^{\text{true}}_{\text{3.9 - 4.0 \text{GeV}}})^{T},$ (8)
where $N^{\text{true}}_{\text{0.0 - 0.1 \text{GeV}}}$ is the number of events
in the bin with ranges 0.0 and 0.1 GeV. The migration matrix $\mathbf{M}$ used
for this analysis is a square matrix and defined such that
$\vec{h}^{\text{measured}}=\mathbf{M}\vec{h}^{\text{true}}$ where
$\vec{h}^{\text{measured}}$ is the expected histogram of reconstructed
quantities in the detector.
With a perfect detector $\mathbf{M}=\text{diag.}(1,1,\ldots,1)$. $\mathbf{M}$
is unitary if and only if it describes only energy smearing. Efficiencies are
included into $\mathbf{M}$ by removing the unitarity constraint.
(a) $\nu_{\mu}$ CC
(b) $\bar{\nu}$ NC
(c) $\bar{\nu}_{\mu}$ CC
(d) $\nu_{e}$ CC
Figure 8: Migration matrices for $\nu_{\mu}$ CC, $\bar{\nu}$ NC,
$\bar{\nu}_{\mu}$ CC, and $\nu_{e}$ CC.
Migration matrices have been computed for $\nu_{\mu}$ CC, $\bar{\nu}_{\mu}$
CC, $\bar{\nu}_{\mu}$ NC, and $\nu_{e}$ CC (See [16]) and can be seen in Fig.
8. The background level of $\nu_{e}$ NC events into the signal window are
negligible compared with $\bar{\nu}_{\mu}$ NC due to the lower energies. These
numbers are derived using a GENIE and Geant4 simulation, described in the
cited text, which are MC method softwares. Statistical fluctuations exist in
the migration matrices due to computational limitations.
Figure 9: The rule rate as a function of observed energy for the appearance
channel $\nu_{e}\to\nu_{\mu}$. Migration matrices are used for $\nu_{\mu}$ CC,
$\bar{\nu}_{\mu}$ CC, $\bar{\nu}_{\mu}$ NC, and $\nu_{e}$ CC.
## 7 Statistics
It is necessary to determine if the number of events observed after cuts
(_i.e._ rule rates) is statistically significant. The experiment must reject
the null hypothesis when accounting for statistical fluctuations.
The hypothesis $H_{0}$ of no oscillations is the null hypothesis and designate
$H_{1}$ to be the alternate hypothesis. These hypotheses have oscillation
parameters associated with them: let $\theta_{0}=\\{\Delta
m^{2}_{41},\theta_{34},\theta_{24},\theta_{14}\\}$ be the oscillation
parameters associated with $H_{0}$, and similarly $\theta_{1}$ for $H_{1}$.
The _test statistic_ $X$ is a function of the experimental observations and
let $W$ be the space of all possible values of $X$. One can divide $W$ into
two regions: the region $w$ for those possible values of $X$ which would
suggest that the null hypothesis $H_{0}$ is not true and the remaining region
$W-w$.
It is desirable to have a small probability of $X$ – by statistical
fluctuations alone – taking a value in $w$ when $H_{0}$ is true. A level of
significance $\alpha$ can be defined:
$P(X\in w|H_{0})=\alpha$ (9)
where $\alpha$ corresponds to, colloquially, “5$\sigma$” when $\alpha\simeq
2.8\times 10^{-7}$ and “10$\sigma$” when $\alpha\simeq 7.6\times 10^{-24}$.
The number of “$\sigma$” correspond to the $p$-value of having a greater than
$n\sigma$ upward fluctuation of a Gaussian centered at zero. No Gaussian
assumptions are made in this analysis.
The test statistic that will be used for hypothesis testing is the likelihood
ratio test. Let there be $N$ observations $\mathbf{X}=\\{X_{1},...,X_{N}\\}$
and a probability distribution function $f(X_{i}|\theta)$. The likelihood
function is:
$\displaystyle L(\mathbf{X}|\mathbf{\theta})$ $\displaystyle=$
$\displaystyle\prod^{N}_{i=1}f(X_{i}|\mathbf{\theta})$ (10) $\displaystyle=$
$\displaystyle\prod_{i}e^{-\lambda_{i}}\lambda^{X_{i}}_{i}/{X_{i}}!$ (11)
where $\lambda_{i}$ is the expected number of background in the bin with
$X_{i}$ events and is a function of $\theta$. The distribution is Poisson
because the background levels are small. The short baseline parameters
$\theta_{1}$ for $H_{1}$ are free to take any value but the parameters
$\theta_{0}$ are fixed to zero by the null hypothesis requiring no
oscillations. The likelihood ratio test defines a test statistic $\lambda$
such that:
$\lambda=\frac{L(\mathbf{X}|\mathbf{\theta_{0}})}{\max_{\theta_{1}}L(\mathbf{X}|\mathbf{\theta_{1}})}$
(12)
where the denominator is maximized with respect to $\theta_{1}$ while the
numerator remains fixed. Using Eq. 11 leads to:
$\lambda=\prod_{i}e^{-\lambda_{i}+X_{i}}\left(\lambda_{i}/X_{i}\right)^{X_{i}}.$
(13)
The $\chi^{2}$ can be defined as $\chi^{2}=-2\ln\lambda$ (See [3]) which is
preferable to using $\lambda$ because of specifics about how multiplication is
performed by a computer. Using this definition, one finds:
$\displaystyle\chi^{2}=-2\ln\lambda=2\sum_{i}\lambda_{i}-X_{i}+X_{i}\ln\left(\frac{\lambda_{i}}{X_{i}}\right)$
(14)
which has two degrees of freedom since the numerator of Eq. 12 has no degrees
of freedom and the denominator has two degrees of freedom.
A test statistic has been defined that allows for determining if an experiment
is sensitive to various oscillation parameters. The $\chi^{2}$ can be computed
in terms of energy bins, with the appropriate definition of $X_{i}$, allowing
for spectral information to be used when computing sensitivities.
## 8 The Appearance Analysis
The parameters to be explored in the appearance analysis are $\Delta
m^{2}_{41}$ and $\sin^{2}(\theta_{e\mu})$. Contours in the neutrino parameter
space $\Delta m^{2}_{41}$ versus $\sin^{2}(\theta_{e\mu})$ can be used to
compare the sensitivities of various proposed short baseline experiments. A
statistics-only $\chi^{2}$ using spectral information is used (Fig.10).
Care must be taken when defining $\chi^{2}(\Delta
m^{2}_{41},\sin^{2}(\theta_{e\mu}))$ to ensure that it is well-defined. In the
(3+1) scenario, the signal $\nu_{e}\to\nu_{\mu}$ depends on the amplitude
$\sin^{2}(\theta_{e\mu})=4|U_{e4}|^{2}|U_{\mu 4}|^{2}$ and frequency $\Delta
m^{2}_{41}$ (See Eq. 2). If there is an appearance signal, then
$|U_{e4}|^{2}|U_{\mu 4}|^{2}\neq 0$ which implies that both $U_{e4}$ and
$U_{\mu 4}$ are nonzero. There is disappearance of the CC and NC backgrounds
(See Eq. 4) which affects the background estimation in the $\chi^{2}$. This
issue is addressed by not oscillating the backgrounds thus overestimating the
backgrounds.
Figure 10: Sterile sensitivity under the appearance channel
$\nu_{e}\to\nu_{\mu}$. This channel is the CPT of the LSND anomaly
$\bar{\nu}_{\mu}\to\bar{\nu}_{e}$. There is $10\sigma$ sensitivity to the LSND
and MiniBooNe 99% confidence interval [8]. Figure 11: A baseline optimization
using a total rates statistics-only $\chi^{2}$, a signal efficiency of 0.5,
and background rejection of charge misidentification and NCs at $10^{-3}$ and
$10^{-4}$. Figure 12: Tuning the NC rejection cut. The NC rejection level is
shown versus the signal efficiency. A charge misidentification background of
$10^{-4}$ is shown to illustrate when NC backgrounds become statistically
significant. A total rates statistics-only $\chi^{2}$ is used. Figure 13:
Tuning the charge misidentification cut. The charge misidentification level is
shown versus the signal efficiency. A NC background of $10^{-4}$ is shown to
illustrate when charge misidentification backgrounds become statistically
significant. A total rates statistics-only $\chi^{2}$ is used. Figure 14: An
optimization between the detector performance and accelerator performance
using the charge misidentification rates and number of muon decays as the
performance metric. IDR refers to the Interim Design Report [4] detector
performance. FODO refers to the FODO lattice design that gives $1.8\times
10^{18}$ useful muon decays whilst FFAG refers to the FFAG design that gives
$4.68\times 10^{18}$ useful muon decays. Both accelerators assume a front-end
of the main injector at 60 GeV/c.
As the cuts-based detector performance section improves and various cost
optimizations are done, there are numerous parameters that can be tuned to
compensate and conserve the physics that can be done with such a facility. For
example, the optimization of baseline and energy (Fig. 11) allows one to
change the baseline depending on site constraints or modify the energy of the
ring if the accelerator gets too expensive. As the cuts-based detector
performance improves, the various background rejections (Fig. 12 and 13) may
allow for a smaller detector or cheaper target station. The tools have been
developed that allow the important accelerator and detector performance
metrics into cost optimizations.
Figure 11 shows that, for a fixed baseline, increasing the muon energy is
always advantageous. This effect arises because the maximum of the $\nu_{e}$
flux is not at the oscillation maximum but rather at a higher energy. At high
energies the oscillation probability is:
$\displaystyle\text{Pr}[\nu_{e}\to\nu_{\mu}]$ $\displaystyle=$
$\displaystyle\sin^{2}(2\theta_{e\mu})\sin^{2}\left(\frac{\Delta
m^{2}_{41}L}{4E}\right)$ (15) $\displaystyle=$
$\displaystyle\sin^{2}(2\theta_{e\mu})\left(\frac{\Delta
m^{2}_{41}L}{4}\right)^{2}E^{-2}.$ (16)
The oscillation probability decreases as $E^{-2}$ for a fixed baseline. The
signal rates increase as $E^{3}$: there is a factor of $E^{2}$ from the solid
angle arising from the $1/\gamma$ opening angle and another factor of $E$ from
the cross section. The conclusion is that raising the stored muon energy will
increase the event rates linearly with energy for a fixed baseline. This
result has been confirmed by similar analyses for other muon-decay based
facilities (See sensitivity work in [4]).
## 9 Conclusion
The sensitivity of $\nu$STORM rules out the LSND 99% confidence interval at
$10\sigma$ using only appearance information. The appearance channel is the
CPT invariant of the observed anti-neutrino LSND anomaly. Optimizations have
been shown to guide future costing and performance work.
## Acknowledgements
The author thanks Alan Bross, John Cobb, and Joachim Kopp for their guidance
and knowledge. The author also thanks Ryan Bayes for the migration matrices
used in this analysis.
## References
* [1] K.N. Abazajian, M.A. Acero, S.K. Agarwalla, A.A. Aguilar-Arevalo, C.H. Albright, et al. Light Sterile Neutrinos: A White Paper. 2012\. 1204.5379.
* [2] C. Athanassopoulos et al. Evidence for nu(mu) —¿ nu(e) neutrino oscillations from LSND. Phys.Rev.Lett., 81:1774–1777, 1998. nucl-ex/9709006.
* [3] Steve Baker and Robert D. Cousins. Clarification of the use of chi-square and likelihood functions in fits to histograms. Nuclear Instruments and Methods in Physics Research, 221(2):437 – 442, 1984.
* [4] S. Choubey et al. International Design Study for the Neutrino Factory, Interim Design Report. 2011\. 1112.2853.
* [5] M. Aguilar-Benitez _et al._ Review of Particle Physics. Physical Review D, 2008.
* [6] Steve Geer, Olga Mena, and Silvia Pascoli. A Low energy neutrino factory for large $\theta_{13}$. Phys.Rev., D75:093001, 2007. hep-ph/0701258.
* [7] Carlo Giunti and Chung W. Kim. Fundamentals of Neutrino Physics and Astrophysics. Oxford University Press, USA, 2007.
* [8] Carlo Giunti and Marco Laveder. Towards 3+1 Neutrino Mixing. 2011\. 1109.4033.
* [9] GNU. GPL 3.0.
* [10] M.C. Gonzalez-Garcia, Michele Maltoni, and Jordi Salvado. Updated global fit to three neutrino mixing: status of the hints of theta13 ¿ 0. JHEP, 1004:056, 2010. 1001.4524.
* [11] Patrick Huber, Joachim Kopp, Manfred Lindner, Mark Rolinec, and Walter Winter. New features in the simulation of neutrino oscillation experiments with GLoBES 3.0: General Long Baseline Experiment Simulator. Comput.Phys.Commun., 177:432–438, 2007. hep-ph/0701187.
* [12] Patrick Huber, M. Lindner, and W. Winter. Simulation of long-baseline neutrino oscillation experiments with GLoBES (General Long Baseline Experiment Simulator). Comput.Phys.Commun., 167:195, 2005. hep-ph/0407333.
* [13] Joachim Kopp. Efficient numerical diagonalization of hermitian $3\times 3$ matrices. Int. J. Mod. Phys., C19:523–548, 2008. physics/0610206.
* [14] Joachim Kopp, Manfred Lindner, Toshihiko Ota, and Joe Sato. Non-standard neutrino interactions in reactor and superbeam experiments. Phys. Rev., D77:013007, 2008. 0708.0152.
* [15] Mark D. Messier. Evidence for neutrino mass from observations of atmospheric neutrinos with super-kamiokande. 1999\. UMI-99-23965.
* [16] $\nu$STORM Collaboration. nuSTORM: Neutrinos from STORed Muons. 2012\. 1206.0294.
* [17] E. A. Paschos and J. Y. Yu. Neutrino interactions in oscillation experiments. Phys. Rev., D65:033002, 2002. hep-ph/0107261.
* [18] Peter Renton. Electroweak Interactions: An Introduction to the Physics of Quarks and Leptons. Cambridge University Press, 1990.
* [19] C. D. Tunnell. https://code.launchpad.net/$\sim$c-tunnell1/+junk/, 5 2012. questions should be directed to the corresponding author.
* [20] Christopher D. Tunnell, John H. Cobb, and Alan D. Bross. Sensitivity to eV-scale Neutrinos of Experiments at a Very Low Energy Neutrino Factory. 2011\. 1111.6550.
* [21] Walter Winter. Optimization of a Very Low Energy Neutrino Factory for the Disappearance Into Sterile Neutrinos. 2012\. 1204.2671.
|
arxiv-papers
| 2012-05-29T11:51:28 |
2024-09-04T02:49:31.326020
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. D. Tunnell",
"submitter": "Christopher Tunnell",
"url": "https://arxiv.org/abs/1205.6338"
}
|
1205.6365
|
arxiv-papers
| 2012-05-29T13:40:57 |
2024-09-04T02:49:31.331988
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Asia Furones",
"submitter": "Asia Furones",
"url": "https://arxiv.org/abs/1205.6365"
}
|
|
1205.6488
|
# Masses, Radii, and Cloud Properties of the HR 8799 Planets
Mark S. Marley NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035;
Mark.S.Marley@NASA.gov Didier Saumon Los Alamos National Laboratory, Mail
Stop F663, Los Alamos NM 87545; dsaumon@lanl.gov Michael Cushing Department
of Physics and Astronomy, The University of Toledo, 2801 West Bancroft Street,
Toledo, OH 43606; michael.cushing@utoledo.edu Andrew S. Ackerman NASA
Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025;
andrew.ackerman@nasa.gov Jonathan J. Fortney Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064;
jfortney@ucolick.org Richard Freedman SETI Institute & NASA Ames Research
Center, MS-245-3, Moffett Field, CA 94035, U.S.A.;
freedman@darkstar.arc.nasa.gov
###### Abstract
The near-infrared colors of the planets directly imaged around the A star HR
8799 are much redder than most field brown dwarfs of the same effective
temperature. Previous theoretical studies of these objects have concluded that
the atmospheres of planets b, c, and d are unusually cloudy or have unusual
cloud properties. Some studies have also found that the inferred radii of some
or all of the planets disagree with expectations of standard giant planet
evolution models. Here we compare the available data to the predictions of our
own set of atmospheric and evolution models that have been extensively tested
against observations of field L and T dwarfs, including the reddest L dwarfs.
We require mutually consistent choices for effective temperature, gravity,
cloud properties, and planetary radius. This procedure thus yields plausible
values for the masses, effective temperatures, and cloud properties of all
three planets. We find that the cloud properties of the HR 8799 planets are
not unusual but rather follow previously recognized trends, including a
gravity dependence on the temperature of the L to T spectral transition–some
reasons for which we discuss. We find the inferred mass of planet b is highly
sensitive to whether or not we include the $H$ and $K$ band spectrum in our
analysis. Solutions for planets c and d are consistent with the generally
accepted constraints on the age of the primary star and orbital dynamics. We
also confirm that, like in L and T dwarfs and solar system giant planets, non-
equilibrium chemistry driven by atmospheric mixing is also important for these
objects. Given the preponderance of data suggesting that the L to T spectral
type transition is gravity dependent, we present an exploratory evolution
calculation that accounts for this effect. Finally we recompute the the
bolometric luminosity of all three planets.
brown dwarfs — planetary systems — stars: atmospheres – stars: low mass, brown
dwarfs – stars: individual (HR 8799)
## 1 INTRODUCTION
Establishing the masses, radii, effective temperatures, and atmospheric
composition of the planets orbiting the A star HR 8799 has been a challenge.
Of the four planets (Marois et al., 2008, 2010) directly imaged orbiting the
star HR 8799, broad photometric coverage (1 – $5\,\rm\mu m$) is available for
three planets, b, c, and d (Marois et al., 2008; Currie et al., 2011), and
some spectral data is available for one planet, b (Barman et al., 2011a).
Efforts to fit the available data with atmosphere and evolution models have
produced mixed results. In some cases the best-fitting models predict radii
and ages that are at odds with other constraints, such as evolution models and
the age of the system. The purportedly unusual cloud properties of the planets
have also received great attention. Here we present an examination of the
properties of HR 8799 b, c, and d using publicly available data as well as our
own evolution and atmosphere models. Our aim is to determine if a set of
planet properties can be derived that simultaneously satisfy all observational
and theoretical constraints and to ascertain the nature of atmospheric
condensate layers in each planet.
We open below with a summary of the model parameters previously derived for
these planets. In the remainder of this section we briefly review what is
known about the atmospheric evolution of brown dwarfs and discuss the issues
that have arisen to date in the study of the HR 8799 planets, particularly
regarding the inferred cloud properties and planet radii. In succeeding
sections we explore the nature of clouds in low-mass objects more deeply and
present model solutions for the masses, effective temperatures $(T_{\rm
eff}$), and cloud properties of the planets. We find, as have all other
previous studies, that clouds are present in the visible atmosphere of these
planets at lower effective temperatures than in typical field brown dwarfs. In
agreement with Barman et al. (2011a) but unlike most other previous studies
(e.g., Bowler et al., 2010; Currie et al., 2011; Madhusudhan et al., 2011) we
find that the clouds of the HR 8799 planets are similar to those found in
field L dwarfs.
### 1.1 Masses and Radii of HR 8799 b,c, and d
In the HR 8799 b, c, and d discovery paper, Marois et al. (2008) derived the
mass and effective temperature of each object in two ways. In the first method
they computed the luminosity of each object and compared that to theoretical
cooling tracks for young giant planets given the constraint of their estimated
age of the primary star. In the second method they fit atmosphere models
derived using the PHOENIX code (Hauschildt et al., 1999) to the available six-
band near-infrared photometry (1 to $2.5\,\rm\mu m$) to constrain $T_{\rm
eff}$ and $\log g$, the two most important tunable parameters of atmosphere
models. Radii of each planet were derived by comparing the model emergent
spectra with the observed photometry and known distance to the target. Notably
only models that included the effects of refractory silicate and iron clouds
were consistent with the data. However the radii estimated by this method were
far smaller than expected for solar metallicity gas giant planets at such
young ages.
A number of followup studies presented new data new data and models in an
attempt to better understand the planets. Barman et al. (2011a) fit a suite of
models to the available photometry (but not the $M$ band (Galicher et al.,
2011) data) and $H$ and $K$ band spectra that they obtained for planet b. By
comparing the integrated flux from their best fitting model atmosphere to the
estimated bolometric luminosity of the planet, they found a small radius for
the planet $R\sim 0.75\,\rm R_{J}$. Galicher et al. (2011) also fit the Barman
atmosphere models to the photometry, including new $M$ band data. They found
somewhat higher gravity solutions than Barman et al. (2011a) but also required
a small radius for planet b, approximately 70%–or about one-third the
volume–expected from planetary evolution models. Such a large discrepancy is
difficult to reconcile with our understanding of both giant planet evolution
and the high pressure equation of state of hydrogen. Instead the most
straightforward interpretation is that the atmosphere models are not
representative of the actual planetary atmosphere and Barman et al. suggest
that higher metallicity models might provide a better fit and give more
plausible radii.
Likewise Bowler et al. (2010) selected the model spectra (from among the
models of Hubeny & Burrows (2007); Burrows et al. (2006); Allard et al.
(2001)) which best fit the available photometry for HR 8799b. Their best
fitting spectra were quite warm, with $T_{\rm eff}$ from 1300 to 1700 K and
thus they required even smaller radii ($\sim 0.4\,\rm R_{J}$) in order to meet
the total luminosity constraint given the photometry available at that time.
In contrast Currie et al. (2011) searched for the best fitting models while
requiring that the planet radii either matched those predicted by a set of
evolution models (Burrows et al., 1997) or were allowed to vary. They found
that what they termed to be “standard” brown dwarf cloud models required
unphysically small planet radii to fit the data. However their “thick cloud”
models could fit the data shortward of $3\,\rm\mu m$ by employing planetary
radii that were within about 10% of the usual evolution model prediction. As
we note below, however, the “standard” cloud model has itself not been
demonstrated to fit cloudy, late L-type dwarfs; thus this exercise does not
necessarily imply the planets’ clouds are “non-standard”. Nevertheless they
were able to fit much of the photometry with planetary radii consistent with
evolution model predictions.
Finally Madhusudhan et al. (2011) explored a set of models similar to those
studied by Currie et al. with yet another cloud model but without the radius
constraint. Their best fits are very similar to those of Currie et al. but
with somewhat lower $T_{\rm eff}$.
The characteristics of the planets as derived in the 2011 publications are
summarized in Table 1. Not all authors report every parameter so some radii
and ages are left blank. Note the diverse set of masses, radii, and effective
temperatures derived by the various studies. Despite the variety some trends
are clear: planet b consistently is found to have the lowest mass and
effective temperature and its derived radius is almost always at odds with the
expectation of evolution and interior models.
We note that at very young ages the model radii of giant planets depends on
the initial conditions of the evolutionary calculation (Stevenson, 1982;
Baraffe et al., 2002; Marley et al., 2007a; Spiegel & Burrows, 2012). However
at ages younger than several hundred million years the planetary radius is
expected to be no smaller than about 1.1 times that of Jupiter regardless of
the formation mechanism. Hence radii derived by Barman et al. (2011a) and
Galicher et al. (2011) are not consistent with evolutionary calculations,
regardless of the initial boundary conditions. Indeed the equation of state
for gas giant planets, even ones enriched in heavy elements, preclude such
radii.
### 1.2 Clouds
#### 1.2.1 Brown Dwarfs
As a brown dwarf ages it radiates and cools. When it is warm, refractory
condensates, including iron and various silicates, form clouds in the visible
atmosphere. Over time the clouds become progressively thicker and more opaque,
leading to ever redder near-infrared colors. As the dwarf cools the cloud
decks are found at higher pressures, deeper in the atmosphere. Eventually the
clouds disappear from the photosphere. Indeed the first two brown dwarfs to be
discovered, GD 165B (Becklin & Zuckerman, 1988) and Gl 229B (Nakajima et al.,
1995), were ultimately understood to represent these two different end cases:
the cloudy L and the clear T dwarfs (see Kirkpatrick (2005) for a review).
Understanding the behavior of clouds in substellar atmospheres and how it
might vary with gravity has become one of the central thrusts of brown dwarf
science.
The earliest models for these objects assumed that the condensates were
uniformly distributed vertically throughout the atmosphere (e.g., Chabrier et
al., 2000). Later, more sophisticated approaches attempted to model the
formation of discrete cloud layers that would result from the gravitational
settling of grains.
With falling effective temperature, $T_{\rm eff}$, the bases of the iron and
silicate cloud decks are found progressively deeper in the atmosphere. Because
of grain settling the overlying atmosphere well above the cloud deck loses
grain opacity and becomes progressively cooler. Thus over time more of the
visible atmosphere becomes grain free and cooler. Cooler temperatures favor
$\rm CH_{4}$ over CO. The removal of the opacity floor that the clouds
provided at higher $T_{\rm eff}$ also allows flux in the water window regions
to escape from deeper in the atmosphere. This leads to a brightening in the
$J$ band and a blueward color shift in the near-infrared. In field brown
dwarfs this color change begins around effective temperature $T_{\rm eff}\sim
1200$ to $1400\,\rm K$ and is complete over a strikingly small effective
temperature range of only 100 to 200 K (see Kirkpatrick (2005) for a review).
This experience led to the presumption that all objects with effective
temperatures below about 1100 K would have blue near-infrared colors, like the
field brown dwarfs.
#### 1.2.2 HR 8799 b, c, and d
The early directly imaged low mass companions confounded these expectations
from the brown dwarf experience. The companion 2MASSW J1207334-393254 b
(hereafter 2M1207 b) has red infrared colors despite its low luminosity and
apparently cool $T_{\rm eff}$ (Chauvin et al., 2004) . Likewise the HR 8799
planets have colors reminiscent of hot, cloudy L dwarfs but their bolometric
luminosities coupled with radii from planetary structure calculations imply
$T_{\rm eff}\sim 1000\,\rm K$ or lower (Marois et al., 2008, 2010).
The red colors, particularly of the HR 8799 planets, spawned a storm of
studies investigating the atmospheric structure of the planets. Essentially
all of these papers concluded that the planets could be best explained by
invoking thick cloud decks. Since this ran counter to expectation, these
clouds were deemed “radically enhanced” when compared to “standard” models
(Bowler et al., 2010). Likewise Currie et al. (2011) compared their data to
the Burrows et al. (2006) model sequence and concluded (their §5) that the HR
8799 planets have much thicker clouds than “…standard L/T dwarf atmosphere
models.” Madhusudhan et al. (2011) state that their fiducial models “…have
been shown to provide good fits to observations of L and T dwarfs (Burrows et
al., 2006)”. They then find that much cloudier models are required to fit the
imaged exoplanets and thus conclude that the cloud properties must be highly
discrepant from those of the field L dwarfs.
Such conclusions, however, seem to overlook that the study of L dwarf
atmospheres is still in its youth. Cloudy atmospheres of all kinds are
challenging to model and the L dwarfs have proven to be no exception. Thus
whether or not the HR 8799 planets have unusual clouds depends on the point of
reference. Indeed while most published models of brown dwarfs are able to
reproduce the spectra of cloudy, early L-type dwarfs and cloudless T dwarfs,
the latest, reddest—and presumably cloudiest—L dwarfs have been a challenge.
The points of comparison for the work of Currie et al. (2011) and Madhusudhan
et al. (2011) were the models described in Burrows et al. (2006). When
compared to the red-optical and near-infrared photometry of L and T dwarfs,
those models did not reproduce the colors of the latest L dwarfs as the models
are too blue (see figure 17 of Burrows et al. (2006)) implying that they
lacked sufficient clouds. Burrows et al. (2006) also presented comparisons of
their models to L dwarf spectra; however the comparisons are only to an L1 and
an L5 dwarf. There are no comparisons to very cloudy late L dwarf spectra in
the paper so the fidelity of their model under such conditions cannot be
judged. For these reasons a comparison of the cloudy HR 8799 planets to the
“standard” L dwarf models, such as presented by Madhusudhan et al. (2011) and
Currie et al. (2011), does not address the question whether the HR 8799
planets are really all that different from the cloudiest late L dwarfs since
those models have apparently do not reproduce the colors of the latest L
dwarfs.
At least one set of atmosphere and evolution models is available that has been
compared against the near- to mid-infrared spectra and colors of latest L
dwarfs. In Cushing et al. (2008) and Stephens et al. (2009) we compared our
group’s models to observed far-red to mid-infrared spectra of L and T dwarfs,
including L dwarfs with IR spectral types as late as L9 (with 7 objects in the
range L7 to L9.5). We found that the models with our usual cloud prescription
fit the spectra of L dwarfs of all spectral classes (including the latest
field dwarfs) well, but not perfectly. In Saumon & Marley (2008) we also
presented a model of brown dwarf evolution that well reproduced the usual
near-infrared color magnitude diagrams of L and T dwarfs, including the
reddest L dwarfs. Here we apply our set of cloudy evolution models to the HR
8799 planet observations in an attempt to better understand these objects.
### 1.3 Chemical Mixing
Shortly after the discovery of Gl 229B, Fegley & Lodders (1996) predicted
that—as in Jupiter—vertical mixing might cause CO to be overabundant compared
to $\rm CH_{4}$ in chemical equilibrium in this object. This was promptly
confirmed by the detection of CO absorption at $4.6\,\rm\mu m$ by Noll et al.
(1997) and Oppenheimer et al. (1998). The overabundance is caused by the slow
conversions of CO to $\rm CH_{4}$ relative to the mixing time scale.
An obvious mechanism for vertical mixing in an atmosphere is convection. Brown
dwarf atmospheres are convective at depth where the mixing time scale is short
(minutes). The overlying radiative zone is usually considered quiescent but a
variety of processes can cause vertical mixing, albeit on much longer time
scales. Since the conversion time scales for $\rm CO\rightarrow CH_{4}$ and
$\rm N_{2}\rightarrow NH_{3}$ range from seconds (at $T\sim 3000\,\rm K$) to
many Hubble times (for $T<1000\,\rm K$), even very slow mixing in the
radiative zone can drive the chemistry of carbon and nitrogen out of
equilibrium. From this basic consideration, it appears that departures from
equilibrium are inevitable in the atmospheres of cool brown dwarfs and indeed
the phenomenon is well established (e.g., Saumon et al., 2000; Geballe et al.,
2001; Hubeny & Burrows, 2007; Geballe et al., 2009; Mainzer et al., 2007;
Saumon et al., 2006; Stephens et al., 2009).
With falling gravity the point at which chemical reactions are quenched occurs
deeper in the atmosphere, where the higher temperature result in a greater
atmospheric abundance of CO (Hubeny & Burrows, 2007; Barman et al., 2011a). At
exoplanet gravities, mixing can even produce CO/$\rm CH_{4}$ ratios in excess
of 1 (Barman et al., 2011a). Thus a complete giant planet exoplanet atmosphere
model must account for such departures from chemical equilibrium as well.
## 2 Gravity, Refractory Clouds and the L/T Transition
### 2.1 Nature of the Transition
Two main causes of the loss of cloud opacity at the L to T transition have
been suggested. In one view the atmospheric dynamical state changes, resulting
in larger particle sizes that fall out of the atmosphere more rapidly, leading
to a sudden clearing or collapse of the cloud (Knapp et al., 2004; Tsuji &
Nakajima, 2003; Tsuji et al., 2004). This view is supported by fits of spectra
to model spectra (Saumon & Marley, 2008) computed with the Ackerman & Marley
(2001) cloud model. In that formalism, a tunable parameter, $f_{\rm sed}$
controls cloud particle sizes and optical depth. Larger $f_{\rm sed}$ yields
larger particles along with physically and optically thinner clouds. Cushing
et al. (2008) and Stephens et al. (2009) have demonstrated that progressively
later dwarfs (L9 to T4) can be fit by increasing $f_{\rm sed}$ across the
transition at a nearly fixed effective temperature. A variation on this
hypothesis is that a cloud particle size change is responsible for the
transition (Burrows et al., 2006).
The second view is inspired by thermal infrared images of the atmospheres of
Jupiter and Saturn at $\sim 5\,\mu$m (e.g. Westphal, 1969; Westphal et al.,
1974; Orton et al., 1996; Baines et al., 2005). Gaseous opacity is low at this
wavelength and the clouds stand out as dark, mottled features against a bright
background of flux emitted from deeper, warmer levels in the atmosphere. Such
images of both Jupiter and Saturn clearly show that the global cloud decks are
not homogenous, but rather are quite patchy. Ackerman & Marley (2001),
Burgasser et al. (2002), and Marley et al. (2010) have suggested that the
arrival of holes in brown dwarf clouds, perhaps due to the clouds passing
through a dynamical boundary in the atmosphere, might also be responsible for
the L to T transition. This view is supported by the discovery of L-T
transition dwarfs that vary in brightness with time with relatively large
near-infrared amplitudes (Artigau et al., 2009; Radigan et al., 2011). Indeed
Radigan (in prep) has found in a survey of about 60 L and T type brown dwarfs
that the most variable dwarfs are the early T’s, which are in the midst of the
$J-K$ color change.
In order to match observations, modern thermal evolution models for the
cooling of brown dwarfs have to impose some arbitrary mechanism, such as
varying sedimentation efficiency or the imposition of cloud holes, by which
the thick clouds in the late L dwarfs dissipate. A uniform cloud layer that
simply sinks with falling $T_{\rm eff}$ as the atmosphere cools turns to the
blue much more slowly than is observed. Application of such a transition
mechanism to reliably reproduce the colors and spectra of late L and early T
dwarfs (e.g., near-infrared color-magnitude diagrams) led to the expectation
that the normal behavior for cooling brown dwarfs–or extrasolar giant
planets–is to turn to the blue at around 1300 K.
However there have been indications that such a narrative is too simplistic
and that gravity plays a role as well. Two brown dwarf companions to young
main sequence stars were found to have unexpectedly cool effective
temperatures for their L-T transition spectral types by Metchev & Hillenbrand
(2006) and Luhman et al. (2007). The analysis of Luhman et al. of the T dwarf
HN Peg B was further supported by additional modeling presented in Leggett et
al. (2008). Dupuy et al. (2009) presented evidence of a gravity dependent
transition $T_{\rm eff}$ on the basis of a dynamical mass determination of an
$\rm M8+L7$ binary. Stephens et al. (2009) fit the model spectra of Marley et
al. (2002) to the 1 – $15\,\rm\mu m$ spectra of L and T dwarfs and found that
L dwarf cloud clearing (as characterized by large $f_{\rm sed}$) occurs at
$T_{\rm eff}\sim 1300\,\rm K$ for $\log g=5.0$ and at $\sim 1100\,\rm K$ for
$\log g=4.5$, although the sample size was admittedly small (Figure 1).
Nevertheless such an association implies a cooler transition temperature at
even lower gravity.
### 2.2 Clouds at Low Gravity
Even if directly imaged planets are not considered, there is already
considerable evidence that the cloud clearing associated with the L to T
transition occurs at lower effective temperatures in lower gravity objects
than in high gravity ones. To understand what underlies this trend it is
necessary to consider three separate questions. First, where does the
optically-thick portion of the cloud lie in the atmosphere relative to the
photosphere, as a function of gravity? An optically-thick cloud lying well
below the photosphere will be essentially invisible whereas the same cloud
lying higher in the atmosphere would be easily detected. Second, how does the
total optical depth of the cloud vary with gravity? This is a complex problem
involving the pressure of the cloud base and the particle size distribution.
Third, how does the mechanism by which clouds dissipate vary with gravity? For
example, do holes form at a different effective temperature in different
gravity objects? In this section we consider only the first two questions and
defer the third question to Section 5.6.
To address the first question we need to understand how atmospheric
temperature $T$ varies with pressure $P$ as a function of gravity. For a fixed
effective temperature, a lower gravity atmosphere is warmer at a fixed
pressure level than a higher gravity one. This is because more atmospheric
mass–and thus greater opacity–overlies a given pressure level at lower
gravity. Figure 2 provides an example using our model profiles. Since at
equilibrium condensation begins at the intersection of the vapor pressure and
thermal profiles, the cloud base occurs at lower pressure (higher in the
atmosphere) in a low gravity object than a high gravity one.
As objects cool with time (at essentially fixed gravity) clouds will persist
at lower pressure and remain visible to cooler effective temperatures in lower
gravity objects than higher gravity ones. For example in Figure 2 the lowest
gravity model shown at $T_{\rm eff}=900\,\rm K$ is hotter at all pressures
greater than a few hundred millibar than a higher gravity $T_{\rm
eff}=1300\,\rm K$ object. As explained below this degeneracy between cooler
low gravity and warmer high gravity temperature profiles lies at the heart of
the problem of simultaneously distinguishing gravity and effective temperature
with a limited photometric dataset.
Addressing the second question requires us to understand how the cloud column
optical depth varies with gravity. This depends both on the amount of
condensible material in the atmosphere available to form clouds and on the
cloud particle size. From basic scaling laws and mass balance Marley (2000)
derived an expression for the wavelength-dependent total column optical depth
$\tau_{\lambda}$ of a cloud in a hydrostatic atmosphere
$\tau_{\lambda}=75\epsilon Q_{\lambda}(r_{\rm
eff})\varphi{\biggl{(}{P_{cl}\over{1\,\rm
bar}}\biggr{)}}{\biggl{(}{10^{5}\,{\rm
cm\,s^{-2}}\over{g}}\biggr{)}}{\biggl{(}{1\,{\rm\mu m}\over{r_{\rm
eff}}}\biggr{)}}{\biggl{(}{1.0\,{\rm g\,cm^{-3}}\over{\rho_{c}}}\biggr{)}}.$
$None$
Here $P_{cl}$, $r_{\rm eff}$ and $\rho_{c}$ refer to the pressure at the cloud
base and the condensate effective (area-weighted) radius111Marley (2000)
employed the mean particle size $r_{c}$ rather than the more rigorous area-
weighted size. and density (see also Eq. 18 of Ackerman & Marley (2001)).
$\varphi$ is the product of the condensing species number mixing ratio and the
ratio of the mean molecular weight of the condensate to that of the
atmosphere. The expression assumes that some fraction $\epsilon$ of the
available mass above the cloud base forms particles with extinction cross
section $Q_{\lambda}$ (which can be computed through Mie theory) . Ackerman &
Marley (2001) also estimate the column optical depth of a cloud with a similar
result. Generalizing their Eq. 16,
$\tau_{\lambda}\propto{P_{cl}\over{gr_{\rm eff}(1+f_{\rm sed})}}.$ $None$
Both Equations (1) and (2) hold that all else being equal–including particle
sizes–we expect $\tau\propto P_{cl}/g$, just because the column mass above a
fixed pressure level is greater at low gravity and there is more material to
condense. Any cloud model which self-consistently computes the column mass of
condensed material should reproduce this result. As shown above, however, the
cloud base is at lower pressure in lower gravity objects, roughly
$P_{cl}\propto g$, thus leading to the expectation that the cloud $\tau$ would
be approximately constant with changing gravity. This is not exactly true
since there is a slope to the vapor pressure equilibrium curve and thus the
actual variation is somewhat weaker, but the effects of gravity and the cloud
base pressure alone do not strongly influence cloud column optical depth.
The second component affecting the column cloud opacity is particle size.
While a cloud model is required for rigorous particle size computation, we can
examine the scaling of size with gravity. At lower gravity particle fall
speeds are reduced, which reduces the downward mass flux carried by
condensates of a given size $r$. Since fall speed is proportional to $r^{2}$
in the Stokes limit (the viscous regime at low Reynolds numbers) while the
mass is proportional to $r^{3}$, the flux scales with $r^{5}$, a slight
increase in particle size can produce the same mass balance in the atmosphere
at lower gravity, and thus $r$ is expected to increase relatively slowly with
decreasing $g$. At large Reynolds number the dependence on fall speed is
weaker than $r^{2}$ and the equivalent result is found. Indeed recasting the
Ackerman & Marley (2001) model equations suggests $r\propto(f_{\rm
sed}/g)^{1/2}$, although the actual dependence is more complex as it depends
upon an integral over the size distribution. Tests with the complete cloud
model coupled to our atmosphere code predict about a factor of 4 increase in
cloud particle radius (25 to $100\,\rm\mu m$) as gravity decreases by an order
of magnitude from 300 to $30\,\rm m\,s^{-2}$, a slightly faster increase than
$\sqrt{g}$. A roughly $r\propto g^{-1/2}$ relationship is also seen in the
cloud model of Cooper et al. (2003) (see their Figures 2, 3, and 4). Returning
to Eq. (1) and combining with the scaling discussed above thus suggests that
all else being equal we expect cloud $\tau\propto\sqrt{g}$.
Figure 3 illustrates all of these effects in model cloud profiles calculated
for three atmosphere models with varying $g$ and $T_{\rm eff}$. The
atmospheric gravity spans two orders of magnitude while the effective
temperature varies from 1200 to 1000 K from the warmest to coolest object. As
expected the cloud particle size indeed varies inversely with gravity($r\sim
g^{-1/2}$) while the cloud base pressure decreases with decreasing gravity.
The choice in the plot of a cooler $T_{\rm eff}$ for the lowest gravity object
counteracts what would otherwise be an even greater difference in the cloud
base pressure. The net result is that the total column optical depth for the
silicate cloud in all three objects is very similar, $\tau\sim 10$. Thus a
cooler, low gravity object has a cloud with a column optical depth that is
almost indistinguishable from that of a warmer, more massive object.
The thicker portion of the lines denoting cloud column optical depth signify
the regions in the atmosphere where the brightness temperatures between
$\lambda=1$ and $6\,\rm\mu m$ are equal to the local temperature. In other
words the thick line represents the near-infrared photosphere. In all three
cases there is substantial cloud optical depth ($\tau_{\lambda}>0.1$) in the
deeper atmospheric regions from which flux emerges in the near-infrared. As a
result clouds play comparable roles in all three objects despite the two order
of magnitude difference in gravity and the 200 K temperature difference. We
thus conclude that the net effect of all of these terms is to produce clouds
in lower gravity objects with optical depths and physical locations relative
to the photosphere comparable to clouds in objects with higher gravity and
higher effective temperature.
## 3 Modeling Approach
To model the atmospheres and evolution of exoplanets we apply our usual
modeling approach which we briefly summarize in this section. We stress that
the fidelity of model fits in previous applications of our method to both
cloudy and clear atmosphere brown dwarfs (Marley et al., 1996, 2002; Burrows
et al., 1997; Roellig et al., 2004; Saumon et al., 2006, 2007; Leggett et al.,
2007a, b; Mainzer et al., 2007; Blake et al., 2007; Cushing et al., 2008;
Geballe et al., 2009; Stephens et al., 2009) validates our overall approach
and provides a basis of comparison to the directly imaged planet analysis. In
addition to brown dwarfs the model has been applied to Uranus (Marley & McKay,
1999) and Titan (McKay et al., 1989) as well.
### 3.1 Atmosphere and Cloud Models
The atmospheric structure calculation is described in McKay et al. (1989);
Marley et al. (1996); Burrows et al. (1997); Marley & McKay (1999); Marley et
al. (2002); Saumon & Marley (2008). Briefly we solve for a radiative-
convective equilibrium thermal profile that carries thermal flux given by
$\sigma T_{\rm eff}^{4}$ given a specified gravity and atmospheric
composition. The thermal radiative transfer follows the source function
technique of Toon et al. (1989) allowing inclusion of arbitrary Mie scattering
particles in the opacity of each layer. Our opacity database includes all
important absorbers and is described in Freedman et al. (2008).
There are, however, two particularly important updates to our opacity database
since Freedman et al. (2008). First we use a new molecular line list for
ammonia (Yurchenko et al., 2011). Secondly we have updated our previous
treatment of pressure-induced opacity arising from collisions of $\rm H_{2}$
molecules with $\rm H_{2}$ and He. This new opacity is discussed in Frommhold
et al. (2010) and the impact on our model spectra and photometry in general is
discussed in Saumon et al. (2012).
The abundances of molecular, atomic, and ionic species are computed for
chemical equilibrium as a function of temperature, pressure, and metallicity
following Fegley & Lodders (1994, 1996); Lodders (1999); Lodders & Fegley
(2002); Lodders (2003); Lodders & Fegley (2006) assuming the elemental
abundances of Lodders (2003). In this paper we explore only solar composition
models.
For cloud modeling we employ the approach of Ackerman & Marley (2001) which
parameterizes the importance of sedimentation relative to upwards mixing of
cloud particles through an efficiency factor, $f_{\rm sed}$. Large values of
$f_{\rm sed}$ correspond to rapid particle growth and large mean particle
sizes. Under such conditions condensates quickly fall out of the atmosphere,
leading to physically and optically thinner clouds. In the case of small
$f_{\rm sed}$ particles grow more slowly resulting in a larger atmospheric
condensate load and thicker clouds. Both our cloud model and chemical
equilibrium calculations are fully coupled with the radiative transfer and the
$(P,T)$ structure of the model during the calculation of a model so that they
are fully consistent when convergence is obtained.
We note in passing that the cloud models employed in previous studies of the
HR 8799 planets have been ad hoc, as straightforwardly discussed in those
papers. Particle sizes, cloud heights, and other cloud properties are fixed at
given values while gravity, $T_{\rm eff}$, and other model parameters are
varied. The methodology used here is distinct since in each case we compute a
consistent set of cloud properties given a specific modeling approach, the
Ackerman & Marley cloud.
The coupled cloud and atmosphere models have been widely compared to spectra
and photometry of L and T dwarfs in the publications cited in the introduction
to this section. We emphasize in particular that Cushing et al. (2008) and
Stephens et al. (2009) show generally good fits between our model spectra and
observations of cloudy L dwarfs. The near-infrared colors of brown dwarfs are
quite sensitive to the choice of $f_{\rm sed}$, a point we will return to in
Section 5.4.
### 3.2 Evolution Model
Our evolution model is described in Saumon & Marley (2008). In fitting the HR
8799 data, we use the sequence computed with a surface boundary condition
extracted from our cloudy model atmospheres with $f_{\rm sed}=2$. As we will
see below, our best fits show that all three planets are cloudy with $f_{\rm
sed}=2$, which justifies this choice of evolution a posteriori. As the three
planets appear to have significant cloud decks (as will be confirmed below),
it is not necessary to use evolution sequences that take into account the
transition explicitly in this comparison with models. Nevertheless, we will
explore the effects of a gravity-dependent transition between cloudy and
cloudless atmospheres in Section 5.4 as this is a topic of growing interest.
The Saumon & Marley (2008) models were computed with what has come to be known
as a traditional or hot-start initial condition. As discussed in Baraffe et
al. (2002), Marley et al. (2007a) and Spiegel & Burrows (2012) however, the
computed radii of young giant planets at ages of 100 Myr and less is highly
dependent on the details of the assumed initial condition. Even assuming very
cold initial conditions, however, computed planetary radii never fall below
$1\,\rm R_{J}$ at ages of less than 1 Gyr. Rather than carrying out the model
fitting for an uncertain set of assumed cold initial conditions, we choose
here to employ the traditional hot-start boundary conditions for the evolution
modeling. In this way we avoid unphysical very small radii ($R<1\,\rm R_{J}$)
while adding an additional constraint to the modeling.
## 4 Application to HR 8799 Planets
### 4.1 Constraints on the HR 8799 System Properties
A number of the properties of the HR 8799 system as a whole help to constrain
the properties of the individual planets. Of foremost importance of course is
the age of the primary star since older ages require greater planetary masses
to provide a fixed observed luminosity. The massive dust disk found outside of
the orbit of the most distant planet, HR 8799 b, constrains the mass of that
planet since a very massive planet would disrupt the disk. Finally dynamical
models of the planetary orbits circumscribe the parameter space of orbits and
masses that are stable over the age of the system. All of these topics have
been discussed extensively in the literature so here we briefly summarize the
current state of affairs. A more thorough review can be found in Sudol &
Haghighipour (2012).
Since the discovery of the first three planets, the age of HR 8799 has been
debated. As summarized initially by the discoverers, most indicators suggest a
young age of 30 to 60 Myr (Marois et al., 2008). However the typical age
metrics are somewhat more in doubt than usual because HR 8799 is a $\lambda$
Boo-type star with an unusual atmospheric and uncertain internal composition.
Moya et al. (2010) review the various estimates of the age of the star prior
to 2010 and argue that most of the applied metrics, including color and
position on the HR diagram, are not definitive. Most recently Zuckerman et al.
(2011) conclude that the Galactic space motion of HR 8799 is very similar to
that of the 30 Myr old Columba association and suggest that it is a member of
that group. They also argue that the $B-V$ color of HR 8799 in comparison to
Pleiades A stars also supports a young age, although the unusual composition
hampers such an argument. Perhaps the fairest summary of the situation to date
would be that most traditional indicators support a young age for the primary,
but that no single indicator is entirely definitive on its own.
One indicator that the age could be much greater than usually assumed is
discussed by Moya et al. (2010). Those authors use the $\gamma$ Doradus g-mode
pulsations of the star to place an independent constraint on the stellar age.
Their analysis is dependent upon the rotation rate of the star and
consequently the unknown inclination angle and thus is also uncertain.
Nevertheless they find model solutions that match the observed properties of
the star in which the stellar age can plausibly be in excess of 100 Myr and in
some cases as large as 1 Gyr or more. They state that their analysis is most
uncertain for inclination angles in the range of 18 to $36^{\circ}$, which
corresponds to the likely inclination supported by observations of the
surrounding dust belt (see below). Thus stellar seismology provides an
intriguing, but likewise still uncertain constraint.
The dust disk encircling the orbits of the HR 8799 planets can in principle
provide several useful constraints on the planetary masses and orbits. First
the inclination of the disk affects the computed orbital stability of the
companions (Fabrycky & Murray-Clay, 2010) if we assume the disk is coplanar
with the planetary orbits. If the rotation axis of the star is perpendicular
to the disk, the inclination also has a bearing on the stellar age since the
seismological analysis in turn depends upon its inclination to our line of
sight (Moya et al., 2010). Hughes et al. (2011) discuss a variety of lines of
evidence that bear on the inclination, $i$, of the HR 8799 dust disk. While
they conclude that inclinations near $20^{\circ}$ are most likely, the
available data cannot rule out a face-on ($i=0^{\circ}$) configuration.
Finally an additional important constraint on the mass of HR 8799 b could be
obtained if it is responsible for truncating the inner edge of the dust disk.
An inner edge at 150 AU is consistent with available data (Su et al., 2009)
and this permits HR 8799 b to have a mass as large as $20\,\rm M_{J}$
(Fabrycky & Murray-Clay, 2010). It is worth noting, however, that this limit
depends upon the model-dependent inner edge of the disk and the dynamical
simulations.
Finally dynamical simulations of the planetary orbits constrained by the
available astrometric data can provide planetary mass limits. In the most
thorough study to date Fabrycky & Murray-Clay (2010) found that if planets c
and d were in a 2:1 mean-motion resonance their masses could be no larger than
about $10\,\rm M_{J}$. However if there were a double resonance in which c, d,
and b participated in a “double 2:1” or 1:2:4 resonance (originally identified
by Goździewski & Migaszewski (2009)) then masses as large as $20\,\rm M_{J}$
are permitted and such systems are stable for 160 Myr (Fabrycky & Murray-Clay,
2010). Such a resonance was found to be consistent with the limited baseline
of astrometric data. HR 8799 b,c, and d have also been identified in an
archived HST image taken in 1998 (Lafrenière et al., 2009; Soummer et al.,
2011). These data continue to allow the possibility of the 1:2:4 mean motion
resonance, a solution which implies a moderate inclination ($i=28^{\circ}$)
for the system. New dynamical models that include both this new astrometric
data and the innermost e planet are now required to fully evaluate the
system’s stability. Sudol & Haghighipour (2012) studied such a system with
masses of 7, 10, 10, and $10\,M_{\rm J}$. They generally found system
lifetimes shorter than 50 Myr for such large masses but at least one system
was found to be stable for almost 160 Myr.
Taken as a whole the age of the system and the available astrometric data and
dynamical models are consistent with a relatively young age (30 to 60 Myr) and
low masses for the planets (below $10\,\rm M_{J}$). However the possibility of
an older system age, as allowed by the asteroseismology, and higher planet
masses, as permitted if the planets are in resonance and by the dust disk
dynamics, cannot be fully ruled out. Given this background we now consider the
planetary atmosphere models.
### 4.2 Data Sources
The available photometric data for each planet is summarized in Table 2 and
shown on Figures 4–6. In addition for planet b we employ $H$ and $K$ band
spectra as tabulated in Barman et al. (2011a). We do not include the narrow
band photometry of Barman et al. (2011a) since this dataset has been
superseded by the spectroscopy. We also do not include very recent 3.3-$\rm\mu
m$ photometry from Skemer et al. (2012) which became available after the
submission of this manuscript although we do plot the point in Figures 4–6.
Below we summarize the sources of the photometry used in the fitting. With the
exception of the Subaru z-band which sits in an atmospheric window, we
included an atmospheric transmission curve when computing the synthetic
magnitudes of the model spectra. The transmission curve was generated with
ATRAN (Lord 1992) at an airmass of 1 with a precipitable water vapor content
of $2\,\rm mm$.
#### 4.2.1 Subaru-$z$ band
The Subaru-$z$-band photometry is from Currie et al. (2011) and was obtained
with the Infrared Camera and Spectrograph (IRCS; Tokunaga et al. (1998)) on
the Subaru Telescope. The filter profile was kindly provided by Tae-Soo Pyo.
No atmospheric absorption was included because the filter sits in a window
that is nearly perfectly transparent.
#### 4.2.2 $J$ band
The $J$ band data were taken from Marois et al. (2008) and Currie et al.
(2011). The former observations were done with the Near-Infrared Camera
(NIRC2) on Keck II which uses a Mauna Kea Observatories Near-Infrared (MKO-
NIR) $J$ band filter. We used the filter transmission profile from Tokunaga et
al. (2002). The latter observations were obtained with the Infrared Camera and
Spectrograph (IRCS; Tokunaga et al. (1998)) on the Subaru Telescope which also
uses a MKO-NIR $J$ band filter.
#### 4.2.3 $H$ and $Ks$ bands
The $H$-band and $K_{s}$-band data were taken from Marois et al. (2008). The
observations were done with the Near-Infrared Camera (NIRC2) on Keck II which
uses MKO-NIR filters. We used the filter transmission profile from Tokunaga et
al. (2002).
#### 4.2.4 [3.3] band
The [3.3]-band data was taken from (Currie et al., 2011). The observations
were done with the Clio camera at the MMT Telescope (Freed et al., 2004;
Sivanandam et al., 2006). The filter is non standard and has a central
wavelength of $3.3\,\rm\mu m$, and half-power points of 3.10 and $3.5\,\rm\mu
m$. The filter transmission profile was provided by Phil Hinz.
#### 4.2.5 $L^{\prime}$ band
The $L^{\prime}$-band data was taken from Currie et al. (2011). The filter is
the $L^{\prime}$ filter in the MKO-NIR system so we used the filter
transmission profile from Tokunaga et al. (2002).
#### 4.2.6 $M^{\prime}$-band
The $M$-band photometry of Galicher et al. (2011) was obtained using the Near-
Infrared Camera (NIRC2) on Keck II. This filter profile is the same as the
$M^{\prime}$ band of the MKO-NIR system. We therefore used the filter
transmission profile from Tokunaga et al. (2002).
### 4.3 Fitting Method
In order to determine the atmospheric properties of the HR 8799 planets, we
compared the observed photometry to synthetic spectra generated from our model
atmospheres. We used a grid of solar metallicity models with the following
parameters: $T_{\rm eff}=600$–$1300\,\rm K$ in steps of 50 K, $\log g=3.5$–5.5
in steps of 0.25 dex, $f_{\rm sed}=1,2$, and eddy mixing coefficient $K_{\rm
zz}=0$, $10^{4}\,\rm cm^{2}\,s^{-1}$. We identify the best fitting model and
estimate the atmospheric parameters of the planets following the technique
described in Cushing et al. (2012, in prep). In brief, we use Bayes’ theorem
to derive the joint posterior probability distribution of the atmospheric
parameters given the data $P(T_{\rm eff},\log g,f_{\rm sed},K_{\rm
zz}|\mathbf{f})$, where $\mathbf{f}$ represents a vector of the flux density
values (or upper limits) in each of the bandpasses. Since the posterior
distribution is only known to within a multiplicative constant, the practical
outcome is a list of models ranked by their relative probabilities.
Estimates and uncertainties for each of the atmospheric parameters can also be
derived by first marginalizing over the other parameters and then computing
the mean and standard deviation of the resulting distribution. For example,
the posterior distribution of $T_{\rm eff}$ is given by,
$P(T_{\rm eff}|\mathbf{f})=\int P(T_{\rm eff},\log g,f_{\rm sed},K_{\rm
zz}|\mathbf{f})\,\,d\,\log g\,\,df_{\rm sed}\,\,d\,K_{\rm zz}$
Since $(T_{\rm eff},\log g)$ values can be mapped directly to $(M,R,L_{\rm
bol})$ values using evolutionary models, we can also construct marginalized
distribution for $M$, $R$, and $L_{\rm bol}$. Figure 7 shows the resulting
distribution of $T_{\rm eff}$, $\log g$, $M$, and $L_{\rm bol}$ for each
planet and indicates the formal solution for these parameters and and their
associated uncertainties.
Finally note that we chose to use a Bayesian formalism rather than the more
common approach of minimizing $\chi^{2}$ because 1) we can marginalize over
model parameters such as the distance and radii of the brown dwarfs, and 2) we
can incorporate upper limits using the formalism described in Isobe et al.
(1986).
### 4.4 Results of Model Fitting
In this section we discuss the individual best fits to each planet. Figures 4
– 6 display the model fits to the observed spectra and photometry. Each panel
of Figure 8 shows contours, denoting integrated probabilities of 68, 95, and
99%, in the $\log g-T_{\rm eff}$ plane. In these figures evolution tracks for
planets and brown dwarfs of various masses are shown. The objects evolve from
right to left across the figures as they cool over time. Isochrones for a few
ages are shown; the kinks arise from deuterium burning. In some cases at a
fixed age a given $T_{\rm eff}$ can correspond to three different possible
masses (e.g., a 1150 K object at 160 Myr). Also shown are contours of constant
$L_{\rm bol}$. Note that the isochrones are derived from the conventional hot-
start giant planet evolution calculation. A different choice of initial
conditions would result in different isochrones.
The best fitting parameters are also shown in Figure 7 as histograms of
probability distribution for $T_{\rm eff}$, $\log g$, $M$ and $L$. For $\log
g$ and $T_{\rm eff}$ the histograms are projections of the contours shown in
Figure 8 onto these two orthogonal axes. The mean of the fit and the size of
the standard deviation is indicated in each panel and also illustrated by the
solid and dashed vertical lines. The third and fourth columns of Figure 7
depict the same information but for the mass and luminosity corresponding to
each $(T_{\rm eff},\log g)$ pair, as computed by the evolution model.
We discuss each set of fits for each planet in turn below.
#### 4.4.1 HR 8799b
HR 8799b is the only one of the three planets considered here for which there
is spectroscopic data and our results are sensitive to whether or not this
data is included in our fit. Contours which show the locus of the best fitting
models for the photometry are shown in the left-hand panel of Figure 8. When
only the photometric data is fit high masses around $\sim 26\,\rm M_{J}$ are
favored. The photometry-only fit finds $T_{\rm eff}=1000\,\rm K$ and $f_{\rm
sed}=2$ while a fit to both the spectroscopy and the photometry results in
$T_{\rm eff}=750\,\rm K$ and $f_{\rm sed}=2$ with a mass of $\sim 3\,\rm
M_{J}$. We reject the low temperature fit for several reasons: the solution
lies at the edge of our model grid, such a planet would be very young, and
such a cold effective temperature is not consistent with the bolometric
luminosity of planet b (see §5.2). These models are illustrated in the top two
panels of Figure 4.
To isolate the effect of the spectroscopy of Barman et al. (2011a) on the
preferred fit, we relaxed the radius and distance constraint on the fitting
and found the model that best reproduces the shape of the spectra. Somewhat
surprisingly this is a cold, very low gravity and very cloudy model ($T_{\rm
eff}=600\,\rm K$, $\log g=3.5$ and $f_{\rm sed}=1$). With a standard radius
such a model is again too young and faint and also lies at the edge of the
model grid.
The reason the derived gravity depends so strongly on the $H$ and $K$ spectra
is that the shape of the emergent flux–and not just the total flux in a given
band–contains information about the gravity. In particular a “triangular” $H$
band shape serves as an indicator of low gravity (see Rice et al. (2011) and
Barman et al. (2011a)). This shape results from the interplay of a continuum
opacity source–either cloud opacity (in a cloudy atmosphere) or the collision-
induced opacity of molecular hydrogen (when cloud opacity is unimportant)–and
a sawtooth-shaped water opacity (discussions in the literature generally only
highlight the latter). At high pressures the continuum hydrogen opacity and/or
the cloud opacity tends to fill in the opacity trough at the minimum of the
water opacity in $H$ band. Since the photosphere of lower gravity objects at
fixed effective temperature is at lower pressures, the $\rm H_{2}$ and cloud
opacity is somewhat less important allowing the angular shape of the water
opacity to more strongly control the emergent flux (see Figure 9 and Figure 6
of Rice et al. (2011)).
Thus we find that the shape of the $H$ band spectrum is responsible for
pulling the preferred model fits to low gravity and low effective temperature.
Weaker methane bands at lower $\log g$ in this $T_{\rm eff}$ range also push
the fit to lower gravity. The greater number of datapoints in the spectra
overwhelms the photometric data which is why the contours for the best overall
fit lie outside of the accepted luminosity range. As we discuss in Section 5.1
our preferred interpretation is that none of our current models match the true
composition, mass, and age of this planet.
The model which best fits the photometry alone in the top panel of Figure 4
fits the $YJHK$ and [3.3]-$\rm\mu m$ (but not the revised Skemer et al. (2012)
[3.3]) photometry to within $1\sigma$. The model is too bright at $L^{\prime}$
and $M^{\prime}$. The photometry plus spectrum fit features a methane band
head at $2.2\,\rm\mu m$ that is too prominent, even with $\log K_{zz}=4$. Both
sets of solutions, are inconsistent with the accepted age of the the star. The
lower mass solution would imply very young ages for the planet, well below 30
Myr. Conversely the higher mass range implies ages in excess of about 300 Myr.
Thus along with the discarded low mass fit the photometry-only, higher mass
fit is problematical since the mass conflicts with the constraints discussed
in Section 4.1
#### 4.4.2 HR 8799c
For planet c there is no available spectroscopy and we fit only to the
photometry. The formal best fitting solution yields $T_{\rm eff}=980\pm
70\,\rm K$ and $\log g=4.33\pm 0.28$ for a mass of $15\pm 8\,\rm M_{\rm J}$.
However in both the contour diagram (Figure 8) and the histogram (Figure 7) we
find two islands or clusters of acceptable fits, one at higher gravity and
effective temperature, and one with lower values for both. The high mass
solution lies at masses greater than $20\,\rm M_{J}$ and $T_{\rm eff}\sim
1100\,\rm K$. Such models are consistent only with ages around 300 My, well in
excess of the preferred age range for the primary and the dynamical
constraints on the mass. The second island of acceptable fits lies at $\log
g\sim 4.25$ and $T_{\rm eff}\sim 950\,\rm K$. Figure 5 illustrates the spectra
for the best fitting model from each case. The lower mass model has $\log
g=4.25$, $f_{\rm sed}=2$, and $\log K_{\rm zz}=4$, implying $M\approx 10\,\rm
M_{J}$ which is consistent with the dynamical mass constraint and represents
our preferred solution and is listed in Table 1. The age predicted by the
evolution of these models is about 160 Myr, consistent with the
asteroseismological age constraint but not the generally favored range of 30
to 60 Myr. However models with modestly lower gravity and slightly smaller
masses also fall within the $1\sigma$ contours seen in Figure 8 do lie within
this age range.
The cooler model fits most of the photometric points to within 2$\sigma$ or
better, but varies most significantly from the data at $[3.3]\,\rm\mu m$ and
$L^{\prime}$, which perhaps imply that despite the disequilibrium chemistry
the models have too much methane. The lower gravity solutions differ from the
high gravity ones most prominently in the red side of $K$ band (where the
cooler model has a much more prominent methane band head) and at 3 to
$4\,\rm\mu m$. By constraining the methane band depth in the $K$ band and, to
a lesser extent, in the $H$ band, spectroscopy has the potential to
distinguish between these two cases. The shape of $H$ band (Figure 9) can also
serve as a gravity discriminator with a more triangular shape indicating lower
gravity.
#### 4.4.3 HR 8799d
Because of larger observational error bars, the model fits for the innermost
of the three planets considered here are the most uncertain. As seen in Figure
8 the best fitting models allow masses ranging from 5 to $60\,\rm M_{J}$ and
$T_{\rm eff}$ between 900 and 1200 K. However the very best fitting models
favor solutions with $\log g$ around 4.25 to 4.50 and $T_{\rm eff}=1000\,\rm
K$ yielding a mass of 10 to $20\,\rm M_{\rm J}$. As with planet c such a
solution is consistent with the dynamical constraint but not the age
constraint. Also as with planet c the lower end of this mass range offers
marginally poorer fits that nevertheless still lie within the $1\sigma$
contour and that do satisfy the age constraint. The best fitting spectrum is
shown in Figure 6.
## 5 Discussion
### 5.1 Implied Masses and Ages
To summarize our findings from the previous section, each of the three planets
considered presents a different challenge to characterize. Some model fits to
planets c and d imply implausibly large masses or ages but other acceptable
fits satisfy all of the available constraints. Both c and d can be
characterized as having masses as low as 7 to $8\,\rm M_{\rm J}$, $T_{\rm
eff}=1000\,\rm K$, and $f_{\rm sed}=2$ which implies ages of around 60 Myr,
within the most commonly cited age range of the primary. Some better fitting
models have slightly larger masses ($10\,\rm M_{\rm J}$) and ages (160 Myr).
This age is greater than the range of ages typically quoted for the primary
star of 30 to 60 Myr, although it is within the range permitted by the
asteroseismology. Evolution models starting from a cooler initial state than
the hot-start models would reach these effective temperatures and gravity at a
younger age than 160 Myr and be more in accord with the usual age range.
For planet b none of the models are satisfactory. Since we do not allow
arbitrary radii models to fit the data (with the exception of the lowermost
panel in Figure 4), we cannot invoke what we judge to be unphysical radii to
produce acceptable fits. The solution which best fits the photometry alone has
$M=26\,\rm M_{\rm J}$, $T_{\rm eff}=1000\,\rm K$, and $f_{\rm sed}=2$, but
this mass clearly violates the constraints discussed in Section 4.1. A fit to
the entire spectral and photometric dataset results in $M\approx 3\,\rm M_{\rm
J}$, $T_{\rm eff}=750\,\rm K$, and $f_{\rm sed}=1$. However we discard this
model as discussed in Section 4.4.1. This effective temperature is cooler than
favored by Barman et al. (2011a) and Currie et al. (2011) but is comparable to
that found by Madhusudhan et al. (2011).
The most likely explanation for the difficulty in fitting this object is that
one of the assumptions of the modeling is incorrect. Barman et al. (2011a)
speculate that a super-solar atmospheric abundance of heavy elements might
explain the departures of the data from the models. Indeed all of the
atmospheres of solar system giant planets are enhanced over solar abundance
with a trend that the enhancement is greater at lower masses. For example
Saturn’s atmosphere is enhanced in methane by about a factor of ten while
Jupiter is only a factor of about three (see Marley et al. (2007b) for a
review). The available data on exoplanet masses and radii suggest that lower
mass planets are more heavily enriched in heavy elements than higher mass
planets (Miller & Fortney, 2011). If the mass of HR 8799b is intermediate
between our two sets of best fits, for example with a mass near 6 or $7\,\rm
M_{J}$, as favored by the discovery paper, and if atmospheric abundance trends
are similar in the HR 8799 system to our own, then it may not be surprising if
planet b has different atmospheric heavy element abundances than c and d. We
will consider non-solar abundance atmosphere models in a future paper. The
full range of model phase space has certainly not yet been explored.
Overall we find that a consistent solution can be found for planets c and d in
which both have similar masses and ages. This is essentially the solution
favored by the discovery paper (Marois et al., 2008) and is within the ranges
of favored solutions presented by Currie et al. (2011) and Madhusudhan et al.
(2011). However we differ from some of these previous studies in our finding
that the radii for planets b and c that are fully consistent with that
expected for their individual masses. Unusual radii are not required.
### 5.2 Bolometric Luminosities
The distance to HR 8799 has been measured as $d=39\pm 1.0\,\rm pc$ (van
Leeuwen, 2007) and thus the bolometric luminosity of each planet can be
computed from the observed photometry. In the discovery paper, Marois et al.
(2008) compare the photometry available at that time to models and brown dwarf
spectra and report the now commonly cited results $\log L_{\rm
bol}/L_{\odot}=-5.1\pm 0.1$ for planet b and $-4.7\pm 0.1$ for c and d.
Since the work of Marois et al. (2008), the photometry of the three planets
has been expanded to cover the SED from $\sim 1$–4.8$\,\mu$m. This better
constrains $L_{\rm bol}$ as $\sim$80% of the flux is emitted at these
wavelengths. In principle, the bolometric luminosity can be obtained by
fitting synthetic photometry to the observations, with a scaling factor chosen
to minimize the residuals. The integrated scaled flux of the model and the
known distance gives $L_{\rm bol}$ (Marois et al., 2008). The fitted model
thus provides an effective bolometric correction to the photometry by
approximating the flux between the photometric bands. The scaling factor
corresponds to $(R/d)^{2}$, where $R$ is the radius of the planet. The
optimized scaling thus corresponds to an optimization of the radius
independent of the physical radius of the planet. As is well known, this
results in radii for the HR 8799 planets that are considerably smaller than be
accounted for with the evolution models (Section 1.1). The approach can also
lead to unrealistic bolometric corrections if the fitted $T_{\rm eff}$
deviates too far from the actual value.
To circumvent this difficulty, here we determine $L_{\rm bol}$ by using the
radius obtained from our evolution sequences, which is consistent with our
approach to fit the photometry. Of course such theoretical radii have their
own uncertainty, including a dependence at young ages – particularly below 100
Myr – on the initial conditions (Baraffe et al., 2002; Marley et al., 2007a;
Spiegel & Burrows, 2012). We neglect the dependence on initial conditions
since planets forming in the ‘cold-start’ calculation of Marley et al. (2007a)
never get as warm or as bright as the HR 8799 planets. Intermediate cases,
such as explored by Spiegel & Burrows (2012) are possible, but we set those
aside for now. Our approach, however, does eliminate unphysical solutions by
constraining the radius to reasonable values (in excess of $1\,\rm R_{J}$).
Thus, for each fitted model $(T_{\rm eff},\log g)$ we obtain a $L_{\rm bol}$
from the radius $R(T_{\rm eff},\log g)$ obtained with the evolution222With
“perfect” atmosphere and evolution models the two methods would give identical
results.. The resulting probability distributions of $L_{\rm bol}$ for each
planet are shown in Fig. 7, along with the mean value and dispersion of each
distribution.
Our fits are based on a model grid with spacing of 50 K and 0.25 dex in
$T_{\rm eff}$ and $\log g$, respectively, which introduces an additional
uncertainty intrinsic to the fitting procedure of about half a grid spacing,
or $\pm 25\,$K and $\pm 0.13\,$dex. We derive the corresponding uncertainty in
$L_{\rm bol}$ as follows. The bolometric luminosity is given by
$L_{\rm bol}=4\pi R^{2}\sigma T_{\rm eff}^{4}={4\pi GM_{\odot}\sigma T_{\rm
eff}^{4}\over g}\Bigl{(}{M\over M_{\odot}}\Bigr{)},$
where the symbols have their usual meaning. From the cloudy evolution of
Saumon & Marley (2008), we find an approximate relation for $M(T_{\rm
eff},\log g)$ in the range of $T_{\rm eff}$ and mass of interest:
$\log{M\over M_{\odot}}=0.746\log g+{T_{\rm eff}\over 5090}-5.35,$
where $T_{\rm eff}$ is in $K$ and $g$ in cm s-2. Thus,
$\log L_{\rm bol}=4\log T_{\rm eff}+{T_{\rm eff}\over 5090}-0.254\log g+A,$
where $A$ is a constant. With the grid spacing uncertainties given above, we
find $\Delta\log L_{\rm bol}=\pm 0.054$, which we round up to 0.06. Combining
quadratically this uncertainty with the dispersion in $L_{\rm bol}$ found in
our fits (Fig. 7), we find the luminosity for planet b to be $\log L_{\rm
bol}/L_{\odot}=-4.95\pm 0.06$, $-4.90\pm 0.10$ for planet c333Note that the
dispersion for planet c is non-Gaussian (Fig. 7)., and $-4.80\pm 0.09$ for
planet d. These values are consistent with those reported by Marois et al.
(2008) although they are 0.1 dex brighter for planet b, 0.2 dex fainter for
planet c, and 0.1 dex fainter for planet d. The quoted uncertainties are lower
limits of course, since they do not account for obvious systematic errors in
the models (Figures 4–6).
### 5.3 Cloud Properties
Although there is a dispersion in the best fitting $\log g$ and $T_{\rm eff}$,
essentially all of the acceptable fits require a cloud sedimentation
efficiency of $f_{\rm sed}=2$. As shown in Figure 1 this value is typical of
the best fitting parameters for most field L dwarfs we have previously studied
(Cushing et al., 2008; Stephens et al., 2009). The persistence of clouds to
lower effective temperatures at low gravity is also apparent from this figure.
By 1000 K most field dwarfs with $\log g\geq 5$ have already progressed to
$f_{\rm sed}\geq 4$ whereas clouds persist much more commonly among lower
gravity objects down to 1000 K. By very cool effective temperatures, however,
the silicate and iron clouds have certainly departed from view as demonstrated
by the one $\log g=4$, $T_{\rm eff}\sim 500\,\rm K$ object (ULAS
J133553.45+113005.2, (Burningham et al., 2008; Leggett et al., 2009)).
As Figure 1 attests, the cloud in planets b, c, and d are unusual not so much
for their global characteristics (the same cloud model that describes L dwarf
clouds fits them as well), but rather for their persistence. At $f_{\rm
sed}=2$ there are three field objects with $T_{\rm eff}\leq 1200\,\rm K$.
These objects are 2MASS 0825196+211552 (Kirkpatrick et al., 2000), SDSS
085758+570851 (Geballe et al., 2002), and SDSS J151643.01+305344.4 (Chiu et
al. (2006); hereafter SDSS 1516+30). Their infrared spectral types are L6, L8,
and T0.5 and the first two are both redder in $J-K$ than is typical for those
spectral types (Stephens et al., 2009).
Figure 3 compares some of the model silicate cloud properties of a low gravity
planet with models for field L6 and T0.5 objects. As expected from the
discussion in Section 2.2, the lower gravity model is marked by a larger
particle size than the higher gravity models, and the column optical depth of
the silicate cloud in all three objects ends up being very similar. More
importantly the range of cloud optical depths that lie in the near-infrared
photosphere are similar for all three objects. Thus a low gravity ($\log
g=3.5$) object with $T_{\rm eff}=1000\,\rm K$ ends up with cloud opacity that
is very similar to a high gravity ($\log g=5.5$) object with $T_{\rm
eff}=1200\,\rm K$ and consequently similar spectra and colors. Indeed Barman
et al. (2011a) has already noted the similarity of SDSS 1516+30 to HR 8799b.
This congruence between lower gravity and higher gravity models led to the
initial surprise that the apparently cool planets seem to have clouds
reminiscent of higher gravity–and warmer–L dwarfs.
The relative contribution of clouds to the opacity in individual photometric
bands is depicted in Figure 10. This figure presents contribution functions
for the $J$, $H$, $K$, $L^{\prime}$, and $M^{\prime}$ bands for six different
combinations of gravity, effective temperature, and cloud treatment. The
contribution functions illustrate the fractional contribution to the emergent
flux as a function of pressure in the atmosphere. In a cloud-free, $T_{\rm
eff}=1000\,\rm K$, $\log g=5.0$ atmosphere (left panel, Figure 10a) the
$L^{\prime}$ flux emerges predominantly near $P=0.6\,\rm bar$ while the
$J$-band flux emerges from near 8 bar. The contribution functions do not
account for the effect of cloud opacity, but rather show for each case where
the flux would emerge from for that particular model if there were no clouds.
The center two panels of Figure 10a and b illustrate the vertical location of
the cloud layers for both $f_{\rm sed}=1$ and 2. The $f_{\rm sed}=2$ clouds
are thinner and the cloud base is deeper since these less cloudy atmospheres
are cooler than the $f_{\rm sed}=1$ case, as seen in the right hand panels. If
the cloud deck lies above or overlaps the plotted contribution function of a
given band then the emergent flux in that band will be strongly affected by
the presence of the cloud. The figure makes clear that regardless of gravity
thicker clouds impact more of the emergent spectra than thinner clouds. Clouds
described by $f_{\rm sed}=2$ strongly impact $J$, $H$, and $K$ bands, but are
less important at $L^{\prime}$, and $M^{\prime}$. We conclude that at least
for the effective temperature range inhabited by HR 8799 b, c, and d that
clouds are most strongly impacting the observed spectra at wavelengths shorter
than about $2.5\,\rm\mu m$ while the longer wavelength flux is primarily
emerging from above the cloud tops. Figures such as this illustrate the value
multi-band photometry has in both constraining not only the total emergent
flux, but also the vertical structure of the clouds.
### 5.4 Evolution with a gravity-dependent L to T transition
The growing evidence that the cloudy to cloudless transition in field brown
dwarfs depends on gravity (§2.1) is complemented by the published analyzes of
the HR 8799 planets (including the present work) which all indicate that their
atmospheres are cloudy and that they have $T_{\rm eff}$ well below the
estimated $\sim 1400\,$K limit of the L dwarf sequence. Thus, it appears that
the atmospheres of lower gravity dwarfs and of imaged exoplanets retain their
clouds to lower $T_{\rm eff}$, which is supported by simple cloud model
arguments (§2.2). As we have argued, this is the simplest interpretation of
the fact that the HR 8799 planets have $T_{\rm eff}$ typical of cloudless T
dwarfs but have evidently cloudy atmospheres. How is the evolution of brown
dwarfs across the transition from cloudy to clear atmosphere affected?
The atmosphere of a brown dwarf largely controls its evolution because it acts
as a surface boundary condition for the interior. A more opaque atmosphere
(more clouds, or higher metallicity, for instance) slows the escape of
radiation and increases the cooling time of the interior. Saumon & Marley
(2008) looked at the evolution of brown dwarfs across the transition by
assuming that the atmosphere was cloudy ($f_{\rm sed}=2$) down to $T_{\rm
eff}=1400\,$K, and clear below 1200 K, with an linear interpolation of the
atmospheric boundary condition in the transition regime. This effectively
corresponds to increasing the sedimentation efficiency across the transition,
one of the proposed explanations for the cloud clearing (§2.1). By converting
the evolution sequences to magnitudes using synthetic spectra ($f_{\rm sed}=1$
for cloudy atmospheres, and $f_{\rm sed}=4$ for “clear” atmospheres444These
are not fully consistent with the values used for the evolution, but the
effect on the evolution of this small difference in $f_{\rm sed}$ is small.) a
good match to the near-infrared color magnitude diagrams of field dwarfs was
found from the cloudless late M dwarfs, along the cloudy L dwarf sequence,
across the L/T transition and down to late T dwarfs.
We now extend this toy model to include a gravity-dependent range of $T_{\rm
eff}$ for the transition to explore the consequences, at the semi-quantitative
level, on the cooling tracks of brown dwarfs and exoplanets. In view of the
success obtained for field dwarfs (of relatively high gravity) with the Saumon
& Marley (2008) toy model, and the requirement that the lower gravity HR 8799
planets be cloudy at $T_{\rm eff}\sim 1000\,$K, we define the transition
region to be $T_{\rm eff}=1400$ to 1200 K at $\log g=5.3$ (cgs) and 900 to 800
K at $\log g=4$ with a linear interpolation in between (Fig. 11). The cloudy
boundary condition above the transition is based on our $f_{\rm sed}=2$
atmosphere models, and our cloudless models below the transition, as in Saumon
& Marley (2008). Synthetic magnitudes are generated from the cooling tracks
using our new $f_{\rm sed}=1$ and cloudless atmosphere models (Saumon et al.,
2012).
The resulting cooling tracks of two low-mass objects of 5 and 20 MJ are shown
in Fig. 12 where the same calculation, but based on a fixed $T_{\rm eff}$
transition (Fig. 11) is also displayed for comparison. It is immediately
apparent that these low-mass objects, which retain their clouds to lower
$T_{\rm eff}$ ($\sim 850\,$K for 5 MJ and $\sim 1050\,$K for 20 MJ) with the
prescribed gravity-dependent transition evolve along the L dwarf sequence
longer and reach the region of the color-magnitude diagram occupied by the HR
8799 planets before they turn to blue $J-K$ colors as the cloud clears. Also
remarkable is that in the transition region where the $J-K$ color changes from
$\sim 2$ to $\sim 0$, the low mass object is fainter in $K$ than the higher
mass object, the reverse of the situation for a transition that is independent
of $T_{\rm eff}$. This effect persists up to a cross over mass of $\sim
60\,$MJ above which the trend reverses (Fig. 11). This implies that low mass
objects that are in the transition region should appear below (i.e. be dimmer)
the field T0–T4 dwarfs, perhaps by up to 1–2 magnitudes. We note that the pile
up of objects in the transition region reported in Saumon & Marley (2008)
still occurs in this new calculation but it is more spread out in $T_{\rm
eff}$, as would be expected from the broader span of the transition in $T_{\rm
eff}$ (Fig. 11).
We emphasize that this evolution calculation is a toy model that has been
loosely adjusted to account for limited observational constraints. It reveals
trends but is not quantitatively reliable. In particular, we have had to use
$f_{\rm sed}=1$ to match the near infrared colors of the HR 8799 planets while
our best fits give $f_{\rm sed}=2$ for all three planets. This reflects the
fact that the models give different best-fit parameters when applied to a
subset of the data, a well-known difficulty with current models (Cushing et
al., 2008; Patience et al., 2012).
### 5.5 Mixing
Given the discussion in Section 1.3 regarding the prevalence of atmospheric
mixing resulting in departures from chemical equilibrium in solar system
giants and brown dwarfs, it is not surprising that mixing is also important in
warm exoplanet atmospheres as well. Barman et al. (2011a) discuss the
influence of non-equilibrium chemistry at low gravity and find that the $\rm
CO/CH_{4}$ ratio can become much larger than 1 in the regimes inhabited by the
HR 8799 planets. Also Barman et al. (2011b) found non-equlibrium chemistry was
likely important in 2M1207b.
We find that all of the best fitting models for each planet, b, c, and d,
include non-equilibrium chemistry. Within our limited grid with $K_{zz}=0$ and
$10^{4}\,\rm cm^{2}\,s^{-1}$, the latter choice was strongly preferred in all
cases providing yet another indication of the importance of chemical mixing in
substellar atmospheres. This also suggests that a fuller range of models with
a greater variety of eddy mixing strengths should be considered in future
studies to better constrain this parameter.
### 5.6 Mechanism for Gravity Dependent Transition
In Section 2.2 we demonstrated that the effect of a given cloud layer, all
else being equal, is greater in a lower mass extrasolar giant planet than in a
more massive brown dwarf of the same effective temperature. If we add
effective temperature as a variable then we find that a cooler low mass object
can have clouds comparable to a warmer high mass object. Such a congruence is
empirically demonstrated by the similar spectra of SDSS 1516+30 and HR 8799 b
(as originally noted by (Barman et al., 2011a)). The former is a $\sim 70\,\rm
M_{J}$, 1200 K field L dwarf while the latter is plausibly a few Jupiter mass,
1000 K young gas giant planet (although the modeling discussed here does not
select this solution). Likewise in Section 5.4 our simple evolution
calculation with a gravity-dependent L to T type transition temperature
illustrates that the location of young objects on the color magnitude diagram
can be understood if clouds remain to lower effective temperatures at lower
gravity. The fact that such behavior is dependent upon gravity is not in
itself surprising as a lower gravity would be expected to alter its behavior.
However the specific question remains, what is the specific mechanism that
results in lower mass objects making the L to T type spectral transition at
lower effective temperatures than higher mass objects? In this section we
offer some speculation while recognizing that a serious analysis is beyond the
scope of this paper.
A possible contributing factor might be found in the relative positions of the
convection zone and the photosphere as a function of gravity (a point also
raised in Barman et al. (2011a, b) and Rice et al. (2011). To illustrate this
effect in Figure 10 the contribution functions for different bandpasses are
shown for two different gravities. At $T_{\rm eff}=1000\,\rm K$ for moderately
cloudy ($f_{\rm sed}=2$) atmospheres the convection zone, regardless of
gravity, penetrates into the cloud layers that control the $J$ and $H$ band
fluxes. For cloudless atmospheres, however, the convection zone for the high
gravity case is quite deep ($P>20\,\rm bar$), well below even the region
probed by the $J$ band (Figure 10a). At lower gravity, however, the convection
zone penetrates higher into the atmosphere to much lower pressure, overlapping
the $J$ band contribution function (Figure 10b). If we imagine that a given
patch of atmosphere begins to clear, perhaps because of more efficient local
sedimentation, in the high gravity case the removal of cloud opacity leads the
atmosphere to become radiative and more quiescent, favoring particle
sedimentation relative to convective mixing and enlarging what began as a
localized clearing. At low gravity however the removal of cloud opacity does
not as dramatically push the atmosphere to a quiescent state. Thus convection
continues to loft cloud particles and the local clearing fills back in. Only
when the clear atmosphere convection zone lies very deep do the clouds
dissipate. Since low gravity atmospheres are more opaque than high gravity
ones this process of the growth of clearings begins at lower effective
temperature at lower gravity.
Another possibility is that detached convection zones play a role in hastening
the L to T transition. Within some effective temperature ranges there are two
atmospheric convection zones, one deeply seated and a detached zone that is
separated by a small radiative zone. This can be seen in the $f_{\rm sed}=1$
temperature profiles in Figure 10. Burrows et al. (2006) and Witte et al.
(2011) have speculated that the interplay of dynamical and cloud microphysics
effects that may occur when the intermediate radiative zone forms or departs
may play a role in the transition. Perhaps at some effective temperature
threshold particles forming in the upper convective zone grow large enough
that they fall all the way through the cloud base and the intermediate
radiative zone before they completely evaporate. Depending on the efficiency
of mixing in the radiative zone this could result in a net transport and
sequestration of condensate away from the near-infrared photosphere. Witte et
al. (2011) discuss a similar idea of the convection “fanning” the fall of
particles away from the upper zone. As seen in Figure 10, however, for both
the $f_{\rm sed}=2$ and the cloudless case there is only one convection zone,
so the potential for multilayered convection is less compelling in this case.
Nevertheless such mechanisms require more sophisticated modeling to ascertain
how they might be affected by gravity and effective temperature.
Arguments such as these that are based upon 1D radiative convective models
only scratch the surface of the underlying complex dynamical problem. For
example Freytag et al. (2010) performed two-dimensional radiation hydrodynamic
simulations of brown dwarf atmospheres to study the effects of clouds on
atmospheric convection. They found that atmospheric mixing driven by cloud
opacity launches gravity waves that in turn play a role in maintaining the
cloud structure. The Freytag et al. study considered a domain a few hundred
kilometers wide by about 100 km deep and only investigated a single gravity
($\log g=5$) so how such effects might vary with gravity is not yet known.
Furthermore how the local clouds might interact with the very large scale
planetary circulation has not been explored. Perhaps clouds form holes or
otherwise dissipate only when most of the cloud optical depth lies deeply
enough to be strongly influenced by global atmospheric circulation. Large
scale global dynamical simulations that capture the relevant physics of
particle and energy vertical and horizontal transport are likely required to
fully describe the L to T transition mechanism.
### 5.7 Future
Our experience in fitting the spectra of planet b in particular points to the
importance of spectra in the analysis. Adding the $H$ and $K$ band spectra to
the analysis results in much lower preferred masses than fitting photometric
data alone. Thus we expect that additional spectral data will further inform
future model fits.
As noted in Section 2.1 one hypothesis for the nature of the L to T transition
is that it involves partial clearing of the assumed global cloud cover. It is
possible that models which include partial cloudiness may better describe the
observed flux and Currie et al. (2011) have explored this possibility. Given
the limited data available today we feel the addition of another free model
parameter is premature and in any event we have found that brown dwarfs with
partial cloud cover have an overall near-infrared spectrum that resembles a
homogeneous dwarf with a thinner, homogenous global cloud (Marley et al.,
2010).
Another method for characterizing these planets and probing atmospheric
condensate opacity in self-luminous planets is by polarization (Marley &
Sengupta, 2011; de Kok et al., 2011). Marley & Sengupta (2011) found that
rapidly rotating, homogenously cloud-covered planets may be sufficiently
distorted to show polarization fractions of a few percent if they are
relatively low mass. de Kok et al. (2011) found that even when partial
cloudiness is considered much larger polarization fractions are unlikely.
However if this level of polarization could be measured in one of the HR 8799
planets this would confirm the presence of clouds and also place an upper
limit on the planetary mass. Objects in this effective temperature range (near
1000 K) and with $\log g>4$ are predicted to exhibit polarization well below
0.2%. Both SPHERE and GPI have polarization imaging modes, but it is not clear
if they would have sufficient sensitivity to place useful upper limits on the
HR 8799 system.
## 6 Conclusions
We have explored the physical properties of three of the planets orbiting HR
8799 by fitting our standard model spectra to the available photometry and
spectroscopy. Unlike some previous studies we have required that models with a
given $\log g$ and $T_{\rm eff}$ have a corresponding radius that is
calculated from a consistent set of evolution models. While the radii of the
planets are not variables, we do include two other free parameters: the cloud
sedimentation efficiency $f_{\rm sed}$ and the minimum value of the
atmospheric eddy mixing coefficient $K_{zz}$.
In agreement with all previous studies we find that the atmospheres of all
three planets are cloudy, which runs counter to the expectation of
conventional wisdom given their relative low effective temperature. However as
we argue in Sections 2.1 and 2.2, finding clouds to be present at lower
effective temperatures in lower gravity objects is fully consistent with
trends already recognized among field L and T dwarfs and from basic
atmospheric theory. We uniformly find that the best fitting value of the
sedimentation efficiency $f_{\rm sed}$ is, in essentially all cases, 2, which
is typical of the value seen in pre-L/T transition field L dwarfs (Fig. 1)
(Cushing et al., 2008; Stephens et al., 2009). In agreement with Barman et al.
(2011a) we thus find that the clouds in these objects are neither “radically
enhanced” (Bowler et al., 2010) nor representative of a “new class”
(Madhusudhan et al., 2011) of atmospheres.
As have some previous authors (Barman et al., 2011a, b) we find that eddy
mixing in nominally stable atmospheric layers is an important process for
altering the chemical composition of all three planets. While we have not
carried out a comprehensive survey of non-equilibrium models, we find that
values of the eddy mixing coefficient near $\log K_{zz}\sim 4$ generally fit
the available data better than models that neglect mixing. Such values are
typical of those found for field L and T dwarfs (e.g., Stephens et al., 2009)
and the stratospheres of solar system giant planets (e.g., see the detailed
discussion for Neptune in Bishop et al. (1995)).
The best fitting values for the primary model parameters $\log g$ and $T_{\rm
eff}$ are less secure. For HR 8799 b the inclusion of the $H$ and $K$ band
spectra of Barman et al. (2011a) drive our fits to low masses of $\sim 3\,\rm
M_{J}$ and effective temperatures, a solution which we discard as discussed in
Section 4.1.1. The photometry alone favors much higher masses, $\sim 25\,\rm
M_{J}$ that are apparently ruled out by dynamical considerations. Thus we find
no plausible model that fits all of the accepted constraints. Fits for the
planets c and d likewise generally favor higher masses, although there are
some solutions that are consistent with masses near or below $\sim 10\,\rm
M_{J}$ with ages consistent with the available constraints. For all three
planets the photometry predicted by the best fitting model is generally
consistent with the observed data within 1 to 2 standard deviations. We stress
that all of these fits have radii that are appropriate for the stated
effective temperature and gravity.
In conclusion the modeling approach that has successfully reproduced the
spectra of field L and T dwarfs seems to also be fully applicable to the
directly imaged planets. Nevertheless a larger range of model parameters,
including non-solar metallicity, must be explored in order to fully
characterize these objects as well as the planets yet to be discovered by the
upcoming GPI, SPHERE, and other coronagraphs.
## 7 Acknowledgements
We thank Travis Barman and Bruce Macintosh for helpful conversations and
Travis Barman for a particularly helpful review. This material is based upon
work supported by the National Aeronautics and Space Administration through
the Planetary Atmospheres and Astrophysics Theory Programs as well as the
Spitzer Space telescope Theoretical Research Program. This research was
supported in part by an appointment to the NASA Postdoctoral Program at the
Jet Propulsion Laboratory, administered by Oak Ridge Associated Universities
through a contract with NASA. Based in part on data collected at Subaru
Telescope, which is operated by the National Astronomical Observatory of
Japan. Observations used here were obtained at the MMT Observatory, a joint
facility of the University of Arizona and the Smithsonian Institution. Some of
the data presented herein were obtained at the W.M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W.M. Keck Foundation.
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Table 1: Summary of Derived Planet Properties Planet | Ref.11B11a=Barman et al. (2011a); C11=Currie et al. (2011); G11 = Galicher et al. (2011); M11=Madhusudhan et al. (2011); M12=this work. | $M$ ($M_{\rm Jup}$) | $\log g$ | $T_{\rm eff}\,\rm(K)$ | $R(R_{\rm J}$) | age (Myr) | $\log L_{\rm bol}/L_{\odot}$
---|---|---|---|---|---|---|---
b22Parameters derived by Bowler et al. (2010) are not listed because of very large scatter depending upon various assumptions. | B11a | $0.1-3.3$ | $3.5\pm 0.5$ | $1100\pm 100$ | 0.63 - 0.92 | $30-300$ | $-5.1\pm 0.1\tablenotemark{3}$
| C11 | $5-15$ | $4-4.5$ | $800-1000$ | $\cdots$ | $30-300$ |
| G11 | 1.8 | $4$ | $1100$ | 0.69 | $\cdots$ |
| M11 | $2-12$ | $3.5-4.3$ | $750-850$ | $\cdots$ | $10-150$ |
| M1244For b this is the formal best fit single model to the photometry alone, for c and d these are the preferred solution ranges as discussed in the text. The b fit is incompatible with the generally accepted constraints as discussed in the text. Formal solutions are shown in Figure 7. | 26 | 4.75 | 1000 | 1.11 | 360 | $-4.95\pm 0.06$
c | C11 | $7-17.5$ | $4-4.5$ | $1000-1200$ | $\cdots$ | $30-300$ | $-4.7\pm 0.1\tablenotemark{3}$
| G11 | 1.1 | $3.5$ | $1200$ | 0.97 | $\cdots$ |
| M11 | $7-13$ | $4-4.3$ | $950-1025$ | $\cdots$ | $30-100$ |
| M12 | 8 – 11 | $4.1\pm 0.1$ | $950\pm 60$ | 1.32 – 1.39 | 40 – 100 | $-4.90\pm 0.10$
d | C11 | $5-17.5$ | $3.75-4.5$ | $1000-1200$ | $\cdots$ | $30-300$ | $-4.7\pm 0.1\tablenotemark{3}$
| G11 | 6 | $4.0$ | 1100 | 1.25 | $\cdots$ |
| M11 | $3-11$ | $3.5-4.2$ | $850-1000$ | $\cdots$ | $10-70$ |
| M12 | $8-11$ | $4.1\pm 0.1$ | $1000\pm 75$ | 1.33 – 1.41 | 30 – 100 | $-4.80\pm 0.09$
Table 2: Photometric Data for the HR 8799 Planets Planet | Band | Abs. Mag. | Ref.11C11=Currie et al. (2011)
M08=Marois et al. (2008)
G11=Currie et al. (2011)
---|---|---|---
b | Subaru-$z$ | $18.24\pm 0.29$ | C11
| $J$ | $16.52\pm 0.14$ | C11
| $H$ | $14.87\pm 0.17$ | M08
| $K_{s}$ | $14.05\pm 0.08$ | M08
| [3.3] | $13.96\pm 0.28$ | C11
| $L^{\prime}$ | $12.68\pm 0.12$ | C11
| $M^{\prime}$ | $13.07\pm 0.30$ | G11
c | Subaru-$z$ | $>16.48$ | C11
| $J$ | $14.65\pm 0.17$ | M08
| $H$ | $13.93\pm 0.17$ | M08
| $K_{s}$ | $13.13\pm 0.08$ | M08
| [3.3] | $12.64\pm 0.20$ | C11
| $L^{\prime}$ | $11.83\pm 0.07$ | C11
| $M^{\prime}$ | $12.05\pm 0.14$ | G11
d | Subaru-$z$ | $>15.03$ | C11
| $J$ | $15.26\pm 0.43$ | M08
| $H$ | $13.86\pm 0.22$ | M08
| $K_{s}$ | $13.11\pm 0.12$ | M08
| [3.3] | $>11.63$ | C11
| $M^{\prime}$ | $11.67\pm 0.35$ | G11
Figure 1: Model parameters $f_{\rm sed}$ and $T_{\rm eff}$ as derived by
various applications of Marley & Saumon atmosphere and evolution models. Size
of dot reflects derived $\log g(\rm cm\,s^{-2})$ and ‘nc’ denotes cloudless
models (note that ‘nc’, which corresponds to $f_{\rm sed}\rightarrow\infty$,
is arbitrarily plotted at $f_{\rm sed}=5$). Points which would otherwise
overlap are slightly offset vertically and the $T_{\rm eff}$ values decrease
to the right to suggest evolution in time. The points for HR 8799 c and d from
the analysis here are labeled with planet designator. Remaining points are
from Geballe et al. (2001); Mainzer et al. (2007); Leggett et al. (2007a,
2008); Geballe et al. (2009); Leggett et al. (2009); Stephens et al. (2009);
Mainzer et al. (2011) although fits to unresolved binaries and objects with
very poorly constrained properties (e.g., Gl 229 B with $\log g$ uncertain by
a full dex) are excluded. SDSS 1516+30 is denoted by ‘1516’. The cross denotes
size of the typical uncertainties in the model fits which are usually $\pm
100\,\rm K$ in effective temperature, $\pm 0.25\,\rm dex$ in $\log g$, and
$\pm 0.5$ in $f_{\rm sed}$, although the uncertainty analysis is not uniform
across the various sources.
Figure 2: Model atmosphere temperature-pressure profiles for cloudy brown
dwarfs and planets assuming $f_{\rm sed}=2$ (Ackerman & Marley, 2001). Each
profile is labeled with $\log g$ and $T_{\rm eff}$ of the model. The
condensation curve for forsterite is shown with a dotted line.
Figure 3: Silicate cloud properties as computed by the Ackerman & Marley
(2001) cloud model for three models. From left to right the the best-fitting
models Stephens et al. (2009) for 2MASS 0825+21 and SDSS 1516+30 are shown
along with a profile for a young, cloudy, three Jupiter mass planet. Labels
underneath each object name denote model $T_{\rm eff}(\rm K)$ / $\log g\,({\rm
cgs})$ / $f_{\rm sed}$. Dashed curves show the effective radius, $r_{\rm eff}$
of the particles on the top axis. The column optical depth as measured from
the top of the atmosphere is shown by the solid lines and the scale on the
bottom axis. Thicker lines denote the region of the cloud which lies within
the $\lambda=1$ to $6\,\rm\mu m$ photosphere. Other modeled clouds are not
shown for clarity.
Figure 4: Observed (black) and model (red, green, purple) photometry and
spectra (see Table 1 and Barman et al. (2011a)) for HR 8799b. Models are
identified in the upper left hand corner of each panel by $T_{\rm eff}/\log
g\,({\rm cgs})/f_{\rm sed}/K_{zz}$. The top panel shows the model that best
fits the photometry alone while the middle panel shows the solution that best
fits both the photometry (excluding $H$ and $K$ bands) and spectroscopy
simultaneously. Model fluxes and photometry have been computed for radii
specific to the $T_{\rm eff}$ and $\log g$ of the atmosphere model at a
distance of 39.4 pc as observed from Earth. The [3.3] $\rm\mu m$ photometry of
Skemer et al. (2012) is shown as a blue star and is not included in the fits
but rather is shown for comparison purposes only. The lower panel shows the
model that best fits the $H$ and $K$-band spectrum alone. However in contrast
to the top two panels where the absolute flux level of the models are set by
the model radii and known distance to HR 8799, the absolute flux level of the
model in the lower panel is determined by minimizing $\chi^{2}$ between the
models and data.
Figure 5: The two best fitting model spectra for HR 8799 c. Observed
photometry (see Table 2) is shown in black, high and low gravity solutions in
green and red, respectively. The two solutions correspond to the centers of
the two best fitting islands in the contour plot shown in the middle panel of
Figure 8. Models are identified in the upper left hand corner by $T_{\rm
eff}/\log g\,({\rm cgs})/f_{\rm sed}/K_{zz}$. The [3.3] $\rm\mu m$ photometry
of Skemer et al. (2012) is shown as a blue star and is not included in the
fits but rather is shown for comparison purposes only.
Figure 6: The best fitting model for HR 8799 d. Observed photometry (see Table
1) is shown in black; model photometry is indicated by the red dots. Model is
identified in the upper left hand corner by $T_{\rm eff}/\log g\,({\rm
cgs})/f_{\rm sed}/K_{zz}$. The 3.3-$\rm\mu m$ photometry of Skemer et al.
(2012) is shown as a blue star and is not included in the fits but rather is
shown for comparison purposes only.
Figure 7: Histograms depicting the probability density distributions of the
various model parameters to planets HR 8799 b, c, and d. For planet b only the
results for the fitting of the photometry are shown. The $T_{\rm eff}$ and
$\log g$ histograms can be thought of as the projection of the contours shown
in Figure 8 onto these two orthogonal axes. In each case the mean of the fit
and the standard deviation are indicated by $\mu$ and $\sigma$, respectively.
These quantities are in turn illustrated by the solid and dashed vertical
lines. For the parameters for planet b, only a single model is identified so
no standard deviation is given. The third and fourth columns of histograms
depict the same information as the first two, but for the mass and luminosity
corresponding to each $(T_{\rm eff},\log g)$ pair, as computed by the
evolution model.
Figure 8: Contours illustrate domain of best-fitting models on the $\log
g-T_{\rm eff}$ plane. For each planet three contours are shown which
correspond to integrated probabilities of 68, 95, and 99% (red, thick to thin
contours). Evolution tracks from Saumon et al. (2007) are shown as labeled
black curves; planets evolve from right to left with time across the diagram
as they cool and contract. Blue curves are isochrones at (bottom to top) 30,
160, and 300 Myr; kinks in the older two isochrones arise from deuterium
burning (objects burning D are substantially hotter than lower mass objects of
the same age). Green curves are constant luminosity curves at (left to right)
$\log L/L_{\odot}=-5,-4.75,-4.5$. For planet b solid contours denote fits to
only the photometry while dashed curves are fits to photometry and H and
K-band spectra. Crosses denote the individual model cases plotted in Figures 4
– 6.
Figure 9: Model spectra at fixed $T_{\rm eff}=900\,\rm K$ and varying
gravities (labeled along right hand side), including several of the cases
shown in Figure 2. Models are shown at a spectral resolution $R=1000$.
Figure 10: Illustration of the effect of gravity and cloud properties on
modeled emergent flux for $T_{\rm eff}=1000\,\rm K$ and $\log g=5.0$ (a) and
3.75 (b). Both plots (a) and (b) consist of four sub-panels. The right-most
sub-panel depicts the $T(P)$ profiles for three atmosphere models with the
indicated $T_{\rm eff}$ and $\log g$. In both cases the profiles are for (left
to right) for cloudless, $f_{\rm sed}=2$, and 1 models. Thick lines denote the
convective regions of the atmosphere models. The dotted line denotes chemical
equilibrium between CO and $\rm CH_{4}$. The dashed lines are the condensation
curves for Fe (right) and $\rm Mg_{2}SiO_{4}$ (left). The cloud base is
expected at the point where the condensation curves cross the $T(P)$ profiles.
Remaining panels show the contribution function (see text) averaged over the
J, H, K, $L^{\prime}$ and $M^{\prime}$ bandpasses (colored lines) for each of
the three model cases. The shaded regions denote the extent of the cloud,
extending from the point where the integrated optical depth from the top of
the model is 0.1 to the cloud base. Thick horizontal dashed line denotes cloud
$\tau=2/3$.
Figure 11: Definition of the transition from cloudy to cloudless surface
boundary condition for the evolution. This represents a toy model of the L/T
transition. In the hybrid toy model of Saumon & Marley (2008), the transition
region is independent of gravity and the cloud clearing occurred between
$T_{\rm eff}=1400$ and 1200 K (lightly hashed area). To the right of the
transition region shown, the surface boundary condition is based on cloudy
atmosphere models; to the left, on cloudless atmospheres; and on a simple
interpolation in the transition region. Here, we present an evolution
calculation where the $T_{\rm eff}$ range of the transition is made gravity
dependent (densely hashed area). Representative cooling tracks are shown in
black and labeled by the mass. Isochrones are the blue dotted lines.
Figure 12: Examples of cooling tracks for objects of 5 MJ (red) and 20 MJ
(blue) in a $M_{K}$ vs. $J-K$ (MKO system) color-magnitude diagram where the
transition from cloudy ($f_{\rm sed}=1$) to cloudless atmospheres is taken
into account explicitly as in Saumon & Marley (2008). Dashed lines show the
evolution when the transition occurs over a fixed range of $T_{\rm eff}$ that
is independent of gravity, solid lines show the evolution for the gravity-
dependent transition (see Fig. 11). The planets in the HR 8799 planets are
shown with green symbols while resolved field objects are shown in black (M
dwarfs), red (L dwarfs) and blue (T dwarfs). The photometry is from Leggett et
al. (2002), Knapp et al. (2004), Marocco et al. (2010) McCaughrean et al.
(2004), Burgasser et al. (2006), and Liu & Leggett (2005). The parallaxes are
from Perryman et al. (1997), Dahn et al. (2002), Tinney et al. (2003), Vrba et
al. (2004), Marocco et al. (2010), and various references in Leggett et al.
(2002).
|
arxiv-papers
| 2012-05-29T20:35:55 |
2024-09-04T02:49:31.342384
|
{
"license": "Public Domain",
"authors": "Mark S. Marley, Didier Saumon, Michael Cushing, Andrew S. Ackerman,\n Jonathan J. Fortney, Richard Freedman",
"submitter": "Mark S. Marley",
"url": "https://arxiv.org/abs/1205.6488"
}
|
1205.6547
|
# Hermite polynomials related to Genocchi, Euler and Bernstein polynomials
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Jong Jin Seo
Department of Applied Mathematics, Pukyong National University, Busan 608-737,
Republic of KOREA seo2011@pknu.ac.kr (Corresponding author) and Mehmet
Acikgoz University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
The objective of this paper is to derive some interesting properties of
Genocchi, Euler and Bernstein polynomials by means of the orthogonality of
Hermite polynomials.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Genocchi numbers and polynomials, Euler numbers and
polynomials, Bernstein polynomials, orhogonality.
## 1\. Introduction
The Genocchi, Euler and Bernstein polynomials possess many interesting
properties and arising in many areas of mathematics. These polynomials have
been studied by many researcher (see [1-17]).
Recently, Kim $et$ $al$., studied on the Hermite polynomials and their
applications associated with Bernoulli an Euler numbers in ”D. S. Kim, T. Kim,
S. H. Rim and S. H. Lee., Hermite polynomials and their applications
associated with Bernoulli and Euler numbers, Discrete Dynamics in Nature and
Society, http://www.hindawi.com/journals/ddns/aip/974632/ (Article in Press).
They derived some interesting properties of Hermite polynomials by using its
orthogonality properties. They also showed that Hermite polynomials related to
Bernoulli and Euler polynomials. It is objective of this paper to derive for
Bernstein, Genocchi and Euler polynomials by using same method of theirs.
We firstly list definition of Euler, Genocchi, Bernstein and Hermite
polynomials as follows:
The ordinary Euler polynomials are defined by the means of the following
generating function:
(1)
$\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}=e^{xt}\frac{2}{e^{t}+1}\text{.}$
Substituting $x=0$ into (1), then we have $E_{n}\left(0\right):=E_{n}$, which
is called the Euler numbers (for details, see [1], [3], [5], [10], [11] and
[12]).
As is well-known, Genocchi polynomials are also defined by
(2)
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=e^{xt}\frac{2t}{e^{t}+1}\text{.}$
Similarly, for $x=0$ in (2), $G_{n}\left(0\right):=G_{n}$ are Genocchi numbers
(for details about this subject, see [12], [16] and [17]).
Let $C\left(\left[0,1\right]\right)$ be the space of continuous functions on
$\left[0,1\right]$. For $C\left(\left[0,1\right]\right)$, the Bernstein
operator for $f$ is defined by
$B_{n}\left(f,x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)B_{k,n}\left(x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)\binom{n}{k}x^{k}\left(1-x\right)^{n-k}$
where $n,$ $k\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,...\right\\}$. Here
$B_{k,n}\left(x\right)$ is called Bernstein polynomials, which are defined by
(3) $B_{k,n}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}\text{,
}x\in\left[0,1\right]$
(see [1], [4] and [17]).
The Hermite polynomials are defined by the generating function as follows:
(4)
$\sum_{n=0}^{\infty}H_{n}\left(x\right)\frac{t^{n}}{n!}=e^{H\left(x\right)t}=e^{2tx-t^{2}}$
with the usual convention about replacing by $H^{n}\left(x\right)$ by
$H_{n}\left(x\right)$.
The Hermite polynomials have the following properties
(5)
$H_{n}\left(x\right)=\left(-1\right)^{n}e^{x^{2}}\frac{d^{n}e^{-x^{2}}}{dx^{n}}\text{.}$
By (4), we have the following
(6) $\frac{dH_{n}\left(x\right)}{dx}=2nH_{n-1}\left(x\right)\text{,
}n\in\mathbb{N}\text{.}$
The Hermite polynomials have the orthogonal properties as follows:
(7)
$\int_{-\infty}^{\infty}e^{-x^{2}}H_{n}\left(x\right)H_{m}\left(x\right)dx=\left\\{\QATOP{2^{n}n!\sqrt{\pi}\text{,
if }n=m}{0\text{, if }n\neq m.}\right.$
By (7), it is not difficult to show that
(8) $\int_{-\infty}^{\infty}e^{-x^{2}}x^{l}dx=\left\\{\QATOP{0\text{, if
}l\equiv
1\left(\mathop{\mathrm{m}od}2\right)}{\frac{l!\sqrt{\pi}}{2^{l}\left(\frac{l}{2}\right)!}\text{,
if }l\equiv 0(\mathop{\mathrm{m}od}2),}\right.$
where $l\in\mathbb{Z}_{+}$. By (8), we can derive the following
(9)
$\int_{-\infty}^{\infty}\left(\frac{d^{n}e^{-x^{2}}}{dx^{n}}\right)x^{m}dx=\left\\{\QATOP{0\text{,
if }n>m\text{ with }n-m\equiv
1(\mathop{\mathrm{m}od}2)}{\frac{m!\left(-1\right)^{n}\sqrt{\pi}}{2^{m-n}\left(\frac{m-n}{2}\right)!}\text{,
if }n\leq m\text{ with }n-m\equiv 0(\mathop{\mathrm{m}od}2).}\right.$
We note that
$H_{o}\left(x\right),H_{1}\left(x\right),H_{2}\left(x\right),...,H_{n}\left(x\right)$
are orthogonal basis for the space
$\mathcal{P}_{n}=\left\\{p\left(x\right)\in\mathbb{Q}\left[x\right]\mid\deg
p\left(x\right)\leq n\right\\}$ with respect to the inner product
(10) $\left\langle
p\left(x\right),q\left(x\right)\right\rangle=\int_{-\infty}^{\infty}e^{-x^{2}}p(x)q\left(x\right)dx\text{.}$
For $p\left(x\right)\in\mathcal{P}_{n}$, the polynomial $p\left(x\right)$ is
given by
(11) $p(x)=\sum_{k=0}^{n}C_{k}H_{k}\left(x\right)$
where
$\displaystyle C_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{k}k!\sqrt{\pi}}\left\langle
p\left(x\right),H_{k}\left(x\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)p\left(x\right)dx\text{.}$
For more informations of Eqs. (4-12), you can refer to [11].
## 2\. Main Results
In this section, we consider families of Bernoulli, Bernstein and Genocchi
polynomials. Then, we discover novel properties for them.
Let us take $p\left(x\right)=G_{n}\left(x\right)$. From (11),
$p\left(x\right)$ can be rewritten as
$G_{n}\left(x\right)=\sum_{k=0}^{n}C_{k}H_{k}\left(x\right)$
where
$C_{k}=\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)G_{n}\left(x\right)dx\text{.}$
Now, let us solve
$\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)G_{n}\left(x\right)dx$
as follows:
$\displaystyle\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)G_{n}\left(x\right)dx$
$\displaystyle=$
$\displaystyle\left(-n\right)\left(-\left(n-1\right)\right)...\left(-\left(n-k+1\right)\right)\int_{-\infty}^{\infty}e^{-x^{2}}G_{n-k}\left(x\right)dx$
$\displaystyle=$ $\displaystyle\frac{n!}{2^{k}k!}\sum_{\underset{l\equiv
0(\mathop{\mathrm{m}od}2)}{0\leq l\leq
n-k}}\frac{G_{n-k-l}}{\left(n-k-l\right)!2^{l}\left(\frac{l}{2}\right)!}\text{.}$
Thus, we procure the following theorem.
###### Theorem 2.1.
For $n\in\mathbb{Z}_{+}$, we have
$G_{n}\left(x\right)=n!\sum_{k=0}^{n}\frac{1}{2^{k}k!}\sum_{\underset{l\equiv
0(\mathop{\mathrm{m}od}2)}{0\leq l\leq
n-k}}\frac{G_{n-k-l}H_{k}\left(x\right)}{\left(n-k-l\right)!2^{l}\left(\frac{l}{2}\right)!}\text{.}$
Now we consider $p\left(x\right)=B_{l,n}\left(x\right)$. From (11), we note
that, $p\left(x\right)$ can be rewritten as
$B_{l,n}\left(x\right)=\sum_{k=0}^{n}C_{k}H_{k}\left(x\right)$
where
$\displaystyle C_{k}$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)B_{l,n}\left(x\right)dx$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\frac{n!}{l!(n-l)!}\left[\sum_{j=0}^{n-l}\frac{\left(n-l\right)!}{j!(n-l-j)!}\left(-1\right)^{j}\left\\{\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)x^{l+j}dx\right\\}\right]$
$\displaystyle=$ $\displaystyle\frac{n!}{k!}\sum_{\underset{k-l-j\equiv
0(\mathop{\mathrm{m}od}2)}{0\leq j\leq
n-l}}\binom{l+j}{l}\frac{\left(-1\right)^{j}}{\left(n-l-j\right)!2^{l+j}\left(\frac{l+j-k}{2}\right)!}\text{.}$
Thus, we have novel properties of Bernstein polynomials with the following
theorem.
###### Theorem 2.2.
For $n\in\mathbb{Z}_{+}$ and $0\leq l\leq n$, we have
$B_{l,n}\left(x\right)=n!\sum_{k=0}^{n}\frac{1}{k!}\sum_{\underset{k-l-j\equiv
0(\mathop{\mathrm{m}od}2)}{0\leq j\leq
n-l}}\binom{l+j}{l}\frac{\left(-1\right)^{j}}{\left(n-l-j\right)!2^{l+j}\left(\frac{l+j-k}{2}\right)!}H_{k}\left(x\right)\text{.}$
In [10], Kim $et$ $al$., derived the following interesting equality:
$\sum_{k=0}^{n}E_{k}\left(x\right)x^{n-k}=\frac{1}{2}\sum_{j=0}^{n-1}\binom{n+1}{j}\left\\{-\sum_{l=j}^{n-1}E_{l-j}+2\right\\}E_{j}\left(x\right)+(n+1)E_{n}\left(x\right)\text{.}$
We now consider as $p\left(x\right)=\sum_{k=0}^{n}E_{k}\left(x\right)x^{n-k}$.
Then, we compute as follows:
$\displaystyle C_{k}$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)p\left(x\right)dx$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\left\\{\begin{array}[]{c}\frac{1}{2}\sum_{j=0}^{n-1}\binom{n+1}{j}\left\\{-\sum_{l=j}^{n-1}E_{l-j}+2\right\\}\left[\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)E_{j}\left(x\right)dx\right]\\\
+(n+1)\int_{-\infty}^{\infty}\left(\frac{d^{k}e^{-x^{2}}}{dx^{k}}\right)E_{n}\left(x\right)dx\end{array}\right\\}$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k}}{2^{k}k!\sqrt{\pi}}\left\\{\begin{array}[]{c}\frac{1}{2}\sum_{j=0}^{n-1}\binom{n+1}{j}\left\\{-\sum_{l=j}^{n-1}E_{l-j}+2\right\\}\left[\frac{\left(-1\right)^{k}j!}{\left(j-k\right)!}\int_{-\infty}^{\infty}e^{-x^{2}}E_{j-k}\left(x\right)dx\right]\\\
+(n+1)\frac{\left(-1\right)^{n}n!}{\left(n-k\right)!}\int_{-\infty}^{\infty}e^{-x^{2}}E_{n-k}\left(x\right)dx\end{array}\right\\}$
After these applications, we easily reach the following
$\displaystyle C_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=0}^{n-1}\binom{n+1}{j}\left\\{-\sum_{l=j}^{n-1}E_{l-j}+2\right\\}\frac{j!}{\left(j-k\right)!}\sum_{\underset{m\equiv
0(\mathop{\mathrm{m}od}2)}{0\leq m\leq
j-k}}\binom{j-k}{m}E_{j-k-m}\frac{m!}{2^{m+k}k!\left(\frac{m}{2}\right)!}$
$\displaystyle+\left(n+1\right)\left(-1\right)^{n+k}\sum_{\underset{s\equiv
0\left(\mathop{\mathrm{m}od}2\right)}{0\leq s\leq
n-k}}\binom{n}{k}\binom{n-k}{s}E_{n-k-s}\frac{s!}{2^{s+k}\left(\frac{s}{2}\right)!}\text{.}$
Consequently, we have the following theorem.
###### Theorem 2.3.
The following identity holds true:
$\displaystyle\sum_{k=0}^{n}E_{k}\left(x\right)x^{n-k}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{H_{k}\left(x\right)}{2}\sum_{j=0}^{n-1}\binom{n+1}{j}\left\\{-\sum_{l=j}^{n-1}E_{l-j}+2\right\\}\frac{j!}{\left(j-k\right)!}$
$\displaystyle\times\sum_{\underset{m\equiv 0(\mathop{\mathrm{m}od}2)}{0\leq
m\leq
j-k}}\binom{j-k}{m}E_{j-k-m}\frac{m!}{2^{m+k}k!\left(\frac{m}{2}\right)!}$
$\displaystyle+\left(n+1\right)\sum_{k=0}^{n}\left(-1\right)^{n+k}\sum_{\underset{s\equiv
0\left(\mathop{\mathrm{m}od}2\right)}{0\leq s\leq
n-k}}\binom{n}{k}\binom{n-k}{s}H_{k}\left(x\right)E_{n-k-s}\frac{s!}{2^{s+k}\left(\frac{s}{2}\right)!}\text{.}$
## References
* [1] A. Bayad, T. Kim, Identities involving values of Bernstein $q$-Bernoulli and $q$-Euler polynomials, Russ. J. Math. Phys. 18 (2011), no. 2, 133-143.
* [2] T. Kim J. Choi, Y. H. Kim, C. S. Ryoo, On $q$-Hermite polynomials, Proc. Jangjeon Math. Soc., 14 (2011), No. 2, 215-221.
* [3] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-Stiriling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16(2009), no. 4, 484-491.
* [4] T. Kim, A note on $q$-Bernstein polynomials, Russ. J. Math. Phys., 18 (2011), 73-82.
* [5] T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161–170.
* [6] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299.
* [7] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008), 598–608.
* [8] T. Kim, $q$-generalized Euler numbers and polynomials, Russ. J. Math. Phys. 13 (2006), no. 3, 293-298.
* [9] T. Kim, The symmetry $p$-adic invariant integral on $\mathbb{Z}_{p}$ for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), 1267-1277.
* [10] D. S. Kim, D. V. Dolgy, T. Kim and S. H. Rim, Some Formulae for the product of Two Bernoulli and Euler Polynomials, Abstr. Appl. Anal., 2012(2012), Art. ID 784307, 14 pages(Article in Press).
* [11] D. S. Kim, T. Kim, S. H. Rim and S. H. Lee., Hermite polynomials and their applications associated with Bernoulli and Euler numbers, Discrete Dynamics in Nature and Society, http://www.hindawi.com/journals/ddns/aip/974632/ (Article in Press).
* [12] H. Jolany and H. Sharifi, Some results for Apostol-Genocchi Polynomials of higher order, In press in Bulletin of the Malaysian Mathematical Sciences Society vol 36, no.2, 2013.
* [13] M. Acikgoz, Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
* [14] S. H. Rim and J. Jeong, A note on the Modified $q$-Euler Numbers and polynomials with weight $\alpha$, International Mathematical Forum, Vol. 6, 2011, no. 65, 3245-3250.
* [15] C. S. Ryoo, A note on the weighted $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), page 47-54
* [16] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, accepted in Bulletin of the Malaysian Mathematical Sciences and Society.
* [17] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
|
arxiv-papers
| 2012-05-30T05:37:40 |
2024-09-04T02:49:31.358265
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Jong Jin Seo and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1205.6547"
}
|
1205.6579
|
# Results on direct $C\\!P$ violation in $B$ decays in LHCb
T.M. Karbach aaaon behalf of the LHCb collaboration
I present three studies from the LHCb experiment on the subject of direct
$C\\!P$ violation in $B^{0}$ and $B^{0}_{s}$ decays. First, we measure the
$C\\!P$ asymmetry in $B^{\pm}\rightarrow\psi K^{\pm}$ decays, with
$\psi={J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu},\psi(2S)$, using
$0.35\mbox{\,fb}^{-1}$ of $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-0.80005ptV}$. We find no evidence for $C\\!P$
violation. Second, using the same data sample, we see the first evidence of
$C\\!P$ violation in the decays of $B^{0}_{s}$ mesons to $K^{\pm}\pi^{\mp}$
pairs, $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\mathrm{(stat)}\pm
0.02\mathrm{(syst)}$ ($3.3\sigma$). Third, using $1.0\mbox{\,fb}^{-1}$ of
data, measurements of $C\\!P$ sensitive observables of the $B^{\pm}\rightarrow
DK^{\pm}$ system are presented. They include the first observation of the
suppressed mode $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$. Combining
several $D$ final states, $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$
decays is observed with a significance of $5.8\sigma$.
## 1 Measurement of $C\\!P$ asymmetries in $B^{\pm}\rightarrow\psi h^{\pm}$
decays
The $B^{\pm}\rightarrow\psi h^{\pm}$ decays, with
$\psi=({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu},\psi(2S))$ and $h=K,\pi$,
receive contributions from both tree and penguin diagrams. If these
contributions have different weak phases, direct $C\\!P$ violation may occur.
The Standard Model predicts that for $b\rightarrow c\bar{c}s$ decays the tree
and penguin contributions have the same weak phase and thus no direct $C\\!P$
violation is expected in $B^{\pm}\rightarrow\psi K^{\pm}$. For $b\rightarrow
c\bar{c}d$ transitions, however, both contributions have different weak
phases, and $C\\!P$ violation in $B^{\pm}\rightarrow\psi\pi^{\pm}$ decays may
occur. Their branching fractions are expected to be about 5% of the favoured
$B^{\pm}\rightarrow\psi K^{\pm}$ modes. In our paper $\\!{}^{{\bf?}}$ we
analyse a data sample of $0.35\mbox{\,fb}^{-1}$ of $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, taken in 2011 with the LHCb
detector. We define the $C\\!P$ asymmetry and the charge-averaged ratio of
branching ratios as
$A^{\psi\pi}=\frac{{\cal B}(B^{-}\rightarrow\psi\pi^{-})-{\cal
B}(B^{+}\rightarrow\psi\pi^{+})}{{\cal B}(B^{-}\rightarrow\psi\pi^{-})+{\cal
B}(B^{+}\rightarrow\psi\pi^{+})}~{},\quad R^{\psi}=\frac{{\cal
B}(B^{\pm}\rightarrow\psi\pi^{\pm})}{{\cal B}(B^{\pm}\rightarrow\psi
K^{\pm})}~{}.$ (1)
The $\psi$ resonance is reconstructed in the $\mu^{+}\mu^{-}$ final state, and
the well known and abundant decay
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ is
used as a control channel. It is crucial to control its cross feed into the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ channel.
Here we benefit from LHCb’s two ring imaging Cherenkov (RICH) detectors that
provide strong $K/\pi$ separation. We obtain the signal yields from a
simultaneous fit to the $B$ candidate invariant mass distribution in eight
independent subsamples, defined by the charge ($\times 2$), the $\psi$ state
($\times 2$) and the flavour of the bachelor hadron ($K,\pi$, $\times 2$). The
fit projections for the $\psi(2S)$ subsamples are shown in Figure 1. The
measured ratios of branching fractions are
$R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}=(3.83\pm 0.11\pm 0.07)\times
10^{-2}$ and $R^{\psi(2S)}=(3.95\pm 0.40\pm 0.12)\times 10^{-2}$, where the
first uncertainty is statistical and the second systematic. $R^{\psi(2S)}$ is
compatible with the one existing measurement $\\!{}^{{\bf?}}$, $(3.99\pm
0.36\pm 0.17)\times 10^{-2}$. The measurement of
$R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is $3.2\sigma$ lower than
the current world average $\\!{}^{{\bf?}}$, $(5.2\pm 0.4)\times 10^{-2}$.
Using the established measurements of the Cabibbo-favoured branching fractions
$\\!{}^{{\bf?}}$, we deduce ${\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm})=(3.88\pm 0.11\pm 0.15)\times 10^{-5}$, ${\cal
B}(B^{\pm}\rightarrow\psi(2S)\pi^{\pm})=(2.52\pm 0.26\pm 0.15)\times 10^{-5}$.
The measured $C\\!P$ asymmetries,
$\displaystyle A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi}_{C\\!P}$
$\displaystyle=0.005\pm 0.027\pm 0.011~{},$ (2) $\displaystyle
A^{\psi(2S)\pi}_{C\\!P}$ $\displaystyle=0.048\pm 0.090\pm 0.011~{},$ (3)
$\displaystyle A^{\psi(2S)K}_{C\\!P}$ $\displaystyle=0.024\pm 0.014\pm
0.008~{},$ (4)
have comparable or better precision than previous results, and no evidence of
direct $C\\!P$ violation is seen.
Figure 1: $B^{\pm}\rightarrow\Psi(2S)h^{\pm}$ invariant mass distributions,
overlaid by the total fitted PDF (thin line). Pion-like events are
reconstructed as ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$ and
enter in the top plots. All other events are reconstructed as
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ enter the bottom plots
(shown in logarithmic scale). $B^{-}$ decays are shown on the left, $B^{+}$ on
the right. The dark [red] curve shows the
$B^{\pm}\rightarrow\psi(2S)\pi^{\pm}$ component, the light [green] curve
represents $B^{\pm}\rightarrow\psi(2S)K^{\pm}$. Partially reconstructed
backgrounds are shaded.
## 2 Direct $C\\!P$ violation in $B^{0}(B^{0}_{s})\rightarrow K^{-}\pi^{+}$
decays
$C\\!P$ violation is well established in the $K^{0}$ and $B^{0}$ meson
systems. Recent results from LHCb have also provided evidence for $C\\!P$
violation in the $D^{0}$ system $\\!{}^{{\bf?}}$. In our paper
$\\!{}^{{\bf?}}$ we report evidence of direct $C\\!P$ violation in the last
neutral meson system, the $B^{0}_{s}$ system. We reconstruct both
$B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays
in $0.35~{}\mathrm{fb}^{-1}$ of data collected with the LHCb detector in 2011.
The considered decays have contributions from both tree and penguin diagrams,
and are sensitive to contribution of new physics in the penguins. The $C\\!P$
asymmetry in the $B^{0}\rightarrow K^{+}\pi^{-}$ is well established
$\\!{}^{{\bf?}}$. The probability or a $b$ quark to decay as
$B^{0}_{s}\rightarrow K\pi$ is about 14 times smaller than that to decay as
$B^{0}\rightarrow K\pi$. However, both tree and penguin diagrams are roughly
of the same magnitude, so $C\\!P$ violation effects can potentially be large.
We define the $C\\!P$ asymmetries as
$A_{C\\!P}(B^{0}_{(s)})=\frac{\Gamma(\overline{B}^{0}_{(s)}\rightarrow\bar{f}_{(s)})-\Gamma(B^{0}_{(s)}\rightarrow
f_{(s)})}{\Gamma(\overline{B}^{0}_{(s)}\rightarrow\bar{f}_{(s)}+\Gamma(B^{0}_{(s)}\rightarrow
f_{(s)})}~{},$ (5)
with $f=K^{+}\pi^{-}$ and $f_{s}=K^{-}\pi^{+}$. To distinguish the
$K^{+}\pi^{-}$ and $K^{-}\pi^{+}$ final states we rely on the RICH particle
identification system. We carefully control the efficiencies and
misidentification rates from data, through large control samples of
$D^{*}\rightarrow D\pi\rightarrow(K\pi)_{D}\pi$ and $\Lambda_{b}\rightarrow
p\pi$ decays. There are cross feeds from $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays, whose line shape we predict from
simulation. We compute a raw asymmetry from the yields of a fit to the
invariant mass distribution in the positive charge and negative charge
subsamples. Figure 2 shows the projections. This raw asymmetry needs to be
corrected for two effects: an inherent detector charge asymmetry (which we
estimate from our $D^{*}$ control samples) and a non-zero production asymmetry
that is further diluted by $B$ mixing (thus it mostly affects the $B^{0}$
channel due to its much slower $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ oscillation). The total
corrections to the raw asymmetry are $\Delta A_{C\\!P}(B^{0})=-0.007\pm 0.006$
and $\Delta A_{C\\!P}(B^{0}_{s})=0.010\pm 0.002$, where the errors are
statistical. The systematic uncertainty of $A_{C\\!P}(B^{0})$ is dominated by
uncertainties due to instrumentation and production asymmetry, while the
systematic uncertainty of $A_{C\\!P}(B^{0}_{s})$ receives a leading
contribution from the combinatorial background description. In conclusion we
obtain the following measurements of the $C\\!P$ asymmetries:
$A_{C\\!P}(B^{0}\rightarrow K\pi)=-0.088\pm 0.011\,\mathrm{(stat)}\pm
0.008\,\mathrm{(syst)}$
and
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)}.$
The result for $A_{C\\!P}(B^{0}\rightarrow K\pi)$ constitutes the most precise
measurement available to date. It is in good agreement with the current world
average $\\!{}^{{\bf?}}$. The significance of the measured deviation from zero
exceeds $6\sigma$. The result for $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is in
agreement with the only measurement previously available $\\!{}^{{\bf?}}$. The
significance computed for $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is
3.3$\sigma$, making this the first evidence for $C\\!P$ violation in the
decays of $B^{0}_{s}$ mesons.
Figure 2: $K\pi$ invariant mass spectra obtained using the event selection
adopted for the best sensitivity on (a, b) $A_{C\\!P}(B^{0}\rightarrow K\pi)$
and (c, d) $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. Plots (a) and (c) represent
the $K^{+}\pi^{-}$ invariant mass whereas plots (b) and (d) represent the
$K^{-}\pi^{+}$ invariant mass. The results of the unbinned maximum likelihood
fits are overlaid. The main components contributing to the fit model are also
shown.
## 3 Observation of $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$
The CKM angle $\gamma=\arg\left(-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*}\right)$ is
the least well known angle of the corresponding unitarity triangle of the CKM
matrix $V$. The angle $\gamma$ can be measured in $B^{\pm}\rightarrow
DK^{\pm}$ decays where the $D$ signifies a $D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson. The amplitude for the
$D^{0}$ contribution is proportional to $V_{cb}$ whilst the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ amplitude depends on $V_{ub}$. If
the $D$ final state is accessible for both $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons, the two amplitudes
interfere and give rise to observables that are sensitive to $\gamma$. Many
different $D$ final states can be used. In our analysis $\\!{}^{{\bf?}}$ of
1.0 $\mbox{\,fb}^{-1}$ of $\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$ data
collected by LHCb in 2011, we use the $C\\!P$ eigenstates $D\rightarrow
K^{+}K^{-}$, $\pi^{+}\pi^{-}$ (often referred to as “GLW” modes
$\\!{}^{{\bf?},{\bf?}}$), and the flavour eigenstate
$D\rightarrow\pi^{-}K^{+}$ (labelled “ADS” mode $\\!{}^{{\bf?},{\bf?}}$). The
latter requires the favoured, $b\rightarrow c$ decay to be followed by a
doubly Cabibbo-suppressed $D$ decay, and the suppressed $b\rightarrow u$ decay
to be followed by a favoured $D$ decay. As a consequence, the interfering
amplitudes are of similar magnitude and hence large interference can occur. In
total, 13 observables are measured: three ratios of partial widths
$R_{K/\pi}^{f}=\frac{\Gamma(B^{-}\rightarrow[f]_{D}K^{-})+\Gamma(B^{+}\rightarrow[f]_{D}K^{+})}{\Gamma(B^{-}\rightarrow[f]_{D}\pi^{-})+\Gamma(B^{+}\rightarrow[f]_{D}\pi^{+})}~{},$
(6)
where $f$ represents $KK$, $\pi\pi$ and the favoured $K\pi$ mode, six $C\\!P$
asymmetries
$A_{h}^{f}=\frac{\Gamma(B^{-}\rightarrow[f]_{D}h^{-})-\Gamma(B^{+}\rightarrow[f]_{D}h^{+})}{\Gamma(B^{-}\rightarrow[f]_{D}h^{-})+\Gamma(B^{+}\rightarrow[f]_{D}h^{+})}~{},$
(7)
and four charge-separated partial widths of the $ADS$ mode relative to the
favoured mode
$R_{h}^{\pm}=\frac{\Gamma(B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm})}~{}.$
(8)
Similar analyses have found evidence of the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$ decay
$\\!{}^{{\bf?},{\bf?},{\bf?}}$. The abundant $B^{-}\rightarrow D\pi^{-}$
decays have limited sensitivity to $\gamma$ and provide a large control sample
from which probability density functions are shaped. The analysis method
benefits greatly from a boosted decision tree, which combines 20 kinematic
variables to effectively suppress combinatorial backgrounds. Charmless
backgrounds are suppressed by exploiting the large forward boost of the $D$
meson through a cut on its flight distance. The signal yields are estimated by
a simultaneous fit to 16 independent subsamples, defined by the charges
($\times 2$), the $D$ final states ($\times 4$), and the $K$ or $\pi$ nature
of the bachelor hadron ($\times 2$). Figures 3 and 4 show the projections of
the $\pi^{+}\pi^{-}$ and suppressed $\pi^{\pm}K^{\mp}$ subsamples,
respectively. It is crucial to control the cross feed of the abundant
$B^{-}\rightarrow D\pi^{-}$ decays into the signal decays. For this we rely on
the two RICH detectors, which allow to place particle identification cuts on
the bachelor hadron. These cuts are $87.6\%$ efficient for kaons at a rate of
$3.8\%$ misidentified pions. Many systematic uncertainties cancel in the
ratios Eqns. 6-8. The remaining systematic uncertainties are dominated by an
intrinsic charge asymmetry of the detector, and by the uncertainty on the
particle identification. From the measured 13 observables the following
established quantities can be deduced (the full set is contained in our
paper$\\!{}^{{\bf?}}$):
$\displaystyle R_{C\\!P+}$ $\displaystyle=\phantom{-}1.007\pm 0.038\pm
0.012~{},$ $\displaystyle A_{C\\!P+}$ $\displaystyle=\phantom{-}0.145\pm
0.032\pm 0.010~{},$ $\displaystyle R_{K}^{-}$
$\displaystyle=\phantom{-}0.0073\pm 0.0023\pm 0.0004~{},$ $\displaystyle
R_{K}^{+}$ $\displaystyle=\phantom{-}0.0232\pm 0.0034\pm 0.0007~{},$
where the first error is statistical and the second systematic; $R_{C\\!P+}$
is computed from $R_{C\\!P+}\approx\langle
R_{K/\pi}^{KK},R_{K/\pi}^{\pi\pi}\rangle/R_{K/\pi}^{K\pi}$ with an additional
$1\%$ systematic uncertainty assigned to account for the approximation;
$A_{C\\!P+}$ is computed as $A_{C\\!P+}=\langle
A_{K}^{KK},A_{K}^{\pi\pi}\rangle$. From the $R_{K}^{\pm}$ we also compute
$\displaystyle R_{ADS(K)}$ $\displaystyle=\phantom{-}0.0152\pm 0.0020\pm
0.0004~{},$ $\displaystyle A_{ADS(K)}$ $\displaystyle=-0.52\pm 0.15\pm
0.02~{},$
as $R_{ADS(K)}=(R_{K}^{-}+R_{K}^{+})/2$ and
$A_{ADS(K)}=(R_{K}^{-}-R_{K}^{+})/(R_{K}^{-}+R_{K}^{+})$.
To summarise, the $B^{\pm}\rightarrow DK^{\pm}$ $ADS$ mode is observed with
$\approx 10\sigma$ statistical significance when comparing the maximum
likelihood to that of the null hypothesis. This mode displays evidence
($4.0\sigma$) of a large negative asymmetry, consistent with previous
experiments $\\!{}^{{\bf?},{\bf?},{\bf?}}$. The combined asymmetry
$A_{C\\!P+}$ is smaller than (but compatible with) previous measurements
$\\!{}^{{\bf?},{\bf?}}$. It is $4.5\sigma$ significant. We compare the maximum
likelihood with that under the null-hypothesis in all three $D$ final states
where the bachelor is a kaon, diluted by the non-negligible correlated
systematic uncertainties. From this we observe, with a total significance of
$5.8\sigma$, direct $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays.
Figure 3: The invariant mass distribution of selected
$B^{\pm}\rightarrow[\pi^{+}\pi^{-}]_{D}h^{\pm}$ candidates. The left plots are
$B^{-}$ candidates, $B^{+}$ are on the right. In the top plots, the bachelor
track passes the kaon RICH cut and the $B$ candidates are reconstructed
assigning this track the kaon mass. The remaining events are placed in the
bottom row and are reconstructed with a pion mass hypothesis. The dark (red)
curve represents the $B\rightarrow DK^{\pm}$ events, the light (green) curve
is $B\rightarrow D\pi^{\pm}$. The shaded contribution are partially
reconstructed events and the thin line shows the total PDF which also includes
a linear combinatoric component. Figure 4: The invariant mass distribution of
selected $B^{\pm}\rightarrow[\pi^{\pm}K^{\pm}]_{D}h^{\pm}$ candidates. See the
caption of Fig. 3 for a full description. The broken line here represents the
partially reconstructed, but Cabibbo favoured, $B^{0}_{s}\rightarrow
D^{0}K^{+}\pi^{-}$ decays where the pion is lost.
## References
## References
* [1] LHCb collaboration, R. Aaij et al., Measurements of the branching fractions and CP asymmetries of $B^{+}\rightarrow J/\psi\pi^{+}$ and $B^{+}\rightarrow\psi(2S)\pi^{+}$ decays, arXiv:1203.3592
* [2] Belle Collaboration, V. Bhardwaj et al., Observation of $B^{\pm}\rightarrow\psi(2S)\pi^{\pm}$ and search for direct CP-violation, Phys. Rev. D78 (2008) 051104, arXiv:0807.2170
* [3] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G G37 (2010) 075021
* [4] LHCb Collaboration, R. Aaij et al., Evidence for CP violation in time-integrated $D^{0}\rightarrow h^{-}h^{+}$ decay rates, Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938
* [5] LHCb collaboration, R. Aaij et al., First evidence of direct CP violation in charmless two-body decays of $B_{s}$ mesons, arXiv:1202.6251
* [6] Heavy Flavor Averaging Group, D. Asner et al., Averages of b-hadron, c-hadron, and tau-lepton properties, arXiv:1010.1589, Updates available online at http://www.slac.stanford.edu/xorg/hfag
* [7] CDF collaboration, T. Aaltonen et al., Measurements of direct CP violating asymmetries in charmless decays of strange bottom mesons and bottom baryons, Phys. Rev. Lett. 106 (2011) 181802, arXiv:1103.5762
* [8] LHCb collaboration, R. Aaij et al., Observation of CP violation in $B^{+}\rightarrow DK^{+}$ decays, arXiv:1203.3662
* [9] M. Gronau and D. London, How to determine all the angles of the unitarity triangle from $B_{d}^{0}\rightarrow DK^{0}_{\rm\scriptscriptstyle S}$ and $B_{s}^{0}\rightarrow D\phi$, Phys. Lett. B253 (1991) 483
* [10] M. Gronau and D. Wyler, On determining a weak phase from $C\\!P$ asymmetries in charged $B$ decays, Phys. Lett. B265 (1991) 172
* [11] D. Atwood, I. Dunietz, and A. Soni, Enhanced CP violation with $B\rightarrow KD^{0}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ modes and extraction of the CKM angle $\gamma$, Phys. Rev. Lett. 78 (1997) 3257, arXiv:hep-ph/9612433
* [12] D. Atwood, I. Dunietz, and A. Soni, Improved methods for observing CP violation in $B^{\pm}\rightarrow KD$ and measuring the CKM phase $\gamma$, Phys. Rev. D63 (2001) 036005, arXiv:hep-ph/0008090
* [13] Belle collaboration, Y. Horii et al., Evidence for the suppressed decay $B^{-}\rightarrow DK^{-},D\rightarrow K^{+}\pi^{-}$, Phys. Rev. Lett. 106 (2011) 231803, arXiv:1103.5951
* [14] Babar collaboration, P. del Amo Sanchez et al., Search for $b\rightarrow u$ transitions in $B^{-}\rightarrow DK^{-}$ and $B^{-}\rightarrow D^{*}K^{-}$ decays, Phys. Rev. D82 (2010) 072006, arXiv:1006.4241
* [15] CDF collaboration, T. Aaltonen et al., Measurements of branching fraction ratios and CP-asymmetries in suppressed $B^{-}\rightarrow D(\rightarrow K^{+}\pi^{-})K^{-}$ and $B^{-}\rightarrow D(\rightarrow K^{+}\pi^{-})\pi^{-}$ decays, Phys. Rev. D84 (2011) 091504, arXiv:1108.5765
* [16] Babar collaboration, P. del Amo Sanchez et al., Measurement of CP observables in $B^{\pm}\rightarrow D_{CP}K^{\pm}$ decays and constraints on the CKM angle $\gamma$, Phys. Rev. D82 (2010) 072004, arXiv:1007.0504
* [17] CDF collaboration, T. Aaltonen et al., Measurements of branching fraction ratios and CP asymmetries in $B^{\pm}\rightarrow D_{CP}K^{\pm}$ decays in hadron collisions, Phys. Rev. D81 (2010) 031105, arXiv:0911.0425
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arxiv-papers
| 2012-05-30T08:26:55 |
2024-09-04T02:49:31.363132
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Till Moritz Karbach (for the LHCb collaboration)",
"submitter": "Till Moritz Karbach",
"url": "https://arxiv.org/abs/1205.6579"
}
|
1205.6590
|
# A note on the Frobenius-Euler Numbers and polynomials associated with
Bernstein polynomials
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com and Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
The present paper deals with Bernstein polynomials and Frobenius-Euler numbers
and polynomials. We apply the method of generating function and fermionic
$p$-adic integral representation on $\mathbb{Z}_{p}$, which are exploited to
derive further classes of Bernstein polynomials and Frobenius-Euler numbers
and polynomials. To be more precise we summarize our results as follows, we
obtain some combinatorial relations between Frobenius-Euler numbers and
polynomials. Furthermore, we derive an integral representation of Bernstein
polynomials of degree $n$ on $\mathbb{Z}_{p}$. Also we deduce a fermionic
$p$-adic integral representation of product Bernstein polynomials of different
degrees $n_{1},n_{2},\cdots$ on $\mathbb{Z}_{p}$ and show that it can be
written with Frobenius-Euler numbers which yields a deeper insight into the
effectiveness of this type of generalizations. Our applications possess a
number of interesting properties which we state in this paper
###### Key words and phrases:
Frobenius-Euler numbers and polynomials, Bernstein polynomials, fermionic
$p$-adic integral on $\mathbb{Z}_{p}$.
###### 2000 Mathematics Subject Classification:
05A10, 11B65, 28B99, 11B68, 11B73.
## 1\. Introduction and Notations
Let $p$ be a fixed odd prime number. Throughout this paper we use the
following notations. By $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by
$\left|p\right|_{p}=\frac{1}{p}\text{.}$
In this paper, we assume $\left|q-1\right|_{p}<1$ as an indeterminate. In
[17-19], let $UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly
differentiable functions on $\mathbb{Z}_{p}$. For $f\in
UD\left(\mathbb{Z}_{p}\right)$, the fermionic $p$-adic integral on
$\mathbb{Z}_{p}$ is defined by T. Kim:
(1.1)
$I_{-1}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-1}\left(\xi\right)=\lim_{N\rightarrow\infty}\sum_{\xi=0}^{p^{N}-1}f\left(\xi\right)\left(-1\right)^{\xi}\text{.}$
From (1.1), we have well known the following equality:
(1.2) $I_{-1}\left(f_{1}\right)+I_{-1}\left(f\right)=2f\left(0\right)$
here $f_{1}\left(x\right):=f\left(x+1\right)$ (for details, see[3-24]).
Let $C\left(\left[0,1\right]\right)$ be the space of continuous functions on
$\left[0,1\right]$. For $C\left(\left[0,1\right]\right)$, the Bernstein
operator for $f$ is defined by
$B_{n}\left(f,x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)B_{k,n}\left(x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)\binom{n}{k}x^{k}\left(1-x\right)^{n-k}$
where $n,$ $k\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,...\right\\}$. Here
$B_{k,n}\left(x\right)$ is called Bernstein polynomials, which are defined by
(1.3) $B_{k,n}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}\text{,
}x\in\left[0,1\right]$
(for more informations on this subject, see [1-6, 11, 14, 15, 17, 21-24])
In [7], as is well known, Frobenius-Euler polynomials are defined by means of
the following generating function:
(1.4)
$\sum_{n=0}^{\infty}H_{n}\left(u,x\right)\frac{t^{n}}{n!}=e^{H\left(u,x\right)t}=\frac{1-u}{e^{t}-u}e^{xt}\text{.}$
where the usual convention about replacing $H^{n}\left(u,x\right)$ by
$H_{n}\left(u,x\right)$. For $x=0$ in (1.4), we have to
$H_{n}\left(u,0\right):=H_{n}\left(u\right)$, which is called Frobenius-Euler
numbers. Then, we can write the following
(1.5)
$e^{H\left(u\right)t}=\sum_{n=0}^{\infty}H_{n}\left(u\right)\frac{t^{n}}{n!}=\frac{1-u}{e^{t}-u}\text{.}$
By (1.4) and (1.5), we easily see the following applications:
$\displaystyle e^{\left(H\left(u\right)+1\right)t}-ue^{H\left(u\right)t}$
$\displaystyle=$ $\displaystyle 1-u$
$\displaystyle\sum_{n=0}^{\infty}\left[\left(H\left(u\right)+1\right)^{n}-uH_{n}\left(u\right)\right]\frac{t^{n}}{n!}$
$\displaystyle=$ $\displaystyle 1-u$
After these applications, we derive the following Lemma.
###### Lemma 1.
For $\left|u\right|>1$ and
$n\in\mathbb{Z}_{+}:=\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$,
we have
(1.6)
$\left(H\left(u\right)+1\right)^{n}-uH_{n}\left(u\right)=\left\\{\QATOP{1-u\text{,
if }n=0}{0\text{, \ \ \ \ \ \ if }n\neq 0.}\right.$
In this paper, we obtained some relations between the Frobenius-Euler numbers
and polynomials and the Bernstein polynomials. From these relations, we derive
some interesting identities on the Frobenius-Euler numbers.
## 2\. On the Frobenius-Euler numbers and polynomials
Let us take $f\left(x\right)=u^{x}e^{tx}$ in (1.1), by (1.2), we see that
(2.1) $\int_{\mathbb{Z}_{p}}u^{\eta}e^{\eta
t}d\mu_{-1}\left(\eta\right)=\frac{2}{1+u}H_{n}\left(-u^{-1}\right)\text{.}$
By (1.4) and (2.1), we have the following theorem.
###### Theorem 1.
(2.2)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(x+\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}H_{n}\left(-u^{-1},x\right)\text{.}$
By applying some combinatorial techniques in (2.2), we derive the following
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(x+\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}\left\\{\int_{\mathbb{Z}_{p}}u^{\eta}\eta^{k}d\mu_{-1}\left(\eta\right)\right\\}\text{.}$
So, from above, we have the well known identity
(2.3)
$H_{n}\left(-u^{-1},x\right)=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}H_{k}\left(-u^{-1}\right)=\left(H\left(-u^{-1}\right)+x\right)^{n}\text{.}$
by using the $umbral$(symbolic) convention
$H^{n}\left(u\right):=H_{n}\left(u\right)$.
The Frobenius-Euler polynomials have to symmetric properties, which is shown
by Choi $et$ $al$. in [7], as follows:
$H_{n}\left(-u^{-1},1-x\right)=\left(-1\right)^{n}H_{n}\left(-u^{-1},x\right)\text{.}$
For $n\in\mathbb{N}$, by (2.3), Choi $et$ $al$. derived the following
equality:
(2.4)
$u^{2}H_{n}\left(-u^{-1},2\right)=u^{2}+u+H_{n}\left(-u^{-1}\right)\text{.}$
From (2.2) and (2.4), we easily see that
$\displaystyle\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}u^{\eta}\left(\eta-1\right)^{n}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\left(-1\right)^{n}H_{n}\left(-u^{-1},-1\right)$
$\displaystyle=$ $\displaystyle\frac{2}{u+1}H_{n}\left(-u^{-1},2\right).$
Thus, we obtain the following Theorem.
###### Theorem 2.
The following identity
(2.6)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}H_{n}\left(-u^{-1},2\right)$
is true.
Let $n\in\mathbb{N}$. By expression of (2.4) and (2.6), we get
(2.7)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right).$
From (2.7), we procure the following corollary.
###### Corollary 1.
For $n\in\mathbb{N}$, we have
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right).$
## 3\. Some identities on the Frobenius-Euler numbers
In this section, we develop Frobenius-Euler numbers, that is, we derive some
interesting and worthwhile relations for studying in Theory of Analytic
Numbers.
Now also, for $x\in\left[0,1\right]$, we rewrite definition of Bernstein
polynomials as follows:
(3.1) $B_{k,n}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}\text{,
where }n,k\in\mathbb{Z}_{+}\text{.}$
By expression of (3.1), we have the properties of symmetry of Bernstein
polynomials as follows:
(3.2) $B_{k,n}\left(x\right)=B_{n-k,n}\left(1-x\right)\text{, (for detail, see
\cite[cite]{[\@@bibref{}{kim 19}{}{}]}).}$
Thus, from Corollary 1, (3.1) and (3.2), we see that
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}B_{n-k,n}\left(1-\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n-l}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right)\text{.}$
For $n$, $k\in\mathbb{Z}_{+}$ with $n>k$, we compute
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right),\text{
if }k>0.}\right.$
Let us take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ on the
Bernstein polynomials of degree $n$ as follows:
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\int_{\mathbb{Z}_{p}}\eta^{k}\left(1-\eta\right)^{n-k}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\binom{n}{k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}H_{l+k}\left(-u^{-1}\right)\text{.}$
Consequently, by expression of (3) and (3), we state the following Theorem:
###### Theorem 3.
The following identity holds true:
$\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}H_{l+k}\left(-u^{-1}\right)=\left\\{\QATOPD..{1+u^{-1}+u^{-2}H_{n}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0\text{,}}{\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(1+u^{-1}+u^{-2}H_{n-l}\left(-u^{-1}\right)\right),\text{
if }k>0\text{.}}\right.$
Let $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k$. Then, we derive
the followings
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\int_{\mathbb{Z}_{p}}\left(1-\eta\right)^{n_{1}+n_{2}-l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left(\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0.}\right.$
Therefore, we obtain the following Theorem:
###### Theorem 4.
For $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k,$ we have
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0.}\right.$
By using the binomial theorem, we can derive the following equation.
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\eta^{2k+l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}H_{2k+l}\left(-u^{-1}\right).$
Thus, we can obtain the following Corollary:
###### Corollary 2.
For $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k,$ we have
$\displaystyle\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}H_{2k+l}\left(-u^{-1}\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{1+u^{-1}+u^{-2}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(1+u^{-1}+u^{-2}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0\text{.}}\right.$
For $\eta\in\mathbb{Z}_{p}$ and $s\in\mathbb{N}$ with $s\geq 2,$ let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk$. Then
we take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ for the
Bernstein polynomials of degree $n$ as follows:
$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{\underbrace{B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)...B_{k,n_{s}}\left(\eta\right)}}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\int_{\mathbb{Z}_{p}}\eta^{sk}\left(1-\eta\right)^{n_{1}+n_{2}+...+n_{s}-sk}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{l+sk}\int_{\mathbb{Z}_{p}}\left(1-\xi\right)^{n_{1}+n_{2}+...+n_{s}-l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$
So from above, we have the following Theorem:
###### Theorem 5.
For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk$. Then
we have
$\displaystyle\int_{\mathbb{Z}_{p}}u^{\eta}\mathop{\displaystyle\prod}\limits_{i=1}^{s}B_{k,n_{i}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$
From the definition of Bernstein polynomials and the binomial theorem, we
easily get
(3.6)
$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{\underbrace{B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)...B_{k,n_{s}}\left(\eta\right)}}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\xi^{sk+l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}H_{sk+l}\left(-u^{-1}\right).$
Therefore, by (3.6), we get novel properties of Frobenius-Euler numbers with
the following corollary:
###### Corollary 3.
For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk.$
Then, we have
$\displaystyle u^{2}$
$\displaystyle\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}H_{sk+l}\left(-u^{-1}\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{u^{2}+u+H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(u^{2}+u+H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$
## References
* [1] Açıkgöz, M. and Araci, S., A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics-Modelling and Simulation, vol.1, no. 1, pp. 10–14, 2010.
* [2] Açıkgöz, M. and Şimşek, Y., A New generating function of $q$-Bernstein type polynomials and their interpolation function, Abstract and Applied Analysis, Article ID 769095, 12 pages, doi: 10.1155/2010/769095.01-313.
* [3] Araci, S., Erdal, D., and Seo, J-J., A study on the Fermionic $p$-adic $q$-integral Representation on $\mathbb{Z}_{p}$ Associated with Weighted $q$-Bernstein and $q$-Genocchi Polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [4] Araci, S. Erdal, D. and Kang, D-J., Some New Properties on the $q$-Genocchi numbers and Polynomials associated with $q$-Bernstein polynomials, Honam Mathematical J. 33 $\left(2011\right)$ no. 2, pp. 261-270
* [5] Araci, S., Acikgoz, M., Qi, F., On the $q$-Genocchi numbers and polynomials with weight $0$ and their applications, http://arxiv.org/abs/1202.2643.
* [6] A. Bayad, T. Kim, Identtities involving values of Bernstein $q$-Bernoulli, and $q$-Euler polynomials, Russ. J. Math. Phys. 18 (2011), no. 2, 133-143.
* [7] J. Choi, D. S. Kim, T. Kim and Y. H. Kim, A note on Some identities of Frobeniu-Euler Numbers and Polynomials, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 861797, 9 pages.
* [8] T. Kim and B. Lee, Some Identities of the Frobenius-Euler polynomials, Abstract and Applied Analysis, Volume 2009, Article ID 639439, 7 pages.
* [9] T. Kim, On the multiple $q$-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486. arXiv:0801.0978v1 [math.NT]
* [10] T. Kim, A Note on the $q$-Genocchi Numbers and Polynomials, Journal of Inequalities and Applications 2007 (2007) doi:10.1155/2007/71452. Article ID 71452, 8 pages.
* [11] T. Kim, A note $q$-Bernstein polynomials, Russ. J. Math. Phys. 18 (2011), 41-50.
* [12] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288-299.
* [13] T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 $\left(2008\right),$ 51-57.
* [14] T. Kim, J. Choi, Y. H. Kim and C. S. Ryoo, On the fermionic $p$-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, J. Inequal. Appl. 2010 $\left(2010\right)$, Art ID 864247, 12pp.
* [15] T. Kim, J. Choi and Y. H. Kim Some identities on the $q$-Bernstein polynomials, $q$-Stirling numbers and $q$-Bernoulli numbers, Adv. Stud. Contemp. Math. 20 $\left(2010\right),$ 335-341.
* [16] T. Kim, An invariant $p$-adic $q$-integrals on $\mathbb{Z}_{p}$, Applied Mathematics Letters, vol. 21, pp. 105-108, 2008.
* [17] T. Kim, J. Choi and Y. H. Kim $q$-Bernstein Polynomials Associated with $q$-Stirling Numbers and Carlitz’s $q$-Bernoulli Numbers, Abstract and Applied Analysis, Article ID 150975, 11 pages, doi:10.1155/2010/150975.
* [18] T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27.
* [19] T. Kim, New approach to $q$-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225.
* [20] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4, 484–491.
* [21] T. Kim, A. Bayad, Y. H. Kim, A Study on the $p$-Adic $q$-Integrals Representation on $\mathbb{Z}_{p}$ Associated with the weighted $q$-Bernstein and $q$-Bernoulli polynomials, Journal of Inequalities and Applications, Article ID 513821, 8 pages, doi:10.1155/2011/513821.
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* [23] H. Y. Lee, N. S. Jung, and C. S. Ryoo, Some Identities of the Twisted $q$-Genocchi Numbers and Polynomials with weight $\alpha$ and $q$-Bernstein Polynomials with weight $\alpha,$ Abstract and Applied Analysis, Volume 2011 (2011), Article ID 123483, 9 pages.
* [24] N. S. Jung, H. Y. Lee and C. S. Ryoo, Some Relations between Twisted ($h$,$q$)-Euler Numbers with Weight $\alpha$ and $q$-Bernstein Polynomials with Weight $\alpha$, Discrete Dynamics in Nature and Society, Volume 2011 (2011), Article ID 176296, 11 pages.
|
arxiv-papers
| 2012-05-30T08:57:49 |
2024-09-04T02:49:31.368810
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1205.6590"
}
|
1205.6666
|
11institutetext: Condensed Matter Physics Laboratory, Heinrich-Heine-
Universität, Universitätsstr.1, 40225 Düsseldorf, Germany
Graphene Electronic transport in mesoscopic systems Scattering by point
defects, dislocations, surfaces, and other imperfections
# Impurity and edge roughness scattering in armchair graphene nanoribbons:
Boltzmann approach
Hengyi Xu Thomas Heinzel
###### Abstract
The conductivity of armchair graphene nanoribbons in the presence of short-
range impurities and edge roughness is studied theoretically using the
Boltzmann transport equation for quasi-one-dimensional systems. As the number
of occupied subbands increases, the conductivity due to short-range impurities
converges towards the two-dimensional case. Calculations of the
magnetoconductivity confirm the edge-roughness-induced dips at cyclotron radii
close to the ribbon width suggested by the recent quantum simulations.
###### pacs:
81.05.ue
###### pacs:
73.23.-b
###### pacs:
72.10.Fk
## 1 Introduction
The theoretical descriptions of electron transport in graphene sheets with
various scattering sources, such as charged impurities in substrates,
microscopic corrugations or short-range resonant scatters, have been
frequently based on the Boltzmann approach. Such studies, in particular the
electron density dependence of the conductivity, are of fundamental interest
since they help identifying the dominant scattering sources. [1, 2] It has
been verified systematically that the Boltzmann approach works quite well to
describe the transport in broad parameter ranges for both single and double
layer graphene. [3] As the width of the graphene strips is decreased, graphene
nanoribbons (GNRs) are fromed, where size-dependent effects, for example the
inhomogeneous electron density [4] or the edge roughness, become relevant for
the transport properties. In this case, the transport has so far been
described within the framework of the Landauer-Büttiker model with the aid of
Green’s function techniques. [5, 6] It is well established that in GNRs, edge
disorder can contribute significantly to the scattering [7, 8, 9, 10] which in
wide structures is governed by a combination of scattering at charged
impurities and resonant scattering at short-range defects. Edge disorder has
been suggested as the source of the transport gap in narrow GNRs around the
charge neutrality point.[10, 11, 12, 13] Furthermore, a typical size effect,
the so-called edge-roughness-induced magnetoconductance dip (ERID) in GNRs has
been studied by numerical quantum simulations, which is interpreted as a
magnetic-field-enhanced diffusive scattering when the electron trajectory
grazes at the edges. [14] On the other hand, the Boltzmann approach has been
applied to treat a variety of scattering sources in conventional quasi-one-
dimensional (Q1D) systems, for example quantum wires. [15, 16, 17]. However,
only a few aspects of transport in GNRs have so far been studied within the
Boltzmann model. [18, 19]
In the present paper, we apply the linear Boltzmann equation to armchair GNRs
and determine its transport properties in the presence of $\delta$-type short-
range impurities and edge roughness. The magnetoconductivity in wide GNRs with
rough edge roughness is studied as well.
## 2 Model and theory
We start with the Dirac Hamiltonian
$H=\hbar v_{F}(\sigma_{x}\tau_{z}k_{x}+\sigma_{y}k_{y})$ (1)
with Fermi velocity $v_{F}\approx 10^{6}m/s$ and Pauli matrices $\sigma_{x,y}$
and $\tau_{z}$ acting on the $A/B$ sublattice and $K/K^{\prime}$ valley
spaces, respectively. The energy spectrum of GNRs depends on the nature of
their edges, namely zigzag or armchair. Within the present work, we restrict
ourselves to metallic armchair GNRs (AGNRs). For this system, the boundary
conditions imposed on the wave function, namely
$\Psi_{A}(x=0)=\Psi_{B}(x=0)=\Psi_{A}(x=W)=\Psi_{B}(x=W)=0$, give rise to the
allowed transverse wave vectors as
$k_{n}=\frac{n\pi}{W}-\frac{4\pi}{3a}$ (2)
with $a=0.246\mathrm{nm}$ being the lattice constant of graphene. $W$ is the
width of AGNRs. The integer $n$ is of the order of $W/a$ for the energetically
lowest modes. Throughout this text, we denote the energy $\epsilon$ normalized
to $\hbar v_{F}$ as $\widetilde{\epsilon}\equiv\epsilon/(\hbar v_{F})$ with
$\widetilde{\epsilon}^{2}=k_{n}^{2}+k^{2}$. The normalized wave function for
the $n$th subband reads [20, 21]
$\Psi(\mathbf{r})=\frac{e^{iky}}{\sqrt{4WL}}\left(\begin{array}[]{c}e^{ik_{n}x}\\\
\frac{k_{n}+ik}{\widetilde{\epsilon}_{nk}}e^{ik_{n}x}\\\ -e^{-ik_{n}x}\\\ \
-\frac{k_{n}+ik}{\widetilde{\epsilon}_{nk}}e^{-ik_{n}x}\end{array}\right)$ (3)
which is a mixture of two Dirac points $\mathbf{K}=(4\pi/(3a),0)=(K,0)$ and
$\mathbf{K^{\prime}}=(-4\pi/(3a),0)=(-K,0)$. Here we choose the wave vectors
in the $x$-direction to be quantized and the transport is oriented along
$y$-direction. $L$ is the length of the system.
To describe the transport properties of GNRs, we adopt the linearized
Boltzmann equation describing the general Q1D system
$-\frac{eE_{y}}{\hbar}\frac{\partial f_{nk}^{0}(\epsilon_{nk})}{\partial
k}=\sum_{n^{\prime}}\sum_{k^{\prime}}\mathcal{W}_{n^{\prime}k^{\prime}nk}\left[f_{n^{\prime}k^{\prime}}-f_{nk}\right]$
(4)
where $E_{y}$ is the applied electric field along the transport direction,
$f_{nk}$ is the distribution function of a state with wave vector $k$ and
energy $\epsilon_{nk}$ in the $n$th subband, and the superscript “0” denotes
the equilibrium distribution. According to Fermi’s Golden Rule, the scattering
probability due to the perturbation potential is given by
$\mathcal{W}_{n^{\prime}k^{\prime}nk}=\frac{2\pi}{\hbar}|\langle
n^{\prime},k^{\prime}|U|n,k\rangle|^{2}\delta(\epsilon_{n^{\prime}k^{\prime}}-\epsilon_{nk})$
(5)
Using the relaxation time approximation, the nonequilibrium distribution
function can be written as
$f_{nk}(\epsilon_{nk})=f_{nk}^{0}(\epsilon_{nk})-eE_{x}v_{n}(\epsilon_{nk})\tau_{n}(\epsilon_{nk})\delta(\epsilon_{nk}-E_{F})$
(6)
with Fermi energy $E_{F}$ and the relaxation time $\tau_{n}$ for the state in
the $n$th subband. The velocity for the $n$th subband
$v_{n}=(1/\hbar){\partial\epsilon_{nk}}/{\partial
k}=v_{F}{k}/{\sqrt{(k_{n})^{2}+k^{2}}}$ for the linear spectrum of graphene.
Inserting Eq. (6) into Eq. (4), the Boltzmann equation at zero temperature can
be written as
$\displaystyle\frac{k}{\widetilde{\epsilon}_{nk}}\delta(\epsilon_{nk}-E_{F})$
$\displaystyle=$
$\displaystyle\sum_{n^{\prime},k^{\prime}}\mathcal{W}_{n^{\prime}k^{\prime}nk}\left[\frac{k}{\widetilde{\epsilon}_{nk}}\tau_{n}(\epsilon_{nk})\delta(\epsilon_{nk}-E_{F})\right.$
(7)
$\displaystyle\left.-\frac{k^{\prime}}{\widetilde{\epsilon}_{n^{\prime}k^{\prime}}}\tau_{n^{\prime}}(\epsilon_{n^{\prime}k^{\prime}})\delta(\epsilon_{n^{\prime}k^{\prime}}-E_{F})\right].$
Multiplying both sides of Eq. (7) by $k$ and summing over $k$, we obtain after
some algebra
$k_{F}^{n}=\frac{2\pi}{\hbar}\sum_{n^{\prime}}\mathcal{T}_{nn^{\prime}}\tau_{n^{\prime}}(E_{F})$
(8)
where $k_{F}^{n}$ is Fermi wave vector in the $n$th subband. The transition
matrix element $\mathcal{T}_{nn^{\prime}}$ is defined as
$\displaystyle\mathcal{T}_{nn^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\pi\hbar
v_{F}}{L}\sum_{k^{\prime}}\sum_{k}\left[\delta_{nn^{\prime}}\sum_{\mu}|\langle\mu,k^{\prime}|U|n,k\rangle|^{2}\right.$
$\displaystyle\times$
$\displaystyle\frac{k^{2}}{\widetilde{\epsilon}_{nk}}\delta(\epsilon_{nk}-E_{F})\delta(E_{\mu
k^{\prime}}-E_{F})$ $\displaystyle-$ $\displaystyle\left.|\langle
n^{\prime},k^{\prime}|U|n,k\rangle|^{2}\frac{kk^{\prime}}{\widetilde{\epsilon}_{n^{\prime}k^{\prime}}}\delta(\epsilon_{nk}-E_{F})\delta(\epsilon_{n^{\prime}k^{\prime}}-E_{F})\right]$
with the summation running over the mode index $\mu$. The Boltzmann
conductivity for GNRs is then given by
$\sigma(E_{F})=\frac{2e^{2}}{h}\frac{\hbar^{2}v_{F}^{2}}{\pi
E_{F}}\frac{1}{W}\sum_{n,n^{\prime}}k_{F}^{n}k_{F}^{n^{\prime}}(\mathcal{T}^{-1})_{nn^{\prime}}$
(10)
For nonzero temperature, the conductivity is obtained from
$\sigma=\int d\epsilon\left(-\frac{\partial
f(\epsilon)}{\partial\epsilon}\right)\sigma(\epsilon)$ (11)
We proceed by describing how the scattering potentials have been implemented
in this formalism. First, we consider $\delta$-type impurities in the form of
$U=\gamma\sum_{j=1}^{N_{I}}\delta(x-x_{j})\delta(y-y_{j})$ (12)
where $\gamma$ and $N_{I}$ are the strength and the number of impurities,
respectively. Thus the matrix element squared of the perturbation is evaluated
as
$\displaystyle|\langle
n^{\prime}k^{\prime}|U|nk\rangle|^{2}=\frac{\gamma^{2}}{4W^{2}L^{2}}$ (13)
$\displaystyle\times$
$\displaystyle\sum_{j=1}^{N_{j}}\cos^{2}\frac{(n-n^{\prime})\pi
x_{j}}{W}\left|1+\frac{(k_{n^{\prime}}-ik^{\prime})(k_{n}+ik)}{\widetilde{\epsilon}_{n^{\prime}k^{\prime}}\widetilde{\epsilon}_{nk}}\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{\gamma^{2}n_{i}}{4WL}(1+\delta_{nn^{\prime}})\left(1+\frac{k_{n^{\prime}}k_{n}+k^{\prime}k}{\widetilde{\epsilon}_{n^{\prime}k^{\prime}}\widetilde{\epsilon}_{nk}}\right)$
with $n_{i}=N_{I}/WL$. The transition matrix elements are finally given by
$\displaystyle\mathcal{T}_{nn^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{2}n_{i}}{4\pi
W}\delta_{nn^{\prime}}\left(\sum_{\mu}\left[(1+\delta_{n\mu})\left(\frac{E_{F}}{\hbar^{2}v_{F}^{2}}+\frac{k_{\mu}k_{n}}{E_{F}}\right)\frac{k_{F}^{n}}{k_{F}^{\mu}}\right]\right.$
(14)
$\displaystyle\left.-\frac{k_{F}^{n^{\prime}}k_{F}^{n}}{{E}_{F}}\right)-\frac{\gamma^{2}n_{i}}{4\pi
W}\frac{k_{F}^{n^{\prime}}k_{F}^{n}}{{E}_{F}}.$
Using Eq. (14) and Eq. (10) we obtain for the conductivity of GNRs the
expression
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{8e^{2}}{h}\frac{(\hbar
v_{F})^{2}}{\gamma^{2}n_{i}}\sum_{n,n^{\prime}}\left\\{\left[\sum_{\mu}\left((1+\delta_{n\mu})(\widetilde{E}^{2}_{F}+k_{\mu}k_{n})\frac{k_{F}^{n}}{k_{F}^{\mu}}\right)\right.\right.$
(15)
$\displaystyle\left.\left.-k_{F}^{n^{\prime}}k_{F}^{n}\right]\delta_{nn^{\prime}}-k_{F}^{n^{\prime}}k_{F}^{n}\right\\}^{-1}_{nn^{\prime}}k_{F}^{n}k_{F}^{n^{\prime}}$
For a large number of subbands, i.e., $\mathcal{N}\gg 1$, we have
$k_{\mu},k_{n}\ll\widetilde{E}_{F}$ and
$k_{F}^{n},k_{F}^{n^{\prime}}\sim\widetilde{E}_{F}$. In this approximation,
Eq. (15) converges to the well-known result for the extended case
$\sigma=\frac{8e^{2}}{h}\frac{\hbar^{2}v_{F}^{2}}{\gamma^{2}n_{i}}$ (16)
which is independent of the carrier concentration. [2, 3]
As a second scattering mechanism, we study the effects of edge roughness in
the absence of magnetic fields. The edge roughness is parametrized by
$\Delta(y){\partial V(x)}/{\partial x}$, an expression which has been applied
before to model rough semiconductor quantum wires and interfaces. [22, 23]
$V(x)$ is the one-dimensional confinement potential which can be modeled by a
finite mass term in the Dirac Hamiltonian. [24] $\Delta(y)$ is a function
describing the potential fluctuation of GNRs and characterized by
$\langle\Delta(y)\rangle=0$ and the autocovariance function
$\langle\Delta(y)\Delta(y^{\prime})\rangle=\Delta^{2}\exp[-(y-y^{\prime})^{2}/\Lambda^{2}]$
with $\Lambda$ being the correlation length. Furthermore,
$\langle\cdots\rangle$ denotes position averaging.
To evaluate the perturbation matrix element, we define the function $\Xi$
related to the $x$-components of the wave functions as
$\Xi_{n^{\prime}n}=\frac{1}{W}\int_{0(W)}^{-\infty(+\infty)}dx\phi_{n^{\prime}}^{*}(x)\frac{\partial
V(x)}{\partial x}\phi_{n}(x)$ (17)
where $\phi_{n}(x)$ denotes one of the components of wave function in Eq. (3).
For the hard-wall confinement potential present in GNRs, this function can be
expressed as
$\Xi_{n^{\prime}n}=\left.-\frac{1}{W}\frac{\hbar
v_{F}}{2\widetilde{E}_{F}}\left[\frac{\partial\phi^{*}_{n^{\prime}}}{\partial
x}\frac{\partial\phi_{n}}{\partial x}\right]\right|_{x=0,W}$ (18)
It is noteworthy that a linear form for the matrix elements of the edge
roughness perturbation has been used, which however neglects the interband
scattering. [25, 19] Using Eq. (18), the square of the matrix element for edge
roughness reads
$\displaystyle|\langle n^{\prime},k^{\prime}|U|n,k\rangle|^{2}$
$\displaystyle=$ $\displaystyle\frac{\pi^{9/2}n^{\prime
2}n^{2}}{8W^{6}}\frac{(\hbar
v_{F})^{2}}{\widetilde{E}^{2}_{F}}\left(1+\frac{k_{n}k_{n^{\prime}}+kk^{\prime}}{\widetilde{\epsilon}_{nk}\widetilde{\epsilon}_{n^{\prime}k^{\prime}}}\right)$
(19)
$\displaystyle\times\frac{\Lambda\Delta^{2}}{L}\exp[-\Lambda^{2}(k-k^{\prime})^{2}/4]$
where we have used the Gaussian integral in the evaluation of part in the
$y$-direction. In the case of small correlation length
$\Lambda\ll\lambda_{F}$, the transition matrix element has the form
$\displaystyle\mathcal{T}_{nn^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{7/2}\Lambda\Delta^{2}}{8W^{6}}\frac{\hbar
v_{F}}{\widetilde{E}_{F}}n^{2}\left[\sum_{\mu}\mu^{2}\left(1+\frac{k_{n}k_{\mu}}{\widetilde{E}^{2}_{F}}\right)\frac{k_{F}^{n}}{k_{F}^{\mu}}\delta_{nn^{\prime}}\right.$
(20) $\displaystyle\left.-n^{\prime
2}\frac{k_{F}^{n^{\prime}}k_{F}^{n}}{\widetilde{E}^{2}_{F}}\right]$
This results in a conductivity given by
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{16e^{2}}{h}\frac{W^{5}}{\pi^{9/2}\Lambda\Delta^{2}}\sum_{n,n^{\prime}}k_{F}^{n}k_{F}^{n^{\prime}}$
$\displaystyle\times$
$\displaystyle\left[n^{2}\sum_{\mu}\mu^{2}\left(1+\frac{k_{n}k_{\mu}}{\widetilde{E}^{2}_{F}}\right)\frac{k_{F}^{n}}{k_{F}^{\mu}}\delta_{nn^{\prime}}-n^{2}n^{\prime
2}\frac{k_{F}^{n^{\prime}}k_{F}^{n}}{\widetilde{E}^{2}_{F}}\right]^{-1}_{nn^{\prime}}.$
Regarding the diagonal contributions of the transition rate matrix, it is
convenient to write the inverse of the relaxation time for the $n$th subband
as
$\frac{1}{\tau_{n}}=\frac{\pi^{9/2}}{4W^{6}}\frac{\hbar
v_{F}^{2}}{E_{F}}\Lambda\Delta^{2}n^{2}\sum_{\mu}\mu^{2}\left[\frac{1}{k_{F}^{\mu}}+\frac{k_{\mu}k_{n}/k_{F}^{\mu}}{\widetilde{E}^{2}_{F}}\right].$
(22)
For a large number of occupied subbands ($\mathcal{N}\gg 1$), the second term
in the bracket can be neglected since $k_{n,\mu}\ll\widetilde{E}_{F}$, and the
summation can be replaced by an integral. Eq.(22) can then be written as
$\frac{1}{\tau_{n}}\approx\frac{\pi^{5/2}}{16W^{3}}\Lambda\Delta^{2}v_{F}\widetilde{E}_{F}n^{2}$
(23)
which shows a striking similarity for the corresponding results reported for
semiconductor quantum wires. [15]. Furthermore, this relaxation time results
in a conductivity in the limit of $\mathcal{N}\gg 1$ of
$\sigma\approx\frac{32e^{2}}{h}\frac{1}{3\pi^{1/2}}\frac{W^{2}}{\Lambda\Delta^{2}\widetilde{E}_{F}}.$
(24)
Up to now, we have looked at the transport with edge roughness in the absence
of magnetic fields $B$ and now continue by including it and discussing its
semiclassical effects. A weak magnetic field may homogenize the contributions
of the occupied subbands to the overall conductivity and result in a reduction
of the magnetoconductivity when the cyclotron radius $r_{c}\sim W$. This
effect has been verified numerically by recent quantum calculations in GNRs
[14] but, to the best of our knowledge, has so far not yet been observed. Here
the maximum reduction of the relaxation time $\tau$ in magnetic fields can be
roughly estimated by averaging over all occupied modes, resulting in
$\frac{1}{\tau(B>0)}\approx\frac{\pi^{1/2}}{48W}\Lambda\Delta^{2}v_{F}\widetilde{E}^{3}_{F}.$
(25)
The magnetoconductivity near the dip is then given by
$\sigma(B>0)\approx\frac{48e^{2}}{h}\frac{1}{\pi^{1/2}}\frac{W}{\Lambda\Delta^{2}\widetilde{E}_{F}^{2}},$
(26)
which holds for $\mathcal{N}\gg 1$.
## 3 Numerical results and discussion
Figure 1: (Colour online) The numerical conductivity of $\delta$-type
impurities in armchair GNRs with $W=180$nm plotted versus the Fermi energy for
two different temperatures T. We have chosen the parameters as
$\overline{\gamma}=0.9$ and $\overline{n}_{i}=0.5$, where
$\overline{\gamma}=\gamma(\hbar v_{F})^{2}/E_{F}$ and
$\overline{n}_{i}=n_{i}\lambda_{F}^{2}$.
In Fig. 1 we show the conductivity for $\delta$-type impurities according to
Eq.(15) as a function of Fermi energy. For different degrees of disorder, the
conductivity shows very similar features while the amplitude of the
conductivity depends on the disorder parameters. Prominent quantum
oscillations at zero temperature are observed, i.e., the conductivity drops
rapidly as a new scattering channel is opened and increases again until the
Fermi energy hits the next subband at larger energies. The oscillations are
smeared by finite temperature to some extent. In the whole range of Fermi
energies studied, the average conductivity remains independent of the carrier
concentration, which is consistent with the two-dimensional case. [2, 3]
Figure 2: (Colour online) The Boltzmann conductivity as a function of the
Fermi energy for different edge roughness in AGNRs. The dashed and solid lines
correspond to zero temperature and a temperature of $T=10\mathrm{K}$,
respectively. The smooth solid lines are calculated from Eq.(24) in the limit
of $\mathcal{N}\gg 1$.
Fig. 2 shows the conductivity for edge roughness as a function of Fermi energy
calculated from Eq. (LABEL:sigma_edge). The parameter values chosen for
$\Lambda$ and $\Delta$ correspond to short-range defects, for instance, a few
atoms missing at the GNR edges, as widely assumed in simulations of edge
disorder [8, 26, 9, 14]. The correlation length ensures that
$\Lambda\ll\lambda_{F}$ over the whole range of Fermi energies. The Boltzmann
conductivity at nonzero temperature (indicated by the solid lines) shows
suppressed quantum fluctuations in comparison with the zero-temperature cases
(indicated by the dashed line). The overall conductivity decreases as the
Fermi energy increases. Since the correlation length $\Lambda$ and edge
position fluctuation amplitude $\Delta$ increase relative to the Fermi
wavelength as $E_{F}$ is increased, this behavior is similar to that one found
in conventional quantum wires [15, 17]. In the case of large number of
subbands, $\mathcal{N}\gg 1$, the results from Eq. (24) (indicated by solid
lines) exhibit the same overall trends and agree well with the exact ones
except the absence of the quantum oscillations.
The Fermi energies in Fig. 1 and 2 correspond to numbers of subbands between
$10$ and $30$. For smaller Fermi energies, i.e. a few occupied modes,
conductivity shows more prominent quantum fluctuations and may deviate
considerably from the asymptotic expressions. Moreover, it should be noted
that our calculations based on the Boltzmann approach is valid for the case
where the interband scattering is rather strong. This is the case when the
mean free path is considerably shorter than the length of the graphene ribbon.
Figure 3: (Colour online) The width dependence of the conductivity for
different edge roughness in AGNRs with $E_{F}=200\mathrm{meV}$ and temperature
$T=10\mathrm{K}$. The solid lines show the exact conductivity from Eq.(11) by
the $\mathcal{T}$ matrix inversion, and dashed lines correspond to the limit
$\mathcal{N}\gg 1$ from Eq.(24).
Fig. 3 shows the Boltzmann conductivity as a function of the GNR width for
different edge roughness parameters. Here, only results for $T>0$ are
presented. The overall conductivity for two roughness levels exhibits a
parabolic dependence on the width, superimposed by quantum oscillations. This
quadratic behavior may be seen more clearly from the analytical expression Eq.
(24), as illustrated by the dashed lines.
Figure 4: (Colour online) The magnetoconductivity around ERID as a function of
the width for different Fermi energies at finite temperature $T=10\mathrm{K}$.
(Note the logarithmic scale.) The zero-field conductivity with
$E_{F}=200\mathrm{meV}$ is also plotted at the top. The solid and dashed lines
correspond to the results from Eq.(LABEL:sigma_edge) and Eq.(26) for
$\mathcal{N}\gg 1$, respectively. $r_{c}$ is the cyclotron radius.
In the following, we give a rough estimate of the GNR conductivity in magnetic
fields with amplitudes close to the position of the ERID, i.e., $r_{c}\approx
W$. A more exact calculation would have to rely upon a calculation of the wave
functions in magnetic fields, which can be obtained by solving the
eigenequation of the Dirac Hamiltonian with magnetic fields included. [27, 28]
This, however, should have only a marginal effect and we limit ourselves to
the qualitative properties of the system close to the ERID.
In Fig. 4, the conductivity at $T=10\mathrm{K}$ is shown in the vicinity of
the ERID for different Fermi energies. The parameters for edge roughness are
fixed to $\Lambda=0.3\mathrm{nm}$ and $\Delta=6\mathrm{nm}$. For comparison,
the corresponding zero-field conductivity with $E_{F}=200\mathrm{meV}$ is
plotted as well. The conductivity around the ERID increases linearly with the
GNR width, in contrast to the parabolic dependence in the absence of a
magnetic field. This linear relationship can be easily identified from Eq.
(26) and is also illustrated by the dashed lines in Fig. 4. As a consequence
of the distinctly different dependencies of $\sigma$ on W, the ERID can be
expected to be more pronounced in wider GNRs. This feature is in qualitative
agreement with our previous quantum simulations. [14].
We conclude this analysis by commenting on the observability of the ERID in
realistic GNRs. First of all, the length of the GNR is irrelevant in the
present treatment since diffusive transport has been assumed. Second, we have
restricted ourselves to the case of rather small correlation lengths for the
edge roughness. It is self-evident that a large correlation length suppresses
the ERID in view the reduced diffusiveness of the scattering at the edges.
Furthermore, the bulk disorder must remain at a sufficiently low level as
indicated by the quantum simulations before, such that it does not mask
completely the edge roughness scattering. We moreover expect that
qualitatively, the ERID does not depend much in the the type of edges, even
though numerical simulations suggest that zigzag GNRs are more robust with
respect to edge disorder. [14] Similar analytical expressions for zigzag GNRs
are possible in principle but more complicated due to the presence of surface
states and the interdependence of the transverse and longitudinal wave
vectors.
In summary, we have studied the transport properties of AGNRs with short-range
impurities and edge roughness within the framework given by the Boltzmann
equation. An edge-roughness-induced magnetoconductivity minimum suggested by
the recent quantum calculations is confirmed by the Boltzmann results and
should become observable experimentally if the correlation length of the edge
roughness is not much larger than the Fermi wavelength and the bulk disorder
is sufficiently low. It has been shown that the ERID induced by the magnetic-
field-enhanced diffusive scattering at rough edges shows a behavior very
similar to that one found in conventional semiconductor quantum wires, despite
the fundamentally different energy dispersion.
###### Acknowledgements.
H.X. and T.H. acknowledge financial support from Heinrich-Heine-Universität
Düsseldorf.
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|
arxiv-papers
| 2012-05-30T13:22:39 |
2024-09-04T02:49:31.375265
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hengyi Xu and Thomas Heinzel",
"submitter": "Hengyi Xu",
"url": "https://arxiv.org/abs/1205.6666"
}
|
1205.6821
|
# Harmonic Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma
Density Gradients
C. Rödel1,2 D. an der Brügge3 J. Bierbach1 M. Yeung4 T. Hahn5 B. Dromey4 S.
Herzer1 S. Fuchs1 A. Galestian Pour1 E. Eckner1 M. Behmke5 M. Cerchez5 O.
Jäckel1,2 D. Hemmers5 T. Toncian5 M. C. Kaluza1,2 A. Belyanin6 G. Pretzler5 O.
Willi5 A. Pukhov3 M. Zepf4 G. G. Paulus1,2 1Institut für Optik und
Quantenelektronik, Friedrich-Schiller-Universität Jena, Germany 2Helmholtz
Institut Jena, Germany 3Institut für Theoretische Physik, Heinrich-Heine
Universität Düsseldorf, Germany 4Centre for Plasma Physics, School of
Mathematics and Physics, Queen’s University Belfast, United Kingdom 5Institut
für Laser- und Plasmaphysik, Heinrich-Heine Universität Düsseldorf, Germany
6Department of Physics, Texas A&M University, College Station TX, United
States
###### Abstract
Harmonic generation in the limit of ultra-steep density gradients is studied
experimentally. Observations demonstrate that while the efficient generation
of high order harmonics from relativistic surfaces requires steep plasma
density scale-lengths ($L_{p}/\lambda<1$) the absolute efficiency of the
harmonics declines for the steepest plasma density scale-length $L_{p}\to 0$,
thus demonstrating that near-steplike density gradients can be achieved for
interactions using high-contrast high-intensity laser pulses. Absolute photon
yields are obtained using a calibrated detection system. The efficiency of
harmonics reflected from the laser driven plasma surface via the Relativistic
Oscillating Mirror (ROM) was estimated to be in the range of $10^{-4}-10^{-6}$
of the laser pulse energy for photon energies ranging from
$20-40\,\mathrm{eV}$, with the best results being obtained for an intermediate
density scale-length.
surface high-harmonic generation, relativistic laser plasma interaction,
attosecond pulse generation
###### pacs:
52.59.Ye, 52.38.-r
††preprint: APS/surface harmonic generation
Ultrashort XUV pulses are a promising tool for a wide range of applications
including attosecond laser physics and seeding of free-electron X-ray lasers.
Typically, they are created by the nonlinear frequency up-conversion of an
intense femtosecond driving laser field in a gaseous medium. Remarkable
progress has been made to the present date with efficiencies reaching the
level of $10^{-4}$ at 20 nm wavelengths Kim2008 ; Sansone2011 . Such
efficiencies are not yet available at shorter wavelengths or for attosecond
pulse generation and the low intensities at which harmonic conversion takes
place in gaseous media, makes harnessing the high peak power in the
$0.1-1\rm{PW}$ regime challenging. High-harmonic generation at a sharp plasma-
vacuum interface via the Relativistically Oscillating Mirror (ROM) mechanism
Gibbon1996 is predicted to overcome these limitations and result in
attosecond pulses of extreme peak power Tsakiris2006 ; Gordienko2004 .
While other mechanisms such as Coherent Wake Emission (CWE) can also emit XUV
harmonics Quere2006 , the ROM mechanism is generally reported to dominate in
the limit of highly relativistic intensities, where the normalized vector
potential $a_{0}^{2}=I\lambda^{2}/(1.37\cdot 10^{18}\rm\mu
m^{2}\,\mathrm{W/cm}^{2})\gg 1$. The efficiency of ROM harmonics is predicted
to converge to a power law for ultra-relativistic intensities Baeva2006 , such
that the conversion efficiency is given by
$\eta\approx(\omega/\omega_{0})^{-8/3}$ up to a threshold frequency
$\omega_{t}\sim\gamma^{3}$, beyond which the spectrum decays exponentially.
Here, $\gamma$ is the maximum value of the Lorentz-factor associated with the
reflection point of the ROM process. While these predictions correspond well
with the observations made in experiments using pulse durations of the order
of picoseconds in terms of highest photon energy up to keV Dromey2007 ;
Norreys1996 and the slope of the harmonic efficiency Dromey2006 , no absolute
efficiency measurements have been reported to date.
The plasma density scale-length plays a critical role in determining the
response of the plasma to the incident laser radiation. In the picosecond
regime, the balance between the laser pressure and the plasma results in the
formation of scale-lengths and density profiles which are close to ideal for
ROM harmonic generation in terms of efficiency for a broad range of laser
pulse contrast. Achieving ultra-short (attosecond) XUV pulses requires lasers
with 10s of femtosecond (few-cycle) duration. Under these conditions, there is
insufficient time to modify the density scale-length significantly and hence
the density gradient and profile become critical control parameters.
Here, we report on the first absolute measurements of the ROM harmonic yield.
The highest yield is observed for intermediate pulse contrast, while the yield
declines again for the highest pulse contrast, consistent with a plasma vacuum
interface approaching step-like conditions. Achieving and verifying such
extreme interaction conditions for relativistic laser intensities is an
essential step towards exploiting the potential of a wide range of phenomena,
such as bright XUV harmonics, radiation pressure driven ion-sources and the
formation of relativistic electron sheets Kulagin2007 .
Experimental setup: High-contrast laser pulses were focused with an f/2 off-
axis parabolic mirror to up to $3\cdot 10^{19}\,\mathrm{W/cm}^{2}$ on a fused
silica or plastic coated substrate at $45^{\circ}$ p-pol. The XUV emission was
recorded with two spectrometers separately (in the presented data only plastic
coated targets are used). The flat-field spectrometer shown in configuration 1
allows a measurement of the beam divergence. The XUV spectrometer system in
configuration 2 has a larger collection angle and was calibrated regarding the
incident photon flux. The black line represents the centroid beam of the laser
steering into the center of the XUV spectrometers.
Two experiments were performed at the 30-fs Titanium-Sapphire laser systems
“Jeti” at the University of Jena and “Arcturus” at the University of
Düsseldorf, which produced similar harmonic spectra. The laser was focused
onto targets made of either glass or photoresist at an incidence angle of
$45^{\circ}$ with an FWHM intensity of $a_{0}=3.5$. At both laser systems the
pulse contrast was controlled by a single plasma mirror with different plasma
mirror targets (PMT). The plasma scale length $L_{p}$ was calculated using the
hydrodynamic simulation code “Multi-fs” RAMIS1988 based on the actual pulse
profile measured with a 3rd order autocorrelator. Details are given in
Ref.Rodel2011 . The highest pulse contrast was achieved with an anti-
reflection (AR) coated PM and resulted in a scale length of
$L_{p}\lesssim\lambda/10$, while the uncoated borosilicate glass PMs produced
an intermediate pulse contrast and $L_{p}\approx\lambda/5$ Behmke2011 . The
harmonics’ pulse energy was determined using an imaging XUV spectrometer with
a $8\,\rm mrad\times 6\,mrad$ acceptance angle which was calibrated at a
synchrotron source Fuchs2012 . The divergence of the harmonic beam was
determined with an angularly resolving XUV spectrometer (see Fig. Harmonic
Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density
Gradients left), which allowed the measured yield to be corrected for the
observed angular divergence and thus to obtain the total photon yield. The
measurement uncertainty of the photon yield is in the order of 70 % (20 %
spectrometer calibration, 50 % uncertainty due to divergence changes). The
absolute efficiency $\eta=E_{\rm XUV}/E_{0}$ was determined by only
considering the fraction of the laser pulse energy $E_{0}$ that contributes to
the harmonic generation process. Given the strong non-linearity Thaury2010
only the fraction of the laser energy that is focused to sufficiently high
intensities ($a_{0}\geq 1$) contributes to the interaction. Measurements of
the intensity distribution of the focal spot with a microscope objective
showed $20$ to $40\,\%$ of the laser pulse energy to be concentrated within
the FWHM of the focus and thus only this fraction is considered when comparing
the measured efficiencies to simulations.
Two typical harmonic spectra from photoresist targets are shown in
Fig.Harmonic Generation from Relativistic Plasma Surfaces in Ultra-Steep
Plasma Density Gradients (right) using high and intermediate contrast
settings, respectively. The angular distribution indicated in Fig.Harmonic
Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density
Gradients (left) reveals a divergence of $18.6\,\rm mrad$ for the 21st
harmonic using photoresist targets and high contrast. The importance of
accounting for the divergence of the harmonic beam when comparing changes in
the other parameters is highlighted by the observed changes in divergence
under conditions where only the pulse contrast was varied. Changing the pulse
contrast from the AR to the glass setting (and hence the scale-length from
$L_{p}\approx\lambda/10$ to $L_{p}\approx\lambda/5$) changes the divergence
from $18.6\,\rm mrad$ to $26\,\rm mrad$ for the 21st harmonic. In addition,
the divergence of different ROM harmonic orders taken from a single
measurement has an almost constant value. This is in excellent agreement with
previous observations Dromey2009 and fits well with the analysis that the
divergence of the ROM harmonic beam is characteristic of a beam with excellent
spatial coherence and is determined by the curvature (’dent’) of the emission
surface imprinted to the target by the light pressure. Since the velocity of
the hole-boring process depends on the plasma density Wilks1992 the longer
scale-length deforms more rapidly and should therefore result in a larger
divergence in agreement with the observations. This implies that the
divergence should be reduced substantially in the limit of larger spots/or
shorter pulses, which would reduce the curvature of the dent at the peak of
the pulse respectively.
Until now, it has generally been accepted that achieving sufficiently steep
density gradients for ROM harmonics is the major challenge and hence one would
expect the harmonic yield to increase as the prepulse level is reduced. For
our conditions and peak intensities, the strongest harmonic emission is
observed for intermediate contrast settings (glass PMTs), suggesting that the
higher contrast setting with AR PMTs has density scale lengths which are even
shorter than the ideal value. The pulse energies $E_{\rm XUV}$ (efficiencies
$\eta$) for individual harmonics are in the order of $3-24\,\rm\mu J$
($0.1-1\times 10^{-4}$) for the 17th harmonic and approximately
$0.3-2.7\,\rm\mu J$ ($0.1-1\times 10^{-5}$) for the 21st using different
contrast settings. Thus, we find for the first time that we have clear
quantitative evidence of ROM generation in the limit of ultra-steep scale-
lengths. While the benefit of a small, but finite, plasma scale-length for ROM
has previously been highlighted by simulations Tarasevitch2007 ; Thaury2010 ,
the experiments performed so-far have required the highest achievable pulse
contrast or shortest possible scale length, respectively, in order to optimize
ROM efficiency and beam quality Dromey2006 ; Dromey2009 .
The influence of the plasma scale length has been studied both for glass
substrates and photoresist targets that have been coated onto the optically
polished glass substrate reducing the density from $2.2\,\rm g/cm^{3}$ to
$\approx 1.1\,\rm g/cm^{3}$ (or from $\approx 400n_{c}$ to $\approx 200n_{c}$
in terms of the critical density $n_{c}$). The harmonic emission for these
high target densities and respective scale lengths is comparable indicating
that the enhanced harmonic emission at intermediate scale lengths is not very
sensitive for such high peak densities. This means that for our parameters the
harmonic emission is enhanced due to the lower density in the plasma gradient
and not by using a lower maximum density. Since the reflection point of the
ROM is located near the critical density at elongated plasma density ramps,
the ROM process is affected by the length of the plasma gradient instead of
the maximum plasma density. The observed dependence of the efficiency on the
scale length can be understood in terms of the plasma dynamics as follows.
First, the denser the plasma and the steeper the gradient, the more the
electric field in the skin layer is reduced. Second, the “spring constant” of
the electron plasma becomes larger for denser and steeper plasmas, making the
ROM harder to drive to the high values of $\gamma$ associated with a more
efficient production of higher harmonic orders.
(a) Surface field $\mathbf{E}_{\mathrm{crit}}$ at the critical density in
units of the incident field $\mathbf{E}_{0}$, as estimated from the equation
in Ref. Kruer2003 (black dashed line) and computed exactly by numerical
integration for an exponential gradient (blue line). (b) Efficiency of SHHG
above the 14th order
$\eta_{ROM}=\int^{\infty}_{14\omega_{0}}I(\omega)d\omega/P_{0}$ for
$a_{0}=3.5$ at different plasma scale lengths from a set of 1D PIC
simulations. Incidence was p-polarized, the plasma ramp is exponential up to a
maximum density of $n_{e}=200n_{c}$.
To make an analytical estimate of the field at the critical density surface we
can consider a laser interacting at normal incidence with the target (oblique
incidence can be treated by switching into the frame of reference, in which
the laser is normally incident Bourdier1983 ). The electrons can gain kinetic
energy only through the $\mathbf{E}$-field of the laser. This field is
tangential to the surface and is attenuated due to the skin effect. We assume
for the moment, that the field is non-relativistic. Evaluating the linear wave
equation, we find the threshold condition for it to become relativistic. For a
perfectly steep plasma edge, it can be calculated analytically by evaluation
of the continuity condition at the plasma edge, yielding
$|\mathbf{E}_{\mathrm{crit}}|/|\mathbf{E}_{0}|=2\omega_{0}/\omega_{p}$. Hence,
for our laser and plasma parameters, the field would not be relativistic for a
perfect step density profile. This is reflected in Fig. Harmonic Generation
from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density Gradients(b):
In the limit $L\rightarrow 0$ there are no relativistic harmonics. The skin
field is however enhanced due to a finite density ramp. As a first estimate
for finite ramps we may consider the calculation found in Ref. Kruer2003 ,
leading to $|\mathbf{E}_{\mathrm{crit}}|/|\mathbf{E}_{0}|\approx
1.4(\omega_{0}L/c)^{1/6}$ at the critical density. This formula becomes exact
for linear and extended ($L\gg\lambda$) gradients. For steep, exponential
ramps, as are expected in the experiments, we find the skin field by numerical
integration of the inhomogeneous wave equation. Results of this computation
are shown in Fig. Harmonic Generation from Relativistic Plasma Surfaces in
Ultra-Steep Plasma Density Gradients(a), along with the simple scaling from
Ref. Kruer2003 . It can be seen that even small scale lengths can considerably
boost the skin field compared to the case of step-like profiles. Already at
$L=\lambda/20$, there is practically no attenuation at the critical density,
but the field can still grow slightly for longer plasma scales. We further
note that the simple sixth-root dependence calculated for a linear gradient
also yields a reasonable estimate for the exponential gradient, only slightly
overestimating the field in comparison to the exact result. Fig. Harmonic
Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density
Gradients(b) shows the integrated efficiency $\eta_{\mathrm{ROM}}$ of ROM
harmonics for the same density gradients and a laser amplitude $a_{0}=3.5$. As
expected from the previous considerations and Fig.Harmonic Generation from
Relativistic Plasma Surfaces in Ultra-Steep Plasma Density Gradients(a), it
can be seen that the ROM efficiency rises quickly as soon as the skin field
becomes relativistic. For the scale lengths $L>\lambda/10$, the integrated
efficiency remains approximately constant at $\eta\approx 7\times 10^{-3}$ as
expected from the ROM model. While the reduction in the critical field
explains the drop to very low efficiencies at very short scale-lengths, it
does not fully explain the efficiency scaling at intermediate scale-lengths.
Since the reflection point oscillates around the immobile ion background due
to the driving relativistic laser field, another contribution must come from
the restoring force due to the quasi-static field generated by the plasma once
the electrons are driven out of equilibrium by the laser field. For a given
mean displacement of the plasma electrons, the restoring force is proportional
to the plasma density. In the limit of a step-like density profile, the
restoring force is directly proportional to the maximum plasma density, while
in the limit of very long density scale-lengths the restoring force is
determined by the critical density. In the intermediate case of relevance
here, the restoring force will depend in a complex fashion on the density
scale-length, peak density and amplitude of the oscillation. While it is not
easily possible to express this dependence in a closed analytical form, it is
clear that one would expect the effective density and hence the restoring
force to be lower for increasingly shallow density gradients resulting in the
dependence shown in Fig. Harmonic Generation from Relativistic Plasma Surfaces
in Ultra-Steep Plasma Density Gradients. Generally, the denser the plasma and
the steeper the density ramp, the harder it is for the laser to drive large
amplitude oscillations at the plasma surface. This in turn leads to a smaller
oscillation amplitude of the ROM and, consequently, to a lower
$\gamma$-factor. An analysis of the electron spring model at ultra-
relativistic intensities can be found in Ref.Gonoskov2011 . In agreement with
our experimental observations a trend towards higher efficiency for moderately
long scale-length or low peak densities is expected.
Experimental efficiencies (circles) are compared to spectral densities from 1D
PIC simulations (lines) for different plasma scale lengths (density
$n_{e}=200n_{c}$, exponential density profile). The experimental efficiencies
have been normalized to a pulse energy of $250\,\rm mJ$ (energy that is
focused to $a_{0}>1$). In the ultra-relativistic limit the efficiencies
converge to the BGP power scaling $\eta\approx(\omega/\omega_{0})^{-8/3}$
Baeva2006 .
Under our experimental conditions ($a_{0}=3.5$) the efficiencies are still
expected to be below the relativistic limit regime where the
$\eta\approx(\omega/\omega_{0})^{-8/3}$ scaling applies. Fig. Harmonic
Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density
Gradients compares our experimental results to a range of efficiencies
predicted by 1D PIC simulations. While the efficiencies are broadly compatible
with the range of efficiencies predicted by the simulations, they appear
somewhat lower than predictions for the nominal density gradients derived from
the measurement of the pulse-contrast and Multi-fs modelling. While, to our
knowledge, Multi-fs is the best suited code to calculate the hydrodynamic
expansion under such conditions, the code has not been validated directly by
measurements of the scale-length under such conditions. Consequently one
possible explanation for the discrepancy may be that the density gradients are
even steeper than predicted. What is clear both experimentally and from
simulations is that the efficiency of the ROM process depends sensitively on
the plasma scale length. The generation of surface waves, which have been
found in 2D simulations, induce high harmonic emission at angular sidebands
Brugge2012 . This may lead to differences between the experimental results and
the 1D simulations. Another important effect that is not considered in our
simulations is the ion motion. In fact, Thaury and Quéré Thaury2010 have
shown that the harmonics efficiency in simulations with mobile ions is
significantly reduced.
In conclusion, we have investigated harmonic generation in the limit of ultra-
steep density gradients and shown first experimental evidence of the absolute
yield reducing for very steep gradients. This demonstrates that relativistic
interactions in the limit of ultra-steep density gradients can be achieved by
a careful control of the laser parameters. Harmonic efficiency is optimized
for intermediate scale-lengths. Our results suggest the generation of intense
attosecond pulse trains with pulse energies exceeding $10\,\rm\mu J$, thus
paving the way towards applications such as nonlinear attosecond experiments
or the seeding of free-electron lasers with surface high-harmonic radiation.
###### Acknowledgements.
This work was funded by the DFG project SFB TR18 and Laserlab Europe. C.R.
acknowledges support from the Carl Zeiss Stiftung. Monika Toncian, Burgard
Beleites and Falk Ronneberger contributed to this work by operating the
Arcturus and Jeti laser facility.
## References
* [1] I. J. Kim, G. H. Lee, S. B. Park, Y. S. Lee, T. K. Kim, C. H. Nam, T. Mocek, and K. Jakubczak. Generation of submicrojoule high harmonics using a long gas jet in a two-color laser field. Applied Physics Letters, 92(2):021125, January 2008.
* [2] G. Sansone, L. Poletto, and M. Nisoli. High-energy attosecond light sources. Nature Photonics, 5(11):656–664, November 2011.
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* [6] F. Quéré, C. Thaury, P. Monot, S. Dobosz, P. Martin, J.-P. Geindre, and P. Audebert. Coherent wake emission of high-order harmonics from overdense plasmas. Physical Review Letters, 96(12):125004, March 2006.
* [7] T. Baeva, S. Gordienko, and A. Pukhov. Theory of high-order harmonic generation in relativistic laser interaction with overdense plasma. Physical Review E, 74(4):046404, October 2006.
* [8] B. Dromey, S. Kar, C. Bellei, D. C. Carroll, R. J. Clarke, J. S. Green, S. Kneip, K. Markey, S. R. Nagel, P. T. Simpson, L. Willingale, P. McKenna, D. Neely, Z. Najmudin, K. Krushelnick, P. A. Norreys, and M. Zepf. Bright multi-kev harmonic generation from relativistically oscillating plasma surfaces. Physical Review Letters, 99(8):085001, August 2007.
* [9] P. A. Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M. Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely, F. N. Walsh, J. S. Wark, and A. E. Dangor. Efficient extreme uv harmonics generated from picosecond laser pulse interactions with solid targets. Physical Review Letters, 76(11):1832–1835, March 1996.
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* [11] V. V. Kulagin, V. A. Cherepenin, M. S. Hur, and H. Suk. Theoretical investigation of controlled generation of a dense attosecond relativistic electron bunch from the interaction of an ultrashort laser pulse with a nanofilm. Physical Review Letters, 99(12):124801, September 2007.
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* [17] B. Dromey, D. Adams, R. Hörlein, Y. Nomura, S. Rykovanov, D. Carroll, P. Foster, S. Kar, K. Markey, P. McKenna, D. Neely, M. Geissler, G. Tsakiris, and M. Zepf. Diffraction-limited performance and focusing of high harmonics from relativistic plasmas. Nature Physics, 5:146–152, 2009.
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|
arxiv-papers
| 2012-05-30T20:08:14 |
2024-09-04T02:49:31.385557
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian R\\\"odel, Daniel an der Br\\\"ugge, Jana Bierbach, Mark Yeung,\n Thomas Hahn, Brendan Dromey, Sven Herzer, Silvio Fuchs, Arpa Galestian Pour,\n Erich Eckner, Michael Behmke, Mirela Cerchez, Oliver J\\\"ackel, Dirk Hemmers,\n Toma Toncian, Malte C. Kaluza, Alexey Belyanin, Georg Pretzler, Oswald Willi,\n Alexander Pukhov, Matthew Zepf and Gerhard G. Paulus",
"submitter": "Christian R\\\"odel",
"url": "https://arxiv.org/abs/1205.6821"
}
|
1205.6842
|
# On the $q$-Hardy-littlewood-type maximal operator with weight related to
fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com and Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
The fundamental aim of this paper is to define weighted $q$-Hardy-littlewood-
type maximal operator by means of fermionic $p$-adic $q$-invariant
distribution on $\mathbb{Z}_{p}$. Also, we derive some interesting properties
concerning this type maximal operator.
###### Key words and phrases:
fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$, Hardy-littlewood theorem,
$p$-adic analysis, $q$-analysis
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. Introduction and Notations
$p$-adic numbers also play a vital and important role in mathematics. $p$-adic
numbers were invented by the German mathematician Kurt Hensel [11], around the
end of the nineteenth century. In spite of their being already one hundred
years old, these numbers are still today enveloped in an aura of mystery
within the scientific community.
The fermionic $p$-adic $q$-integral are originally constructed by Kim [4]. Kim
also introduced Lebesgue-Radon-Nikodym Theorem with respect to fermionic
$p$-adic $q$-integral on $\mathbb{Z}_{p}$. The fermionic $p$–adic $q$-integral
on $\mathbb{Z}_{p}$ is used in Mathematical Physics for example the functional
equation of the $q$-Zeta function, the $q$-Stirling numbers and $q$-Mahler
theory of integration with respect to the ring $\mathbb{Z}_{p}$ together with
Iwasawa’s $p$-adic $q$-$L$ function.
In [9], Jang also defined $q$-extension of Hardy-Littlewood-type maximal
operator by means of $q$-Volkenborn integral on $\mathbb{Z}_{p}$. Next, in
previous paper [10], Araci and Acikgoz added a weight into Jang’s $q$-Hardy-
Littlewood-type maximal operator and derived some interesting properties by
means of Kim’s $p$-adic $q$-integral on $\mathbb{Z}_{p}$. Now also, we shall
consider weighted $q$-Hardy-Littlewood-type maximal operator on the fermionic
$p$-adic $q$-integral on $\mathbb{Z}_{p}$. Moreover, we shall analyse
$q$-Hardy-Littlewood-type maximal operator via the fermionic $p$-adic
$q$-integral on $\mathbb{Z}_{p}$.
Assume that $p$ be an odd prime number. Let $\mathcal{\mathbb{Q}}_{p}$ be the
field of $p$-adic rational numbers and let $\mathcal{\mathbb{C}}_{p}$ be the
completion of algebraic closure of $\mathcal{\mathbb{Q}}_{p}$.
Thus,
$\mathcal{\mathbb{Q}}_{p}=\left\\{x=\sum_{n=-k}^{\infty}a_{n}p^{n}:0\leq
a_{n}<p\right\\}.$
Then $\mathbb{Z}_{p}$ is an integral domain, which is defined by
$\mathcal{\mathbb{Z}}_{p}=\left\\{x=\sum_{n=0}^{\infty}a_{n}p^{n}:0\leq
a_{n}\leq p-1\right\\},$
or
$\mathcal{\mathbb{Z}}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq
1\right\\}.$
In this paper, we assume that $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}<1$ as an indeterminate.
The $p$-adic absolute value $\left|.\right|_{p}$, is normally defined by
$\left|x\right|_{p}=\frac{1}{p^{r}}\text{,}$
where $x=p^{r}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and
$r\in\mathcal{\mathbb{Q}}$.
A $p$-adic Banach space $B$ is a $\mathbb{Q}_{p}$-vector space with a lattice
$B^{0}$ ($\mathcal{\mathbb{Z}}_{p}$-module) separated and complete for
$p$-adic topology, ie.,
$B^{0}\simeq\lim_{\overleftarrow{n\in\mathbb{N}}}B^{0}/p^{n}B^{0}\text{.}$
For all $x\in B$, there exists $n\in\mathcal{\mathbb{Z}}$, such that $x\in
p^{n}B^{0}$. Define
$v_{B}\left(x\right)=\sup_{n\in\mathbb{N}\cup\left\\{+\infty\right\\}}\left\\{n:x\in
p^{n}B^{0}\right\\}\text{.}$
It satisfies the following properties:
$\displaystyle v_{B}\left(x+y\right)$ $\displaystyle\geq$
$\displaystyle\min\left(v_{B}\left(x\right),v_{B}\left(y\right)\right)\text{,}$
$\displaystyle v_{B}\left(\beta x\right)$ $\displaystyle=$ $\displaystyle
v_{p}\left(\beta\right)+v_{B}\left(x\right)\text{, if
}\beta\in\mathbb{Q}_{p}\text{.}$
Then, $\left\|x\right\|_{B}=p^{-v_{B}\left(x\right)}$ defines a norm on $B,$
such that $B$ is complete for $\left\|.\right\|_{B}$ and $B^{0}$ is the unit
ball.
A measure on $\mathcal{\mathbb{Z}}_{p}$ with values in a $p$-adic Banach space
$B$ is a continuous linear map
$f\mapsto\int
f\left(x\right)\mu=\int_{\mathbb{Z}_{p}}f\left(x\right)\mu\left(x\right)$
from $C^{0}\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$,
(continuous function on $\mathcal{\mathbb{Z}}_{p}$) to $B$. We know that the
set of locally constant functions from $\mathcal{\mathbb{Z}}_{p}$ to
$\mathcal{\mathbb{Q}}_{p}$ is dense in
$C^{0}\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$ so.
Explicitly, for all $f\in
C^{0}\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$, the
locally constant functions
$f_{n}=\sum_{i=0}^{p^{n}-1}f\left(i\right)1_{i+p^{n}\mathbb{Z}_{p}}\rightarrow\text{
}f\text{ in }C^{0}\text{.}$
Now if
$\mu\in\mathcal{D}_{0}\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{Q}}_{p}\right)$,
set
$\mu\left(i+p^{n}\mathcal{\mathbb{Z}}_{p}\right)=\int_{\mathbb{Z}_{p}}1_{i+p^{n}\mathcal{\mathbb{Z}}_{p}}\mu$.
Then $\int_{\mathcal{\mathbb{Z}}_{p}}f\mu$ is given by the following “Riemann
sums”
$\int_{\mathbb{Z}_{p}}f\mu=\lim_{n\rightarrow\infty}\sum_{i=0}^{p^{n}-1}f\left(i\right)\mu\left(i+p^{n}\mathcal{\mathbb{Z}}_{p}\right)\text{.}$
T. Kim defined $\mu_{-q}$ as follows:
$\mu_{-q}\left(\xi+dp^{n}\mathcal{\mathbb{Z}}_{p}\right)=\frac{\left(-q\right)^{\xi}}{\left[dp^{n}\right]_{-q}}$
and this can be extended to a distribution on $\mathcal{\mathbb{Z}}_{p}$. This
distribution yields an integral in the case $d=1$.
So, $q$-Volkenborn integral was defined by T. Kim as follows:
(1.1)
$I_{-q}\left(f\right)=\int_{\mathcal{\mathbb{Z}}_{p}}f\left(\xi\right)d\mu_{q}\left(\xi\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{n}-1}\left(-1\right)^{\xi}f\left(\xi\right)q^{\xi}\text{
}$
Where $\left[x\right]_{q}$ is a $q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{,}$
note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ cf. [2], [3], [4], [5],
[9].
Let $d$ be a fixed positive integer with $\left(p,d\right)=1$. We now set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{n}}\mathcal{\mathbb{Z}}/dp^{n}\mathcal{\mathbb{Z}},$
$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle\mathbb{Z}_{p},$
$\displaystyle X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathcal{\mathbb{Z}}_{p},$
$\displaystyle a+dp^{n}\mathcal{\mathbb{Z}}_{p}$ $\displaystyle=$
$\displaystyle\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}p^{n}\right)\right\\},$
where $a\in\mathcal{\mathbb{Z}}$ satisfies the condition $0\leq a<dp^{n}$. For
$f\in UD\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$,
$\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)=\int_{X}f\left(x\right)d\mu_{-q}\left(x\right),$
(for details, see [8]).
By the meaning of $q$-Volkenborn integral, we consider below strongly $p$-adic
$q$-invariant distribution $\mu_{-q}$ on $\mathbb{Z}_{p}$ in the form
$\left|\left[p^{n}\right]_{-q}\mu_{-q}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)-\left[p^{n+1}\right]_{-q}\mu_{-q}\left(a+p^{n+1}\mathcal{\mathbb{Z}}_{p}\right)\right|<\delta_{n},$
where $\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$ and $\delta_{n}$ is
independent of $a$. Let $f\in
UD\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$, for any
$a\in\mathcal{\mathbb{Z}}_{p}$, we assume that the weight function
$\omega\left(x\right)$ is defined by $\omega\left(x\right)=\omega^{x}$ where
$\omega\in\mathbb{C}_{p}$ with $\left|1-\omega\right|_{p}<1$. We define the
weighted measure on $\mathcal{\mathbb{Z}}_{p}$ as follows:
(1.2)
$\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)=\int_{a+p^{n}\mathcal{\mathbb{Z}}_{p}}\omega^{\xi}f\left(\xi\right)d\mu_{-q}\left(\xi\right)$
where the integral is the fermionic $p$-adic $q$-integral. By (1.2), we easily
note that $\mu_{f,-q}^{\left(\omega\right)}$ is a strongly weighted measure on
$\mathbb{Z}_{p}$. Namely,
$\displaystyle\left|\left[p^{n}\right]_{-q}\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)-\left[p^{n+1}\right]_{-q}\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n+1}\mathcal{\mathbb{Z}}_{p}\right)\right|_{p}$
$\displaystyle=$
$\displaystyle\left|\sum_{x=0}^{p^{n}-1}\left(-1\right)^{x}\omega^{x}f\left(x\right)q^{x}-\sum_{x=0}^{p^{n}}\left(-1\right)^{x}\omega^{x}f\left(x\right)q^{x}\right|_{p}$
$\displaystyle\leq$
$\displaystyle\left|\frac{f\left(p^{n}\right)\left(-1\right)^{p^{n}}\omega^{p^{n}}q^{p^{n}}}{p^{n}}\right|_{p}\left|p^{n}\right|_{p}$
$\displaystyle\leq$ $\displaystyle Cp^{-n}$
Thus, we get the following proposition.
###### Proposition 1.
For $f,g\in UD\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$,
then, we have
$\mu_{\alpha f+\beta
g,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)=\alpha\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)+\beta\mu_{g,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)\text{.}$
where $\alpha,\beta$ are positive constants. Also, we have
$\left|\left[p^{n}\right]_{-q}\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)-\left[p^{n+1}\right]_{-q}\mu_{f,-q}^{\left(\omega\right)}\left(a+p^{n+1}\mathcal{\mathbb{Z}}_{p}\right)\right|\leq
Cp^{-n}$
where $C$ is positive constant.
Let
$\mathcal{P}_{q}\left(x\right)\in\mathbb{C}_{p}\left[\left[x\right]_{q}\right]$
be an arbitrary $q$-polynomial. Now also, we indicate that
$\mu_{\mathcal{P},-q}^{\left(\omega\right)}$ is a strongly weighted fermionic
$p$-adic $q$-invariant measure on $\mathbb{Z}_{p}$. Without a loss of
generality, it is sufficient to evidence the statement for
$\mathcal{P}\left(x\right)=\left[x\right]_{q}^{k}$.
(1.3)
$\mu_{\mathcal{P},-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)=\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q}}\sum_{i=0}^{p^{m-n}-1}w^{a+ip^{n}}\left[a+ip^{n}\right]_{q}^{k}\left(-q\right)^{a+ip^{n}}\text{.}$
where
$\displaystyle\left[a+ip^{n}\right]_{q}^{k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\binom{k}{j}\left[a\right]_{q}^{k-j}q^{aj}\left[p^{n}\right]_{q}^{j}\left[i\right]_{q^{p^{n}}}^{j}$
$\displaystyle=$
$\displaystyle\left[a\right]_{q}^{k}+k\left[a\right]_{q}^{k-1}q^{a}\left[p^{n}\right]_{q}\left[i\right]_{q^{p^{n}}}+...+q^{ak}\left[p^{n}\right]_{q}^{k}\left[i\right]_{q^{p^{n}}}^{k}\text{.}$
and
(1.5)
$w^{a+ip^{n}}=w^{a}\sum_{l=0}^{ip^{n}}\binom{ip^{n}}{l}\left(w-1\right)^{l}\equiv
w^{a}\left(\mathop{\mathrm{m}od}p^{n}\right)\text{.}$
Similarly,
(1.6)
$\left(-q\right)^{a+ip^{n}}=\left(-q\right)^{a}\sum_{l=0}^{ip^{n}}\binom{ip^{n}}{l}\left(-1\right)^{l}\left(q+1\right)^{l}\equiv\left(-q\right)^{a}\left(\mathop{\mathrm{m}od}p^{n}\right)\text{.}$
By (1.3), (1), (1.5) and (1.6), we have the following
$\displaystyle\mu_{\mathcal{P},-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)$
$\displaystyle\equiv$
$\displaystyle\left(-1\right)^{a}\omega^{a}q^{a}\left[a\right]_{q}^{k}\left(\mathop{\mathrm{m}od}p^{n}\right)$
$\displaystyle\equiv$
$\displaystyle\left(-1\right)^{a}\omega^{a}q^{a}\mathcal{P}\left(a\right)\left(\mathop{\mathrm{m}od}p^{n}\right)\text{.}$
For $x\in\mathcal{\mathbb{Z}}_{p}$, let $x\equiv
x_{n}\left(\mathop{\mathrm{m}od}p^{n}\right)$ and $x\equiv
x_{n+1}\left(\mathop{\mathrm{m}od}p^{n+1}\right)$, where $x_{n}$,
$x_{n+1}\in\mathcal{\mathbb{Z}}$ with $0\leq x_{n}<p^{n}$ and $0\leq
x_{n+1}<p^{n+1}$.
Then, we procure the following
$\left|\left[p^{n}\right]_{-q}\mu_{\mathcal{P},-q}^{\left(\omega\right)}\left(a+p^{n}\mathcal{\mathbb{Z}}_{p}\right)-\left[p^{n+1}\right]_{-q}\mu_{\mathcal{P},-q}^{\left(\omega\right)}\left(a+p^{n+1}\mathcal{\mathbb{Z}}_{p}\right)\right|\leq
Cp^{-n}\text{,}$
where $C$ is positive constant and $n>>0$.
Let $UD\left(\mathcal{\mathbb{Z}}_{p},\mathcal{\mathbb{C}}_{p}\right)$ be the
space of uniformly differentiable functions on $\mathcal{\mathbb{Z}}_{p}$ with
supnorm
$\left\|f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|f\left(x\right)\right|_{p}.$
The difference quotient $\Delta_{1}f$ of $f$ is the function of two variables
given by
$\Delta_{1}f\left(m,x\right)=\frac{f\left(x+m\right)-f\left(x\right)}{m},\text{
for all }x\text{, }m\in\mathbb{Z}_{p}\text{, }m\neq 0\text{.}$
A function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}$ is said to be a
Lipschitz function if there exists a constant $M>0$ $\left(\text{the Lipschitz
constant of }f\right)$ such that
$\left|\Delta_{1}f\left(m,x\right)\right|\leq M\text{ for all
}m\in\mathbb{Z}_{p}\backslash\left\\{0\right\\}\text{ and
}x\in\mathbb{Z}_{p}.$
The $\mathbb{C}_{p}$ linear space consisting of all Lipschitz function is
denoted by $Lip\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$. This space is a
Banach space with the respect to the norm
$\left\|f\right\|_{1}=\left\|f\right\|_{\infty}\mathop{\textstyle\bigvee}\left\|\Delta_{1}f\right\|_{\infty}$
(for more informations, see [1], [2], [3], [4], [5], [6], [9]). The objective
of this paper is to introduce weighted $q$-Hardy Littlewood type maximal
operator on the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$. Also, we
show that the boundedness of the weighted $q$-Hardy-littlewood-type maximal
operator in the $p$-adic integer ring.
## 2\. The weighted $q$-Hardy-littlewood-type maximal operator
In view of (1.2) and the definition of fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$, we now consider the following theorem.
###### Theorem 1.
Let $\mu_{-q}^{\left(w\right)}$ be a strongly fermionic $p$-adic $q$-invariant
on $\mathbb{Z}_{p}$ and $f\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$.
Then for any $n\in\mathbb{Z}$ and any $\xi\in\mathbb{Z}_{p}$, we have
$(1)$
$\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}f\left(\xi\right)\left(-q\right)^{-\xi}d\mu_{-q}\left(\xi\right)=\frac{\left(-1\right)^{a}\omega^{a}}{\left[p^{n}\right]_{-q}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)$,
$(2)$
$\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}d\mu_{-q}\left(\xi\right)=\frac{\omega^{a}\left(-q\right)^{a}}{\left[p^{n}\right]_{-q}}\frac{2}{1+\omega^{p^{n}}q^{p^{n}}}$.
###### Proof.
(1) By using (1.1) and (1.2), we see the followings applications
$\displaystyle\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}f\left(\xi\right)\left(-q\right)^{-\xi}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m+n}\right]_{-q}}\sum_{\xi=0}^{p^{m}-1}\omega^{a+p^{n}\xi}f\left(a+p^{n}\xi\right)\left(-q\right)^{-\left(a+p^{n}\xi\right)}q^{a+p^{n}\xi}\left(-1\right)^{a+p^{n}\xi}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{a}\omega^{a}\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q^{p^{n}}}\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{m}-1}\omega^{\xi}\left(-q\right)^{-p^{n}\xi}f\left(a+p^{n}\xi\right)\left(-q^{p^{n}}\right)^{\xi}$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{a}\omega^{a}}{\left[p^{n}\right]_{-q}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right).$
(2) By the same method of (1), then, we easily derive the following
$\displaystyle\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m+n}\right]_{-q}}\sum_{\xi=0}^{p^{m}-1}\omega^{a+\xi
p^{n}}\left(-q\right)^{a+\xi p^{n}}$ $\displaystyle=$
$\displaystyle\frac{\omega^{a}\left(-q\right)^{a}}{\left[p^{n}\right]_{-q}}\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q^{p^{n}}}}\sum_{\xi=0}^{p^{m}-1}\left(\omega^{p^{n}}\right)^{\xi}\left(-q^{p^{n}}\right)^{\xi}$
$\displaystyle=$
$\displaystyle\frac{\omega^{a}\left(-q\right)^{a}}{\left[p^{n}\right]_{-q}}\lim_{m\rightarrow\infty}\frac{1+\left(\omega^{p^{n}}q^{p^{n}}\right)^{p^{m}}}{1+\omega^{p^{n}}q^{p^{n}}}$
$\displaystyle=$
$\displaystyle\frac{\omega^{a}\left(-q\right)^{a}}{\left[p^{n}\right]_{-q}}\frac{2}{1+\omega^{p^{n}}q^{p^{n}}}$
Since $\underset{m\rightarrow\infty}{\lim}q^{p^{m}}=1$ for
$\left|1-q\right|_{p}<1,$ our assertion follows.
We are now ready to introduce definition of weighted $q$-Hardy-littlewood-type
maximal operator related to fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ with a strong fermionic $p$-adic $q$-invariant distribution
$\mu_{-q}$ in the $p$-adic integer ring.
###### Definition 1.
Let $\mu_{-q}^{\left(\omega\right)}$ be a strongly fermionic $p$-adic
$q$-invariant distribution on $\mathbb{Z}_{p}$ and $f\in
UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$. Then, $q$-Hardy-littlewood-type
maximal operator with weight related to fermionic $p$-adic $q$-integral on
$a+p^{n}\mathbb{Z}_{p}$ is defined by the following
$\mathcal{M}_{p,q}^{\left(\omega\right)}f\left(a\right)=\underset{n\in\mathbb{Z}}{\sup}\frac{1}{\mu_{1,-q}^{\left(w\right)}\left(\xi+p^{n}\mathbb{Z}_{p}\right)}\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}\left(-q\right)^{-\xi}f\left(\xi\right)d\mu_{-q}\left(\xi\right)$
for all $a\in\mathbb{Z}_{p}$.
We recall that famous Hardy-littlewood maximal operator $\mathcal{M}_{\mu}$,
which is defined by
(2.1) $\mathcal{M}_{\mu}f\left(a\right)=\underset{a\in
Q}{\sup}\frac{1}{\mu\left(Q\right)}\int_{Q}\left|f\left(x\right)\right|d\mu\left(x\right)\text{,}$
where $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a locally bounded
Lebesgue measurable function, $\mu$ is a Lebesgue measure on
$\left(-\infty,\infty\right)$ and the supremum is taken over all cubes $Q$
which are parallel to the coordinate axes. Note that the boundedness of the
Hardy-Littlewood maximal operator serves as one of the most important tools
used in the investigation of the properties of variable exponent spaces (see
[9]). The essential aim of Theorem 1 is to deal with the weighted
$q$-extension of the classical Hardy-Littlewood maximal operator in the space
of $p$-adic Lipschitz functions on $\mathbb{Z}_{p}$ and to find the
boundedness of them. By the meaning of Definition 1, then, we state the
following theorem.
###### Theorem 2.
Let $f\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ and
$x\in\mathbb{Z}_{p}$, we get
(1)
$\mathcal{M}_{p,q}^{\left(\omega\right)}f\left(a\right)=\frac{\left(-1\right)^{a}}{2q^{a}}\underset{n\in\mathbb{Z}}{\sup\left(1+\omega^{p^{n}q^{p^{n}}}\right)}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(x+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)$,
(2)
$\left|\mathcal{M}_{p,q}^{\left(\omega\right)}f\left(a\right)\right|_{p}\leq\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|1+\omega^{p^{n}}q^{p^{n}}\right|_{p}\left\|f\right\|_{1}\left\|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}$,
where
$\left\|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}=\int_{\mathbb{Z}_{p}}\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)$.
###### Proof.
(1) Because of Theorem 1 and Definition 1, we see
$\displaystyle M_{p,q}^{\left(\omega\right)}f\left(a\right)$ $\displaystyle=$
$\displaystyle\underset{n\in\mathbb{Z}}{\sup}\frac{1}{\mu_{1,-q}^{\left(\omega\right)}\left(\xi+p^{n}\mathbb{Z}_{p}\right)}\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}\left(-q\right)^{-\xi}f\left(\xi\right)d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{a}}{2q^{a}}\underset{n\in\mathbb{Z}}{\sup\left(1+\omega^{p^{n}q^{p^{n}}}\right)}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(x+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)\text{.}$
(2) On account of (1), we can derive the following
$\displaystyle\left|M_{p,q}^{\left(\omega\right)}f\left(a\right)\right|_{p}$
$\displaystyle=$
$\displaystyle\left|\frac{\left(-1\right)^{a}}{2q^{a}}\underset{n\in\mathbb{Z}}{\sup}\left(1+\omega^{p^{n}}q^{p^{n}}\right)\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(x+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)\right|_{p}$
$\displaystyle\leq$
$\displaystyle\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|\left(1+\omega^{p^{n}}q^{p^{n}}\right)\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(x+p^{n}\xi\right)\left(-q\right)^{-p^{n}\xi}d\mu_{-q^{p^{n}}}\left(\xi\right)\right|_{p}$
$\displaystyle\leq$
$\displaystyle\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|1+\omega^{p^{n}}q^{p^{n}}\right|_{p}\int_{\mathbb{Z}_{p}}\left|f\left(a+p^{n}\xi\right)\right|_{p}\left|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\xi}\right|_{p}d\mu_{-q^{p^{n}}}\left(\xi\right)$
$\displaystyle\leq$
$\displaystyle\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|1+\omega^{p^{n}}q^{p^{n}}\right|_{p}\left\|f\right\|_{1}\int_{\mathbb{Z}_{p}}\left|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\xi}\right|_{p}d\mu_{-q^{p^{n}}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|1+\omega^{p^{n}}q^{p^{n}}\right|_{p}\left\|f\right\|_{1}\left\|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}\text{.}$
Thus, we complete the proof of theorem.
We note that Theorem 2 (2) shows the supnorm-inequality for the $q$-Hardy-
Littlewood-type maximal operator with weight on $\mathbb{Z}_{p}$, on the other
hand, Theorem 2 (2) shows the following inequality
(2.2)
$\left\|\mathcal{M}_{p,q}^{\left(\omega\right)}f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|\mathcal{M}_{p,q}^{\left(\omega\right)}f\left(x\right)\right|_{p}\leq\mathcal{K}\left\|f\right\|_{1}\left\|\left(\frac{-q^{p^{n}}}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}$
where
$\mathcal{K}=\left|\frac{\left(-1\right)^{a}}{2q^{a}}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|1+\omega^{p^{n}}q^{p^{n}}\right|_{p}$.
By the equation (2.2), we get the following Corollary, which is the
boundedness for weighted $q$-Hardy-Littlewood-type maximal operator with
weight on $\mathbb{Z}_{p}$.
###### Corollary 1.
$\mathcal{M}_{p,q}^{\left(\omega\right)}$ is a bounded operator from
$UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ into
$L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$, where
$L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ is the space of all
$p$-adic supnorm-bounded functions with the
$\left\|f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|f\left(x\right)\right|_{p}\text{,}$
for all $f\in L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$.
## References
* [1] T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$, Russ. J. Math. Phys. 19 (2012) (in press)
* [2] T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic $q$-Volkenborn distribution on $\mu_{q}$, Appl. Math. Comp. 187 (2007), 266–271.
* [3] T. Kim, S. D. Kim, D.W. Park, On Uniformly differntiabitity and $q$-Mahler expansion, Adv. Stud. Contemp. Math. 4 (2001), 35–41.
* [4] T. Kim, $q$-Volkenborn integration, Russian J. Math. Phys. 9 (2002) 288–299.
* [5] T. Kim, On a $q$-analogue of the $p$-adic log Gamma functions and related integrals, Journal of Number Theory 76 (1999), 320-329.
* [6] T. Kim, Note on Dedekind-type DC sums, Advanced Studies in Contemporary Mathematics 18(2) (2009), 249-260.
* [7] T. Kim, A note on the weighted Lebesgue-Radon-Nikodym Theorem with respect to $p$-adic invariant integral on $\mathbb{Z}_{p}$, J. Appl. Math. & Informatics, Vol. 30(2012), No. 1, 211-217.
* [8] T. Kim, Non-archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli Polynomials, Russ. J. Math Phys. 10 (2003) 91-98.
* [9] L-C. Jang, On the $q$-extension of the Hardy-littlewood-type maximal operator related to $q$-Volkenborn integral in the $p$-adic integer ring, Journal of Chungcheon Mathematical Society, Vol. 23, No. 2, June 2010.
* [10] S. Araci and M. Acikgoz, A note on the weighted $q$-Hardy-littlewood-type maximal operator with respect to $q$-Volkenborn integral in the $p$-adic integer ring, http://arxiv.org/abs/1202.1969.
* [11] K. Hensel, Theorie der Algebraischen Zahlen I. Teubner, Leipzig, 1908.
* [12] N. Koblitz, $p$-adic Numbers, $p$-adic Analysis and Zeta Functions, Springer-Verlag, New York Inc, 1977.
|
arxiv-papers
| 2012-05-30T21:45:29 |
2024-09-04T02:49:31.392136
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1205.6842"
}
|
1205.6970
|
# Sixteen years of Collaborative Learning through Active Sense-making in
Physics (CLASP) at UC Davis
Wendell Potter David Webb Emily West Cassandra Paul Mark Bowen Brenda
Weiss University of California at Davis, Department of Physics, Davis, CA
95616 webb@physics.ucdavis.edu
###### Abstract
The introductory physics series for bioscience students at the University of
California, Davis is described. A central feature of the class is sense-making
by the students during organized discussion/labs in which the students take
part in peer-peer discussions, argumentation, and presentations of ideas.
Differences in outcomes (MCAT scores and upper division GPAs) of students
taking this class and students taking the standard physics series that this
class supplanted are discussed.
## I Introduction
In 1995 two faculty members, a postdoctoral student, and 3 graduate students
at the University of California at Davis (UC Davis) began teaching a series of
introductory physics coursesFootnote00 that rapidly became the standard
introduction to physics for bioscience (and many agriculture) majors here at
Davis. These people were strongly influenced by research in educationOsbWitt ;
OsbFrey ; DrivAsoko including research in physics education.Hest87 As the
references show, these instructors’ efforts were informed by the
constructivist idea that each student builds their own knowledge through
discussion and argumentation. Toward this end these curriculum developers’
specific goals were to develop a way to keep the discussions and argumentation
at a high intellectual level and to keep the students’ main focus on concepts
rather than calculations, on genuine understandingAusubel rather than rote
memorization, and on the few big ideas in physics rather than the very many
small details. Essentially all aspects of the class including the physics
topics covered in the class, the organization of those topics, the delivery of
instruction, and the exam questions and the grading of those questions were
changed from the standard introductory physics series that this class
supplanted. In the subsequent years many instructors made contributions and
this physics series evolved as experience changed ideas of what can be and
should be accomplished with a group of students at this stage in their
academic careers . However, the general purpose and the general structure of
the course has been consistent throughout its existence. This paper is a
description of this series of introductory courses. The series of courses that
we are discussing are Physics 7A, 7B, and 7C at UC Davis but we will often
refer to these reformed courses generically as our CLASP (Collaborative
Learning through Active Sense-making in Physics) program.
The paper begins with a discussion of the general structure and pedagogy of
the course paying particular attention to how the course is designed to allow
peer-peer communication and to keep this communication at the level of making
sense of physics and using the ideas, concepts, and models of physics rather
than at the level of simple memorization or learning algorithms without
understanding. We then include enough detail of the actual course to make it
clearer what the class really looks like to an instructor or a student.
Finally, we include a short discussion of some of the things we have
discovered or measured in trying to assess the effects of the course. In
particular we discuss our students’ learning of physics, their understanding
of physics and science in general, and their abilities in science in general.
## II General Features of CLASP
### II.1 Content organization
#### II.1.1 Organized around a set of models describing phenomena
One of the most distinct differences between CLASP and the series of courses
it succeeded is in the order and organization of the physical ideas that we
include in the curriculum. The courses that CLASP supplanted were organized in
the relatively standard way (starting with small ideas like particle position
and velocity, building to include the ideas of Newton and, eventually,
building a logical structure including i) broader ideas like the major
conservation laws and ii) extensions of the basic ideas to more complex
phenomena such as wave motion and the force laws of E&M.) This type of course
may not be as useful as we hope to studentsKim02 who are learning the subject
for the first time and who are judging “what is important to really
understand” by the number of problems that they have to solve using the
various ideas and by the overwhelming number of algorithms that seem
important. As noted above, the originators of CLASP wanted to keep each
student’s focus on the main ideas of physics (i.e. exactly on “what is
important to really understand”) and the unity inherent in the structure of
physics rather than on the immense number of detailed specific examples and
algorithms that are inevitably used to show how these main ideas play out in
the world.Hest87 Toward this end, the three courses in the series are
organized around a set of about two dozen modelsMag02 that physicists use to
describe the major features of how the world works. Of these models, it is
probably fair to say that about a half dozen of them are the most important
overarching models.Footnote01 These models, and this organization of ideas,
are prominent in all of the work that the students do so we will briefly
describe their location in the course.
#### II.1.2 Which models and in what order in our intro-physics course for
bio-science majors
In an attempt to build on the student’s familiarity with chemistry, the series
of courses begins with _conservation of energy_ (both internal energies and
mechanical energies). This is immediately followed by the _statistical
properties of systems of large numbers of atoms_ and so completes the
discussion of _Thermodynamics_. Next, conservation of energy ideas are used to
analyze _fluid flow_ and _electrical charge flow_. After this is a shift to
two other conservation laws, _conservation of momentum_ and _conservation of
angular momentum_ (this is the introduction to _Newtonian mechanics_).
Following these discussions in mechanics, we introduce _wave models_ ,
_interference_ , and _optics_. Finally, the students discuss _fields_ (mainly
E&M) and _quantum mechanics_.
#### II.1.3 Pictures, diagrams, etc. help keep discussions at a high
intellectual level
One important reason for dealing so explicitly with models in CLASP is the
hope that the students who learn to work with these models end up building a
conceptual structure that allow them to make some progress in (i.e. begin)
understanding almost any physical situation in the real world. To help the
students build and use this conceptual structure, each model comes with a
pictorial (diagrammatic) or graphical representation that gives them a way to
begin working with the model (i.e. begin understanding a particular physical
situation) beforeKohl07 writing down equations and doing complicated
algebra.Wheeler The students very readily use these diagrammatic and
graphical representations in their discussionsStone12 and the representations
clearly help them structure the presentations of their ideas to the entire
class. Schematic pictures of two of these diagrammatic representations (one
for energy conservation and one for momentum conservation) are shown in Figure
1.
Figure 1: The diagram on the left is constructed by the student to help them
think and talk about the energy exchanges that occur when an ice-cube is added
to a large container of liquid nitrogen. The diagram on the right is
constructed by the student to help them think and talk about the final motion
of two objects (initially moving in different directions) that stick together
after a collision (in this particular situation forces exerted on the two
objects by their surroundings can be neglected compared to forces between the
two objects during the interval of the collision).
### II.2 General characteristics and pedagogy of content presentation
A regular offering of a Physics 7 course at UC Davis includes a lecture which
meets once a week for about 80 minutes and a discussion/laboratory (DL) that
meets twice a week for about 140 minutes each time. So about 1/5 of the class
time is spent in lecture with the rest spent in the DLs mostly in
intellectually intensive discussions, in small groups (typically 5 students),
concerned with either i) making sense of the models or ii) using the models to
make sense of various important physical situations.
#### II.2.1 Lectures
Lectures in the CLASP courses generally begin the presentation of material but
they do not have to be nearly as complete or as self contained as lectures in
a standard physics course would be. For instance, in a CLASP lecture, the
lecturer may define the appropriate technical words, describe the appropriate
physical concepts and models, and ask the students to use them in real-world
examples. However, the lecturer does not need to work out any example problems
for the students because the students will be working hard on applying the
models in different (example) situations during their approximately 5 hours of
DL time each week. Many lecturers consider the lecture time to be essentially
a “bonus time” rather than the time when the students must see all of the
material. This is not only quite different from the usual view of lecture, it
also liberates the lecturer to engage the class in whatever activities the
lecturer thinks are most useful.
#### II.2.2 Discussion/laboratories
The discussions in DL are what places our CLASP course in the category of
“interactive engagement” classesHake and they are as student-centered as we
have been able to make them. The discussions are facilitated by an instructor
who helps each group of students figure things out for themselves whenever
they get stuck. The intent is that the pace of these small group discussions
is completely controlled by the students and that the discussions are carried
out primarily in the student’s voice even when an instructor is present. We
provide instructors with notes for each activity, and some of these notes
remind the instructor that they are supposed to be a “guide on the side” not a
“sage on the stage”.Footnote02
#####
In the discussion/laboratory (DL) the students work in small groups on
activities aimed at helping each student build their own personal
understanding of the constructs of any particular model and the way in which
these constructs are used within the model. The activities are intended to
help our students become fully literate in a particular model so the
activities generally ask the students to discuss specific physical situations
in their own words, discuss them using the technical words and concepts of the
appropriate model, diagram the situation using one of the representations
discussed above, and, sometimes, translate these discussions into the
mathematical language of the model. We leave much of the mathematics and
almost all of the arithmetic for the students to do at home. Attendance in our
DLs is required so the DL activities require that _each_ student spends time
making sense of the ideas (hence, CLASP).
#####
The pace of the discussions is determined approximately by the majority of the
students in the class. After a reasonable number of the small groups (half of
them or more) have come to their conclusions about the activities that we
asked them to work on, the instructor stops the small group discussions and
leads a whole class discussion on the activity. Ideally, this whole class
discussion is also carried out in the voice of the students (i.e. student-
student discussions of the ideas).
We have many goals in our introduction of a discussion with the whole class at
the end of an activity. The first goal will be clear to any teacher. The whole
class discussion aims to leave each student, at a minimum, with a basic
understanding of what ideas needed to be used in the activity, how they needed
to be used, where these ideas fit into the field of Physics, and how the
activities relate to other activities that they have done. However, beyond
this learning of physics concepts, we hope that our class gives our students a
(somewhat) realistic viewDrivAsoko of how science proceeds and we see no
reason that this cannot be done in concert with the first goal. For instance,
a whole class discussion may result in some groups advocating for one way of
thinking about things and other groups advocating for another way (this is
actually not uncommon when 5 or 6 groups work on their activities relatively
independently) and then the discussion can bring out differing assumptions,
differing viewpoints, and (of course) genuine conceptual misunderstandings.
The whole class discussion also gives the students a chance to practice
developing proper scientific discussions.Footnote03 This practice at
generating proper scientific arguments should not only help build confidence
that they can do well on our exams but also confidence in performing well in
their other classes.
### II.3 Assessments of physics understanding
We will discuss the details of assessments and grading in a separate paper
but, briefly, the culture in the CLASP series at UCDavis is that there are
many short quizzes (something like a 20 minute quiz every week or a 30 minute
quiz every two weeks) and a final exam. There are two main reasons for very
frequent quizzes. The first is that the course emphasizes understanding of
physical ideas and their application so we want to give the students a way to
monitor their understanding of each idea or set of ideas.BlackDylan The
second is so that students have many chances to learn how to produce a
scientifically correct argument and a complete discussion of a problem.
Finally, there is also a culture regarding the types of exam questions in the
CLASP series at UCDavis that is followed by many of the instructors (perhaps
18-20 of the 25-30 instructors each year). This culture: i) values exam
questions and problems that are significantly different from those that the
students have already seen and, also, are not amenable to algorithmic solution
and ii) prizes the quality of a written scientific discussion given by a
student above the algebraic correctness of a mathematical answer.
## III Implementation of a CLASP class at UC Davis
At UC Davis over 1700 students each year complete the Physics 7A, 7B, 7C
series. Most of these students are in bioscience/agriculture majors where this
physics series (or its equivalent) is a major requirement. We offer about 50
DL sections each term divided into about 10 sections for each of five Physics
7 (A, or B, or C) classes. Thus, each DL section has about 30 students who
typically work in about six groups of five students each. Each of these five
classes is taught by two co-instructors along with four or five graduate
teaching assistants (TAs) so that the entire series has 10 instructors and
20-25 TAs associated with it each term. Usually 30-50% of the instructors are
regular faculty and the rest are either temporary lecturers or advanced
graduate students who are known to be excellent CLASP TAs and would like to
gain broader teaching experience.
### III.1 Co-Instructors
The two co-instructors divide up the teaching times and responsibilities in
any way they decide to. The most common way is for one instructor to give the
two identical 80 minute lectures each week (approximately half of the 300+
students in each lecture) and to handle the major administrative duties of the
class and for the other instructor to teach the first discussion/lab section,
run two TA meetings each week, and deal with the administrative issues
associated with the discussion/lab. Because no instructor teaches alone, it
turns out that this course is a good way to introduce new instructors to
teaching CLASP (in particular, it provides the perfect course for an advanced
graduate student to practice lecturing under the mentorship of an experienced
faculty member teaching the course with them). Both instructors are
responsible for the final grades so both work on writing and grading the
exams.
### III.2 DL Instructors and TA professional development
The role of the DL instructor (either a faculty member or a graduate student)
is largely to facilitate the discussions that students have about their
assigned activities and/or homework problems and to keep the students on task.
The actual DL activities are usually packaged with course notes and purchased
by the students at the beginning of the quarter but, sometimes, activities are
given out to the students during the quarter.
The DLs are where the students do much of their thinking and get much of their
practice with the material so they are the most important part of the course.
Graduate TAs lead over 90% of these DL sections and we have found that new
graduate TAs must rapidly learn about both teaching and learning. For this
reason, we have a significant professional development program focused on our
new graduate students teaching this course for their first time. For new
graduate TAs, this includes:
i) a mandatory 3-day introduction to teaching in CLASP which, besides dealing
with nuts and bolts of the job, also puts the new TAs in the roles of students
working on CLASP activities, followed by putting the TAs in the roles of
getting ready to teach DL, followed by putting the new TAs in the roles of
teachers (with reflections/comments on teaching and learning after each TA is
finished). These activities are interspersed with more general discussions of
teaching and learning, which are taught in the CLASP (group discussion) style.
ii) a mandatory 1-hour per week TA training course during the first term that
the new graduate student is enrolled at UC Davis and begins teaching a DL.
This class is generally aimed at the theory and practice of teaching an
interactive engagement type of class.Ishikawa Among other things, in this
class the new TAs visit DLs of senior TAs and comment on what they have seen,
discuss and practice grading, discuss teaching and learning, discuss the use
of models in science, work on improving their whole class discussions as well
as their small group discussions, and monitor one meeting of one of their
fellow new graduate student’s class (using a computer programRIOT to quantify
how their fellow TA spends their time in class).
iii) We also offer (non-mandatory) TA professional development classes after
the Fall term. These are generally aimed at studying and improving each TAs
teaching skills and/or the CLASP activities.
### III.3 Thoughts of UCD faculty who have taught large traditional
introductory Physics classes as well as a CLASP course
One feature of our CLASP course is that it introduces Physics faculty to
interactive engagement classes. Unfortunately, we don’t have a measure of how
much our faculty has been changed by the CLASP courses.Fairweather2010
However, in 2002, five faculty who had taught in Physics 7 but who had not
been involved in its development were asked for some of their thoughts about
the course. They were very happy with all the interactive engagement aspects
of the course and all of them (except one who has retired) continue to teach
it. The main negative comments were about course materials issues that no
longer apply so we will skip them and just present these instructors’ analyses
of the some of the successes of the course (as well as the prompts which led
to the comments).
In answer to the prompt: ‘What were your expectations coming into Physics 7?’
One faculty member: I came into Physics 7 with a very negative attitude. I
…had taught Physics 5 (the traditional intro-physics for bioscience course at
UCD) three times through. I had a student …who would come in every day and
tell me what a fiasco this whole Physics 7 was…. Rather than going to a
faculty meeting and in complete ignorance try to stop this disaster, I figured
the only honest thing to do was to try to teach it myself so that I could then
draw my own conclusions. When I did that I had sort of the complete opposite
experience than [my student] had. I really enjoyed it. I really felt that it
was a much more dynamic learning environment…. Getting the students up
presenting their responses to the class really forced them to think their
ideas out very clearly.
In answer to the prompt: ‘What did you find most surprising about teaching
Physics 7?’ A different faculty member: How much fun it was. It’s a riot! I
mean you really get to meet …six hours …that’s more time than most people
spend with their kids at this age, you know, or younger. And you get to know
them all. And that’s kind of fun…it’s the methodology of the activities. How
you work in little groups, and how the groups present their stuff to the
larger group. It’s not the activity per se, but how we go about investigating
it and sharing it (I hate that word) …telling other people it.
In a discussion following the prompt ‘What do you think of the attitudes that
students have in Physics 7?’ A third faculty member: The attitude may be
unchanged, but because it forces them to talk and participate …at least it
draws out something in them that they don’t get drawn out in a lecture class.
I was actually pleasantly surprised at what a large fraction would actually
talk, would ask questions, puzzle on things…. I think that’s why this is a
successful class, because it forces some mental activity on the students’ part
that is always lacking if they are sitting taking notes in lecture.
## IV Measurements of student learning and transfer
In this section we will discuss some data that we have examined over the years
and that help us judge the success of this course. First, we directly compare
students who took the Physics 7 series with those who took our previous
physics series (Physics 5). Then we discuss scores on concept inventories.
Finally, we discuss our students’ more general understandings of what science
(specifically, physics) is.
### IV.1 Direct comparison between Physics 7 students and Physics 5 students
We expect the Physics 7 series to help prepare students for later work. In
checking this we have examined two main things, student’s work in later
courses and students’ MCAT scores.
#### IV.1.1 Preparation for later courses
In the few years after the introduction of this CLASP course we had a chance
to compare the students taking the Physics 7 (CLASP) series with those who
took the previous UCD intro-physics series for bioscience students (Physics
5). As a proxy for the upper division major GPA we calculate a student’s GPA
during the 7 quarters (just over two years) that preceded their graduation and
use those to compare different groups of students. We only include students
who had completed at least 65 quarter units (about 1.5 years of a normal class
load) and we did not include any students who started their intro-physics
series less than 5 quarters before their graduation. Finally, we remove any
intro-physics grade points and units that they received in the 7 quarters
before graduation. There are two kinds of comparisons that we have done, i) a
direct comparison between Physics 5 students and Physics 7 students who had
overlapping graduation years and ii) a comparison between the students who
took their intro-physics at UCD in either Physics 5 or Physics 7 and those
students who did not take either of those and so must have taken another
intro-physics course (most students in this group have transferred into a
biosci major after two years at a community college).
We can directly compare the GPAs for the graduating classes of 1998 and 1999,
which are the only years with significant overlap between Physics 5 and
Physics 7 students. We find mean GPAs ofFootnote04 3.068 $\pm$ 0.018 for the
755 students who took Physics 5 (and met the criteria discussed above) and
3.127 $\pm$ 0.018 for the 666 students who took Physics 7. We conclude that
the Physics 7 students were (statistically) significantly better in their
major courses in this direct comparison.Footnote05 This direct comparison
might be criticized because these students have made a decision (either
directly or indirectly) as to which Physics series to take. This is the reason
for our second comparison.
In the second comparison we calculate the same (essentially upper-division)
graduating GPAs for four groups of students in two sets of years, 1993-1997
(biosci students could only take Physics 5) and 1999-2003 (only Physics 7 was
available). The groups of students are: a) students who took our intro-physics
series for biosci majors (either Physics 5 or 7 depending on the graduation
years we choose) because this course was a requirement for their biosci major,
ii) students who graduated with these biosci majors requiring intro-physics
but who did not take our intro course (almost all of these took intro-physics
at another college and almost all are transfer students), iii) students who
completed non-bioscience majors at UCD and were admitted as
Freshmen,Footnote05A and iv) students who completed non-bioscience majors at
UCD and transferred to UCD after completing their lower division work at
another college.Footnote05B
The differences between the GPAs of these third and fourth groups of students
will be used as a measure of the academic strength of our transfer students
compared to the students admitted as Freshmen. For each of these sets of years
the non-biosci students admitted as Freshmen had an average (graduating) GPA
that was 0.02 $\pm$ 0.01 higher than the average for the non-biosci transfer
students. In other words, in each of these sets of years, transfer students
fared about as well as the 4-year students at UCD and the difference did not
change from one set of years to the other.
This near equivalence between those students admitted to UC Davis as Freshmen
and those admitted as transfer students allows us to compare the GPAs of the
bio-sci students graduating in the different sets of years. For instance, bio-
sci students who took Physics 5 and who graduated in the years 1993-1997 had
an average GPA that was 0.057 $\pm$ 0.015 larger than those students
graduating with the same majors but without having taken our intro-physics
courses. We compare this with bio-sci students who took Physics 7, who
graduated in the years 1999-2003, and who had an average GPA 0.115 $\pm$ 0.014
higher than those students graduating with the same majors in those same years
but without having taken our intro-physics courses. This much larger GPA gap
is another piece of information suggesting that students taking Physics 7 were
better prepared for their major classes than students who took Physics 5. As
an aside, we note that about 25% of this increase came from an increased
average GPA of males and about 75% of it came from the increased GPAs of
females.
#### IV.1.2 MCAT scores
Another indication of a positive effect due to Physics 7 came from an analysis
of UC Davis students’ performance on the Medical College Admissions Test
(MCAT). We used about five years of data centered on the point at which we
stopped teaching Physics 5 and began teaching Physics 7. We compared our
students’ performance (N = 386 for students who took Physics 5 and N = 347 for
students who took Physics 7) on both the Physical Science and Biological
portions of the MCAT. For the Biological Science part of the test the scores
ranged from 3-15 with an average of 9.71 $\pm$ 0.10 whether the students took
Physics 5 or Physics 7. The Physical Science part of the test had a similar
range of scores and an average of 9.26 $\pm$ 0.10 for the students who took
Physics 5 and 9.42 $\pm$ 0.11 for the students who took Physics 7. This gap of
0.16 $\pm$ 0.15 suggests that the Physics 7 students were better prepared for
the MCAT. For the MCAT scores, almost all of the increase in performance came
from female students whose mean Physical Science MCAT scores were larger by
0.27 $\pm$ 0.2 for the students taking Physics 7 (with male students having
the same mean score whether they took Physics 5 or 7).
### IV.2 Conceptual understanding of force and motion
The Force Concept Inventory (pre-test at beginning of 7A and post-test at the
end of 7B) was given to four different groups of students (total of 898
students) in 1999-2001 resulting in an average pretest score of 31% correct
and an average normalized gain of 0.39 $\pm$ 0.01. It is probably not
surprising that this is well above the range associated by HakeHake with
traditional courses and in the middle of the range of “interactive engagement”
courses.
### IV.3 Attitudes toward physics
Over the past two decades, researchRedishMPEX ; AdamsCLASS has shown that a
majority of students leave introductory physics classrooms not only confused
about the conceptual content of physics, but also about nature of scientific
knowledge. These ideas are epistemological in nature and the implicit
epistemological message sent in many traditional classrooms is apparently not
what we wantSchommer our students to learn. Indirectly, the students appear
to be encouraged toward approaches to learning such as rote memorization of
many specific algorithms (rather than learning to use broad general
principles) and dissuaded from reconciling their everyday experiences with the
content presented in the course to form a coherent worldview.
Several epistemological surveys have been developedRedishMPEX ; AdamsCLASS to
categorize these beliefs. In general, these surveys consist of a set of
statements, such as “Knowledge in physics consists of many pieces of
information, each of which applies primarily to a specific situation.”, with
which the student is asked to agree or disagree. Student responses are coded
as favorable (matching what an expert would say), as unfavorable, or as
neutral. The surveys are given twice, once at the beginning of instruction and
once at the end and movement in student responses are classified as towards or
away from expert-like beliefs (gains or losses).
Results from these sorts of surveys have been collected from many large
lecture classes, across a variety of educational institutions, and from both
‘reformed’ and traditional introductory physics classes. Results are fairly
consistent. In general, unless the class has focused explicitly on addressing
epistemologies, after one semester of physics instruction, student populations
tend to move away from the experts in their opinions (even for most “reformed”
classes).RedishMPEX ; AdamsCLASS
In the Fall quarter of 2008 we administered the MPEX-II (Maryland Physics
Expectations Survey)RedishHammerMPEXII to two separate lecture sections (for
a total of about 600 students) of Physics 7A. The results showed that the
student epistemologies in this set of classes were statistically unchanged
over the course of the quarter: favorable fraction of responses changed from
0.46 $\pm$ 0.01 to 0.47 $\pm$ 0.01 and the unfavorable fraction changed from
0.27 $\pm$ 0.01 to 0.28 $\pm$ 0.01. Thus, unlike most standard classes and
even many reformed Physics classes whose students seem to end the course with
less expert epistemologies, this CLASP class seems to leave the students
epistemological ideas unchanged on average.
## V Summary and conclusions
Though a radical departure from traditional instruction, the series of
introductory physics courses discussed in this paper has been fully
institutionalized. It has outlasted the people who originally developed it, it
is positively received by faculty, and it provides a good training arena both
for new instructors and for new graduate TAs. It is also positively received
by our administration, partly because we use no more resources than other
introductory courses even though our exams ask the students to discuss their
ideas in writing. Over 1700 students take the course each year. These students
leave the series of courses better prepared for their later studies, with
better conceptual physics knowledge, and with more expert epistemologies than
they would have left our previous introductory series. For all of these
reasons, we consider our CLASP series to be a successful addition to our
curriculum.
The CLASP series of courses is a work in progress but most of the recent work
has been not on how to change the activities to improve learning but, instead,
on how to adjust the amounts of time spent on the various models of physics.
We have been focused on the types of physics that our bioscience students will
have to understand in their later courses (the ideal introductory physics
course for these students should, perhaps, be an introductory biophysics
course) and we are currently using some of the recent reports on undergraduate
education for bio-scienceBIO2010 and premedAAMCHHMI students in order to
help us with this thinking. Because our activities (rather than a textbook)
drive the course we can readily change the course and rapidly adjust the
weights given to the various parts of the course according to the results that
we get.
Finally, as the activity development efforts wind down, our Physics Education
Group is freed to use the CLASP series as a learning laboratory to investigate
fundamental issues of teaching and learning. In other words, we are free to
make small modifications in activities, exams, lectures, DL culture, etc. and
study the results of those changes or to spend more of our time on TA training
and other activities that are likely associated with learning.
At this point, the CLASP series is published by Hayden McNeil as “College
Physics: A Models Approach.” Anyone interested in learning more about or
obtaining some CLASP materials may contact David Webb
(webb@physics.ucdavis.edu) or Wendell Potter (potter@physics.ucdavis.edu).
###### Acknowledgements.
The authors thank the National Science Foundation for providing the intial
funding for this work under Grant No. DUE-9354528 and they also thank that
many faculty, postdoctoral students, and graduate students who have helped
improve the CLASP series.
## References
* (1) Initial support was provided by the National Science Foundation under Grant No. DUE-9354528.
* (2) R.J. Osborne and M.C. Wittrock, “The generative learning model and its implications for science education,” Studies in Science Education 12, 59–87 (1985).
* (3) R. Osborne and P. Freyberg, Learning in Science: The implications of children’s science (Heinemann, Aukland 1985).
* (4) Rosalind Driver, Hilary Asoko, John Leach, Philip Scott and Eduardo Mortimer, “Constructing Scientific Knowledge in the Classroom,” Educational Researcher, 23, 5-129 (1994).
* (5) David Hestenes, “Toward a modeling theory of physics instruction,” American Journal of Physics, 55, 440-454 (1987).
* (6) David Ausubel, Educational Psychology: A Cognitive View. (Holt, Rinehart & Winston, New York, 2nd ed. 1978).
* (7) Eunsook Kim and Sung J. Pak, “Students do not overcome conceptual difficulties after solving 1000 traditional problems,” American Journal of Physics, 70, 759-765 (2002).
* (8) For a discussion of the uses of models in science see: Model-Based Reasoning: Science, Technology, Values, ed. by Lorenzo Magnani and Nancy Nersessian, (Kluwer, Dordrecht 2002).
* (9) As an example of this kind of organization see the “Six Ideas that Shaped Physics” series of texts.
* (10) For a study of the effects of doing considerable conceptual thinking prior to solving a problem see: Patrick Kohl, David Rosengrant, and Noah Finkelstein, “Strongly and weakly directed approaches to teaching multiple representation use in physics”, Physical Review Special Topics - Physics Education Research, 3, 010108 (2007).
* (11) The spirit behind WHEELER’S FIRST MORAL PRINCIPLE – “Never make a calculation until you know the answer.” (from Chap.1 of Spacetime Physics by Taylor and Wheeler) is somewhat appropriate here since we insist that the students do a lot of thinking, drawing, and/or diagramming before any algebraic calculation.
* (12) Antoinette Stone, Wendell Potter, and David Webb, “Diagrammatic representations as reasoning tools and their impact on model-based reasoning approaches in a reformed college physics classroom”, in preparation.
* (13) As defined by: Richard Hake, “Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses”, American Journal of Physics, 66, 64-74 (1998).
* (14) Later in this paper we describe more about our instructor training and professional development.
* (15) We may even motivate the students to develop these skills by reminding them that they are practicing the argument skills that they will use on exams.
* (16) Paul Black and William Dylan, “Assessment and classroom learning,” Assessment in Education: Principles, Policy & Practice, 5, 7-74 (1998).
* (17) Catherine M. Ishikawa, Wendell H. Potter, and William E. Davis, “Beyond this week’s lab: integrating long-term professional development with short-term preparation for science graduate students”, Journal of Graduate Teaching Assistant Development, 8, 133-138 (2001).
* (18) This is a modified version of the RIOT described in Emily West, Cassandra Paul, David Webb, and Wendell Potter, submitted to Physical Review Special Topics - Physics Education Research.
* (19) J. Fairweather, “Linking Evidence and Promising Practices in Science, Technology, Engineering, and Mathematics (STEM) Undergraduate Education: A Status Report for The National Academies National Research Council Board of Science Education (BOSE)”, (BOSE, Washington, DC, 2010).
* (20) In all of the following quantitative measures, the error estimates that we quote are the standard error of the mean.
* (21) This comparison is also complicated by the fact that the developers of the Physics 7 curriculum had tried “interactive engagement” laboratories in two or three classes of Physics 5 during 1993-94 so as many as 20
* (22) It was not always possible to decide who was admitted as a Freshman so we have used a cutoff of 170 UCD quarter units or more (at graduation) as a proxy for ”admitted as a Freshman”. In the years for which we can compare the proxy definition with the actual status, we find that fewer than 1% of the students in this group are transfer students.
* (23) It was not always possible to decide who was admitted as a transfer student so we have used a cutoff of 115 UCD quarter units or fewer (at graduation) as a proxy for ”admitted as a Junior transfer student”. In the years for which we can compare the proxy definition with the actual status, we find that fewer than 1% of the students in this group were admitted as a Freshman.
* (24) Edward F. Redish, Jeffery M. Saul, and Richard N. Steinberg, “Student expectations in introductory physics”, American Journal of Physics, 66, 212-224 (1998).
* (25) W. K. Adams, K. K. Perkins, N. S. Podolefsky, M. Dubson, N. D. Finkelstein, and C. E. Wieman, “New instrument for measuring students beliefs about physics and learning physics: The Colorado Learning Attitudes about Science Survey”, Physical Review Special Topics - Physics Education Research, 2, 1-14 (2006).
* (26) Marlene Schommer, “Effects of beliefs about the nature of knowledge on comprehension”, Journal of Educational Psychology, 82, 498-504 (1990).
* (27) E. F. Redish and D. Hammer, “Reinventing college physics for biologists: explicating an epistemological curriculum,” American Journal of Physics, 77, 629-642 (2009).
* (28) BIO 2010 - Transforming Undergraduate Education for Future Research Biologists, Committee on Undergraduate Biology Education to Prepare Research Scientists for the 21st Century, (National Academies Press, Washington DC, 2003).
* (29) Scientific Foundations for Future Physicians, SFFP Committee, (Association of American Medical Colleges, Washington DC, 2009).
|
arxiv-papers
| 2012-05-31T12:28:01 |
2024-09-04T02:49:31.401662
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wendell Potter, David Webb, Emily West, Cassandra Paul, Mark Bowen,\n Brenda Weiss, Lawrence Coleman, Charles De Leone",
"submitter": "David Webb",
"url": "https://arxiv.org/abs/1205.6970"
}
|
1205.7051
|
DESY 12-091 arXiv:1205.7051 [math.NT]
On Multiple Zeta Values of Even Arguments
Michael E. Hoffman111Supported by a grant from the German Academic Exchange
Service (DAAD) during the preparation of this paper. The author also thanks
DESY for providing facilities and financial support for travel.
Dept. of Mathematics, U. S. Naval Academy
Annapolis, MD 21402 USA
and
Deutsches Elektronen-Synchrotron DESY
Platanenalle 6, D-15738 Zeuthen, Germany
meh@usna.edu
June 8, 2012
Keywords: multiple zeta values, symmetric functions, Bernoulli numbers
MR Classifications: Primary 11M32; Secondary 05E05, 11B68
For $k\leq n$, let $E(2n,k)$ be the sum of all multiple zeta values with even
arguments whose weight is $2n$ and whose depth is $k$. Of course $E(2n,1)$ is
the value $\zeta(2n)$ of the Riemann zeta function at $2n$, and it is well
known that $E(2n,2)=\frac{3}{4}\zeta(2n)$. Recently Z. Shen and T. Cai gave
formulas for $E(2n,3)$ and $E(2n,4)$ in terms $\zeta(2n)$ and
$\zeta(2)\zeta(2n-2)$. We give two formulas for $E(2n,k)$, both valid for
arbitrary $k\leq n$, one of which generalizes the Shen-Cai results; by
comparing the two we obtain a Bernoulli-number identity. We also give an
explicit generating function for the numbers $E(2n,k)$.
## 1 Introduction and Statement of Results
For positive integers $i_{1},\dots,i_{k}$ with $i_{1}>1$, we define the
multiple zeta value $\zeta(i_{1},\dots,i_{k})$ by
$\zeta(i_{1},\dots,i_{k})=\sum_{n_{1}>\dots>n_{k}\geq
1}\frac{1}{n_{1}^{i_{1}}\cdots n_{k}^{i_{k}}}.$ (1)
The multiple zeta value (1) is said to have weight $i_{1}+\dots+i_{k}$ and
depth $k$. Many remarkable identities have been proved about these numbers,
but in this note we will concentrate on the case where the $i_{j}$ are even
integers. Let $E(2n,k)$ be the sum of all the multiple zeta values of even-
integer arguments having weight $2n$ and depth $k$, i.e.,
$E(2n,k)=\sum_{\begin{subarray}{c}\text{$i_{1},\dots,i_{k}$ even}\\\
i_{1}+\dots+i_{k}=2n\end{subarray}}\zeta(i_{1},\dots,i_{k}).$
Of course
$E(2n,1)=\zeta(2n)=\frac{(-1)^{n-1}B_{2n}(2\pi)^{2n}}{2(2n)!},$ (2)
where $B_{2n}$ is the $2n$th Bernoulli number, by the classical formula of
Euler. Euler also studied double zeta values (i.e., multiple zeta values of
depth 2) and in his paper [2] gave two identities which read
$\displaystyle\sum_{i=2}^{2n-1}(-1)^{i}\zeta(i,2n-i)$
$\displaystyle=\frac{1}{2}\zeta(2n)$
$\displaystyle\sum_{i=2}^{2n-1}\zeta(i,2n-i)$ $\displaystyle=\zeta(2n)$
in modern notation. From these it follows that
$E(2n,2)=\frac{3}{4}\zeta(2n),$
though Gangl, Kaneko and Zagier [3] seem to be the first to have pointed it
out in print. Recently Shen and Cai [10] proved the formulas
$\displaystyle E(2n,3)$
$\displaystyle=\frac{5}{8}\zeta(2n)-\frac{1}{4}\zeta(2)\zeta(2n-2),\ n\geq 3$
(3) $\displaystyle E(2n,4)$
$\displaystyle=\frac{35}{64}\zeta(2n)-\frac{5}{16}\zeta(2)\zeta(2n-2),\ n\geq
4.$ (4)
Identity (3) was also proved by Machide [9] using a different method.
This begs the question whether there is a general formula of this type for
$E(2n,k)$. The pattern
$\frac{3}{4},\quad\frac{3}{4}\cdot\frac{5}{6}=\frac{5}{8},\quad\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{7}{8}=\frac{35}{64}$
of the leading coefficients makes one curious. In fact, the general result is
as follows.
###### Theorem 1.
For $k\leq n$,
$E(2n,k)=\frac{1}{2^{2(k-1)}}\binom{2k-1}{k}\zeta(2n)\\\
-\sum_{j=1}^{\lfloor\frac{k-1}{2}\rfloor}\frac{1}{2^{2k-3}(2j+1)B_{2j}}\binom{2k-2j-1}{k}\zeta(2j)\zeta(2n-2j).$
The next two cases after (4) are
$\displaystyle E(2n,5)$
$\displaystyle=\frac{63}{128}\zeta(2n)-\frac{21}{64}\zeta(2)\zeta(2n-2)+\frac{3}{64}\zeta(4)\zeta(2n-4)$
$\displaystyle E(2n,6)$
$\displaystyle=\frac{231}{512}\zeta(2n)-\frac{21}{64}\zeta(2)\zeta(2n-2)+\frac{21}{256}\zeta(4)\zeta(2n-4).$
We prove Theorem 1 in §3 below, using the generating function
$F(t,s)=1+\sum_{n\geq k\geq 1}E(2n,k)t^{n}s^{k}.$
In §2 we establish the following explicit formula.
###### Theorem 2.
$F(t,s)=\frac{\sin(\pi\sqrt{1-s}\sqrt{t})}{\sqrt{1-s}\sin(\pi\sqrt{t})}.$
Our proof uses symmetric functions. We define a homomorphism
$\mathfrak{Z}:\operatorname{Sym}\to\mathbf{R}$, where $\operatorname{Sym}$ is
the algebra of symmetric functions, and a family
$N_{n,k}\in\operatorname{Sym}$ such that $\mathfrak{Z}$ sends $N_{n,k}$ to
$E(2n,k)$. We then obtain a formula for the generating functions
$\mathcal{F}(t,s)=1+\sum_{n\geq k\geq
1}N_{n,k}t^{n}s^{k}\in\operatorname{Sym}[[t,s]]$
and apply $\mathfrak{Z}$ to get Theorem 2.
From the form of $\mathcal{F}(t,s)$ we show that it satisfies a partial
differential equation (Proposition 1 below), which is equivalent to a
recurrence for the $N_{n,k}$. From the latter we obtain a formula for the
$N_{n,k}$ in terms of complete and elementary symmetric functions, to which
$\mathfrak{Z}$ can be applied to give the following alternative formula for
$E(2n,k)$.
###### Theorem 3.
For $k\leq n$,
$E(2n,k)=\frac{(-1)^{n-k-1}\pi^{2n}}{(2n+1)!}\sum_{i=0}^{n-k}\binom{n-i}{k}\binom{2n+1}{2i}2(2^{2i-1}-1)B_{2i}.$
Note that the sum given by Theorem 3 has $n-k+1$ terms, while that given by
Theorem 1 has $\lfloor\frac{k-1}{2}\rfloor+1$ terms. Yet another explicit
formula for $E(2n,k)$ can be obtained by setting $d=1$ in Theorem 7.1 of
Komori, Matsumoto and Tsumura [7]. That formula expresses $E(2n,k)$ as a sum
over partitions of $k$, and it is not immediately clear how it relates to our
two formulas.
Comparison of Theorems 1 and 3 establishes the following Bernoulli-number
identity.
###### Theorem 4.
For $k\leq n$,
$\sum_{i=0}^{\lfloor\frac{k-1}{2}\rfloor}\binom{2k-2i-1}{k}\binom{2n+1}{2i+1}B_{2n-2i}=\\\
(-1)^{k}2^{2k-2n}\sum_{i=0}^{n-k}\binom{n-i}{k}\binom{2n+1}{2i}(2^{2i-1}-1)B_{2i}.$
It is interesting to contrast this result with the Gessel-Viennot identity
(see [1, Theorem 4.2]) valid on the complementary range:
$\sum_{i=0}^{\lfloor\frac{k-1}{2}\rfloor}\binom{2k-2i-1}{k}\binom{2n+1}{2i+1}B_{2n-2i}=\frac{2n+1}{2}\binom{2k-2n}{k},\quad
k>n.$ (5)
Note that the right-hand side of equation (5) is zero unless $k\geq 2n$.
## 2 Symmetric Functions
We think of $\operatorname{Sym}$ as the subring of
$\mathbf{Q}[[x_{1},x_{2},\dots]]$ consisting of those formal power series of
bounded degree that are invariant under permutations of the $x_{i}$. A useful
reference is the first chapter of Macdonald [8]. We denote the elementary,
complete, and power-sum symmetric functions of degree $i$ by $e_{i}$, $h_{i}$,
and $p_{i}$ respectively. They have associated generating functions
$\displaystyle E(t)$
$\displaystyle=\sum_{j=0}^{\infty}e_{j}t^{j}=\prod_{i=1}^{\infty}(1+tx_{i})$
$\displaystyle H(t)$
$\displaystyle=\sum_{j=0}^{\infty}h_{j}t^{j}=\prod_{i=1}^{\infty}\frac{1}{1-tx_{i}}=E(-t)^{-1}$
$\displaystyle P(t)$
$\displaystyle=\sum_{j=1}^{\infty}p_{j}t^{j-1}=\sum_{i=1}^{\infty}\frac{x_{i}}{1-tx_{i}}=\frac{H^{\prime}(t)}{H(t)}.$
As explained in [5] and in greater detail in [6], there is a homomorphism
$\zeta:\operatorname{Sym}^{0}\to\mathbf{R}$, where $\operatorname{Sym}^{0}$ is
the subalgebra of $\operatorname{Sym}$ generated by $p_{2},p_{3},p_{4},\dots$,
such that $\zeta(p_{i})$ is the value $\zeta(i)$ of the Riemann zeta function
at $i$, for $i\geq 2$ (in [5, 6] this homomorphism is extended to all of
$\operatorname{Sym}$, but we do not need the extension here). Let
$\mathcal{D}:\operatorname{Sym}\to\operatorname{Sym}$ be the degree-doubling
map that sends $x_{i}$ to $x_{i}^{2}$. Then
$\mathcal{D}(\operatorname{Sym})\subset\operatorname{Sym}^{0}$, so the
composition $\mathfrak{Z}=\zeta\mathcal{D}$ is defined on all of
$\operatorname{Sym}$. (Alternatively, we can simply think of $\mathfrak{Z}$ as
sending $x_{i}$ to $1/i^{2}$: see [8, Ch. I, §2, ex. 21].) Note that
$\mathfrak{Z}(p_{i})=\zeta(2i)\in\mathbf{R}$. Further, $\mathfrak{Z}$ sends
the monomial symmetric function $m_{i_{1},\dots,i_{k}}$ to the symmetrized sum
of multiple zeta values
$\frac{1}{|\operatorname{Iso}(i_{1},\dots,i_{k})|}\sum_{\sigma\in
S_{k}}\zeta(2i_{\sigma(1)},2i_{\sigma(2)},\dots,2i_{\sigma(k)}),$
where $S_{k}$ is the symmetric group on $k$ letters and
$\operatorname{Iso}(i_{1},\dots,i_{k})$ is the subgroup of $S_{k}$ that fixes
$(i_{1},\dots,i_{k})$ under the obvious action.
Now let $N_{n,k}$ be the sum of all the monomial symmetric functions
corresponding to partitions of $n$ having length $k$. Of course $N_{n,k}=0$
unless $k\leq n$, and $N_{k,k}=e_{k}$. Then $\mathfrak{Z}$ sends $N_{n,k}$ to
$E(2n,k)$. Also, if we define (as in the introduction)
$\mathcal{F}(t,s)=1+\sum_{n\geq k\geq 1}N_{n,k}t^{n}s^{k},$
then $\mathfrak{Z}$ sends $\mathcal{F}(t,s)$ to the generating function
$F(t,s)$. We have the following simple description of $\mathcal{F}(t,s)$.
###### Lemma 1.
$\mathcal{F}(t,s)=E((s-1)t)H(t)$.
###### Proof.
Evidently $\mathcal{F}(t,s)$ has the formal factorization
$\prod_{i=1}^{\infty}(1+stx_{i}+st^{2}x_{i}^{2}+\cdots)=\prod_{i=1}^{\infty}\frac{1+(s-1)tx_{i}}{1-tx_{i}}=E((s-1)t)H(t).$
∎
###### Proof of Theorem 2.
Using the well-known formula for $\zeta(2,2,\dots,2)$ [4, Cor. 2.3],
$\mathfrak{Z}(e_{n})=\zeta(\underbrace{2,2,\dots,2}_{n})=\frac{\pi^{2n}}{(2n+1)!}.$
(6)
Hence
$\mathfrak{Z}(E(t))=\frac{\sinh(\pi\sqrt{t})}{\pi\sqrt{t}},$
and the image of $H(t)=E(-t)^{-1}$ is
$\mathfrak{Z}(H(t))=\frac{\pi\sqrt{-t}}{\sinh(\pi\sqrt{-t})}=\frac{\pi\sqrt{t}}{\sin(\pi\sqrt{t})}.$
Thus from Lemma 1 $F(t,s)=\mathfrak{Z}(\mathcal{F}(t,s))$ is
$\mathfrak{Z}(E((s-1)t)H(t))=\frac{\sinh(\pi\sqrt{(s-1)t})}{\pi\sqrt{(s-1)t}}\frac{\pi\sqrt{t}}{\sin(\pi\sqrt{t})}=\frac{\sin(\pi\sqrt{(1-s)t})}{\sqrt{1-s}\sin(\pi\sqrt{t})}.$
∎
Taking limits as $s\to 1$ in Theorem 2, we obtain
$F(t,1)=\frac{\pi\sqrt{t}}{\sin\pi\sqrt{t}}$
and so, taking the coefficient of $t^{n}$, the following result.
###### Corollary 1.
For all $n\geq 1$,
$\sum_{k=1}^{n}E(2n,k)=\frac{2(2^{2n-1}-1)(-1)^{n-1}B_{2n}\pi^{2n}}{(2n)!}.$
Another consequence of Lemma 1 is the following partial differential equation.
###### Proposition 1.
$t\frac{\partial{\mathcal{F}}}{\partial{t}}(t,s)+(1-s)\frac{\partial{\mathcal{F}}}{\partial{s}}(t,s)=tP(t)\mathcal{F}(t,s).$
###### Proof.
From Lemma 1 we have
$\displaystyle\frac{\partial{\mathcal{F}}}{\partial{t}}(t,s)$
$\displaystyle=(s-1)E^{\prime}((s-1)t)H(t)+E((s-1)t)H^{\prime}(t)$
$\displaystyle\frac{\partial{\mathcal{F}}}{\partial{s}}(t,s)$
$\displaystyle=tE^{\prime}((s-1)t)H(t)$
from which the conclusion follows. ∎
Now examine the coefficient of $t^{n}s^{k}$ in Proposition 1 to get the
following.
###### Proposition 2.
For $n\geq k+1$,
$p_{1}N_{n-1,k}+p_{2}N_{n-2,k}+\dots+p_{n-k}N_{k,k}=(n-k)N_{n,k}+(k+1)N_{n,k+1}.$
It is also possible to prove this result directly via a counting argument like
that used to prove the lemma of [6, p. 16].
The preceding result allows us to write $N_{n,k}$ explicitly in terms of
complete and elementary symmetric functions as follows.
###### Lemma 2.
For $r\geq 0$,
$N_{k+r,k}=\sum_{i=0}^{r}(-1)^{i}\binom{k+i}{i}h_{r-i}e_{k+i}.$
###### Proof.
We use induction on $r$, the result being evident for $r=0$. Proposition 2
gives
$\sum_{i=1}^{r+1}p_{i}N_{k+r+1-i,k}=(r+1)N_{k+r+1,k}+(k+1)N_{k+r+1,k+1},$
which after application of the induction hypothesis becomes
$\sum_{i=1}^{r+1}\sum_{j=0}^{r+1-j}(-1)^{j}p_{i}\binom{k+j}{j}h_{r+1-i-j}N_{k+j,k+j}=\\\
(r+1)N_{k+r+1,k}+(k+1)\sum_{j=0}^{r}\binom{k+1+j}{j}h_{r-j}N_{k+1+j,k+1+j}.$
The latter equation can be rewritten
$\sum_{j=0}^{r}(-1)^{j}\binom{k+j}{j}N_{k+j,k+j}\sum_{i=1}^{r+1-j}p_{i}h_{r+1-i-j}=\\\
(r+1)N_{k+r+1,k}-(k+1)\sum_{j=1}^{r+1}(-1)^{j}\binom{k+j}{j-1}h_{r+1-j}N_{k+j,k+j}.$
Now the inner sum on the left-hand side is $(r+1-j)h_{r+1-j}$ by the
recurrence relating the complete and power-sum symmetric functions, so we have
$(r+1)N_{k+r+1,k}-(r+1)N_{k,k}h_{r+1}=\\\
\sum_{j=1}^{r+1}(-1)^{j}h_{r+1-j}N_{k+j,k+j}\left((r+1-j)\binom{k+j}{j}+(k+1)\binom{k+j}{j-1}\right),$
and the conclusion follows after the observation that
$(k+1)\binom{k+j}{j-1}=j\binom{k+j}{j}$. ∎
###### Proof of Theorem 3.
Rewrite Lemma 2 in the form
$N_{n,k}=\sum_{i=0}^{n-k}\binom{n-i}{k}(-1)^{n-k-i}h_{i}e_{n-i}$
and apply the homomorphism $\mathfrak{Z}$, using equation (6) and
$\mathfrak{Z}(h_{i})=\frac{2(2^{2i-1}-1)(-1)^{i-1}B_{2i}\pi^{2i}}{(2i)!}.$
∎
## 3 Proof of Theorems 1 and 4
From the introduction we recall the statement of Theorem 1:
$E(2n,k)=\frac{1}{2^{2(k-1)}}\binom{2k-1}{k}\zeta(2n)\\\
-\sum_{j=1}^{\lfloor\frac{k-1}{2}\rfloor}\frac{1}{2^{2k-3}(2j+1)B_{2j}}\binom{2k-2j-1}{k}\zeta(2j)\zeta(2n-2j).$
We note that Euler’s formula (2) can be used to write the result in the
alternative form
$E(2n,k)=\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^{j}\pi^{2j}\zeta(2n-2j)}{2^{2k-2j-2}(2j+1)!}\binom{2k-2j-1}{k}$
(7)
which avoids mention of Bernoulli numbers.
We now expand out the generating function $F(t,s)$. We have
$F(t,s)=\frac{1}{\sqrt{1-s}\sin\pi\sqrt{t}}\sin(\pi\sqrt{t}\sqrt{1-s})\\\
=\frac{\pi\sqrt{t}}{\sin\pi\sqrt{t}}\sum_{j=0}^{\infty}\frac{(-1)^{j}\pi^{2j}t^{j}(1-s)^{j}}{(2j+1)!}=\sum_{k=0}^{\infty}s^{k}G_{k}(t),$
where
$G_{k}(t)=(-1)^{k}\frac{\pi\sqrt{t}}{\sin\pi\sqrt{t}}\sum_{j\geq
k}\frac{(-1)^{j}\pi^{2j}t^{j}}{(2j+1)!}\binom{j}{k}.$ (8)
Then Theorem 1 is equivalent to the statement that
$G_{k}(t)=\sum_{n\geq
k}t^{n}\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^{j}\pi^{2j}\zeta(2n-2j)}{2^{2k-2j-2}(2j+1)!}\binom{2k-2j-1}{k}$
for all $k$. We can write the latter sum as
$\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}\sum_{n\geq
j+1}\zeta(2n-2j)t^{n-j}-\\\
\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}\sum_{n=j+1}^{k-1}\zeta(2n-2j)t^{n-j}=\\\
\frac{1}{2}(1-\pi\sqrt{t}\cot\pi\sqrt{t})\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}-\\\
\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}\sum_{n=j+1}^{k-1}\zeta(2n-2j)t^{n-j},$
(9)
where we have used the generating function
$\frac{1}{2}(1-\pi\sqrt{t}\cot\pi\sqrt{t})=\sum_{i=1}^{\infty}\zeta(2i)t^{i}.$
Note that the last sum in (9) is a polynomial that cancels exactly those terms
in
$\frac{1}{2}(1-\pi\sqrt{t}\cot\pi\sqrt{t})\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}$
(10)
of degree less than $k$. Thus, to prove Theorem 1 it suffices to show that
$G_{k}(t)=\text{terms of degree $\geq k$ in expression (\ref{expr}).}$
From equation (8) it is evident that
$G_{k}(t)=\frac{\pi\sqrt{t}}{\sin\pi\sqrt{t}}\cdot\frac{(-t)^{k}}{k!}\cdot\frac{d^{k}}{dt^{k}}\left(\frac{\sin\pi\sqrt{t}}{\pi\sqrt{t}}\right).$
(11)
We use this to obtain an explicit formula for $G_{k}(t)$.
###### Lemma 3.
For $k\geq 0$,
$G_{k}(t)=P_{k}(\pi^{2}t)\pi\sqrt{t}\cot\pi\sqrt{t}+Q_{k}(\pi^{2}t),$
where $P_{k},Q_{k}$ are polynomials defined by
$\displaystyle P_{k}(x)=$
$\displaystyle-\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4x)^{j}}{2^{2k-1}(2j+1)!}\binom{2k-2j-1}{k}$
$\displaystyle Q_{k}(x)=$
$\displaystyle\sum_{j=0}^{\lfloor\frac{k}{2}\rfloor}\frac{(-4x)^{j}}{2^{2k}(2j)!}\binom{2k-2j}{k}.$
###### Proof.
In view of equation (11), the conclusion is equivalent to
$f^{(k)}(t)=(-1)^{k}k!t^{-k}P_{k}(\pi^{2}t)\cos\pi\sqrt{t}+(-1)^{k}k!t^{-k}Q_{k}(\pi^{2}t)f(t),$
where $f(t)=\sin\pi\sqrt{t}/\pi\sqrt{t}$. Differentiating, one sees that the
polynomials $P_{k}$ and $Q_{k}$ are determined by the recurrence
$\displaystyle(k+1)P_{k+1}(x)$
$\displaystyle=kP_{k}(x)-xP_{k}^{\prime}(x)-\frac{1}{2}Q_{k}(x)$
$\displaystyle(k+1)Q_{k+1}(x)$
$\displaystyle=\frac{2k+1}{2}Q_{k}(x)-xQ_{k}^{\prime}(x)+\frac{x}{2}P_{k}(x)$
together with the initial conditions $P_{0}(x)=0$, $Q_{0}(x)=1$. The
recurrence and initial conditions are satisfied by the explicit formulas
above. ∎
###### Proof of Theorem 1.
Using Lemma 3, we have
$G_{k}(t)=-\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-1}(2j+1)!}\binom{2k-2j-1}{k}\pi\sqrt{t}\cot\pi\sqrt{t}\\\
+\sum_{j=0}^{\lfloor\frac{k}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k}(2j)!}\binom{2k-2j}{k}=\\\
\frac{1}{2}(1-\pi\sqrt{t}\cot\pi\sqrt{t})\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-4\pi^{2}t)^{j}}{2^{2k-2}(2j+1)!}\binom{2k-2j-1}{k}\\\
+\text{terms of degree $<k$},$
and this completes the proof. ∎
###### Proof of Theorem 4.
Using Theorem 1 in the form of equation (7), eliminate $\zeta(2n-2j)$ using
Euler’s formula (2) and then compare with Theorem 3 to get
$\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^{n-1}\pi^{2n}B_{2n-2j}}{2^{2k-2n-1}(2n-2j)!(2j+1)!}\binom{2k-2j-1}{k}=\\\
\frac{(-1)^{n-k-1}\pi^{2n}}{(2n+1)!}\sum_{i=0}^{n-k}\binom{n-i}{k}\binom{2n+1}{2i}2(2^{2i-1}-1)B_{2i}.$
Now multiply both sides by $(-1)^{n-1}2^{2k-2n-1}\pi^{-2n}(2n+1)!$ and rewrite
the factorials on the left-hand side as a binomial coefficient. ∎
## References
* [1] W. Y. C. Chen and L. H. Sun, Extended Zeilberger’s algorithm for identities on Bernoulli and Euler polynomials, _J. Number Theory_ 129 (2009), 2111-2132.
* [2] L. Euler, Meditationes circa singulare serierum genus, _Novi Comm. Acad. Sci. Petropol._ 20 (1775), 140-186; reprinted in _Opera Omnia_ , ser. I, vol. 15, B. G. Teubner, Berlin, 1927, pp. 217-267.
* [3] H. Gangl, M. Kaneko, and D. Zagier, Double zeta values and modular forms, in _Automorphic Forms and Zeta Functions_ , S. Böcherer et. al. (eds.), World Scientific, Singapore, 2006, pp. 71-106.
* [4] M. E. Hoffman, Multiple harmonic series, _Pacific J. Math._ 152 (1992), 275-290.
* [5] M. E. Hoffman, The algebra of multiple harmonic series, _J. Algebra_ 194 (1997), 477-495.
* [6] M. E. Hoffman, A character on the quasi-symmetric functions coming from multiple zeta values, _Electron. J. Combin._ 15 (2008), res. art. 97.
* [7] Y. Komori, K. Matsumoto and H. Tsumura, A study on multiple zeta values from the viewpoint of zeta-functions of root systems, preprint arXiv:1205.0182.
* [8] I. G. Macdonald, _Symmetric Functions and Hall Polynomials_ , 2nd ed., Oxford Univ. Press, New York, 1995.
* [9] T. Machide, Extended double shuffle relations and the generating function of triple zeta values of any fixed weight, preprint arXiv:1204.4085.
* [10] Z. Shen and T. Cai, Some formulas for multiple zeta values, _J. Number Theory_ 132 (2012), 314-323.
|
arxiv-papers
| 2012-05-31T17:31:45 |
2024-09-04T02:49:31.410737
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Michael E. Hoffman",
"submitter": "Michael E. Hoffman",
"url": "https://arxiv.org/abs/1205.7051"
}
|
1205.7067
|
We present detailed $16$-GHz interferometric observations using the
Arcminute Microkelvin Imager (AMI) of 19 clusters with $L_X > 7\times10^{37}$ W ($h_{50}=1$)
selected from the Local Cluster
Substructure Survey (LoCuSS; $0.142 \le z \le
0.295$) and of Abell 1758b, which is in the field of view of Abell 1758a.
We detect and resolve Sunyaev-Zel'dovich (SZ) signals towards 17 clusters, with
peak surface brightnesses between 5 and 23$\sigma$.
We use a fast, Bayesian cluster analysis to
obtain cluster parameter
estimates in the presence of radio point sources, receiver
noise and primordial CMB anisotropy. We fit isothermal $\beta$-models to our data and assume the clusters are virialized
(with all the kinetic energy in gas internal energy). Our gas temperature, $T_{\rm{AMI}}$,
is derived from AMI SZ data and not from X-ray spectroscopy. Cluster parameters internal to
$r_{500}$ are derived under the assumption of hydrostatic equilibrium.
We find the following.
(i) Different gNFW parameterizations yield significantly different parameter degeneracies.
(ii) For $h_{70}=1$, we find the classical virial
radius, $r_{200}$, to be typically 1.6$\pm$0.1 Mpc and the total mass
$M_{\rm{T}}(r_{200})$ typically to be 2.0-2.5$\times$ $M_{\rm{T}}(r_{500})$.
(iii) Where we have found $M_{\rm{T}}(r_{500})$
and $M_{\rm{T}}(r_{200})$ X-ray and weak-lensing values in the literature, there
is good agreement between weak-lensing and AMI estimates (with $M_{\rm{T},\rm{AMI}}/M_{\rm{T},WL}=1.2^{+0.2}_{-0.3}$ and $=1.0\pm 0.1$ for
$r_{500}$ and $r_{200}$, respectively). In comparison, most Suzaku/Chandra
estimates are higher than for AMI (with $M_{\rm{T}, X}/M_{\rm{T},\rm{AMI}}=1.7 \pm {0.2}$ within $r_{500}$), particularly for the stronger mergers.
(iv) Comparison of $T_{\rm{AMI}}$ to $T_X$ sheds light on high X-ray masses: even at
large radius, $T_X$ can substantially exceed $T_{\rm{AMI}}$ in mergers.
The use of these higher $T_X$ values will give higher X-ray masses. We stress
that large-radius $T_{\rm{AMI}}$ and $T_X$ data are scarce and must be increased.
(v) Despite the paucity of data, there is an indication of a relation between merger activity and SZ ellipticity.
(vi) At small radius (but away from any cooling flow) the SZ signal (and $T_{\rm{AMI}}$) is less sensitive to ICM disturbance
than the X-ray signal (and $T_X$) and, even at high radius, mergers affect $n^2$-weighted X-ray data more than
$n$-weighted SZ, implying that significant shocking or clumping or both occur in even the outer parts of mergers.
cosmology: observations – cosmic microwave background –
galaxies: clusters – Sunyaev–Zel'dovich X-ray – galaxies: clusters:
individual (Abell 586, Abell 611, Abell 621, Abell 773, Abell 781, Abell 990, Abell 1413, Abell 1423,
Abell 1704, Abell 1758a, Abell 1758b, Abell 2009, Abell 2111, Abell 2146, Abell 2218, Abell 2409, RXJ0142+2131, RXJ1720.1+2638,
Zw0857.9+2107, Zw1454.8+2233)
§ INTRODUCTION
The virtues of galaxy clusters are often extolled
as, for example, being the largest gravitationally bound systems in the
Universe, or being excellent samplers of the matter field on large scales, or simply
as being of fundamental importance to astrophysics and cosmology (see e.g., 155, 40 and 65).
To make full use of these virtues one needs observations that, amongst others things, reach large distances
away from cluster centres. It would often be very useful to reach the classical
virial radius $\approx$ $r_{200}$ of a cluster, internal to which the average
density is 200 times the closure density.
Studying clusters on these scales is important for many reasons. First, these measurements
are needed to characterize the entire cluster volume. Second, they can be key for any attempt at precision cosmology, including calibrating scaling relations [66],
as they are believed to be less susceptible to the complicated physics of the core region from
e.g., star formation, energy feedback from active galactic nuclei and gas cooling.
Third, the virial radius marks the transition between the accreting matter and the gravitationally-bound, virialized gas of
clusters and thus contains information on the current processes responsible for large-scale structure formation.
However, there are few such observations due to the difficulties of obtaining a signal far away from the cluster centre.
We now comment on four methods of estimating cluster masses (see 3 for a recent, overall review):
* Spectroscopic measurements of the velocity dispersion of cluster members
require very high sensitivity at moderate to high redshift, and confusion
becomes worse as redshift increases and as distance on the sky from the cluster
centre increases. Cluster masses have recently been obtained this way in e.g., [129] and [140].
* X-ray observations of the Bremsstrahlung (free-free radiation) from the intracluster plasma
(by convention referred to as `gas') have delivered a great deal of information
on cluster physics on a large number of clusters (see e.g., 38, 24 and 72).
Observations are, of course, difficult at high redshift due to cosmic dimming, and because the X-ray signal is
$\propto \int n^{2} f(T) dl$, where $n$ is the
electron density, $T$ is electron temperature, $f(T)$ is a
weak function of $T$, and $l$ is the line of sight through the cluster, there is
significant bias to gas concentration, which makes reaching a high radius
difficult – however, at low to intermediate redshift there is a small but
growing number of observations that approach or reach $r_{200}$ mainly with the Suzaku satellite, though the sky
background subtraction is challenging (e.g., 46, 61).
* Gravitational lensing of background galaxies
gives the distribution of all the matter in the cluster directly,
without relying on assumptions obout the dynamical state of the cluster. Any mass concentrations along the line-of-sight not
associated with the cluster will lead to an overestimate of the weak lensing cluster mass.
But the `shear'
signal is proportional to the rate of change with radius of the
gravitational potential, which changes increasingly slowly with radius at large radius,
so reaching large radius is difficult.
Confusion also bears strongly on this difficulty, and measurement is of course harder
as redshift increases. Example weak-lensing cluster studies include [114] and [34], for analyses of individual high-mass clusters,
and [86] and [133] for analyses of stacked lensing profiles for many low-mass clusters.
* The Sunyaev Zel'dovich (SZ; 146; see e.g 18 and 30 for
reviews) signal from inverse Compton scattering of the CMB by the cluster gas
has relatively little bias to gas concentration since it is $\propto \int n T dl$, and has remarkably little sensitivity to redshift
over moderate to high redshift; both of these properties make the SZ
effect extremely attractive. The problem with SZ is that it is intrinsically very faint. The
first generation of SZ telescopes, including the OVRO 40-m (see e.g.,
20), the OVRO 5-m (see e.g., 58), the
OVRO/BIMA arrays <cit.> and the Ryle Telescope (see e.g.,
52) had to integrate for a very long time to get a
significant SZ detection of a single known cluster. The new generation,
including ACT (see e.g., 59 and 89), AMI
(see e.g., 11 and 10), AMiBA (see e.g., 79 and 159), MUSTANG (see e.g., 70 and 107), OCRA (see e.g.,
27 and 76), Planck (see e.g 148 and 119), SPT (see e.g 31 and 158) and SZA (see e.g., 29 and 109) are all much more sensitive.
The new generation of SZ facilities include two types of instrument:
ACT, Planck and SPT are instruments with
wide fields of view (FoV) optimized for detecting CMB imprints in large sky
areas in a short amount of time – this is a very important ability but, for the
imaging of a particular cluster, a wide FoV is of no benefit; in contrast, AMI,
AMiBA, MUSTANG, OCRA and SZA are designed to go deep and to measure
the masses of the majority of clusters.
In [12], we reported initial SZ observations of seven X-ray clusters
(selected to have low radio flux-densities to limit confusion) that approach or reach
$r_{200}$. In this paper we report on resolved, interferometric SZ observations with arcminute resolution
that approach or reach $r_{200}$ in a substantial sample of X-ray clusters selected
above an X-ray flux-density limit (plus a radio flux-density limit) and over a
limited redshift-range (which limits the effects of cosmic evolution); as far as we
are aware, this is the first time such SZ observations of a large cluster sample have been undertaken. These
measurements are timely since complementary large-$r$ X-ray data have recently been
obtained with Suzaku (e.g., 17, 61 and 68).
These early Suzaku measurements, despite the large
model uncertainties, are already showing that ICM profiles on these scales appear to disagree with
predictions from hydrodynamical cluster simulations (e.g., 46) and have drawn attention
to possible causes such as ICM clumping [111] and the breakdown of assumptions such as
hydrostatic equilibrium (e.g., 41), which can bias the X-ray masses (e.g., 125, 100 and 42).
We stress that SZ observations, like those in the optical/IR and X-ray, also have their contaminants and
systematics, and all four methods are also hampered by projection effects.
Studying large samples of clusters using
multiple techniques is important for building a thorough understanding of cluster physics.
Well-calibrated mass-observable relations are
crucial for current and future cosmological studies – see e.g., [3].
To our knowledge, this is the largest cluster-by-cluster study
for which masses have been derived from SZ targeted observations out to the virial radius. The results
from this work will be very valuable for detailed comparisons of cluster mass estimates.
This paper is organized as follows.
In Sec. 2
we describe the sample selection. The data and instrument are introduced in Sec. 3, while Sec. 4 focuses on the methods applied
for mapping the data, identifying radio source foregrounds and removing them from the maps. The analysis of the cluster + radio sources
environment is outlined in Sec. 5. Given the difficulty of comparing cluster mass measurements from different data,
we provide considerable detail in our results section, Sec. 6. In particular, we present: maps; details on the radio source environment towards
the clusters; full cluster parameter
posterior distributions internal to two overdensities, $r_{500}$ and $r_{200}$; an investigation of contaminating radio
sources (our main source of systematic error); and we compare our $\beta$-model parameterization
with several generalized Navarro-Frenk-White (gNFW) parameterizations. In Sec. 7 we illustrate the ability of our methodology
to recover the cluster mass even for a cluster with a challenging source environment. In Sec. 7
we discuss our results, in particular, the morphology and dynamical state of the
clusters and the comparison of SZ-, weak lensing- and X-ray-derived cluster masses and large $r$ X-ray and SZ temperatures. The conclusions of our
study are summarized in Sec. 9.
Throughout, we assume a concordance $\Lambda$CDM cosmology
with $\Omega_{\rm{m},0}=0.3$, $\Omega_{\Lambda,0}=0.7$, $\Omega_k=0$,
$\Omega_{b,0}=0.041$, $w_0=-1$, $w_a=0$ and $\sigma_8=0.8$. For the probability
distribution plots and the tables, we take $h=H_0/100$ km s$^{-1}$ Mpc$^{-1}$;
elsewhere we take $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ as the default value and also
refer when necessary to $h_{X}=H_0/X$ kms$^{-1}$ Mpc$^{-1}$. All
coordinates are at epoch J2000.
§ THE LOCUSS CATALOGUE AND OUR SUB-SAMPLE
LoCuSS [143, 144] is a multi-wavelength survey of 164 X-ray luminous
($L_{\rm{X}}$ $\geq 2\times10^{37}$ W over the 0.1-2.4 keV band in the
cluster rest frame (38 and 39, $h_{50}=1$)
galaxy clusters. The narrow range of redshifts $z$ ($0.142 \le z \le
0.295$ minimises cosmic evolution. The clusters have been selected from the ROSAT All-Sky Survey [38, 39, 22] without taking
into account their structures or dynamical states. Relevant LoCuSS
papers include [90] and [163].
In this work, we study a sub-sample of 19 clusters from the LoCuSS catalogue and Abell 1758b[Abell 1758b was serendipitously
observed in the field of view of Abell 1758a, a LoCuSS cluster.]
(Tab. <ref>) using 16-GHz interferometric AMI data with arcminute resolution.
Our sub-sample includes only those clusters with $\delta >20^{\circ}$. AMI can observe down to
lower declinations but suffers from poorer $uv$-coverage and satellite interference at $\delta <20^{\circ}$.
We also applied an X-ray luminosity cut, $L_X > 7\times10^{37}$ W (0.1-2.4 keV restframe, $h_{50}=1$),
lower-luminosity clusters tend to be fainter in SZ. Contamination from radio sources at 16 GHz can significantly affect our
SZ detections. For this reason, we have chosen to exclude
clusters with sources brighter than 10 mJy $\rm{beam}^{-1}$ within 10$\arcmin$
of the cluster X-ray centre.
Several studies of the LoCuSS sample of clusters are ongoing. These include both ground-based (Gemini, Keck, MMT, NOAO, Palomar, Subaru, SZA, UKIRT and VLT)
as well as space-based (Chandra, HST, GALEX, XMM-Newton and Spitzer) facilities.
Our AMI SZ data are complementary to other data taken towards these clusters as they probe the large-scale gas structure, are sensitive
to gas from destroyed density peaks and are particularly beneficial for obtaining robust cluster masses since the SZ signal
has long been recognised as a good mass proxy (see e.g., 105).
Cluster details. It should be noted that Abell 1758b is not part of LoCuSS.
Cluster Right ascension Declination Redshift X-ray
luminosity Alternative cluster names
(J2000) (J2000) /$
10^{37}$ W
($h_{50}=1$; see text)
Abell 586 07 32 12 +31 37 30 0.171 11.1
Abell 611 08 00 56 +36 03 40 0.288 13.6
Abell 621 08 11 09 +70 02 45 0.223 7.8
Abell 773 09 17 54 +51 42 58 0.217 13.1
Abell 781 09 20 25 +30 31 32 0.298 17.2
Abell 990 10 23 39 +49 08 13 0.144 7.7
Abell 1413 11 55 18 +23 24 29 0.143 13.3
Abell 1423 11 57 18 +33 36 47 0.213 10.0
Abell 1704 13 14 18 +64 33 27 0.216 7.8
Abell 1758a 13 32 45 +50 32 31 0.280 11.7
Abell 1758b 13 32 29 +50 24 42 0.280 7.3
Abell 2009 15 00 21 +21 22 04 0.153 9.1
Abell 2111 15 39 40 +34 26 00 0.229 10.9
Abell 2146 15 55 58 +66 21 09 0.234 9.0
Abell 2218 16 35 45 +66 13 07 0.171 9.3
Abell 2409 22 00 57 +20 57 50 0.147 8.1
RXJ0142+2131 01 42 03 +21 31 40 0.280 9.9
RXJ1720.1+2638 17 20 10 +26 37 31 0.164 16.1
Zw0857.9+2107 09 00 39 +20 55 17 0.235 10.8
Zw1454.8+2233 14 57 15 +22 20 34 0.258 13.2
§ INSTRUMENT AND OBSERVATIONS
AMI consists of two aperture-synthesis interferometric arrays located
near Cambridge.
The Small Array (SA) is optimized for SZ imaging while the Large Array (LA) is
used to observe radio sources
that contaminate the SZ effect in the SA observations.
AMI's $uv$-coverage is well-filled all the way down to $\approx 180\lambda$, corresponding to a maximum angular scale of $\approx 10\arcmin$.
AMI is described in
detail in AMI Consortium: Zwart et al. (2008)[The observing frequency range given in AMI Consortium: Zwart et al. has been altered
as described in AMI Consortium: Franzen et al. 2010.].
SA pointed observations of all the clusters were taken between 2007 and
2010 while LA raster observations, which were mostly 61+19 pt hexagonal
rasters[A 61+19 point raster observation consists of 61 pointings
with separations of 4$\arcmin$, of which the central 19 pointings have lower noise
levels, see e.g., [7] for example LA maps.]
centred on the cluster X-ray position, were made between 2008 and 2010.
each cluster was observed for 20-80 hours with the SA and for 10-25 hours with the LA.
The thermal noise levels for the SA ($\sigma_{\rm{SA}}$) and for the LA
($\sigma_{\rm{LA}}$) were obtained by
applying the aips[http://www.aips.nrao.edu] task imean
on a
section of the map far down the primary beam and free from any significant
contamination. In Tab. <ref> we provide central thermal noise
estimates for the SA and LA observations; they reflect
the amounts of data remaining after flagging. A series of algorithms has been developed to
remove (or `flag-out') bad data points arising from interference,
shadowing, hardware and other errors. This is a stringent process that typically results in $\approx$ 30-50$\%$ of
data being discarded before the analysis.
A primary-beam correction factor has been applied, as the thermal noise level is dependant on the distance
from the pointing centre.
The raw visibility files were put through our local data reduction
pipeline, reduce, described in detail in
[5], and exported in fits format.
Bi-daily observations of 3C286 and 3C84 were used for flux calibration while
interleaved calibrators
selected from the Jodrell Bank VLA Survey [118, 26, 157]
were observed every hour for phase calibration.
§ MAPPING AND SOURCE DETECTION AND SUBTRACTION
Our LA map-making and source-finding procedures follow [10]. We
applied standard aips tasks to image the continuum and
individual-channel uvfits data output from reduce.
At 16 GHz, the dominant contaminants to the SZ decrements are radio sources.
In order to recover the SZ signal, the contribution of these radio sources to the data
need to be removed; this is done as follows.
* First, the cleaned LA continuum maps[The LA continuum maps were cleaned down to $3\sigma_{\rm{LA}}$ with no boxes.] were put
through the AMI-developed source-extraction software sourcefind [6] to
identify and characterize radio sources on the LA maps above a certain signal-to-noise. sourcefind provides estimates for the
right ascension $x_s$, declination $y_s$, flux density[We catalogue
the peak flux of the source,
unless the source is extended, in which case we
integrate the source surface brightness over its projected solid angle to give its integrated flux density (see e.g., 6).],
$S_0$, and spectral index $\alpha$[We adopt the convention $S\propto \nu^{-\alpha}$.] at the
central frequency $\nu_0$ for identified radio sources. We impose a detection threshold such that
we select only those radio sources with a flux density $\geq 4\sigma_{LA_p}$ on
the cleaned LA continuum maps, where $\sigma_{\rm{LA_p}}$ refers to
pixel values on the LA noise maps.
The number of $\geq$4$\sigma_{LA_p}$ sources
detected in our LA observations of each cluster is given in Tab.
Second, prior to any source subtraction, we run our cluster-analysis software, which fits for the position, flux and spectral index of the
sourcefind-detected radio sources using the source parameters obtained by sourcefind as priors. For some of the
less contaminating radio sources, our cluster-analysis software uses delta-priors for the source parameters centred at the LA estimates (see Sec. <ref> for further details) .
Third, the source parameters given by the cluster analysis were used to perform source subtraction on the SA maps. This was done using in-house software,
muesli, which is an adaptation of the standard aips task
uvsub optimized for processing AMI data.
The flux-density contributions from detected radio sources were subtracted from each SA channel uvfits file using either
the mean values for their position, spectral index and flux density derived
from our Bayesian analysis, when these parameters are not given
delta-function priors, or, otherwise, using the LA estimates for these
source parameters. Details of the
priors assigned to each of the sources labelled on the SA maps can be found in
Sec. <ref> and Tab. <ref>.
Fourth, after source subtraction, the SA maps were cleaned with a
box around the SZ signal. In contrast, the LA maps and SA maps before source subtraction
were cleaned with a single box comprising the entire map.
Both the SA and the LA maps were cleaned down to $3\sigma$.
Observational details. SA and LA noise levels, $\sigma_{\rm{SA}}$ and $\sigma_{\rm{LA}} \geq 4\sigma$ and the number of sources detected above $4\sigma_{\rm{LA_p}}$ on the LA rasters for each cluster.
Abell 1758b is not part of LoCuSS.
Cluster $\sigma_{SA}$ $\sigma_{LA}$ Number of LA
$4\sigma_{LA}$ sources
(mJy) (mJy)
Abell 586 0.17 0.09 23
Abell 611 0.11 0.07 23
Abell 621 0.11 0.09 13
Abell 773 0.13 0.09 9
Abell 781 0.12 0.07 24
Abell 990 0.10 0.08 20
Abell 1413 0.13 0.09 17
Abell 1423 0.08 0.07 31
Abell 1704 0.09 0.06 13
Abell 1758a 0.12 0.08 14
Abell 1758b 0.13 0.08 14
Abell 2009 0.11 0.14 18
Abell 2111 0.09 0.07 22
Abell 2146 0.15 0.06 15
Abell 2218 0.07 0.10 15
Abell 2409 0.14 0.05 15
RXJ0142+2131 0.11 0.06 22
RXJ1720.1+2638 0.08 0.10 17
Zw0857.9+2107 0.13 0.12 13
Zw1454.8+2233 0.10 0.10 16
§ ANALYSIS
We use our own Bayesian analysis package, McAdam, to
estimate cluster parameters internal to $r_{500}$ and $r_{200}$ from AMI data in the presence
of radio point sources, receiver noise and primordial CMB anisotropy.
The cluster and radio sources are parameterized in our analysis (see below) while the remaining
components are included in a generalized noise covariance matrix; we note that these are the
only significant noise contributions because large-scale emission from e.g., foreground galactic emission is resolved
out by our interferometric observations.
McAdam was originally
developed by [92] and [45] adapted it to work on AMI data. The latest
McAdam uses MultiNest (43, 44) as its inference engine to
allow Bayesian evidence and posterior
distributions to be calculated efficiently, even for posterior distributions
with large (curved) degeneracies and/or mutiple peaks. This addition has been key to
our analysis since the posteriors of AMI data often have challenging dimensionalities, $>30$, primarily as a result
of the presence of a large number of radio sources in the AMI observations.
§.§ Model
We have modelled the cluster density profile assuming
spherical symmetry using a $\beta$-model [33]:
\begin{equation}
\rho_{\rm{g}}(r)=\frac{\rho_{\rm{g}}(0)}{\left[ 1 + \left( \frac{r}{r_{\rm{c}}}
\right)^2 \right]^{\frac{3\beta}{2}}},
\label{eq:den}
\end{equation}
where gas mass density $\rho_{\rm{g}}(r)=\mu n(r)$, $\mu=1.14m_{\rm{p}}$ is the gas mass per electron and $m_{\rm{p}}$ is
the proton mass.
The core radius $r_{\rm{c}}$ gives the density profile a flat top at low
$\frac{r}{r_{\rm{c}}}$ and $\rho_{\rm{g}}$ has a logarithmic slope of
$3\beta$ at large $\frac{r}{r_{\rm{c}}}$.
We choose to model the gas as isothermal, using the virial mass-temperature
relation and assuming that all kinetic energy is in gas internal energy:
\begin{align}
\rm{k}_{\rm{B}}T(r_{200}) &= \frac{\rm{G}\mu M_{\rm{T}}(r_{200})}{2r_{200}} \\
\frac{\rm{G}\mu}{2\left(\frac{3}{4\pi\left(200\rho_{\rm{crit}}\right)}\right)^{1/3}}M_{\rm{T}}^{2/3}(r_{200})
\\
&= 8.2
\textrm{keV}\left(\frac{M_{\rm{T}}(r_{200})}{10^{15}h^{-1}\rm{M}_{\odot}}\right)^{2/3}\left(\frac{H(z)}{H_{0}}\right)^{2/3}.
\label{eq:virtemp}
\end{align}
$M_{\rm{T}}(r_{200})$ and $T(r_{200})$
refer to the total mass and gas temperature within $r_{200}$ (see e.g.,
153). This relation allows cluster parameters within
$r_{200}$ to be inferred without
assuming hydrostatic equilibrium; note that, in our methodology, parameters
describing the cluster at smaller $r$ (e.g., $r_{500}$) do, however,
assume hydrostatic equilibrium.
Further details of the cluster analysis can be found in [9] and
[8]. The good agreement between mass estimates from weak-lensing and AMI data on 6 clusters in [7]
supports the use of this $M-T$ relation in our analysis.
§.§ Priors
§.§.§ Cluster priors
The cluster model parameters
$\vect{\Theta}_{\rm{c}}=(x_{\rm{c}}$, $y_{\rm{c}}$,
$M_{\rm{T}}(r_{200})$, $f_{\rm{g}}(r_{200})$, $\beta$, $r_{\rm{c}}$, $z$) have
priors that are assumed to be separable. $x_{\rm{c}}, y_{\rm{c}}$ are the cluster position (RA and Dec, respectively) and $f_{\rm{g}}$ is the gas fraction, which is defined as
\begin{equation}
f_{\rm{g}} =\frac{M_{\rm{g}}}{M_{\rm{T}}}.
\label{eq:fg}
\end{equation}
Further details on these priors are given in Tab. <ref>.
This set of sampling parameters has proved sufficient for our cluster detection
algorithm (10) and to describe the physical cluster parameters.
We emphasize that this way of analysing the data is different from the
way used traditionally, in which an X-ray spectroscopic temperature is used as
an input parameter. The difficulty with this use of an X-ray temperature is
that, in practice, the temperature measurement usually applies to gas relatively
close to the cluster centre (but any cooling flow is excised). By sampling from $M_{\rm{T}}$ and using the $M-T$ relation (Eq. <ref>), the
temperature of each cluster is derived from SZ data only and is
averaged over the angular scale of the SZ observation, which is typically larger
than the angular scale of the X-ray temperature measurement. This way, although our analysis
does not yield $T(r)$, it gives and uses a temperature which is representative of the cluster volume
we are investigating.
Summary of the priors for the sampling parameters in each
model. The value for the redshift and position priors have not been
included in this table since they are cluster specific. Instead, they are
given in Tab. <ref> for each cluster.
Parameter Prior Type Values Origin
$x_{\rm{c}},y_{\rm{c}} {''}$ Gaussian at
cluster position
[39]
$\beta$ uniform $0.3-2.5$ [92]
$M_{\rm{T}}(r_{200})/h^{-1}\rm{M}_\odot$ uniform in log
$1\times10^{13.5}-5\times10^{15}$ physically reasonable, e.g., 160
$r_{\rm{c}}/h^{-1}\textrm{kpc}$ uniform $10-1000$ physically
reasonable e.g., 160
$z$ delta cluster redshift [39]
$f_{\rm{g}}(r_{200})/h^{-1}$ Gaussian, $\sigma= 0.0216$ $0.0864$
[78, 163]
§.§.§ Source priors
Radio sources detected on the LA maps using sourcefind are modelled
by four source parameters,
$\vect{\Theta}_{\rm{S}}$ = ($x_s$, $y_s$, $S_0$, $\alpha$). Priors on these
parameters are based on LA measurements,
discussed in Sec. <ref>.
Sources on the source-subtracted SA maps are labelled according to Tab.
<ref>. Delta-function priors on all the
source parameters tend to be given to those sources whose flux density
is $<4\sigma_{SA}$ and to those outside the $10\%$ radius of the SA power
beam. The remaining sources are usually assigned a delta-function prior on
position and Gaussian priors on $\alpha$
and $S_0$. However, in a few cases we replace delta-function priors on the
source parameters with Gaussian priors as this can increase the accuracy of
the source
subtraction. These wider priors can be necessary to account for discrepancies
between the LA and SA measurements. Reasons for these differences include: a
poor fit of our
Gaussian model for the power primary beam far from the
pointing centre, correlator artifacts, source variability and source
Priors on position, spectral index and flux density given to detected
sources. The symbols correspond to the labels in the SA
source-subtracted maps. The Gaussian priors are centred
on the LA measurements. $\sigma$ values for the Gaussian priors are assigned as follows:
for the Gaussian prior on the flux-density of each radio source, $\sigma$ is set to 40% of the source flux density;
for the spectral index $alpha$, $\sigma$ is set to the LA error on $\alpha$ and for
the source position, $\sigma$ is set to $60\arcsec$.
Symbol $\Pi(S_0)$ $\Pi(\alpha)$ $\Pi(x_s, y_s)$
+ delta delta delta
$\times$ Gaussian Gaussian delta
$\triangle$ Gaussian Gaussian Gaussian
§ RESULTS AND COMMENTARY
Out of the 20 clusters listed in Tab. <ref>, we detect SZ
decrements towards 17. For these clusters we
present SA maps before and after source subtraction as well as
posterior distributions for some cluster and source parameters
and mean values of
selected cluster parameters (Tab. <ref>), with the exception of
Abell 2409, which was found
to have a local environment which renders it unsuitable for robust
parameter estimation (see Sec. <ref>).
For the posterior distributions all ordinates and abscissae in these plots are linear,
the $y$-axis for the 1-d marginals is the probability density and $h$ is short for $h_{100}$.
It is important to note that, while the posterior
probability distributions for large-scale cluster parameters reflect the
uncertainty in the
McAdam-derived flux-density estimates, the radio source-subtracted maps
do not, as they simply use a single value (the mean) for each source parameter.
The effect of our priors on the results
has been thoroughly tested in a previous study by [8], which
found that the priors used in this parameterization do not to lead
to any strong biases in the cluster parameter estimates.
The SA maps have labels indicating the position of detected radio
sources and their priors (Tab. <ref>); the square box in these plots
indicates the best-fit cluster position determined by
McAdam. No primary-beam
correction has been applied to the SA maps presented in this paper, unless stated otherwise. The contour levels on the SA maps, unless otherwise stated, start at $2\sigma_{SA}$ and
increase linearly from 2 to $10\sigma_{SA}$. On radio-only images, positive contours are shown as solid lines and negative contours as dashed lines,
but on radio+X-ray images, negative radio contours are shown as solid lines and X-ray shown as greyscale. The bottom-left ellipses on the SA maps are the FWHMs of the synthesized beams.
A $0.6$-k$\lambda$ taper
was applied to the SA source-subtracted maps to downweight
long-baseline visibilities with the purpose of increasing the
signal-to-noise of the large-scale structure; this typically leads to a
$\approx 20\%$ increase in the noise.
The X-ray images are obtained from archive ROSAT and Chandra data.
We remind readers that when looking at a radio map – necessarily with a particular $uv$-weighting – a near-circular image does not mean that the SA failed to resolve the SZ signal. Investigating angular structure / size requires assessment in $uv$-space,
which can be done with a selection of maps made over different $uv$ ranges but is optimally done here in $uv$-space with McAdam. In fact, all the SZ decrements in this paper are resolved.
Source properties for detected sources within 5$\arcmin$ of the
SZ mean central position. The number next to each cluster name denotes the source number; this label is used in the plots showing the marginalized posterior distributions for the source fluxes.
$S_0{\rm{McA}}$ is the McAdam-derived best-fit source flux at 16 GHz. $\alpha$ is the source spectral index estimated by McAdam and centred at the McAdam-derived mean frequency and
the last column contains the distance between the cluster SZ centroid (as determined by McAdam) and the source.
Name Right ascension Declination $S_0{\rm{McA}}$
$\alpha$ Distance from SZ centroid
(hh:mm:ss , J2000) ($^{\circ}$:$\arcmin$:$\arcsec$ , J2000) mJy $\arcsec$
A586_0 07:32:20.5 +31:38:02.8 0.26 1.20 28
A586_1 07:32:19.1 +31:40:25.6 0.86 0.28 171
A586_2 07:32:11.0 +31:39:47.6 0.86 1.20 178
A586_3 07:32:04.5 +31:39:09.6 0.41 1.53 222
A586_4 07:32:35.4 +31:35:35.5 1.03 -0.47 227
A586_5 07:32:21.2 +31:41:26.3 7.44 0.46 232
A586_6 07:32:42.7 +31:38:37.1 0.53 0.16 293
A611_0 08:01:07.0 +36:02:18.9 0.32 -0.27 108
A611_1 08:00:52.6 +36:06:14.2 0.44 2.18 199
A611_2 08:01:17.0 +36:04:27.8 0.5 -0.71 229
A621_3 08:11:12.8 +70:02:27.2 7.18 1.34 25
A621_4 08:11:19.3 +70:00:48.4 0.6 -1.31 127
A621_5 08:11:35.2 +70:04:25.6 0.16 0.13 166
A621_6 08:10:38.0 +70:04:09.3 0.09 0.01 181
A781_0 09:20:24.7 +30:31:49.9 0.24 -0.04 4
A781_1 09:20:23.3 +30:29:49.3 8.97 0.97 126
A781_2 09:20:08.4 +30:32:15.8 1.49 -0.18 213
A781_3 09:20:14.0 +30:28:60.0 2.12 0.63 223
A990_0 10:23:47.3 +49:11:25.5 0.44 -0.21 208
A990_1 10:24:02.1 +49:06:51.8 2.78 2.18 239
A1413_0 11:55:15.4 +23:23:59.4 0.47 1.04 59
A1413_1 11:55:08.8 +23:26:16.6 3.1 0.98 222
A1423_2 11:57:17.1 +33:36:30.6 0.54 -0.19 63
A1423_3 11:57:28.5 +33:35:31.0 0.26 2.16 134
A1423_4 11:57:19.7 +33:39:58.3 0.39 0.73 171
A1423_5 11:57:35.2 +33:37:21.8 0.19 0.58 176
A1423_6 11:57:40.5 +33:35:10.1 0.18 -0.40 270
A1423_7 11:57:20.5 +33:41:57.8 0.73 -0.22 290
A1423_8 11:57:39.0 +33:34:03.3 0.24 1.47 290
A1704_0 13:14:02.3 +64:38:29.5 0.2 -0.03 273
A1704_1 13:14:52.6 +64:37:59.9 0.81 0.86 287
A1758a_0 13:32:53.3 +50:31:40.6 7.08 0.5 54
A1758a_1 13:32:38.6 +50:33:37.7 0.77 0.36 150
A1758a_2 13:33:02.2 +50:29:26.4 1.43 0.28 190
A1758a_3 13:32:39.6 +50:34:31.1 0.3 1.45 192
A1758a_4 13:32:41.5 +50:26:47.7 0.46 -0.76 294
A1758b_0 13:32:33.1 +50:22:35.1 0.23 0.11 90
A1758b_1 13:32:41.5 +50:26:47.7 0.51 -0.68 196
A2009_0 15:00:19.7 +21:22:12.6 1.85 3.14 58
A2009_1 15:00:28.6 +21:22:45.8 0.18 0.66 133
A2009_2 15:00:19.6 +21:22:11.3 1.97 2.64 77
A2009_3 15:00:28.6 +21:22:45.7 0.18 0.66 135
A2111_0 15:39:30.1 +34:29:05.5 0.5 0.68 222
A2111_1 15:39:56.7 +34:29:31.8 0.81 -1.47 297
A2146_0 15:56:04.2 +66:22:13.0 5.94 0.55 43
A2146_1 15:56:14.0 +66:20:53.5 1.82 1.03 59
A2146_2 15:56:15.4 +66:22:44.5 0.15 0.34 89
A2146_3 15:55:57.4 +66:20:03.1 1.67 -0.22 106
A2146_4 15:56:27.1 +66:19:43.8 0.1 0.64 164
A2146_5 15:55:25.7 +66:22:04.0 0.48 -0.22 249
A2218_0 16:35:47.4 +66:14:46.1 2.86 0.07 100
A2218_1 16:35:21.8 +66:13:20.6 5.99 0.23 141
A2218_2 16:36:15.6 +66:14:24.0 1.77 0.72 200
A2409_0 22:00:39.7 +20:58:55.0 0.75 1.9 241
A2409_1 22:01:11.2 +20:54:56.8 3.12 0.1 275
RXJ0142+2131_0 01:42:09.2 +21:33:23.4 1.09 0.7 117
RXJ0142+2131_1 01:42:11.0 +21:29:45.3 1.16 1.52 156
RXJ0142+2131_2 01:42:23.3 +21:30:46.7 0.3 0.03 273
RXJ1720+2638_0 17:20:10.0 +26:37:29.7 6.92 1.24 46
RXJ1720+2638_1 17:20:01.2 +26:36:32.3 2.05 0.57 105
RXJ1720+2638_2 17:19:58.4 +26:34:19.6 1.22 1.46 203
RXJ1720+2638_3 17:20:25.5 +26:37:57.2 0.88 0.89 234
Zw0857.9+2107_0 09:00:36.9 +20:53:41.4 1.22 0.31 102
Zw0857.9+2107_1 09:00:55.5 +20:57:21.2 0.96 1.37 259
Zw0857.9+2107_2 09:00:52.8 +20:58:36.5 5.57 0.09 274
Zw1454.8+2233_0 14:57:14.8 +22:20:34.2 1.64 0.28 14
Zw1454.8+2233_1 14:57:08.2 +22:20:08.6 1.55 1.89 108
Zw1454.8+2233_2 14:57:10.6 +22:18:45.6 1.49 0.94 137
Zw1454.8+2233_3 14:56:58.9 +22:18:49.6 8.36 0.17 258
Zw1454.8+2233_4 14:57:04.3 +22:24:11.9 0.83 -0.63 260
Zw1454.8+2233_5 14:57:24.8 +22:24:52.6 0.13 -0.25 281
Zw1454.8+2233_6 14:57:35.7 +22:19:46.8 1.04 1.67 285
Mean and 68$\%$-confidence uncertainties for some McAdam-derived large-scale cluster parameters.
Cluster name $M_{T}(r_{200})$ $M_{T}(r_{500})$ $M_{g}(r_{200})$ $M_{g}(r_{500})$ $r_{200}$ $r_{500}$ $T_{\rm{AMI}}$ $Y(r_{200})$ $Y(r_{500})$
$\times10^{14}h_{100}^{-1}M_{\odot}$ $\times10^{14}h_{100}^{-1}M_{\odot}$ $\times10^{13}h_{100}^{-2}M_{\odot}$ $\times10^{13}h_{100}^{-2}M_{\odot}$ $h_{100}^{-1}$ Mpc $\times10^{-1}h_{100}^{-1}$ Mpc keV $\times 10^{-5}\rm{arcmin}^2$ $\times 10^{-5}\rm{arcmin}^2$
A586 $5.1 \pm 2.1$ $2.1 \pm 0.9$ $4.3 \pm 1.7$ $2.6 \pm 0.7$ $1.2 \pm 0.2$ $6.6 \pm 1.0$ $5.2 \pm 1.4$ $3.6_{-2.1}^{+2.0}$ $2.7 \pm 1.4$
A611 $4.0_{-0.8}^{+0.7}$ $2.0 \pm 0.5$ $3.5 \pm 0.6$ $2.8 \pm 0.3$ $1.1 \pm 0.1$ $6.3 \pm 0.5$ $4.5 \pm 0.6$ $2.2 \pm 0.5$ $2.1 \pm 0.4$
A621 $4.8_{-1.8}^{+1.7}$ $1.4 \pm 0.9 $ $4.1 \pm 1.0$ $1.5 \pm 0.8$ $1.2_{-0.1}^{+0.2}$ $5.3_{-0.1}^{+0.2}$ $5.0 \pm 1.2$ $3.1_{-1.6}^{+1.5}$ $1.9 \pm 1.0$
A773 $3.6 \pm 1.2$ $1.7 \pm 0.6$ $3.1_{-0.9}^{+1.0}$ $2.1 \pm 0.4$ $1.1 \pm 0.1$ $6.0 \pm 0.7$ $4.1_{-1.0}^{+0.9}$ $1.9 \pm 0.9$ $1.6_{-0.7}^{+0.6}$
A781 $4.1 \pm 0.8$ $2.0 \pm 0.5$ $3.6 \pm 0.6$ $2.9 \pm 0.4$ $1.1 \pm 0.1$ $6.3 \pm 0.5$ $4.5 \pm 0.6$ $2.3 \pm 0.6$ $2.2 \pm +0.5$
A990 $2.0_{-0.1}^{+0.4}$ $1.1 \pm 0.2$ $1.8 \pm 0.3$ $1.6 \pm 0.2$ $0.9 \pm 0.1$ $5.5 \pm 0.3$ $2.8 \pm 0.3$ $0.7 \pm 0.2$ $0.7 \pm 0.15$
A1413 $4.0 \pm 1.0$ $1.9 \pm 0.6$ $3.5 \pm 0.8$ $2.7 \pm 0.4$ $1.1 \pm 0.1$ $6.6 \pm 0.6$ $4.4 \pm 0.7$ $2.2_{-0.8}^{+0.7}$ $2.1 \pm 0.6$
A1423 $2.2 \pm 0.8$ $1.1 \pm 0.4$ $1.9 \pm 0.7$ $1.5 \pm 0.4$ $0.9 \pm 0.1$ $5.3 \pm 0.6$ $3.0_{-0.7}^{+0.8}$ $0.9 \pm 0.5$ $0.8 \pm 0.4$
A1758a $4.1_{-0.8}^{+0.7}$ $2.5 \pm 0.4$ $3.6 \pm 0.5$ $3.4 \pm 0.4$ $1.1 \pm 0.1$ $6.8 \pm 0.4$ $4.5 \pm 0.5$ $2.3 \pm 0.5$ $2.3 \pm 0.4$
A1758b $4.4 \pm 1.9$ $2.2 \pm 1.0$ $3.7_{-1.5}^{+1.6}$ $2.2 \pm 0.5$ $1.1 \pm 0.2$ $6.4_{-1.0}^{+1.1}$ $4.6 \pm 1.4$ $2.7 \pm 1.7$ $1.6 \pm 0.6$
A2009 $4.6 \pm 1.5$ $2.0_{-0.6}^{+0.2}$ $3.9 \pm 0.7$ $2.4_{-0.3}^{+0.2}$ $1.2 \pm 0.1$ $6.5_{-0.6}^{+0.4}$ $4.8_{-0.6}^{+0.4}$ $2.8_{-1.4}^{+1.3}$ $2.2 \pm 0.9$
A2111 $4.2 \pm 0.9$ $1.8 \pm 0.5$ $3.6 \pm 0.7$ $2.5 \pm 0.3$ $1.1 \pm 0.1$ $6.2 \pm 0.6$ $4.6 \pm 0.6$ $2.4_{-0.7}^{+0.6}$ $2.2 \pm 0.5$
A2146 $5.0 \pm 0.7$ $2.7 \pm 0.5$ $4.4 \pm 0.5$ $3.7 \pm 0.4$ $1.2 \pm 0.1$ $7.1 \pm 0.5$ $5.2 \pm 0.5$ $3.2 \pm 0.5$ $3.1 \pm 0.4$
A2218 $6.1 \pm 0.9$ $2.7 \pm 0.6$ $5.4_{0.7}^{+0.6}$ $4.3 \pm 0.4$ $1.3 \pm 0.1$ $7.3 \pm 0.5$ $5.9 \pm 0.6$ $4.5 \pm 0.7$ $4.4 \pm 0.7$
RXJ0142+2131 $3.7_{-1.2}^{+1.1}$ $1.7 \pm 0.6$ $3.1_{-1.0}^{+0.9}$ $2.1 \pm 0.4$ $1.0 \pm 0.1$ $5.9_{0.7}^{+0.8}$ $4.2 \pm 0.9$ $2.0_{-0.9}^{+0.8}$ $1.5_{-0.5}^{+0.6}$
RXJ1720+2638 $2.0 \pm 0.4$ $1.2 \pm 0.2$ $1.7 \pm 0.3$ $1.6 \pm 0.3$ $0.9 \pm 0.1$ $5.6 \pm 0.4$ $2.8 \pm 0.4$ $0.7 \pm 0.2$ $0.7 \pm 0.2$
§.§ Comparison with gNFW parameterizations
The adequacy of different profiles, such as the $\beta$, Navarro Frenk and White (NFW), generalized NFW (gNFW, 112) and other
hybrid profiles (e.g., 108, 4, 116)
is still very much under debate.
We attempt to illustrate the impact that the choice of some of these profiles may
have on the parameter estimates by comparing the results obtained from five gNFW parameterizations and from our $\beta$ parameterization (see Sec.
for two clusters: Abell 611 (see Sec. <ref>) and Abell 2111 (see Sec. <ref>).
For this analysis we sample from the cluster position parameters ($x_{\rm{c}}, y_{\rm{c}}$), $\theta_S=r_S/D_{A}$,
and $Y_{\theta}=Y_{\rm{tot}}/D^2_A$. $r_s$ is the scale radius, $D_{A}$ the angular diameter distance and $Y_{\rm{tot}}=Y_{\rm{sph}}(5r_{500})$,
where $Y_{\rm{sph}}$ is the integrated Compton $y$ parameter within $5r_{500}$ ([13] take $5r_{500}$
as the radius where the pressure profile flattens). Assuming a spherical geometry, $Y_{\rm{sph}}$ is calculated by integrating the plasma pressure within a spherical volume of radius $r$:
\begin{equation}
Y_{\rm{sph}}(r) = \frac{\sigma_{\rm{T}}}{m_{\rm{e}}c^2} \int^{r}_0 P_{\rm{e}}(r')4\pi r'^2 \rm{d}r',
\label{eq:ytotsph}
\end{equation}
where $\sigma_{\rm{T}}$ is the Thomson scattering cross-section, $m_{\rm{e}}$ is the electron mass, $c$ is the speed of light and $P_{\rm{e}}(r)$ is
the electron pressure at radius $r$.
The following priors were given for the sampling parameters: an exponential
prior between 1.3$\arcmin$ and 45$\arcmin$ for $\theta_S$ and a power law prior between
0.0005 and 0.2 arcmin$^2$ for $Y_{\rm{sph}}/D^2_A$, with a power law index of 1.6 . We note that for the purposes
of this exercise – to show the
different degeneracies for different, plausible sets of gNFW-profile
parameters – such wide priors are acceptable; naturally, where appropriate, the
prior ranges can be refined.
We choose to use a pressure profile for this parameterization since
has been shown to hold best for this quantity and pressure profiles
have low cluster-to-cluster scatter (e.g., 110). The gNFW
profile is given by
\begin{equation}
P_{\rm{e}}(r)= \frac{P_{\rm{e},i}}{(r/r_s)^{c}\left[ 1+ (r/r_s)^{a}
\right]^{\frac{b-c}{a}}}.
\label{eq:pgNFW}
\end{equation}
$P_{e,i}$, the overall normalisation coefficient of the pressure profile,
is calculated by computing Eq. (<ref>) for $5r_{500}$; once $P_{e,i}$ has been found, $y$ can be obtained.
The shape of the gNFW profile is governed by $c$ in the inner cluster
regions ($r<<r_s$), by $a$ at intermediate radii ($r\approx r_s$), and
by $b$ on the cluster outskirts ($r>r_s$). These parameters, together
with the concentration parameter, $c_{500}$, are fixed in most analyses to some
best-fit values (e.g., 108). With $c_{500}$, $r_s$ can be expressed in terms of $r_{500}$: $r_s =r_{500}/c_{500}$,
which is a common reparameterization (see e.g., 13 and 110).
We ran our analysis using a gNFW
profile with parameters defined by Nagai et al. as $\rm{gNFW}_N$, another
defined by Arnaud et al. as the `universal' profile $\rm{gNFW}_A$, and three other
combinations for the slope parameters and
$c_{500}$ that were found to provide the best fit for some clusters in Arnaud
et al.; $\rm{gNFW}_1$, $\rm{gNFW}_2$ and $\rm{gNFW}_3$[The parameters for the three other gNFW profiles all lie within
3$\sigma$ of the average value of
each parameter obtained using the cluster sample in Arnaud et al. .]. The gNFW
parameters for our five choices are given in Tab. <ref>.
Parameters for the gNFW pressure profile. The parameters for gNFW$_A$
and gNFW$_N$ have been taken from 13 and
106, respectively. The values in
106 are the corrected values for the results published by
Profile Label $a$ $b$ $c$ $c_{500}$
gNFW$_1$ 1.37 5.49 0.035 2.16
gNFW$_2$ 0.33 5.49 0.065 0.17
gNFW$_3$ 2.01 5.49 0.860 1.37
gNFW$_A$ 1.0620 5.4807 0.3292 1.156
gNFW$_N$ 0.9 5.0 0.4 1.3
The 2D-marginalized posterior distributions of $Y_{\rm{sph}}(r_{500})$
against $r_{500}$ obtained for each of the five parameterizations, as well as
the $\beta$ parameterization from Sec. <ref>,
for Abell 611 and Abell 2111 are shown in
Fig. <ref> .
We test for possible biases in our results from the
choice of priors by running the analysis without data; the results indicate the constraints imposed by our priors.
We find no evidence for significant biases, as shown in Fig. <ref>.
When the shape parameters of the $\beta$ profile are fitted to the SZ data instead of being set to the X-ray value (typically derived
for data sensitive to smaller scales than AMI data) we find the mean values for $Y_{\rm{sph}}(r_{500})$ and
$r_{500}$ derived from the $\beta$ analysis to be consistent (within 1-2$\sigma$) with those from $\rm{gNFW}_A$
and $\rm{gNFW}_{N}$ – the averaged gNFW profiles. For these two clusters we find all gNFW parameterizations
yield lower values for $Y_{\rm{sph}}(r_{500})$ than for the $\beta$ analysis; this is not the case for $r_{500}$, for which
no systematic difference is seen. The constraints on $Y_{\rm{sph}}(r_{500})$ are similar for most of the gNFW models (with
the exception of $\rm{gNFW}_2$) and the $\beta$ model, while those for $r_{500}$ appear to be tighter for the $\beta$ model. One striking
difference between the two types of parameterizations is the shape and orientation of the $Y_{\rm{sph}}-r_{500}$ degeneracy.
The resolution and limited spatial dynamic range of
the AMI data do not allow profile selection to be made robustly, as indicated by the small difference in evidence values between the different parameterizations (Tab. <ref>). Hence,
our $\beta$ parameterization provides a comparable fit to that of the commonly used, averaged gNFW profiles, $\rm{gNFW}_A$, and $\rm{gNFW}_N$.
It is clear from Fig. <ref> that the distribution for the
$Y_{\rm{sph}}(r_{500})-r_{500}$ degeneracies is very sensitive to the choice
for the slope parameters (and $c_{500}$ for gNFW). Cluster parameters for a cluster with a profile described by e.g., a $\rm{gNFW}_2$ recovered using a $\rm{gNFW}_A$
paramaterization will be biased.
2-D marginalized distributions for $Y_{\rm{sph}}(r_{500})$ against
$r_{500}$ obtained using the $\beta$-based cluster parameterization
and five gNFW-based cluster
parameterizations with slope parameters and $c_{500}$ given in
Tab. <ref>.
The crosses denote the McAdam-derived mean values.
The results are for Abell 611
(left) and Abell 2111 (right). The blue filled
show the results of the $\beta$ parameterization.
2-D marginalized distribution for $Y_{\rm{sph}}(r_{500})$ against
$r_{500}$ obtained using the $\beta$-based cluster parameterization
without any data.
Log$_e$ evidences for five cluster parameterizations applied to Abell 2111 and Abell 611
Abell 2111 Abell 611
Profile Label
gNFW$_1$ 23198.88 21114.08
gNFW$_2$ 23199.51 21114.40
gNFW$_3$ 23198.64 21114.74
gNFW$_A$ 23198.94 21114.50
gNFW$_N$ 23198.76 21114.65
$\beta$ 23194.92 21112.05
§.§ Abell 586
Results for Abell 586 are given in Figs. <ref> and <ref>.
This cluster has a complex source environment, with 7 sources within $5\arcmin$ from the cluster SZ centroid, which include two radio sources
of $\approx 260$ and $744$ $\mu$Jy at $0.5\arcmin$ and
$4\arcmin$ from the pointing centre.
After source-subtraction there are only $\approx 1\sigma$
residuals left on the map. Uncertainties in the source fluxes are carried
through into the posterior distributions for the cluster parameters. From
Fig. <ref>,
it can be seen that there is no strong degeneracy between the source flux
densities and the cluster mass.
Abell 586 has been studied extensively in the X-ray band (e.g., 2
and 156). A recent
analysis of the temperature profile [35] shows how the
temperature falls from $\approx9$ keV at the cluster centre to
$\approx 5.5$ keV at a radius $\approx 280\arcsec$. Cypriano et al. have used
the Gemini Multi-Object Spectrograph together with X-ray data taken from the
Chandra archive to measure the properties of Abell 586.
They compare mass estimates derived from the velocity distribution and from
the X-ray temperature profile and find that both give very similar results,
$M_{\rm{g}}\approx 0.48\times10^{14}M_{\odot}$ (for $h_{70}=1$) within 1.3$h_{70}^{-1}$ Mpc.
They suggest that the cluster is spherical and relaxed with no recent mergers.
It is less clear whether this cluster has a cool core or not, with [1]
reporting its existence and [91] saying otherwise.
The peak X-ray and SZ emissions are consistent with each other and the AMI
SZ decrement shows some signs of being extended towards the SW (Fig. <ref> B and C); there are no contaminating sources in the vicinity of this SZ-`tail'.
The SZ effect from Abell 586 has previously been observed with OVRO/BIMA by
[77] and [24]. LaRoque et al. apply an isothermal
$\beta$-model to SZ and Chandra X-ray observations
and find $M_{\rm{g}}(r_{2500})=2.49\pm{0.32}\times10^{13}M_{\odot}$
, respectively (using
$h_{70}=1$ and excising the inner 100 kpc from the X-ray data).
In addition, they determine an X-ray
spectroscopic temperature of the cluster gas of $\approx 6.35$ keV between a
radius of 100 kpc and $r_{2500}$. In comparison, [115] use
Subaru to calculate the cluster mass from weak lensing by applying
a Navarro, Frenk & White (NFW; 112) profile. They find
$M_{\rm{T}}(r_{2500})=2.41^{+0.45}_{-0.41} \times10^{14}M_{\odot}$
$M_{\rm{T}}(r_{500})=4.74^{+1.40}_{-1.14} \times10^{14}M_{\odot}$ (using
In this work, we find $M_{\rm{T}}(r_{500})=3.0\pm 1.3
\times10^{14}M_{\odot}$, where $r_{500}=0.94\pm 0.14$ Mpc and $h_{70}=1$. Note
that the fluxes of the radio sources $S_1$ and $S_5$ are degenerate in our
analysis of Abell 586 (see Fig. <ref>); this is because these sources
are separated by only 66$\arcsec$ and their individual fluxes cannot be
disentangled in the analysis of the AMI SA data.
Abell 586
A D
B E
Results for Abell 586. Panels A and B show the SA map before and after source subtraction, respectively; a 0.6 k$\lambda$ taper has been applied in B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. Panel C shows the smoothed Chandra X-ray map overlaid with contours from B. D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. The $y$-axis for the 1-d marginals is the probability density and for all the posterior distributions plots in this paper $h$ refers to $h_{100}$. In D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times 10^{14}M_{\rm{\odot}}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV. The slight rise in the distribution for $r_{200}$ at large $r$ is a result of a binning artifact and, in fact,
this distribution does tail off smoothly, as expected.
1 and 2-D marginalized posterior distributions for the flux densities, in Jys, of sources detected within $5\arcmin$ of the SZ centroid of Abell 586 (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$, in units of $h_{100}^{-1}M_{\odot}\times 10^{14}$.
§.§ Abell 611
Results for Abell 621 are presented in Fig. <ref>. Our methodology is able to model the radio sources + cluster enviroment well,
as demonstrated by the good constraints on the mass and other parameters and
the lack of degeneracies between the sources closest to the cluster and the
cluster mass (Fig. <ref> D, E and F). We do not expect any significant contamination from
radio sources nor from extended emission since GMRT observations by
[151] found no evidence for a radio halo associated with Abell 611
at 610 MHz.
The decrement on the source-subtracted maps appears to be circular, in
agreement with the X-ray surface brightness from the Chandra achive data
shown in Fig. <ref> C, which also appears to be
smooth and whose peak is close to the position of the brightest cluster
galaxy and the SZ peak. These facts might be taken to imply the cluster is relaxed but,
it does not seem to have a cool core [91]. Abell 611 has also previously been observed in the SZ at 15 GHz by
[54], AMI Consortium: Zwart et al. (2010) and [7], and at 30 GHz by [23], [24] and [77].
From the analysis in [36]
the cluster mass was estimated to be 9.32–11.11$\times 10^{14} \rm{M_{\odot}}$ (within a radius of 1.8$\pm$0.5 Mpc) by fitting different cluster models
to X-ray data and between 4.01–6.32$\times 10^{14} \rm{M_{\odot}}$ (within a radius of 1.5$\pm$0.2 Mpc) when fitting different models to the
lensing data; all estimates use $h_{70}=1$.
Several other analyses of Chandra
data produce comparable mass estimates (e.g., 138,
103, 104 and
[131] perform a weak-lensing analysis of Abell 611 using data from
the Large Binocular Telescope; with an NFW profile they estimate
$M_{\rm{T}}(r_{200})$ = 4–7$\times 10^{14} \rm{M_{\odot}}$ for $h_{70}=1$.
These are in agreement with the values
obtained from Subaru weak lensing observations by Okabe et al.. AMI Consortium: Hurley-Walker et al. estimate the total mass for this system within $r_{200}$.
Using lensing data they find it is $4.7\pm 1.2 \times 10^{14} h_{70}^{-1}M_{\odot}$ and using AMI SZ data they find it is $6.0\pm 1.9 \times 10^{14} h_{70}^{-1}M_{\odot}$.
We find $M_{\rm{T}}(r_{200})$ = $5.7\pm 1.1\times 10^{14}\rm{M_{\odot}}$, where
$r_{200}=1.6\pm 0.1$ and $h_{70}=1$; this value is significantly smaller than the result given
in AMI Consortium: Zwart et al. (2010); this is due to their mass measurements being biased
high, as they said, and is further discussed in [8].
A D
B E
C F
Results for Abell 611. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. C shows the smoothed Chandra X-ray map overlaid with contours from B. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities, in Jys, (within $5\arcmin$ of the cluster SZ centroid, see Tab. <ref>) and $M_{\rm{T}}(r_{200})$, in $h_{100}^{-1}\times 10^{14}M_{\odot}$. In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Abell 621
Fig. <ref> contains our results for Abell 621.
Out of the 13 radio sources detected on the LA raster for Abell 621, three lie
near the edge of the cluster decrement in the source-subtracted map and one, which has a flux
density $\approx 7$ $\rm{mJy}$, is coincident with the best-fit cluster
position, as indicated
by the box in Fig. <ref> A.
However, whatever reasonable source subtraction we try makes almost no difference to the inferred cluster mass.
The ROSAT HRI X-ray image presented in Fig. <ref> C appears to
quite uniform and circular and the offset between the X-ray and SZ cluster
centroids is small. We find the cluster mass to be $M_{\rm{T}}(r_{200})$ =
4.8$^{+1.7}_{-1.8}$$\times 10^{14}h_{100}^{-1}\rm{M_{\odot}}$ from our analysis;
at $6\sigma$, this is one of our less significant
The data for the probability distributions
in Fig. <ref> E have been binned relatively finely to avoid misleading
features, in particular towards the lower limits of our plots. As a result,
the noise in these bins is higher, which makes the distributions appear
less smooth. For some combinations of cluster parameters, there is nowhere in
cluster density estimation that the density of the gas
reaches $500\rho_{\rm{crit}}$. In these cases, where there is no physical
for $r_{500}$, we set $r_{500}=0$. This leads to
sharp, meaningless peaks at small radius in the distributions for some cluster parameters at $r_{500}$ (Fig.
<ref> E). These features have also been discussed in [12].
3cAbell 621
Results for Abell 621. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, the other symbols are in Tab. <ref>. The smoothed ROSAT HRI X-ray map overlaid with contours from B is given in panel C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) within $5\arcmin$ of the cluster SZ centroid (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ in $h_{100}^{-1}\times 10^{14}M_{\odot}$. In D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Abell 773
Results for Abell 773 are shown in Fig. <ref>.
Abell 773 has few associated radio sources, all of which are $\gtrsim
10\arcmin$ away from the pointing centre, weak ($\lesssim
3$ mJy), and are subtracted well from our data (Fig.<ref> B).
We do not find any evidence for extended positive emission in our maps. Observations by
[47] revealed
the presence of a radio halo with a luminosity of $2.8\times
10^{24}$ WHz$^{-1}$ at 1.4 GHz; this result has been confirmed with
the VLA by [49]. Given the typical steep spectral index of radio
halos, we do not expect our SZ signal to be affected at 16 GHz.
Our observations clearly show the SZ image is extended along the NW-SE direction, contrary to the
X-ray image from Chandra observations, which appears to be elongated in an approximately perpendicular
direction. As might be expected from a
disturbed system, Abell 773 appears to not have a cool core [1].
[15] present a comprehensive study of Abell 773 from
the Telescopio Nazionale Galileo (TNG) telescope
and X-ray data from the Chandra data archive. They
find two peaks in the velocity distribution of the cluster members which are
by 2$\arcmin$ along the E-W direction. Two peaks can also be seen in the
X-ray, although these are along the NE-SW direction. Barrena et al. estimate
the virial mass of the main cluster to be
$M_{\rm{T}}(r_{\rm{vir}})=1.0-2.5\times 10^{15}$ $h_{70}^{-1}\rm{M_{\odot}}$
$M_{\rm{T}}$ = 1.2-2.7$\times 10^{15}h_{70}^{-1}\rm{M_{\odot}}$ for the entire
system, using the virial theorem, dispersion velocity measurements and a galaxy King-like
distribution. Assuming an NFW profile they estimate the mass for the system to be
$M_{\rm{T}}(< r=1 h_{70}^{-1}\rm{Mpc})=5.9-11.1\times 10^{14} h_{70}^{-1}M_{\odot}$.
A further analysis of Chandra data by [50] yielded a mean
temperature of 7.5$\pm$0.8 keV within a radius of 800 kpc ($h_{70}=1$).
Another X-ray study of this cluster by [162] using XMM-Newton
found $M_{\rm{T}}(r_{500})$ = 8.3 $\pm$2.5 $\times 10^{14}\rm{M_{\odot}}$
assuming isothermality, spherical symmetry and
The SZ effect associated with Abell 773 has been observed several times
29, 137, 24, LaRoque et al. 2006). Most recently,
AMI Consortium: Zwart et al. (2010) observed the cluster and found a cluster mass of
$M_{\rm{T}}(r_{200})$ = 1.9$^{+0.3}_{-0.4}$$\times 10^{15}\rm{M_{\odot}}$
using $h_{100}=1$; however, their $M_{\rm{T}}$ estimates are biased high, as they say,
and we find $M_{\rm{T}}(r_{200}) = 3.6 \pm 1.2 \times 10^{14} h_{100} \rm{M_{\odot}}$.
Abell 773
A D
B E
Results for Abell 773. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is given in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
No plots of the degeneracy between cluster mass and source flux densities are shown
since all detected sources are $>5'$ from the cluster SZ centroid and thus should not have a strong impact on the marginalized distribution for the cluster mass.
§.§ Abell 781
Fig. <ref> contains our results for Abell 781.
It is evident from inspection of Figs. <ref> A and F and Tab. <ref> that
there is strong emission from radio sources lying on the decrement. One of the
sources with a flux density of $9$ mJy lies on top of the McAdam best-fit cluster position. The
difficulty of accurately disentangling the signal contributions from this source and the cluster is translated into a
degeneracy between the source's flux density and the cluster mass: Fig. <ref> F.
No extended emission was detected on the LA
maps and, after source subtraction, the residuals on the maps are $\lesssim
2\sigma$ (Fig. <ref> B). [134]
have found evidence in WENSS data at 327 MHz of diffuse emission from a radio galaxy and some other
unknown source
with a flux within a radius of 500 kpc of 40 mJy, while
[151] estimate diffuse emission at the centre to be $\approx
15-20$ mJy
using 325-MHz GMRT data. Assuming a typical steep spectral index
for radio halos, in the range of $1.2-1.4$ (e.g., 56), even as far as 16 GHz, we
would expect to find an $\approx 170$ $\mu$Jy signal around
the cluster and $\lesssim 85$ $\mu$Jy at the centre. The GMRT contour map in
Venturi et al. identifies the relic at a similar location to that of some
unsubtracted positive
emission in our maps at $\approx$ RA 09:30:00, Dec 30:28:00.
X-ray observations with Chandra and XMM-Newton
(139) imply that Abell 781 is a
complex cluster merger: the main cluster is surrounded by three smaller
clusters, two to the East of the main cluster and one to
the West. Sehgal et al. estimate the mass of Abell 781 within $r_{500}$
assuming a NFW matter density profile to be
$5.2^{+0.3}_{-0.7}$$\times$$10^{14}\rm{M_{\odot}}$ from X-ray data
and $2.7^{+1.0}_{-0.9}$$\times$$10^{14}\rm{M_{\odot}}$ from the Kitt Peak
Mayall 4-m telescope lensing observations, using $h_{71}=1$.
results from XMM-Newton by Zhang et al. yield
$M_{\rm{T}}(r_{500})$ = 4.5$\pm$1.3 $\times 10^{14}\rm{M_{\odot}}$
assuming isothermality, spherical symmetry and $h_{70}=1$. We obtain
$M_{\rm{T}}(r_{500})$ =
2.9$\pm$0.6$\times 10^{14}\rm{M_{\odot}}$ and
$M_{\rm{T}}(r_{200})$ = 5.9$\pm$1.1$\times 10^{14}\rm{M_{\odot}}$ for $h_{70}=1$.
A D
B E
C F
Results for Abell 781. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, other symbols are in Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is given in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) within $5\arcmin$ of the cluster SZ centroid (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Abell 990
Results for Abell 990 are given in Fig. <ref>. We detected 20 sources towards Abell 990. Those
detected above $4\sigma_{\rm{LA}}$ within $10\arcmin$ from the pointing
were found to have flux densities $<2.8$ mJy, not to be extended
with respect to the LA synthesized beam (Tab. <ref>),
and none to lie on the SZ decrement, as seen in the source-subtracted map (Fig.
<ref> B).
The subtraction
has worked well and there are only low-level ($\approx 1-2\sigma$) residuals. [134] do not detect any
significant amount of diffuse emission within a radius of 500 kpc in 327 MHz WENSS data; given the
steep falling spectrum associated with this emission, we do not expect it to
contaminate our SZ signal.
The imaged decrement is fairly circular but extended along the NE-SW direction
coincident with the distribution of the X-ray signal. Our spherical cluster
model provides a good fit and the parameter distributions are tightly
The low resolution X-ray map shown in Fig. <ref> C
provides tentative evidence that the X-ray emitting cluster gas
has a clumpy distribution.
3cAbell 990
Results for Abell 990. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed ROSAT HRI X-ray map overlaid with contours from B is shown in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities in Jy (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ (in $\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Abell 1413
In Fig. <ref> we present results for Abell 1413.
It can be seen from Figs. <ref> A and B
that there are two of sources on the decrement
with flux densities of $0.47$ and $3.1$ mJy (in Tab. <ref>). The brightest source in
our LA maps has a flux density of 14 mJy
but, since it is 700$''$ from the cluster X-ray centre, it does not contaminate
our SZ signal.
Some residual flux is seen on the source-subtracted SA maps; the strongest
are not associated with sources in the LA data,
suggesting they could be extended emission resolved out from the LA maps.
[51] find tentative ($\approx 3\sigma$)
evidence in FIRST data at 1.4 GHz for a weak mini halo with a luminosity of
$1.0\times 10^{23}$ W Hz$^{-1}$.
The peak signal from this mini halo is offset to the East with respect to the
central cD
galaxy, similarly to our SZ peak, which is slightly offset to the SE of the
X-ray centroid. Abell 1413 does seem to be a relaxed cluster; this is
supported by the
smooth X-ray distribution, the good agreement between the X-ray and SZ
centroids, the circular appearance of the projected SZ signal and the presence
of a cool core [1].
We therefore expect our model to provide a good fit to the AMI data towards this cluster.
Abell 1413 has been observed in the X-ray by XMM-Newton (e.g.,
123), Chandra
(e.g., 152 and 24) and most recently by the Suzaku
satellite [61]; SZ images have been made with
the Ryle Telescope at 15 GHz
(53) and with OVRO/BIMA at 30 GHz (LaRoque et al. and
24). These analyses
indicate that Abell 1413 seems indeed to be a relaxed cluster with no evidence of recent
merging. Different temperature and
density profiles obtained from X-ray data are in good agreement out
to half the virial radius. Hoshino et al. measure the variation of
temperature with radius, finding a
temperature of 7.5 keV near the centre and of 3.5 keV at $r_{200}$; they
assume spherical symmetry, an NFW density profile
and hydrostatic equilibrium to calculate $M_{\rm{T}}(r_{200})$ =
6.6$\pm$2.3$\times 10^{14}$ $h_{70}^{-1}\rm{M_{\odot}}$.
Zhang et al. use
XMM-Newton and find
$M_{\rm{T}}(r_{500})$ = 5.4 $\pm$1.6 $\times 10^{14}\rm{M_{\odot}}$
; they assume isothermality,
spherical symmetry and $h_{70}=1$.
We determine $M_{\rm{T}}(r_{200})$ to be $5.7\pm 1.4\times 10^{14}$ $\rm{M_{\odot}}$ for $h_{70}=1$.
Abell 1413
A D
B E
C F
Results for Abell 1413. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B in presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) within $5\arcmin$ of the cluster SZ centroid (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Abell 1423
Results for Abell 1423 are shown in Fig. <ref>.
The source environment for Abell 1423 is challenging (see Fig. <ref> A)
– 23 sources have been detected
within 10$\arcmin$ of the X-ray cluster centroid, of which 4 lie on the
decrement, as seen from the source-subtracted map. We find
no evidence for extended emission, in agreement with the lack of
diffuse emission towards this cluster at 327 MHz reported by
[134] and
the results in [132].
The sources closest to the cluster all have flux densities $<
1.3$ mJy (Tab. <ref>) and only small positive residuals remain after
source subtraction. As shown in Fig. <ref> F, the flux densities for
some of the sources close to the cluster centroid
manifest degeneracies with the cluster mass.
The details on the dynamics of Abell 1423 are largely unknown. The lack of
strong radio halo emission is indicative of a system without very significant
dynamical activity [28], as is the
good agreement between the X-ray and SZ emission peak positions.
On the other hand, the X-ray data in Fig. <ref> C shows signs of
substructure and our SZ image is be elongated along the SE–NW direction. [136] find that the
logarithmic gradient for the gas density profile of Abell 1423
at $0.04r_{500}$ is $\alpha \approx -0.98$ –
a key signature of cooling core clusters ([160] suggest
$\alpha<-0.7$ for strong cooling flows). In their study clusters with small
offsets at $r_{500}$ between the X-ray and the Brightest Cluster Galaxy (BCG)
are tightly correlated with large, negative spectral indices, an indication
that the strength of cooling cores tends to drop in more disturbed systems, but
Abell 1423 is an unsual outlier in this trend with a small offset
and a steep $\alpha$.
§.§ Abell 1704
Abell 1704 has been observed with ROSAT HRI and
PSPC [130]. These observations show a shift in
between the peak emission and the cluster centroid and distinct signs of
elongations in the gas distribution. Further analysis of X-ray observations
suggest the presence of a cooling flow [1].
[29] attempted to
detect an SZ effect using the OVRO array at 30 GHz towards this cluster but found no
convincing SZ signal.
The NVSS map at $1.4$ GHz shows complex, extended emission (Fig.
<ref>). These
features are detected on our SA maps but a significant portion of the
emission is resolved out on our LA maps (see Tab. <ref> for
more details on these sources).
Our model is not sophisticated enough to deal
properly with extended structure and significant residual emission can
be seen in the source-subtracted SA map, Fig. <ref>. Consequently, we
are not able
to convincingly detect an SZ effect towards Abell 1704.
3cAbell 1423
Results for Abell 1423. Panels A and B show the SA map before and after surce-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) within $5\arcmin$ of the cluster SZ centroid (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
1cAbell 1704
A: source-subtracted SA map produced using a
0.6-k$\lambda$ taper. The contours increase linearly in units of
$\sigma_{SA}$. B: 1.4-GHz NVSS map towards Abell 1704.
§.§ Abell 1758
Results for Abell1758a and b are given in Fig. <ref>-Fig <ref>.
It is clear from Fig. <ref> A, B and C that Abell 1758 is a complex system
comprising two gravitationally-bound main clusters,
Abell 1758a and Abell 1758b, separated by $8\arcmin$ (130 and
David & Kempner find no conclusive evidence for interaction between
these two main clusters, yet each of them is undergoing major mergers –
Abell 1758a between two 7-keV clusters and Abell 1758b between two 5-keV
clusters; since
both sets of mergers are between clusters of approximately equal mass, provided
each of the primary clusters was virialized pre-merger,
we might expect the average temperature to be higher by some $25\%$ when all the
gas mass of the subcluster has merged with that of the primary cluster.
To map the full extent of this system we took raster observations with the SA, which
are presented in Fig. <ref> B and C.
From <ref> Cii it can be seen that the SZ signal follows the X-ray
emission but there seems to be a hint of an SZ signal connecting these two clusters;
note that the clusters have identical redshifts. No connecting X-ray signal would be
expected and indeed none is seen.
A recent analysis of Spitzer/MIPS 24$\mu$m data by [55]
classifies Abell 1758 as the most
active system they have observed at that wavelength. They also identify
numerous smaller mass peaks and filamentary structures, which
are likely to indicate the presence of infalling galaxy groups, in support of
the David & Kempner observations.
For Abell 1758a we obtain
$M_{\rm{T}}(r_{500})$ = 2.5 $\pm 0.4\times 10^{14}h_{100}\rm{M_{\odot}}$
and $M_{\rm{T}}(r_{200})$ = $4.1^{0.7}_{0.8}\times 10^{14}h_{100}\rm{M_{\odot}}$.
Zhang et al. studied Abell 1758a using XMM-Newton and found
$M_{\rm{T}}(r_{500}) = 1.1\pm 0.3\times 10^{15}\rm{M_{\odot}}$
; they assumed isothermality, spherical symmetry and $h_{70}=1$.
Abell 1758a
A D
B E
C.i C.ii
Panels A. and B. show the SA map before and after source subtraction
(the latter map has had $0.6$-k$\lambda$ taper applied to it). The boxes in panels A and B indicates SZ centroid for each cluster, for the other symbols see Tab. <ref>.
The maps shown here are primary beam corrected signal-to-noise maps cut off at 0.3 of the primary beam. The noise
level is $\approx$ 115$\mu$Jy towards the upper cluster (Abell 1758a) and
$\approx$ 130$\mu$Jy towards the lower cluster (Abell 1758b).
The source-subtracted SA maps from B are overlaid with the the Chandra map in Ci and with ROSAT PSPC X-ray map in Cii.
D and E show the marginalized posterior distributions for sampling and derived parameters, respectively. In panel D, $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}M_{\odot}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
Abell 1758b
Abell 1758b. Left panel: 1 and 2-D marginalized posterior distributions for the cluster sampling parameters. $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. Right panel: 1-D marginalized
posterior distributions for the cluster derived parameters. $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
1 and 2-D marginalized posterior distributions for
$M_{\rm{T}}(r_{200})$ and sources within $5\arcmin$ from the cluster X-ray
centroid for Abell 1758a. Source flux densities are given in units of Jys and $M_{\rm{T}}(r_{200})$ in units of $h_{100}^{-1}M_{\odot}\times10^{14}$.
§.§ Abell 2009
Results for Abell 2009 are given in Fig. <ref>.
Eighteen sources were detected above $4\sigma_{\rm{LA}}$ in our LA maps. Given
that all of the sources, except one, are further away than one arcminute from
the pointing centre and have
flux densities $<2$ mJy, the source environment should not
significantly contaminate the SZ signal on the SA maps. The source-subtraction has worked well and there are only 2$\sigma$ residuals
(Fig. <ref>, B); the most
prominent residual is likely to be associated with some extended emission seen in
the SA map before source subtraction.
We find the SZ image is extended in an approximately NS direction.
[115] fit an NFW profile to weak lensing data from the
Subaru/Suprime-Cam and find
(with $h_{72}=1.0$).
We find $M_{\rm{T}}(r_{200})=4.6 \pm 1.5\times10^{14}h_{100}^{-1}\rm{M}_{\odot}$.
The misleading sharp peaks at small radius in the distributions for cluster parameters at
$r_{500}$ (Fig. <ref> F)
are discussed in Sec. <ref>.
3cAbell 2009
Results for Abell 2009. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) given in Tab. <ref> and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV
§.§ Abell 2111
Results for Abell 2111 are presented in Fig. <ref>.
The source environment in the vicinity of Abell 2111 does not present a problem
in our analysis:
all the sources are located on the edge of the decrement or beyond and have
flux densities $\lesssim 3$ mJy (Fig. <ref> A and Tab, <ref>).
Some residual flux with a peak surface brightness $\approx 700$ $\mu$Jy beam$^{-1}$
remains in our source-subtracted map but is sufficiently far ($\approx
45\arcsec$) that it has a negligible effect on our SZ detection (Fig. <ref> B).
X-ray studies of ROSAT PSPC and HRI data by [154]
reveal Abell 2111 has substructure on small scales but
appears to be reasonably relaxed on larger scales away from the core.
Wang et al. identify a main X-ray emitting
component and a hotter subcomponent and conclude that Abell 2111 is most likely to be a head-on
merger between two subclusters; this is supported by [57]
using ASCA data. A disturbed nature of Abell 2111
might also be indicated by the apparently clumpy X-ray emission and X-ray-SZ offset seen in Fig. <ref> C.
Recent investigations by [129] find the virial mass for Abell 2111
to be
$M_{\rm{T}}(r_{100})=4.01\pm0.41\times10^{14}M_{\odot}$ using $h_{70}=1.0$ from
average of 90 member redshifts within $r_{100}$. [94]
fit a modified version of the standard 1D isothermal $\beta$-model to Chandra data with $h_{70}=1.0$ to compute
obtain a value of $M_{g}(r_{500})= 2.5 \pm
Previously, [77] fitted an isothermal $\beta$-model to Chandra data (excising the
$r<100$ kpc
from the core) and OVRO/BIMA data and found a gas mass
$M_{\rm{g}}(r_{2500})=2.15\pm{0.42}\times10^{13}M_{\odot}$ (for
$h_{70}=1.0$); they also found an X-ray
spectroscopic temperature of $\approx8.2$ keV. On larger scales, at
$r_{200}$, we obtain a lower temperature, $4.6\pm 0.6$ keV, which suggests
the average cluster temperature falls with
radius. Moreover, Henriksen et al. report a radially decreasing
temperature structure for Abell 2111 and parameterize it by a polytropic
index $\gamma\approx1.45$. On larger scales [7] estimate $M_{\rm{T}}(r_{200})=6.9\pm 1.1\times10^{14}h_{70}^{-1}M_{\odot}$
from lensing data and $M_{\rm{T}}(r_{200})=6.3 \pm 2.1\times 10^{14}h_{70}^{-1}M_{\odot}$ from AMI SZ data; they also find that a circular geometry
is a slightly better fit to the data than an elliptical geometry. Our results,
$M_{\rm{T}}(r_{200})=4.2\pm 0.9\times10^{14}h_{100}^{-1}M_{\odot}$ are in very good agreement.
3cAbell 2111
Results for Abell 2111. Panels A and B show the SA map before and after source-subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the higher-resolution source-subtracted map (no taper). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV
2-D and 1-D marginalized posterior distributions for $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}M_{\odot}\times 10^{14}$) and source flux densities (in Jys) within $5\arcmin$ from the cluster X-ray centroid for Abell 2111.
§.§ Abell 2146
We have re-analysed the AMI data used in [9]
with the cluster parameterization described in Sec. <ref>,
which is slightly different to theirs; our results are presented in Fig.
They obtain $M_{\rm{g}}(r_{200})= 4.9 \pm
0.5 \times 10^{13} h_{100}^{-2}\rm{M}_\odot$
and $T=4.5 \pm 0.5$ keV while our results give
$M_{\rm{g}}(r_{200})= 4.4 \pm 0.6
\times 10^{13} h_{100}^{-2}\rm{M}_\odot$
and $T=5.2 \pm 0.5$ keV. Given the similarities between the
two analyses and the fact the same data were used for both, we would indeed expect this good
between these sets of results. We have further investigated the effect of
sources in this cluster
and have found a slight degeneracy between the cluster mass and the flux
density of the source lying closest to the cluster centre – see Fig.
<ref> F –
which had not been seen for the brighter, $\approx 6$ mJy beam$^{-1}$
source lying a few arcseconds away from the cluster centroid.
Chandra data analysed by [135] have revealed that Abell 2146
is undergoing a rare merger event
similar to that of “Bullet-cluster” [88], with two
shock fronts with Mach numbers $M\approx2$, and strong non-uniformities in the temperature
profile. Note the different – essentially 90$^{\circ}$ – orientaions between the X-ray and the SZ
extensions. To understand this we have to consider collision geometry, mass ratio and, especially, time
of snapshot since the merger start – see Sec. <ref>.
3cAbell 2146
Results for Abell 2146. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
2-D and 1-D marginalized posterior distributions for $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$) and source flux densities (in Jys; see Tab. <ref>) within $5\arcmin$ from the cluster X-ray centroid for Abell 2146.
§.§ Abell 2218
Results for Abell 2218 are shown in Fig. <ref>.
There is substantial radio emission towards Abell 2218, most of which is
subtracted from our maps to leave a $470$ $\mu$Jy beam$^{-1}$ positive feature to
the West of the decrement, which could be extended emission. Rudnick et al. detect diffuse emission from a radio halo with a flux of $0.05$ Jy within
a 500 kpc radius at 327 MHz, from which one might expect a $\leq
200$ $\mu$Jy signal at 16 GHz (for a typical halo spectral index, see e.g.,
Several observations in the X-ray (e.g., 87, 50, 83), optical (e.g., 48), SZ (e.g.,
64) and
lensing (e.g., 142 and 144) have suggested that Abell 2218 is
a complex,
disturbed system. High-resolution ROSAT [87] and
Chandra [50, 83] data
show signs of substructure, particularly on
small scales. Moreover, lensing studies by [142] and [144] have
revealed a bi-modal
mass distribution and associated
elongated structures in the mass distribution. Abell 2218 also shows signs of
strong temperature
variations (50 and 81). All of these results
are indicative that the cluster is not relaxed.
SZ observations towards Abell 2218 have been made with the Ryle Telescope
[64] at 15 GHz, at 36 GHz using the Nobeyama Telescope
[149]
and with OCRA-p at 30 GHz [75]. Earlier SZ
observations towards this cluster include [19],
[20], [80], [69], [63] and [21].
Pratt et al. find from XMM-Newton data
that $T(r)$ falls from 8 keV near the centre to 6.6 keV at
700 kpc. [162] calculate a cluster mass estimate from the
XMM-Newton data; using
$h_{70}=1.0$, they obtain
and $f_{\rm{g}}(r_{500})=0.15\pm0.09$. We find $M_{\rm{T}}(r_{500})=2.7 \pm
The Chandra X-ray image shown in Fig. <ref> C appears to be
along the N–S direction on arcminute scales and along the
$\approx$SE–NW direction on scales
$\approx2\arcmin$. On the other hand, the distribution of the X-ray signal on
scales, $\approx3\arcmin$, tends to be more circular.
On the untapered, source-subtracted SA map, Fig. <ref> F, the SZ
signal towards Abell 2218 is clearly extended.
There is no significant degeneracy between the cluster mass and the source flux densities, as seen from Fig. <ref>.
3cAbell 2218
Results for Abell 2218. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is presented in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV. F shows the higher-resolution source-subtracted map (no taper). Contours of the map in panel C are not the same as in panel B.; they range from -1.388 to
-0.188 mJy$\,$beam$^{-1}$ in steps of $+$0.15 mJy$\,$beam$^{-1}$.
2-D and 1-D marginalized posterior distributions for
$M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$) and source flux densities (in Jys) within $5\arcmin$ from the cluster X-ray
centroid for Abell 2218 (Tab. <ref>).
§.§ Abell 2409
We detect a $12\sigma_{SA}$ SZ effect towards Abell 2409 in the
tapered, source-subtracted SA maps,
Fig. <ref> B. Despite the high SNR we
are not able to obtain sensible parameter estimates for this
cluster. As shown in Fig. <ref> A, the effect of some
emission close to the
pointing centre is to give the decrement a shape that cannot be
well approximated by a spherical $\beta$-profile with free shape parameters. The
parameter estimates from McAdam are thus not reliable and we
present only the AMI SA map. Fixing the shape of the profile can improve the
fit to this cluster. Cluster parameters
for Abell 2409 from AMI data have been obtained using a
gNFW parameterization – see [121].
The nature of the residual emission around the cluster is
uncertain. Pointed LA observations towards the location of these
sources of positive flux were made in an attempt to detect possible
sources lying just below our detection threshold. Despite the noise at these
locations on the LA map
reaching $\approx 50\mu$Jy beam$^{-1}$, no additional sources were detected; it
seems likely that
this is (at this resolution) extended emission with relatively low surface brightness. However, no
evidence for extended
emission was found in either the NVSS 1.4 GHz or in the VLSS 74 MHz maps.
Abell 2409
Results for Abell 2409. A: Source-subtracted SA map produced using a 0.6-k$\lambda$
taper. B: SA map from A. overlaid onto Chandra X-ray image. Contours increase linearly in units of $\sigma_{SA}$.
§.§ RXJ0142+2131
The maps and parameters for RXJ0142+2131 are presented in Fig. <ref>. The source
is not expected to contaminate our SZ detection, with the brightest source
having a flux density
of $\approx 2$ mJy and lying several arcminutes away from the
cluster centroid; residual
emission after source subtraction is seen on the SA maps at the $1\sigma$
The composite image of the SZ and X-ray data
reveals good agreement between the emission peaks of these two datasets.
A photometric and spectroscopic study of RXJ0142+2131 by
[14] finds the velocity dispersion of this cluster
($\sigma_x=1278\pm134$ km s$^{-1}$) to be surprisingly large, given its
X-ray luminosity (Tab. <ref>). This study indicates that galaxies
in this cluster have older
luminosity-weighted mean ages than expected, which could be explained
by a short increase in the star formation rate, possibly due to a
cluster-cluster merger. Moreover, RXJ0142+2131 shows signs of not
being fully virialized since the brightest cluster galaxy was
found to be displaced by 1000 km s$^{-1}$ from the systemic
cluster redshift.
[115] fitted an NFW profile for the mass density to
Subaru/Suprime-Cam data and assumed a spherical geometry for the cluster
to derive
$M_{\rm{T}}(r_{200})=3.86^{+0.98}_{-0.82}\times10^{14}h_{72}^{-1}\rm{M}_{\odot}$ (using $h_{72}=1.0$).
From our analysis, we find $M_{\rm{T}}(r_{500})=1.7 \pm
0.6\times10^{14}h_{100}^{-1}\rm{M}_{\odot}$ and $M_{\rm{T}}(r_{200})=3.7^{+1.1}_{-1.2}\times10^{14}h_{100}^{-1}\rm{M}_{\odot}$.
Results for RXJ0142+2131. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is shown in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) within $5\arcmin$ of the cluster SZ centroid (see Tab. <ref>) and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ RXJ1720.1+2638
Results for RXJ1720.1+2638 are given in Fig. <ref>.
At 16 GHz the source environment around the cluster is challenging: in our LA
data we detect a 3.9 mJy source at the same position as the
cluster, and several other
sources with comparable flux densities within $4\arcmin$ from the cluster
centre. The difficulty of modelling this system is clear from the degeneracies between
some of the source flux densities and the cluster mass (Fig. <ref> F).
However, we always recover a similarly asymmetric SZ decrement.
RXJ1720.1+2638 has been studied by [95] and [97]
through Chandra observations. This cool-core cluster has two cold fronts
within $100\arcsec$ of the X-ray
centroid and a regular morphology away from the core region; the authors
attribute the dynamics of this cluster to the sloshing scenario, in agreement
with later work by [117] using optical spectroscopy.
Merger activity has also been suggested by Okabe et al. whose weak lensing
data reveal a second mass concentration to the North of the main cluster, while
the analysis of SDSS data by [101] finds no evidence of
substructure. Our data reveal a strong abundance of radio emission towards
this cluster
, including some extended emission, which might support the suggestion in
Mazzotta et al. (2001) that this cluster contains a low-frequency radio halo that did not
disappear after the merger event.
Mazzotta et al. 2001 determined the mass profile for the cluster assuming
hydrostatic equilibrium to be $M_{\rm{T}}(< r=1000\rm{kpc})= 5^{+3}_{-2}\times
10^{14}h^{-1}_{50}\rm{M_{\odot}}$. We find
$M_{\rm{T}}(r_{500}) = 1.2\pm 0.2\times 10^{14}h_{100}^{-1}\rm{M_{\odot}}$.
Results for RXJ1720.1+2638. Panels A and B show the SA map before and after source subtraction, respectively; a $0.6$ k$\lambda$ taper has been applied to B. The box in panels A and B indicates the cluster SZ centroid, for the other symbols see Tab. <ref>. The smoothed Chandra X-ray map overlaid with contours from B is shown in image C. Panels D and E show the marginalized posterior distributions for the cluster sampling and derived parameters, respectively. F shows the 1 and 2-D marginalized posterior distributions for source flux densities (in Jys) given in Tab. <ref> and $M_{\rm{T}}(r_{200})$ (in $h_{100}^{-1}\times 10^{14}M_{\odot}$). In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§.§ Zw0857.9+2107
Top panel: Source-subtracted SA map for Zw0857.9+2107 produced using a 0.6-k$\lambda$
taper. The contours increase linearly in units of
$\sigma_{SA}$. Middle panel: LA signal-to-noise map. Contours start at $3\sigma$ and increase linearly to $10\sigma$,
where $\sigma=97 mu$Jy Beam$^{-1}$.
Bottom panel: SA contours overlaid onto the Chandra
X-ray image. The SA contours are the same as in the upper
We report a null detection of an SZ
signal towards this cluster, despite the low noise levels on our SA maps and a
seemingly benign source environment. We reached a noise level ($1\sigma$) of
97$\mu$Jy beam$^{-1}$ on the LA map (Fig. <ref>, middle panel) and found no evidence
for sources below our $4\sigma_{LA}$ detection threshold.
We detect a $1.4$ mJy radio source at the location of the peak
signal (see the electronic version in the ACCEPT Chandra
data archive for a higher resolution X-ray image) but we seem to be able to
subtract it well from the SA maps.
Zw0857.9+2107 is not a well-studied cluster.
There are two temperature measurements for the cluster gas in
Zw0857.9+2107 from the ACCEPT Chandra data archive
[32]: $T\approx 3\pm4$ keV between
$\approx 10<r<100$ kpc and
$T\approx4.2\pm2.2$ keV between
$\approx 100<r<600$ kpc. One might expect the average temperature for
the cluster to be even lower at larger radii, such that $T(r_{200})<3$ keV.
The absence of an
SZ signal could be explained by a sharp radial drop in $T$ or, perhaps, this
cluster is particularly
dense and compact such that it is X-ray bright but does not produce a strong SZ
signal on the scales
AMI is sensitive to (Alastair Edge, private communication). Fig. <ref> illustrates what the marginalized
parameter distributions look like for non-detections such as this.
§.§ Zw1454.8+2233
We detect no SZ effect in the AMI data towards Zw1454.8+2233, despite the low
noise levels of our SA maps.
We detect several sources close to the cluster centre, including ones with a
flux density of 1.64 mJy, 1.55 mJy and
8.4 mJy (at 13$\arcsec$, $1.8\arcmin$ and $4.3\arcmin$
away from the pointing centre, respectively).
The SA maps and derived parameters are shown in Fig. <ref>. The
derived parameters for this non-detection are as expected: we find that
$M_{\rm{T}}(r_{200})$ approaches our lower prior limit and that $M_g$ shows similiar behaviour (see e.g., Fig. <ref>).
Zhang et al. found $M_{\rm{T}}(r_{500})= 2.4\pm
0.7\times 10^{14}\rm{M_{\odot}}$ using XMM-Newton, assuming
isothermality, spherical symmetry and $h_{70}=1$. Chandra
X-ray observations by [16] suggest the cluster has a cooling flow and
[150]
find from 610-MHz GMRT observations that the cluster
has a core-halo radio source.
A D
BE C
The null detection of Zw1454.8+2233 in SZ. Panel A
shows the SA map before subtraction, which reveals the challenging source environment towards this cluster.
The SA map after source subtraction is shown in panel B; no convincing SZ decrement
is visible. Image C shows the Chandra X-ray map overlaid with SA contours from panel B. Panels D and E show the distributions for the sampling and derived parameters respectively;
such distributions are consistent with a null detection. In panel D $M_{\rm{T}}$ is given in units of $h_{100}^{-1}\times10^{14}M_{\odot}$ and $f_{\rm{g}}$ in $h_{100}^{-1}$; both parameters are estimated within $r_{200}$. In E $M_{\rm{g}}$ is in units of $h_{100}^{-2}M_{\odot}$, $r$ in $h_{100}^{-1}$Mpc and $T$ in KeV.
§ SOURCE-SUBTRACTION SIMULATION
Extracting robust cluster parameters for a system like Abell 2146 with
bright sources lying at or very close to the cluster is extremely
challenging. Many factors can affect the reliability of
the detection and of the recovered parameters. Aside from model
assumptions, other important factors are:
the SNR of the decrement in our maps, the $uv$-coverage, the size of the
cluster and the distance of the sources from the cluster, their
flux-densities and their morphologies. From Sec. <ref>,
one can appreciate
that at 16 GHz the SZ signal is potentially strongly contaminated by
radio sources.
We have examined some of the effects of these sources in a controlled
environment through
simulations. For this purpose, we generated mock visibilities between hour
angles -4.0 to 4.0,
with an RMS noise per channel per baseline per second of 0.54 Jy.
Noise contributions from a CMB realisation
and from confusion from faint sources lying below our subtraction limit were included;
for the former we used a $\Lambda$CDM
model and for the latter we
integrated the 10C LA source counts from 10$\mu$Jy to 300$\mu$Jy. A cluster at
$z=0.23$ was simulated
using an isothermal $\beta$-profile to model the
gas distribution, with a central electron density of
$9\times 10^{3}$ m$^{-3}$, $\beta=1.85$, $r_{\rm{c}}=440h_{70}^{-1}$ kpc and
$T=4.8$ keV. Integrating the density profile out to $r_{200}$ (Eq.
<ref>) assuming a spherical cluster geometry yields
10^{13}h_{70}^{-2}$M$_{\odot}$. From the virial $M-T$ relation given in
Eq. <ref> $M_{\rm{T}}(r_{200})=5.70\times
10^{14}h_{70}^{-1}$M$_{\odot}$ and using these two estimates and
Eq. <ref>, we find $f_{\rm{g}}(r_{200})=0.11h^{-1}_{70}$.
Three point sources were included into the simulation. Their positions,
flux-densities and spectral indices are given in Tab. <ref>.
Source parameters for the three simulated sources.
Source RA (h m s) Dec ($^{o}$ $^{'}$ $^{''}$) $S_{16}$ (Jy)
Spectral index
1 15 56 04.23 66 22 12.94 5.92 0.6
2 15 56 14.30 66 20 53.45 1.83 0.1
3 15 55 57.42 66 20 03.11 1.65 -0.2
The map of the data is shown in Fig. <ref>.
SA contour map of simulated data containing thermal
noise + confusion noise + CMB + cluster + resolved point sources. Contours increase linearly in units of $\sigma_{\rm{SA}}$.
The data for the simulation were run through the same analysis as described in
Sec. <ref>. In this case the source
priors were centred on the simulated values (Tab. <ref>) and the
cluster priors were the same of those in
Tab. <ref>, with the delta-prior on $z$ set to 0.23.
The 1-D and 2-D marginalized posterior distributions
for the sampling parameters are presented in Fig. <ref>.
It can be seen that the cluster position and gas fraction are
recovered well by the sampler;
the core radius and $\beta$ cannot be constrained by AMI data alone, thus,
as expected, the agreement between the input and output mean
values for these paramaters is poor; the total cluster mass, on
the other hand, is very well-constrained and the
recovered value is consistent with the input value. Hence, despite the
challenging source environment, and the degeneracies between the cluster mass
and the source flux densities,
our analysis is able to provide robust cluster mass estimates.
Left: One and two-dimensional marginalized posterior distributions for
the cluster sampling parameters from our simulation. $M_{\rm{T}}$ and $f_{\rm{g}}$ are estimated within $r_{200}$ and $M_{\rm{T}}$ is given in units of $\times 10^{14}$. The green crosses
in the 2-D marginals denote the mean of the distribution.
Right: One and two-dimensional marginalized posterior distributions for the
source flux densities and $M_{\rm{T}}(r_{200})$ for our simulation. Red lines
indicate the mean of the marginalized
distribution and the blue lines represent the input value.
§ DISCUSSION
Of the 20 target clusters, we have detected SZ towards 17, all of which are resolved,
and with “peak” detections between 5$\sigma_{SA}$ and 23$\sigma_{SA}$.
The analysis
has produced robust parameter extraction for 16 of the 17 – this was not possible for Abell 2409
because of nearby extended radio emission that distorts the SZ signal and gives an unacceptable
fit for a spherical $\beta$-model. The three null detections are of Abell 1704 (difficult source environment),
Zw0857.9+2107 (it is unclear to us why we have not detected this), and Zw1458.8+2233 (difficult source environment).
§.§ Cluster morphology and dynamics
The images frequently show significant differences in position of the SZ peak (and of the SZ centroid)
and the X-ray peak, indicating that the densest part of a cluster is not at the centre
of the large-scale gas distribution.
In Abell 773 and Abell 2146, both mergers, there is evidence of SZ extension
perpendicular to the X-ray emission. Abell 1758a and Abell A1758b are both
major mergers and there is a hint of an SZ signal between a & b.
Unlike what one might naively expect, there are cases of
SZ extensions in non-mergers and cases of near-circular SZ map structures
in mergers.
To attempt to quantify the cluster morphology from the AMI data, we ran our analysis with
an ellipsoidal model for the cluster geometry. This model simply fits for two additional parameters: an ellipticity parameter, $\eta$,
which is the ratio between the semi major and semi minor axes and an angle $\theta$ measured anticlockwise from the West; these values
are given in Tab. <ref>. For further details on this model see e.g., [7].
As a check that switching from spherical to ellipsoidal SZ analysis does not itself introduce significant bias in mass, we give in Tab. <ref> the
ratios $M_{SZ, sph}/M_{SZ, ellip}$ within $r_{200}$ and $r_{500}$ and $T_{AMI, sph}/T_{AMI, ellip}$: no significant bias is evident. Of course,
elsewhere in this paper we use spherical SZ estimates because the X-ray and almost all the optical total cluster mass estimates also
assume spherical symmetry.
Median, mean and standard deviation for $M_{SZ, sph}/M_{SZ, ellip}$ within $r_{200}$ and $r_{500}$ and $T_{AMI, sph}/T_{AMI, ellip}$. Data
for all clusters in Tab. <ref> were included, except for Abell 1758a and b. Ratios for each cluster at these two overdensities are given in Tab. <ref>.
Median Mean Standard deviation
$M_{SZ, sph}/M_{SZ, ellip}$ within $r_{500}$ 0.96 0.96 0.16
$M_{SZ, sph}/M_{SZ, ellip}$ within $r_{200}$ 0.97 0.99 0.16
$T_{AMI, sph}/T_{AMI, ellip}$ 0.98 0.98 0.10
Tab. <ref> also includes other possible indicators of dynamical state.
The presence of cooling cores (CC) is associated with relaxed clusters since it is widely accepted that merger events tend to disrupt cooling flows, e.g., [98].
We have used Chandra data from the ACCEPT database, where available, to compute three CC indicators described in [62]: the central entropy,
the central cooling time and the ratio of (approximately) the central cluster temperature to the virial temperature; Tab. <ref> also includes other assessments of dynamical
state that we have found in the literature.
The projected separation of the brightest cluster galaxy (BCG) and the peak of the X-ray emission has been shown to correlate
with the dynamical equilibrium state of the the host cluster (67 and 136). Similarly, the offset between the SZ centroid and the X-ray peak
can also be a diagnostic for cluster disturbance. For this purpose, the separation between the AMI SZ centroid, X-ray peak cluster position and the position of the BCG are
given in Tab. <ref>; in Tab. <ref> some sample statistics are provided. Large offsets between these measurements have been reported in observations
(e.g., 93, 99, and 71) and in simulations (e.g., [102]).
Examination of Tabs. <ref>-<ref> indicates that even
for well-studied clusters there are conflicting indications as to whether the cluster is a merger or not, e.g., Abell 773 does not
appear to have a CC, has high degree of ellipticity, the X-ray and SZ signals appear to be oriented quasi-perpendicularly to
each other and yet the relatively small position offsets in Tab. <ref>
might suggest the cluster is relaxed.
Dynamical indicators: $\theta$, the angle measured anticlockwise from the West, $\eta$ the ratio
between the semi major and semi minor axes (these values arise from fitting the SZ data with an elliptical geometry (see text)).
Cooling core information: CC denotes the presence of a cooling core and NCC the lack of ($`-'$ means this information is not clear or not known);
$\rm{Core}_1$ is a result from this study obtained by using three CC indicators described in Hudson et al. 2010 – the central entropy,
the central cooling time and the ratio of approximately the central cluster temperature to the virial temperature, $T_0/T_{\rm{vir}}$, where all
the data have been taken from the Chandra ACCEPT database; Core$_2$ is cooling core information on the cluster available from other studies.
$\dag$: the core type of Abell 1423 is unclear; the ratio of $T_0/T_{\rm{vir}}$ taken from ACCEPT suggests it is not a cool-core cluster but, a CC
cannot be ruled out due to the large uncertainty in the X-ray temperature measurements; the central entropy and cooling time are unclear.
Cluster Name $\theta$ $\eta$ Core$_1$ Core$_2$
Abell 586 136 $\pm$ 33 0.73 $\pm$ 0.13 NCC CC [2], NCC [91]
Abell 611 79 $\pm$ 39 0.80 $\pm$ 0.12 NCC NCC [91]
Abell 621 64 $\pm$ 61 0.73 $\pm$ 0.13 - -
Abell 773 41 $\pm$ 12 0.59 $\pm$ 0.10 NCC NCC [2]
Abell 781 132 $\pm$ 32 0.70 $\pm$ 0.13 - -
Abell 990 109 $\pm$ 40 0.78 $\pm$ 0.13 - -
Abell 1413 101 $\pm$ 21 0.75 $\pm$ 0.12 CC CC [2], [127]
Abell 1423($\dag$) 66 $\pm$ 28 0.70 $\pm$ 0.14 NCC? $\&$ CC? [136]
Abell 1704 - - - CC [2]
Abell 1758a 72 $\pm$ 31 0.73 $\pm$ 0.14 NCC -
Abell 1758b 85 $\pm$ 49 0.77 $\pm$ 0.13 - -
Abell 2009 88 $\pm$ 49 0.78 $\pm$ 0.12 - -
Abell 2111 90 $\pm$ 29 0.77 $\pm$ 0.12 NCC -
Abell 2146 126 $\pm$ 4 0.56 $\pm$ 0.05 - `Bullet-like merger' [135]
Abell 2218 107 $\pm$ 80 0.87 $\pm$ 0.07 NCC NCC [127]
Abell 2409 - - - -
RXJ0142+2131 87 $\pm$ 41 0.77 $\pm$ 0.13 - -
RXJ1720.1+263 26 $\pm$ 12 0.58 $\pm$ 0.07 CC CC [127]
Zw0857.9+2107 - - - -
Zw1454.8+2233 - - CC [16]
X-ray cluster position; SZ centroids from our analysis; position of the BCG from SDSS (the BCG was identified as the brightest galaxy in the central few hundred kpc from the cluster X-ray position. For clusters labelled with (*) the BCG could not be identified unambiguously.
Entries filled with a '-' indicate there is no available information.
Cluster Name BCG X-ray SZ Position offsets ($\arcsec$)
RA (Deg) Dec (Deg) RA (Deg) Dec (Deg) RA (Deg) Dec (Deg) SZ-X-ray SZ-SDSS X-ray-SDSS
Abell 586 113.0844 31.6334 113.0833 31.6328 113.0833 31.6264 23.0 25.7 4.5
Abell 611 120.2367 36.0563 120.2458 36.0503 120.7958 36.0531 10.1 34.9 39.4
Abell 621 - - 122.8000 70.0408 122.7875 70.0458 48.5 - -
Abell 773 139.4724 51.7270 139.4666 51.7319 139.4667 51.7331 4.0 29.9 27.3
Abell 781(*) 140.1073 30.4941 140.1083 30.5147 140.1000 30.5314 67.0 136.7 74.2
Abell 990 155.9161 49.1438 155.9208 49.1439 155.9125 49.1369 39.0 27.9 16.9
Abell 1413 178.8250 23.4050 178.8250 23.4078 178.8250 23.3894 66.0 55.8 10.2
Abell 1423 179.3222 33.6110 179.3416 33.6319 179.3375 33.6189 49.2 62.2 103.0
Abell 1704 198.6025 64.5753 198.5917 64.5750 - - - - 39.0
Abell 1758a(*) 203.1189 50.4697 203.1500 50.4806 179.3375 50.5264 209.3 209.3 118.6
Abell 1758b - - - - 203.1250 50.4003 209.3 209.3 -
Abell 2009 225.0833 21.3678 225.0811 21.3692 225.0875 21.3553 55.2 47.5 9.5
Abell 2111 234.9333 34.4156 234.9187 34.4240 234.9125 34.4331 39.4 97.9 60.8
Abell 2146 - - 239.0291 66.3597 239.0250 66.3589 15.1 - -
Abell 2218 - - 248.9666 66.2139 248.9375 66.2186 106.1 - -
Abell 2409 330.2189 20.9683 330.2208 20.9606 - - - - 28.5
RXJ0142+2131 - - 25.51250 21.5219 25.51667 21.5303 33.6 - -
RXJ1720.1+2638 260.0418 26.6256 260.0416 26.6250 260.0333 26.6125 54.0 56.0 2.1
Zw0857.9+2107 135.1536 20.8943 135.1583 20.9158 - - - - 79.5
Zw1454.8+2233 224.3130 22.3428 224.3125 22.3417 - - - - 4.3
Mean, standard deviation and median for the differences in X-ray and SZ cluster centroids and the position of the BCG from SDSS maps.
Abell 1758 (a and b) has been excluded from this analysis due to its exceptionally disturbed state.
Mean ($\arcsec$) Standard Deviation ($\arcsec$) Median ($\arcsec$)
SZ - Xray 43.9 26.8 43.6
SZ - SDSS 51.6 35.3 43.6
Xray - SDSS 27.9 32.2 35.7
§.§ SZ temperature, large-radius X-ray temperature, and dynamics
In Fig. <ref> we compare the AMI SA observed cluster temperatures within $r_{200}$ ($T_{\rm{AMI}}$) with large-radius X-ray values ($T_{X}$) from Chandra or Suzaku that we have been able to find in the literature. We use large-radius ($\approx$ 500 kpc) X-ray temperature values to be consistent with the angular scales measured by AMI. (For Abell 611 we have plotted two X-ray values from Chandra data – one from the ACCEPT archive (32), which is higher than our AMI SA measurement, and a second X-ray measurement from Chandra (Donnarumma et al.), which is consistent with our measurement).
There is reasonable correspondence between SZ and X-ray temperatures at lower X-ray luminosity,
with excess (over SZ) X-ray temperatures at higher X-ray luminosity. The mean, median and standard deviation for the
ratio of $T_{\rm{AMI}}/T_X$ were found to be 0.7, 0.8 and 0.2, respectively, when considering all the cluster in Fig. <ref> )except for
Abell 1758a, due to it being a complex double-merger). The numbers are obviously small, but the two
systems that are strong mergers by clear historical consensus – Abell 773 and Abell 1758a – are unambiguously clear outliers with much
higher large-radius X-ray temperatures than SZ temperatures.
[144] investigate the
scatter between lensing
masses within $\leq 500$ kpc with Chandra X-ray temperatures
averaged over $0.1-2$ Mpc for ten clusters and also find that
disturbed systems have higher temperatures. However, [90] measure
the relationship between SZ-$Y_{\rm{sph}}$ and lensing masses within
$350$ kpc for 14 clusters and find no segregation between disturbed
and relaxed systems. [74] analysed a cluster sample extracted from cosmological simulations
and noticed that X-ray temperatures of disturbed clusters were biased high, while the X-ray analogue of SZ-$Y_{\rm{sph}}$,
did not depend strongly on cluster structure.
Taken together, these results suggest
that, even at small distances from the core, SZ-based mass (or temperature) is
a less sensitive indicator of disturbance disturbance than is X-ray-based mass.
Major mergers in our sample have large-radius X-ray temperatures (at
$\approx 500$ kpc) higher than the SZ temperatures (averaged over the whole cluster).
This suggests that the mergers affect the
$n^2$-weighted X-ray temperatures more than the $n$-weighted SZ temperatures
and do so out to large radius. This is evidence for shocking or clumping or both
at large radius in mergers. Indications that clumping at large $r$ might have
a significant impact on X-ray results have been found by e.g., [68], who find
a flattening of the entropy profile around the virial radius, contrary to the theoretical predictions (e.g., 161).
Hydrodynamical simulations by [111] have shown that gas clumping
can indeed introduce a large bias in large-$r$ X-ray measurements and could help explain the results by e.g., Kawaharada et al.
It should be noted, however, that [96] expect X-ray temperatures to be lower than mass-weighted temperatures
for clusters with temperature structure since the detectors of Chandra and XMM-Newton are more efficient on the soft bands, which leads to an upweighting of the cold gas.
However, in simulations by [126] mass-weighted temperatures were shown to be larger than X-ray temperatures
for the vast majority of their clusters, particularly for very the most disturbed clusters in their sample.
Examination of Fig. <ref> given Tab. <ref> is suggestive of another relation, again with obviously small numbers. Fig. <ref>,
shows AMI cluster temperature versus large-scale X-ray temperature but with each cluster X-ray luminosity replaced by AMI ellipicity, $\eta$, and its error (note that
we have removed the two Abell 611 points because of their apparently discrepant X-ray values). With one exception (Abell 2146), the clusters with
large-radius X-ray temperature $\geq 6$ keV have $\eta$ values $\leq 0.70$, whereas the first two outliers to the right (RXJ1720.1+2638 and Abell 773)
have significantly smaller values of AMI ellipticity. The rightmost outlier (Abell 1758a) itself has the ellipticity value $0.73\pm 0.14$ but this will be
misleadingly high if we should instead be considering the ellipticity of the Abell 1758a+b taken as a merging pair. The true relationship
between SZ ellipticity and merger state is bound to be influenced by the collision geometry, the time since the start of the merger (Fig. 1 in 113 illustrates
how SZ $\eta$ and $\theta$ can vary with merger evolution), the mass ratio, and so on. Far more data, including data on clusters not selected by X-ray luminosity, are essential.
The AMI mean temperature within $r_{200}$ versus the X-ray temperature. Each point is labelled with the cluster name and X-ray luminosity. Most of the X-ray measurements are large-radius temperatures from the ACCEPT archive (32) with 90% confidence bars. The radius of the measurements taken from the ACCEPT archive are 400-600 kpc for Abell 586, 300-700 kpc for Abell 611, 300-600 kpc for Abell 773, 450-700 kpc for Abell 1423, 500-1000 kpc for Abell 2111, 450-550 for Abell 2218 and for RXJ1720.1+2638 r = 550-700 kpc. The Abell 611* temperature is the 450-750 kpc value with 68$\%$ confidence bars (36). The Abell 2146 temperature measurement is from Russell et al. 2010 (with 68$\%$ confidence bars). The Abell 1413 X-ray temperature is estimated from the 700-1200 kpc measurements made with the Suzaku satellite (61), this value is consistent with 152 and 145. The ACCEPT archive temperature for Abell 1758A is 16$\pm$7 keV at r= 475-550 kpc, and with SZ temperature 4.5$\pm$0.5, is off the right-hand edge of this plot. Abell 611 has been plotted using dashed blue lines to emphasize that this cluster has two X-ray-derived large-$r$ temperatures. The black diagonal solid line is the 1:1 line.
Plot analogous to Fig. <ref> but with the X-ray luminosity values replaced by SZ $\eta$ (ellipticity).
§.§ Comparison of masses within $r_{500}$ and within the virial radius ($\approx r_{200}$)
The classical virial radius, $\approx r_{200}$, found is typically 1.2$\pm$0.1 Mpc. Values for $M_{\rm{T}}(r_{200})$ range from
$2.0^{+0.4}_{-0.1} \times 10^{14}h^{-1}_{70}M_{\odot}$ to $6.1 \pm 0.9 \times 10^{14}h^{-1}_{70}M_{\odot}$
and are typically 2.0-2.5$\times$ larger than $M_{\rm{T}}(r_{500})$. In Fig. <ref> and <ref> AMI
mass estimates at two overdensity radii are compared with other published mass estimates. The scarcity of mass measurements at large $r$ is apparent from these figures.
* For $M_{\rm{T}}(r_{500})$, there is good agreement between optical and AMI (HSE) mass estimates. In contrast, the X-ray (HSE) estimates tend to be higher, sometimes substantially so.
* For $M_{\rm{T}}(r_{200})$, there is very good agreement between optical estimates, the Suzaku X-ray (HSE) estimate, and the AMI ($M-T$) estimates.
Good agreement between AMI and optical masses has previously been reported by [7].
* From our sample we cannot determine whether the disagreement of masses is a function of radius.
* The discrepancy between the X-ray and AMI masses for Abell 1413 is reduced at $r_{200}$, with the X-ray mass being larger than the AMI mass by $\approx 50\%$ at $r_{500}$ and smaller than the AMI mass by $\approx 10\%$ at $r_{200}$.
* The largest discrepancies between mass measurements in SZ, optical and X-ray correspond to the strongest mergers within our sample but the
X-ray masses are always higher than our SZ masses, even for the few relaxed clusters in our sample. Given that the lensing masses agree well with our SZ estimates,
this might be an indication of a stronger bias in masses estimated from X-ray data than from SZ or lensing data, especially for disturbed systems. However, most recent
simulations and analyses indicate that X-ray HSE masses are underestimated with respect to lensing masses (e.g., 110, 100, 126).
§.§.§ Related results from the literature
To illustrate some of the issues in mass estimation, we bring together some of the other results in the literature.
* X-ray and weak-lensing masses
Observational studies by [84], [162] and [163] find systematic differences between X-ray HSE-derived and weak-lensing masses, with the lensing masses
typically exceeding the X-ray masses. Madhavi et al. report a strong radial dependence for this difference, with weak-lensing masses being $\approx 3\%$ smaller within $r_{2500}$ but
$\approx 20\%$ larger within $r_{500}$ than the X-ray masses, yet find no correlation between the difference level and the presence of cool cores. Zhang et al. (2010) find that X-ray masses seem
underestimated by $\approx 10\%$ for undisturbed systems and overestimated by $\approx 6\%$ for disturbed clusters within $r_{500}$. For relaxed clusters, they find the discrepancy is reduced at
larger overdensities.
The underestimate of HSE X-ray masses with respect to lensing masses has been widely produced in simulations (e.g., 110, 100)
and 126.
In Tab. <ref> we follow Mahdavi et al. to calculate a weighted best-fit ratio of two mass estimates at different overdensities for different data. The simulations by Rasia et al.
and Meneghetti et al. yield significantly lower $M_X/M_{WL}$ at $r_{500}$ than the observational data. [141] suggest that a higher incidence of temperature substructure in the
simulations might be responsible for this effect. It is interesting to see how the mass agreement for the study by Zhang et al. seems to weaken when excluding disturbed systems.
What is very different from the literature is that we find HSE X-ray masses to be consistently higher than our HSE SZ masses within $r_{500}$.
Modelling our clusters with an elliptical model for the cluster geometry does not substantially improve the agreement.
* SZ Y with X-ray and lensing masses
[25] find good agreement between $Y_{\rm{sph}}(r_{500})$ estimated from a joint SZ and X-ray analysis and from SZ data alone, in support of results by [120].
For their sample of massive, relaxed clusters there appears to be no significant systematics affecting the ICM pressure measurements from X-ray or SZ data. But, of course, this result
might not be reproduced for a sample of disturbed clusters.
[90] measure the scaling between $Y_{\rm{SZ}}$ and weak lensing mass measurements within 350 kpc ($\approx r_{4000-8000}$) for 14 LoCuSS clusters. They find it
behaves consistently with the self-similar predictions, has considerably less scatter than the relation between lensing mass and $T_X$
and does not depend strongly on the dynamical state of the cluster. They suggest SZ parameters derived from observations near the
cluster cores may be less sensitive to the complicated physics of these regions than those in X-ray. A later study by [91]
comparing two $Y_{\rm{SZ}}-M$ scaling relations using weak-lensing masses and X-ray (HSE) masses at $r_{2500}, r_{1000}$ and $r_{500}$ indicates the latter has more scatter and is more sensitive to cluster
morphology, with the mass estimates of undisturbed clusters exceeding those of disturbed clusters at fixed $Y_{\rm{sph}}$ by $\approx 40\%$ at large overdensities.
However, this division is not predicted by comparing SZ and true masses from simulations and is
could due to the use of a simple spherical lens model. Moreover, recently, [126] have shown through simulations that selecting relaxed clusters for weak-lensing studies
based on X-ray morphology is not optimal since there can be mass from, e.g., filaments not associated to X-ray counterparts biasing the lensing mass estimates even for
systems which appear to be regular in X-rays.
* Simulations
Simulations of cluster mergers have shown these events generate turbulence, bulk flows and complex temperature structure, all of which can
result in cluster mass biases (e.g., 122). Predominantly, simulations indicate that X-ray HSE masses tend to be underestimated (e.g., 73)
particularly in disturbed clusters, though the amount of the bias varies depending on the the simulation details, particularly on the physical processes taken into consideration.
Projections effects, model assumptions and the dynamical state of the cluster are some of the factors affecting how well the true cluster mass can be measured.
As shown by e.g., [147], even mass estimates for spherical X-ray systems are not always recovered well. Recent simulations by
[113] have investigated in detail the evolution of the non-thermal support bias as function of radius and of the merger stage.
They reveal a very complex picture: the
HSE bias appears to vary in amplitude and direction radially and as the merger evolves
(and the shocks propagate through); for the most part, the HSE bias leads to an underestimate for the mass, there
are times when it has the opposite effect.
From simulations there appear to be two main, competing effects that can lead to a mass bias from the effects of a merger.
Firstly, the merger event can boost the X-ray luminosity and temperature (e.g., 128) such that if the cluster is observed during this period its X-ray mass will be overestimated.
Secondly, the increase in non-thermal pressure support during the merger can
lead to X-ray (HSE) cluster masses being underestimated (e.g., 124).
The cluster sample derived from simulations studied by [74]
showed that the X-ray temperatures were biased high for disturbed clusters, unlike $Y_X$, the product of the gas mass and temperature as deduced from X-ray observations
(the X-ray analogue of the SZ $Y$) which did not appear to depend strongly on cluster structure.
Best-fit mass ratios calculated following Mahdavi et al. 2008. R12 are the results from simulations by Rasia et al. 2012, ME10 are the simulations from Meneghetti et al. 2010, Z10 from Zhang et al. 2010 and MA10 from Mahdavi et al 2008. For our results we have used for simplicity sph to denote our SZ masses derived using a spherical geometry and ellip when assuming an elliptical model. We have excluded Abell 1758 (A and B) from the analysis, given its abnormally disturbed and complex nature.
$r_{500}$ $r_{200}$
R12- full sample $0.75\pm 0.02$ -
R12- regular clusters $0.75\pm 0.04$ -
ME10- full sample $0.88\pm 0.02$ -
Z10- full sample $0.99\pm 0.07$ -
Z10- relaxed $0.91\pm 0.06$ -
MA10- all $0.78\pm 0.09$ -
This work
$M_X/M_{SZ, sph}$ $1.7 \pm 0.2$ -
$M_X/M_{SZ,ellip}$ $1.6\pm 0.3$ -
$M_{SZ,sph}/M_{WL}$ $1.2^{+0.2}_{-0.3}$ $1.0 \pm 0.1$
$M_{SZ, ellip}/M_{WL}$ $1.2^{+0.2}_{-0.3}$ $0.9 \pm 0.1$
Comparison of AMI $M_{\rm{T}}(r_{500})$ measurements with others. Methods used for estimating $M_{\rm{T}}(r_{500})$ are given in the legend.
The line of gradient one has been included to aid the comparison. The references are as follows: Abell 586 [115]; Abell 611 [115]; Abell 773 [163]; Abell 781 (139 and 163); Abell 1413 [163], Abell 1758A [163], Abell 2218 [163] and RXJ0142+2131 [115]. AMI values are given in Tab. <ref>. These were the $M(r_{500})$ from X-ray and weak lensing data that we found in the literature.
Comparison of AMI $M_{\rm{T}}(r_{200})$ measurements with others. Methods used for estimating $M_{\rm{T}}(r_{500})$ are given in the legend. Mass is given in units of $\times 10^{14}M_{\odot}$.
The line of gradient one has been included to aid comparison. The references are as follows: Abell 586 [115]; Abell 611 (115, 131 and 7); Abell 1413 [61]; Abell 2111 [7] and RXJ0142+2131 [115]. AMI values are given in Tab. <ref>.
Comparison of cluster masses at $r_{500}$ and $r_{200}$ for a spherical and an elliptical model for the cluster geometry. Ratio refers to the ratio between spherical and elliptical $M_{\rm{T}}$
$M_{\rm{T}}(r_{200})/\times10^{14}M_{\odot}$ $M_{\rm{T}}(r_{500}/\times10^{13}M_{\odot}$)
Cluster Name Spherical Elliptical Ratio Spherical Elliptical Ratio
A586 $7.3 \pm 3.0$ $7.5^{+3.0}_{-3.1}$ $0.97 \pm 0.59$ $3.0 \pm 1.3$ $3.1 \pm 1.4$ $0.97 \pm 0.65$
A611 $5.7 \pm 1.1$ $5.8 \pm 1.2$ $0.98 \pm 0.30$ $2.9 \pm 0.7$ $2.9 \pm 0.7$ $1.00 \pm 0.34$
A621 $6.8^{+2.4}_{-2.5}$ $7.2^{+2.3}_{-2.4}$ $0.94 \pm 0.53$ $2.0 \pm 1.3$ $2.2^{+1.3}_{-1.4}$ $0.91^{+0.97}_{-1.00}$
A773 $5.1 \pm 1.7$ $7.4 \pm 2.3$ $0.69 \pm 0.66$ $2.4 \pm 0.9$ $3.1^{+1.4}_{-1.3}$ $0.77^{+0.76}_{-0.73}$
A781 $5.9 \pm 1.1$ $7.2 \pm 1.8$ $0.82 \pm 0.38$ $2.9 \pm 0.6$ $3.2 \pm 1.0$ $0.91 \pm 0.41$
A990 $2.9_{-0.1}^{+0.6}$ $2.9 \pm 0.6$ $1.00^{+0.29}_{-0.21}$ $1.6 \pm 0.3$ $1.6 \pm 0.3$ $1.00 \pm 0.27$
A1413 $5.7 \pm 1.4$ $5.8 \pm 1.5$ $0.98 \pm 0.36$ $2.7 \pm 0.9$ $2.8 \pm 0.8$ $0.96 \pm 0.46$
A1423 $3.1 \pm 1.1$ $4.3 \pm 1.8$ $0.72 \pm 0.76$ $1.6 \pm 0.6$ $2.0 \pm 0.9$ $0.80 \pm 0.73$
A1758a $5.9_{-1.1}^{+1.0}$ $6.2 \pm 1.2$ $0.95^{+0.27}_{-0.28}$ $3.6 \pm 0.6$ $3.7 \pm 0.7$ $0.97 \pm 0.26$
A1758b $6.3 \pm 2.7$ $5.8 \pm 2.5$ $1.09 \pm 0.56$ $3.1 \pm 1.4$ $2.8 \pm 1.2$ $1.11 \pm 0.56$
A2009 $6.6 \pm 2.1$ $5.7^{+2.6}_{-2.9}$ $1.16^{+0.48}_{-0.52}$ $2.9_{-0.9}^{+0.3}$ $2.0^{+1.2}_{-1.4}$ $1.45^{+0.42}_{-0.53}$
A2111 $6.0 \pm 1.3$ $6.0 \pm 1.4$ $1.00 \pm 0.32$ $2.6 \pm 0.7$ $2.7 \pm 0.9$ $0.96 \pm 0.44$
A2146 $7.1 \pm 1.0$ $7.5 \pm 1.5$ $0.95 \pm 0.26$ $3.9 \pm 0.7$ $3.8 \pm 0.8$ $1.03 \pm 0.27$
A2218 $8.7 \pm 1.3$ $9.0^{+1.6}_{-1.5}$ $0.97^{+0.24}_{-0.23}$ $3.9 \pm 0.9$ $4.0^{+1.0}_{-0.9}$ $0.97^{+0.35}_{-0.33}$
RXJ0142+2131 $5.3_{-1.7}^{+1.6}$ $5.4^{+1.8}_{-1.9}$ $0.98^{+0.46}_{-0.49} $ $2.4 \pm 1.0$ $2.5 \pm 1.0$ $0.96 \pm 0.6$
RXJ1720+2638 $2.9 \pm 0.6$ $3.6 \pm 0.7$ $0.81 \pm 0.35$ $1.7 \pm 0.3$ $2.1 \pm 0.4$ $0.81 \pm 0.32$
§ CONCLUSIONS
We observe 19 LoCuSS clusters with $L_X > 7\times 10^{37}$ W ($h_{50}=1.0$) and present SZ images before and after source subtraction for 16 of them (and for Abell 1758b, which was found within the FoV of Abell 1758a).
We do not detect SZ effects towards Zw1458.8+2233 and Abell 1704, due to difficult source environments, nor towards Zw0857.9+2107, for reasons unclear to us. We have produced marginalized posterior distributions at $r_{500}$ and $r_{200}$ for 16 clusters (since Abell 2409 can not be fitted adequately by our model).
* Measurements of $M_{\rm{T}}(r_{200})$ are not common in the literature but are very important for testing large-radius scaling relations and understanding the physics in the outskirts of clusters. Consequently, the 16 measurements presented here, from a sample with narrow redshift-range, represent a significant increment to what already exists.
* For the clusters studied, we find values for $M_{\rm{T}}(r_{200})$ span $2.0-6.1\pm 0.9 \times 10^{14}h^{-1}_{70}M_{\odot}$ and are typically 2-2.5 times larger than $M_{\rm{T}}(r_{500})$; we find $r_{200}$ is typically $1.1 \pm 0.1h_{70}^{-1}$ Mpc.
* AMI measurements of $M_{\rm{T}}(r_{500})$ are consistent with published optical results for 3 out of 4 clusters in our sample, with the weighted best-fit ratio[With the exception of Abell 1758a+b]
between AMI SZ masses and lensing masses being $1.2^{+0.2}_{-0.3}$ within $r_{500}$ and $1.0 \pm 0.1$ within $r_{200}$. They are systematically lower than existing X-ray measurements
of $M_{\rm{T}}(r_{500})$ for 6 clusters with available
X-ray estimates and are only consistent with one of these measurements. The more discrepant masses correspond to the stronger mergers of the sample. The ratio of the X-ray masses to the AMI SZ
masses is $1.7 \pm 0.2$ for the sample. The agreement with optical measurements improves for $M_{\rm{T}}(r_{200})$, though there are few data. We have investigated the AMI vs X-ray discrepancy by comparing
$T_{\rm{AMI}}$ estimates with $T_{X}$ estimates, when available, at $r \approx$ 500 kpc. There tends to be good agreement in less X-ray luminous clusters and in non-mergers but large-radius $T_{X}$ can be
substantially larger than $T_{\rm{AMI}}$ in mergers. This explains why some X-ray mass estimates are significantly higher than the AMI estimates: the use of a higher
temperature will give a consequently higher mass in the hydrostatic equilibrium model used. Another implication of a higher large-radius $T_{X}$ than $T_{\rm{AMI}}$ (given the respective $n^2$ and $n$ emission weightings) is that, even at around $r_{500}$, the gas is clumped or shocked or both. There is a clear need for more large-scale measurements.
* We have investigated the effects of our main contaminant, radio sources, by searching for degeneracies in the posterior distributions of source flux densities for sources within $5\arcmin$ of the cluster SZ centroids. We find small or negligible degeneracies between source flux densities and cluster mass for all clusters, with the exceptions of Abell 781, Abell 1758a and RXJ1720.1+2638, which have
sources with flux densities of 9, 7 and 7 mJy at $\lesssim 2\arcmin$ from the cluster SZ centroids. By simulating a cluster with a challenging source environment, we have shown that our AMI analysis can approximately recover the true mass, even in a degenerate scenario.
* We often find differences in the position of SZ and X-ray peaks, with an average offset of $35\arcsec$, a median of $34\arcsec$ and a sample standard deviation of $24\arcsec$ for the entire sample (excluding Abell 1758a+b), confirming what has been seen in previous observational studies
and in simulations.
We emphasize that our sample size is small, but we find
no clear relation (except for Abell 1758) between position difference and merger activity. There is however an indication of a relation between merger activity and SZ ellipticity.
* We have analysed the AMI data for two clusters: Abell 611 and Abell 2111, with a $\beta$ parameterization and with five gNFW parameterizations,
including the widely used [13] “universal” and the [110] ones. This has revealed very different degeneracies in $Y_{\rm{sph}}(r_{500})-r_{500}$
for the two types of cluster parameterization. For both clusters, the $\beta$ parameterization, which allows the shape parameters to be fitted,
yielded stronger constraints on $r_{500}$ than any of the gNFW
paramaterizations. The Nagai et al. and Arnaud et al. gNFW parameters produced consistent results, with the latter giving slightly better constraints. Setting the gNFW parameters to different, but
reasonable, values altered the degeneracies significantly.
This illustrates the risks of using a single set of fixed, averaged profile shape parameters to model all clusters.
§ ACKNOWLEDGMENTS
We thank an anonymous referee for very quick and helpful comments
and suggestions, and Alastair Edge for helpful discussion.
We are grateful to the staff of the Cavendish Laboratory and the Mullard Radio
Observatory for the maintenance and operation of AMI.
We acknowledge support from Cambridge University and
STFC for funding and supporting AMI. MLD, TMOF, MO, CRG, MPS and
TWS are grateful for support from STFC studentships. This work was
carried out using
the Darwin Supercomputer of Cambridge University High
Performance Computing Service (http://www.hpc.cam.ac.uk/), provided
by Dell Inc. using Strategic Research Infrastructure Funding from the
Higher Education Funding Council for England and the Altix 3700
supercomputer at DAMTP, Cambridge University, supported by HEFCE
and STFC. We thank Stuart Rankin and Andrey Kaliazin for their
computing support.
This research has made use of data from the Chandra Data
Archive (ACCEPT) [32].
We acknowledge the use of NASA's SkyView facility
(http://skyview.gsfc.nasa.gov) located at NASA Goddard
Space Flight Center.
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|
arxiv-papers
| 2012-05-31T18:47:04 |
2024-09-04T02:49:31.419125
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The AMI Consortium: Carmen Rodr\\'iguez-Gonz\\'alvez, Timothy W.\n Shimwell, Matthew L. Davies, Farhan Feroz, Thomas M. O. Franzen, Keith J. B.\n Grainge, Michael P. Hobson, Natasha Hurley-Walker, Anthony N. Lasenby, Malak\n Olamaie, Guy Pooley, Richard D. E. Saunders, Anna M. M. Scaife, Michel P.\n Schammel, Paul F. Scott, David J. Titterington, Elizabeth M. Waldram",
"submitter": "Carmen Rodriguez-Gonzalvez",
"url": "https://arxiv.org/abs/1205.7067"
}
|
1206.0232
|
11institutetext: LMAM & School of Mathematical Sciences, Peking University
11email: dailiyun@pku.edu.cn xbc@math.pku.edu.cn
# Non-Termination Sets of Simple Linear Loops
Liyun Dai Bican Xia
Corresponding author
###### Abstract
A simple linear loop is a simple while loop with linear assignments and linear
loop guards. If a simple linear loop has only two program variables, we give a
complete algorithm for computing the set of all the inputs on which the loop
does not terminate. For the case of more program variables, we show that the
non-termination set cannot be described by Tarski formulae in general.
###### Keywords:
Simple linear loop, termination, non-termination set, eigenvalue, Tarski
formula
## 1 Introduction
Termination of programs is an important property of programs and one of the
main research topics in the field of program verification. It is well known
that the following so-called “uniform halting problem” is undecidable in
general.
Using only a finite amount of time, determine whether a given program will
always finish running or could execute forever.
However, there are some well known techniques for deciding termination of some
special kinds of programs. A popular technique is to use ranking functions. A
ranking function for a loop maps the values of the loop variables to a well-
founded domain; further, the values of the map decrease on each iteration. A
linear ranking function is a ranking function that is a linear combination of
the loop variables and constants. Some methods for the synthesis of ranking
functions and some heuristics concerning how to automatically generate linear
ranking functions for linear programs have been proposed, for example, in
Colón and Sipma [3], Dams et al. [4] and Podelski and Rybalchenko [6].
Podelski and Rybalchenko [6] provided an efficient and complete synthesis
method based on linear programming to construct linear ranking functions. Chen
et al. [2] proposed a method to generate nonlinear ranking functions based on
semi-algebraic system solving. The existence of ranking function is only a
sufficient condition on the termination of a program. There are programs,
which terminate, but do not have ranking functions. Another popular technique
based on well-orders, presented in Lee et al. [5], is size-change principle.
The well-founded data can ensure that there are no infinitely descents, which
guarantees termination of programs.
For linear loops, some other methods based on calculating eigenvectors of
matrices have been proposed. Tiwari [7] proved that the termination problem of
a class of linear programs (simple loops with linear loop conditions and
updates) over the reals is decidable through Jordan form and eigenvector
computation. Braverman [1] proved that it is also decidable over the integers.
Xia et al. [8] considered the termination problems of simple loops with linear
updates and polynomial loop conditions, and proved that the termination
problem of such loops over the integers is undecidable. In [9], Xia et al.
provided a novel symbolic decision procedure for termination of simple linear
loops, which is as efficient as the numerical one given in [7].
A counter-example to termination is an infinite program execution. In program
verification, the search for counter-examples to termination is as important
as the search for proofs of termination. In fact, these are the two folds of
termination analysis of programs. Gupta et al. [10] proposed a method for
searching counter-examples to termination, which first enumerates lasso-shaped
candidate paths for counter-examples and proves the feasibility of a given
lasso by solving the existence of a recurrent set as a template-based
constraint satisfaction problem. Gulwani et al. [11] proposed a constraint-
based approach to a wide class of program analyses and weakest precondition
and strongest postcondition inference. The approach can be applied to
generating most-general counter-examples to termination.
In this paper, we consider the set of all inputs on which a given program does
not terminate. The set is called NT throughout the paper. For simple linear
loops, we are interested in whether the NT is decidable and how to compute it
if it is decidable. Similar problems was also considered in [12]. Our
contributions in this paper are as follows. First, for homogeneous linear
loops (see Section 2 for the definition) with only two program variables, we
give a complete algorithm for computing the NT. For the case of more program
variables, we show that the NT cannot be described by Tarski formulae in
general.
The rest of this paper is organized as follows. Section 2 introduces some
notations and basic results on simple linear loops. Section 3 presents an
algorithm for computing the NT of homogeneous linear loops with only two
program variables. The correctness of the algorithm is proved by a series of
lemmas. For linear loops with more than two program variables, it is proved in
Section 4 that the NT is not a semi-algebraic set in general, i.e., it cannot
be described by Tarski formulae in general. The paper is concluded in Section
5.
## 2 Preliminaries
In this paper, the domain of inputs of programs is ${\mathbb{R}}$, the field
of real numbers. A simple linear loop in general form over ${\mathbb{R}}$ can
be formulated as
${\tt P1}:\quad{\rm while}\ \left({B\vec{x}>\vec{b}}\right)\
\left\\{{\vec{x}:=A\vec{x}+\vec{c}}\right\\}$
where $\vec{b},\vec{c}$ are real vectors, $A_{n\times n},B_{m\times n}$ are
real matrices. $B\vec{x}>\vec{b}$ is a conjunction of $m$ linear inequalities
in $\vec{x}$ and $\vec{x}:=A\vec{x}+\vec{c}$ is a linear assignment on the
program variables $\vec{x}$.
###### Definition 1
[7] The non-termination set of a program is the set of all inputs on which the
program does not terminate. It is denoted by NT in this paper.
In particular,
${\rm NT}({\tt P1})=\\{\vec{x}\in{\mathbb{R}}^{n}|{\tt P1}\ {\rm does\ not\
terminate\ on}\ \vec{x}\\}\enspace.$
We list some related results in [7].
###### Proposition 1
[7] For a simple linear loop P1, the following is true.
* •
The termination of P1 is decidable.
* •
If $A$ has no positive eigenvalues, the NT is empty.
* •
The NT is convex.
In this paper, only the following homogeneous case is considered.
${\tt P2}:\quad{\rm while}\ ({B\vec{x}>0})\ \\{\vec{x}:=A\vec{x}\\}\enspace.$
Let $B_{1},\ldots,B_{m}$ be the rows of $B$. Consider the following loops
$L_{i}:\quad{\rm while}\ (B_{i}\vec{x}>0)\ \\{\vec{x}:=A\vec{x}\\}\enspace.$
Obviously, NT(P2)=$\bigcap_{i=1}^{m}{\rm NT}(L_{i}).$ Therefore, without loss
of generality, we assume throughout this paper that $m=1$, i.e., there is only
one inequality as the loop guard. The following is a simple example of such
loops.
${\rm while}\ (4x_{1}+x_{2}>0)\quad\left\\{\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right):=\left(\begin{array}[]{cc}-2&4\\\
4&0\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)\right\\}\enspace.$
That is $B=(4,1),A=\left(\begin{array}[]{cc}-2&4\\\
4&0\end{array}\right)\enspace.$
## 3 Two-variable case
To make things clear, we restate the problem for this two-variable case as
follows.
For a given homogeneous linear loop P2 with exactly two program variables and
only one inequality as the loop guard, compute NT(P2).
For simplicity, we denote the program variables by $x_{1},x_{2}$ and use NT
instead of NT(P2) in this section. If $\vec{\alpha}$ is a non-zero point in
the plane, we denote by $\overrightarrow{\vec{\alpha}}$ a ray starting from
the origin of plane and going through the point $\vec{\alpha}$.
###### Proposition 2
NT must be one of the following:
(1) an empty set;
(2) a ray starting from the origin;
(3) a sector between two rays starting from the origin.
###### Proof
We view an input $(x_{1},x_{2})$ as a point in the real plane with origin $O$.
If there exists a point $M(x_{1},x_{2})\in$ NT, any point $\vec{P}$ on the ray
$\overrightarrow{\vec{OM}}$ can be written as $\vec{P}=kM=(kx_{1},kx_{2})$ for
a positive number $k$. So
$BA^{n}(kx_{1},kx_{2})^{T}=k^{n}BA^{n}(x_{1},x_{2})^{T}>0$ for any
$n\in{\mathbb{N}}$. That means $\vec{P}\in{\rm NT}$. Therefore, it is clear
from the item 3 of Proposition 1 that the conclusion is true.
By the above proposition, the key point for computing the NT is to compute the
ray(s) which is (are) the boundary of NT. We give the following algorithm to
compute the ray(s) (and thus the NT) for P2 if the NT is not empty. The
algorithm, as can be expected, is mainly based on the computation of
eigenvalues and eigenvectors of $A$. The correctness of our algorithm will be
proved by a series of lemmas following the algorithm.
Input: Matrices $A_{2\times 2}$ and $B_{1\times 2}$.
Output: The NT of P2 with $A$ and $B$.
1 if _$A={\bf 0}$ or $B={\bf 0}$_ then
2 return $\emptyset$;
3Compute the eigenvalues of $A$ and denote them by $\lambda_{1},\lambda_{2}$;
4 if _$\lambda_{1}\ngtr 0\wedge\lambda_{2}\ngtr 0$_ then
5 return $\emptyset$; // Proposition 1
6Take $\vec{\alpha_{0}}\in{\mathbb{R}}^{2}\setminus\\{0\\}$ such that
$B\vec{\alpha_{0}}=0$ and $BA\vec{\alpha_{0}}\geq 0$;
7 if _$BA\vec{\alpha_{0}}=0$_ then
8 choose $\vec{\xi}$ such that $B\vec{\xi}>0$
9 if _$B(A\vec{\xi}) >0$_ then
10 return $\\{\vec{x}|\vec{x}\in{\mathbb{R}}^{2},B\vec{x}>0\\}$ // Lemma 4
11 else
12 return $\emptyset$ // Lemma 5
13
14if _$\lambda_{1}=0\vee\lambda_{2}=0$_ then
15 return $\\{\vec{x}|\vec{x}\in{\mathbb{R}}^{2},B\vec{x}>0,BA\vec{x}>0\\}$;
// Lemma 6
16Suppose $\lambda_{1}\geq\lambda_{2}$
17 if _$\lambda_{1}\geq\lambda_{2} >0$_ then
18 choose an eigenvector $\vec{\beta_{2}}$ related to $\lambda_{2}$ such that
$B\vec{\beta_{2}}\geq 0$;
19 return
$\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha_{0}}+k_{2}\vec{\beta_{2}},k_{1}\geq
0,k_{2}>0\\}$; // Lemmas 7 and 8
20if _$\lambda_{1} >0\wedge\lambda_{2}<0$_ then
21 if _$\lambda_{1}\geq|\lambda_{2}|$_ then
22 let $\vec{\alpha_{-1}}=A^{-1}\vec{\alpha_{0}}$ and return
$\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha_{0}}+k_{2}\vec{\alpha_{-1}},k_{1}>0,k_{2}>0\\}$;
23 if _$\lambda_{1} <|\lambda_{2}|$_ then
24 choose an eigenvector $\vec{\beta}$ related to $\lambda_{1}$ such that
$B\vec{\beta}>0$ and
25 return $\\{\vec{x}|\vec{x}=k\vec{\beta},k>0\\}$ // Lemma 10
26
Algorithm 1 NonTermination Figure 1: Lemma 1
###### Lemma 1
Suppose NT is not empty and $\partial{\rm NT}$ is the boundary of NT. If
$\vec{x}\in\partial{\rm NT}$ and $B\vec{x}\neq 0$, then
$A\vec{x}\in\partial{\rm NT}$.
###### Proof
Obviously, $B$ is a linear map from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}$ .
Because $B\vec{y}>0$ for all $\vec{y}\in{\rm NT}$, we have $B\vec{x}\geq 0$.
And thus $B\vec{x}>0$ by the assumption that $B\vec{x}\neq 0$. Hence, there
exists an open ball $o_{1}(\vec{x},r_{1})$ such that $B\vec{y}>0$ for all
$\vec{y}\in o_{1}(\vec{x},r_{1}).$
Let $F$ be the linear map from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$ that
$F(\vec{y})=A\vec{y}$ for any $\vec{y}\in{\mathbb{R}}^{2}$ and hence $F$ is
continuous. So for any neighborhood $o(A\vec{x},r)$ of $A\vec{x}$, there
exists a positive real number $r_{2}$ such that $o_{2}(\vec{x},r_{2})\subseteq
o_{1}(\vec{x},r_{1})$ and $F(o_{2}(\vec{x},r_{2}))\subseteq o(A\vec{x},r).$
Because $\vec{x}\in\partial{\rm NT}$, there exist $\vec{y},\vec{z}\in
o_{2}(\vec{x},r_{2})$ such that $\vec{y}\in{\rm NT}$ and $\vec{z}\notin{\rm
NT}$. Then $A(\vec{y})$, $A(\vec{z})\in o(A\vec{x},r)$, $A\vec{(y})\in{\rm
NT}$ and $A(\vec{z})\notin{\rm NT}$. It is followed that there are both
terminating and non-terminating inputs in any neighborhood of $A\vec{x}$.
Therefore, $A\vec{x}\in\partial{\rm NT}$.
Figure 2: Lemma 2
###### Lemma 2
Suppose NT is neither empty nor a ray and $\partial{\rm NT}\
\cap\\{\vec{x}|B\vec{x}=0\\}=\\{(0,0)\\}$. If $B\vec{y}=0$ and $BA\vec{y}>0$,
then $A\vec{y}\in{\rm NT}$.
###### Proof
By Proposition 2, $\partial{\rm NT}$ consists of two rays. Let $l_{1},l_{2}$
be the two rays. Since neither $l_{1}$ nor $l_{2}$ is on $Bx=0$, $l_{1}$ and
$l_{2}$ are not collinear. So we can choose two points $\vec{z}\in l_{1}$ and
$\vec{v}\in l_{2}$ such that $B\vec{z}>0$, $B\vec{v}>0$ and
$\vec{y}=t_{1}\vec{z}+t_{2}\vec{v}$ for some
$t_{1}\in{\mathbb{R}},t_{2}\in{\mathbb{R}}$. By Lemma 1, $A\vec{z}$ and
$A\vec{v}$ must be on the boundary of NT, i.e., $l_{1}$ or $l_{2}$. Thus, we
have at most four possible cases as follows.
* (1)
$A\vec{z}=k_{1}\vec{z},A\vec{v}=k_{2}\vec{v},$ (i.e., $A\vec{z}\in
l_{1},A\vec{v}\in l_{2}$)
* (2)
$A\vec{z}=k_{1}\vec{z},A\vec{v}=k_{2}\vec{z},$ (i.e., $A\vec{z}\in
l_{1},A\vec{v}\in l_{1}$)
* (3)
$A\vec{z}=k_{1}\vec{v},A\vec{v}=k_{2}\vec{v},$ (i.e., $A\vec{z}\in
l_{2},A\vec{v}\in l_{2}$)
* (4)
$A\vec{z}=k_{1}\vec{v},A\vec{v}=k_{2}\vec{z},$ (i.e., $A\vec{z}\in
l_{2},A\vec{v}\in l_{1}$)
where $k_{1}>0,k_{2}>0$.
Case (1). Because $B\vec{y}=t_{1}B\vec{z}+t_{2}B\vec{v}=0$ and
$BA\vec{y}=BA(t_{1}\vec{z}+t_{2}\vec{v})=t_{1}k_{1}B\vec{z}+t_{2}k_{2}B\vec{v}>0,$
we have $t_{1}t_{2}<0$. Without loss of generality, assume that $t_{1}>0$ and
$t_{2}<0$. We denote $t_{1}B\vec{z}$ by $P$. Note that $P>0$ and
$t_{2}B\vec{v}=-P$. Since $BA\vec{y}=(k_{1}-k_{2})P>0$, we have
$k_{1}>k_{2}>0$ and
$BA^{n}(A\vec{y})=k_{1}^{n+1}t_{1}B\vec{z}+k_{2}^{n+1}t_{2}B\vec{v}=k_{1}^{n+1}P-k_{2}^{n+1}P>0$
for any $n\in\mathbb{N}$. By the definition of ${\rm NT}$, $A\vec{y}\in{\rm
NT}$.
Case (2). Because $BA\vec{y}=(t_{1}k_{1}+t_{2}k_{2})B\vec{z}>0$, we have
$BA^{n}(A\vec{y})=k_{1}^{n}(t_{1}k_{1}+t_{2}k_{2})B\vec{z}>0$
for any $n\in\mathbb{N}.$ By the definition of NT, we have $A\vec{y}\in{\rm
NT}$.
Case (3). Similarly as Case (2), we can prove $A\vec{y}\in{\rm NT}$.
Case (4). We shall show that this case cannot happen. Let
$S=\\{\vec{x}|\vec{x}=r_{1}\vec{y}+r_{2}A\vec{y},r_{1}>0,r_{2}>0\\}$
be the sector between the two rays $\overrightarrow{\vec{y}}$ and
$\overrightarrow{\vec{Ay}}$. For any $\vec{w}\in S$, we have
$B\vec{w}=r_{1}B\vec{y}+r_{2}BA\vec{y}=r_{2}BA\vec{y}>0$.
Because
$A^{2}\vec{y}=A(t_{1}k_{1}\vec{v}+t_{2}k_{2}\vec{z})=t_{1}k_{1}k_{2}\vec{z}+t_{2}k_{1}k_{2}\vec{v}=k_{1}k_{2}\vec{y},$
we have
$A\vec{w}=r_{1}A\vec{y}+r_{2}A^{2}\vec{y}=r_{1}A\vec{y}+r_{2}k_{1}k_{2}\vec{y}\in
S$. Therefore, $\vec{w}\in{\rm NT}$ and $S\subseteq{\rm NT}$. As
$\overrightarrow{\vec{y}}$ is a boundary of $S$ and $B\vec{y}=0$,
$\overrightarrow{\vec{y}}$ is contained in $\partial{\rm NT}$, which
contradicts with the assumption of the lemma. So (4) cannot happen.
In summary, $A\vec{y}\in{\rm NT}$.
Figure 3: Lemma 3
###### Lemma 3
If $\partial{\rm NT}$ is composed of two rays $l_{1}$ and $l_{2}$, then either
$l_{1}$ or $l_{2}$ is on $B\vec{x}=0$.
###### Proof
Assume neither $l_{1}$ nor $l_{2}$ is on $B\vec{x}=0$. Choose a point
$\vec{y}$ such that $\vec{y}\neq\bf{0}$ , $B\vec{y}=0$ and $BA\vec{y}\geq 0$.
Suppose $BA\vec{y}=0$. As ${\rm NT}$ is not empty, there exists
$\vec{z}\in{\rm NT}$. Hence $A\vec{y}$ can be rewritten as
$A\vec{y}=h_{1}\vec{z}+h_{2}\vec{y}$ for some
$h_{1}\in{\mathbb{R}},h_{2}\in{\mathbb{R}}$. As a result of
$BA\vec{y}=h_{1}B\vec{z}+h_{2}B\vec{y}=h_{1}B\vec{z}=0$, $h_{1}=0$. Note that
$A^{n}\vec{y}=h_{2}^{n}\vec{y},BA^{n}\vec{y}=h_{2}^{n}B\vec{y}=0\enspace.$ (1)
According to Eq.(1) and $\vec{z}\in{\rm NT}$, we have
$BA^{n}(k_{1}\vec{z}+k_{2}\vec{y})=k_{1}BA^{n}\vec{z}+k_{2}BA^{n}\vec{y}=k_{1}BA^{n}\vec{z}>0$
for any $k_{1}>0,n\in\mathbb{N}$. Hence
$\\{\vec{x}|\vec{x}=k_{1}\vec{z}+k_{2}\vec{y},k_{1}>0\\}\subseteq{\rm NT}$.
Therefore, $\\{\vec{x}|B\vec{x}=0\\}=\partial{\rm NT}$, which contradicts with
the assumption.
If $BA\vec{y}>0$, $A\vec{y}\in{\rm NT}$ follows from Lemma 2. Let
$S=\\{\vec{x}|k_{1}\vec{y}+k_{2}A\vec{y},k_{1}>0,k_{2}>0\\}$. And we have
$BA^{n}\vec{z}=k_{1}BA^{n}y+k_{2}BA^{n+1}\vec{y}>0$ for any $n\in\mathbb{N}$,
$\vec{z}\in S$. Thus $\vec{z}\in{\rm NT}$ and $S\subseteq{\rm NT}$. By the
method of choosing $\vec{y}$, $\overrightarrow{\vec{y}}\subseteq\partial{\rm
NT}$. That means $\overrightarrow{\vec{y}}$ is $l_{1}$ or $l_{2}$, which
contradicts with the assumption.
###### Lemma 4
Suppose $A$ has positive eigenvalues and has an eigenvector $\vec{\alpha}$
satisfying $B\vec{\alpha}=0$. If $\vec{\xi}$ is a vector such that
$B\vec{\xi}>0$ and $BA\vec{\xi}>0$, then ${\rm NT}=\\{\vec{x}|B\vec{x}>0\\}$.
###### Proof
For any $\vec{y}\in\\{\vec{x}|B\vec{x}>0\\}$, it can be written as
$\vec{y}=k_{1}\vec{\xi}+k_{2}\vec{\alpha}$ for some
$k_{1}\in{\mathbb{R}},k_{2}\in{\mathbb{R}}$. As
$B\vec{y}=k_{1}B\vec{\xi}+k_{2}B\vec{\alpha}=k_{1}B\vec{\xi}>0$, we have
$k_{1}>0$. Thus
$BA\vec{y}=k_{1}BA\vec{\xi}+k_{2}BA\vec{\alpha}=k_{1}BA\vec{\xi}>0$ and
$A\vec{y}\in\\{\vec{x}|B\vec{x}>0\\}$. By the definition of ${\rm NT}$, we
have $\\{\vec{x}|B\vec{x}>0\\}\subseteq{\rm NT}$ and hence ${\rm
NT}=\\{\vec{x}|B\vec{x}>0\\}$.
###### Lemma 5
Suppose $A$ has positive eigenvalues and has an eigenvector $\vec{\alpha}$
satisfying $B\vec{\alpha}=0$. If there is a vector $\vec{\xi}$ such that
$B\vec{\xi}>0$ and $BA\vec{\xi}\leq 0$, then ${\rm NT}=\emptyset$.
###### Proof
For any $\vec{y}\in\\{\vec{x}|B\vec{x}>0\\},$ it can be written as
$\vec{y}=k_{1}\vec{\alpha}+k_{2}\vec{\xi}$ for some
$k_{1}\in{\mathbb{R}},k_{2}\in{\mathbb{R}}$. Since
$B\vec{y}=k_{2}B\vec{\xi}>0$, we have $k_{2}>0$. And because
$BA\vec{y}=k_{2}BA\vec{\xi}\leq 0$, ${\rm NT}=\emptyset$.
###### Lemma 6
Suppose $A$ has a positive eigenvalue and a zero eigenvalue. If $\vec{\gamma}$
is an eigenvector related to the positive eigenvalue such that
$B\vec{\gamma}>0$, then ${\rm NT}=\\{\vec{x}|B\vec{x}>0,BA\vec{x}>0\\}.$
###### Proof
Let $\vec{\beta}$ be an eigenvector with respect to eigenvalue 0 and $\lambda$
be the positive eigenvalue. Let $S$ be the set
$\\{\vec{x}|B\vec{x}>0,BA\vec{x}>0\\}$. For any $\vec{y}\in S$, it can be
written as $k_{1}\vec{\beta}+k_{2}\vec{\gamma}$ for some
$k_{1}\in{\mathbb{R}},k_{2}\in{\mathbb{R}}$. We have $BA\vec{y}=k_{2}\lambda
B\vec{\gamma}>0$, thus $k_{2}>0$. Note that
$BA^{n}\vec{y}=k_{2}\lambda^{n}\vec{\gamma}>0$ for any $n\in\mathbb{N}$, hence
$S\subseteq{\rm NT}$. Because $\\{\vec{x}|B\vec{x}\leq 0\vee BA\vec{x}\leq
0\\}\cap{\rm NT}=\emptyset$, ${\rm NT}=\\{\vec{x}|B\vec{x}>0,BA\vec{x}>0\\}$.
###### Lemma 7
Suppose $A$ has two positive eigenvalues $\lambda_{1}>\lambda_{2}>0$ and two
eigenvectors $\vec{\beta_{1}}$ and $\vec{\beta_{2}}$ related to $\lambda_{1}$
and $\lambda_{2}$, respectively, such that
$B\vec{\beta_{1}}>0,B\vec{\beta_{2}}>0$. If $\vec{\alpha}$ is a vector such
that $B\vec{\alpha}=0$ and $BA\vec{\alpha}>0$, then ${\rm
NT}=\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha}+k_{2}\vec{\beta_{2}},k_{1}\geq
0,k_{2}>0\\}.$
###### Proof
It is easy to know $\vec{\beta_{1}},\vec{\beta_{2}}\in{\rm NT}$, thus NT is
neither empty nor a ray. By Lemma 3 there is a
$\overrightarrow{\vec{y}}\subseteq\partial{\rm NT}$ and $\vec{y}$ satisfies
$B\vec{y}=0$. Since for any $\vec{z}\in\partial{\rm NT}$, we have
$BA\vec{z}\geq 0$. So $BA\vec{y}\geq 0$ and hence
$\overrightarrow{\vec{\alpha}}=\overrightarrow{\vec{y}}$. In other word,
$\overrightarrow{\vec{\alpha}}$ is one ray of $\partial{\rm NT}$. Let the
other ray of $\partial{\rm NT}$ be $l$. As $-BA\vec{\alpha}<0$,
$\overrightarrow{\vec{-\alpha}}$ is not $l$. By Lemma 1, we have
$Al\in\partial{\rm NT}$. So $l$ is one of
$\overrightarrow{\vec{\beta_{1}}},\overrightarrow{\vec{\beta_{2}}}$ and
$\overrightarrow{\vec{A^{-1}}\alpha}$. By directly checking, we know
$\overrightarrow{\vec{\beta_{2}}}$ is $l$ and so ${\rm
NT}=\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha}+k_{2}\vec{\beta_{2}},k_{1}\geq
0,k_{2}>0\\}$.
###### Lemma 8
Assume that $A$ has one positive eigenvalue $\lambda$ with multiplicity $2$
and only one eigenvector $\vec{\beta}$ satisfying $B\vec{\beta}>0$. If
$\vec{\alpha}$ is a vector such that $B\vec{\alpha}=0$ and $BA\vec{\alpha}>0$,
then ${\rm NT}=\\{\vec{x}|\vec{x}=h_{1}\vec{\alpha}+h_{2}\vec{\beta},k_{1}\geq
0,k_{2}>0\\}$.
###### Proof
By the theory of Jordan normal form in linear algebra, there exists a vector
$\vec{\beta_{1}}$ such that
$A\vec{\beta_{1}}=\vec{\beta}+\lambda\vec{\beta_{1}}$ and $\vec{\beta}$ and
$\vec{\beta_{1}}$ are linearly independent.
Let $\vec{\alpha_{1}}=A\vec{\alpha}$. We claim that
$\forall n\in\mathbb{N}.(BA^{n}\vec{\alpha_{1}}>0\wedge\exists
h_{2}>0.(A^{n}\vec{\alpha_{1}}=h_{1}\vec{\beta}+h_{2}\vec{\beta_{1}})).$ (2)
To prove this claim we use induction on the value of $n$.
Suppose $\vec{\alpha}=h_{1}\vec{\beta}+h_{2}\vec{\beta_{1}}$. If $n=0$, then
$\vec{\alpha_{1}}=A\vec{\alpha}=(h_{1}\lambda+h_{2})\vec{\beta}+h_{2}\lambda\vec{\beta_{1}}$.
Because $B\vec{\alpha_{1}}=\lambda
B\vec{\alpha}+h_{2}B\vec{\beta}=h_{2}B\vec{\beta}>0$, we have $h_{2}>0$.
Now assume that the claim is true for $n-1$. Let
$A^{n-1}\vec{\alpha_{1}}=h_{1}\vec{\beta}+h_{2}\vec{\beta_{1}}$ where
$h_{2}>0$. Because $A^{n}\vec{\alpha_{1}}=A(A^{n-1}\vec{\alpha_{1}})=(\lambda
h_{1}+h_{2})\vec{\beta}+\lambda h_{2}\vec{\beta_{1}}$, we have $\lambda
h_{2}>0$ and $BA^{n}\vec{\alpha_{1}}=\lambda
BA^{n-1}\vec{\alpha_{1}}+h_{2}B\vec{\beta}>0$. So the claim is true for any
$n\in\mathbb{N}$ and we have $\vec{\alpha_{1}}\in{\rm NT}$.
Obviously, $\vec{\beta}\in{\rm NT}$ and $\vec{\beta}$ and $\vec{\alpha_{1}}$
are linearly independent, so NT is not a ray. By Lemma 3,
$\overrightarrow{\vec{\alpha}}\subseteq\partial{\rm NT}$.
Let the other ray of $\partial{\rm NT}$ be $l$. As $-BA\vec{\alpha}<0$,
$\overrightarrow{\vec{-\alpha}}$ is not $l$. By Lemma 1, $Al=l$ or
$Al=\overrightarrow{\vec{\alpha}}$. So $l$ must be
$\overrightarrow{\vec{\beta}}$ or $\overrightarrow{\vec{A^{-1}\alpha}}$. By
directly checking, we know $l$ is $\overrightarrow{\vec{\beta}}$ and thus
${\rm NT}=\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha}+k_{2}\vec{\beta},k_{1}\geq
0,k_{2}>0\\}$.
###### Lemma 9
Suppose $A$ has a positive eigenvalue $\lambda_{1}$ and a negative eigenvalue
$\lambda_{2}$ with $\lambda_{1}\geq|\lambda_{2}|$ and two eigenvectors
$\vec{\beta_{1}}$ and $\vec{\beta_{2}}$ related to $\lambda_{1}$ and
$\lambda_{2}$, respectively, such that
$B\vec{\beta_{1}}>0,B\vec{\beta_{2}}>0$. Suppose $\vec{\alpha}$ is a vector
such that $B\vec{\alpha}=0$ and $BA\vec{\alpha}>0$. Let
$\vec{\alpha_{-1}}=A^{-1}\vec{\alpha}$, $\vec{\alpha_{1}}=A\vec{\alpha}$. Then
${\rm NT}=\\{k_{1}\vec{\alpha}+k_{2}\vec{\alpha_{-1}},k_{1}>0,k_{2}>0\\}$.
###### Proof
Let $\vec{\alpha_{-1}}=h_{1}\vec{\beta_{1}}+h_{2}\vec{\beta_{2}}$. So
$\vec{\alpha}=A\vec{\alpha_{-1}}=h_{1}\lambda_{1}\vec{\beta_{1}}+h_{2}\lambda_{2}\vec{\beta_{2}}$
and
$\vec{\alpha_{1}}=A\vec{\alpha}=h_{1}\lambda_{1}^{2}\vec{\beta_{1}}+h_{2}\lambda_{2}^{2}\vec{\beta_{2}}$.
Because $B\vec{\alpha}=0$ and $B\vec{\alpha_{1}}>0$, $h_{1}$, $h_{2}$ and
$A\vec{\alpha_{-1}}$ are all positive.
Note that
$\vec{\alpha_{1}}=(-\lambda_{1}\lambda_{2})\vec{\alpha_{-1}}+(\lambda_{1}+\lambda_{2})\vec{\alpha}$
where $-\lambda_{1}\lambda_{2}>0$ and $\lambda_{1}+\lambda_{2}\geq 0$. Let
$S=\\{\vec{x}|\vec{x}=k_{1}\vec{\alpha}+k_{2}\vec{\alpha_{-1}}$, $k_{1}>0$,
$k_{2}>0\\}$. Since $B\vec{y}=k_{2}B\vec{\alpha_{-1}}>0$ and
$A\vec{y}=(k_{2}+k_{1}(\lambda_{1}+\lambda_{2}))\vec{\alpha}-k_{1}\lambda_{1}\lambda_{2}\vec{\alpha_{-1}}\in
S$ for any $\vec{y}\in S$, we have ${\rm NT}\supseteq S$.
Let $\vec{y}=k_{1}\vec{\alpha}+k_{2}\vec{\alpha_{-1}}$. Because
$B\vec{y}=k_{2}B\vec{\alpha_{-1}}\leq 0$ for any $k_{2}\leq 0$ and
$BA\vec{y}=k_{1}B\vec{\alpha_{1}}\leq 0$ for any $k_{1}\leq 0$, we have ${\rm
NT}=S$.
###### Lemma 10
Suppose A has a positive eigenvalue $\lambda_{1}$ and a negative eigenvalue
$\lambda_{2}$ such that $\lambda_{1}<|\lambda_{2}|$. If there are two
eigenvectors $\vec{\beta_{1}}$ and $\vec{\beta_{2}}$ related to $\lambda_{1}$
and $\lambda_{2}$, respectively, such that $B\vec{\beta_{1}}>0$ and
$B\vec{\beta_{2}}>0$, then ${\rm
NT}=\\{\vec{x}|\vec{x}=k\vec{\beta_{1}},k>0\\}$.
###### Proof
Consider any
$\vec{\beta}=k_{1}\vec{\beta_{1}}+k_{2}\vec{\beta_{2}}\in{\mathbb{R}}^{2}$.
If $k_{2}\neq 0$, because
$A^{n}(k_{1}\vec{\beta_{1}}+k_{2}\vec{\beta_{2}})=k_{1}\lambda_{1}^{n}\vec{\beta_{1}}+k_{2}\lambda_{2}^{n}\vec{\beta_{2}}$
and
$BA^{n}(k_{1}\vec{\beta_{1}}+k_{2}\vec{\beta_{2}})BA^{n+1}(k_{1}\vec{\beta_{1}}+k_{2}\vec{\beta_{2}})<0$
when $n$ is large enough, $k_{1}\vec{\beta_{1}}+k_{2}\vec{\beta_{2}}\notin{\rm
NT}$.
If $k_{2}=0$, obviously, ${\rm
NT}\supseteq\\{\vec{x}|\vec{x}=k\vec{\beta_{1}},k>0\\}$ and
$Bk\vec{\beta_{1}}\ \not\in{\rm NT}$ for any $k\leq 0$.
So ${\rm NT}=\\{\vec{x}|\vec{x}=k\vec{\beta_{1}},k>0\\}$.
Now, the correctness of our algorithm NonTermination can be easily obtained as
follows.
###### Theorem 3.1
The algorithm NonTermination is correct.
###### Proof
First, the termination of NonTermination is obvious because there are no loops
and no iterations in it. Second, it is also clear that the algorithm discusses
all the cases of eigenvalues of $A$, respectively. According to Lemmas 4-10
(each of them corresponds to a certain case in the algorithm as commented in
the algorithm), the output of the algorithm in each case is correct.
###### Example 1
Compute the NT of the following loop.
${\rm while}~{}(4x_{1}+x_{2}>0)\quad\left\\{\left(\begin{array}[]{c}x_{1}\\\
x_{2}\\\ \end{array}\right)=\left(\begin{array}[]{cc}-2&4\\\ 4&0\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\
\end{array}\right)\right\\}$
Herein, $B=(4,1),A=\left(\begin{array}[]{cc}-2&4\\\ 4&0\\\
\end{array}\right).$
The computation of NonTermination on the loop is:
Line 1. $B\neq 0$ and $A\neq 0$.
Line 4. $A$ has a positive eigenvalue $-1+\sqrt{17}$.
Line 6. Let
$\vec{\alpha_{0}}=(-1,4)^{T},\vec{\alpha_{1}}=A\vec{\alpha_{0}}=(18,-4)^{T}$.
Line 7. $B\vec{\alpha_{1}}=68\neq 0$.
Line 13. The two eigenvalues of $A$ are $-1+\sqrt{17},-1-\sqrt{17}$,
respectively. Neither of them is $0$.
Line 19. $A$ has two eigenvalues, of which one is positive and the other
negative.
Line 20. The absolute value of the negative eigenvalue is greater than the
positive eigenvalue.
Line 22. The eigenvector with respect to the positive eigenvalue is
$\vec{\beta}=(1,\frac{\sqrt{17}+1}{4})^{T}$ and $B\vec{\beta}>0$. Return
$\\{\vec{x}|\vec{x}=k\vec{\beta},k>0\\}$.
## 4 More variables
###### Theorem 4.1
In general, NT is not a semi-algebraic set.
###### Remark 1
All Tarski formulae are in the form of conjunctions or/and disjunctions of
polynomial equalities and/or inequalities, so, in other words, semi-algebraic
sets are exactly the sets defined by Tarski formulae. By Theorem 4.1, we can
conclude that the non-termination sets of linear loops with more than two
variables cannot be defined by Tarski formulae in general.
###### Remark 2
It should be noticed that all polynomial invariants are semi-algebraic sets.
In order to prove the above theorem, we give an example to demonstrate its NT
is not a semi-algebraic set.
###### Proposition 3
Let a linear loop with three program variables be as follows.
${\tt P3:}\ {\rm while}\ (x_{1}+2x_{2}+x_{3}\geq
0)\quad\left\\{\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ x_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}2&0&0\\\ 0&3&0\\\ 0&0&5\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ x_{3}\\\
\end{array}\right)\right\\}.$
Then NT(P3) is not a semi-algebraic set.
The conclusion can be proved by using the following lemmas. For simplicity,
NT(P3) is denoted by NT in this section.
###### Lemma 11
Denote by $\tau$ the following set
$\\{9(x_{1}^{2}+x_{2}^{2})-x_{3}^{2}<0,x_{3}>0\\},$
then $\tau\subseteq{\rm{\rm NT}}.$
###### Proof
For any $(x_{1},x_{2},x_{3})\in\tau$, we have $x_{3}>3|x_{1}|,x_{3}>3|x_{2}|$
and thus $x_{1}+2x_{2}+x_{3}>0.$ Because
$A(x_{1},x_{2},x_{3})^{T}=(2x_{1},3x_{2},5x_{3})^{T}$ and
$9(4x_{1}^{2}+9x_{2}^{2})-25x_{3}^{2}<0$, $A(x_{1},x_{2},x_{3})^{T}\in\tau$.
Therefore $\tau\subseteq{\rm{\rm NT}}$.
###### Lemma 12
$\partial{\rm NT}\subseteq{\rm NT}.$
###### Proof
Because the loop guard is of the form $B(x_{1},x_{2},x_{3})^{T}\geq 0$, NT is
a closed set. So the conclusion is correct. Furthermore, for any
$(x_{1},x_{2},x_{3})\in\partial{\rm NT},x_{1}+2x_{2}+x_{3}\geq 0.$
###### Lemma 13
If $(x_{1},x_{2},x_{3})\in{\rm NT}$ and
$A(x_{1},x_{2},x_{3})^{T}\in\partial{\rm NT}$, then
$(x_{1},x_{2},x_{3})\in\partial{\rm NT}.$
###### Proof
Let $\vec{x}=(x_{1},x_{2},x_{3})$. If the conclusion is not true, there exists
a ball $o(\vec{x},r)\subseteq{\rm NT}$. Because $A\vec{x}^{T}\in\partial{\rm
NT}$, there exists $\vec{x^{\prime}}$ such that
$|A\vec{x}-\vec{x^{\prime}}|<r$ and $\vec{x^{\prime}}$ is not in NT.
Since $|A^{-1}\vec{x^{\prime}}-\vec{x}|<|\vec{x^{\prime}}-A\vec{x}|<r$,
$A^{-1}\vec{x^{\prime}}\in o(\vec{x},r)$. So $A^{-1}\vec{x^{\prime}}\in{\rm
NT}$ and thus $\vec{x^{\prime}}\in{\rm NT}$, which is a contradiction.
###### Lemma 14
$\\{(\frac{1}{2^{n}},-\frac{1}{3^{n}},\frac{1}{5^{n}})\\}_{n=0}^{\infty}\subseteq\partial{\rm
NT}.$
###### Proof
Let $\vec{p}_{n}=(\frac{1}{2^{n}},-\frac{1}{3^{n}},\frac{1}{5^{n}}),n\geq 0.$
We use induction on the value of $n$.
When $n=0$, because $B\vec{p}_{0}=B(1,-1,1)^{T}=0$ and
$BA^{k}\vec{p}_{0}=2^{k}-2\times 3^{k}+5^{k}>0\ ~{}~{}{\rm for\ any}\
k\in{\mathbb{N}}^{+},$
we have $\vec{p}_{0}\in\partial{\rm NT}.$
Now assume that the conclusion holds for $n-1$. So,
$A\vec{p}_{n}=\vec{p}_{n-1}\in\partial{\rm NT}\subseteq{\rm NT}.$ By Lemma 13,
$\vec{p}_{n}\in\partial{\rm NT}$.
###### Lemma 15
For any non-zero polynomial
$f(x_{1},x_{2},x_{3})\in{\mathbb{R}}[x_{1},x_{2},x_{3}]$, there exists an $N$
such that $f(\frac{1}{2^{n}},-\frac{1}{3^{n}},\frac{1}{5^{n}})\neq 0$ for all
$n>N$.
###### Proof
Assume that the conclusion does not hold. Then there exists a subsequence
$\\{((\frac{1}{2})^{n_{k}},-(\frac{1}{3})^{n_{k}},(\frac{1}{5})^{n_{k}})\\}_{k=1}^{\infty}$
such that $f$ vanishes on each point of it.
Let
$f=b_{1}x_{1}^{\alpha_{1}}x_{2}^{\beta_{1}}x_{3}^{\gamma_{1}}+...+b_{s}x_{1}^{\alpha_{s}}x_{2}^{\beta_{s}}x_{3}^{\gamma_{s}}$
where $b_{i}\in\mathbb{R},b_{i}\neq
0,\alpha_{i}\in\mathbb{N},\beta_{i}\in\mathbb{N},\gamma_{i}\in\mathbb{N},$ and
$(\alpha_{i},\beta_{i},\gamma_{i})\neq(\alpha_{j},\beta_{j},\gamma_{j})$ for
$i\neq j.$
Obviously $s\geq 1$ because $f\not\equiv 0$. Let
$t_{i}=(\frac{1}{2})^{\alpha_{i}}(\frac{1}{3})^{\beta_{i}}(\frac{1}{5})^{\gamma_{i}}$.
It is an obvious fact that $2^{\alpha_{j}}3^{\beta_{j}}5^{\gamma_{j}}\neq
2^{\alpha_{i}}3^{\beta_{i}}5^{\gamma_{i}}$ for $i\neq j.$ Hence
$t_{1},t_{2},...,t_{s}$ are pairwise distinct. Without loss of generality, let
$t_{1}>t_{2}>...>t_{s}.$
For every $j>1,$ we have
$\lim\limits_{k\to\infty}{(\frac{t_{j}}{t_{1}})^{n_{k}}}=0$. Thus
$\lim\limits_{k\to\infty}{|\frac{f((\frac{1}{2})^{n_{k}},-(\frac{1}{3})^{n_{k}},(\frac{1}{5})^{n_{k}})}{((\frac{1}{2})^{\alpha_{1}}(\frac{1}{3})^{\beta_{1}}(\frac{1}{5})^{\gamma_{1}})^{n_{k}}}|=|b_{1}|}\neq
0\enspace.$
This contradicts with
$f((\frac{1}{2})^{n_{k}},-(\frac{1}{3})^{n_{k}},(\frac{1}{5})^{n_{k}})=0$.
Therefore the conclusion follows.
Using the above lemmas, we can now prove Theorem 4.1.
###### Proof
Denote by $S$ the sequence
$\\{(\frac{1}{2})^{n},-(\frac{1}{3})^{n},(\frac{1}{5})^{n})\\}$. By Lemma 14,
$S\subseteq\partial{\rm NT}.$
Assume ${\rm NT}$ is a semi-algebraic set. Then there exist finite many
polynomials $f_{i,j}\in\mathbb{R}[x_{1},x_{2},x_{3}]$ and
$\triangleleft_{i,j}\in\\{<,=\\}$ for $i=1,...,s$ and $j=1,...,r_{i}$ such
that
${\rm
NT}=\bigcup\limits_{i=1}^{s}\bigcap\limits_{j=1}^{r_{i}}\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}|f_{i,j}\triangleleft_{i,j}0\\}.$
(3)
Because $S\subseteq\partial{\rm NT}\subseteq\\{f_{i,j}=0\\}_{i,j}$, for any
$x\in S$, there exists a polynomial $f_{i,j}$ such that $f_{i,j}(x)=0$. By
pigeonhole principle there exists an $f_{i,j}$ and a subsequence $S_{1}$ of
$S$ such that $f_{i,j}$ vanishes on $S_{1}$, which contradicts with Lemma 15.
## 5 Conclusion
In this paper, we consider whether the NT of a simple linear loop is decidable
and how to compute it if it is decidable. For homogeneous linear loops with
only two program variables, we give a complete algorithm for computing the NT.
For the case of more program variables, we show that the NT cannot be
described by Tarski formulae in general.
## Acknowledgements
The work is partly supported by NNSFC 91018012 and the EXACTA project from ANR
and NSFC.
## References
* [1] M. Braverman: Termination of Integer Linear Programs. CAV 2006, LNCS 4114, 372–385, 2006.
* [2] Y. Chen, B. Xia, L. Yang, N. Zhan and C. Zhou: Discovering Non-linear ranking functions by Solving Semi-algebraic Systems. LNCS 4711, 34–49, 2007.
* [3] M. A. Colón and H. B. Sipma: Synthesis of linear ranking functions. TACAS 01, LNCS 2031, 67–81, 2001.
* [4] D. Dams, R. Gerth, and O. Grumberg: A heuristic for the automatic generation of ranking functions. Workshop on Advances in Verification (WAVe 00), 1–8, 2000.
* [5] C. S. Lee, N. D. Jones and A. M. Ben-Amram: The size-change principle for program termination. POPL, 81–92, 2001.
* [6] A. Podelski and A. Rybalchenko: A complete method for the synthesis of linear ranking functions. VMCAI, LNCS 2937, 465–486, 2004.
* [7] A. Tiwari: Termination of Linear Programs. CAV 2004, LNCS 3114, 70–82, 2004.
* [8] B. Xia and Z. Zhang: Termination of linear programs with nonlinear constraints, Journal of Symbolic Computation, 45: 1234–1249, 2010.
* [9] B. Xia, L. Yang, N. Zhan and Z. Zhang: Symbolic decision procedure for termination of linear programs. Formal Aspects of Computing, 23:171–190, 2011.
* [10] A. Gupta, T. Henzinger, R. Majumdar, A. Rybalchenko and R.-G. Xu: Proving non-termination, POPL, 147–158, 2008.
* [11] S. Gulwani, S. Srivastava and R. Venkatesan: Program analysis as constraint solving, POPL, 281–292, 2008.
* [12] S. Zhao and D. Chen: Decidability Analysis on Termination Set of Loop Programs. The International Conference on Computer Science and Service System(CSSS), 3124–3127, 2011.
|
arxiv-papers
| 2012-05-31T11:28:23 |
2024-09-04T02:49:31.460881
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liyun Dai and Bican Xia",
"submitter": "Dai Liyun",
"url": "https://arxiv.org/abs/1206.0232"
}
|
1206.0279
|
# Magnetic Neutron Scattering of Thermally Quenched K-Co-Fe Prussian Blue
Analogue Photomagnet
Daniel M. Pajerowski NIST Center for Neutron Research, Gaithersburg, MD
20899-6012, USA Department of Physics and National High Magnetic Field
Laboratory, University of Florida, Gainesville, FL 32611-8440, USA V. Ovidiu
Garlea Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831-6393, USA Elisabeth S. Knowles Department of Physics and
National High Magnetic Field Laboratory, University of Florida, Gainesville,
FL 32611-8440, USA Matthew J. Andrus Department of Chemistry, University of
Florida, Gainesville, FL 32611-7200, USA Matthieu F. Dumont Department of
Physics and National High Magnetic Field Laboratory, University of Florida,
Gainesville, FL 32611-8440, USA Department of Chemistry, University of
Florida, Gainesville, FL 32611-7200, USA Yitzi M. Calm Department of Physics
and National High Magnetic Field Laboratory, University of Florida,
Gainesville, FL 32611-8440, USA Stephen E. Nagler Quantum Condensed Matter
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA Xin
Tong Instrument and Source Design Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831-6393, USA Daniel R. Talham Department of Chemistry,
University of Florida, Gainesville, FL 32611-7200, USA Mark W. Meisel
Department of Physics and National High Magnetic Field Laboratory, University
of Florida, Gainesville, FL 32611-8440, USA
###### Abstract
Magnetic order in the thermally quenched photomagnetic Prussian blue analogue
coordination polymer K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O has
been studied down to 4 K with unpolarized and polarized neutron powder
diffraction as a function of applied magnetic field. Analysis of the data
allows the onsite coherent magnetization of the Co and Fe spins to be
established. Specifically, magnetic fields of 1 T and 4 T induce moments
parallel to the applied field, and the sample behaves as a ferromagnet with a
wandering axis.
###### pacs:
75.50.Xx, 75.25.-j, 75.30.Gw, 75.50.LK
## I Introduction
Manipulating magnetization with photons is now a major research focus because
it may yield materials capable of dense information storage. An epitomic
example of a photomagnetic coordination polymer is potassium cobalt
hexacyanoferrate, KαCo[Fe(CN)6]${}_{\beta}\cdot$nH2O (from now on referred to
as Co-Fe, with the crystal structure shown in Fig. 1), which displays magnetic
order and an optical charge transfer induced spin transition (CTIST).Sato1996
The details of the magnetism in Co-Fe have been investigated with bulk probes
such as magnetization,Sato1999 and AC-susceptibility,Pejakovic2002 as well
as atomic level probes such as X-ray magnetic circular dichroism
(XMCD),Champion2001 and muon spin relaxation ($\mu$-SR).Salman2006 However,
we utilize neutron scattering because it is capable of extracting the magnetic
structure, including the length and direction of the magnetic moments
associated with different crystallographic positions.
Neutron scattering research has been important in understanding the structure
of materials similar to Co-Fe. For example, neutron diffraction has been used
to elucidate the location of water molecules, to identify the long-range
magnetic order, and to explore the spin delocalizetion in Prussian
blue.Buser1977 ; Herren1980 ; Day1980 Later work used similar techniques to
investigate hydrogen adsorption in Cu3[Co(CN)6]2, along with vibrational
spectroscopy,Hartman2006 and neutron vibrational spectroscopy was also
measured in Zn3[Fe(CN)6]2.Adak2010 Likewise, magnetic structure determination
with neutrons was used to explore negative magnetization in
Cu0.73Mn0.77[Fe(CN)${}_{6}]\cdot$nH2O,Kumar2008 and to extract on-site
moments in Berlin greenKumar2005 ; Kumar2004 and in (NixMn1-x)3[Cr(CN)6]2
molecule-based magnets.Mihalik2010
Figure 1: (Color online) Co-Fe unit cell. Crystallographic positions of atoms
within the unit cell are illustrated with cubes for K (cyan), Co (red), Fe
(blue), C (white), N (black), and coordinated D2O, O positions (yellow), while
the interstitial D2O density is displayed using contoured isosurfaces (green)
and Fe-C bonds are displayed as tubes (white). Details of structure
determination are presented in Section III.
To this end, we have performed neutron powder diffraction (NPD) on deuterated
Co-Fe samples in magnetic states resulting from thermal quenching. Briefly, at
room temperature, the photomagnetic Co-Fe with optimal iron vacancies is
paramagnetic, with a transition to the diamagnetic low-spin state when cooling
below nominally 200 K. It is below 100 K that applied light may convert
molecules from diamagnetic to paramagnetic and back, and below around 20 K
where this effect is most striking due to the large susceptibility of the
magnetically ordered state. However, a magnetically ordered state may also be
achieved at low temperatures by thermally quenching,Park2007 ; Chong2011
where the paramagnetic 300 K state is cooled so quickly to below 100 K that it
does not relax to the diamagnetic ground state. It is this magnetic, thermally
quenched state that we study with NPD as a function of magnetic field, while
complementary magnetization, transmission electron microscopy (TEM), Fourier
transform infrared spectroscopy (FT-IR), and elemental analysis have also been
performed on the sample. We find that Co-Fe possesses a correlated spin glass
ground state that is driven via magnetic field to behave as a ferromagnet with
a wandering axis.
## II Experimental DetailsNISTdisclaimer
### II.1 Synthesis
To begin preparation of KαCo[Fe(CN)6]${}_{\beta}\cdot$nD2O powder, a 75 mL
solution of 0.1 mol/L KNO3 in D2O was added to a 75 mL solution of
20$\times$10-3 mol/L K3[Fe(CN)6] in D2O, and stirred for ten minutes. While
continuing to stir, a 300 mL solution of 5$\times$10-3 mol/L CoCl2 in D2O was
added drop-wise over the course of two hours. Stirring of the final solution
was allowed to continue two additional hours subsequent to complete mixing.
Next, the precipitate was collected by centrifugation at 2000 rpm (210 rad/s)
for 10 min (600 s) and dried under vacuum. This procedure was repeated 14
times until 4.37 g of powder was collected. Potassium ferricyanide, anhydrous
cobalt chloride, and potassium nitrate were all purchased from Sigma-Aldrich.
To remove water, the potassium nitrate was heated in an oven to 110 ∘C (383 K)
for 4 h before use. All other reagents and chemicals were used without further
purification. Deuterium oxide was purchased from Cambridge Isotope
Laboratories, Inc.
### II.2 Instrumentation
Neutron powder-diffraction experiments were conducted using the HB2A
diffractometer at the High Flux Isotope Reactor,Garlea2010 using the Ge[113]
monochromator with $\lambda$ = 2.41 $\mathrm{\AA}$ (0.241 nm). Sample
environment on HB2A utilized an Oxford 5 T, vertical-field magnet with helium
cryogenics. Neutron polarization was achieved with a 3He cell that produced
79$\%$ polarization at the beginning of the experiment and decayed to 63$\%$
polarization after 20 hours at the end of the experiment, to give an average
polarization of 71$\%$ for both up and down polarization measurements, and we
did not perform polarization analysis after the sample but instead followed
established methods for powder diffraction with polarized
neutrons.Lelievre2010 ; Wills2005 The flipping difference spectra were
obtained by subtracting the diffraction data, measured with the incident
neutron polarization parallel to the applied field and magnetization, from the
data recorded with the incident polarization antiparallel to the field.
Magnetic measurements were performed using a Quantum Design MPMS XL
superconducting quantum interference device (SQUID) magnetometer. Infrared
spectra were recorded on a Thermo Scientific Nicolet 6700 spectrometer. Energy
dispersive X-ray spectroscopy (EDS) and TEM were conducted on a JEOL 2010F
super probe by the Major Analytical Instrumentation Center at the University
of Florida (UF). The UF Spectroscopic Services Laboratory performed combustion
analysis.
### II.3 Analysis Preparations
For NPD, 4.37 g of powder were mounted in a cylindrical aluminum can. Thermal
quenching to trap the magnetic state was achieved by filling the cryostat bath
with liquid helium and directly inserting a sample stick from ambient
temperature. To avoid hydrogen impurities, the powder was wetted with
deuterium oxide, and to avoid sample movement in magnetic fields, an aluminum
plug was inserted above the sample. To measure magnetization, samples heavier
than 10 mg were mounted in gelcaps and held in plastic straws. Thermal
quenching in the SQUID was achieved by equilibrating the cryostat to 100 K,
and directly inserting the sample stick from ambient temperature. For
measurements in 10 mT, samples are cooled through the ordering temperature in
10 mT, and for 1 T and 4 T measurements, there is no observed thermal
hysteresis. For FT-IR, less than 1 mg amounts of sample were suspended in an
acetone solution and deposited on KBr salt plates and allowed to dry. For EDS
and TEM, acetone suspensions of the powder were deposited onto 400 mesh copper
grids with an ultrathin carbon film on a holey carbon support obtained from
Ted Pella, Inc.
### II.4 Diffraction analysis scheme
Intensities were fit to the standard powder diffraction equation with a
correction for absorption,
$\displaystyle I(\theta)$ $\displaystyle=$ $\displaystyle
A_{0}\frac{m_{hkl}|F(hkl)|^{2}}{\sin\theta\,\sin
2\theta}\;\eta(\theta)~{}~{}~{},$ (1)
with
$\displaystyle\eta(\theta)$ $\displaystyle=$ $\displaystyle
e^{-(1.713-0.037\sin^{2}\theta)\mu
R+(0.093+0.375\sin^{2}\theta)\mu^{2}R^{2}},$ (2)
where $A_{0}$ is an overall scale factor, $m_{hkl}$ is the multiplicity of the
scattering vector, $F$ is the structure factor, $\theta$ is the scattering
angle, $\mu$ is the linear attenuation coefficient, and $R$ is the radius of
the sample cylinder. For our sample and experimental arrangement, $\mu
R=0.17$, which has little effect on the observed intensities aside from scale.
The structure factor has nuclear ($F_{N}$) and magnetic ($F_{M}$)
contributions, and for unpolarized neutrons
$|F|^{2}~{}=~{}|F_{N}+F_{M}|^{2}~{}=~{}|F_{N}|^{2}+|F_{M}|^{2}~{}~{}~{}.$ (3)
On the other hand, $F_{N}$ and $F_{M}$ can coherently interfere for polarized
neutrons such that, for moments co-linear with $P$,
$|F|^{2}~{}=~{}|F_{N}+F_{M}|^{2}~{}=~{}|F_{N}|^{2}+|F_{M}|^{2}\pm
2PF_{N}F_{M}~{}~{}~{},$ (4)
where $P$ is the neutron polarization fraction and the sign of the final term
depends upon up or down neutron polarization.Schweizer2006 For nuclear
scattering,
$F_{N}(hkl)~{}=~{}\sum_{j}{n_{j}b_{j}e^{iG\cdot d_{j}}e^{-W_{j}}}~{}~{}~{},$
(5)
where the sum is over all atoms in the unit cell, $n$ is related to the
average occupancy, $b$ is the coherent nuclear scattering length, $G$ is the
$hkl$ dependent reciprocal lattice vector, $d$ is the direct space atomic
position, and $W=BQ^{2}/16\pi^{2}$ is the Debye-Waller factor. For magnetic
scattering, all coherent scattering is modeled to be along the applied field,
which is perpendicular to the scattering plane, so that
$F_{M}(hkl)~{}=~{}\frac{\gamma r_{0}}{2}\sum_{j}{m_{j}(Q)e^{iG\cdot
d_{j}}e^{-W_{j}}}~{}~{}~{},$ (6)
where $\frac{\gamma r_{0}}{2}~{}=~{}2.695$ fm, and the magnetization can be
written as
$\displaystyle m_{j}(Q)$ $\displaystyle=$ $\displaystyle\langle
L_{z}\rangle_{j}f_{L,j}(Q)~{}+~{}2\langle S_{z}\rangle_{j}f_{S,j}(Q)$ (7)
$\displaystyle=$ $\displaystyle g_{J,j}\langle J_{z}\rangle_{j}f_{J,j}(Q)$
$\displaystyle=$ $\displaystyle\langle
J_{z}\rangle_{j}(g_{L,j}f_{L,j}(Q)~{}+~{}g_{S,j}f_{S,j}(Q))~{}~{}~{},$
where $\langle J_{z}\rangle$ is the average total angular momentum, $\langle
L_{z}\rangle$ is the average orbital angular momentum, $\langle S_{z}\rangle$
is the average spin angular momentum, $f_{J}(Q)$ is the magnetic form factor
for the total angular momentum, $f_{L}(Q)$ is the magnetic form factor for the
orbital angular momentum, $f_{S}(Q)$ is the magnetic form factor for the spin
angular momentum, and $g_{J}$, $g_{L}$ and $g_{S}$ may be determined by
Wigner’s formula.Sakurai1994 The tabulated form factor values within the
dipole approximation are used for the spin and orbital form
factors.Clementi1974 Squared differences between observed and calculated
intensities were minimized using a Nelder-Mead simplex algorithm. Open-source
Python 2.7 libraries were utilized to aid in plotting routines, matplotlib
1.0.1 and Mayavi2, and computation, NumPy 1.6.1 and SciPy 0.7.2. Reported
uncertainties of fit parameters are the square root of the diagonal terms in
the covariance matrix multiplied by the standard deviation of the residuals.
Figure 2: (Color online) Neutron powder diffraction of Co-Fe at $T$ = 40 K. Observed scattering is shown as open circles (obs), a full fit including sample mount contributions is shown as a black line (calc’d), the diffuse background is illustrated with a magenta line (bgr), and the residuals of the fit are shown below the zero line with a green line (resid). The signal due to Co-Fe is emphasized with a red filling. Experimental uncertainties derived from counting statistics are smaller than the plotting symbols. Table 1: Atomic coordinates and occupancies for Co-Fe at $T$ = 40 K. atom | position | $n$ | x | y | z
---|---|---|---|---|---
Co | 4a | 1 | 0.5 | 0.5 | 0.5
Fe | 4b | 0.73 | 0 | 0 | 0
C | 24e | 0.73 | 0.212 | 0 | 0
N | 24e | 0.73 | 0.313 | 0 | 0
K | 8c | 0.135 | 0.25 | 0.25 | 0.25
O | 24e | 0.27 | 0.243 | 0 | 0
D | 96k | 0.135 | 0.303 | 0.060 | 0.060
Figure 3: (Color online) Magnetic neutron powder diffraction of Co-Fe at $T$ =
4 K as a function of applied magnetic field. The difference between the $T$ =
40 K diffractogram and the $T$ = 4 K diffractogram (open circles) along with
profile fits to intensities (red line from model #1 for 4 T and 1 T data in
Table II) are shown for (a) 4 T, (b) 1 T, and (c) 10 mT. Additionally, the
difference between the $T$ = 4 K up-neutron-polarization diffractogram and the
$T$ = 4 K down-neutron-polarization diffractogram (open circles) along with
profile fits to intensity (red line from model #5 for polarized data in Table
II) are shown for (d) 1 T. Uncertainty bars on experimental data points are
statistical in nature representing one standard deviation from the mean, using
counting statistics.
## III Results and Analyses
The nuclear crystal structure of Co-Fe can be modeled with space group
$Fm\overline{3}m$ (No. 225), where ferricyanide molecules and cobalt ions are
alternately centered on the high symmetry points of the unit cell, with heavy-
water bound to cobalt when ferricyanide is absent, and potassium ions and
heavy-water molecules filling in voids.Herren1980 ; Buser1977 ; Hanawa2003
This structure is used as a starting point to fit the $T$ = 40 K thermally
quenched Co-Fe contribution to the measured intensity profile, Fig. 2, which
also has sample mount contributions due to $P63/mmc$ (No. 194) D2O and
$Fm\overline{3}m$ (No. 225) aluminum.Dowell1960 ; Hull1917 Incomplete
trapping of the high-temperature state in Co-Fe gives rise to a highly
microstrained nuclear structure,Hanawa2003 and to account for this effect
during refinement, we use an asymmetric double sigmoidal peak shape, namely
$y_{a2s}~{}=~{}\frac{I}{2.49w}\left(1-\frac{1}{1+e^{-\frac{\theta-\theta_{c}}{3.43w}}}\right)\left(\frac{1}{1+e^{-\frac{\theta-\theta_{c}}{w}}}\right)~{}~{}~{},$
(8)
where $I$ is the intensity, $w$ is the width, and $\theta_{c}$ is the center
of the reflection, and these fits yield an effective lattice constant of 10.23
$\mathrm{\AA}$. Observed Co-Fe reflections that can be clearly separated from
sample holder reflections are used to extract structure factors. In modeling
the unit cell, the cobalt to iron ratio was determined with EDS, while the
room temperature oxidation states with FT-IR. The carbon and nitrogen content
were established with combustion analysis, and the potassium ions provide
charge balance. Finally, the heavy-water concentration and positions were
refined along with the scale factor to fit the structure factors.
Table 2: Comparison of the eight magnetic models, as described in the text, numbered (#) $1-8$ for Co-Fe at $T$ = 4 K in different magnetic fields tabulated as “cond.”, which is shorthand for experimental condition, where “$P$” designates the data acquired with polarized neutrons. Here, “align.” is short for “moment alignment,” where + denotes parallel alignment of moments and - denotes antiparallel alignment of moments. The units of $m_{z,Co}$ and $m_{z,Fe}$ are $\mu_{B}$, and the units of M are $\mu_{B}$ mol-1. The sum of the residuals are normalized to model 1 for each experimental condition. # | cond. | align. | $g_{S,Co}$ | $g_{L,Co}$ | $g_{S,Fe}$ | $g_{L,Fe}$ | $J_{z,Co}$ | $J_{z,Fe}$ | $m_{z,Co}$ | $m_{z,Fe}$ | M | $\sum_{j}{residual^{2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 4 T | + | 10/3 | 1 | 2/3 | 4/3 | 0.63 $\pm$ 0.02 | 0.13 $\pm$ 0.03 | 2.7 | 0.3 | 2.6 | 1.000
2 | 4 T | - | 10/3 | 1 | 2/3 | 4/3 | 0.62 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 2.7 | 0.0 | 2.4 | 1.019
3 | 4 T | + | 10/3 | 1 | 2 | 0 | 0.64 $\pm$ 0.02 | 0.12 $\pm$ 0.04 | 2.8 | 0.2 | 2.6 | 1.000
4 | 4 T | - | 10/3 | 1 | 2 | 0 | 0.63 $\pm$ 0.02 | 0.00 $\pm$ 0.11 | 2.7 | 0.0 | 2.5 | 1.020
5 | 4 T | + | 2 | 0 | 2/3 | 4/3 | 1.08 $\pm$ 0.03 | 0.16 $\pm$ 0.03 | 2.2 | 0.3 | 2.1 | 1.028
6 | 4 T | - | 2 | 0 | 2/3 | 4/3 | 1.04 $\pm$ 0.03 | 0.00 $\pm$ 0.03 | 2.1 | 0.0 | 1.9 | 1.049
7 | 4 T | + | 2 | 0 | 2 | 0 | 1.09 $\pm$ 0.03 | 0.12 $\pm$ 0.04 | 2.2 | 0.2 | 2.1 | 1.030
8 | 4 T | - | 2 | 0 | 2 | 0 | 1.05 $\pm$ 0.03 | 0.00 $\pm$ 0.03 | 2.1 | 0.0 | 1.9 | 1.048
1 | 1 T | + | 10/3 | 1 | 2/3 | 4/3 | 0.37 $\pm$ 0.03 | 0.20 $\pm$ 0.09 | 1.6 | 0.4 | 1.7 | 1.000
2 | 1 T | - | 10/3 | 1 | 2/3 | 4/3 | 0.37 $\pm$ 0.02 | 0.00 $\pm$ 0.04 | 1.6 | 0.0 | 1.4 | 1.020
3 | 1 T | + | 10/3 | 1 | 2 | 0 | 0.38 $\pm$ 0.03 | 0.12 $\pm$ 0.07 | 1.6 | 0.2 | 1.6 | 1.004
4 | 1 T | - | 10/3 | 1 | 2 | 0 | 0.38 $\pm$ 0.05 | 0.00 $\pm$ 0.11 | 1.6 | 0.0 | 1.5 | 1.019
5 | 1 T | + | 2 | 0 | 2/3 | 4/3 | 0.64 $\pm$ 0.04 | 0.20 $\pm$ 0.09 | 1.3 | 0.4 | 1.4 | 1.001
6 | 1 T | - | 2 | 0 | 2/3 | 4/3 | 0.64 $\pm$ 0.10 | 0.00 $\pm$ 0.21 | 1.3 | 0.0 | 1.2 | 1.022
7 | 1 T | + | 2 | 0 | 2 | 0 | 0.65 $\pm$ 0.04 | 0.13 $\pm$ 0.07 | 1.3 | 0.3 | 1.3 | 1.006
8 | 1 T | - | 2 | 0 | 2 | 0 | 0.63 $\pm$ 0.09 | 0.00 $\pm$ 0.08 | 1.3 | 0.0 | 1.1 | 1.021
1 | $P$ | + | 10/3 | 1 | 2/3 | 4/3 | 0.38 $\pm$ 0.01 | 0.06 $\pm$ 0.02 | 1.6 | 0.1 | 1.5 | 1.000
2 | $P$ | - | 10/3 | 1 | 2/3 | 4/3 | 0.40 $\pm$ 0.02 | 0.00 $\pm$ 0.01 | 1.7 | 0.0 | 1.5 | 1.058
3 | $P$ | + | 10/3 | 1 | 2 | 0 | 0.36 $\pm$ 0.02 | 0.18 $\pm$ 0.05 | 1.6 | 0.4 | 1.6 | 1.004
4 | $P$ | - | 10/3 | 1 | 2 | 0 | 0.40 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 1.7 | 0.0 | 1.5 | 1.065
5 | $P$ | + | 2 | 0 | 2/3 | 4/3 | 0.68 $\pm$ 0.02 | 0.06 $\pm$ 0.02 | 1.4 | 0.1 | 1.3 | 0.995
6 | $P$ | - | 2 | 0 | 2/3 | 4/3 | 0.71 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 1.4 | 0.0 | 1.3 | 1.058
7 | $P$ | + | 2 | 0 | 2 | 0 | 0.66 $\pm$ 0.03 | 0.16 $\pm$ 0.06 | 1.3 | 0.3 | 1.3 | 1.005
8 | $P$ | - | 2 | 0 | 2 | 0 | 0.74 $\pm$ 0.03 | 0.00 $\pm$ 0.05 | 1.5 | 0.0 | 1.3 | 1.079
To begin, refinement yielded interstitial heavy-water pseudo-atoms at the 8c
position ($n$ = 0.618, $B$ = 5) and the 32f position (x = 0.3064, $n$ = 0.333,
$B$ = 20), after which all other parameters were fixed (Table 1) and Fourier
components of the heavy-water were further refined to give the interstitial
distribution shown in Fig. 1. These refinements give a chemical formula of
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O. Moreover, at room
temperature, the more complete chemical formula with oxidation states of the
metal ions included is
K0.27Co${}^{2+}_{0.94}$Co${}^{3+}_{0.06}$[Fe3+(CN)6]0.58
[Fe2+(CN)6]0.15[D2O6]${}_{0.27}\cdot$1.42D2O, or more compactly represented by
Co${}^{2+}_{0.94}$Co${}^{3+}_{0.06}$Fe${}^{3+}_{0.58}$Fe${}^{2+}_{0.15}$.
Having highly ionic wavefunctions, the magnetic ground states of Co and Fe in
Co-Fe are well described with ligand field theory.Figgis2000 As displayed in
Fig. 1, the iron atoms are octahedrally coordinated by carbon atoms that
introduce a ligand field splitting parameter ($\Delta_{\mathrm{Fe}}$) of
approximately 0.70 aJ (35,000 cm-1 or 4.3 eV), and typical Fe Racah parameters
put $d^{5}-$Fe3+ into a ${}^{2}T_{2g}$ ground state, and $d^{6}-$Fe2+ into a
diamagnetic ${}^{1}A_{1g}$ ground state. Similarly, cobalt atoms are
octahedrally coordinated with oxygen and nitrogen atoms to give
$\Delta_{\mathrm{Co}^{3+}}~{}\approx~{}$0.46 aJ (23,000 cm-1 or 2.9 eV) for
$d^{6}-$Co3+ that has a diamagnetic ${}^{1}A_{1g}$ ground state, and
$\Delta_{\mathrm{Co}^{2+}}~{}\approx~{}$0.20 aJ (10,000 cm-1 or 1.2 eV) for
$d^{7}-$Co2+ that has a ${}^{4}T_{1g}$ ground state, using typical Co Racah
parameters. At temperatures much less than the spin-orbit coupling energy,
only the lowest energy total angular momentum levels are appreciably occupied,
so that the relevant states are Fe3+[$J=1/2$, $g_{J}=(2+4k)/3$, $g_{L}=4k/3$,
$g_{S}=2/3$] and Co2+[$J=1/2$, $g_{J}=(10+2Ak)/3$, $g_{L}=2Ak/3$,
$g_{S}=10/3$], where $A$ is expected to be nearly 1.5 due to the weak ligand
field, and $k$ is the orbital reduction parameter. It is worth noting that
analogous orbitally degenerate terms have been observed for
$d^{5}-$Fe3+(${}^{2}T_{2g}$) in K3Fe(CN)6,Figgis1969 and
$d^{7}-$Co2+(${}^{4}T_{1g}$) in K1.88Co[Fe(CN)6]${}_{0.97}\cdot$3.8H2O and
Na1.52K0.04Co[Fe(CN)6]${}_{0.89}\cdot$3.9H2O.Matsuda2010 Alternatively, if
interaction with the lattice drastically quenches the orbital moment, spin-
orbit coupling no longer splits the ground states and the magnetic parameters
become Fe3+[$J=1/2$, $g_{J}=2$, $g_{L}=0$, $g_{S}=2$] and Co2+[$J=3/2$,
$g_{J}=2$, $g_{L}=0$, $g_{S}=2$]. To estimate the relative proportion of the
different oxidation states in the thermally quenched state, the effective
paramagnetic moment is linearized as a function of temperature for the 300 K
and 100 K states,PajerowskiPHD and the measured lattice constant is compared
to a weighted average of quenched and ground-state lattice constants,Chong2011
to self-consistently give
Co${}^{2+}_{0.90}$Co${}^{3+}_{0.10}$Fe${}^{3+}_{0.54}$Fe${}^{2+}_{0.19}$ for
the magnetic quenched state at 100 K and below that is analyzed in detail
herein.
Cooling the sample further, subsequent to quenching, the bulk magnetization
measured in 10 mT showed the well-documented upturn at around 15 K
corresponding to the onset of magnetic order. Therefore, additional NPD was
performed at 4 K in applied fields of 10 mT, 1 T, and 4 T, Fig. 3, to compare
to the scattering in the paramagnetic state. Furthermore, polarized NPD was
performed at 4 K in an applied field of 1 T, Fig. 3 (d), where the difference
between diffractograms for up and down incident neutron polarizations
increases signal to noise of the measured magnetic structure at reflections
with large nuclear contributions. For each of the three experimental
conditions where magnetic scattering is observed, we compare the results of
eight plausible but different models that all have moments along the applied
field. Specifically, each possible case considers various combinations of the
parallel or antiparallel alignment of Co and Fe moments when each ion
possesses either spin-only or orbitally degenerate magnetic states, Table 2.
The analyses indicate that most magnetism resides on the Co 4a site for all
models, with a parallel alignment of Fe and Co moments giving the best fits
and $\chi^{2}$ surfaces suggesting a reduced but present orbital moment on
both ions. No magnetic scattering is observed in 10 mT, and increased coherent
magnetic scattering appears with increasing field, which is consistent with
the presence of significant random anisotropy, where a correlated spin glass
(CSG) is the ground state and sufficiently large fields cause entrance into a
ferromagnetic phase with wandering axis (FWA) state or at even larger fields a
nearly collinear (NC) state.Chudnovsky1986
Analytical expressions for the magnetization process for magnets with random
anisotropy are availableChudnovsky1986 for the Hamiltonian
$\displaystyle\mathcal{H}~{}=~{}-\mathcal{J}\sum_{i,j}{S_{i}\cdot
S_{j}}-D_{r}\sum_{i}{(\hat{n_{i}}\cdot S_{i})^{2}}$ $\displaystyle-
D_{c}\sum_{i}{(S_{i}^{z})^{2}}-g\mu_{B}\sum_{i}{H\cdot S_{i}}~{}~{}~{},$ (9)
where $\mathcal{J}$ is the superexchange constant, $S$ is the spin operator,
$D_{r}$ is strength of the random anisotropy, $\hat{n}$ is the direction of
the random on-site anisotropy, $D_{c}$ is the strength of the coherent
anisotropy, $g$ is the Land$\acute{\mathrm{e}}$ factor, and $H$ is the applied
field. In the FWA regime where the applied field energy is larger than the
random anisotropy energy but much less than the exchange field, the low
temperature magnetization is
$\displaystyle M_{FWA}$ $\displaystyle=$ $\displaystyle
M_{S}-\frac{6\sqrt{2}D_{r}^{2}\Omega
M_{S}}{5\pi^{2}a^{3}(z\mathcal{J})^{3/2}(H+H_{C})^{1/2}}$ (10)
$\displaystyle=$ $\displaystyle
M_{S}\left(1-\frac{D_{FWA}^{1/2}}{(H+H_{C})^{1/2}}\right)~{}~{}~{},$
where $M_{S}$ is the saturation magnetization, $\Omega$ is the integrated
local anisotropy correlation function, $z$ is the number of magnetic
neighbors, $a$ is the mean distance between neighboring spin sites, $H_{C}$ is
the coherent anisotropy field, and $D_{FWA}$ is a measure of the random
anisotropy to superexchange strengths. For the NC phase that is reached when
the applied field and coherent anisotropy field are much larger than the
exchange field, the low temperature magnetization is
$\displaystyle M_{NC}$ $\displaystyle=$ $\displaystyle
M_{S}-\frac{4D_{r}^{2}M_{S}}{15a^{6}(H+H_{C})^{2}}$ (11) $\displaystyle=$
$\displaystyle M_{S}\left(\frac{1-D_{NC}^{2}}{(H+H_{C})^{2}}\right)~{}~{}~{},$
where $D_{NC}$ is a measure of the random anisotropy. Based upon the magnetic
ordering temperature, Co-Fe is expected to be in an FWA-like phase, and
although both FWA and NC expressions may be fit to the low temperature
magnetization data, Fig. 4, the parameters extracted by the NC fit are not
consistent with the derivation limit for Eq. 11, further suggesting an FWA-
like state.
Figure 4: (Color online) SQUID magnetization of Co-Fe at $T$ = 2 K.
Experimental magnetization is shown as open circles (SQUID magnetization) and
model fits as a red line (model fit), where FWA and NC are visually
indistinguishable; $M(1~{}\mathrm{T})=1.1\pm 0.1~{}\mu_{B}$ mol-1 and
$M(4~{}\mathrm{T})=1.6\pm 0.2~{}\mu_{B}$ mol-1. The units of $M_{S}$ are
$\mu_{B}$ mol-1, and $H_{C}$, $D_{FWA}$, and $D_{NC}$ are all units of Tesla.
Uncertainty bars represent one standard deviation from the mean, where
statistics are generated by measuring the magnetization of the 14 synthesis
batches required to generate 4.37 g for NPD. Figure 5: (Color online) An
illustration of magnetic structure for different magnetic field regimes. Here,
the magnetic field points towards the top of the page, short arrows represent
iron moments, and long arrows represent cobalt moments. (a) The Co-Fe sample
cooled in zero field has a CSG-like state with no average on-site moment, as
shown for the measurement of magnetic scattering in 10 mT, Fig. 3 (c). (b) The
application of magnetic field cants the moments towards the field (Fig. 3 (b)
and (d) ), and (c) larger fields induce larger average moments (Fig. 3 (a) ).
(d) More complicated mesoscopic states that contain texture are also possible,
but are not unambiguously determined with our data.
## IV Discussion
We have presented neutron diffraction and bulk magnetization measurements of
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O that suggest a CSG ground
state that enters a FWA-like state in applied magnetic fields of the order 1 T
and larger (Fig. 5). This conclusion is based upon the field dependence of the
magnetization, and particularly the diffraction experiments that show an
absence of long range order in the 10 mT data and ordered moments that are
induced by the applied field, ruling out a high-field domain magnetization
process. This random-anisotropy-based magnetization process explains the
appreciable slope observed for Co-Fe even in fields of 7 T at 2 K (Fig. 4), in
a way similar to the magnetization process at low temperature and high
magnetic fields for the vanadium tetracyanoethylene molecular magnet prepared
using solvent based methods.Zhou1993 This magnetization process is different
than for other reported cubic complex cyanide systems that have magnetically
ordered ground states and saturate magnetization at 2 K and 7 T.Mihalik2010 ;
Kumar2004 ; Kumar2005 The CSG ground state is consistent with previous AC-
susceptibility measurements of K1-2xCo1+x[Fe(CN)6]$\cdot y$H2O (0.2
$\leq~{}x~{}\leq$ 0.4, $y~{}\approx$ 5) that showed glassy
behavior,Pejakovic2002 although the relative orientation we find for Co and
Fe at 1 T is contrary to the XMCD experiment that reported antiparallel Co and
Fe at 1 T in Rb1.8Co4[Fe(CN)6]${}_{3.3}\cdot$13H2O and
K0.1Co4[Fe(CN)6]${}_{2.7}\cdot$18H2O.Champion2001 For fields of 1 T and 4 T,
we find a parallel alignment of Co and Fe moments minimizes the residuals
between model and data, and this alignment is clearly seen for the low-angle 4
T peaks, Fig. 6. However, the lack of coherent scattering for CSG means we
cannot strictly discuss the nature of the superexchange interaction in the
ground state, although we infer some significant ferromagnetic character based
upon the high-field regime. One must also be careful about applying
qualitative Goodenough-Kanamori rules to this system,Goodenough1955 ;
Goodenough1958 ; Kanamori1959 as the single-electron states are not only
mixed from electrostatic interactions, but also due to the aforementioned
presence of spin-orbit coupling. A band structure calculation using a full
potential linearized augmented plane wave method resulted in an
antiferromagnetic ground state for Co-Fe, with +0.296 $\mu_{B}$ on the Co site
and -0.280 $\mu_{B}$ on the Fe site,takegahara2002 although such values do
not agree with experimental findings.
For the Co-Fe presented, care was taken to ensure that the average particle
size was greater than $\approx$100 nm (as measured by TEM) to avoid finite
size effects,Pajerowski2007 and the FWA-like phase can explain the previously
reported changes in low temperature high field magnetization with particle
size as a tuning of the local random anisotropy with size, larger particles
requiring higher fields to saturate at base-temperature as they are deeper in
a glassy phase. The saturation value for bulk
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot 1.42$D2O, $M_{S}=2.7\pm
0.3~{}\mu_{B}$ mol-1, is comparable to a variety of similar states with
different degrees of orbital reduction on Co and Fe sites; $exempli~{}gratia$,
considering complete orbital moments on both Co and Fe gives $M_{S}\approx
2.5~{}\mu_{B}$ mol-1 and spin-only moments gives $M_{S}\approx 3.2~{}\mu_{B}$
mol-1.
The NPD experiments show a model-independent increase in coherent magnetic
scattering as a function of field, and a slightly form-factor dependent ratio
of Co to Fe moments, Table 2. A parallel alignment of Co and Fe is found for
both 1 T and 4 T, and when an antiparallel alignment is forced, Fe moments go
to zero to achieve best fits. The moment ratio is heavily dictated by the
low-Q peaks where the form-factor has little effect, while the scale of the
moments is different depending upon the presumed shape of the scatterer.
Previous neutron diffraction measurements have shown covalency effects, due to
$\sigma$-bonding and $\pi$-back-bonding with CN, to be important in the
chemically similar CsK2[Fe(CN)6].Figgis1990 Covalency can increase the
direct-space size of the moments, thereby decreasing the reciprocal-space size
even in the presence of orbital moments. For Co-Fe, the smaller reciprocal-
space form-factors give magnetizations most similar to those determined by
SQUID, although covalency makes assignment of orbital and spin magnetism based
upon spatial distribution inconclusive. We do not refine the form-factor in
this manuscript because high parameter covariance is introduced. Finally,
small quantitative differences between SQUID and NPD moment values may also be
due to sample inhomogeneity overestimating the NPD moments, and unitemized
experimental uncertainties due to the complicated and highly un-stoichiometric
formulation of the Co-Fe material, but our conclusions remain robust with
respect to such perturbations. The line-widths for magnetic and nuclear NPD
are similar, suggesting comparable domain sizes for the scattering objects,
but it is possible that the induced moments have a texture over some other
length scale, Fig. 5 (d), a possibility suggested by cluster-glass behavior in
AC susceptibility studies.Pejakovic2002 As shown in the Appendix, regions of
coherent magnetization at an angle $\xi$ from the applied field would rescale
the measured NPD longitudinal moment by
$\cos\xi\sqrt{\frac{2}{\cos^{2}\xi+1}}$, but the similarity between the
polarized and unpolarized magnetic diffraction further suggests that such an
effect is small in our samples.
Figure 6: (Color online) Visual comparison of magnetic configurations for Co-
Fe. Here, the low angle H = 4 T data (open circles) are set side by side with
model #1 for 4 T from Table II (thick-solid, red line), model #2 for 4 T from
Table II (dotted, black line), a ferrimagnetic structure with a 3:1 Co:Fe spin
ratio (dashed, blue line), and a ferromagnetic structure with a 3:1 Co:Fe spin
ratio (thin-solid, black line). Uncertainty bars are representative of one
standard deviation from the mean, using counting statistics.
## V Conclusions
The neutron diffraction and bulk magnetization measurements of Co-Fe suggest a
magnetization process that evolves from a correlated spin glass to a quasi-
ferromagnetic state with increasing magnetic field, where average Co and Fe
moments are induced to lie along the applied field. When considering memory
storage applications of molecule based magnetic materials, structure-property
relationships that may give rise to coherent and random anisotropy will be
important to consider.
###### Acknowledgements.
DMP acknowledges support from the NRC/NIST post-doctoral associateship
program. Research at High Flux Isotope Reactor at ORNL was sponsored by the
Scientific User Facilities Division, Office of Basic Energy Sciences, U. S.
Department of Energy. This work was supported, in part, by NSERC, CFI, and NSF
through Grants No. DMR-1005581 (DRT) and No. DMR-0701400 (MWM).
*
## Appendix A Effect of Canting on Intensity
A powder sample consisting of domains canted at an angle $\xi$ away from the
applied field, with random rotational distribution, may give the same
unpolarized NPD signal as domains along the field, but with a different
magnetic moment. With the magnetic field along the z-axis, and the scattering
vector along the x-axis, the uncanted magnetic moment is simply
$\mathbf{M_{u}}~{}=~{}(0,0,M)~{}~{}~{},$ (12)
and the canted moment can be expressed as
$\mathbf{M_{canted}}~{}=~{}M(\sin\xi\cos\phi,\sin\xi\sin\phi,\cos\xi)~{}~{}~{},$
(13)
where $\xi$ is the canting angle, $\phi$ is the rotation angle about the
field, and $M$ is the magnitude of the magnetic moment. The interaction
vectorSchweizer2006
$\left|\mathbf{M_{\bot}}\right|^{2}~{}=~{}\sum_{\alpha,\beta}{\left(\delta_{\alpha\beta}-\widehat{Q}_{\alpha}\widehat{Q}_{\beta}\right)M^{*}_{\alpha}M_{\beta}}~{}~{}~{},$
(14)
then gives a dependence of the intensity on the canting angle such that
$\left|\mathbf{M_{u,\bot}}\right|^{2}~{}=~{}M^{2}~{}~{}~{},$ (15)
and
$\left|\mathbf{M_{canted,\bot}}\right|^{2}~{}=~{}M^{2}{\frac{\cos^{2}\xi+1}{2}}~{}~{}~{},$
(16)
where random $\phi$-angles have been averaged over. Therefore, an uncanted
model with a z-component (relevant to compare with longitudinal magnetization)
of $M$ can give the same unpolarized NPD intensity as a canted model with a
z-component of $M\cos\xi\sqrt{\frac{2}{\cos^{2}\xi+1}}$.
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|
arxiv-papers
| 2012-06-01T19:24:30 |
2024-09-04T02:49:31.469322
|
{
"license": "Public Domain",
"authors": "D. M. Pajerowski, V. O. Garlea, E. S. Knowles, M. J. Andrus, M. F.\n Dumont, Y. M. Calm, S. E. Nagler, X. Tong, D. R. Talham, M. W. Meisel",
"submitter": "Daniel Pajerowski",
"url": "https://arxiv.org/abs/1206.0279"
}
|
1206.0339
|
# Ferromagnetism of cobalt-doped anatase TiO2 studied by bulk- and surface-
sensitive soft x-ray magnetic circular dichroism
V. R. Singh vijayraj@wyvern.phys.s.u-tokyo.ac.jp Department of Physics,
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan K. Ishigami Department
of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan V. K. Verma
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
G. Shibata Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo
113-0033, Japan Y. Yamazaki Department of Physics, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan T. Kataoka Department of Physics,
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan A. Fujimori Department
of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan F.-H. Chang
National Synchrotron Radiation Research Center (NSRRC), Hsinchu 30076, Taiwan,
Republic of China D.-J. Huang National Synchrotron Radiation Research Center
(NSRRC), Hsinchu 30076, Taiwan, Republic of China H.-J. Lin National
Synchrotron Radiation Research Center (NSRRC), Hsinchu 30076, Taiwan, Republic
of China C. T. Chen National Synchrotron Radiation Research Center (NSRRC),
Hsinchu 30076, Taiwan, Republic of China Y. Yamada Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan T. Fukumura Department
of Chemistry, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan PRESTO,
Japan Science and Technology Agency, Kawaguchi 332-0012, Japan M. Kawasaki
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
WPI-AIM Research, Tohoku University, Sendai 980-8577, Japan Quantum-Phase
Electronics Center and Department of Applied Physics, University of Tokyo,
Tokyo 113-8656, Japan CREST, Japan Science and Technology Agency, Tokyo
102-0075, Japan
###### Abstract
We have studied magnetism in anatase Ti1-xCoxO2-δ (x = 0.05) thin films with
various electron carrier densities, by soft x-ray magnetic circular dichroism
(XMCD) measurements at the Co $L_{2,3}$ absorption edges. For electrically
conducting samples, the magnetic moment estimated by XMCD was $<$ 0.3
$\mu_{B}$/Co using the surface-sensitive total electron yield (TEY) mode,
while it was 0.3-2.4 $\mu_{B}$/Co using the bulk-sensitive total fluorescence
yield (TFY) mode. The latter value is in the same range as the saturation
magnetization 0.6-2.1 $\mu_{B}$/Co deduced by SQUID measurement. The
magnetization and the XMCD intensity increased with carrier density,
consistent with the carrier-induced origin of the ferromagnetism.
Semiconductors partially substituted with magnetic ions are called diluted
magnetic semiconductors (DMSs) and are expected to be useful in spintronics
devices, where electron spins can be controlled by electric field and/or by
photons. Ferromagnetic DMS’s with Curie temperatures ($T_{C}$’s) higher than
room temperature are highly desirable for the development of spintronic
devices. To date, much work in this area has been done, mainly on II-VI and
III-V compounds doped with magnetic ions such as (Cd,Mn)Te 1 and (Ga,Mn)As 2
; 3 , but their $T_{C}$’s are far below room temperature. Ferromagnetism was
observed in Mn-based zinc-blende II-VI compounds such as (Cd,Mn)Te after the
result of carrier induced ferromagnetism 90 . Kuroda et al.50 ; 95 reported
that Cr rich phases of (Zn,Cr)Te showed room temperature ferromagnetism,
causing a stimulation of II-VI DMS. Ferromagnetism was also observed in
(Ga,Mn)As 2 ; 3 . It was theoretically suggested 40 that the co-doping of
magnetic semiconductors with shallow impurities affects the self-assembly of
magnetic nanocrystals during epitaxy, and therefore modifies both the global
and local magnetic behavior of the material. This concept was also
qualitatively corroborated by experimental data for (Cd, Mn, Cr)Te 45 and
(Ga, Mg, Fe)N 60 ; 70 ; 80 . However, origin of ferromagnetism at room-
temperature is controversial so far. Recently, Matsumoto et al.100 ; 105
reported the occurrence of room temperature ferromagnetism in Co-doped anatase
TiO2 films. According to Fukumura et al. 5 ; 7 the high electron carrier
densities and Co content favor the ferromagnetic phase in Co-doped rutile TiO2
at 300 K. Room-temperature ferromagnetism was also reported in such materials
as (Ga,Mn)N 22 and (Al,Cr)N 23 . The near edge x-ray absorption fine
structure study of Co-doped TiO2 by Griffin et al. 24 claims that
ferromagnetism is due to $d$-$d$ double exchange mediated by tunneling of $d$
electrons within the impurity band. Some studies that also claim the
ferromagnetism of Co-doped TiO2 is due to Co metal clusters 25 ; 26 ; 27 ; 28
. The recent theoretical study by Calderon et al. 29 , electric field-induced
anomalous Hall effect (AHE) study by Yamada et al. 15 and x-ray photoemission
spectroscopy study by Ohtsuki et al. 30 suggested the ferromagnetism of Co-
doped TiO2 is due to carrier mediated. However, direct information about the
magnetization as a function of carrier density has been lacking. Soft x-ray
magnetic circular dichroism (XMCD) at the Co 2$p\rightarrow 3d$ absorption (Co
$L_{2,3}$) edge is a powerful technique to clarify this issue because it is an
element-specific magnetic probe 28 . A previous XMCD study on rutile Co-doped
TiO2 by Mamiya et al. has revealed that the ferromagnetism is not due to
segregated Co metal clusters but is due to Co2+ ions in the TiO2 matrix 6 .
However, the XMCD signal intensities were an order of magnitude lower than
that expected from the bulk magnetization 6 . In a more recent work 7 , we
performed x-ray absorption spectroscopy (XAS) and XMCD studies on rutile Co-
doped TiO2 not only by the surface-sensitive total electron yield (TEY) mode
but also the bulk-sensitive total fluorescence yield (TFY) mode and found that
Co ions in the bulk indeed have a large magnetic moment of 0.8-2.2
$\mu_{B}$/Co.
In this work we have extended the same approach to anatase Co-doped TiO2 and
studied correlation between magnetism and transport properties. Magnetization
measurements of anatase Ti1-xCoxO2-δ thin films reveal ferromagnetic
hysteresis behavior in the M-H loop at room temperature with a saturation
magnetization. In the bulk region probed by the TFY mode, strong XMCD spectra
with similar spectral line shapes were obtained for all the samples. The
magnetization and the XMCD intensity increased with carrier density,
consistent with the carrier-induced origin of the ferromagnetism.
Anatase Ti1-xCoxO2-δ epitaxial thin films with $x$ = 0.05 were synthesized by
the pulsed laser deposition method on LaAlO3 (001) substrates at 523 K and
oxygen pressures ($P_{{\rm O}_{2}}$) of 5 $\times$ $10^{-7}$, 1 $\times$
$10^{-6}$ and 2 $\times$ $10^{-6}$ Torr. The resistivity increases in this
order and these samples are hereafter referred to metallic, intermediate,
insulating samples, respectively. The carrier densities $n_{e}$ were 4.1
$\times$ $10^{19}$, 1.1 $\times$ $10^{19}$ and 4.0 $\times$ $10^{18}$ cm-3,
respectively. Segregation of secondary phases were not observed under careful
inspections by x-ray diffraction (XRD) and transmission electron microscopy
(TEM) 15 of $\sim$40 nm thick films 15 . Reflection high-energy electron
diffraction was monitored during the in-situ growth. An intensity oscillation
was observed at the initial stage of the growth. We have confirmed that Co
distribution along the film thickness direction in our films is uniform using
TEM 15 , unlike the inhomogeneous distribution in films prepared on Si
demonstrated using atom probe tomography by Larde et al 31 . Ferromagnetism at
room temperature was confirmed by Hall-effect measurements and magnetization
measurements. XAS and XMCD measurements were performed at the BL-11A beamline
of the National Synchrotron Radiation Research Center, Taiwan. In XMCD
measurements, magnetic fields (H) were applied parallel to the direction of
anatase (001). XAS and XMCD spectra were obtained in the TEY and TFY modes and
probing depths were $\sim$5 and 100 nm, respectively.
Figure 1: (Color online) M-H curves of Ti0.95Co0.05O2-δ at 300 K. (a)
Metallic, intermediate and insulating anatase samples. (b) Metallic rutile
sample.
Figure 1(a) shows the magnetization curves of anatase Ti1-xCoxO2-δ ($x$ =
0.05) at 300 K for various carrier densities ($n_{e}$). The $n_{e}$ for
metallic, intermediate and insulating samples were 4.1 $\times$ $10^{19}$, 1.1
$\times$ $10^{19}$ and 4.0 $\times$ $10^{18}$ cm-3, respectively. That of
metallic rutile thin films which has the carrier density of 7 $\times$
$10^{21}$ cm-3 is also shown in Fig 1(b). The saturation magnetization of the
anatase sample is 0.6-2.1 $\mu_{B}$/Co with a coercive force of $\sim$100 to
200 Oe. In the M(H) measurements, magnetic field was applied parallel to the
the direction of anatase (001). Anomalous Hall-effect (AHE) measurements for
anatase Ti1-xCoxO2-δ with various $n_{e}$ also show similar magnetic field
dependences 15 . From Fig. 1, it is clear that the magnetization of the
anatase thin films is larger than the rutile thin films, which may be
attributed to the fact that anatase films in this study have a mobility
$\sim$2-11 cm2V-1s-1 which is two orders of the magnitude higher than the
mobility of rutile thin films 5 .
Figure 2: (Color online) Co $L_{2,3}$-edge of anatase Ti0.95Co0.05O2-δ taken
in the TEY mode at T = 300 K and H = 1 T. (a) XAS. (b),(c) XAS and XMCD
spectra of the metallic anatase Ti0.95Co0.05O2-δ sample. The XAS and XMCD
spectra of Co metal by Kim et al.28 are shown for comparison.
In Fig. 2(a), we show the Co $L_{2,3}$-edge XAS (metallic, intermediate and
insulating thin films) and Fig. 2(b)-(c) XAS and XMCD spectra of (metallic
thin film) anatase Ti1-xCoxO2-δ obtained in the TEY mode. In the figure,
$\mu_{+}$ and $\mu_{-}$ refer to the absorption coefficients for photon
helicity parallel and antiparallel to the Co majority spin direction,
respectively. The XMCD spectra $\Delta\mu$ = $\mu_{+}$ \- $\mu_{-}$ have been
corrected for the degree of circular polarization. The XAS and XMCD spectra of
the metallic anatase Ti1-xCoxO2-δ sample showed multiplet features as shown by
Fig 2(a)-(c), which is similar to Mamiya et al. 6 and agree with our $D_{2h}$
high-spin crystal-field symmetry cluster model calculations using the
parameter values : Charge-transfer energy ($\Delta$)= 4 eV, On-site 3$d$-3$d$
Coulomb energy ($U_{dd}$)=5 eV, 3$d$-2$p$ Coulomb energy ($U_{dc}$)= 7 eV,
Hopping integral between the Co 3$d$ and O 2$p$ orbitals of Eg symmetry
($V_{{\rm E}_{g}}$)= 1.1 eV and Crystal-field splitting (10Dq)= 0.9 eV. The
multiplet features of the XMCD spectra show almost one-to-one correspondence
to those in the XAS spectra. The spectral line shapes of the XAS and XMCD
spectra for the metallic and intermediate anatase Ti1-xCoxO2-δ samples are
also similar to those of rutile Co-doped TiO2 results which were reported in
our previous work 6 ; 7 . For the insulating sample, we observed an XAS
spectrum similar to those of the metallic and intermediate samples. The
estimated magnetic moments for all samples obtained from XMCD in the TEY mode
were $<$ 0.3 $\mu_{B}$/Co. These values are larger than the 0.1 $\mu_{B}$/Co
which is reported by Mamiya et al. 6 , but they are still smaller than the
saturation magnetic moments 0.6-2.1 $\mu_{B}$/Co deduced from magnetization
measurements. The XAS and XMCD spectra of Co metal is also shown at the bottom
of Fig. 2 (a)-(c) for comparison. It is demonstrated that the present XAS and
XMCD spectra of Co-doped TiO2 are distinctly different from Co metal.
Figure 3: (Color online) Co $L_{2,3}$-edge of anatase Ti0.95Co0.05O2-δ taken in the TFY mode at T = 300 K and H = 1 T. (a) XAS. (b),(c) XAS and XMCD spectra of anatase Ti0.95Co0.05O2-δ for metallic sample in the TFY mode. The XAS and XMCD spectra of Co metal by Kim et al. 28 are shown for comparison. (d),(e) Comparison of XAS and XMCD spectra shown in (b) and (c) with cluster-model calculation 34 . Table 1: Electronic structure parameters for anatase Co-doped TiO2 thin film used in the cluster-model calculations in units of eV to analyze. Crystal-field symmetry | Spin | $\Delta$ | $U_{dd}$ | $U_{dc}$ | $V_{{\rm E}_{g}}$ | 10Dq | Weight(%)
---|---|---|---|---|---|---|---
$D_{2h}$ | Low | 4 | 5 | 7 | 1.1 | 1.1$-$1.2 | 35
$O_{h}$ | Low | 3 | 6 | 7.5 | 1.1 | 1.1$-$1.2 | 35
$O_{h}$ | High | 2 | 5 | 7.5 | 1.1 | 0.8$-$0.9 | 30
Figures 3(a),(b) and (c) show the Co $L_{2,3}$ XAS and XMCD spectra of the
same samples taken in the TFY mode. From the figure, it is clear that the XMCD
intensities are much higher than those taken in the TEY mode. The large
difference between the bulk-sensitive TFY mode with $\sim$100 nm probing depth
and the surface-sensitive TEY mode with $\sim$5 nm probing depth suggests that
there is a magnetically dead layer of $\sim$5 nm thickness or more at the
surface of the samples as in the case of rutile 6 ; 7 . The presence of a
surface dead layer of $\sim$5 nm thickness is consistent with the recent
measurements of the film-thickness dependence of AHE 32 . The spectral line
shapes of the XAS and XMCD spectra of all the samples taken in the TFY mode
show broad features with spectral line shapes similar to those of rutile Co-
doped TiO2 7 . Both magnetization and XMCD intensity increased with carrier
density. This is consistent with spin alignment arises due to the interaction
of local spins with the spin polarized free carriers, in which carrier-
mediated ferromagnetism and ferromagnetic ordering is realized. Yamada et
al.15 have also demonstrated electrically induced ferromagnetism at room-
temperature in anatase Ti1-xCoxO2-δ, by means of electric double layer gating
resulting in high density electron accumulation ($>$1014 cm-2). By applying a
gate voltage of a few volts, a low-carrier paramagnetic state was transformed
to a high-carrier ferromagnetic state. This also supports theoretically as
well as experimentally the idea that the ferromagnetism originates from a
carrier-mediated mechanism 15 ; 29 . The broadening of the TFY spectra may be
due to the randomly displaced positions of Co atoms, which leads to in various
local structures as suggested by the anomalous X-ray scattering study of
Matsumura et al. 33 . The experimental XAS and XMCD spectra are distinctly
different from Co metal and show qualitatively good agreement with the
calculated spectra for the Co2+ in random crystal fields 34 , where the
calculations were done using the various electronic structure parameters as
listed in Table I.
Figure 4: (Color online) Magnetization as a function of magnetic field
obtained from the XMCD intensities of anatase Ti0.95Co0.05O2-δ compared with
M-H curves obtained using a SQUID.
Figure 4 shows magnetization versus magnetic field curves estimated from the
XMCD spectra obtained in the TEY and TFY modes using sum rules 6 , as compared
with the M-H curves measured using a SQUID. We have divided the obtained spin-
magnetic moment by a correction factor of 0.92 given by Teramura et al.35 .
The Co magnetic moment is found to be obviously much larger in the bulk region
than in the surface region. These results are also consistent with the x-ray
photoemission spectroscopy study by Yamashita et al. 14 . Since we know that
TFY suffers from self-absorption and therefore it will saturate the XAS
signal. This saturated XAS signal will reduce XMCD signal. Because of this
very fact, we can conclude that the real value of magnetic moment in bulk
should be even higher than the measured TFY value in metallic and intermediate
samples which are reported in the present work. Accordingly, our observation
by using the TEY and TFY modes are validated. The magnetic moment obtained
from cluster-model calculation (Fig.3) is 1.6 $\mu_{B}$/Co, which is similar
to the magnetization of $\sim$2 $\mu_{B}$/Co deduced from the TFY results and
the SQUID measurement. These results suggest that the Co ions in the bulk
region are responsible for the ferromagnetism in anatase Ti1-xCoxO2-δ.
In conclusion, we have studied the ferromagnetism of cobalt-doped anatase TiO2
thin films using element-specific XMCD at the Co $L_{2,3}$ edges in both the
surface-sensitive TEY and bulk-sensitive TFY modes. The large magnetic moment
of the Co ions, 0.6-2.4 $\mu_{B}$/Co, was observed by the TFY method. The
carrier-induced origin of ferromagnetism at room-temperature in anatase
Ti1-xCoxO2-δ is supported by the XMCD study of the samples with different
carrier concentration. According to the spectra taken in the TFY mode, the
positions of Co2+ atoms seem to be displaced from the regular Ti4+ sites,
resulting in random crystal fields. Good agreement is demonstrated not only in
magnetization and AHE but also in the magnetic field dependences of XMCD. The
magnetic moment values deduced with the TEY mode was $<$ 0.3 $\mu_{B}$/Co,
indicating the presence of a magnetically dead layer of $\sim$5 nm thickness
at the sample surfaces.
This work was supported by a Grant-in-Aid for Scientific Research in Priority
Area “Creation and Control of Spin Current” (19048012) from MEXT, Japan,
Grant-in-Aid for Scientific Research (S 22224005) from JSPS and TF was
supported by the Funding Program for Next Generation World-Leading
Researchers.
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|
arxiv-papers
| 2012-06-02T03:05:20 |
2024-09-04T02:49:31.478414
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. R. Singh, K. Ishigami, V. K. Verma, G. Shibata, Y. Yamazaki, T.\n Kataoka, A. Fujimori, F.-H. Chang, D.-J. Huang, H.-J. Lin, C. T. Chen, Y.\n Yamada, T. Fukumura, M. Kawasaki",
"submitter": "Vijay Raj Singh Dr",
"url": "https://arxiv.org/abs/1206.0339"
}
|
1206.0540
|
Quasilocal energy-momentum for tensor V in small regions
Lau Loi So
Department of Physics, National Central University, Chung-Li 320, Taiwan
Department of Physics, Tamkang University, Tamsui 251, Taiwan
###### Abstract
The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal
energy-momentum in a small sphere. Using a pseudotensor approach to evaluate
the energy-momentum in a half-cylinder, we find that $B$ and $V$ have
different values, not proportional to the ‘Bel-Robinson energy-momentum’.
Furthermore, even if we arrange things so that we do get the same ‘Bel-
Robinson energy-momentum’ value, the angular momentum gives different values
using $B$ and $V$ in half-cylinder. In addition, we find that $B$ and $V$ have
a different number of independent components. The fully trace free property of
$B$ and $V$ implies conservation of pure ‘Bel-Robinson energy-momentum’ in
small regions, and vice versa.
## 1 Introduction
In attempts to identify a good physical expression for the local distribution
of gravitational energy-momentum there have been many different approaches
which are similar to Einstein’s [1]. For example, those of Landau-Lifshitz
[2], Bergmann-Thomson [3], Papapetrou [4] and Weinberg [5]. Most of them deal
with the Einstein equation: $G_{\mu\nu}=\kappa{}T_{\mu\nu}$, where $\kappa$ is
a constant, $G_{\mu\nu}$ and $T_{\mu\nu}$ are the Einstein and stress tensors.
One can define a superpotential with a suitable anti-symmetry
$U_{\alpha}{}^{\mu\nu}\equiv{}U_{\alpha}{}^{[\mu\nu]}$ and remove a divergence
of $U_{\alpha}{}^{\mu\nu}$ from $G_{\mu\nu}$ to define the gravitational
energy-momentum density
$2\kappa{}\mathbf{t}_{\alpha}{}^{\mu}:=\partial_{\nu}U_{\alpha}{}^{[\mu\nu]}-2\sqrt{-g}\,G_{\alpha}{}^{\mu}.$
(1)
Note that $\mathbf{t}_{\alpha}{}^{\mu}$ is a pseudotensor [6]. Using the
Einstein equation, we have a total energy-momentum density which satisfies
$\partial_{\nu}U_{\alpha}{}^{[\mu\nu]}=2\kappa{\cal{}T}_{\alpha}{}^{\mu}=2\kappa(\mathbf{T}_{\alpha}{}^{\mu}+\mathbf{t}_{\alpha}{}^{\mu}),$
(2)
where $\mathbf{T}_{\alpha}{}^{\mu}=\sqrt{-g}\,T_{\alpha}{}^{\mu}$ and hence,
due to the antisymmetry of $U_{\alpha}{}^{[\mu\nu]}$, is automatically
conserved, i.e., has a vanishing divergence.
The proposed criteria for testing quasilocal expressions include: (i) limit to
good weak field values (i.e., linearized gravity). (ii) good asymptotic values
both at spatial and null infinity. We emphasize that the criteria for these
two are not very restrictive; they only test the quasilocal expression to
linear order. (iii) positivity (i.e., globally) is a strong test but is not
easy to achieve, (iv) small region inside of matter: the quasilocal energy-
momentum expression should, by the equivalence principle, reduce to the
material source terms. Most classical pseudotensors pass this test. (v) small
region in vacuum: positivity for the first non-vanishing parts of the
quasilocal expression. This depends on the gravitational field non-linearly,
and hence it can give a discriminating test of the expression; it is quite
non-trivial but not impossibly difficult.
Positive quasilocal gravitational energy should hold not only on a large scale
but also on the small scale [7]. However it is generally not at all easy to
prove that a particular expression enjoys this property. A good test case is
the small region limit. This will be our concern in this work. Here we
consider specifically the pseudotensor expressions. For a small region, one
can expand the energy-momentum density in Riemann normal coordinates (RNC)
about the origin:
$\displaystyle{\cal{}T}_{\alpha}{}^{\beta}(x)$ $\displaystyle=$
$\displaystyle{\cal{T}}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}{\cal{}T}_{\alpha}{}^{\beta}|_{0}x^{\mu}+\frac{1}{2}\partial^{2}_{\mu\nu}{\cal{T}}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}+...$
(3) $\displaystyle=$
$\displaystyle\mathbf{T}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}\mathbf{T}_{\alpha}{}^{\beta}|_{0}x^{\mu}+...+\mathbf{t}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}+\frac{1}{2}\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}+....$
By construction $\mathbf{t}_{\alpha}{}^{\beta}|_{0}$ and
$\partial_{\mu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}$ vanish in vacuum.
Consequently, for small $x^{\mu}$ inside of matter the
$\mathbf{T}_{\alpha}{}^{\beta}$ and
$\partial_{\mu}\mathbf{T}_{\alpha}{}^{\beta}$ terms dominate (this is a
reflection of the equivalence principle). In vacuum regions all the
$\mathbf{T}_{\alpha}{}^{\beta}$ terms vanish, then the lowest order non-
vanishing term is
$\frac{1}{2}\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}$.
This is the object on which we focus our attention in this work. It turns out
that for all the proposed pseudotensors and quasilocal energy-momentum
expressions this fourth rank tensor is quadratic in the Riemann (equivalent in
empty space regions to the Weyl) tensor. That is why the quadratic curvature
expressions become interesting and important (i.e.,
$\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}\simeq{}R_{....}R_{....}$).
Normally, the expansion of a pseudotensor expression up to second order can
only be some linear combination of three tensors $\\{B,S,K\\}$ or
$\\{B,V,S\\}$ [6, 8, 9] which are each certain quadratic expressions in the
curvature.
According to a review article (4.2.2 in [7]): “Therefore, in vacuum in the
leading $r^{5}$ order any coordinate and Lorentz-covariant quasilocal energy-
momentum expression which is non-spacelike and future pointing must be
proportional to the Bel-Robinson ‘momentum’
$B_{\mu\lambda\xi\kappa}t^{\lambda}t^{\xi}t^{\kappa}$.” Note that here
$t^{\alpha}$ is timelike unit vector and ‘momentum’ means 4-momentum (see
(28)). This is a strong test. The Bel-Robinson tensor $B$ has many nice
properties such as fully symmetric, traceless and divergence free [10]. It is
known that $B$ contributes positivity in a small sphere region and perhaps it
may thought that it is the only one. However, we recently proposed an
alternative $V$ (see (18)) which has the identical ‘Bel-Robinson momentum’ at
the same limit, i.e.,
$(B_{\mu\lambda\xi\kappa}-V_{\mu\lambda\xi\kappa})t^{\lambda}t^{\xi}t^{\kappa}\equiv
0$. Confined to a small spherical or cubical regions [11], $B$ and $V$ cannot
be distinguished. One may suspect that $V$ is redundant because $B$ can manage
all the jobs. But we claim not.
As the basic requirement for the quasilocal energy is any closed 2-surface, we
examined the energy-momentum and angular momentum in other regions (see Table
1). We find for the energy in a small half-cylinder when $h\neq\sqrt{3}a$ give
different values if substituting $\mathbf{t}$ by $B$ and $V$, which means that
they are distinguishable. Only for one particular ratio $h=\sqrt{3}a$, $B$ and
$V$ both give the same ‘Bel-Robinson momentum’ value, however we lose the
distinction between them again. Therefore we turn to examining the angular
momentum in a small half-cylinder, and show that when replacing $\mathbf{t}$
by $B$ and $V$ in the angular momentum expression they contribute different
values, thereby clarifying that the two tensors are really distinguishable.
Here we remark that some components of the angular momentum in a hemi-sphere
show that $B$ contributes a null result while $V$ gives non-zero values (see
section 3.2). The reason comes from the fully symmetric property of $B$, while
$V$ only has some certain symmetry property (see (19)). Consequently, $V$ is
non-replaceable.
## 2 Technical background
Using a Taylor series expansion, the metric tensor can be written as
$\displaystyle
g_{\alpha\beta}(x^{\lambda})=g_{\alpha\beta}|_{x^{\lambda}_{0}}+\partial_{\mu}g_{\alpha\beta}|_{x^{\lambda}_{0}}(x^{\mu}-x^{\mu}_{0})+\frac{1}{2}\partial^{2}_{\mu\nu}g_{\alpha\beta}|_{x^{\lambda}_{0}}(x^{\mu}-x^{\mu}_{0})(x^{\nu}-x^{\nu}_{0})+...,$
(4)
where the metric signature is $+2$. For simplicity, let $x^{\lambda}_{0}=0$
and at the origin in RNC
$\displaystyle g_{\alpha\beta}|_{0}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta},\quad\quad\partial_{\mu}g_{\alpha\beta}|_{0}=0,$
(5) $\displaystyle-3\partial^{2}_{\mu\nu}g_{\alpha\beta}|_{0}$
$\displaystyle=$ $\displaystyle
R_{\alpha\mu\beta\nu}+R_{\alpha\nu\beta\mu},\quad\quad-3\partial_{\nu}\Gamma^{\mu}{}_{\alpha\beta}|_{0}=R^{\mu}{}_{\alpha\beta\nu}+R^{\mu}{}_{\beta\alpha\nu}.$
(6)
Three basic tensors [6, 8, 9] that commonly occurred in pseudotensors are:
$\displaystyle
B_{\alpha\beta\mu\nu}\equiv{}B_{(\alpha\beta\mu\nu)}:=R_{\alpha\lambda\mu\sigma}R_{\beta}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\nu\sigma}R_{\beta}{}^{\lambda}{}_{\mu}{}^{\sigma}-\frac{1}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(7) $\displaystyle
S_{\alpha\beta\mu\nu}\equiv{}S_{(\alpha\beta)(\mu\nu)}\equiv{}S_{\mu\nu\alpha\beta}:=R_{\alpha\mu\lambda\sigma}R_{\beta\nu}{}^{\lambda\sigma}+R_{\alpha\nu\lambda\sigma}R_{\beta\mu}{}^{\lambda\sigma}+\frac{1}{4}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(8) $\displaystyle
K_{\alpha\beta\mu\nu}\equiv{}K_{(\alpha\beta)(\mu\nu)}\equiv{}K_{\mu\nu\alpha\beta}:=R_{\alpha\lambda\beta\sigma}R_{\mu}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\beta\sigma}R_{\nu}{}^{\lambda}{}_{\mu}{}^{\sigma}-\frac{3}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(9)
where $\mathbf{R}^{2}=R_{\rho\tau\xi\kappa}R^{\rho\tau\xi\kappa}$.
It may be worthwhile to mention that $B$ has a very good analog with the
electromagnetic energy-momentum tensor $\mathbf{T}^{\mu\nu}$. In Minkowski
coordinates $(t,x,y,z)$:
$\displaystyle\mathbf{T}^{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(E_{a}E^{a}+B_{a}B^{a}),$ (10)
$\displaystyle\mathbf{T}^{0i}$ $\displaystyle=$
$\displaystyle\delta^{ij}\epsilon_{jab}E^{a}B^{b},$ (11)
$\displaystyle\mathbf{T}^{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\delta^{ij}(E_{a}E^{a}+B_{a}B^{a})-2(E^{i}E^{j}+B^{i}B^{j})\right].$
(12)
where $\vec{E}$ and $\vec{B}$ refer to the electric and magnetic field
density. In order to appreciate the nice properties of $B$, we compare the
energy density with $S$ and $K$
$\displaystyle
B_{0000}=E^{2}_{ab}+H^{2}_{ab},\quad{}S_{0000}=2(E^{2}_{ab}-H^{2}_{ab}),\quad{}K_{0000}=-E^{2}_{ab}+3H^{2}_{ab},$
(13)
where the evaluation has used the electric part $E_{ab}$ and magnetic part
$H_{ab}$, defined in terms of the Weyl tensor [12]: $E_{ab}:=C_{a0b0}$ and
$H_{ab}:=*C_{a0b0}$, where $*C_{\alpha\beta\mu\nu}$ means its dual. Likewise
for the linear momentum density (i.e., Poynting vector)
$B_{000i}=2\epsilon_{ijk}E^{jd}H^{k}{}_{d},\quad{}S_{000i}=0,\quad{}K_{000i}=2\epsilon_{ijk}E^{jd}H^{k}{}_{d}.$
(14)
Finally, the stress,
$\displaystyle B_{00ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}(E_{ab}E^{ab}+H_{ab}H^{ab})-2(E_{id}E_{j}{}^{d}+H_{id}H_{j}{}^{d}),$
(15) $\displaystyle S_{00ij}$ $\displaystyle=$
$\displaystyle-2\left[\delta_{ij}(E_{ab}E^{ab}-H_{ab}H^{ab})+2(E_{id}E_{j}{}^{d}-H_{id}H_{j}{}^{d})\right],$
(16) $\displaystyle K_{00ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}(5E_{ab}E^{ab}-3H_{ab}H^{ab})-4E_{id}E_{j}{}^{d}.$
(17)
We observe that summing up $S$ and $K$ has exactly the same energy as $B$:
$(B_{0000}-S_{0000}-K_{0000})\equiv{}0\equiv(B_{00ij}-S_{00ij}-K_{00ij})\delta^{ij}$.
It is natural to define the alternative 4th rank tensor [9] as follows
$V:=S+K\equiv{}B+W,$ (18)
where
$W_{\alpha\beta\mu\nu}:=\frac{3}{2}S_{\alpha\beta\mu\nu}-\frac{1}{8}(5g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu})\mathbf{R}^{2}$.
For a comparison of $B$ and $V$, we find that it is more convenient to use
$(B+W)$ instead of $(S+K)$ for the representation of $V$. Both $V$ and $W$
satisfy the following properties:
$\displaystyle
X_{\alpha\beta\mu\nu}\equiv{}X_{(\alpha\beta)(\mu\nu)}\equiv{}X_{\mu\nu\alpha\beta},\quad{}X_{\alpha\beta\mu}{}^{\mu}\equiv{}0\equiv{}X_{\alpha\mu\beta}{}^{\mu}.$
(19)
However, unlike $B$ (see (7)), they are not fully symmetric. Intuitively,
referring to (18), $V$ may contain more non-trivial independent components
than $B$ and indeed it is the case (see section 3.3).
In our work, we are mainly dealing with expression of the 4th rank which are
quadratic in the curvature tensor. There are four tensors which form a basis
with appropriate symmetries [13], we use
$\displaystyle\tilde{B}_{\alpha\beta\mu\nu}$ $\displaystyle:=$ $\displaystyle
R_{\alpha\lambda\mu\sigma}R_{\beta}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\nu\sigma}R_{\beta}{}^{\lambda}{}_{\mu}{}^{\sigma},\quad{}\tilde{S}_{\alpha\beta\mu\nu}:=R_{\alpha\mu\lambda\sigma}R_{\beta\nu}{}^{\lambda\sigma}+R_{\alpha\nu\lambda\sigma}R_{\beta\mu}{}^{\lambda\sigma},\quad$
(20) $\displaystyle\tilde{K}_{\alpha\beta\mu\nu}$ $\displaystyle:=$
$\displaystyle
R_{\alpha\lambda\beta\sigma}R_{\mu}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\beta\sigma}R_{\nu}{}^{\lambda}{}_{\mu}{}^{\sigma},\quad\tilde{T}_{\alpha\beta\mu\nu}:=-\frac{1}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2}.$
(21)
They are designed to describe the gravitational energy expression based on the
pseudotensor (see (22)) and are manifestly symmetric in the last two indices,
i.e., $\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{\alpha\beta(\mu\nu)}$. Then
$\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{(\alpha\beta)\mu\nu}$ and it also
naturally turns out
$\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{\mu\nu\alpha\beta}$.
## 3 Energy-momentum tensors of $B$ and $V$
### 3.1 Alternative gravitational energy-momentum tensor $V$
Let $x^{\mu}=(t,x,y,z)$ and using a RNC Taylor expansion around any point,
consider all the possible combinations of the small region in vacuum. The
total energy-momentum density pseudotensor is in general expressed as
${\cal{T}}_{\alpha}{}^{\beta}=\kappa^{-1}G_{\alpha}{}^{\beta}+(a_{1}\tilde{B}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{2}\tilde{S}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{3}\tilde{K}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{4}\tilde{T}_{\alpha}{}^{\beta}{}_{\xi\kappa})x^{\xi}x^{\kappa}+{\cal{}O}(\mbox{Ricci},x)+{\cal{}O}(x^{3}),$
(22)
where $a_{1}$ to $a_{4}$ are constants. Since our concern is the vacuum case,
so $G_{\alpha\beta}=0=T_{\alpha\beta}$. Then the first order linear in Ricci
terms ${\cal{}O}({\mbox{Ricci}},x)$ vanish. The lowest order non-vanishing
term is of second order, and compared to this in the small region limit we
ignore the third order terms ${\cal{}O}(x^{3})$. It should be noted that
${\cal{T}}_{\alpha}{}^{\beta}$ in (2) or (22) is a pseudotensor, but in the
Taylor expansion on the right hand side in (22) the coefficients of the
various powers of $x$ are tensors. As argued in [13],
$\partial^{2}_{\mu\nu}{\cal{T}}_{\alpha}{}^{\beta}(0)$ must be some linear
combination of 4 tensors, here we use {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$,
$\tilde{T}$}. From now on, we only keep the second order term and drop the
others. There are two physical conditions which can constrain the unlimited
combinations between {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$, $\tilde{T}$}:
4-momentum conservation and positivity, both considered in the small region
vacuum limit (i.e., not restricted to a 2-sphere).
First condition: energy-momentum conservation. Consider (2) and (22) in vacuum
$\displaystyle
0=4\,\partial_{\beta}\mathbf{t}_{\alpha}{}^{\beta}=(a_{1}-2a_{2}+3a_{3}-a_{4})g_{\alpha\beta}x^{\beta}\mathbf{R}^{2}.$
(23)
Therefore, the constraint for the conservation of the energy-momentum density
is
$a_{4}=a_{1}-2a_{2}+3a_{3}.$ (24)
No single element from {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$, $\tilde{T}$}
can satisfy (23), however certain linear combinations of them can. Eliminate
$\tilde{T}$ which is absorbed by $\tilde{B}$, $\tilde{S}$ or $\tilde{K}$,
comparing (2) and using (24), rewrite (22)
$\displaystyle\mathbf{t}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\left[a_{1}(\tilde{B}_{\alpha\beta\xi\kappa}+\tilde{T}_{\alpha\beta\xi\kappa})+a_{2}(\tilde{S}_{\alpha\beta\xi\kappa}-2\tilde{T}_{\alpha\beta\xi\kappa})+a_{3}(\tilde{K}_{\alpha\beta\xi\kappa}+3\tilde{T}_{\alpha\beta\xi\kappa})\right]x^{\xi}x^{\kappa}$
(25) $\displaystyle=$
$\displaystyle(a_{1}B_{\alpha\beta\xi\kappa}+a_{2}S_{\alpha\beta\xi\kappa}+a_{3}K_{\alpha\beta\xi\kappa})x^{\xi}x^{\kappa}$
$\displaystyle=$
$\displaystyle\left[a_{1}B_{\alpha\beta\xi\kappa}+a_{3}V_{\alpha\beta\xi\kappa}+(a_{2}-a_{3})S_{\alpha\beta\xi\kappa}\right]x^{\xi}x^{\kappa}.$
Consider all the possible expressions for the pseudotensors (some of which
explicitly included the flat metric), there indeed does appear linear
combinations of these three tensors [6, 8, 9]. Explicitly one can use either
$\\{B,S,K\\}$ or $\\{B,V,S\\}$.
Second condition: non-negative gravitational energy. For simplicity, we use a
small sphere. For any quantity at $t=t_{0}$ we consider the limiting value for
the radius $r:=\sqrt{x^{2}+y^{2}+z^{2}}$. The 4-momentum at time $t=0$ is
$\displaystyle
2\kappa{}P_{\mu}=\int{}\mathbf{t}^{\rho}{}_{\mu\xi\kappa}x^{\xi}x^{\kappa}d\Sigma_{\rho}=\mathbf{t}^{0}{}_{\mu{}ij}\int{}x^{i}x^{j}d^{3}x=\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}\,\frac{4\pi{}r^{5}}{15}.$
(26)
Thus, from (25)
$P_{\mu}=(-E,\vec{P})=-\frac{r^{5}}{60G}\left[a_{1}B_{\mu{}0ij}+a_{3}V_{\mu{}0ij}+(a_{2}-a_{3})S_{\mu{}0ij}\right]\delta^{ij}.$
(27)
The energy-momentum values associated with $\\{B,V,S\\}$ are
$\displaystyle
B_{\mu{}0ij}\delta^{ij}\equiv{}V_{\mu{}0ij}\delta^{ij}=(E^{2}_{ab}+H^{2}_{ab},2\epsilon_{cab}E^{ad}H^{b}{}_{d}),~{}{}S_{\mu{}0ij}\delta^{ij}=-10(E^{2}_{ab}-H^{2}_{ab},0).$
(28)
Here we emphasize that in a small sphere region, the energy-momentum of $B$ or
$V$ is inside the light cone, $-P_{0}\geq|\vec{P}|\geq 0$. Observing (27),
basically we are considering positive energy, $B$ and $V$ already satisfy this
condition and the remaining job is to find $\\{a_{2},a_{3}\\}$. Equation (28)
shows that $S_{\mu{}0ij}\delta^{ij}$ cannot ensure positivity, since we should
allow for any magnitude of $|E_{ab}|$ and $|H_{ab}|$. The only possibility for
(27) to guarantee positivity is to require $a_{1}+a_{3}\geq{}10|a_{2}-a_{3}|$.
However, if we insist on the pure ‘Bel-Robinson momentum’ [7], obviously, we
only have one choice $a_{2}=a_{3}$.
### 3.2 Computing energy-momentum and angular momentum
The Papapetrou pseudotensor [9] gives a certain linear combination of $B$ and
$V$:
$2\kappa{}P^{\alpha\beta}=\frac{1}{9}(4B^{\alpha\beta}{}_{\xi\kappa}-V^{\alpha\beta}{}_{\xi\kappa})x^{\xi}x^{\kappa}$.
The energy using (26) in a small sphere is
$P_{0}=-\frac{r^{5}}{540G}(4B_{00ij}-V_{00ij})\delta^{ij}\equiv-\frac{r^{5}}{180G}B_{00ij}\delta^{ij},$
(29)
where $(B_{00ij}-V_{00ij})\delta^{ij}\equiv 0$. Before we proceed, one might
question that perhaps $V$ is superfluous since $B$ and $V$ have so far shown
no distinction. We claim that $B$ and $V$ are distinct because they are
constructed from different basic quadratic curvatures
$\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$: $B=\tilde{B}+\tilde{T}$ and
$V=\tilde{S}+\tilde{K}+\tilde{T}$. Strictly speaking, we claim $B$ and $V$ are
fundamentally different [9]. But this raises a question regarding how to see
the distinction clearly. We realize that it is impossible to distinguish $B$
and $V$ if we consider 4-momentum or angular momentum in a small sphere. So we
change our strategy to evaluating these physical quantities in other
quasilocal volume elements (see Table 1).
We claim $B$ and $V$ can have different energy values, for instance, in a
small box with different dimensions. Here we give a concrete example: let
$a=b$, $c=a+\Delta$ and $|\Delta|<<a$. The energy for substituting
$\mathbf{t}$ by $B$ is
$P^{B}_{0}\simeq\frac{a^{5}}{12}(B^{0}{}_{0ij}\delta^{ij}+\frac{2\Delta}{a}B^{0}{}_{033})$.
Similarly for $V$,
$P^{V}_{0}\simeq\frac{a^{5}}{12}(V^{0}{}_{0ij}\delta^{ij}+\frac{2\Delta}{a}V^{0}{}_{033})$.
Thus, generally, $B$ and $V$ are separable:
$P^{V}_{0}-P^{B}_{0}\simeq\frac{a^{4}\Delta}{6}W^{0}{}_{033}\neq 0$. Following
the restriction that the quasilocal energy-momentum must be a multiple of
‘Bel-Robinson momentum’ [7]. We can fulfill this requirement using either $B$
or $V$ in a small region for a perfect sphere or a box with
$a\equiv{}b\equiv{}c$, i.e., a cube [11], for a cylinder or half-cylinder we
need $h\equiv\sqrt{3}a$. These are desirable results, but unfortunately, we
lose the distinction between $B$ and $V$ again.
Is it possible to keep a multiple of ‘Bel-Robinson momentum’ and still able to
tell the difference between $B$ and $V$ naturally? Yes, it is possible: we
turn to examining the angular momentum (see, e.g., $\S$20.3 in [8]) which can
be defined as follows
$\displaystyle J^{\mu\nu}:=\int(x^{\mu}\mathbf{t}^{\nu
0}{}_{\xi\kappa}-x^{\nu}\mathbf{t}^{\mu
0}{}_{\xi\kappa})x^{\xi}x^{\kappa}d^{3}x,$ (30)
where $\mathbf{t}$ can be $B$ or $V$. According to Table 1, we observe that
the angular momentum vanishes for a perfect sphere, ellipsoid, box or
cylinder. Conversely, both hemi-sphere and half-cylinder ($h\equiv\sqrt{3}a$)
have non-vanishing angular momentum. In these regions, the angular momentum
values for $B$ and $V$ are distinguishable, i.e., $V$ is no longer
superfluous. Moreover, we remark that for a hemi-sphere, if we substitute
$\mathbf{t}$ by the completely symmetric $B$,
$J^{12}_{B}=\frac{\pi}{12}(B_{1023}-B_{2013})a^{6}\equiv{}0$. However, if
consider $V$, $J^{12}_{V}=\frac{\pi}{12}(V_{1023}-V_{2013})a^{6}\neq 0$
generally. Thus, the difference between $B$ and $V$ becomes sharply manifest,
showing that in this case $V$ is essential, not redundant.
Perfect- | $P_{\mu}=\frac{4\pi}{15}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{5}$, $r\in[0,a],~{}\theta\in[0,\pi],~{}\phi\in[0,2\pi]$
---|---
sphere | $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Ellipsoid | $P_{\mu}=\frac{4\pi}{15}(\mathbf{t}^{0}{}_{\mu{}11}a^{2}+\mathbf{t}^{0}{}_{\mu{}22}b^{2}+\mathbf{t}^{0}{}_{\mu{}33}c^{2})abc$, $x\in[-a,a],~{}y\in[-b,b],~{}z\in[-c,c]$
| $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Hemi- | $P_{\mu}=\frac{2\pi}{15}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{5}$, $r\in[0,a],~{}\theta\in[0,\pi/2],~{}\phi\in[0,2\pi]$
sphere | $J^{0m}=\frac{\pi}{24}(2\mathbf{t}^{0}{}_{013},2\mathbf{t}^{0}{}_{023},\mathbf{t}^{0}{}_{0ij}\delta^{ij}+\mathbf{t}^{0}{}_{033})a^{6}$
|
$J^{12}=\frac{\pi}{12}(\mathbf{t}^{1}{}_{023}-\mathbf{t}^{2}{}_{013})a^{6}$,
$J^{13}=\frac{\pi}{24}(\mathbf{t}^{1}{}_{0ij}\delta^{ij}+\mathbf{t}^{1}{}_{033}-2\mathbf{t}^{3}{}_{013})a^{6}$,
|
$J^{23}=\frac{\pi}{24}(\mathbf{t}^{2}{}_{0ij}\delta^{ij}+\mathbf{t}^{2}{}_{033}-2\mathbf{t}^{3}{}_{023})a^{6}$
Box | $P_{\mu}=\frac{1}{12}(\mathbf{t}^{0}{}_{\mu 11}a^{2}+\mathbf{t}^{0}{}_{\mu 22}b^{2}+\mathbf{t}^{0}{}_{\mu 33}c^{2})abc$ , $x\in[-\frac{a}{2},\frac{a}{2}],~{}y\in[-\frac{b}{2},\frac{b}{2}],~{}z\in[-\frac{c}{2},\frac{c}{2}]$
| $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Cylinder | $P_{\mu}=\frac{\pi}{4}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{4}h+\frac{\pi}{12}\mathbf{t}^{0}{}_{\mu 33}(h^{2}-3a^{2})a^{2}h$, $\rho\in[0,a],~{}\varphi\in[0,2\pi],~{}z\in[-\frac{h}{2},\frac{h}{2}]$
| $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Half- | $P_{\mu}=\frac{\pi}{8}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{4}h+\frac{\pi}{24}\mathbf{t}^{0}{}_{\mu 33}(h^{2}-3a^{2})a^{2}h$, $\rho\in[0,a],~{}\varphi\in[0,\pi],~{}z\in[-\frac{h}{2},\frac{h}{2}]$
cylinder | $J^{01}=\frac{4}{15}\mathbf{t}^{0}{}_{012}a^{5}h$, $J^{02}=\frac{1}{18}\mathbf{t}^{0}{}_{033}a^{3}h^{3}+\frac{2}{15}(\mathbf{t}^{0}{}_{011}+2\mathbf{t}^{0}{}_{022})a^{5}h$, $J^{03}=\frac{1}{9}\mathbf{t}^{0}{}_{023}a^{3}h^{3}$
|
$J^{12}=\frac{1}{18}\mathbf{t}^{1}{}_{033}a^{3}h^{3}+\frac{2}{15}(\mathbf{t}^{1}{}_{011}+2\mathbf{t}^{1}{}_{022}-2\mathbf{t}^{2}{}_{012})a^{5}h$,
$J^{13}=\frac{1}{9}\mathbf{t}^{1}{}_{023}a^{3}h^{3}-\frac{4}{15}\mathbf{t}^{3}{}_{012}a^{5}h$
|
$J^{23}=\frac{1}{18}(2\mathbf{t}^{2}{}_{023}-\mathbf{t}^{3}{}_{033})a^{3}h^{3}-\frac{2}{15}(\mathbf{t}^{3}{}_{011}+2\mathbf{t}^{3}{}_{022})a^{5}h$
Table 1: Energy-momentum and angular momentum in different small regions,
$\mathbf{t}$ can be $B$ or $V$
### 3.3 Counting the independent components of $B$, $V$ and $W$
Basically $B$, $V$ and $W$ are fourth rank tensor and could have 256
components. However, by symmetry, they only have a relatively small number of
independent components. The counting of the number of independent components
of $B$ has already been done, here we claim there is no common term between
$B$ and $W$, i.e., $\\{B\\}\bigcap\,\\{W\\}=\\{\emptyset\\}$. We verify this
statement as follows:
First, we count the components of $B$. In principle, $B$ is fully symmetric,
by explicit examination it reduces to 35. There is a formula that directly
gives this number. A $k$th rank totally symmetric tensor in $n$ dimensional
space has $C^{n+k-1}_{k}$ components. For our case $C^{4+4-1}_{4}=35$. Since
$B$ is completely tracefreeness, there are 10 additional constraints which
reduce the number of components. Therefore, we have left only 25 for $B$ (for
another argument see [14]).
Next we count the number of independent components of $V$. $V$ does not have
the totally symmetric property, but as mentioned in (19) that
$V_{\alpha\beta\mu\nu}\equiv{}V_{(\alpha\beta)(\mu\nu)}\equiv{}V_{\mu\nu\alpha\beta}$.
This reduces $V$ to 55 components. However, the completely traceless condition
gives two extra constraints indicated in (19) again:
$V^{\alpha}{}_{\alpha\mu\nu}\equiv{}0\equiv{}V^{\alpha}{}_{\mu\alpha\nu}$.
Consequently, we have $55-10-10=35$ for $V$.
Finally, we count the number of independent components of $W$. Observing that
$V$ and $W$ are similar. Referring to (19), there should thus be at most 35
components. However, take care an extra condition
$W_{\alpha(\beta\mu\nu)}\equiv{}0$ which gives 25 more constraints. Hence we
find $35-25=10$ for $W$.
### 3.4 Physical meaning of the fully tracefreeness property
It is easy to check that $B$ and $V$ are fully trace free. We are going to
verify that this mathematical property and the physical conservation laws are
in a 1-1 correspondence in the quasilocal limit. Consider a linear combination
between $\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$, let
$A:=a_{1}\tilde{B}+a_{2}\tilde{S}+a_{3}\tilde{K}+a_{4}\tilde{T}.$ (31)
We observe that there are only two distinct traces because of the symmetry:
$\displaystyle
8A^{\alpha}{}_{\mu\alpha\nu}\equiv(a_{1}-2a_{2}+3a_{3}-a_{4})g_{\mu\nu}\mathbf{R}^{2},\quad{}2A^{\alpha}{}_{\alpha\mu\nu}\equiv(a_{1}+a_{2}-a_{4})g_{\mu\nu}\mathbf{R}^{2}.$
(32)
The totally traceless condition requires that the above two equations vanish
simultaneously:
$\displaystyle 0=a_{1}-2a_{2}+3a_{3}-a_{4},\quad{}0=a_{1}+a_{2}-a_{4}.$ (33)
The first equation in (33) is the same as (24), which indicates one of the
mathematical conditions identical to the energy-momentum conservation
criterion: solving the equations in (33), we obtain $a_{2}=a_{3}$, and this is
proportional to the ‘Bel-Robinson momentum’ requirement found from (27); we
have noted that the fully tracefreeness property is related to some physical
conditions.
## 4 Conclusion
For describing positivity, the Bel-Robinson tensor is the best, and perhaps
has been thought to be the only possibility. We recently proposed an
alternative $V$ in such a way that it shares the same energy-momentum as $B$
does in the small sphere limit. One might think that $B$ and $V$ cannot be
distinguished, but we claim they can. After examining the energy found from
other 2-surfaces such as in ellipsoid, box, cylinder and half-cylinder
($h\neq\sqrt{3}a$), we demonstrate that $V$ is not redundant because $B$ and
$V$ are distinguishable. However, if we insist to achieve a multiple of pure
‘Bel-Robinson momentum’ from Szabados’s argument in Living Review, the
distinction between $B$ and $V$ will be lost once more. For a shape such that
both $B$ and $V$ give a multiple of the pure ‘Bel-Robinson momentum’ we can
turn to investigate the angular momentum. Thus when replacing $\mathbf{t}$ by
either $B$ or $V$, indeed they do lead to different angular momentum values
for a hemi-sphere or half-cylinder with $h=\sqrt{3}a$. Moreover, we emphasize
that some of the components of the angular momentum give a null result for $B$
and a non-vanishing result for $V$. The reason is based on the elegant
completely symmetric property of $B$, while $V$ is not fully symmetric. Thus
$V$ can play an essential irreplaceable role.
The tensors $B$ and $V$ are constructed from different fundamental quadratic
curvatures $\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$. As a double check,
we counted the independent components of $B$ and $V$ and find that they are
not the same. Finally, we discover the necessary and sufficient conditions for
$B$ and $V$: fully tracefreeness and conservation of future pointing non-
spacelike pure ‘Bel-Robinson momentum’ in the small region limit.
## Acknowledgment
The author would like to thank Dr. Peter Dobson, Professor Emeritus, HKUST,
for reading the manuscript and providing some helpful comments. This work was
supported by NSC 95-2811-M-032-008, NSC 96-2811-M-032-001, NSC
97-2811-M-032-007 and NSC 98-2811-M-008-078.
## References
* [1] Trautman A 1962 in An introduction to Current Research, ed L Witten (New York: Wiley) p169-198
* [2] Landau L D and Lifshitz E M 1962 The classical theory of fields, 2nd edition (Reading, MA: Addison-Wesley) (Oxford: Pergamon, 1975)
* [3] Bergmann P G and Thomson R 1953 Phys. Rev. 89 400
* [4] Papapetrou A 1948 Proc. Roy. Irish. Acad. A52 11-23
* [5] Weinberg S 1972 Gravitation and Cosmology, (New York: Wiley) p371
* [6] So L L and Nester J M 2009 Phys. Rev. D 79 084028
* [7] Szabados L B 2009 Living Rev. Relativity 12 4
* [8] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman)
* [9] So L L 2009 Class. and Quantum Grav. 26 185004
* [10] Senovilla J M M 2000 Class. Quantum Grav. 17 2799
* [11] Garecki J 1977 Acta Phys. Pol. B8 159
* [12] Carmeli M 1982 Classical Fields General relativuty and Gauge Theory (John Wiley $\&$ Sons)
* [13] Deser S, Franklin J S and Seminaea D 1999 Class. Quantum Grav. 16 2815
* [14] Gomez-Lobo A G P 2008 Class. Quantum. Grav. 25 015006
|
arxiv-papers
| 2012-06-04T08:11:53 |
2024-09-04T02:49:31.490741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Lau Loi So",
"submitter": "Lau Loi So",
"url": "https://arxiv.org/abs/1206.0540"
}
|
1206.0719
|
11institutetext: MCTI/Laboratório Nacional de Astrofísica, Rua Estados Unidos,
154, Bairro das Nações, CEP 37.504-364, Itajubá, MG, Brazil
11email: mfaundez@lna.br, mabans@lna.br 22institutetext: Universidade do Vale
do Paraíba - UNIVAP. Av. Shishima Hifumi, 2911 - Urbanova CEP: 12244-000 - São
José dos Campos, SP, Brazil. 22email: angela.krabbe@gmail.com,
irapuan@univap.br 33institutetext: UNIFEI, Instituto de Engenharia de Produção
e Gestão, Av. BPS 1303 Pinheirinho, 37500-903 Itajubá, MG, Brazil
44institutetext: UEFS, Departamento de Física, Av. Transnordestina, S/N, Novo
Horizonte, Feira de Santana, BA, Brazil, CEP 44036-900 55institutetext: UEFS,
Observatório Astronômico Antares, Rua da Barra, 925, Jardim Cruzeiro, Feira de
Santana, BA, Brazil, CEP 44015-430
55email: paulopoppe@gmail.com, vmartin1963@gmail.com, irafbear@gmail.com
# A study of the remarkable galaxy system AM 546-324
(the core of Abell S0546) ††thanks: Based on observations made at the Gemini
Observatory, under the identification number GS-2010B-Q-7.
M. Faúndez-Abans 11 A. C. Krabbe 22 M. de Oliveira-Abans 1133 P. C. da Rocha-
Poppe
I. Rodrigues 445522 V. A. Fernandes-Martin 4455 I. F. Fernandes 4455
(Received 17 February 2012 / Accepted 7 May 2012)
###### Abstract
Aims. We report first results of an investigation of the tidally disturbed
galaxy system AM 546-324, whose two principal galaxies 2MFGC 04711 and AM
0546-324 (NED02) were previously classified as interacting doubles. This
system was selected to study the interaction of ellipticals in a moderately
dense environment. We provide spectral characteristics of the system and
present an observational study of the interaction effects on the morphology,
kinematics, and stellar population of these galaxies.
Methods. The study is based on long-slit spectrophotometric data in the range
of $\sim$ 4500-8000 $\AA$ obtained with the Gemini Multi-Object Spetrograph at
Gemini South (GMOS-S). We have used the stellar population synthesis code
STARLIGHT to investigate the star formation history of these galaxies. The
Gemini/GMOS-S direct r-G0303 broad band pointing image was used to enhance and
study fine morphological structures. The main absorption lines in the spectra
were used to determine the radial velocity.
Results. Along the whole long-slit signal, the spectra of the Shadowy galaxy
(discovered by us), 2MFGC 04711, and AM 0546-324 (NED02) resemble that of an
early-type galaxy. We estimated redshifts of z= 0.0696, z= 0.0693 and z=
0.0718, corresponding to heliocentric velocities of 20 141 km s-1, 20 057 km
s-1, and 20 754 km s-1 for the Shadowy galaxy, 2MFGC 04711 and AM 0546-324
(NED02), respectively. The central regions of 2MFGC 04711 and AM 0546-324
(NED02) are completely dominated by an old stellar population of $2\times
10^{9}<\rm t\leq 13\times 10^{9}$ yr and do not show any spatial variation in
the contribution of the stellar-population components.
Conclusions. The observed rotation profile distribution of 2MFGC 04711 and AM
0546-324 (NED02) can be adequately interpreted as an ongoing stage of
interaction with the Shadowy galaxy as the center of the local gravitational
potential-well of the system. The three galaxies are all early-type. The
extended and smooth distribution of the material in the Shadowy galaxy is a
good laboratory to study direct observational signatures of tidal friction in
action.
###### Key Words.:
galaxies: general – galaxies: interacting group – individual: AM 0546-324 and
2MFGC 04711 – galaxies: spectroscopy – galaxies: stellar synthesis
††offprints: Max Faúndez-Abans; max@lna.br
## 1 Introduction
Galaxy interactions and mergers are fundamentally important in the formation
and evolution of galaxies. Hierarchical models of galaxy formation and various
observational evidence suggest that elliptical galaxies are, like disk
galaxies, embedded in massive dark-matter halos. Lenticular and elliptical
galaxies, called early-type galaxies have been thought to be the end point of
galaxy evolution. These systems have shown uniform red optical colors and
display a tight red sequence in optical color-magnitude diagrams (e.g. Baldry
et al. b2004 (2004)). Their color separation from star-forming galaxies is
thought to be due to a lack of fuel for star formation, which must have been
consumed, destroyed or removed on a reasonably short timescale (e.g. Faber et
al. f2007 (2007)). In addition, numerical simulations have shown that the
global characteristics of the binary merger remnants of two equal-mass spiral
galaxies, called major mergers, resemble those of early-type galaxies (Toomre
& Toomre tt1972 (1972); Hernquist & Barnes hb1991 (1991); Barnes b1992 (1992);
Mihos et al. m1995 (1995); Springel s2000 (2000); Naab & Burkert nb2003
(2003); Bournaud, Jog & Combes bjc2005 (2005)). Remnants with properties
similar to early-type objects can also be recovered through a multiple minor
merger process, the total accreted mass of which is at least half of the
initial mass of the main progenitor (Weil & Hernquist wh1994 (1994), wh1996
(1996); Bournaud, Jog & Combes bjc2007 (2007)). This scenario of early-type
formation through accretion and merging of bodies would fit well within the
frame of the hierarchical assembly of galaxies provided by cold dark matter
cosmology.
Interactions between early-type galaxies are less spectacular than those
observed in spiral galaxies. While impressive tidal tails, plumes, bridges,
and shells are observed in tidally disturbed spirals, the effects of the
interaction are less easily recognized in elliptical galaxies, since they have
little gas and dust, and are dominated essentially by old stellar populations.
Evidence for recent merger-driven star formation (Rogers et al. r09 (2009))
and morphological disturbances such as shells, ripples, and rings have been
observed in early-type galaxies (Kaviraj et al. kv10 (2010) and Wenderoth et
al. wen2011 (2011)) .
The peculiar Ring Galaxies (pRGs) show a wide variety of ring and bulge
morphologies and were classified by Faúndez-Abans & de Oliveira-Abans (foa98a
(1998)) into five families, following the general behavior of galaxy-ring
structures. From these categories eight morphological subdivisions are
highlighted. One of these morphological subdivisions is a basic structure
called Solitaire. The pRG Solitaire is described as an object with the bulge
on the ring, or very close to it, resembling a one-diamond finger ring (single
knotted ring). In these objects, the ring generally looks smooth and thinner
on the opposite side of the bulge (as archetypes FM 188-15/NED02, AM
0436-472/NED01), ESO 202-IG45/NED01 and ESO 303-IG11/NED01). Although the
statistics are as yet poor, the Solitaire type is probably produced by the
interaction between elliptical-like galaxies and/or gas-poor S0 galaxies with
an elliptical companion. In a forthcoming paper, a list of Solitaire-type pRGs
and a preliminary study and statistics will be presented.
There are no reports of Solitaires in early stages of formation in the
literature yet; so a few pairs of galaxies were selected as early-stage
candidates (see one of them in Wenderoth et al. wen2011 (2011)). Even though
one of the selected candidates, AM 546-324, originally extracted from Arp &
Madore’s catalog (Arp & Madore am1977 (1977), am1986 (1986); category 2,
interacting doubles), seems to be morphologically different from an expected
Solitaire in the early stage, it is remarkable enough to be studied as an
almost isolated “spherical/elliptical and S0 interacting objects” in centrally
sparse clusters of galaxies.
In this paper, we report new results for the tidally disturbed galaxy system
AM 546-324 based on data obtained from long-slit spectrophotometric
observations at Gemini Observatory, in Chile. Values of $H_{\rm o}$ = 70 km
s${}^{-1}{\rm Mpc}^{-1}$, $\Omega_{matter}=0.27$ and $\Omega_{vaccum}=0.73$
have been adopted throughout this work (Freedman et al. f2001 (2001); Astier
et al. a2006 (2006), and Spergel et al. s2003 (2003)).
## 2 AM 0546-324 review
The existing information on this object comprises: (1) the Arp-Madore catalog
(Arp & Madore am1986 (1986)) referred to it as “Category 2: interacting
doubles”, which are objects consisting of two galaxies that, by their apparent
magnitude and spacing, appear to be associated; (2) the redshifts of 165
Southern rich Abell cluster of galaxies by Quintana & Ramírez (qr1995 (1995)),
in which the galaxy system AM 546-324 is a member of Abell S0546; (3) the
2MASS-selected flat galaxy catalog by Mitronova et al. (mit2004 (2004)); (4)
the catalog of near-infrared properties of LEDA galaxies using the full-
resolution images from the DENIS survey (Paturel et al. pat2005 (2005)); and
(5) the entry from the 6dFGS-NVSS data by Mauch & Sadler (ms2007 (2007)). In
Abell et al. (aco1989 (1989)), S0546 is quoted as irregular following the
cluster classification in Abell’s (a1965 (1965)) system, with 23 cluster
members. Using the data quoted in Abell et al. (aco1989 (1989)) in a
magnitude-redshift relation for the Abell southern clusters, and using the
S0546 distance class $m_{10}=$5 result in an approximate radial velocity of
cz= 20 893 km s-1, which agrees with our results (using the non-relativistic
velocity formula, see Table 1).
Figure 1 displays the AM 546-324 system in a 5 minute-exposure GMOS-S pointing
image in the r-G0303 filter (effective wavelength of 6300 $\AA$). Table 1
displays the new velocity and z values, together with some early information
on the principal members of AM0̇546-324, Table 2 displays some information on
relevant objects in and around the system.
Figure 1: System AM 0546-324. Optical 5-min exposure GMOS-S image in the
r-G0303 filter, enhanced by a median filter kernel of 300$\times$300 pixels
(see Faúndez-Abans & de Oliveira-Abans foa98b (1998) for details on the
method). The slit position PA = 157$\degr$ is also displayed as two short
white lines to preserve the image of the objects. The letters stand for: K,
“the Knot” (Quintana & Ramírez qr1995 (1995)); S, the Shadowy galaxy; C, a
companion galaxy; and P, a probable Polar Ring galaxy. Table 1: Basic
properties of the principal galaxies of the system AM 0546-324
Parameter | 2MFGC 04711 | Shadowy galaxy | AM 0546-324 (NED02) | Ref.
---|---|---|---|---
R.A. (2000) | 05 48 34.1 | 05 48 34.7 | 05 48 35.1 | this work
Dec. (2000) | -32 39 30.9 | -32 39 46.2 | -32 40 01.0 | this work
Morphological classification | Elliptical | Cd? | Elliptical | this work
$z$ | 0.0693 | 0.0696 | 0.0718 | this work
$V$(km s-1) (a) | 20 057 $\pm$10 | 20 141 $\pm$10 | 20 754 $\pm$10 | this work
$V$(km s-1) (b) | 20 793 $\pm$10 | 20 880 $\pm$10 | 21 526 $\pm$10 | this work
$z$ | 0.0692 | | 0.0721 | NED (q)
$V$ (km s-1) | 20 749 $\pm$40 | | 21 615 $\pm$26 | NED (q)
Magnitude | 15.0 R | | | NED
Other designations | 2MASX J05483415-3239306 | | 2MASX J05483518-3240006 | NED
Distance (Mpc) | 292.6 | 293.8 | 303.0 | this work
Distance (Mpc) | 293.0 | | 305.0 | NED
$\sigma_{v}$ (km/s) | 312 | 365 | 197 | this work
Mass (lower limit) | 1.63$\times 10^{11}$ M☉ | 5.24$\times 10^{11}$ M☉ | 1.60$\times 10^{11}$ M☉ | this work
U-shaped base (r) | 2$\aas@@fstack{\prime\prime}$0 | | 1$\aas@@fstack{\prime\prime}$54 | this work
Major axis (lower limit) (r) | 9$\aas@@fstack{\prime\prime}$42 | 21$\aas@@fstack{\prime\prime}$73 | 8$\aas@@fstack{\prime\prime}$53 | this work
J - H | 0.404 | | 0.732 | NED
H - K | 0.142 | | $-$0.050 | NED
J - K | 0.546 | | 0.641 | NED
* a
Note: (a): extracted for high velocities (Lang lang1999 (1999)), see also
Lindengren & Dravins (ld2003 (2003)); (b): non-relativistic velocity using the
standard formula (Lang lang1999 (1999)); (r): measurements on the
Gemini/GMOS-S r-G0303-filter; (q): original data from Quintana & Ramírez
(qr1995 (1995)).
Table 2: Relevant objects in and around system AM 0546-324.
Object | R.A.(2000) | Dec.(2000) | $z_{\rm abs}$ | $V$(km s-1) | Distance (Mpc) | Ref.
---|---|---|---|---|---|---
J054832.5-323954.1 | 05 48 32.5 | -32 39 54.1 | | | | Polar Ring? this work
J054834.2-323944.5 | 05 48 34.2 | -32 39 44.5 | 0.0698 | 20 923 | 298.9 | The Knot (a)
J054835.4-324011.5 | 05 48 35.4 | -32 40 11.5 | 0.0685 | 20 550 $\pm$40 | 293.6 | C, this work (b)
* a
Note: (a) Quintana & Ramírez (qr1995 (1995)); (b) the compact companion C
close to AM 0546-324 (NED02).
## 3 Observations and data reduction
The spectroscopic observations were performed with the 8.1-m Gemini South
telescope (Chile) (ID program GS-2010B-Q-7). We used the GMOS-S spectrograph
in long-slit mode (Hook et al. h04 (2004))111A description of the instrument
can be found at http://www.gemini.edu/sciops/instruments/gmos.. The R400+G5325
grating was centered at 6 250 $\AA$ and used with a long-slit 1.5 arcsec x 375
arcsec. The data were binned by 2 in the spatial dimension and 2 in the
spectral dimension, producing a spectral resolution of $\sim$5.1$\AA$ FWHM,
sampled at 0.68 $\AA$ pix-1. The seeing throughout the observations was
0$\aas@@fstack{\prime\prime}$54 and the binned pixel scale was
0$\aas@@fstack{\prime\prime}$145 pix-1. The wavelength range was $\sim$
4500-8000 $\AA$. The spectrophotometric standard star H 600 was observed using
the same experimental set up. The long-slit spectra were taken at one position
angle on the sky, PA = 157$\degr$, to encompass the three objects in one shot,
and it almost crossed the center of each object.
The standard Gemini-IRAF routines were used to carry out bias subtraction,
flat-fielding, and cosmic ray subtraction. The data were then wavelength
calibrated with an accuracy $\leqslant$ 0.3 $\AA$. The binned 2-D spectra were
then flux-calibrated using the photometric standard star H 600. The 2-D
spectra were then extracted into 1-D spectra, which were sky-subtracted and
binned in the spatial dimension. We have cross-correlated our observed spectra
with three galaxy and star templates with good signal-to-noise. These results
were checked with the composite absorption-line template “fabtemp97”
distributed by RVSAO222The RVSAO IRAF (Radial Velocity Package for IRAF)
external package was developed at the Smithsonian Astrophysical Observatory.
Full documentation of this software, including numerous examples of its use,
in on-line at http://tdc-www.harvard.edu/iraf/rvsao/./IRAF external package.
We adopted the redshift value from the best highest correlated coefficient
template.
The spectral apertures were extracted with the APALL/IRAF package and three
methods: (1) the standard IRAF procedure; (2) overlapping a shifted sample
with steps $<5\arcsec$, which causes oversampling; (3) and the event-covering
method, for which we used the aperture step as a mapping event process (Wong &
Chiu wc1987 (1987)333The event-covering method is defined as a strategy of
selecting a subset of statistically independent events in a set of variable-
pairs, regardless of their statistical independence.; see also Schafer s1997
(1997) and Wu & Barbara wb2002 (2002)), which also causes oversampling. The
idea of the last two procedures was to use the aperture size as a filter to
detect kinematical structures in the long-slit velocity map. The results of
the first two procedures have been used in this work. The third method was
mainly used when the original data were either corrupted or incomplete. Its
results were not different from those of the second method because of the data
completeness.
## 4 Analysis and results
### 4.1 The field around AM 0546-324
As can be seen in Fig. 1, this system of galaxies is seen almost edge-on, with
five prominent objects in the field. According to the estimated distances,
they may be physically associated. In Fig. 1, from NW to SE, these objects are
(1) the almost spherical galaxy 2MFGC 04711; (2) the galaxy “Knot”, K, as
labeled by Quintana & Ramírez (qr1995 (1995)); (3) the object that we have
named the Shadowy galaxy (hereafter the S galaxy), whose center is enhanced in
the figure; (4) AM 0546-324 (NED02) (hereafter NED02), which is almost
spherical; and (5) the compact anonymous spherical galaxy “C”. Another
relevant object to the SW in the field (better seen in Fig. 2, bottom panel)
is a probable Polar Ring galaxy, named here “P”, whose redshift value is not
known yet.
To extract as much information as possible from the GMOS-S r-G0303 pointing
image, we used different spatial filtering to find fine morphological
structures in the frame. The top panel of Fig. 2 displays the result of
applying a median filter kernel of 100$\times$100 pixels, where the S galaxy
appears elongated, its center and a few rims having also been enhanced. A few
thin filaments in the K object, with one of them pointing to the center of the
S galaxy, have also been enhanced. Furthermore, the galaxies K, 2MFGC 04711,
and NED02 appear to be slightly deformed by the tidal interaction. The lower
panel of Fig. 2 displays a median filter kernel of 500$\times$500 pixels where
the deformation of the main objects and some faint dwarf structures apparently
bound to this system have been enhanced. The rims and the center of S galaxy
are still evident. The upper panel of Fig. 3 shows the center and the rims of
the S galaxy, after a low-pass filter was used. It highlights the deformation
of the elliptical galaxies and enhances a few notable dwarf satellites around
this system. The bottom panel of Fig. 3 is a zoom on the S galaxy to better
illustrate its center and a few very faint rims, after using a Gaussian
filter.
To determine the central ellipticity of the galaxies, we used the ELLIPSE
STSDAS-task444Space Telescope Science Data Analysis Software Package, which
fits elliptical isophotes to galaxy direct images. Then we created a 2-D
noiseless model image using the BMODEL STSDAS-task built from the results of
the isophotal analysis. Table 3 lists rough estimates of the “non perturbed”
elliptical-class section and the whole major axis-diameter in kpc for the
quoted galaxies.
$\begin{array}[]{cc}\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig02a.eps}}\\\
\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig02b.eps}}\end{array}$
Figure 2: From top to bottom: (a) median filtering with a 100$\times$100-pixel
kernel after the original image subtraction, (b) same as first panel for a
500$\times$500-pixel kernel. The clear patches are artifacts of the method,
but the fine structures are preserved. The candidate for a Polar Ring, which
has been discovered in this work, is marked by the letter “P”.
$\begin{array}[]{cc}\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig03a.eps}}\\\
\leavevmode\resizebox{256.0748pt}{}{\includegraphics{fig03b.eps}}\end{array}$
Figure 3: From top to bottom: (a) resulting image after using a low-pass filter (see text), (b) same as first panel for a Gauss filter, zooming on the S galaxy. Table 3: Ellipticity and major axis diameter. Object | “Bulge-like” section | Major diameter
---|---|---
| (elliptical class) | (kpc)
2MFGC 04711 | E1 | 13.7
Shadowy | E3/4:: | 31.6:
NED02 | E0 | 12.4
The Knot | E1: | 9.5
C companion | E1 | 3.6
### 4.2 The spectra and kinematics
We report the first dedicated long-slit spectroscopic results for the three
main galaxies of the AM 0546-324 system. A sample of the spectra of these
galaxies are shown in Fig. 4. The $\lambda\lambda 5000-7750$ Å spectral
section of 2MFGC 04711 and NED02 are displayed, while the S galaxy’s is
displayed only in the $\lambda\lambda 5000-5800$ Å spectral section. They show
the main stellar absorption features identified in the nuclear region:
H$\beta$, MgIb$\lambda 5174$, MgH$\lambda 5269$, NaID$\lambda 5892$, the TiO
bands $\lambda\lambda 6250,7060$, and the O2 atmospheric band. The column
density and positions of these lines were determined by fitting a Gaussian to
the observed profile. This procedure was reviewed using the RVSAO/XCSAO
package. The spectra in the regions $\lambda\lambda 7750-8000$ Å (for 2MFGC
04711 and NED02) and $\lambda\lambda 5800-8000$ Å (for the S galaxy) were not
taken into account because of the high noise. The characteristics of the
spectra are discussed below.
$\begin{array}[]{cc}\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig04a.eps}}\\\
\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig04b.eps}}\\\
\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig04c.eps}}\end{array}$
Figure 4: Optical nuclear absorption features of 2MFGC 04711, the Shadowy
galaxy and AM 0546-324 (NED02), in the upper, middle and lower panels,
respectively. The spectra are displayed in units of $\times 10^{-16}$ erg
sec-1cm${}^{-2}\AA^{-1}$.
$\begin{array}[]{cc}\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig05a.eps}}\\\
\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig05b.eps}}\end{array}$
Figure 5: (a) upper panel: the U-shaped rotation profile of 2MFGC 04711 along
the nucleus and bulge in the total observed long-slit $-$3″ to
$+$4″distribution (filled circles are data from NaID lines, open circles from
MgIb and open triangles stand for H$\beta$ lines); (b) lower panel: same as
first panel, for the central $-$1″ to $+$1″core features, but using a shifted
aperture sample with steps $<2\arcsec$ according to method (2) cited in item
§3.
$\begin{array}[]{cc}\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig06a.eps}}\\\
\leavevmode\resizebox{433.62pt}{}{\includegraphics{fig06b.eps}}\end{array}$
Figure 6: From top to bottom: (a) the U-shaped rotation profile of AM 0546-324
(NED02) along the nucleus and bulge in the total observed slit length $-$3″ to
$+$4″(filled circles are data from NaID lines, open circles from MgIb); (b)
same as first panel, for the central $-$1″to $+$1″core features, but using a
shifted aperture sample with steps $<2\arcsec$ according to method (2) cited
in item §3.
#### 4.2.1 2MFGC 04711
No emission lines were detected along the slit. The spectral profile resembles
that of an early-type galaxy. Using the NaID, MgIb, and H$\beta$ absorption
lines, we cross-correlated our observed spectra with templates with good
signal-to-noise and these results were checked with the composite absorption-
line template “fabtemp97” distributed by RVSAO. We derived z=0.0693 and a
heliocentric radial velocity of $V$= 20 057 $\pm$10 km s-1 . The velocity
dispersion $\sigma_{v}$ values were estimated by cross-correlation with K and
M star templates, with statistical error in $\sigma_{v}$ in the range of
10%$-$15% (XCSAO/RVSAO, also tested with XCOR/STSDAS). The dynamical masses
were estimated based on a virial relation, the effective radius and the
velocity dispersion of each galaxy. The derived effective radius and dynamical
mass fit within $\leq 1\sigma$ with the size-mass relation presented by
Bezanson et al. (beza2011 (2011)), predicts values close to those derived here
for each galaxy. The calculated distance and dynamical mass are 292.6 Mpc and
1.63$\times 10^{11}$ M☉, respectively. Figure 5 displays the 2MFGC 04711
distribution of radial velocities measured from the H$\beta$, MgIb and NaID
absorption lines, along the slit-section (upper panel) and central region
(lower panel). The errors of the individual velocity measurements do not
exceed 10 km s-1 in the central region and increase to 20$-$40 km s-1 on its
periphery.
#### 4.2.2 AM 0546-324 (NED02)
NED02 also shows an early-type spectral profile. We estimated z=0.0718 and a
heliocentric radial velocity of $V$= 20 754 $\pm$10 km s-1. The calculated
distance and dynamical mass are 303 Mpc and 1.60$\times 10^{11}$ M☉,
respectively. Figure 6 displays the NED02 distribution of radial velocities
measured from the MgIb and NaID absorption lines, along the slit-section
(upper panel) and central region (lower panel). For clarity, the H$\beta$ data
are not shown in the first panel because they are mostly coincident with the
NaID points at r $>\pm$1″ . The errors of the individual velocity measurements
do not exceed 10 km s-1 in the central region and increase to 20$-$40 km s-1
on its periphery.
#### 4.2.3 The Shadowy galaxy
We estimated z=0.0696 and a heliocentric radial velocity of $V$= 20 141
$\pm$10 km s-1, from an integrated central spectral-section of
3$\aas@@fstack{\prime\prime}$5\. On the other hand, the calculated velocities
for the NW and SE sections of the galaxy are 20 193 $\pm$20 km s-1 (NW) and 20
081 $\pm$20 km s-1 (SE), respectively (integrated NW$-$SE spectral-section of
6$\aas@@fstack{\prime\prime}$1). The calculated distance and lower limit of
the dynamical mass are 293.8 Mpc and 5.24$\times 10^{11}$ M☉, respectively.
The distribution of the radial velocities of 2MFGC 04711 and NED02 shows the
U-shaped rotation profile (first panel of Figs. 5 and 6). This shape has been
reported in studies of interacting binary-disturbed elliptical galaxies (see
Borne b1990 (1990); Borne & Hoessel bh1985 (1985),bh1988 (1985); Bender et al.
bpn1991 (1991); Madejsky mad1991 (1991) and Madejsky et al. madall1991
(1991)). The U-shaped profiles are common in strongly interacting elliptical
galaxies, and the physical interpretation, given by Borne et al. (borne1994
(1994)) is that there is a tidal coupling between the orbit of the companion
and the resonant prograde rotating stars in the kinematically disturbed galaxy
(Borne bor1988 (1988); Borne & Hoessel bh1988 (1985); Bacells et al.
bacells1989 (1989)). The coupling of NED02 and 2MFGC 047111 with the S galaxy
and the U-shaped rotation profile of these galaxies are thus a direct
observational signature of tidal friction in action within this system, in
agreement with the physical interpretation of Borne et al. (borne1994 (1994)).
Based on the merging times of simulations performed for a low-mass galaxy
falling on to a massive elliptical by Leeuwin & Combes (lc1997 (1997)) and
adopting for the S galaxy a rmax = 30 kpc, our rough estimate for the decay
times for 2MFGC 04711, NED02 and C galaxy are 4$\times 10^{8}$ yr, 3.6$\times
10^{8}$ yr, and 5.6$\times 10^{8}$ yr, respectively. With the suitably simple
model reported by Leeuwin & Combes (lc1997 (1997)) the amount of friction
should accelerate in the decay time, which could be the scenario for the S
satellite with a decay time shorter than 1.0$\times 10^{9}$ yrs.
The errors of the individual velocity measurements do not exceed 10 km s-1 in
the central region of the galaxies and increase to 20$-$40 km s-1 at their
periphery. There is a significant dispersion in the radial velocity
distribution of both 2MFGC and NED02 (see first panels of Figs. 5 and 6,
respectively). The velocity spread is $\pm$40$-$110 km s-1 around
$\pm$1″$-$3″.
### 4.3 Stellar population synthesis
The detailed study of star formation in tidally perturbed galaxies provides
important information not only on the age distribution of the stellar
population, but also helps to better understand several aspects related to the
interacting process and its effects on the properties of the individual
galaxies and their evolution.
To investigate the star formation history of NED02 and 2MFGC, we used the
stellar population synthesis code STARLIGHT (Cid Fernandes et al. cid04 (2004,
2005); Asari et al.asari07 (2007)). This code has been extensively discussed
in Cid Fernandes et al. (cid04 (2004, 2005)) and is built upon computational
techniques originally developed for empirical population synthesis with
additional ingredients from evolutionary synthesis models. This method was
also used by Krabbe et al. (krabbe2011 (2011)) and has been successful in
describing the stellar population in interacting galaxies.
The code fits an observed spectrum $O_{\lambda}$ with a combination of
$N_{\star}$ single stellar populations (SSPs) from the Bruzual & Charlot
(bruzual03 (2003)) models. These models are based on a high-resolution library
of observed stellar spectra, which allows for detailed spectral evolution of
the SSPs at a resolution of 3 Å across the wavelength range of 3 200-9 500
$\AA$ with a wide range of metallicities. We used the Padova (1994) tracks as
recommended by Bruzual & Charlot (bruzual03 (2003)), with the initial mass
function of Salpeter (salpeter55 (1955)) between 0.1 and 100 $M_{\sun}$.
Extinction is modeled by STARLIGHT as due to foreground dust, using the Large
Magellanic Cloud average reddening law of Gordon et al. (gordon03 (2003)) with
RV= 3.1, and parametrized by the V-band extinction AV. The SSPs used in this
work cover 15 ages namely, t = 0.001 , 0.003 , 0.005 , 0.01 , 0.025 , 0.04 ,
0.1 , 0.3 , 0.6 , 0.9 , 1.4 , 2.5 , 5 , 11 , and 13 Gyr, as well as three
metallicities, Z = 0.2 Z☉, 1 Z☉, and 2.5 Z☉, adding to 45 SSP components. The
fitting is carried out using a simulated annealing plus Metropolis scheme,
with bad pixel regions excluded from the analysis.
Prior to the modeling, the SSPs models were convolved to the same resolution
of the observed spectra; the observed spectra were shifted to their rest-
frame, corrected for foreground Galactic reddening of $E(B-V)=0.036$ mag taken
from Schlegel et al. (sc98 (1998)) and normalized to $\lambda\,5870\,$Å. The
error in $O_{\lambda}$ considered in the fitting was the continuum rms with a
$S/N\geq 10$, where $S/N$ is the signal-to-noise ratio per $\AA$ in the region
around $\lambda_{0}=5870\,\AA$. In addition, the fitting was performed only in
spectra with absorption lines.
Figures 7 and 8 show an example of the observed spectrum corrected by
reddening and the model stellar population spectrum for 2MFGC 04711 and AM
0546-324 (NED02) galaxies, respectively. The results of the synthesis to the
very central region are summarized in Table 4 for the individual spatial bins
in each galaxy, stated as the perceptual contribution of each base element to
the flux at $\lambda\,5\,870$ Å. Following the prescription of Cid Fernandes
et al. (cid05 (2005)), we have defined a condensed population vector by
binning the stellar populations according to the flux contributions into
young, $x_{\rm Y}$ ($\rm t\leq 5\times 10^{7}$ yr); intermediate-age, $x_{\rm
I}$ ($5\times 10^{7}<\rm t\leq 2\times 10^{9}$ yr); and old, $x_{\rm O}$ (
$2\times 10^{9}<\rm t\leq 13\times 10^{9}$ yr) components. The same bins were
used to represent the mass components of the population vector $m_{\rm Y}$,
$m_{\rm I}$, and $m_{\rm O}$). The metallicity (Z), one important parameter to
characterize the stellar population content, is weighted by light fraction.
The quality of the fitting result is measured by the parameters $\chi^{2}$ and
$adev$. The latter gives the perceptual mean deviation
$|O_{\lambda}-M_{\lambda}|/O_{\lambda}$ over all fitted pixels, where
$O_{\lambda}$ and $M_{\lambda}$ are the observed and model spectra,
respectively.
The spatial variation in the contribution of the stellar population components
for 2MFGC. NED02 is completely dominated by an old stellar population.
Table 4: Stellar-population synthesis results
Pos. | $x_{\rm Y}$ | $x_{\rm I}$ | $x_{\rm O}$ | $m_{\rm Y}$ | $m_{\rm I}$ | $m_{\rm O}$ | $Z_{\star}$[1] | $\chi^{2}$ | $\rm adev$ | $\rm A_{v}$
---|---|---|---|---|---|---|---|---|---|---
(arcsec) | (per cent) | (per cent) | (per cent) | (per cent) | (per cent) | (per cent) | | | (mag) |
2MFGC 04711 |
-1.04 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.020 | 1.9 | 2.24 | 0.59
-0.90 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.24 | 0.07
-0.77 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.0 | 1.24 | 0.07
-0.65 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.25 | 0.07
-0.52 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.028 | 2.0 | 1.21 | 0.07
-0.39 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.0 | 1.24 | 0.07
-0.27 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 1.9 | 1.24 | 0.07
-0.14 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.028 | 2.0 | 1.22 | 0.07
-0.05 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.0 | 1.23 | 0.07
0.0 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.023 | 1.8 | 1.30 | 0.00
0.02 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.0 | 1.22 | 0.07
0.11 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.0 | 1.22 | 0.07
0.20 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.0 | 1.26 | 0.07
0.30 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.24 | 0.07
0.41 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.23 | 0.07
0.54 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.24 | 0.07
0.66 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.24 | 0.07
0.79 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 1.9 | 1.23 | 0.07
0.91 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.0 | 1.25 | 0.07
1.06 | 0.0 | 15.0 | 85.0 | 0.0 | 4.9 | 95.1 | 0.041 | 1.9 | 1.93 | 0.00
1.18 | 0.0 | 21.0 | 79.0 | 0.0 | 5.6 | 94.4 | 0.034 | 1.8 | 2.02 | 0.00
Integrated | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 1.9 | 1.23 | 0.06
AM 0546-324 (NED02) |
-0.68 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.41 | 0.02
-0.56 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.43 | 0.02
-0.44 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.025 | 2.2 | 1.36 | 0.02
-0.33 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.2 | 1.43 | 0.02
-0.21 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.2 | 1.41 | 0.02
-0.10 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.39 | 0.02
0.00 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.025 | 2.2 | 1.24 | 0.02
0.10 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.3 | 1.35 | 0.02
0.21 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.34 | 0.02
0.30 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.027 | 2.2 | 1.44 | 0.02
0.40 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.43 | 0.02
0.51 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.025 | 2.2 | 1.35 | 0.02
0.63 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.025 | 2.2 | 1.42 | 0.02
0.73 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.026 | 2.2 | 1.38 | 0.02
0.84 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.024 | 2.4 | 1.63 | 0.17
Integrated | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.020 | 1.7 | 1.54 | 0.00
[1] Abundance by mass with Z☉=0.02
Figure 7: Stellar population synthesis for 2MFGC 04711. Central bin observed
spectrum corrected for reddening (in black, shifted up by a constant) and the
synthesized spectrum, in red. Figure 8: Stellar population synthesis for
NED02. Central bin observed spectrum corrected for reddening (in black,
shifted up by a constant) and the synthesized spectrum, in red.
## 5 Discussion
The galaxy cluster Abell S0546 has 23 cluster members between $m_{3}$ and
$m_{3}+2$ (Abell a1965 (1965)), and the whole AM 0546-324 system is the core
of the S0546. There are quite a few dwarf satellites around it, as expected
for a local gravitational potential well defined by the AM 0546-324 system
(see lower panel of Figs. 2 and 9). The main galaxies quoted in the literature
are 2MFGC 04711, NED02, and the knot (K), which is described as an almost E1
galaxy (Quintana & Ramírez qr1995 (1995)). Other members quoted in our study
are the S galaxy and the C companion, which have been confirmed as members of
this cluster. The derived radial velocity of the S galaxy is 20 141 km s-1,
which is our estimate of the radial velocity of the Abell S0546 core. The
derived radial velocity of S using the non-relativistic formula agrees with
the S0546 distance class $m_{10}=$ 5 (cz = 20 893 km s-1) extracted from Abell
et al. (aco1989 (1989)). Figure 10 displays the spatial distribution of the
main galaxies of the AM 0546-324 system centered on the S galaxy (see also
Fig. 11).
Along the whole slit, the spectra of the three main galaxies show absorption
lines characteristic of early-type objects. No star-forming regions and no
nuclear ionization sources were detected. The whole AM 0546-324 system seems
to be tidally bound with radial velocity differences that range between
56$-$613 km s-1. Adopting the S galaxy as the center of the system, the pair-
velocity-difference combinations of the main objects are displayed in Table 5.
The nearby C galaxy was partially inside the GMOS-S slit and the estimated
redshift z=0.0685 corresponds to a heliocentric radial velocity of $V$= 19 834
$\pm$30 km s-1, with a calculated distance of 289.2 Mpc and a dynamical mass
of 3.74$\times 10^{10}$ M☉. The radial velocity of the K galaxy (20 923 km
s-1) was extracted from Quintana & Ramírez (qr1995 (1995)). From the quoted
mass, 2MFGC and NED02 each have approximately 31% of the mass of the S galaxy,
and the C galaxy has almost 7%. Evidence of the tidal interaction of this
system are seen in the external deformations of 2MFGC 04711, NED02, and K
galaxies, as well as the rims in the S galaxy (displayed in Figs. 1, 2, and
3).
In this section, we propose the following kinematical behavior for the AM
0546-324 system based on spectroscopy and the direct image: (A) the S-galaxy
is the center of the system, its SE-section is approaching and the NW-section
is receding from us; (B) the 2MFGC 04711 galaxy is approaching us, and seems
to be embedded in the peripheral material of the S-galaxy, its motion is
retrograde from de S-galaxy apparent rotation; (C) NED02 is receding from us
and its motion is also retrograde from the S-galaxy apparent rotation; (D) the
K-galaxy is receding from us, and seems to be embedded in S-galaxy material
and coupled prograde with the S-galaxy motion; (E) the C-galaxy is approaching
us and coupled prograde with the S-galaxy motion.
Both 2MFGC 04711 and NED02 show a U-shaped rotation profile. In addition to
the argument that tidal coupling in ellipticals with no net rotation will
result in a U-shaped rotation profile with the galaxy core at the base of the
U (Borne & Hoessel bh1988 (1985); Borne et al. borne1994 (1994)), it was also
suggested that when the galaxy has a low degree of internal rotation, the
tidal coupling should produce this U-type profile (Borne et al. borne1994
(1994)). The lower panels of Figs. 5 and 6 show the rotation profile in the
interval $\pm$1″ for the 2MFGC 04711 and NED02, respectively. The oversampled
data are displayed in both figures to show signatures around the kinematical
center. These figures suggest that (1) 2MFGC 04711 seems to have a low degree
of perturbation and there is maybe a very slow internal rotation, its
kinematical center is almost centered at the U-shaped-base profile; (2) NED02
seems to show no internal rotation in its U-shaped profile, but there is a
break indicated by the MgIb and H$\beta$ sampled data, for which we do not
have a reliable explanation. The kinematical center is off-centered a few pc
to the SE-direction in its U-shaped-base profile. This phenomenon is also seen
in other pair interaction of Solitaire-type galaxies (see Faúndez-Abans et al.
fa_oa2010 (2010)).
Table 5: Radial velocity differences in the galaxy system. Pair | $\bigtriangleup$V (km/s)
---|---
Shadowy$-$2MFGC 04711 | 84
Shadowy$-$Knot | 56
Shadowy$-$NED02 | 613
Shadowy$-$C galaxy | 307
2MFGC 04711$-$Knot | 140
2MFGC 04711$-$NED02 | 697
NED02$-$C galaxy | 920
Figure 9: Reproduction of the image of AM 0546-324 with different contrast to
highlight the dwarf objects crowding the system.
Figure 10: Spatial distribution of the main galaxies of the AM 0546-324 system
centered on the Shadowy galaxy: the line-of-sight in kpc (X-axis); the
calculated distance from us in Mpc (Y-axis); and the relative distance between
the objects in the sky-plane in kpc (Z-axis). The S galaxy is displayed as a
dot inside a circle.
Figure 11: Calculated radial velocity in km s-1 versus the distribution of the
main objects in the line-of-sight centered on the Shadowy galaxy. The open
circles are the center, the SE and NW sections of the S galaxy, respectively.
Figures 2 (lower panel) and 9 show some dwarf objects around the AM 0546-324
system. A few of those objects may be aligned with the visible AM 0546-324
structure. This sparse system has an intrinsic tidal field, which could be an
interesting laboratory for studying the relationship between the central
components and the dark matter halo in weak fields (see simulations for a
denser cluster environment by Pereira & Bryan pb2010 (2010)). A non-negligible
anonymous galaxy is to the east between the field galaxy 2MASX
J05481766-3239441 and the AM 0546-324 system. The coordinates of the centroid
(J2000), as calculated differentially from the centroid of 2MASX
J05481766-3239441, are $\alpha=$05h 48m 25$\aas@@fstack{m}$81 , $\delta=$
$-$32° 39′ 49$\aas@@fstack{\prime\prime}$5, with no previously reported
redshift in the literature. This edge-on anonymous galaxy lies almost aligned
with the linear distribution of the AM 0546-324 system members. Is this object
a candidate for tidal alignment? (see recent discussion on tidal alignment
model of intrinsic galaxy alignments by Blazek et al. bmac2011 (2011))
The stellar formation history of 2MFGC 04711 and AM 0546-324 (NED02) galaxies
were well represented by the stellar population synthesis code STARLIGHT (see
Figs. 7 and 8). The synthesis results in flux fraction as a function of the
distance to the center of each galaxy do not show any spatial variation in the
contribution of the different stellar population components. Both galaxies are
dominated by an old stellar population with age between $2\times 10^{9}<\rm
t\leq 13\times 10^{9}$ yr in all apertures.
## 6 Conclusions
We reported optical band spectroscopy observations of the AM 0546-324 system,
which is the core of Abell S0546 cluster of galaxies. Morphological
substructures were found in an enhanced r-image of this system. This suggests
that the members are presently undergoing early stages of tidal interaction.
Below is a summary of our main results:
* •
The AM 0546-324 system is composed of four main galaxies: 2MFGC 04711, AM
0546-324 (NED02), the K galaxy, and the one named S galaxy by us. Adopting the
S galaxy as the center of this gravitationally bound system, the radial
velocity differences between the different quoted members vary from 43 to 646
km $s^{-1}$.
* •
Within 1.2 arcmin of AM 0546-324 there are a few relevant field companions
such as the C galaxy in the SE direction and a new Polar Ring galaxy candidate
in the SW. Several dwarf objects in and surrounding this system are close
enough to be candidate members of this system, but no quoted redshift for
these objects was found in the literature.
* •
The S galaxy seems to be large enough to wrap up all principal companions with
its smooth distribution of material.
* •
The spectra of 2MFGC 04711, NED02, S, and the C galaxy resemble those of
early-type galaxies and no emission lines were detected. No star-forming
regions and no nuclear ionization sources were detected in the observed
regions of the four main galaxies.
* •
The calculated heliocentric radial velocity for the S galaxy is 20 141 $\pm$
10 km$s^{-1}$ (z = 0.0696), which agrees with the radial velocity of the Abell
S0546 cluster (cz = 20 893 km s-1); for 2MFGC 04711, it is 20 057 $\pm$ 10
km$s^{-1}$ (z = 0.0693); and for NED02, it is 20 754 $\pm$ 10 km$s^{-1}$ (z =
0.0718), both in agreement with quoted values in NED.
* •
The C galaxy, cz = 19 834 $\pm$ 40 km$s^{-1}$ (z = 0.0685), and the K galaxy,
cz = 20 197 (z = 0.0698), are both bound members of the AM 0546-324 system.
* •
From the calculated mass lower limit, both 2MFGC 04711 and NED02 have
$\sim$31% of the mass of the S galaxy, and the C galaxy, almost 7%.
* •
The rotation profiles of 2MFGC 04711 and NED02 are typical of tidal coupling
in ellipticals with no net rotation, which results in a U-shaped rotation
profile with the galaxy core at the base of the U. Both galaxies are
gravitationally coupled directly with the proposed central object of the
cluster, the S galaxy. The U-shaped structure is a direct observational
signature of tidal friction with the extended material of the S galaxy.
* •
Internally, the no-net rotation core in the U-shaped rotation profile of both
2MFGC 04711 and NED02 seems to be slightly perturbed by the tidal interaction
with the S galaxy, which lies in the center of the local gravitational
potential-well of this system.
* •
2MFGC 04711 and AM 0546-324 (NED02) are completely dominated by an old stellar
population with age between $2\times 10^{9}<\rm t\leq 13\times 10^{9}$ yr.
In summary, AM 0546-324 is a system where signatures of tidal perturbations
and friction are clearly visible. The deformity detected in the 2MFGC 04711,
NED02, and K galaxies is due to large tidal forces exerted principally by the
S galaxy (like the deformation and dynamical friction between two elliptical
galaxies$-$Prugniel & Combes pc1992 (1992)). Simultaneously, the S galaxy is
perturbed by the whole interaction with all principal objects of the system.
Two questions still remain to be answered: (1) is the S galaxy environment the
starting point for the birth of a future cD galaxy? and (2) what is the origin
of the S galaxy?
###### Acknowledgements.
This work was partially supported by the Ministerio da Ciência, Tecnologia e
Inova cão (MCTI), Laboratório Nacional de Astrofísica, and Universidade do
Vale do Paraíba - UNIVAP. A. C. Krabbe thanks the support of FAPESP, process
2010/01490-3. We also thank Ms. Alene Alder-Rangel and M. de Oliveira-Abans
for editing the English in this manuscript. Based on observations obtained at
the Gemini Observatory, which is operated by the Association of Universities
for Research in Astronomy, Inc., under a cooperative agreement with the NSF on
behalf of the Gemini partnership: the National Science Foundation (United
States), the Science and Technology Facilities Council (United Kingdom), the
National Research Council (Canada), CONICYT (Chile), the Australian Research
Council (Australia), Ministerio da Ciência, Tecnologia e Inova cão (Brazil)
and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). The
observations were performed under the identification number GS-2010B-Q-7. This
publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the Infrared
Processing and Analysis Center/California Institute of Technology, funded by
the National Aeronautics and Space Administration and the National Science
Foundation. This research also used the NASA/ IPAC Infrared Science Archive,
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration.
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|
arxiv-papers
| 2012-06-04T19:40:21 |
2024-09-04T02:49:31.501863
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Fa\\'undez-Abans, A. C. Krabbe, M. Oliveira-Abans, P. C. da Rocha\n Poppe, I. Rodrigues, V. A. Fernandes Martin, I. F. Fernandes",
"submitter": "Vera Aparecida Fernandes Martin",
"url": "https://arxiv.org/abs/1206.0719"
}
|
1206.0759
|
# Near-IR Variability in young stars in Cygnus OB7
Thomas S. Rice11affiliation: Department of Astronomy, Harvard University, 60
Garden Street, Cambridge, MA 02138. , Scott J. Wolk22affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138. ,
Colin Aspin33affiliation: Institute for Astronomy, University of Hawaii at
Manoa, 640 N Aohoku Pl, Hilo, HI 96720.
###### Abstract
We present the first results from a 124 night $J$, $H$, $K$ near-infrared
monitoring campaign of the dark cloud L 1003 in Cygnus OB7, an active star-
forming region. Using 3 seasons of UKIRT observations spanning 1.5 years, we
obtained high-quality photometry on 9,200 stars down to $J$=17 mag, with
photometric uncertainty better than 0.04 mag. On the basis of near-infrared
excesses from disks, we identify 30 pre-main sequence stars, including 24
which are newly discovered. We analyze those stars and find the NIR excesses
are significantly variable.
All 9,200 stars were monitored for photometric variability; among the field
star population, $\sim$160 exhibited near-infrared variability (1.7% of the
sample). Of the 30 YSOs (young stellar objects), 28 of them (93%) are variable
at a significant level. 25 of the 30 YSOs have near-infrared excess consistent
with simple disk-plus-star classical T Tauri models. Nine of these (36%) drift
in color space over the course of these observations and/or since 2MASS
observations such that they cross the boundary defining the NIR excess
criteria; effectively, they have a transient near-infrared excess. Thus, time-
series $JHK$ observations can be used to obtain a more complete sample of
disk-bearing stars than single-epoch $JHK$ observations. About half of the
YSOs have color-space variations parallel to either the classical T Tauri star
locus (Meyer et al., 1997), or a hybrid track which includes the dust
reddening trajectory. This indicates that the NIR variability in YSOs that
possess accretion disks arises from a combination of variable extinction and
changes in the inner accretion disk: either in accretion rate, central hole
size and/or the inclination of the inner disk. While some variability may be
due to stellar rotation, the level of variability on the individual stars can
exceed a magnitude. This is a strong empirical suggestion that protoplanetary
disks are quite dynamic and exhibit more complex activity on short timescales
than is attributable to rotation alone or captured in static disk models.
accretion, accretion disks – stars: formation – stars: pre-main sequence –
stars: variables – infrared: stars
## 1 Introduction
Near-infrared studies of young stars allow for the direct detection of
optically thick disks around these stars via excess $K$-band flux (Lada &
Adams, 1992; Lada et al., 2000). The intrinsic colors of these disk-bearing
young stars occupy a well-defined locus in $(J-H)$ vs. $(H-K)$ two-color space
(henceforth called $JHK$ space), and their position along this locus is
determined by physical parameters such as their inclination angle, disk inner
hole size, and accretion rate (Meyer et al., 1997; Robitaille et al., 2006).
This technique of identifying young stellar objects (YSOs) using their near-
infrared colors has been used extensively to characterize populations of young
stars associated with star-forming regions in Orion, Taurus, Ophiuchus,
Chamaeleon, and others for nearly two decades (e.g. Strom et al., 1993, 1995;
Itoh et al., 1996; Hillenbrand et al., 1998; Oasa et al., 1999; Haisch et al.,
2000; Robberto et al., 2010).
Disk-bearing young stars, also known as classical T Tauri stars (cTTs), have
long been identified as optically variable (Joy, 1945; Herbig, 1962), with
this variability. At a minimum the variability is due to a combination of (I)
cold starspots, (II) hot accretion spots, and (III) circumstellar dust
occultations (Herbst et al., 1994). Optical variability can be used to
effectively identify young low-mass stars, even in regions lacking other
tracers of star formation such as molecular clouds (Briceño et al., 2005).
The variability of T Tauri stars in the near-infrared has been less thoroughly
studied. One of the first projects was carried out by Skrutskie et al. (1996)
who sampled bright T-Tauri stars, mostly members of the Taurus cloud, over the
course of a few nights. They found nearly all the stars varied significantly
and the amplitude of K-band source variability was weakly correlated with
(K$-$L) excess – a reliable disk diagnostic. From a multi-epoch study of the
Serpens cloud core, Kaas (1999) found the IR-variability a strong indicator of
youth. Variability across 14 epochs was used to reveal new YSOs in the $\rho$
Oph cluster (Alves de Oliveira & Casali, 2008). In the early part of the last
decade, Carpenter et al. (2001) used 16 observations of a $0\fdg 84\times
6\arcdeg$ region over 19 days to study NIR variability toward the Orion Nebula
Cluster, and Carpenter et al. (2002) used 15 observations of a similarly sized
field in the Chamaeleon I star-forming region over 4 months; the typical peak-
to-peak amplitude of variability seen was around 0.2 magnitudes in each band.
These studies found near-IR (NIR) variability most commonly arose from cold
starspots, hot accretion spots, and variable extinction. However, among stars
with a near-infrared excess ($\sim 25\%$ of the variable stars possessed NIR
excess), changes in the accretion disk were required to explain the observed
variability. Eiroa et al. (2002) combined optical and NIR data of 18 bright
stars and found 12 had correlated optical and NIR variability trends,
suggestive of a common physical origin such as spots and/or variable
extinction. The lack of correlation in the other objects was taken as a sign
that in those stars, distinct processes wereAn investigation of long-term NIR
variability in a large sample of T Tauri stars using 2-3 epochs covering
baselines of 6-8 years was carried out by Scholz et al. (2009) (see also
Scholz, 2012). They find the fraction of large-amplitude variables increases
for progressively longer baselines. They derived a 2500-year upper limit on
the duty cycle for large-scale episodic accretion events. The YSOVAR survey of
an 0.9 $\textrm{deg}^{2}$ region of the Orion Nebula Cluster (Morales-Calderón
et al., 2011) obtained 81 epochs of mid-IR Spitzer photometry over 40
consecutive days, in conjunction with 32 epochs of $J$-band images from UKIRT
and 11 epochs of $K_{s}$ photometry from CFHT. In that study of disk-bearing
young stars, various phenomena were observed, including periodic variability
and disk occultation events.
In this paper, we present a survey in which the photometric variability of
objects in the Braid Nebula star-forming region within Cygnus OB7 was
monitored for nearly two years. Our goal is to detect high-confidence YSO
candidates with precision photometry, study their variability, and analyze the
stability of the near-infrared disk diagnostic.
The constellation Cygnus contains several rich and complex star-forming
regions including Cygnus X as well as the North American and Pelican nebulae
(Reipurth & Schneider, 2008). Nine OB associations have been found in Cygnus,
with Cygnus OB7 the nearest at a distance of around 800 pc (Aspin et al.,
2009, distance modulus $\mu=9.5$). Within Cygnus OB7 lies a complex of several
dark clouds collectively known as Kh 141 (Khavtasi, 1955) that have been
individually identified in the Lynds catalog (Lynds, 1962). The dark cloud LDN
1003 in Cyg OB7 has been identified as a site of active star formation, having
first been studied in the optical by Cohen (1980) who found a diffuse red
nebula he named RNO 127. This nebula was later determined to be a bright
Herbig-Haro (HH) object by Melikian & Karapetian (2001, 2003, HH 428). Further
study in the optical and near-infrared identified a number of Herbig-Haro
objects (Devine et al., 1997; Movsessian et al., 2003) and multiple IRAS
sources (Dobashi et al., 1996) that reveal the presence of a young stellar
population and significant star formation activity.
The presence of two FUors in this region (Reipurth & Aspin, 1997; Greene et
al., 2008; Aspin et al., 2011) that has come to be known as the Braid Nebula
region (Movsessian et al., 2006) led to a focused, multi-wavelength effort to
study the young stars in this dark cloud, which is currently ongoing. A
narrowband optical and near-infrared survey of HH outflows observed using the
Subaru 8 m telescope (Magakian et al., 2010) found 12 outflows that have
identifiable originating/exciting sources and many more nebulous objects not
yet associated with identified sources. Aspin et al. (2009) carried out a
near-infrared integral field spectroscopic survey to identify actively
accreting sources such as T Tauri stars, which studied 16 sources in the field
and identified 12 young stars among them. Using the Caltech Submillimeter
Observatory, a 1.1 mm map of cold gas and dust associated with these young
stars was obtained (Aspin et al., 2011), identifying 55 cold dust clumps, 11
of which were associated with IRAS sources. The wide range of evolutionary
states encountered in this region, from starless clumps to optically visible T
Tauri stars, suggests that star formation here is an ongoing process rather
than a one-time occurrence.
In this paper we describe a $JHK$ monitoring survey of the Braid Nebula star-
forming region in Cygnus OB7. We plan to present first results from this
survey in two papers; this is paper 1 of 2. Our goal in Paper I is to identify
disk-bearing young stars, to broadly characterize the variability properties
of these stars using sensitive, long-baseline, nightly-cadence time-series
photometry to investigate the reliability of the near-infrared excess
criterion itself with respect to time. In the second paper (Wolk et al. 2012,
in prep, hereafter known as Paper II) we will describe the specific phenomena
seen in the lightcurves, carefully analyze the variability and periodic nature
of specific stars and present variability statistics for the field star
population.
In §2 we outline the UKIRT observations and data reduction procedures. In §3
we describe our method of identifying stars with near-infrared excesses and
studying their variability. In §4 our results are presented, and in §5 we
discuss the broad implication of these results on infrared studies of young
stars.
## 2 Data
### 2.1 Observations and Data Processing
The $J$, $H$, and $K$ observations of a $0.9^{\circ}\times 0.9^{\circ}$ region
in Cygnus OB7 were obtained using the Wide Field Camera (WFCAM) instrument on
the United Kingdom InfraRed Telescope (UKIRT), an infrared-optimized 3.8 m
telescope atop Mauna Kea, Hawaii at 13,800 feet elevation. These data consist
of WFCAM observations taken from 26-April-2008 to 11-October-2009 (Figure 1)
in three observing seasons as part of a special observation program described
in Aspin et al. 2012 (in preparation). The first season was in spring of 2008
and covered 26 nights. The second season was during fall 2008 and lasted 71
days. The third season was approximately one year later and lasted 75 days.
During the monitoring runs, data were taken once per night on a total of 124
nights in all three bands. Atmospheric seeing was between $0\farcs 5$ and
$1\farcs 6$ on any given night. The $J$, $H$, $K$ filter bands on WFCAM comply
with the $JHK$ broadband filters from the Mauna Kea Observatories Near-
Infrared filter set (Casali et al., 2007; Tokunaga et al., 2002); WFCAM’s
photometric system is described in Hewett et al. (2006).
Figure 1: Schematic representation of when data were taken for this study.
Nights with high errors have been removed (See §2.2).
Detailed specifications for the WFCAM instrument are given in Casali et al.
(2007). The instrument has four 2048 $\times$ 2048 Rockwell Hawaii-II PACE
arrays with a scale of 0.4 arcseconds per pixel, giving a combined solid angle
of 0.21 square degrees per exposure. The four detectors are widely spaced, at
94% of one detector’s width apart (see Fig. 2). We used a standard effective
integration time of 40 seconds per pointing. In a stepping pattern, WFCAM can
scan a nearly-complete square degree of the sky in four pointings (A, B, C, D
in Fig. 2). The pointings have a minor overlap on their edges such that stars
in the overlapping region were observed twice per night.
Data from the survey were pipeline-reduced and processed by the WFCAM Science
Archive System (Irwin, 2008; Hambly et al., 2008), which is also used for the
UKIDSS survey described in Lawrence et al. (2007). Near-infrared observations
are calibrated against 2MASS sources in the field which have extinction-
corrected color $0.0\leq J-K\leq 1.0$ and 2MASS signal-to-noise ratio $>10$ in
each filter. For the target stars, total uncertainties in photometry are
typically 2% down to $J$=16.5, $H$=16, and $K$=15, and errors are less than 4%
at $J$=17 (Hodgkin et al., 2009). In processing such a wide field of view, a
large number of data quality issues arise and are typically dealt with by the
pipeline by assigning photometric error flags for issues such as bad pixels,
deblending, saturation, and other effects.
Figure 2: The footprint of WFCAM consists of four detectors spaced by (94%)
of their width, covering a non-contiguous 0.21 square degrees; four pointings
are required to fill the observing field. In this figure we show how the 16
tiles in this observing field are imaged: the 4 tiles marked “A”, called
“footprint A”, are observed in a simultaneous imaging, then the telescope is
slewed south by $15\arcmin$ to observe the “B” tiles, west by $15\arcmin$ to
observe “C” tiles, and finally back north $15\arcmin$ to observe the “D”
tiles. Underlaid is a map of star counts across the field, showing clearly the
structure of the dark cloud L 1003, which causes the mean extinction in each
tile to vary.
### 2.2 Data Retrieval & Cleaning
We retrieved the processed photometry data from the WFCAM Science Archive
website via an SQL interface. All 124 nights of data were retrieved, cross-
matched, and merged together into a single catalog containing columns for
object ID, observation date, sky coordinates, $JHK$ photometry, and various
photometric processing flags. Our initial query was for data that satisfied
the criteria $J$, $H$, $K<18$; $J$, $H$, $K>9$; and photometric uncertainty
$\sigma_{(J,H,K)}<0.1$, in order to include all possibly relevant data on
young stars in this region (see §3.2 on magnitude cuts) while keeping the
downloaded catalog file at a manageable size.
Hodgkin et al. (2009) present an empirically derived correction from the
pipeline-estimated photometric error to the true, measured error:
$M^{2}=cE^{2}+s^{2}$ (1)
where $M$ is the measured total error, $E$ is the estimated photometric
uncertainty given by the pipeline, the constant of proportionality $c=1.082$,
and the systematic component $s=0.021$. We applied this update to the
estimated photometric uncertainties after retrieving the data and confirming
that night-to-night variations at the 2% level were typical even for high
signal-to-noise stars.
To assess the photometric integrity of this dataset, we calculated the mean
$JHK$ colors for each pointing on each night, averaged over all stars. We
excluded observations where the mean colors showed significant deviations. For
each night, we computed the mean $J-H$ and $H-K$ color of all reliable stars
within each detector footprint (here, “reliable” denotes stars with
photometric uncertainties less than 0.1 magnitude in each band and no
processing error flags, while avoiding bright stars). In practice, this
translated to stars between $13<J<18$, $12<H<17$, and $11.5<K<16.5$.
Typically, 25,000 stars met this criterion every night. We find that
systematic night-to-night color deviations of the ensemble on each footprint
are indeed about two percent (as expected), but a significant minority ($\sim
15\%$) of nights exhibit large offsets in color space (see Fig. 3), likely due
to non-uniform extinction from thin clouds. We applied a form of iterative
outlier clipping to select and remove anomalous nights from our analysis,
leaving only the nights whose mean colors lay within $3\sigma$ of the outlier-
clipped, time-averaged global mean color.
We also note that the mean color in each footprint is significantly different.
This is because each footprint samples regions of different visual extinction,
as seen in Fig. 2. Instrumental effects are not expected to cause this, as all
4 detectors are included in each footprint.
Out of the 124 nights in the original survey, 24 nights were rejected due to
significant deviations in the mean color, leaving 100 nights for our analysis
(Fig. 1).
Figure 3: An illustration of our procedure to reject suspicious observations.
On each night, we calculated the mean color of a large sample of stars in each
footprint. Each nightly mean color is plotted as one point for footprints A,
B, C, and D (see Fig. 2). The dashed blue ellipses enclose the nightly mean
colors for observations considered “reliable”. Nights with mean colors outside
that region were rejected.
## 3 Disk Identification and Analysis
Our scientific goal in this project is to detect young stars that possess
disks by their $K$-band excess, to briefly characterize the variability of
these disked stars, and to investigate the stability of these $K$-band
excesses with respect to time.
### 3.1 The Near-Infrared Excess
To detect optically thick disks around young stars, we use the near-infrared
excess criterion developed by Lada & Adams (1992) and the classical T Tauri
star (CTTS) locus reported by Meyer et al. (1997). We consider stars to have a
near-infrared excess consistent with an optically thick disk at $K$-band
(hereafter referred to simply as a “$K$-band excess”) if their colors fall
significantly to the right of $JHK$ space demarcated by the main sequence
reddening band and within the CTTS locus. To ensure the disk signatures are
significant, we require the sources to be 4$\sigma$ red–ward of the reddening
vector associated with the reddest, non-disk bearing stars (Lada & Adams,
1992):
$(J-H)\leq 1.714\times(H-K)$ (2)
and not more than 4 $\sigma$ below an empirically derived locus of the de-
reddened location of about 30 CTTS on the near IR color-color diagram (Meyer
et al., 1997):
$(J-H)\geq 0.58\times(H-K)+0.52$ (3)
Figure 4: A JHK color-color diagram showing the mean colors of the 9200 stars
included in our analysis plotted as black points. The meaning of regions “P”,
“D”, and “E” are explained in the text at §3.1. The thirty stars that we
identify as disked are plotted as red circles; three stars (10%) lie, on
average, in region “P”, but are considered disk-bearing due to their observed
variability that moves them into region “D” (see §3.4). The solid line is the
locus of main-sequence stars (Koornneef, 1983) and the CTTS locus (Meyer et
al., 1997). Dashed lines are parallel to the reddening vector (Rieke &
Lebofsky, 1985), and a reference reddening vector corresponding to 5
magnitudes of visual extinction ($A_{V}=5$) is shown as a solid arrow.
We show the distribution of stars within $JHK$ space in Figure 4. Underplotted
is the locus of main-sequence star intrinsic colors (Koornneef, 1983) as a
solid curved line, and the classical T Tauri star locus (Meyer et al., 1997)
as a solid straight line that terminates near (1.0, 1.0) in $JHK$ space,
corresponding to the highest accretion rates and smallest disk hole sizes
found by Meyer et al. (1997). Reddening vectors using the extinction law
presented in Rieke & Lebofsky (1985) are plotted out from the tip of the CTTS
locus and the main-sequence curve. Loosely following Itoh et al. (1996), we
partition the inhabited areas of $JHK$ space into 3 regions: “P”, “D”, and
“E”, meaning “photosphere”, “disk”, and “extreme” respectively, demarcated by
these reddening vectors.
Region “P” is inhabited by stars whose NIR emission is dominated by their
photosphere. This includes main sequence stars, giants, and some pre-main
sequence stars with a small or negligible $K$-band excess, including CTTS with
small $K$-band excess, as well as weak T Tauri stars (wTTs). Single-epoch or
time-averaged near-infrared colors cannot distinguish between main sequence
stars and YSOs that lie in this region.
Region “D” is occupied by stars whose NIR emission originates from both a
photosphere and a disk, and is consistent with simple models of an accreting,
optically thick disk at $K$-band (Meyer et al., 1997). All stars in “D” are
definite disk-bearing young stars, but disked stars can also occupy Regions
“P” or “E”, so stars in Region “D” are not a complete sample of disk-bearing
stars.
Region “E” contains stars with more excess at $K$-band than can be accounted
for by an accreting, geometrically flat disk. These stars will be hereafter
referred to as “extreme $K$-excess stars”, and are expected to be less-
evolved; their redder colors may be due to emission from a circumstellar
envelope. Class I protostars have been found to inhabit this region due to
their redder colors (Lada & Adams, 1992; Robitaille et al., 2006).
### 3.2 Study Depth
Our goal is to search for high-confidence pre-main sequence stars in Cygnus
OB7. We chose a J=17 brightness cut. This limits errors in $J$ to about 4%
with similar errors in $H$ and $K$ for typical stellar colors. This reduced
our input catalog to 9,200 stars. At the published distance of Cyg OB7 (800
pc; distance modulus $\mu=9.5$), we can estimate to what stellar mass depth
this survey reaches by using pre-main sequence isochrones calculated from
Siess et al. (2000). For these isochrones we assume a typical YSO age of
$10^{6}$ yr. The most extinguished YSO in this sample is seen through about
11.5 magnitudes of visual extinction, as estimated by tracing its $JHK$ color
back to the CTTS locus and measuring the resulting color offset in units of
$A_{V}$. Assuming a maximum extinction of $A_{V}=12$, this survey is reaches a
nominal depth of 0.3 $M_{\sun}$, and in less extinguished regions where
$A_{V}<7$ we should reach down to the hydrogen-burning limit ($\sim
0.1M_{\sun}$). However, since the deepest part of the clouds have not been
penetrated by the survey we have no knowledge of the maximum extinction, nor
any depth to which we can be assured we are complete.
### 3.3 Variability
We identify a star as “variable” if it is seen to change at a level greater
than its photometric noise. To quantitatively select stars that are variable
in this dataset, we use the Stetson variability index $S$ (Stetson, 1996;
Carpenter et al., 2001). The Stetson index is useful for multi-wavelength
simultaneous observations, as it assumes that true variability will cause
observations at different wavelengths to rise or fall in unison; its
usefulness as a criterion for variability has been established by multiple
time-series studies (e.g. Carpenter et al., 2001, 2002; Plavchan et al., 2008;
Morales-Calderón et al., 2011). The Stetson index identifies variables even
among stars whose variability is comparable to photometric noise without any
assumptions about the type of variability seen, except that true variability
should cause all channels to vary.
The Stetson index is computed by the following equation:
$S={\sum_{i=1}^{p}\textrm{sgn}\left(P_{i}\right)\sqrt{\left|P_{i}\right|}}$
(4)
where $p$ is the number of pairs of simultaneous observations of a star.
$P_{i}=\delta_{j(i)}\delta_{k(i)}$ is the product of the relative error of two
observations.The relative error is defined as:
$\delta_{i}=\sqrt{\frac{n}{n-1}}\frac{m_{i}-\bar{m}}{\sigma_{i}}$ (5)
for a given band. The size of the bias is $\sqrt{{(n-1)}/{n}}$ where $n$ is
the total number of observations contributing to the mean. The second term is
the standard error term, where $m_{i}$ is the measure magnitude, $\bar{m}$ the
mean magnitude and $\sigma_{i}$ the intrinsic error of the individual
measurement.
Formally, the Stetson index is designed to identify stars as variable when
$S>1$ if photometric uncertainties are properly estimated. After applying the
error correction described in §2.2 and calculating $S$ for all 9,200 stars, we
find the outlier-clipped mean $S$ value to be 0.2, with the outlier-clipped
distribution having a standard deviation of 0.16. Therefore, stars with $S\geq
1$ can be considered $5\sigma$ variables, and we use $S\geq 1$ as our
criterion for variability (Fig 5).
All stars brighter than $J$=17 with no photometric processing error flags were
analyzed for variability. Among these 9,200 field stars, we recover $\sim$160
that are variable according to the Stetson index $S>1$. The positions of these
160 stars are plotted in Fig. 6 as blue squares. In this paper, we focus on
the identification and variability characteristics of the disked population;
the variability characteristics seen in the non-disked stars will be discussed
in Paper II.
Figure 5: The value of the Stetson index for all 9200 stars. As a function of
H magnitude the distribution is flat with a typical value of about 0.2. The
threshold of 1 is about a 4 $\sigma$ deviation and is exceeded by $\sim$160
sources. Figure 6: The spatial distribution of disked and variable stars
detected in our analysis. Disked stars are plotted as red circles; variable
stars that lack $K$-band excess are plotted as blue squares. Most (90%) disked
stars lie within the boundaries of the dark cloud, while variables are found
uniformly in the field.
### 3.4 Transient Excesses
If a star exhibits a $K$-band excess in only a fraction of its observations,
we consider its $K$-band excess to be transient. We do not expect the
circumstellar disks of such stars to actually disappear and reappear; rather,
the disks in such systems are likely undergoing physical changes that cause
their $H-K$ colors to vary back and forth across the line demarcating
unambiguous disked stars (region “D”) from ambiguous main sequence stars
(region “P”). Such a change could feasibly be induced by, star spots (hot or
cool), impulsive heating events such as stellar flares, changes in (inner)
disk inclination, local extinction, central hole size, or a varying accretion
rate (Bouvier & Bertout, 1989; Meyer et al., 1997; Scholz, 2012).
To identify and characterize stars with transient $K$-band excess, all data
satisfying our quality filter were evaluated against Equations 2 and 3 (see
§3.1). Stars that showed a $K$-band excess according to these criteria were
tallied, producing a table of stars with $K$-band excess in at least one
observation, along with the number of times that star was observed and the
fraction of nights that the star displayed a $K$-band excess.
We find 528 stars that show a $K$-band excess on at least one night. Given our
$4\sigma$ cutoff, we expect a substantial number of single-night false
positives due to photometric noise assuming Gaussian statistics in $\sim
920,000$ individual observations. We filter most of these false positives by
removing all stars that show a near-infrared excess in fewer than 15% of
nights or those that met our measurement criteria on 25 or fewer nights. (See
Figure 7, inset.) This cut makes us insensitive to any YSOs who genuinely
possess a disk that contributes to a significant $K$-band excess in less than
15% of observations, but it filters out virtually all false positives while
allowing us to remain sensitive to stars with small and moderate, but stable,
$K$-band excesses.
These criteria identify 42 disked candidates out of the original 9,200 stars.
We individually inspected the remaining lightcurves. If a star was selected by
these criteria but (a) had no photometric variability greater than noise (i.e.
$S<1$), (b) had a $JHK$ color trajectory consistent with Gaussian noise around
a mean value, and (c) on average, lay on or to the left of the boundary
between region “P” and “D” in $JHK$ color space (see Fig. 4), then we
concluded it was not clearly a CTTS that possessed a $K$-band excess, and
removed it from our analysis. 12 stars were removed this way.
Figure 7: A histogram showing how steady the near-infrared excess was in each
of the 30 YSOs. Inset is the raw sample showing the 528 stars with a $K$-band
excess on at least one night. All stars with $K$ excess on less than 15% of
nights were rejected as false positives (dotted line). Most of the confirmed
YSOs show a consistent NIR excess on every night or nearly every night, but a
significant minority of the YSO sample (seven stars) exhibit a transient
$K$-band excess.
## 4 Results
After applying these criteria we recover 30 pre-main sequence stars, whose
properties we present in Tables 1 and 2. We designate them RWA 1–30.
### 4.1 Identification of pre-main sequence stars
Based on the method presented in §3, we report the identification of 30 young
stellar objects that possess a near-infrared excess consistent with an
optically thick disk at $K$-band (a “$K$-band excess”). Of these, 5 were
previously reported as actively accreting YSOs by Aspin et al. (2009) based on
Br$\gamma$ emission and other spectral signatures, and one was reported as a
possible but unconfirmed YSO; the remaining 24 are new discoveries.
The positions of stars identified with $K$-band excess were checked against
the IPAC database; all had a 2MASS counterpart within 0.2 arcsec, except for
RWA 28 which had no counterpart within 2″. Two stars were also found as IRAS
sources, and seven are AKARI sources. Six stars have been discussed by Aspin
et al. (2009). 2MASS photometry corroborate the presence of a $K$-band excess
at a significant level for 15 stars. In 10 more, the colors are within 2.5
$\sigma$ of the line separating regions “P” and “D”, so would be considered
ambiguous. Four stars have 2MASS colors indicating a significant _lack_ of
$K$-band excess. The YSOs CN 3S (RWA 5) and CN 7 (RWA 13) did not show NIR
excess at 2MASS epoch but were identified as $K$-band excess sources in these
observations; the classification of CN 3S was inconclusive based on its
spectrum at $1.4-2.5\mu$m, but our identification of it as a variable star
that possesses a $K$-band excess in these 2008-2009 observations, supports its
status as a YSO.
With the exception of source CN 3N, we recovered all of the YSOs identified in
Aspin et al. (2009) that our search was sensitive to – the only other stars
that we missed were either too bright or too faint for our search, or did not
show a NIR excess at the time of the 2MASS observations presented in Aspin et
al. (2009). The recovery of spectroscopically confirmed young stars in our
analysis provides a useful indication that our search is finding real YSOs.
However, this is not expected to be a complete sample of all of the young
stars in the field for three reasons. First, not all stars that possess an
accreting circumstellar disk (Class II stars) are identifiable in a $JHK$
color-color diagram, especially those seen at unfavorable inclination angles,
low accretion rates, and/or large inner disk holes (Meyer et al., 1997). Many
of these disked stars can be recovered using longer-wavelength observations
(Haisch et al., 2000; Lada et al., 2000). Second, the brightness cutoffs used
in this study to guarantee reliable photometry exclude the brighter PMS stars
and, if they exist, fainter or more substantially extincted ($A_{V}>7$) low-
mass stars. Three stars in this field (CN 3N, Cyg 19, IRAS 15N) have been
confirmed as YSOs based on spectroscopic and 2MASS observations (Aspin et al.,
2009), but are saturated in the WFCAM images; two confirmed PMS stars (the
Braid Star and IRAS 14) are likewise fainter than our cutoff. Finally, in this
analysis we removed all stars that showed any photometric error flags, such as
from deblending or bad pixels, that may in fact have useful photometry
sufficient to identify a $K$-band excess. Nonetheless, our goal in this paper
was not a complete determination of the YSO population, but rather a high-
confidence sample of $K$-excess stars whose NIR variability properties could
be reliably studied.
### 4.2 Variability of pre-main sequence stars
Of the 30 YSOs, 28 (93%) are variable at a significant level. Values of $S$
among these stars range from $S=2$ to $S=60$. Variable stars typically vary in
all 3 bands, with most stars also showing color variations. $J$-band RMS
variability in these stars ranged widely from 0.02 mag to 0.70 mag, peak-to-
trough variability ranging from near the photometric noise limit to greater
than two magnitudes at $J$ (Table 3). The median $J$-band RMS on these
variable YSOs was $\sim$0.1 mag, corresponding to a median variability index
$S\sim 10$. YSOs varied significantly on all timescales studied. Many varied
noticeably from one night to the next. However, the manner of the variations
differed among the stars with some showing slow and steady changes, while
others were more abrupt. Figure 8 shows the $K$-band lightcurves from season 2
for two “typical” stars, RWA 15 and RWA 17. In the case of RWA 15, the global
range is about 0.75 mag with night to night changes of nearly 30%. The data
also seem to have a pattern of peaks and troughs separated by about 10 days
(these will be discussed further in paper II). RWA 17 is a little chaotic in
the beginning, but in general, it shows a slow steady increase of 25% over the
course of a month, followed by a decline.
Figure 8: K band light curves for 2 sample stars RWA15 and RWA 17. Data are
from season 2. The $Stetson$ index for each star is given for season 2.
The only two $K$-excess sources which are not identified as variable, RWA 28
and 30, are both at the faintest end of our search near $J=17$ with the
largest photometric uncertainties, typically around 4% at $J=17$; hence the 2%
variability noted in some brighter stars could go undetected here. RWA 30 is
plausibly variable under its photometric noise: its $S$ value is 0.76,
$3.5\sigma$ higher than the mean $S=0.2$ value for non-variable stars, and its
observed $J$-band RMS value (while dominated by the photometric uncertainty)
is higher than four disked stars identified as variable. RWA 28, on the other
hand, shows no indications of any true variability: its variability index
$S=0.18$ is consistent with stars whose observed variations arise purely from
photometric noise. RWA 28’s photometric noise causes it to drift near the
border between “P” and “D” (in similar fashion to the 12 stars rejected from
our source list as described in §3.4). It was not excluded from our source
list because unlike the 12 excluded stars, RWA 28’s mean $JHK$ colors lie
squarely in region “D”. No other stars in our source list are suspicious in
this way. Overall, over 90% of the YSOs we identify are variable at a
significant level.
### 4.3 Extreme NIR excess.
Twenty-five stars lie in region “D” for the color-color diagram for at least
part of the observations, The remaining 5 stars, which lie in “E”, have more
excess at $K$-band than can be accounted for by accreting T-Tauri stars, like
those in Taurus which were used by Meyer et al. (1997) to derive the cTTSs
locus. We refer to these as “extreme $K$-excess stars”. Four of the five
extreme $K$-excess stars (RWA 2, 15, 19, and 26) exhibit extreme variability
as well, with variability index $S>15$. These stars may be younger, more
active counterparts to the relatively quiet classical T Tauri stars that
inhabit region “D”; the extra $K$-band excess may arise from warm, infalling
circumstellar material that is not in a disk. Indeed, three of these stars are
detected as AKARI ($9-200\mu$m) sources, and the brighter two are also IRAS
($25-100\mu$m) sources. Spectral energy distribution fits of these mid and far
IR data following Robitaille et al. (2006) support the interpretation that
they are less-evolved “Class I” protostars. Two of these stars also give
indications of being eruptive variables (Wolk et al. 2012, in prep).
### 4.4 Transient NIR excesses.
Of the 25 simple $K$-excess stars, seven vary in color space such that they
spend more than 15% of their time in region “P”, and would not be detected by
near-infrared excess criterion at these epochs (see Fig. 7). Figure 9 shows
examples of two such stars. Further, 3 of these 7 stars have mean colors that
lay in region “P”; these YSOs would be undetected in a search of time-averaged
$JHK$ color. Finally, comparison with 2MASS data show two stars that possess a
$K$-band excess in all of these UKIRT observations but show no significant
$K_{s}$-band excess at the 2MASS epoch. These nine stars (nearly 1/3 of our
sample), have been identified as exhibiting a transient $K$-band excess.
Among the variable CTTS candidates, many show $JHK$ color-color variability
parallel to either the CTTS locus (Meyer et al., 1997) or to a combination of
the CTTS locus plus extinction; the two remaining show chaotic behavior in
$JHK$ space. In Figure 10 we show the trajectories of the thirty YSOs. The
upper-left panel shows the simple disked variables which show small ($<$ 0.5
mag in color) variations which, for the most part, appear to move the star
parallel to the main sequence track or directly along the CTTS locus. The
lower-left panel shows the extreme variables, plus a few stars which move
parallel to the main sequence. The upper-right panel shows eight trajectories
which appear to be indicative of systematic changes in the disk structure.
Theoretical models of the CTTS locus (Meyer et al., 1997) derive its slope as
owing to different accretion rates, disk hole sizes, and inclination angles.
Among the extreme $K$-excess stars, color-space variability is largely
chaotic, but in two cases seems to roughly follow the same pattern of positive
color slope that seems to contain contributions from the CTTS track and from
the dust reddening track. Of course, there are more than just 3 parameters
(accretion rates, disk hole size, and inclination angle) which determine the
final location. By varying 14 parameters in their radiative transfer–based
models, Robitaille et al. (2006) calculate 200,000 model SEDs in evolutionary
stages.111 Robitaille et al. (2006) use “stages” as a theoretical equivalent
to the observational “classes” but the mapping is not exact since stages 0-III
cover Classes 0-II. The Class of an object can depend both on Stage and, for
example, viewing angle. Additional model parameters that appear susceptible to
short timescale variations include the effective stellar temperature, which
can change due to flares or spots, as well as parameters relating to the disk
structure such as the scale height of the inner disk.
While we see stars regularly cross between “P” and “D”, no stars cross between
“D” and “E”. Our sample is very small and not cleanly defined in terms of
Class. thus, the results are more open to speculation than interpretation. The
YSOs in this sample seem to separate into simple-disked Class II stars that
inhabit regions “P” and “D”, and more extreme sources that inhabit region “E”.
Models describe all but one of the stars in the “E” region as stage I
(Robitaille et al., 2006). There is a paucity of stage I models which occupy
the “D” region. This supports speculation that these are Class I sources and
we infer from Robitaille et al. (2006) that changes in the various accretion
parameters in these stars lead to changes in $J-H$ and $H-K$ color which are
mediated by an envelope which is more complex than the thin disk surrounding
Class II (Stage III) objects.
Figure 9: $JHK$ color trajectories for RWA 4 and 23, two of the nine YSOs
identified in this analysis as having a transient near-infrared excess. RWA 4
has a significant near-infrared excess in only 41% of observations, and its
time-averaged mean $JHK$ colors lie in region “P”. RWA 23 exhibits a
significant NIR excess in 59% of observations. Colored circles indicate the
progression of time from early 2008 (dark blue) to late 2009 (dark red). Solid
line: CTTS locus. Dashed line: reddening vector. The plus (+) in the bottom-
right corner illustrates the typical uncertainty on each individual $JHK$
measurement. Figure 10: The color trajectories of 30 YSOs over the course of
the observations. Each star s trajectory is plotted in a different color; some
colors are repeated. Color trajectories can be broadly divided into three
groups: small systematic variations (upper-left), large systematic variation
(upper-right), and others – including large stochastic variables and a few
stars that parallel the cool main sequence (lower-left). All are overlaid in
the lower-right. Nine YSOs drift between regions P (photosphere) and D (disk)
and could be missed by single-epoch observations. A few lie, on average, in
region P and would likely be invisible to a search of time-averaged JHK color.
The small plus (+) in the bottom-right of each panel illustrates the typical
uncertainty on each individual measurement.
## 5 Discussion
### 5.1 YSOs are variable in the near-infrared
As found in §4.2, virtually all of the detected YSOs showing a $K$-band excess
also exhibit near-infrared variability. Importantly, our search did not
include variability as a selection criterion except to disambiguate close
cases. As noted in §3.2, our study is not complete. For example, only 6 of the
12 YSOs discussed in Aspin et al. (2009) were recovered by our study. Further,
the stars in our sample were subject to both brightness and faintness cuts to
ensure sensitivity to photometric variations on the order of $\sim$2%.
Nonetheless, it is clear that near-infrared variability is a behavior common
to all disked pre-main sequence stars bright enough to be measured at $>$2%
accuracy and possess a $K$-band excess. This is consistent with previous NIR
variability studies of young stars (Carpenter et al., 2001, 2002) and also
consistent with optical studies (Herbst et al., 1994; Briceño et al., 2005).
It seems likely that NIR variability could be used alone to identify young
stars, as seen in the optical (e.g. Briceño et al., 2005; Parihar et al.,
2009). Parihar et al. (2009) noted that long term monitoring increased the
variability detection rate in the optical by about 50% for periodic variables.
We do not find the effect of extended monitoring as pronounced. Twenty-six of
the 30 stars were found to be variable via the Stetson index in the 26 night,
first observing season. For the 70+ nights of seasons two and three, the
results were 25/30 and 27/30 respectively. Even the inclusion of all three
seasons only lead to the detection of 28/30 as variables. Because it was
consistently the same stars which were detected as non-variable (RWA 6, 8, 11,
28 and 30), it appears the detection of variability on the longer datasets was
not an effect of long term periodicity, but rather the increase in signal to
noise enabled by the additional data. We suspect that using the specific
trajectories in $JHK$ color space seen in these 30 stars (§4.4) as an
additional selection criterion could be useful in detecting disked stars
within the “P” portion of the reddening band. This would also be consistent
with variability seen in Carpenter et al. (2001) where stars that lack near-
infrared excess, but that are associated with the Orion A molecular cloud, are
seen to vary at a level significantly higher than field stars. This means that
Class III stars should be detected as NIR variables (e.g. Parihar et al.,
2009; Wolk et al., $in~{}prep.$).
### 5.2 On the variability of the NIR disk diagnostic
While other studies have used time-series $JHK$ photometry to investigate
young stars with disks, the use of time-averaged NIR colors to identify disked
stars (e.g. Carpenter et al., 2002) will still miss some YSOs. In this study,
we found three stars whose time-averaged colors showed no infrared excess, but
whose variability carried them into region “D” of $JHK$ space in 20%-50% of
observations, revealing the presence of a circumstellar disk around these
stars. Therefore, simply searching through time-averaged colors is not a
sufficient YSO detection technique in time-series observations.
$JHK$ observations are known not to be sensitive to 100% of disks around young
stars. The CTTS locus is partially degenerate with reddened main sequence
colors in $JHK$ space (Meyer et al., 1997), and previous infrared studies of
young stellar populations show $L$-band (3.5 $\mu$m) observations can detect
disks around $\sim 85\%$ of young stars at age $\sim 0.3-1$ Myr, while
$JHK$-only single-epoch surveys see disks around only $\sim 50-60\%$ of the
same sample of stars (Haisch et al., 2000; Lada et al., 2000). We summed up
the probability of seeing a disk around each of the RWA stars on a given
night. The probabilities range from $\sim$ 18% through 80% with many stars
that always showed disks (100%). We then calculated an expectation value of
how many disked stars we expect to see on a single night. For our data this
came out to about 25. So on an average night we expect to see 25 of the 30 RWA
stars in the “D” region of the diagram. Multiple observations gave us 30
stars, i.e. an $\sim$20% increase. If our results are typical, then a direct
consequence of this study is that 20$\pm$ 8% more disked stars may be found by
using multiple $JHK$ observations spread out over about a month, increasing
$JHK$ disk sensitivity to roughly $60\%-70\%$. In situations where it is
significantly more practical to obtain multiple $JHK$ observations than to
acquire $L$-band imaging or to carry out a spectroscopic survey to investigate
accretion, this approach could prove a useful way to simultaneously increase
the number of identified circumstellar disks and study variability of young
stars.
### 5.3 On the underlying cause for NIR variability in YSOs
Of the 30 YSOs, 28 are variable at a significant level. As seen in the upper
portion of Figure 10, about half of these vary along linear tracks. Some YSO’s
parallel the CTTS locus of Meyer et al. (1997), others seem follow a somewhat
steeper slope. As presented in §4.4, the aggregate color-domain variability
behavior is consistent with changes in mass accretion rate, inner hole size,
and inclination angle, in some cases combined with changes in extinction or
starspot coverage. Other variability mechanisms exist. These were summarized
recently by Scholz et al. (2009). The dominant process can be indicated by the
range of magnitude and color changes exhibited by the stars (summarized in
Table 3). Among these mechanisms are rotationally modulated changes due to
cool spots or hot spots on the stellar surface, extinction changes, and
changes in the inner disk. Our goal in this section is to discuss possible
factors that may induce the observed variability, not to distinguish among
them.
Changes in the overall extinction may be the simplest to imagine. Perhaps
induced by the disk, extinction can cause unlimited changes in the apparent
flux of the stars. However, such changes should move the star in the direction
of the reddening vector. Figure 10 shows no pure examples of this. However,
there are many cases where the data appear to move predominantly in this
direction (see Figure 10 upper-right). RWA 17 and RWA 26 show some of the
clearest examples of changes in reddening (Fig. 11). However, it is clear from
the color–magnitude plots that the observed changes are not due to reddening
alone.
Cool spots, like those on the Sun, were first identified as a contributor to
the variability of PMS stars in the 1980s (Vrba et al., 1985). Even static
stellar spots induce variability because of the rotation of the star.
Starspots have been used regularly as a method of measuring stellar periods
(e.g. Attridge & Herbst, 1992). But there is a limit to the variability cool
spots can induce, since the spot is typically only 1000-1500K cooler than the
nominal photosphere. In the I band, the luminosity change is typically $<$ 15%
(Cohen, Herbst, & Williams, 2004). The implied color change due to a lower
effective temperature is $<$ 5%. All the stars in our sample exceed a color
range of 9% in $J-K$ (Table 3).
Hot spots, thought to arise from accretion, can cause a larger signal than
cool spots since the temperature difference is typically larger (a factor of 2
or 3 hotter than the surrounding photosphere). These can induce signals as
high at 1 magnitude at $J$ and color changes of 40% in $J-K$, even with a
filling factor as small as 1% (Scholz et al., 2009). However, over 1/3 of our
sample exceeds this color range, so hot spots alone cannot account for this
variability. Of the remaining 20 stars, half of them have color changes in
excess of of 25% in $J-K$, indicative of very active hot spots or a
combination of variable hot spot and other effects.
Figure 11: Season 2 color data for RWA 17 (left) and RWA 26 (right). While the
data generally track the reddening vectors, the significant width of the
tracks indicates a secondary cause of the variations. “Time” refers to days
since the first observation – April 26, 2008
The CTTS locus is derived from models of T Tauri stars with accretion rates
spanning two orders of magnitude, disk hole sizes spanning $1-10R_{\star}$,
and a full range of observable inclination angles (see Meyer et al., 1997;
Robitaille et al., 2006, esp. Fig. 3 and Fig. 18 respectively). That most of
the $JHK$ variability we see in CTTS candidates is focused along this track is
evidence that changes in the overall accretion structure – disk inclination,
hole size and accretion rate (the size of the hot spots) – are the primary
cause for $JHK$ variability in about half of the stars. This is especially
true for the subset of stars in Figure 10 upper-left which move right along
this track and those in Figure 10 upper-right which appear to follow a hybrid
of this track plus reddening.
Because of the degeneracy between the 3 parameters (disk inclination, hole
size and accretion rate), it is not possible to easily disentangle the
contributions from each of these 3 parameters. That said, it is easy to
imagine ways that they might co-vary. For example, a decreasing (or
increasing) inner disk hole size might naturally be simultaneous with an
increasing (or decreasing) accretion rate. The line-of-sight inclination of
the innermost edge of the disk – not the inclination of the entire disk –
might reasonably vary due to warping in response to a strong, misaligned
stellar magnetic field and the rotation of the star. Observational and
theoretical evidence for warped accretion disks has been provided by Bouvier
et al. (2003) and Espaillat et al. (2011). For the data presented here, we do
not attempt to model the individual stars to identify the specific mechanisms
of variability.
### 5.4 Individual variability
In addition to the aggregate color-domain variability just analyzed, we have
identified a number of striking pattens of variability in individual stars’
lightcurves. Periodic, quasi-periodic and eruptive variability is seen among
the identified YSOs, mirroring previously studied classes of variable YSOs
such as the periodic disk eclipses of AA Tau (Bouvier et al., 2003), and the
eruptive, large-scale accretion events of EX Lup and V1118 Ori (Aspin et al.,
2010; Audard et al., 2010). Many classes of variability including eclipsing
and contact binaries, and “long-period” ($P\sim$ weeks) pulsating stars are
seen among the $\sim 160$ variable field stars. In one case one of the disked
stars appears to be part of an eclipsing system. A detailed investigation of
these variable stars will appear in Paper II.
## 6 Summary
We observed a star-forming region in the dark cloud L 1003 in Cyg OB7 on more
than 100 nights spanning 1.5 years using NIR wide-band photometry. Using the
$K$-band excess diagnostic, we found 30 candidate PMS stars, including 25
disked objects (CTTS candidates) and 5 young stars with extreme $K$-band
excess (Class I candidates). Among the 25 CTTS candidates, nine (36%) cross
the main sequence reddening band cutoff, indicating that single-epoch
observations are insufficient to identify all YSOs that show $K$-band excess.
Even time-series observations may miss some stars if they only select using
time-averaged $JHK$ colors. Additionally, the pattern of variability in color
space seen in the variable CTTS candidates is a strong indication that NIR
variability in young stars arises from a combination of variable extinction
and changes in the inner accretion disk. While some of the variability may be
due to rotationally modulated starspots other possibilities include changes in
accretion rate, inner hole size, and/or disk inclination. None of the extreme
$K$-band excess stars are seen to cross into the “disked” region of the NIR
color-color space.
To summarize our results:
(1) The 30 pre-main sequence stars discussed in this paper include 24 newly
identified YSOs.
(2) Overall, $>$90% of the young stars with disks are variable. Over 80% are
variable on a time scale of about 1 month.
(3) YSOs can be separated into “simple-disk” or “extreme” classes based on
degree of $K$-band excess.
(4) 36% of “simple-disk” $K$-band excess sources have a transient $K$-band
excess.
(5) The color behavior of many of the “simple-disk” YSOs is consistent with
changes in disk geometry and/or accretion rate.
In this paper we have presented an analysis of a unique dataset: containing
multi–season NIR monitoring for variability of young stars. A follow-up paper
(Wolk et al., $in~{}prep.$) will discuss the variability of field stars, and a
phenomenological categorization of NIR variability seen in YSOs. Further
observations, at both IR and X-ray wavelengths, are planned to better
characterize the overall pre-main sequence population in this field.
## 7 Acknowledgements
Thanks to Joseph Hora and David Charbonneau for useful comments as this
research project was being developed. Thanks also to Mike Read and Nicholas
Cross for assistance with the data retrieval. Thanks to Bo Reipurth for
stimulating discussions.
S.J.W. is supported by NASA contract NAS8-03060 (Chandra). T.S.R. was
supported by Grant #1348190 from the Spitzer Science Center. Thanks also to
the NSF REU program for funding part of this research via NSF REU site grant
#0757887. This research has made use of the NASA/ IPAC Infrared Science
Archive, which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. The United Kingdom Infrared Telescope is operated by the
Joint Astronomy Centre on behalf of the Science and Technology Facilities
Council of the U.K. We thank A. Nord, L. Rizzi, and T. Carroll for assistance
in obtaining these observations. We also thank the University of Hawaii Time
Allocation Committee for allocating the nights during which these observations
were made. The authors wish to recognize and acknowledge the very significant
cultural role and reverence that the summit of Mauna Kea has always had within
the indigenous Hawaiian community. We are most fortunate to have the
opportunity to conduct observations from this sacred mountain.
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Table 1: Object Coordinates, Identification, and Median Photometry
| | | | Median WFCAM Photometric Values |
---|---|---|---|---|---
Object ID | R.A. (J2000) | Decl. (J2000) | 2MASS ID | $J$ | $H$ | $K$ | Other ID
RWA 1 | 21:00:16.570 | +52:26:23.105 | 21001656+5226230 | 12.886$\pm$0.021 | 11.796$\pm$0.021 | 10.982$\pm$0.021 | CN 2aaFrom Aspin et al. 2009.
RWA 2 | 21:02:05.456 | +52:28:54.477 | 21020547+5228544 | 16.487$\pm$0.028 | 14.155$\pm$0.022 | 11.851$\pm$0.021 | 21005+5217bbAKARI ID
| | | | | | | 2102055+522854ccIRAS ID
RWA 3 | 20:59:04.194 | +52:21:44.936 | 20590419+5221448 | 14.404$\pm$0.022 | 13.098$\pm$0.021 | 12.208$\pm$0.021 |
RWA 4 | 20:58:59.790 | +52:22:18.340 | 20585978+5222182 | 12.403$\pm$0.021 | 11.420$\pm$0.021 | 10.823$\pm$0.021 |
RWA 5 | 21:00:05.086 | +52:34:04.916 | 21000508+5234049 | 12.875$\pm$0.021 | 11.601$\pm$0.021 | 10.778$\pm$0.021 | CN 3SaaFrom Aspin et al. 2009.
RWA 6 | 21:02:43.791 | +52:23:48.278 | 21024378+5223484 | 14.865$\pm$0.022 | 13.605$\pm$0.021 | 12.741$\pm$0.021 |
RWA 7 | 21:00:02.165 | +52:35:16.205 | 21000217+5235160 | 15.361$\pm$0.023 | 13.872$\pm$0.021 | 12.646$\pm$0.021 | 2100019+523515ccIRAS ID
RWA 8 | 21:02:14.992 | +52:32:40.380 | 21021498+5232405 | 16.114$\pm$0.025 | 15.107$\pm$0.023 | 14.400$\pm$0.023 |
RWA 9 | 21:02:38.301 | +52:17:40.841 | 21023831+5217408 | 14.758$\pm$0.022 | 13.620$\pm$0.021 | 12.872$\pm$0.021 |
RWA 10 | 21:00:55.768 | +52:25:22.066 | 21005576+5225221 | 14.783$\pm$0.022 | 13.833$\pm$0.021 | 13.188$\pm$0.021 |
RWA 11 | 20:59:46.192 | +52:33:21.470 | 20594619+5233216 | 13.159$\pm$0.021 | 11.881$\pm$0.021 | 10.961$\pm$0.021 |
RWA 12 | 20:59:51.844 | +52:40:20.613 | 20595184+5240205 | 14.190$\pm$0.021 | 12.132$\pm$0.021 | 10.739$\pm$0.021 | 2059518+524020ccIRAS ID
RWA 13 | 21:00:17.257 | +52:28:25.488 | 21001725+5228253 | 12.100$\pm$0.021 | 11.199$\pm$0.021 | 10.632$\pm$0.021 | CN 7aaFrom Aspin et al. 2009.
RWA 14 | 21:02:01.476 | +52:54:35.758 | 21020145+5254357 | 15.476$\pm$0.023 | 14.632$\pm$0.022 | 14.087$\pm$0.022 |
RWA 15 | 21:00:35.175 | +52:33:24.410 | 21003517+5233244 | 15.128$\pm$0.022 | 13.883$\pm$0.021 | 12.432$\pm$0.021 | CN 1aaFrom Aspin et al. 2009., 20590+5221bbAKARI ID
| | | | | | | 2100352+523324ccIRAS ID
RWA 16 | 20:58:49.664 | +52:19:46.513 | 20584965+5219465 | 12.749$\pm$0.021 | 11.862$\pm$0.021 | 11.249$\pm$0.021 |
RWA 17 | 20:58:50.100 | +52:32:54.427 | 20585009+5232543 | 14.221$\pm$0.021 | 11.957$\pm$0.021 | 10.383$\pm$0.021 | 2058500+523255
RWA 18 | 20:59:45.016 | +52:17:49.147 | 20594500+5217492 | 12.756$\pm$0.021 | 11.832$\pm$0.021 | 11.115$\pm$0.021 |
RWA 19 | 20:59:40.710 | +52:34:13.470 | 20594071+5234135 | 14.679$\pm$0.022 | 12.460$\pm$0.021 | 10.655$\pm$0.021 | CN 6aaFrom Aspin et al. 2009.
| | | | | | | 2059408+523414ccIRAS ID
RWA 20 | 21:00:19.042 | +52:27:28.307 | 21001903+5227281 | 11.622$\pm$0.021 | 10.631$\pm$0.021 | 9.794$\pm$0.021 | CN 8aaFrom Aspin et al. 2009.
| | | | | | | 2100191+522728ccIRAS ID
RWA 21 | 21:00:36.443 | +52:12:03.049 | 21003643+5212030 | 12.748$\pm$0.021 | 11.807$\pm$0.021 | 11.164$\pm$0.021 |
RWA 22 | 20:59:08.952 | +52:22:39.320 | 20590895+5222392 | 13.197$\pm$0.021 | 12.066$\pm$0.021 | 11.244$\pm$0.021 |
RWA 23 | 20:57:43.462 | +52:51:00.401 | 20574345+5251004 | 14.892$\pm$0.022 | 14.040$\pm$0.021 | 13.476$\pm$0.021 |
RWA 24 | 20:59:19.282 | +52:25:43.857 | 20591928+5225437 | 13.657$\pm$0.021 | 12.077$\pm$0.021 | 11.008$\pm$0.021 |
RWA 25 | 21:01:02.586 | +52:27:08.640 | 21010258+5227086 | 11.859$\pm$0.021 | 11.085$\pm$0.021 | 10.610$\pm$0.021 |
RWA 26 | 21:01:02.575 | +52:24:00.053 | 21010256+5223599 | 16.913$\pm$0.036 | 15.199$\pm$0.023 | 13.707$\pm$0.022 |
RWA 27 | 21:01:16.921 | +52:28:32.587 | 21011691+5228325 | 14.704$\pm$0.022 | 13.514$\pm$0.021 | 12.832$\pm$0.021 |
RWA 28 | 20:59:51.642 | +52:41:32.479 | not detected | 16.922$\pm$0.037 | 15.837$\pm$0.028 | 15.080$\pm$0.026 |
RWA 29 | 20:59:10.396 | +52:07:44.327 | 20591038+5207442 | 16.115$\pm$0.025 | 15.124$\pm$0.023 | 14.258$\pm$0.022 |
RWA 30 | 21:01:01.073 | +52:10:42.787 | 21010106+5210427 | 17.049$\pm$0.039 | 15.275$\pm$0.024 | 13.775$\pm$0.022 |
Note. — Median photometric values were extracted from 100 $JHK$ observations
of each star.
Table 2: Variability Characteristics
| Observed RMS | Color RMS | Stetson index | P/D/E | Transient excess?
---|---|---|---|---|---
Object ID | $J$ | $H$ | $K$ | $J-H$ | $H-K$ | $S$ | (on average) |
RWA 1 | 0.473 | 0.379 | 0.278 | 0.099 | 0.103 | 42.96 | D | no
RWA 2 | 0.698 | 0.746 | 0.493 | 0.191 | 0.361 | 59.95 | E | no
RWA 3 | 0.022 | 0.022 | 0.019 | 0.021 | 0.019 | 3.78 | D | no
RWA 4 | 0.086 | 0.085 | 0.087 | 0.028 | 0.041 | 8.59 | P | yes
RWA 5 | 0.064 | 0.051 | 0.054 | 0.023 | 0.028 | 7.72 | D | yesaaTransient $K$-excess classification based on 2MASS data
RWA 6 | 0.020 | 0.023 | 0.031 | 0.013 | 0.013 | 2.23 | D | yesaaTransient $K$-excess classification based on 2MASS data
RWA 7 | 0.241 | 0.276 | 0.295 | 0.059 | 0.068 | 30.81 | D | no
RWA 8 | 0.101 | 0.056 | 0.067 | 0.050 | 0.068 | 2.77 | D | yes
RWA 9 | 0.066 | 0.078 | 0.094 | 0.017 | 0.021 | 8.65 | D | no
RWA 10 | 0.028 | 0.032 | 0.043 | 0.016 | 0.021 | 3.33 | D | no
RWA 11 | 0.029 | 0.027 | 0.041 | 0.014 | 0.016 | 3.40 | D | no
RWA 12 | 0.154 | 0.133 | 0.134 | 0.037 | 0.042 | 14.45 | D | no
RWA 13 | 0.093 | 0.087 | 0.091 | 0.030 | 0.042 | 7.95 | D | yes
RWA 14 | 0.265 | 0.201 | 0.184 | 0.050 | 0.072 | 25.05 | P | yes
RWA 15 | 0.248 | 0.199 | 0.184 | 0.104 | 0.110 | 16.58 | E | no
RWA 16 | 0.127 | 0.112 | 0.091 | 0.028 | 0.048 | 11.50 | D | no
RWA 17 | 0.287 | 0.194 | 0.137 | 0.098 | 0.070 | 24.21 | D | no
RWA 18 | 0.135 | 0.115 | 0.112 | 0.037 | 0.053 | 11.53 | D | no
RWA 19 | 0.135 | 0.182 | 0.145 | 0.067 | 0.057 | 16.15 | E | no
RWA 20 | 0.128 | 0.126 | 0.144 | 0.030 | 0.032 | 10.80 | D | no
RWA 21 | 0.245 | 0.190 | 0.136 | 0.062 | 0.068 | 19.91 | D | no
RWA 22 | 0.062 | 0.067 | 0.093 | 0.019 | 0.034 | 7.59 | D | no
RWA 23 | 0.054 | 0.056 | 0.073 | 0.022 | 0.043 | 6.74 | D | yes
RWA 24 | 0.052 | 0.048 | 0.061 | 0.014 | 0.023 | 5.51 | D | no
RWA 25 | 0.075 | 0.055 | 0.067 | 0.032 | 0.042 | 6.50 | D | yes
RWA 26 | 0.270 | 0.244 | 0.205 | 0.097 | 0.078 | 15.87 | E | no
RWA 27 | 0.247 | 0.181 | 0.132 | 0.070 | 0.058 | 34.58 | P | yes
RWA 28 | 0.035 | 0.019 | 0.015 | 0.036 | 0.025 | 0.18 | D | no
RWA 29 | 0.037 | 0.042 | 0.052 | 0.022 | 0.020 | 3.61 | D | no
RWA 30 | 0.033 | 0.020 | 0.022 | 0.032 | 0.019 | 0.76 | E | no
Note. — Typical photometric errors are $\sim 2\%$ Refer to Table 1 for more
details.
Table 3: Variability extrema Object ID | Median $K$ | $\Delta J$ | $\Delta K$ | $\Delta J-H$ | $\Delta H-K$ | $\Delta J-K$
---|---|---|---|---|---|---
RWA 1 | 10.98 | 1.85 | 1.13 | 0.45 | 0.50 | 0.93
RWA 2 | 11.90 | 2.74 | 1.78 | 1.19 | 1.36 | 1.64
RWA 3 | 12.21 | 0.35 | 0.30 | 0.13 | 0.13 | 0.12
RWA 4 | 10.82 | 1.23 | 0.55 | 1.10 | 0.51 | 0.84
RWA 5 | 10.78 | 0.67 | 0.48 | 0.19 | 0.22 | 0.40
RWA 6 | 12.74 | 0.09 | 0.11 | 0.07 | 0.06 | 0.09
RWA 7 | 12.69 | 0.81 | 0.93 | 0.20 | 0.25 | 0.39
RWA 8 | 14.40 | 0.43 | 0.32 | 0.47 | 0.44 | 0.52
RWA 9 | 12.87 | 0.26 | 0.36 | 0.10 | 0.10 | 0.14
RWA 10 | 13.19 | 0.15 | 0.31 | 0.09 | 0.11 | 0.19
RWA 11 | 10.96 | 0.14 | 0.18 | 0.07 | 0.12 | 0.12
RWA 12 | 10.74 | 0.73 | 0.57 | 0.28 | 0.24 | 0.35
RWA 13 | 10.63 | 0.58 | 0.43 | 0.20 | 0.25 | 0.44
RWA 14 | 14.07 | 1.49 | 0.73 | 0.42 | 0.70 | 0.99
RWA 15 | 12.43 | 1.27 | 0.89 | 0.47 | 0.59 | 1.02
RWA 16 | 11.25 | 0.61 | 0.41 | 0.18 | 0.22 | 0.39
RWA 17 | 10.38 | 0.94 | 0.51 | 0.37 | 0.25 | 0.59
RWA 18 | 11.12 | 0.66 | 0.61 | 0.18 | 0.22 | 0.40
RWA 19 | 10.61 | 0.71 | 0.60 | 0.26 | 0.40 | 0.40
RWA 20 | 9.76 | 0.87 | 0.89 | 0.17 | 0.15 | 0.20
RWA 21 | 11.17 | 1.20 | 0.67 | 0.28 | 0.32 | 0.60
RWA 22 | 11.24 | 0.73 | 0.45 | 0.51 | 0.14 | 0.52
RWA 23 | 13.48 | 0.48 | 0.35 | 0.12 | 0.25 | 0.34
RWA 24 | 11.01 | 0.28 | 0.29 | 0.07 | 0.12 | 0.12
RWA 25 | 10.61 | 0.49 | 0.32 | 0.22 | 0.22 | 0.31
RWA 26 | 13.71 | 1.33 | 1.05 | 0.49 | 0.35 | 0.71
RWA 27 | 12.83 | 1.99 | 1.37 | 0.33 | 0.50 | 0.79
RWA 28 | 15.08 | 0.17 | 0.10 | 0.22 | 0.13 | 0.19
RWA 28 | 14.24 | 0.17 | 0.23 | 0.16 | 0.10 | 0.25
RWA 30 | 13.78 | 0.22 | 0.11 | 0.22 | 0.11 | 0.20
Median | | 0.66 | 0.46 | 0.22 | 0.23 | 0.40
Maximum | | 2.74 | 1.78 | 1.19 | 1.36 | 1.64
Minimun | | 0.09 | 0.10 | 0.07 | 0.06 | 0.09
|
arxiv-papers
| 2012-06-04T20:36:34 |
2024-09-04T02:49:31.512234
|
{
"license": "Public Domain",
"authors": "Thomas S. Rice (1), Scott J. Wolk (1) and Colin Aspin (2) ((1)\n Harvard-Smithsonian Center for Astrophysics, Cambridge, MA (2) Institute for\n Astronomy, University of Hawai'i, Hilo, HI)",
"submitter": "Scott J. Wolk",
"url": "https://arxiv.org/abs/1206.0759"
}
|
1206.0762
|
# Demonstration of images with negative group velocities
Ryan T. Glasser∗, Ulrich Vogl, and Paul D. Lett
###### Abstract
We report the experimental demonstration of the superluminal propagation of
multi-spatial-mode images via four-wave mixing in hot atomic vapor, in which
all spatial sub-regions propagate with negative group velocities. We
investigate the spatial mode properties and temporal reshaping of the fast
light images, and show large relative pulse peak advancements of up to 64 $\%$
of the input pulse width. The degree of temporal reshaping is quantified and
increases as the relative pulse peak advancement increases. When optimized for
image quality or pulse advancement, negative group velocities of up to
$v_{g}=-\frac{c}{880}$ and $v_{g}=-\frac{c}{2180}$, respectively, are
demonstrated when integrating temporally over the entire image. The present
results are applicable to temporal cloaking devices that require strong
manipulation of the dispersion relation, where one can envision temporally
cloaking various spatial regions of an image for different durations.
Additionally, the modes involved in a four-wave mixing process similar to the
present experiment have been shown to exhibit quantum correlations and
entanglement. The results presented here provide insight into how to tailor
experimental tests of the behavior of these quantum correlations and
entanglement in the superluminal regime.
Quantum Measurement Division, National Institute of Standards and Technology,
and
Joint Quantum Institute, NIST and the University of Maryland, Gaithersburg, MD
20899 USA
∗rglasser@nist.gov
(190.0190,190.4380,190.4350,190.5530,350.5500)
## References
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## 1 Introduction
Optical pulse propagation with group velocities larger than the speed of light
in vacuum, $c$, or negative, have been demonstrated theoretically and
experimentally in a variety of systems [1, 2, 3, 4, 5, 6]. The anomalous
dispersion required for generating “fast” light occurs near the center of
absorption lines and on the wings of gain lines [7, 8, 9, 10, 11, 12, 13]. It
has recently been shown that the seed and conjugate pulses involved in the
four-wave mixing (4WM) process in hot rubidium vapor may exhibit large
negative group velocities [14].
The group velocity $v_{g}$ is connected to the slope of the frequency
dependent index of refraction $n(\nu)$ by:
$\displaystyle
v_{g}=\frac{c}{n_{g}}=\frac{c}{n(\nu)+n(\nu)\frac{dn(\nu)}{d\nu}},$ (1)
where $n_{g}$ is the group index and $\nu$ is the optical frequency.
Operationally, the group velocity may be identified with the propagation speed
of the peak of an optical pulse [15]. Pulses propagating through a dispersive
medium experience a relative delay of $\Delta T=\frac{L}{v_{g}}-\frac{L}{c}$,
where $L$ is the length of the medium [13]. When the group velocity is
negative, $\Delta T$ is also negative, corresponding to a relative advancement
of the pulse after traveling through the medium. It is well established (cf.
[4, 5, 6]), that while dispersive media can alter the group velocity of a
pulse, no superluminal transfer of information can be achieved. In the present
experiment we demonstrate the advancement of two-dimensional images carried by
optical pulses that propagate through a region of anomalous dispersion, which
is the complementary fast light analogue of the slow light image experiments
performed by Camacho, et al. [16]. Using the inherently multi-spatial-mode 4WM
process, we investigate the spatial variations of the group velocity and
relative advancement of an image propagating with a negative group velocity. A
relative pulse peak advancement of 64 $\%$ of the input pulse full-width at
half-maximum (FWHM) is shown. Additionally, we analyze the degree of temporal
pulse reshaping as the relative pulse peak advancement is varied.
The ability to impart images on an optical pulse propagating through a region
of anomalous dispersion has a number of interesting applications. Similar 4WM
double-lambda schemes in rubidium have been shown to exhibit multi-spatial-
mode entanglement [17]. By sending one of the two entangled images through the
present fast light medium, the spatial properties of the multi-spatial-mode
entanglement propagating through anomalously dispersive media may be
investigated. Additionally, by spatially controlling the anomalous dispersion
and relative pulse advancement, one can envision combining the present fast
light system with a similar slow light system to develop a multi-spatial-mode
temporal cloak analogous to the single-spatial-mode system in [18].
Figure 1: Schematic showing (1a) the double-lambda level scheme, (1b) the
typical probe gain lineshape with probe detuning indicated, and (1c) the
experimental setup. The pump beam is detuned $\approx$ 400 MHz to the blue of
the 85Rb D1 line, with the probe blue-detuned $\approx$ 3.0 GHz relative to
the pump. The probe’s center frequency is set to be on the blue wing of the
probe gain line. After the 4WM interaction, the probe pulses are imaged onto a
gated, intensified CCD camera. The pulses are able to be time-resolved down to
2.44 ns temporal bins.
## 2 Experimental setup
The 4WM process used here is pumped with a linearly-polarized continuous-wave
laser detuned $\Delta\,\approx$ 400 MHz to the blue of the Rb D1 line at
$\lambda\,\approx\,795$ nm,
$\left|5S_{1/2},F=2\right\rangle\rightarrow\left|5P_{1/2}\right\rangle$, as
shown in Fig. 1(a). The pump beam is produced by spatially filtering the
output of a semiconductor tapered amplifier that is seeded with a tunable,
single-mode diode laser. The probe beam is generated by double-passing a 1.5
GHz acousto-optic modulator (AOM) with light split off from the output of the
same diode laser that produces the pump. The probe beam is then pulsed by
sending 200 ns electronic square pulses into a second AOM with a $1/e$ rise
time of $\approx\,190$ ns, resulting in nearly Gaussian FWHM 200 ns optical
pulses. The probe and pump beams are orthogonally polarized and combined on a
polarizing beam splitter, at an angle of $\approx$ 1∘. A conjugate pulse is
created via the interaction and propagates at an angle satisfying the phase-
matching condition. While previous experiments have shown that this conjugate
pulse may also propagate superluminally [14], we focus here only on the probe
mode and characterize the influence of the anomalous dispersion medium on the
transverse modes of the pulse. The probe is tuned such that its center
frequency is on the blue wing of the gain line peak as indicated in Fig. 1(b),
in order to produce maximal relative pulse peak advancement.
Figure 2: Negative group velocity of a carrier pulse resulting in an advanced
pulse peak with a spatially multi-mode image, in this case the letter “c”. The
green curve is the detected probe pulse, integrated over the image, when the
pump is not present. The red curve is the detected superluminal amplified
probe pulse, integrated over the image, when the pump is turned on. A relative
advancement of $\approx$ 50 ns, corresponding to a group velocity of
$v_{g}=-\frac{c}{880}$, is shown. The probe pulse in this measurement was
shaped with the letter c , and the arrival time was monitored with a gating
width of 3 ns. The snapshots across the top of the graph show the cross
section of the beam at equidistant times between 120 ns and 480 ns (top row:
reference, lower row: superluminal pulse). The two insets show the full time-
integrated images. The advanced image (left inset) shows distortion due to
inhomogeneous gain, Kerr-lensing and leaked pump light, but the principal
shape clearly persists. The superluminal pulse group velocity can be
determined pixel-wise for the image, as well as integrated over the whole
image. The peak gain of the unnormalized superluminal pulse is $\approx$ 2.1.
The $220$ mW pump beam is focused into the cell, resulting in an elliptical
focal spot size of $\approx\,750\,\mu$m x $950\,\mu$m at the center of the
cell. The input probe beam is lightly focused into the cell with a nearly
Gaussian FWHM spot size of $\approx\,600\,\mu$m, and a peak power of 10 $\mu$W
maximum. The 85Rb cell is 1.7 cm long and kept at a constant temperature of
$115\,^{\circ}$C, corresponding to a number density of $\approx 1.2\times
10^{13}\,$cm-3. After the 4WM interaction, the probe pulses are detected with
a gated, intensified CCD camera. The camera allows us to examine $\approx$ 2.5
ns time slices through the temporal envelope of the incident pulses, on each
pixel. Reference pulses are taken with the pump beam blocked, and correspond
to pulses propagating at the speed of light in vacuum $c$, to within our
experimental uncertainty (that is, the arrival time of reference pulses
propagating along the optical path through the cell with no pump beam present
and those propagating along the optical path when the cell is removed show no
arrival time difference to within our experimental uncertainty).
Figure 3: Gain (3a) and relative pulse peak advancement (3b) of the
superluminal probe pulses for the input probe with a Gaussian spot. Each
superpixel corresponds to a 12$\times$12 binning of pixels on an intensified
CCD camera. The pump beam is slightly elliptical, with a waist of $\approx
750\,\mu$m$\times 950\,\mu$m. Relative pulse peak advancement is seen to
increase from $\approx$ 40 ns to $\approx$ 100 ns from the right-hand side to
the left-hand side of the probe spot. The gain also varies spatially, with the
highest gain regions corresponding to the lowest relative pulse peak
advancements. The ellipses correspond to the $1/e^{2}$ intensity of the
detected amplified probe spots. Uncertainties in the relative pulse peak
advancement are largest toward the edges of the image ($\approx$ 10 ns), due
to statistical uncertainties from a decreased signal-to-noise ratio resulting
from the lower intensities. The uncertainty in the relative advancement near
the inner region of the image is $\approx$ 3 ns, resulting from the minimum
detector gating time.
## 3 Results
Due to the inherently multi-spatial-mode nature of the 4WM without the
presence of a cavity, the probe pulse is able to carry an image throughout the
process. In Fig. 2, we show an image of a “c” on the probe pulse. The
amplified probe pulse exhibits a relative pulse peak advancement of $\approx$
50 ns on a 200 ns optical pulse, when integrated over the entire image,
corresponding to a group velocity of $v_{g}=-\frac{c}{880}$. Smaller
subsections of the image all propagate with negative group velocities, with
advancements varying from $\approx$ 10 ns to $>$ 50 ns. The superluminal image
is distorted, but is clearly visible. The current experimental limitations
that result in degraded image quality are primarily the available pump and
probe powers. With sufficient powers, it should be possible to enlarge the
pump and probe beams such that they are collimated and nearly uniform across
the interaction region, resulting in equal advancements across the image. The
relative pulse peak advancements are found by taking the peak of the output
pulse and comparing its arrival time to the arrival time of the peak of a
pulse propagating at $c$. The uncertainty of the pulse peak advancement in the
outer regions of the images is due primarily to the difficulty of determining
the pulse peak due to low optical power in these regions, and is roughly 10
ns. The uncertainty of the pulse peak advancement near the inner regions of
the images where higher intensities are present is limited by the minimum
detector gating time, and is $\approx\,3$ ns.
In order to better analyze the spatial properties of superluminal images, we
use a nearly Gaussian spatial image. The probe beam’s center frequency is set
at the point of optimal advancement of the integrated spatial profile of the
pulses. If the probe and pump beams were perfectly collimated with a uniform
intensity, all spatial regions of the probe are expected to be advanced
uniformly. In order to create a spatially varying group velocity profile, one
can take advantage of angular dispersion [19], which allows for the
translation of a phase-mismatch of the probe and pump beams into a spatially
varying group-index [20]. To accomplish this we adjust the probe focus so that
the beam waist crosses the pump beam slightly off-center. Due to the strong
dispersion of the gain line, as outlined in [20], the phase-matching condition
is fulfilled for a spread of angles determined by $\sqrt{\lambda/L}$ (where
$\lambda$ is the wavelength and $L$ is the length of the medium),
corresponding to 7 mrad ($\approx$ 0.4∘) in our case. The k-vectors of the
pump and probe beams now intersect with a varying angle across the interaction
volume in the cell, so that the phase-matching changes across the beam. As
seen in Fig. 3(a), this results in a spatial gradient of the gain.
Additionally, this results in the spatially-varying group index profile shown
in Fig. 3(b). In the data shown the k-vectors follow a nearly linear gradient
in the horizontal direction and the observed change in group advancement
$\Delta T(x)$ follows the relation $\Delta T(x)\sim x\frac{\partial}{\partial
x}(\frac{\partial k(x,\omega)}{\partial\omega})$, where $x$ denotes the
horizontal transverse beam direction [19]. Data is shown as a function of
superpixel position, which are pixels binned into 12$\times$12 groups. This
provides more light on each superpixel and increases the signal-to-noise (the
ellipses in the figures show the approximate $1/e^{2}$ intensity of the
amplified probe spots). Spatially-varying gain averaged over individual
superpixels ranges from $\approx$ 2 to $\approx$ 12\. Relative pulse peak
advancements range from $\approx$ 40 ns to $\approx$ 95 ns. This demonstrates
another degree of freedom that can be used to control the relative pulse peak
advancement. In principle, one can engineer how the probe is focused into the
cell to produce different spatially-varying pulse advancements across the
image.
Having analyzed some of the spatial properties of the superluminal pulses, we
now turn to the temporal profile. The amount of temporal reshaping that a
superluminal pulse experiences in general gets larger with increasing
advancement. To characterize this distortion, we employ a metric similar to
that used in [21], as it is a convenient measure to use in an experimental
setting without the need for employing phase-sensitive measurements. The
degree of distortion is defined as:
$D=\sqrt{\int^{\infty}_{-\infty}\left|\frac{|E^{\prime}(z+L,t)|^{2}}{\int|E^{\prime}(z+L,t)|^{2}dt}-\frac{|E(z,t-\Delta
T)|^{2}}{\int|E(z,t-\Delta T)|^{2}dt}\right|dt}.$ (2)
Here $E^{\prime}$ and $E$ are the output and reference pulse envelopes,
respectively.
To quantify the temporal pulse reshaping, we use Eq. (2) to analyze the
measured pulses after normalization. We choose Eq. (2) to quantify reshaping
that is due only to propagation through the region of anomalous dispersion as
it is zero for identically shaped pulses, even if gain is present, and is
nonzero for any reshaping. This is similar to other metrics for measuring
distortion, although others may purposely not normalize the advanced and
reference pulses, and thus are nonzero when only gain is present [4]. In order
to see the effect of relative advancement on the pulse distortion, we vary the
relative pulse peak advancement from $\approx$ 5 ns to $\approx$ 75 ns by
changing the input probe power as in [14]. The degree of pulse reshaping
increases over this range of advancements from D $\approx$ 0.45 to D $\approx$
0.6. The principle reshaping can best be described as a narrowing of the
pulse, with a steeper rising edge than falling edge, albeit with some ringing
on the falling edge at the largest advancements. Reshaping results from
contributions of the higher order terms in the expansion of $n(\nu)$, such as
group velocity dispersion [15]. The rising edges are advanced by a smaller
amount than the falling edges, consistent with the analysis performed in [22].
Additionally, the pulse temporal advancement varies spatially across the probe
spot as shown in Fig. 3(b). The fundamental temporal reshaping of the pulse,
however, is relatively uniform across the spot. For the 2$\times$2 fastest
binned spot, the rising and falling edges at FWHM of the 200 ns pulses are
advanced by $\approx\,38$ ns and $\approx\,156$ ns, respectively, and displays
a pulse peak advancement of $\approx$ 100 ns. Integrating over the entire
probe spot, the rising and falling edges at FWHM are advanced by $\approx\,24$
ns and $\approx\,124$ ns, respectively, and an integrated pulse peak
advancement of $\approx$ 80 ns is measured. This relative advancement is
rather large considering the amount of peak gain and absorption theoretically
required to achieve comparable advancements in a similar but somewhat
constrained system, where a constant background gain of e64 and an absorption
at the line-center of e-32 is required to achieve a relative advancement of
$2\sqrt{2}$ times the input pulse width[23].
Figure 4: Plot of the advanced pulse versus time when the system is optimized
for maximum advancement rather than image quality. The red and green curves
are the advanced pulse and reference pulse intensities integrated over the
entire Gaussian spot. A pulse peak advancement of 124 ns is shown,
corresponding to a relative pulse peak advancement of 64 $\%$ compared to the
input pulse FWHM, and a group velocity of $v_{g}=-\frac{c}{2180}$. The gain in
this case is $\approx$ 5, and the relative degree of reshaping is D $\approx$
0.8. Ringing on the trailing edge of the advanced pulse is seen, as expected
for very large relative advancements.
According to [22], the rising and falling edges of a temporally Gaussian pulse
will be advanced, respectively, by:
$\displaystyle T_{adv,\uparrow}=T_{adv}-\tau_{a}+\frac{\tau_{a}}{\beta}$ (3)
$\displaystyle T_{adv,\downarrow}=T_{adv}-\tau_{a}-\frac{\tau_{a}}{\beta}.$
(4)
Here $T_{adv,\uparrow}$, $T_{adv,\downarrow}$ and $T_{adv}$ are the
advancement of the leading edge, the advancement of the falling edge and the
center-of-gravity advancement of the pulse (which we take to be the pulse peak
for consistency with the analysis above), $\tau_{a}$ is the half-width of the
pulse at the point where the values are calculated and $\beta$ is the factor
by which the intensity profile of the pulse is narrowed. Analyzing these
values at the FWHM of the pulses, we find for the total Gaussian spot for the
rising and falling edges, $\beta_{\uparrow}=1.67$ and
$\beta_{\downarrow}=2.04$, respectively. We also examine the region of the
Gaussian spot that exhibits the largest advancement, the 2$\times$2 area of
superpixels (1,5) to (2,6) in Fig. 3. Similar analysis for the 2$\times$2
binned spot mentioned above results in $\beta_{\uparrow}=2.3$ and
$\beta_{\downarrow}=2.68$. As expected, the 2$\times$2 binned spot exhibits
more significant reshaping accompanying the larger relative pulse advancement.
In all cases, the falling edge is advanced more than the rising edge, as
expected [22].
We are able to increase the advancement, at the cost of increased pulse and
image distortion, by decreasing the size of the probe and pump spots in the
cell. Figure 4 shows such a case, where a relative pulse peak advancement of
$>$60$\,\%$, compared to the input pulse FWHM, is obtained using a gain of
$\approx 5$. This corresponds to a group velocity of $v_{g}=-\frac{c}{2180}$.
The degree of pulse reshaping is D $\approx$ 0.84, and ringing after the main
peak is seen, as theoretically predicted [22]. The pulse is again narrowed,
with rising and falling edge advancements of $\approx\,60$ ns and
$\approx\,190$ ns, respectively, and a pulse peak advancement of $\approx$ 124
ns on a 200 ns pulse. This is, to our knowledge, the largest relative pulse
peak advancement of an optical pulse demonstrated experimentally. While the
advancement demonstrated here is significant compared to the best previous
relative pulse advancement (to our knowledge) of 42 $\%$ [3, 6], in a
molecular absorption system, it exhibits larger distortion and less
advancement when compared to a system with an optimized transfer function [4].
As shown in [4], a fast light system with an optimal transfer function can
allow for 100 $\%$ relative pulse advancement with a peak gain of 84, and a
distortion of 15 $\%$ using a similar definition to Eq. (2). Limitations to
the relative pulse peak advancement in the present scheme are primarily due to
the pump and probe powers. Additionally, limitations resulting in image
distortion through the region of anomalous dispersion are due primarily to
phase-matching constraints, over which there is a finite allowable bandwidth
where appreciable 4WM gain can take place. The group velocity dispersion in
the present system is determined by the lineshape resulting from the 4WM
process. The bandwidth of anomalous dispersion resulting from the 4WM gain
line limits the usable probe pulse widths to being larger than roughly 75 ns.
Finally, noise added due to the phase-insensitive nature of the process should
also be considered in future experiments if one is interested in investigating
the behavior of quantum correlations in such a system.
## 4 Conclusion
We have experimentally demonstrated images propagating with negative group
velocities. We have investigated the spatial variation of the relative pulse
peak advancement and gain on pulses that have negative group velocities due to
anomalous dispersion resulting from the 4WM process. The entirety of a nearly
Gaussian spatial spot exhibits large negative group velocities in all spatial
subregions. Three knobs may be tuned to vary the amount of relative pulse peak
advancement, the pump power, the input probe power, and the k-vector of the
probe relative to the pump. Additionally, we have analyzed the degree of
temporal reshaping that the pulses exhibit after having traversed the fast
light medium. Finally, we show relative pulse peak advancements of $>$60$\,\%$
relative to the 200 ns FWHM input pulses, corresponding to a group velocity of
$v_{g}=-\frac{c}{2180}$. These results should prove to be beneficial when
trying to engineer fast light systems to exhibit specific desired properties,
such as the amount of gain, advancement and reshaping.
The flexibility of the present system should allow for investigations into the
effects of negative group velocities on quantum correlations and squeezing, as
well as implementations of a temporal cloak over multiple spatial modes. The
ability to vary the group velocity of optical pulses spatially is a step
toward the demonstration of a spatially-varying temporal cloak. One
implementation of temporal cloaking utilizes a “split time-lens,” in which a
pulse is effectively split in time to create a temporal gap where the original
pulse resided [18]. The split pulse is then closed, and whatever event
occurred in the time gap is hidden. The ability to manipulate the group
velocity to advance pulses by different amounts in different spatial regions
as demonstrated here could allow the temporal cloaking of different regions of
spatially multimode pulses by different durations, which is not possible when
using single-mode fibers as in [18]. Additionally, our results are applicable
to the investigation of the effects of superluminal propagation on bipartite
entangled states. A nearly identical 4WM setup to the one used here has shown
that the probe and conjugate modes can exhibit quantum correlations and
entanglement [24, 25]. In analogy to the experiment in [26], it should be
possible with the present setup to explore the behavior of quantum
correlations under conditions when superluminal propagation occurs. By taking
advantage of the spatially varying group index, one can measure the cross-
correlation as a function of advancement, pixel-by-pixel.
## Acknowledgments
This work was supported by the Air Force Office of Scientific Research. This
research was performed while Ryan Glasser held a National Research Council
Research Associateship Award at NIST. Ulrich Vogl would like to thank the
Alexander von Humboldt Foundation.
|
arxiv-papers
| 2012-06-04T20:54:55 |
2024-09-04T02:49:31.524033
|
{
"license": "Public Domain",
"authors": "Ryan T. Glasser and Ulrich Vogl and Paul D. Lett",
"submitter": "Ryan Glasser",
"url": "https://arxiv.org/abs/1206.0762"
}
|
1206.0981
|
# An Informed Model of Personal Information Release in Social Networking
Sites
Anna Squicciarini College of Information Science and Technology
Penn State University
University Park, PA 16802
E-mail: asquicciarini@psu.edu Christopher Griffin Applied Research Laboratory
Penn State University
University Park, PA 16802
E-mail: griffinch@ieee.org
###### Abstract
The emergence of online social networks and the growing popularity of digital
communication has resulted in an increasingly amount of information about
individuals available on the Internet. Social network users are given the
freedom to create complex digital identities, and enrich them with truthful or
even fake personal information. However, this freedom has led to serious
security and privacy incidents, due to the role users’ identities play in
establishing social and privacy settings.
In this paper, we take a step toward a better understanding of online
information exposure. Based on the detailed analysis of a sample of real-world
data, we develop a deception model for online users. The model uses a game
theoretic approach to characterizing a user’s willingness to release, withhold
or lie about information depending on the behavior of individuals within the
user’s circle of friends. In the model, we take into account both the
heterogeneous nature of users and their different attitudes, as well as the
different types of information they may expose online.
## I Introduction
Online social networks (OSNs) such as Facebook, Myspace, and Google+ allow
individuals to present themselves and establish or maintain connections with
others. Users articulate their social networks by creating and managing
content, social connections, and a possibly large amount of personal
information. A typical OSN in fact allows users to create connections to
friends , thereby sharing with them a wide variety of personal information.
These connections are often based on the alleged identities and properties of
the individuals populating the OSN. Users of social media sites can, however,
generate accounts containing unverified information. On the one hand, this
allows the users to avoid identification and surveillance or observation of
any kind. On the other one, the ability to generate unverified accounts on
most of these sites, renders social relationships potentially weak, if based
on fake identities. Further, unverified accounts may and are often used by
malicious users to carry out disruptive activities hidden behind fake
identities [10]. To date, while some work has studied the incentives behind
information disclosures in OSNs [9, 26, 18], little is known about identities
misrepresentations.
In this paper, we speculate that information revelation in OSN is a complex
process where multiple contrasting influences are in play: not only privacy
attitudes, but also social pressure and personal attitudes are at stake.
Focusing on three types of users’ behavior related to information revelation:
truthful information sharing, information withholding and deception, we study
the effect of misrepresentation in these environments by means of a game
theoretical model.
To ground our model, we conducted an extensive empirical study, collecting
data about users’ common behavior and their attitude toward personal
information disclosure. The study involved almost 300 subjects, all active
social network users.
Our study reveals important insights on users’ attitudes and practices. In
particular, our results show that users’ decisions to lie or withhold
information are not strongly influenced by privacy concerns. Rather, results
show strong correlations between peer-pressure and attitudes toward lying.
Users who feel peer-pressured are less likely to withhold their information,
especially their whereabouts. The quest for gaining or maintaining popularity
also seems to play an important role, in particular with respect to the amount
of information users choose to reveal. Also, we identified that users’
identity information is managed differently depending upon the perceived
sensitivity of the information. For example, for basic demographic
information, users tend not to lie in the main social network account, as this
is typically revealed in the course of social interactions and may be easy to
verify by social network peers. On the other hand, information that is closer
to the users’ personal sphere, for example, social relationships, whereabouts,
etc. is revealed mostly by users who are in search or popularity and/or are
searching for self-affirmation in the network. In addition, we find that
misrepresentation interacts with measures of morality, suggesting that users
do not associate lies in social networks with unethical behavior, and that,
where lying is considered unethical, they are more likely to withhold
information, as a form of boundary control. Finally, we found that users’
behavior is mostly influenced by inner circles of close online friends,
regardless of the actual number of social connections users have.
The analysis of the responses is used as input to inform our qualitative model
of user information sharing, withholding and deception. In particular,
building on the finding that users treat information differently, the model
presupposes that individuals release, withhold or lie about certain classes of
information differently, and that each user behaves according to an individual
payoff function. The payoff function is constructed to take into account the
identified influential decision factors: morality, peer pressure, privacy and
popularity. The output of the function is also affected by the behavior of a
circle of close friends - as we found strong evidence of self-validation and
peer influence in our study. We provide an example model using evolutionary
dynamics, which we posit influences a users’ behavior as he interacts with his
OSN overtime and more accurately understands the true nature of his (personal)
objective function.
The paper is organized as follows. Next section reviews relevant literature.
Section III discusses our empirical study. Section IV presents our model. We
illustrate the various types of users and information in our examples in
Section V. We conclude the paper in Section VI with pointers to future
research directions.
## II Literature review
Digital identity constitutes one of the building blocks of Web 2.0
technologies, ranging from social networking to e-commerce. Problems related
to digital identity management and protection have been tackled by both the
computer science community and by information scientists. From the
computational standpoint, a variety of digital identity and trust management
mechanisms have been developed to allow users to create and maintain complex
digital personas [3, 9] although there has been little work on the topic of
digital identity validation and trust in the context of social computing.
From a social science perspective, various studies have explored identity
sharing behavior in social network sites and the risk of over exposure
(notable examples are [9, 26, 2, 18]). Research studies have shown that users
in online environments rely on a variety of cues to make determinations about
one another; however, all these cues are not deemed equally credible. For
instance, Goffman [12] notes that identity cues can be intentionally given or
unintentionally revealed, and that humans are more likely to place greater
weight on those cues that are perceived to be unintentional as opposed to
strategically constructed. This ability to engage in deceptive self-
presentation online is compounded when users do not share a social network and
therefore have less access to information triangles such as mutual friends who
might confirm or deny information. Donath [10] argues that a shared social
network can provide explicit or implicit verification of identity claims.
Therefore, as highlighted in [18], a highly connected network such as Facebook
should encourage more truthful profiles, or misrepresentations that are
playful or ironic as opposed to being intentionally deceitful.
Burke et al. [7] studied user motivations for contributing in social
networking sites, based on server log data from Facebook. They found that
newcomers who see their friends contributing go on to share more content
themselves. Furthermore, those who were initially inclined to contribute,
receiving feedback and having a wide audience, were also predictors of
increased sharing.
Complementary to the body of work on identification and information
revelation, is the work on anonymity in social network sites [27, 5, 19]. The
emphasis in these works is however on algorithmic approaches for non-
disclosure and anonymity preservation, rather than on actual revelation.
Finally, parallel to this body of work is the work on reputation [29, 14].
Reputation of digital identities and trust in online environments have been
investigated by multiple research communities ranging from computer science
[22] to economics [6, 24].
With respect to our methodology, our work employs analytical models.
Analytical models for various security topics based on game, information and
decision theories are rapidly growing in interest [15]. In particular, game
theoretic approaches to reputation and trust first emerged in the economics
literature (a typical example is [11]) and were then applied to online
settings [1, 17, 21]. However, to the best of our knowledge, the only work
analyzing social identities using analytical tools is from Alpcan and
colleagues [4]. Alpcan’s work focuses on reputation and trust, where
strategies are defined in terms of opinion, quantified through a simple cost
function. As we discuss in the next sections, our focus is on validation and
individual attitudes toward deception, rather than lies. Additionally, an
interesting economically inspired work dealing with users’ privacy is
discussed by Papadimitrous and colleagues [17], who propose a precise estimate
of the value of the private information disclosed by a set of individuals, and
a compensation for such information release that may induce users to release
richer information. Yet, the model applies to a different set of applications,
such as online surveys and e-commerce applications.
This work is part of our research effort on deception and information
revelation in social networking sites. In [25] we studied the interaction of
users and servers at the time of user registration, and used a game
theoretical framework to describe a simple two-player general sum game
describing the behavior of a server system (like Facebook) that provides
utility to user. We showed that in the presence of a binding agreement to
cooperate, most players will agree to share information. In [13], we
investigated a simpler game model in which rewards for releasing information
and costs for withholding information and lying were represented by
arbitrarily chosen concave and convex functions. We showed for a specific
instance of a payoff function that a symmetric Nash equilibria existed and was
related to the automorphism class of the graph describing the interaction
graph of the social network. This work substantially extends our previous work
by more accurately modeling the qualitative nature of the user’s objective
function through the incorporation of information in our survey. Our previous
work was purely theoretical, and used a overly simplified the notion of
identity. Identity was mainly considered as an atomic value, and therefore was
focused on different aspects of information sharing in social networking sites
(for example the registration of new users). We also incorporate a model of
evolutionary dynamics to explain a user’s choices as he interacts with his
social network and is exposed to his friend’s choices.
## III Informing the model through an exploratory study
In order to understand typical social network users’ attitudes and actions
with respect to information disclosure, we conducted an exploratory study
using real-world data. The specific aim of our study was to gain a deeper
understanding of users identity-revealing actions, the peculiar features of
average users, and the perceived understanding of identity on social sites.
### III-A Methods
We conducted a web-based survey, collecting a total of 296 responses.
Respondents were recruited from two different undergraduate courses in the
college of Information Science and Technology at the Pennsylvania State
University. One extra credit point for the course was awarded for their
participation in the study. The survey was constructed to study three specific
aspects of users’ behavior: (1) privacy awareness, (2) attitude toward
information withholding and practices (3) attitude toward lies and
misrepresentation.
The respondents were aged between 20 and 35 ($\mu$=23, sd=2.34). The
respondents were 65% male and 35% female. 99.3% of them declared to have at
least one account on social sites, and 12% declared to have more than one
account on the same OSN. Participants were asked to indicate the social
network they most often accessed: 95.3% most often accessed Facebook, while
the remaining participants were distributed among Google+, Linkedin (6%) and
Twitter. In terms of network usage frequency, 94% of the respondents accessed
social network sites at least once a week, and 83.6% of those were daily
users.
Considering that Facebook is one of the social networks that most heavily
promotes personal information disclosure, our sample was deemed appropriate
for this study. While the overall sample reflects a specific subset of the
population, we notice that most of the active users in Facebook, according to
recent statistics, are below 26 years old (and specifically in the 21-24 age
range) 111http://www.socialbakers.com/facebook-statistics.
The instrument also included five broad types of measures of perceived
privacy, social pressure, and popularity (or social capital), which serve as
dependent variables.
### III-B Measures
* •
Deception was our independent measure, and was measured by two sets of 4 items
each. The first set focused on deceptive activities, and was measured on a a
frequency rating scale (1=all of the time to 5=never). The second set of items
related to the perception about deception (lies and withholding information on
social networking sites). An example item is “Lying in social network is
unethical”. These items were also rate using a Likert scale (5-point rating
scale, where 1= strongly agree and 5=strongly disagree).
* •
Usage was measured using 6 different items. Three of the items where focused
on frequency of usage and number of connections. The remaining items where
added to ascertain the extent to which the participant engages in certain
types of social interactions, e.g., posting images, giving feedback to other’s
posts or images, sharing a url, tagging a video or an image. For these items,
we used a frequency rating scale (1=never to 5=once or a few times a day)
* •
Privacy Concerns. Individual differences in privacy perceptions can be
significant [30, 32]. Thus, we need to establish a baseline understanding
about the awareness of and attitudes toward privacy protection by
participants. In our survey, we included five questions to ask participants
about their information disclosure behaviors and privacy concerns in Social
Networking sites (Cronbach $\alpha=.71)$, rated on a Likert scale (5-point
rating scale, where 1= strongly agree and 5=strongly disagree). An example
item is ”I have had concerns about the privacy of my data on Social Networks”.
* •
Pressure was measured using 5 items (Cronbach $\alpha=.823$), focusing on
pressure of updating information (e.g. “I feel peer pressured to constantly
update my Social Network profile”) and uploading content. The items measured
perceived pressure from the social networking and from peers (e.g. “I need to
update my profile often to be popular among my friends”).
* •
Perceived Popularity was measured by 6 items (Cronbach $\alpha=.732$),
focusing on the impact on one’s popularity (or social capital) upon passively
being involved in the social interactions listed in the usage measures.
ID | Question | Average | Standard Dev.
---|---|---|---
Q1 | I have put false information in my main social network account (1=strongly agree, 5=strongly disagree) | 1.87 | 0.4
Q2 | I have withheld information from my main social network account (1=strongly agree, 5=strongly disagree) | 1.91 | 0.732
Q3 | Putting false information about myself and my whereabouts on my profile can help me be more popular | 2.45 | 0.453
TABLE I: Descriptive Variables concerning Deceptive activities
### III-C Findings
The main purpose of this study was to examine users attitudes towards
deception in social networking sites.
Leveraging results from previous research studies [10, 18], we hypothesized
that (h1) users in fact deceive in social networking sites, but mostly choose
to portray truthful portions of their basic identity, which could be validated
offline (e.g. the name or gender), and deceive or withhold information which
may be deemed too personal or inappropriate for disclosure to the social
network audience. We also hypothesized that participants decision to withhold
information or lie would be influenced by (h2) their privacy inclination, (h3)
their perceived pressure to be active on the social network site and (h4)
their wish to be popular among peers. Finally, we were interested in learning
whether users’ decision to withhold or lie would be connected with ethical
choices. Here, we did not have an initial hypothesis, but were interested in
exploring the correlations between morality and deception. We discuss our
results in a detailed manner in the following.
#### III-C1 h1: Frequency of Deception
We began with identifying whether deception is in fact significant. Table I
presents the descriptive statistics of some of the study’s variables related
to deception. As reported, a vast majority of the participants admit to having
lied at least once, and also chose to withhold information (94% of respondents
either agree or strongly agreed to have lied -the exact statistics are
reported in the table). Figure 1 highlights the specific types of attributes
users most often lie about. We further determined that users who are more
likely to be involved in discussions and are therefore active in the social
networking site report a lower frequency of lying (Pearson r=0.277, p=0.033),
therefore reinforcing the well-known signaling theory identified by Donath
[10]. The relationship with the “withholding question” shows a similar trend,
but it is not statistically significant, therefore a conclusive statement on
this relationship is not possible. Users also report it is easy to detect lies
of their close social connections with whom they often interact with
($\mu=2.61$, sd. 0.912), again confirming that self-validation is effective in
social networking sites.
Figure 1: Data Items most frequently misrepresented
#### III-C2 h1: Types of Information Revealed
To further explore which pieces of information users are likely to withhold or
lie about, we asked users to indicate their preferred action for six different
personal pieces of information: location, gender, GPA, relationship status,
telephone number, current occupation. We select properties that would be
considered important and potentially sensitive for our participants, who were
mostly students. Users were given the option to indicate for each attribute
one of three choices: tell the truth, provide false information, do not put
anything.
Figure 2: Responses breakdown by attribute
The responses, organized by attribute, are reported in Figure 2. In our
survey, most of the participants claimed to misrepresent only specific pieces
of information. In particular, our analysis confirms that highly
interconnected users are likely to reveal basic identity properties, such as
gender, age, etc. truthfully (Pearson .436, r=0.012). Information commonly
deemed as private, such as telephone number and GPA, is instead mostly
withheld, or misrepresented. Finally, there is some interesting variability
with respect to location, current occupation and relationship status, where
there is not a predominant choice. These results confirm our hypothesis (h1).
#### III-C3 Influential factors of information sharing
The analysis of the factors influencing information sharing activities
resulted in the following findings.
* •
h2: Privacy We first analyzed the responses related to privacy awareness, to
get a sense of the respondents attitude toward information revelation and
leakage in social networking sites. An initial notable result is that, despite
the fact that most respondents maintained a detailed profile on their favorite
social network site, many of them also demonstrated relatively high levels of
privacy concern. The responses to the statement “I maintain a detailed profile
on my main social network account” confirm that they maintain rich profiles
($\mu$=1.97, std=0.96), and that they reveal their main identity for the most
part ($\mu$=2.01, std=0.45). Nevertheless, their responses to the statement
“There is a high potential for loss involved in sharing personal information
on Social Networks like Facebook” indicate their awareness of potential
information leakage ($\mu=1.80$, std=0.81). We then tested our first
hypothesis, i.e. whether lying or withholding information was related at all
to the respondents level of privacy awareness. We conducted an exploratory
least-squares multiple regression analysis, regressing their responses to
question Q1, with their responses related to privacy concerns as predictors.
None of these appeared to be strongly related.
The results lead to interesting findings. First, in general participants are
concerned with their privacy on social networking sites and are aware of the
potential loss of privacy; second, the results confirm the existence of
phenomenon known as the privacy paradox [31], in which individuals state that
they have privacy concerns, but behave in ways that seemingly contradict these
statements by providing detailed information about themselves. Finally, our
results show that privacy is not indicative of their choice to deceive, or
withhold information.
* •
h3: Pressure Next, we investigated whether participants feeling peer pressured
are more likely to deceive or withhold information. We first tested whether
feeling peer pressured would be correlated with the amount of personal
information displayed on the social network site. We conducted a simple
regression analysis, using the answer to the question “I feel peer pressured
to constantly update my profile” as an independent variable, and their self-
declared level of detailed social network profile as a dependent variable. The
test shows that the more users agree to feeling pressured to update their
profile, the more they claim to display a detailed profile (Pearson r=0.433,
p=.034). We then studied whether the information being revealed upon being
pressured is truthful or not. We correlated Q1 and Q2 with our items related
to popularity. Our results show that there is not a significant correlation
between users’ perceived peer-pressure and their choice to deceive. However,
there is a clear correlation between their choice to withhold information and
their feeling of being peer pressured (Pearson r=0.163, p=0.03). That is to
say, the more users feel peer pressured, the less likely they are to withhold
information.
* •
h4: Popularity When correlated with measures relative to popularity, we
obtained the following results. First, we correlated participants’ frequency
of sharing content (e.g., images) in the social networking site with their
perceived popularity gain by doing so. We obtained a significant correlation
(Pearsons r=0.272, p=.032). In line with previous studies in this space [18],
this finding shows that the more users perceive certain social interactions to
benefit their social capital, the more likely they are to pursue them. With
respect to deception, the majority of participants disagreed to have lied to
gain popularity or portray a different “self” (only 25% of respondents either
agreed or strongly agreed to the question ”I have put false information to
appear different from my original self”). However, we discovered a significant
correlation between their quest for popularity and their deceptive activities
(Pearson r=-0.2449, r=0.015), therefore confirming our hypothesis.
ID | Question | Avg | St. Dev.
---|---|---|---
Q4 | I consider lying in social network sites unethical | 3.30 | 0.943
Q5 | I consider withholding information in social network sites unethical | 4.10 | 0.842
TABLE II: Morality and Deception (Likert Scale: 1=strongly agree, 5=strongly
disagree)
In summary, this study confirms that a social network user’s tendency to
deceive for certain data types is highly correlated with his or her desire to
portray a successful social image, and not statistically related to privacy
concerns. In other terms, the perceived usefulness of the social network
service increases online users willingness to disclose their personal
information.
Figure 3: Linear regression for questions correlating Q1 and Q4 (z(x)) and Q1
and Q5 (y(x))
#### III-C4 Morality in Social Networking Sites
Our results show non-obvious relation between lying or withholding information
on a social networking site and morality.
The results of correlating social network lying (corresponding to Q1) with Q4
and Q5 are interesting, as they show opposite effects of thinking that lying
on a social network is unethical and withholding information on a social
network is unethical. A higher value for Q4 (meaning an individual disagrees
that lying is unethical) predicts a higher frequency of putting up false
information on a social networking site (Q1). However, a higher value for the
withholding information questions (Q5) predicts a lower frequency of putting
up false information. The linear equations obtained through regression
analysis are reported in Figure 3. This result seems to reinforce the notion
that lying on a social networking website and withholding information function
as two completely different actions and that a user will choose one or the
other based on an internal utility function.
ID | Question | Scale | Avg | St. Dev.
---|---|---|---|---
Q7 | How many friends do you think typically check your profile | 1=“$>$150”, 2=[100,150], 3=[50,100], 4=$<$50 | 3.63 | 0.658
Q8 | How many friends do you check typically | 1=“$>$150”, 2=[100,150], 3=[50,100], 4=$<$50 | 3.53 | 0.760
Q9 | How many social connections do you have? | 1=“$>$150”, 2=[100,150], 3=[50,100], 4=“$<$50” | 1.48 | .901
Q10 | How often do you visit your favorite social network site? | 1=Once or a few times a day 2=Once or a few times a week 3= Once or a few times a month 4= Never | 1.21 | .464
TABLE III: Social network usage
#### III-C5 Inner Circles
Some other interesting findings were related to the existence and importance
users give to inner circles within their social network. Despite the complex
social connections tying users together, users are most strongly influenced by
a small set of connections with whom they interact regularly and whose opinion
counts to them. By correlating Q7 and Q8 (Table III), we discovered that
regardless of the number of social connections users have, most users check
and believe their profile is checked by a parallel number of users (Pearson r=
.521, p$<$0.001). Most of the actions (e.g., comments and feedback) users
perform involve inner-circle users, who are the ones influencing users
decisions about lying and not lying.
## IV Game Theoretic Model of User Behavior
We build on the game theoretic approach to modeling users’ actions in a social
network begun in [25, 13] to qualitatively explain the behavior observed in
our experimental results.
We have identified that users treat types of information differently with
respect to whether they disclose, withhold or deceive. Furthermore, we know
that the behavior of users is highly dependent on the behavior of a small
group of their immediate network neighbors.
Let $G=(V,E)$ be a user graph for a social network and suppose we have several
classes of information $\mathcal{I}=\\{1,\dots,m\\}$. Let
$x^{(j)}_{i}\in[0,1]$ be the proportion of information type $i$ that Player
$j$ will release and let $y^{(j)}_{i}\in[0,1]$ be the proportion of
information type $i$ about which Player $j$ withholds. Then
$z^{(j)}_{i}\in[0,1]$ is the proportion of information type $i$ that Player
$j$ lies about. Then we have:
$x^{(j)}_{i}+y^{(j)}_{i}+z^{(j)}_{i}=1$ (1)
Let:
$\bar{x}^{(j)}_{i}=\frac{1}{|N(j)|}\sum_{k\in N(j)}x^{(k)}_{i}$ (2)
where $N(i)$ is the neighborhood of Player $j$ in $G$. We make similar
definitions for $\bar{y}^{(j)}_{i}$ and $\bar{z}^{(j)}_{i}$. These are the
average network level of releasing, withholding and lying about information
type $i$. In the presence of popularity measures (with respect to a given
user’s circle of friends) Equation 2 can be modified to be a popularity
weighted average with form:
$\hat{x}^{(j)}_{i}=\frac{1}{\sum_{k\in N(j)}p_{k}}\sum_{k\in
N(j)}p_{k}x^{(k)}_{i}$
Here $p_{k}$ is the popularity weight for player $k$.
In [13], we assumed the existence of functions returning the reward for
releasing information and costs for group deception and individual deception.
We assumed these functions were concave, convex and convex (respectively), but
provided no way to isolate their structure. We propose a richer model than the
one in [13], which incorporates our observations from the empirical
evaluation:
The more users interact with the network, the less likely they are to deceive
(see Sect. III-C1))
Users perceive interactions with their social network as a mechanism for
gaining popularity, a form of social capital (see Sect. III-C3). For the
remainder of this section, assume that we’ve fixed an information type (e.g.
location, interest, age etc). Again leveraging our analysis (Section III-C3),
we assume that five elements make up each user’s objective function:
Social capital gained from sharing information within the group,
Personal benefit gained from maintaining information privacy,
Personal cost from the discovery of deceptive information,
Moral cost from deceiving a group and
The cost associated with admitting information (in exchange for social
capital). The easiest way to understand the relationship of these elements is
as a token based model in which each action, causes a token (or fraction
thereof) to be deposited into a specific revenue or cost bucket. The proposed
model for this system is illustrated in the Petri net [8] shown in Figure 4.
Given space constraints, we cannot formally define Petri nets. In short, they
are graphical token models in which transitions move and spread within the
vertices of a graph structure. The interested reader should see Chapter 1 of
[8].
Figure 4: A Petri net model of the accumulation of various components of the
payoff associated to interacting in an online setting.
In general, we can think of the Truth, Withhold and Lie transitions as being
controlled by the user with all other transitions being uncontrolled or
controlled by nature. (Moody [20] discusses control in Petri nets.)
Alternatively, as shown in Figure 4, we can think of the user controlling the
fractional weights on the transitions leading to the Truth, Withhold and Lie
transitions. If the Petri net is continuous, then we can think of the
weightings leaving the controlled transitions as providing the benefit or cost
for each token (or fraction thereof). We let $X^{(j)}$, $Y^{(j)}$ and
$Z^{(j)}$ be correlated random variables whose dynamics are chosen by Player
$j$. In general, $X^{(j)}$ is $1$ only if Player $j$ releases a piece of
information, $Y^{(j)}$ is $1$ only if Player $j$ withholds information a piece
of information and $Z^{(j)}$ is $1$ only if Player $j$ deceives about a piece
of information. Naturally, only one of these elements can be $1$ at any given
time $t$ (thus we can think of these as being the outputs of a single discrete
distribution) chosen by the player. At time $t$, Player $j$’s stochastic
payoff function is:
$\Pi^{(j)}(t)=w_{1}\alpha(X^{(j)}(t),Z^{(j)}(t),\bar{x}^{(j)},\bar{z}^{(j)},t,\tau(t))+\\\
w_{2}\left(\beta(Y^{(j)},\bar{y}^{(j)},t)Y^{(j)}(t)+\eta(Z^{(j)}(t))\right)-\\\
w_{3}\gamma(Z^{(j)},\bar{z}^{(j)},t,\tau(t))-w_{4}\zeta(Z^{(j)}(t))-\\\
w_{5}\theta(X^{(i)}(t))$ (3)
In this expression:
$\alpha(X^{(j)},Z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)},t)$ is a social capital
function that provides the reward obtained by releasing a piece of information
(true or false).
$\beta(Y^{(j)},\bar{y}^{(j)},t)$ is privacy capital function that provides the
reward obtained by keeping a piece of information private.
$\gamma(Z^{(j)},\bar{z}^{(j)},t)$ is a cost function that yields the social
price of lying about a piece of information.
$\theta$ is an admissions cost function for each piece of true information
revealed.
$\zeta$ is a moral cost function associated with each lie told.
$\eta$ is a privacy gain function associated to each lie (since a lie may
protect privacy irrespective of any other moral judgement.
Finally $\tau(t)$ is the probability that a lie will be discovered by the
social group.
Over a period of time, the complete stochastic payoff function for Player $j$
is:
$\Pi^{(j)}=\sum_{t=0}^{T}\rho^{t}\Pi^{(j)}(t)$ (4)
The variables $w_{i}$ ($i=1,\dots,5$) are the relative weights Player $j$
places on each component of his objective function. We can also think of
$\tau$ as being a function of the total quantity of information (true and
false) that has been released to the network:
$Q^{(j)}(s)=\sum_{t=0}^{s}X^{(j)}(t)+Z^{(j)}(t)$ (5)
This provides consistency with two observations reported in Section III-C:
Users who engage in their social network more frequently, tend to deceive less
and
The more information available about a user, the easier it is for him to be
trapped in a lie. The parameter $\rho$ in Equation 4 is a discount factor
chosen in the set $(0,1]$. It is worth noting that $\rho$ is only important if
we wish to consider the limiting dynamics as $T\rightarrow\infty$. When
$\rho<1$, the user recognizes that future rewards have less value than rewards
more immediately. To relate the parameters $x^{(j)}$, $y^{(j)}$ and $z^{(j)}$
to Equation 4, we need to compute the expected value
$\mathbb{E}\left(\Pi^{(j)}\right)$. In the form given, this may be complex,
since the functions defined in Equation 3 maybe non-linear, meaning we cannot
pass the expectation operator through the expression.
The solution to the game is then defined by the simultaneous optimization
problem:
$\forall j\left\\{\begin{aligned}
\max\;\;&\mathbb{E}\left(\Pi^{(j)}(\mathbf{x}(t),\mathbf{y}(t),\mathbf{z}(t))\right)\\\
s.t.\;\;&x^{(j)}(t)+y^{(j)}(t)+z^{(j)}(t)=1\quad\forall t\\\
&x^{(j)}(t),y^{(j)}(t),z^{(j)}(t)\geq 0\quad\forall t\end{aligned}\right.$ (6)
where $\mathbf{x}(t)$, $\mathbf{y}(t)$, $\mathbf{z}(t)$ are the vectors of
decision variables for the players. Let
$\Omega=\prod_{j,t}\left\\{(x^{(j)}(t),y^{(j)}(t),z^{(j)}(t))\in[0,1]^{3}:\right.\\\
x^{(j)}(t)+y^{(j)}(t)+z^{(j)}(t)=1,\\\
\left.x^{(j)}(t),y^{(j)}(t),z^{(j)}(t)\geq 0\right\\}$ (7)
This is the complete strategy space for all players over the course of time
$t\in[0,T]$. Any Nash equilibrium will be chosen from this strategy space.
Theorem 1 of [23] provides the following (uninteresting) result:
###### Proposition IV.1.
Suppose that
$\mathbb{E}\left(\Pi^{(j)}(\mathbf{x}(t),\mathbf{y}(t),\mathbf{z}(t))\right)$
is concave for all $j$ then there is a Nash equilibrium in $\Omega$ for this
game.
###### Remark IV.2.
The uniqueness of a Nash equilibrium in this case is completely a function of
the structure of specific objective functions.
We noted above that the structure of
$\mathbb{E}\left(\Pi^{(j)}(\mathbf{x}(t),\mathbf{y}(t),\mathbf{z}(t))\right)$
maybe complex. By way of simplification, we can write a specific form of
Equation 3, as:
$\Pi^{(j)}(t)=\\\
w_{1}\alpha(x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)},t)\left(X^{(j)}(t)+(1-\tau(t))Z^{(j)}(t)\right)\\\
+w_{2}\left(\beta(y^{(j)},\bar{y}^{(j)},t)Y^{(j)}(t)+\eta Z^{(j)}(t)\right)\\\
-w_{3}\gamma(z^{(j)},\bar{z}^{(j)},t)\tau(t)Z^{(j)}(t)-w_{4}\zeta
Z^{(j)}(t)\\\ -w_{5}\theta X^{(i)}(t)$ (8)
Here we replace the functions from Equation 3 with piecewise constant
multipliers. We can then relate Equation 4 to the parameters $x^{(j)}$,
$y^{(j)}$ and $z^{(j)}$, we note that:
$\mathbb{E}\left(\Pi^{(j)}\right)=\sum_{t=0}^{T}\rho^{t}\cdot\\\
\left[w_{1}\alpha(x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)},t)\left(x^{(j)}(t)+(1-\tau(t))z^{(j)}(t)\right)\right.\\\
+w_{2}\left(\beta(y^{(j)},\bar{y}^{(j)},t)y^{(j)}(t)+\eta
z^{(j)}(t)\right)-\\\
w_{3}\gamma(z^{(j)},\bar{z}^{(j)},t)\tau(t)z^{(j)}(t)-\\\ \left.w_{4}\zeta
z^{(j)}(t)-w_{5}\theta x^{(i)}(t)\right]$ (9)
That is, a user’s expected payoff after engaging in this game, is a function
of the proportion of time he releases information, withholds information and
lies about information. Moreover, because we assume the reward/cost
multipliers ($\alpha$, $\beta$ and $\gamma$) are dependent on the group
average rates of releasing, withholding and lying about information, the
payoff to Player $j$ is dependent on the choices of all other players in his
circle of friends. Thus, a game dynamic is established. We study this
simplified game form in the remainder of the paper.
## V Evolutionary Dynamics and Example
A critical problem with the game defined in the previos section is that
individuals will never optimize their behavior according to it. An individual
can estimate many of the parameters in the model, but will never make
decisions based on the long run objective of maximizing his utility function.
It will simply be impossible for an individual to chose an optimizing strategy
ab initio, particularly without having a clear understanding of the strategies
of other players. This is especially true if (as is likely the case) multiple
Nash equilibria exist.
However, a user may engage in a more evolutionary model of decision making
[28]222Weibull’s book, [28] is an introduction to evolutionary game theory,
which inspires the approach described herein. We are not proposing a classical
evolutionary game. Instead, we are proposing an evolutionary mechanism applied
to a game theoretic context that describes user learning.:
At any time $t$, Player $j$ has a strategy
$(x^{(j)}(t),y^{(j)}(t),z^{(j)}(t))$ with an initial strategy
$(x^{(j)}(0),y^{(j)}(0),z^{(j)}(0))$
At each time $t$, Player $j$ will solve the one-stage game derived from
Equation 9 by finding a maximizing strategy
$\left(\hat{x}^{(j)},\hat{y}^{(j)},\hat{z}^{(j)}\right)$ with respect to the
current observed strategies of the other players and the current parameters in
the model.
Each player’s strategy is updated by the rule:
$\displaystyle
x^{(j)}(t+1)=x^{(j)}(t)+\epsilon^{(j)}\left(\hat{x}^{(j)}-x^{(j)}(t)\right)$
(10) $\displaystyle
y^{(j)}(t+1)=y^{(j)}(t)+\epsilon^{(j)}\left(\hat{y}^{(j)}-y^{(j)}(t)\right)$
(11) $\displaystyle
x^{(j)}(t+1)=z^{(j)}(t)+\epsilon^{(j)}\left(\hat{z}^{(j)}-z^{(j)}(t)\right)$
(12)
Here $\epsilon^{(j)}$ is a learning rate associated to the player, and is
assumed to be small – that is, $\epsilon^{(j)}\ll 1$. The dynamics given in
Equations 10 \- 12 are a discrete variation of the Jacobi iteration for
finding equilibria in games (see e.g., [16]). At its core, this is just a form
of gradient ascent.
Intuitively, each time a user makes a decision about a piece of information,
he considers his knowledge of the other players and computes an optimal move
for this time period. However, instead of changing his strategy completely, he
modifies his strategy (learns) by a small amount in the direction of
optimality. This is consistent with an individual who learns the average
behavior of the social network.
### V-A Example
By way of example, assume we have a small clique of three friends on a social
network (this is the graph governing Equation 2). Consider the following
multiplier definitions for use in Equation 9. These functions are derived from
a qualitative analysis of the data collected in the experiment described in
the previous sections, and are intended to be simple token counting margin
functions.
$\alpha(x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)})=\begin{cases}1&x^{(j)}+z^{(j)}\leq\bar{x}^{(j)}+\bar{z}^{(j)}\\\
0&\text{otherwise}\end{cases}$ (13)
In this case, the social value of information is non-zero only if the
information provided is in some way lower than the mean information provided
by the group. That is to say, you can social capital only if you’re not
posting more information than the group average. However, if the group average
is high, you will accrue social capital the more you post.
$\beta(y^{(j)},\bar{y}^{(j)})=\begin{cases}1&y^{(j)}\geq\bar{y}^{(j)}\\\
0&\text{otherwise}\end{cases}$ (14)
Here, the privacy value is non-zero only if the amount of information that is
to be admitted is larger than average amount of information being admitted.
Similarly, we can define:
$\gamma(z^{(j)},\bar{z}^{(j)})=\begin{cases}1&z^{(j)}\geq\bar{z}^{(j)}\\\
0&\text{otherwise}\end{cases}$ (15)
In this case, the cost of a lie is only non-zero if the lie is in some sense
more egregious than the average level of dishonesty. Finally, we can set:
$\zeta=1$ and $\theta=1$.
Variations in the players behavior can then be created by modifying
$w_{1},\dots,w_{5}$ in Equation 9. Finally, we assume that $\tau$ increases
linearly in time from $0.1$ to $0.9$. Recall, $\tau$ is the probability that a
lie can be detected by the social network. Thus, as time proceeds, it becomes
more likely that a falsehood is detected because more information is available
about each player 333In a fully formalized model, we believe that $\tau$ will
be a function of $Q$ (defined in Equation 5), however for our simple studies,
this is sufficient.
We study three specific examples of the dynamics produced by this model to
illustrate that even these (simple) dynamics are capable of qualitatively
reproducing behavior observed in our study. In the first example, social
capital is deemed more important than privacy ($w_{1}=1$, $w_{2}=0.25$ and
$w_{5}=0.125$), but there is a stronger sense of morality ($w_{3}=0.5$,
$w_{4}=1$). We assume three identical players connected by a complete graph
with three vertices. Player evolution is illustrated in Figure 5
Figure 5: Evolutionary output of a game with three identical players in which
$w_{1}=1$, $w_{2}=0.25$, $w_{3}=0.5$, $w_{4}=1$ and $w_{5}=0.125$, suggesting
that social capital is much more important than the gain associated with
privacy.
When we start the game with three players, each playing the strategy
$x^{(j)}=0.7$, $y^{(j)}=0.2$ and $z^{(j)}=0.1$, we see deception is removed
from the system relatively quickly, while information hiding increases
(replacing deception in the system). Notice the system converges to a
stationary strategy near $x^{(j)}=2/3$, $y^{(j)}=1/3$ and $z^{(j)}=0$. In
simpler terms, these equilibrium points are consistent with our findings that
social capital is much more important than the gain associated with privacy,
because we see that there is a preference toward sharing information
truthfully. This was observed in Section III-C3. It is also worth noting, that
it can be shown numerically these are limiting Nash equilibria for this
example.
By way of comparison, we can construct a game with less morality and even more
importance associated to social capital (being obtained by any means
necessary) with $w_{1}=2$, $w_{2}=0.25$, $w_{3}=0.25$, $w_{4}=0.125$ and
$w_{5}=0.125$. Note, $w_{4}=0.125$ indicates a low moral penalty for lying.
The evolution of the players is illustrated in Figure 6.
Figure 6: Evolutionary output of a game with three identical players in which
$w_{1}=2$, $w_{2}=0.25$, $w_{3}=0.25$, $w_{4}=0.125$ and $w_{5}=0.125$,
suggesting that social capital is much more important than the gain associated
with privacy and morality is of little concern.
The dynamics in this game are different, than those of the first game. There
is an initial, substantial, increase in the level of deception (when it is
easy to lie) in order to obtain social capital. As it becomes more difficult
to lie, the players return to a more truthful scenario that is easier to
support. This example confirms the identified effects of the signaling theory
in our dataset: users are less likely to deceive when they are heavily
involved in social interactions (Section III-C1 and III-C2).
Figure 7: Evolutionary output of a game with three identical players in which
$w_{1}=0.5$, $w_{2}=5$, $w_{3}=2$, $w_{4}=100$ and $w_{5}=3$, suggesting that
morality and privacy are paramount to this user.
In our final example, we consider a highly moral player who puts less emphasis
on social capital and substantial emphasis on information privacy. In this
case, we have: $w_{1}=0.5$, $w_{2}=5$, $w_{3}=2$, $w_{4}=100$ and $w_{5}=3$.
The results are illustrated in Figure 7. This objective function models a user
who is highly moral deciding whether to release an information type that may
be sensitive, such as GPA or dating status and illustrates the ability of the
model to capture the various qualitative results observed in the survey. In
particular, this result is consistent with the finding reported in Section
III-C4: withholding is used as a form of control when deception is considered
unethical.
The proposed approach can also be used to study richer scenarios, including
those with players that begin with different strategies and games that are
played on distinct graph structures as was discussed in [13].
## VI Future Directions
In this work, we have shown an informed model on deception and
misrepresentation in OSNs. The model is derived from a token counting approach
modeled in a stochastic, continuous Petri net. This net is then used to derive
an objective function for each player in which the payoff to a given player is
a function not only of his decisions but also the decisions of his circle of
friends. This leads to a game theoretic framework. We show that while this
game has at least one Nash equilibria, it is more interesting to consider an
evolutionary game dynamic in which players learn over time and converge to a
stationary strategy.
We are left with a number of unsolved questions, that we plan to explore in
the near future. First, we are interested in collecting more detailed data
from real-world users, to deepen our understanding of users’ interactions and
identity revelation processes. For example, in the current study we did not
focus on the users’ actions, that result in identity disclosure. What are the
typical passive social transactions (post an item on your page which may be
silently consumed by those who’ve been given access to it) or active
transactions (sharing, commenting on other’s content or status updates, give
feedback) that lead to information revelation and/or to deception? How do
different outcomes of such transactions affect social capital, and therefore
result in truthful and untruthful information sharing? How do secondary
(friend-of-friend, triad) relationships influence information sharing? Results
obtained from these studies will guide the next step of our research on the
model.
For example, using this information, we would like to determine the structural
characteristics of the benefit and cost functions in Equation 4. In addition
to this, it would be useful to identify whether the dynamics described force
the players to converge to an equilibria of some type. We expect that
convergence to an equilibrium point should occur, but it is not clear if this
is a global property of all well-behaved payoff functions. As noted in [13]
there can be an interaction between the properties of the graph on which the
game is played and the number and type of symmetric equilibria. We have not
explored this using the model presented in this paper, but we believe this is
a necessary step in understanding the behavior of user dynamics in information
expression in social networks.
## Ackowledgement
Portions of Dr. Griffin’s work were supported by the Army Research Office
under Grant W911NF-11-1-0487.
## References
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* [24] C. Shapiro. Consumer information, product quality, and seller reputation, 1982.
* [25] A. C. Squicciarini, S. Sundareswaran, and C. Griffin. A game theoretical perspective of users’ registration in online social platforms. In Accepted to Third IEEE International Conference on Privacy, Security, Risk and Trust, October 9 - 11 2011.
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* [28] J. W. Weibull. Evolutionary Game Theory. MIT Press, 1997.
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|
arxiv-papers
| 2012-06-05T16:25:52 |
2024-09-04T02:49:31.538009
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anna Squicciarini and Christopher Griffin",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1206.0981"
}
|
1206.1030
|
# Einstein Equations and MOND Theory from Debye Entropic Gravity
A. Sheykhi1,2 111 sheykhi@uk.ac.ir and K. Rezazadeh Sarab3 1 Center for
Excellence in Astronomy and Astrophysics (CEAA-RIAAM) Maragha, P. O. Box
55134-441, Iran
2 Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454,
Iran
3 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
###### Abstract
Verlinde’s proposal on the entropic origin of gravity is based strongly on the
assumption that the equipartition law of energy holds on the holographic
screen induced by the mass distribution of the system. However, from the
theory of statistical mechanics we know that the equipartition law of energy
does not hold in the limit of very low temperature. Inspired by the Debye
model for the equipartition law of energy in statistical thermodynamics and
adopting the viewpoint that gravitational systems can be regarded as a
thermodynamical system, we modify Einstein field equations. We also perform
the study for Poisson equation and modified Newtonian dynamics (MOND).
Interestingly enough, we find that the origin of the MOND theory can be
understood from Debye entropic gravity perspective. Thus our study may fill in
the gap existing in the literature understanding the theoretical origin of
MOND theory. In the limit of high temperature our results reduce to their
respective standard gravitational equations.
keywords: entropic; gravity; Debye model.
## I Introduction
Thermodynamics of black holes reveals that geometrical quantities such as
horizon area and surface gravity are related to the thermodynamic quantities
such as entropy and temperature. The first law of black hole thermodynamics
implies that the entropy and the temperature together with the energy (mass)
of the black hole satisfy $dE=TdS$ HB . In $1995$ Jacobson Jac put forwarded
a new step and suggested that the hyperbolic second order partial differential
Einstein equation for the spacetime metric has a predisposition to
thermodynamic behavior. He disclosed that the Einstein field equation is just
an equation of state for the spacetime and in particular it can be derived
from the proportionality of entropy and the horizon area together with the
fundamental relation $\delta Q=TdS$. Following Jacobson, however, several
recent investigations have shown that there is indeed a deeper connection
between gravitational dynamics and horizon thermodynamics. The deep connection
between horizon thermodynamics and gravitational dynamics, help to understand
why the field equations should encode information about horizon
thermodynamics. These results prompt people to take a statistical physics
point of view on gravity.
A next great step put forwarded by Verlinde Verlinde who claimed that the
laws of gravity are not fundamental and in particular they emerge as an
entropic force caused by the changes in the information associated with the
positions of material bodies. According to Verlinde proposal when a test
particle with mass $m$ approaches a holographic screen from a distance
$\triangle x$, the change of entropy on the holographic screen is
$\triangle S=2\pi\frac{m}{\hbar}\triangle x,$ (1)
where we have set $k_{B}=c=1$ for simplicity, through this paper. The entropic
force can arise in the direction of increasing entropy and is proportional to
the temperature,
$F=T\frac{\triangle S}{\triangle x}.$ (2)
Verlinde’s derivation of Newton’s law of gravitation at the very least offers
a strong analogy with a well understood statistical mechanism. Therefore, this
derivation opens a new window to understand gravity from the first principles.
The study on the entropic force has raised a lot of attention recently (see
Cai4 ; Other ; newref ; sheyECFE ; Ling ; Modesto ; Yi ; Sheykhi2 and
references therein).
Verlinde’s proposal on the entropic origin of gravity is based strongly on the
assumption that the equipartition law of energy holds on the holographic
screen induced by the mass distribution of the system, namely,
$E=\frac{1}{2}NT$. However, from the theory of statistical mechanics we know
that the equipartition law of energy does not hold in the limit of very low
temperature. By low temperature, we mean that the temperature of the system is
much smaller than Debye temperature, i.e. $T\ll T_{D}$. It was demonstrated
that the Debye model is very successful in interpreting the physics at the
very low temperature. Hence, it is expected that the equipartition law of
energy for the gravitational systems should be modified in the limit of very
low temperature (or very weak gravitational field).
It is important to note that Verlinde got the Newton’s law of gravitation,
Einstein equations and Poisson equation with the assumption that each bit on
holographic screen is free of interaction. It should be more general that the
bits on holographic screen interact each others. In such case, one could
anticipate that the Newton’s law of gravitation, Einstein equation and Poisson
equation must be modified. For example, Gao Gao studied three dimensional
Debye model and modified the entropic force and henece Friedmann equations.
Such modification can interpret the current acceleration of the universe
without invoking any kind of dark energy Gao . In this paper we use the Debye
model to modify the entropic gravity. We find that this modified entropic
force affects on the law of gravitations and modify them accordingly.
This paper is structured as follows. In the next section we derive Einstein
field equations from Debye entropic gravity. The theoretical origin of MOND
theory is discussed in the framework of Debye entropic gravity in section III.
Sec. IV is devoted to the derivation of the Poisson equation from Debye
entropic force scenario. We finish our paper with conclusions which appear in
Sec. V.
## II Einstein equations from Debye entropic gravity
Following Verlinde’s scenario, gravity may have a statistical thermodynamics
origin. Thus, any modification of statistical mechanics should modify the laws
of gravity accordingly. In this section we use the modified equipartition law
of energy to obtain the modified Einstein equations.
We consider a system that its boundary is not infinitely extended and forms a
closed surface. We can take the boundary as a storage device for information,
i.e. a holographic screen. Assuming that the holographic principle holds, the
maximal storage space, or total number of bits $N$, is proportional to the
area $A$,
$N=\frac{A}{G\hbar}.$ (3)
Suppose there is a total energy $E$ present in the system. Let us now just
make the simple assumption that the energy is divided evenly over the bits.
Each bit on the holographic screen has one dimensional degree of freedom,
hence we can use the one dimensional equipartition law of energy. The
equipartition law of energy which is valid in all range of temperatures is
$E=\frac{1}{2}NTD(x),$ (4)
where T is the temperature of the screen and $D(x)$ is the one dimensional
Debye function defined as
$D(x)\equiv\frac{1}{x}\int_{0}^{x}\frac{y}{e^{y}-1}dy,$ (5)
and $x$ is related to the temperature
$x\equiv\frac{T_{D}}{T},$ (6)
where $T_{D}$ is the Debye temperature. Using the equivalence between mass and
energy, $E=M$, as well as Eq. (3), we can rewrite Eq. (4) in a more general
form,
$M=\frac{1}{2G\hbar}\oint_{S}TD(x)dA,$ (7)
where the integration is over the holographic screen. For temperature, we use
the Unruh temperature formula on the holographic screen,
$T=\frac{\hbar a}{2\pi},$ (8)
where $a$ denotes the acceleration. The acceleration has relation with the
Newton’s potential and in general relativity it may be written as
$a^{b}=-\nabla^{b}\phi,$ (9)
where $\phi$ is the natural generalization of Newton’s potential in general
relativity and for it we have Wald ,
$\phi=\frac{1}{2}Ln(-\xi^{a}\xi_{a}),$ (10)
where $\xi^{a}$ is a global time like Killing vector. The exponent $e^{\phi}$
represents the redshift factor that relates the local time coordinate to that
at a reference point with $\phi=0$, which we will take to be at infinity. We
choose the holographic screen $S$ as a closed equipotential surface or in
other words, a closed surface of constant redshit $\phi$. Therefore Eq. (8)
may be written as Verlinde
$T=\frac{\hbar}{2\pi}e^{\phi}N^{a}\nabla_{a}\phi,$ (11)
where $N^{a}$ is the unit outward pointing vector that is normal to the
equipotential holographic screen $S$ and time like Killing vector $\xi^{b}$.
We inserted a redshift factor $e^{\phi}$, because the temperature $T$ is
measured with respect to the reference point at infinity. Because $N^{a}$ is
normal to the equipotential holographic screen, for it we have
$N^{a}=\frac{\nabla^{a}\phi}{(\nabla^{b}\phi\nabla_{b}\phi)^{1/2}}.$ (12)
Therefore we can rewrite Eq. (11) as
$T=\frac{\hbar}{2\pi}e^{\phi}(\nabla^{a}\phi\nabla_{a}\phi)^{1/2}.$ (13)
Substituting Eq. (11) in Eq. (7), we get
$M=\frac{1}{4\pi G}\oint_{S}e^{\phi}N^{a}\nabla_{a}\phi D(x)dA.$ (14)
Following the same logic of Wald , we can obtain
$M=-\frac{1}{8\pi G}\oint_{S}\nabla^{a}\xi^{b}D(x)dS_{ab},$ (15)
where $dS_{ab}$ is the two-surface element Poisson . On the other hand,
according to the Stokes theorem, we have Poisson
$\oint_{S}B^{ab}dS_{ab}=2\int_{\Sigma}\nabla_{b}B^{ab}d\Sigma_{a},$ (16)
where $B^{ab}$ is an antisymmetric tensor field and $S$ is the two dimensional
boundary of the hypersurface $\Sigma$. $d\Sigma_{a}$ is a directed surface
element on $\Sigma$ and for it we have
$d\Sigma_{a}=\varepsilon n_{a}d\Sigma,$ (17)
where $n^{a}$ is the unit normal of the hypersurface $\Sigma$ and
$\varepsilon$ is equal to -1 or 1 if the hypersurface is spacelike or
timelike, respectively. Now we apply the Stokes theorem (16) for Eq. (15) and
get
$\displaystyle M$ $\displaystyle=-\frac{1}{4\pi
G}\int_{\Sigma}\nabla_{b}[\nabla^{a}\xi^{b}D(x)]d\Sigma_{a}$
$\displaystyle=-\frac{1}{4\pi
G}\int_{\Sigma}[D(x)\nabla_{b}\nabla^{a}\xi^{b}+\nabla^{a}\xi^{b}\nabla_{b}D(x)]d\Sigma_{a}$
$\displaystyle=-\frac{1}{4\pi
G}\int_{\Sigma}[-D(x)\nabla_{b}\nabla^{b}\xi^{a}+\nabla^{a}\xi^{b}\nabla_{b}D(x)]d\Sigma_{a},$
(18)
where in the last step we have used the Killing equation,
$\nabla^{a}\xi^{b}+\nabla^{b}\xi^{a}=0.$ (19)
Now we use the relation Wald
$\nabla^{a}\nabla_{a}\xi^{b}=-R^{b}_{a}\xi^{a},$ (20)
which is implied by the Killing equation for $\xi^{a}$, and get
$\displaystyle M$ $\displaystyle=-\frac{1}{4\pi
G}\int_{\Sigma}[R_{ab}\xi^{b}D(x)+\nabla_{a}\xi^{c}\nabla_{c}D(x)]d\Sigma^{a}$
$\displaystyle=-\frac{1}{4\pi
G}\int_{\Sigma}[R_{ab}\xi^{b}D(x)+e^{-2\phi}(-\xi^{b}\xi_{b})\nabla_{a}\xi^{c}\nabla_{c}D(x)]d\Sigma^{a}$
$\displaystyle=\frac{1}{4\pi
G}\int_{\Sigma}[R_{ab}D(x)-e^{-2\phi}\xi_{b}\nabla_{a}\xi^{c}\nabla_{c}D(x)]n^{a}\xi^{b}d\Sigma,$
(21)
where in the second line we have used Eq. (10). In the last line we have used
$d\Sigma^{a}=-n^{a}d\Sigma$, because the hypersurface $\Sigma$ is spacelike.
On the other hand, $M$ can be expressed as an integral over the enclosed
volume of certain components of stress energy tensor $\mathcal{T}_{ab}$ Wald ,
$M=2\int(\mathcal{T}_{ab}-\frac{1}{2}\mathcal{T}g_{ab})n^{a}\xi^{b}d\Sigma.$
(22)
Equating Eqs. (II) and (22), we find
$D(x)R_{ab}-e^{-2\phi}\xi_{b}\nabla_{a}\xi^{c}\nabla_{c}D(x)=8\pi
G(\mathcal{T}_{ab}-\frac{1}{2}\mathcal{T}g_{ab}).$ (23)
The above equation is the modified Einstein equations resulting from
considering the Debye correction to the equipartition law of energy in the
framework of entropic gravity scenario. This equation is now valid for all
range of temperature, since we have assumed the general equipartition law of
energy. Therefore, we see that in Verlinde’s approach, any modification of
first principles such as equipartition law of energy will modify the
gravitational field equations. The question whether the modified term in
Einstein equation can be detectable practically or not needs more
investigations in the future. One needs to first specify the Debye function
$D(x)$ and then try to solve the field equations (23). The resulting solutions
should be checked with experiments or observations. It is clear that the
correction term only plays role in very low temperature, in which the
curvature of spacetime tends to zero and it becomes flat.
It is instructive to examine the modified Einstein equations in the high
temperatures limit. According to the Unruh temperature formula we have
$g=\frac{2\pi}{\hbar}T,$ (24)
where $g$ is the norm of the gravitational acceleration. Therefore, the
strength of the gravitational field is proportional to the temperature. Also,
we can define the Debye acceleration relating to the Debye temperature as
$g_{D}=\frac{2\pi}{\hbar}T_{D}.$ (25)
Therefore, if the temperature is larger than the Debye temperature, i.e.
$T>T_{D}$, then the norm of the gravitational acceleration is larger than the
Debye acceleration, i.e. $g>g_{D}$. In other words, the limit of high
temperatures compared to the Debye temperature, is corresponding to the strong
gravitational fields. In this case we have $T\gg T_{D}$, thus for $x$ and $y$
in the definition of the Debye function (5), we have $x\ll 1$ and consequently
$y\ll 1$. Therefore we can use the approximation $e^{y}\approx 1+y$ in the
integral of Eq. (5) and as a result, the one dimensional Debye function
reduces to
$D(x)\approx\frac{1}{x}\int_{0}^{x}dy=1.$ (26)
Substituting this result ($D(x)=1$) in the modified Einstein equations (23),
leads to
$R_{ab}=8\pi G(\mathcal{T}_{ab}-\frac{1}{2}\mathcal{T}g_{ab}).$ (27)
Therefore, in the temperatures extremely larger than the Debye temperature
(very strong gravitational fields), one obtains the standard Einstein field
equations as expected.
## III MOND theory from Debye entropic gravity
Modified Newtonian dynamics (MOND) was proposed to explain the flat rotational
curves of spiral galaxies. A great variety of observations indicate that the
rotational velocity curves of all spiral galaxies tend to some constant value
Trimble . Among them are the Oort discrepancy in the disk of Milky Way Bahcall
, the velocity dispersions of dwarf Spheroidal galaxies Vogt and the flat
rotation curves of spiral galaxies Rubin . These observations are in
contradiction with the prediction of Newtonian theory because Newtonian theory
predicts that objects that are far from the galaxy center have lower
velocities.
The most widely adopted way to resolve these difficulties is the dark matter
hypothesis. It is assumed that all visible stars are surrounded by massive
nonluminous matters. Another approach is the MOND theory which was suggested
by M. Milgrom in 1983 Milgrom . This theory appears to be highly successful
for explaining the observed anomalous rotational-velocity. In fact, the MOND
theory is (empirical) modification of Newtonian dynamics through modification
in the kinematical acceleration term ‘$a$’ (which is normally taken as
$a=v^{2}/r$ ) as effective kinematic acceleration $a_{\rm
eff}=a\mu(\frac{a}{a_{0}})$,
$a\mu(\frac{a}{a_{0}})=\frac{GM}{R^{2}},$ (28)
where $\mu=1$ for usual-values of accelerations and $\mu=\frac{a}{a_{0}}$($\ll
1$) if the acceleration ‘$a$’ is extremely low, lower than a critical value
$a_{0}=10^{-10}$ $m/s^{2}$. At large distance, at the galaxy out skirt, the
kinematical acceleration ‘$a$’ is extremely small, smaller than $10^{-10}$
$m/s^{2}$ , i.e., $a\ll a_{0}$, hence the function
$\mu(\frac{a}{a_{0}})=\frac{a}{a_{0}}$. Consequently, the velocity of star on
circular orbit from the galaxy-center is constant and does not depend on the
distance; the rotational-curve is flat, as it observed.
Although MOND theory can explain the flat rotational curve, however its
theoretical origin remains un-known. Thus, it is well motivated to establish a
gravitational theory which can results MOND theory naturally. In this section,
we are able to show that the MOND theory can be extracted completely from the
Debye entropic gravity. This derivation further support the viability of Debye
entropic gravity formalism.
Again, we consider a spherical holographic screen with radius $R$ as the
boundary of the system. Combining Eqs. (3) and (4), and using the equivalence
between mass and energy as well as relation $A=4\pi R^{2}$, we obtain
$\frac{2\pi}{\hbar}TD(x)=\frac{GM}{R^{2}}.$ (29)
Using the Unruh temperature formula (8), the above equation may be written as
$aD(x)=\frac{GM}{R^{2}}.$ (30)
Also, if we use the Unruh temperature formula in the definition of $x$, i.e.
Eq. (6), and define $a_{0}$ as
$a_{0}\equiv\frac{12T_{D}}{\pi\hbar},$ (31)
then we obtain
$x=\frac{\pi^{2}a_{0}}{6a}.$ (32)
Using the above result in Eq. (30) gives
$aD(\frac{\pi^{2}a_{0}}{6a})=\frac{GM}{R^{2}}.$ (33)
This is the MOND theory resulting from Debye entropic gravity. If we compare
this equation with well-known Eq. (28), we see that we can define $\mu$
function as
$\mu(\frac{a}{a_{0}})\equiv D(\frac{\pi^{2}a_{0}}{6a}).$ (34)
In what follows we show that this function satisfies the conditions similar to
those of $\mu$ function in Eq. (28). Let us examine Eq. (33) in two limits of
temperatures. First, we consider the limit corresponding to the temperatures
large relative to the Debye temperature. In this case $x\ll 1$ ($a\gg a_{0}$)
we have $D(x)=1$. Thus Eq. (33) reduces to
$a=\frac{GM}{R^{2}}.$ (35)
Therefore, for strong gravitational fields, Eq. (33) turns into the standard
Newtonian dynamics. As we discussed, for $a\gg a_{0}$ we have also
$\mu(\frac{a}{a_{0}})=1$. We conclude that in the limit of $a\gg a_{0}$ both
$D(x)$ and $\mu(x)$ have the same behavior and become equal to 1.
The second limit corresponds to the temperatures extremely smaller than the
Debye temperature, $T\ll T_{D}$, that is to say in the weak gravitational
fields. In this limit, we have $x\gg 1$ ($a\ll a_{0}$), and the Debye function
can be expanded as
$D(x)=\frac{1}{x}\int_{0}^{\infty}\frac{y}{e^{y}-1}dy\approx\frac{\pi^{2}}{6x}.$
(36)
If we use the approximation (36) in Eq. (33), we obtain
$a\left(\frac{a}{a_{0}}\right)=\frac{GM}{R^{2}}.$ (37)
Therefore, the Newtonian dynamics is modified for weak gravitational fields,
e.g. at large distance from the galaxy center, namely at the galaxy out skirt.
Thus the origin of the MOND theory can be understood completely in the
framework of Debye entropic gravity. In this way we fill in the gap existing
in the literature understanding the theoretical origin of MOND theory.
## IV Poisson equation from Debye entropic force
Finally, we obtain the modified Poisson equation by taking into account the
Debye correction to the equipartition law of energy. We choose a holographic
screen $S$ corresponding to an equipotential surface with fixed Newtonian
potential $\phi_{0}$. We assume that the entire mass distribution given by
$\rho(\vec{x})$ is contained inside the volume enclosed by the screen and
there are some test particles outside this volume. To identify the temperature
of the holographic screen, we take a test particle and move it close to the
screen and measure its local acceleration. The local acceleration is related
to the Newton potential as
$\vec{a}=-\vec{\nabla}\phi.$ (38)
Substituting this relation into Unruh temperature formula, we get
$T=\frac{\hbar|\vec{\nabla}\phi|}{2\pi}.$ (39)
Using the above equation in the definition of $x$, we have
$x\equiv\frac{T_{D}}{T}=\frac{2\pi T_{D}}{\hbar|\vec{\nabla}\phi|}.$ (40)
Inserting (39) in Eq. (7), after using Eq. (3) for the number of bits on the
holographic screen, we obtain
$M=\frac{1}{4\pi G}\oint_{S}D(x)\vec{\nabla}\phi.d\vec{A}.$ (41)
Using the divergence theorem we can rewrite Eq. (41) as
$M=\frac{1}{4\pi G}\int_{V}\vec{\nabla}.[D(x)\vec{\nabla}\phi]dV.$ (42)
On the other hand, for the mass distribution $M$ inside the closed surface
$S$, we have the relation
$M=\int_{V}\rho(\vec{x})dV.$ (43)
Equating Eqs. (42) and (43), we get
$\vec{\nabla}.[D(x)\vec{\nabla}\phi]=4\pi G\rho(\vec{x}).$ (44)
This is the modified Poisson equation which is valid in all range of
temperatures. For high temperatures. i.e. strong gravitational field ($x\ll
1$) and hence $D(x)=1$. In this case Eq. (44) reduces to the standard Poisson
equation,
$\nabla^{2}\phi=4\pi G\rho(\vec{x}).$ (45)
Thus, considering the gravitational system as a thermodynamical system and
taking into account the Debye model for the modified equipartition law of
energy, we see that not only Einstein equation and MOND theory but also the
Poisson equation is modified accordingly. Clearly the modification of Poisson
equation leads to modified Newton’s law of gravitation.
## V Conclusions
In his work, Verlinde applied the equipartition law of energy as
$E=\frac{1}{2}NT$ on the holographic screen induced by the mass distribution
of the system, and obtained the Einstein equations, Newton’s law of
gravitation and the Poisson equation. But we know from statistical mechanics
that the equipartition law of energy does not hold at very low temperatures
and it should be corrected. In this paper, we considered the Debye correction
to the equipartition law of energy as $E=\frac{1}{2}NTD(x)$, where $D(x)$ is
the Debye function. Following Verlinde’s strategy on the entropic origin of
gravity, we obtained the modified form of the Einstein equations, MOND theory
and the modified Poisson equation. Interestingly enough, we found that the
origin of MOND theory can be understood from the Debye entropic gravity
scenario. Since the MOND theory is an acceptable theory for explanation of the
galaxy flat rotation curves, thus the studies on its theoretical origin is of
great importance. This result is impressive and show that the approach here is
powerful enough for deriving the modified gravitational field equations from
Debye model. We also showed that in the temperatures extremely larger than the
Debye temperature (very strong gravitational fields), the obtained modified
equations turn into their respective well-known standard equations. The
results obtained here further support the viability of Verlinde’s formalism.
###### Acknowledgements.
This work has been supported financially by Center for Excellence in Astronomy
and Astrophysics of IRAN (CEAAI- RIAAM) under research project No. 1/2782-77.
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|
arxiv-papers
| 2012-06-04T06:30:20 |
2024-09-04T02:49:31.549876
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, K. Rezazadeh Sarab",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1206.1030"
}
|
1206.1294
|
# A Simple Classification of Solitons
Yousef Yousefi and Khikmat Kh. Muminov
Physical-Technical Institute named after S.U.Umarov
Academy of Sciences of Republic of Tajikistan
Aini Ave 299/1, Dushanbe, Tajikistan
###### Abstract
In this report, fundamental educational concepts of linear and non-linear
equations and solutions of nonlinear equations from the book High-Temperature
Superconductivity: The Nonlinear Mechanism and Tunneling Measurements (Kluwer
Academic Publishers, Dordrecht, 2002, pages 101-142) is given. There are a few
ways to classify solitons. For example, there are topological and
nontopological solitons. Independently of the topological nature of solitons,
all solitons can be divided into two groups by taking into account their
profiles: permanent and timedependent. For example, kink solitons have a
permanent profile (in ideal systems), while all breathers have an internal
dynamics, even, if they are static. So, their shape oscillates in time. The
third way to classify the solitons is in accordance with nonlinear equations
which describe their evolution. Here we discuss common properties of solitons
on the basis of the four classification.
## 1 Introduction
For a long time linear equations have been used for describing different
phenomena. For example, Newton’s, Maxwell’s and Schrödinger’s equations are
linear, and they take into account only a linear response of a system to an
external disturbance. However, the majority of real systems are nonlinear.
Most of the theoretical models are still relying on a linear description,
corrected as much as possible for nonlinearities which are treated as small
perturbations. It is well known that such an approach can be absolutely wrong.
The linear approach can sometimes miss completely some essential behaviors of
the system.
Nonlinearity has to do with thresholds, with multistability, with hysteresis,
with phenomena which are changed qualitatively as the excitations are changed.
In a linear system, the ultimate effect of the combined action of two
different causes is merely the superposition of the effects of each cause
taken individually. But in a nonlinear system adding two elementary actions to
one another can induce dramatic new effects reflecting the onset of
cooperativity between the constituent elements. To understand nonlinearity,
one should first understand linearity. Consider linear waves. In general, a
wave may be defined as a progression through matter of a state of motion.
Characteristic properties of any linear wave are:
(i) the shape and velocity of a linear wave are independent of its amplitude;
(ii) the sum of two linear waves is also a linear wave; and
(iii) small amplitude waves are linear. Fig.1a shows an example of a periodic
linear wave.
Large amplitude waves may become nonlinear. The fate of a wave travelling in a
medium is determined by properties of the medium. Nonlinearity results in the
distortion of the shape of large amplitude waves, for example, in turbulence.
However, there is another source of distortion—the dispersion of a wave.
More than 100 years ago the mathematical equations describing solitary waves
were solved, at which point it was recognized that the solitary wave, shown in
Fig.1b, may exist due to a precise balance between the effects of nonlinearity
and dispersion. Nonlinearity tends to make the hill steeper (see Fig. 1b),
while dispersion flattens it.
Figure 1: Sketch of (a) a periodic linear wave, and (b) a solitary wave.
The solitary wave lives “between” these two dangerous, destructive “forces.”
Thus, the balance between nonlinearity and dispersion is responsible for the
existence of the solitary waves. As a consequence, the solitary waves are
extremely robust. Solitary waves or solitons cannot be described by using
linear equations. Unlike ordinary waves which represent a spatial periodical
repetition of elevations and hollows on a water surface, or condensations and
rarefactions of a density, or deviations from a mean value of various physical
quantities, solitons are single elevations, such as thickenings etc., which
propagate as a unique entity with a given velocity. The transformation and
motion of solitons are described by nonlinear equations of mathematical
physics. The history of solitary waves or solitons is unique. The first
scientific observation of the solitary wave was made by Russell in 1834 on the
water surface[1]. One of the first mathematical equations describing solitary
waves was formulated in 1895. And only in 1965 were solitary waves fully
understood! Moreover, many phenomena which were well known before 1965 turned
out to be solitons! Only after 1965 was it realized that solitary waves on the
water surface, nerve pulse, vortices, tornados and many others belong to the
same category: they are all solitons! That is not all, the most striking
property of solitons is that they behave like particles!. Other important
properties of soliton are:
1\. It does not change shape.
2\. In a region of space is limited.
3\. After dealing with other solitons, keep its shape.
Mathematically, there is a difference between “solitons” and “solitary waves.”
Solitons are localized solutions of integrable equations, while solitary waves
are localized solutions of non-integrable equations. Another characteristic
feature of solitons is that they are solitary waves that are not deformed
after collision with other solitons. Thus the variety of solitary waves is
much wider than the variety of the “true” solitons. Some solitary waves, for
example, vortices and tornados are hard to consider as waves. For this reason,
they are sometimes called soliton-like excitations. To avoid this bulky
expression we shall often use the term soliton in all cases.
## 2 Classification of solitons
There are a few ways to classify solitons[33]. For example, as we known, there
are topological and nontopological solitons. Independently of the topological
nature of solitons, all solitons can be divided into two groups by taking into
account their profiles: permanent and timedependent. For example, kink
solitons have a permanent profile (in ideal systems), while all breathers have
an internal dynamics, even, if they are static. So, their shape oscillates in
time. The third way to classify the solitons is in accordance with nonlinear
equations which describe their evolution. Here we discuss common properties of
solitons on the basis of the four classification.
### 2.1 Classical and quantum solitons
A rough description of a classical soliton is that of a solitary wave which
shows great stability in collision with other solitary waves. A solitary wave,
as we have seen, does not change its shape, it is a disturbance $u(x-ct)$
which translating along the x-axis with speed c. [2]
A remarkable example for this type is soliton solution for linear dispersion
less equation or KdV equation.
Figure 2: Classical soliton.
Quantum solitons for physical systems governed by quantum attractive nonlinear
Schrödinger model and quantum Sine-Gordon model. These solitons are coherent
states or eigenvalues of annihilation operator $\hat{a}$.
The one-dimensional quantum NLS equation, in term of quantum fields
$\hat{\psi}(x,t),\hat{\psi}^{+}(x,t)$ is
$\displaystyle+i\bar{h}\frac{\partial\hat{\psi}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\bar{h}^{2}}{2m}\frac{\partial^{2}\hat{\psi}}{\partial
x^{2}}+2c\hat{\psi}^{+}\hat{\psi}^{2}$
$\displaystyle-i\bar{h}\frac{\partial\hat{\psi}^{+}}{\partial t}$
$\displaystyle=$
$\displaystyle-\frac{\bar{h}^{2}}{2m}\frac{\partial^{2}\hat{\psi}^{+}}{\partial
x^{2}}+2c(\hat{\psi}^{+})^{2}\hat{\psi}$ (1)
Also for second model, Sine-Gordon model, equation is
$\displaystyle\frac{\partial^{2}\hat{\phi}}{\partial
x^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\hat{\phi}}{\partial
t^{2}}=m^{2}\bar{h}^{-2}c^{4}\hat{\phi}$ (2)
### 2.2 topological and non-topological solitons
In renormalize relativistic local field theories all solitary waves are either
non-topological or topological [3,4].
In non-topologically soliton, for example the water canal solitary solution to
the KdV equation means that the boundary conditions at infinity are
topologically the same for the vacuum as for the soliton. The vacuum can be
non-degenerate but an additive conservation law is required.
But topologically soliton need a degenerate vacuum. The boundary conditions at
infinity are topologically different for the solitary wave than for a physical
vacuum state. The solitary of topological soliton is due to the distinct
classes of vacuum at the boundaries where these boundary conditions are
characterized by a particular correspondence (mapping) between the group space
and coordinate space, and because these mappings are not continuously
deformable into one another they are topologically distinct.
In mathematics and physics, a topological soliton or a topological defect is a
solution of a system of partial differential equations or of a quantum field
theory homotopically distinct from the vacuum solution; it can be proven to
exist because the boundary conditions entail the existence of homotopically
distinct solutions. Typically, this occurs because the boundary on which the
boundary conditions are specified has a non-trivial homotopy group which is
preserved in differential equations; the solutions to the differential
equations are then topologically distinct, and are classified by their
homotopy class. Topological defects are not only stable against small
perturbations, but cannot decay or be undone or be de-tangled, precisely
because there is no continuous transformation that will map them
(homotopically) to a uniform or ”trivial” solution.
Various different types of topological defects are possible, with the type of
defect formed being determined by the symmetry properties of the matter and
the nature of the phase transition. They include:
Domain walls, two-dimensional membranes that form when a discrete symmetry is
broken at a phase transition. These walls resemble the walls of closed-cell
foam, dividing the universe into discrete cells.
Cosmic strings are one-dimensional lines that form when an axial or
cylindrical symmetry is broken.
Monopoles, point-like defects that form when a spherical symmetry is broken,
are predicted to have magnetic charge, either north or south (and so are
commonly called ”magnetic monopoles”).
Textures form when larger, more complicated symmetry groups are completely
broken. They are not as localized as the other defects, and are unstable.
Other more complex hybrids of these defect types are also possible.
Topological defects, of the cosmological type, are extremely high-energy
phenomena and are likely impossible to produce in artificial Earth-bound
physics experiments, but topological defects that formed during the universe’s
formation could theoretically be observed.
No topological defects of any type have yet been observed by astronomers,
however, and certain types are not compatible with current observations; in
particular, if domain walls and monopoles were present in the observable
universe, they would result in significant deviations from what astronomers
can see. Theories that predict the formation of these structures within the
observable universe can therefore be largely ruled out.
In condensed matter physics, the theory of homotopy groups provides a natural
setting for description and classification of defects in ordered systems.
Topological methods have been used in several problems of condensed matter
theory. Poénaru and Toulouse used topological methods to obtain a condition
for line (string) defects in liquid crystals can cross each other without
entanglement. It was a non-trivial application of topology that first led to
the discovery of peculiar hydrodynamic behavior in the A-phase of superfluid
Helium-3.
Unlike in cosmology and field theory, topological defects in condensed matter
can be experimentally observed. Ferromagnetic materials have regions of
magnetic alignment separated by domain walls. Nematic and bi-axial nematic
liquid crystals display a variety of defects including monopoles, strings,
textures etc. Defects can also been found in biochemistry, notably in the
process of protein folding.
In quantum field theory, a non-topological soliton (NTS) is a field
configuration possessing, contrary to a topological one, a conserved Noether
charge and stable against transformation into usual particles of this field
for the following reason. For fixed charge Q, the mass sum of Q free particles
exceeds the energy (mass) of the NTS so that the latter is energetically
favorable to exist.
The interior region of an NTS is occupied by vacuum different from surrounding
one. Thus a surface of the NTS represents a domain wall, which also appears as
a topological defect in field theories with broken discrete symmetry. If
infinite, the domain walls cause contradiction with cosmology. But the surface
of an NTS is a closed finite wall so, if it exists in the Universe, it does
not cause those contradictions. Another point is that if the topological
domain wall is closed, it shrinks because of wall tension. As for the NTS
surface, it does not shrink since the decreasing of the NTS volume would
increase its energy.
Quantum field theory has been developed to describe the elementary particles.
However in the middle seventieth it was found out that this theory predicts
one more class of stable compact objects: non-topological solitons. The NTS
represents an unusual coherent state of matter, called also bulk matter.
Models were suggested for the NTS to exist in forms of stars, quasars, the
Dark matter and nuclear matter.
An NTS configuration is the lowest energy solution of classical equations of
motion possessing a spherical symmetry. Such a solution has been found for a
rich variety of field Lagrangians. One can associate the conserved charge with
global, local, Abelian and non-Abelian symmetry. It appears to be possible the
NTS configuration with bosons as well as with fermions to exist. In different
models either one and the same field carries the charge and binds the NTS, or
there are two different fields: charge carrier an binding field.
The spatial size of the NTS configuration may be elementary small or
astronomically large: depending on a model, i.e. the model fields and
constants. The NTS size could increase with its energy until the gravitation
complicates its behavior and finally causes the collapse. Although in some
models the NTS charge is bounded by the stability (or metastability).
### 2.3 Classification of solitons in bases of shape
#### 2.3.1 bell soliton
the soliton solution of KdV equation have a bell shape and a low frequency
solitons. This soliton referred to as non-topological solitons.
Figure 3: bell soliton, (a): solution of Kdv equation , (b): solution of HLS
equation
the soliton solution of NLS equation have a bell shaped hyperbolic secant
envelope modulated a harmonic (cosine) wave. This solution does not depend on
the amplitude and high frequency soliton.
#### 2.3.2 Kink soliton
The solutions of SC equation are called kink or anti-kink solitons, and
velocity does not depend on the wave amplitude. This soliton referred to as
topological solitons.
Figure 4: Kink-antikink soliton solutions to the Sine-Gordon equation Figure
5: Bloch wall between two ferromagnetic domains
A good physical example of a kink solution is a Bloch wall between two
magnetic domains in a ferromagnet. The magnetic spins rotate from say, spin
down in one domain to spin up in the adjacent domain. The transition region
between down and up is called the Bloch wall. Under the influence of an
applied magnetic field, the Bloch wall can propagate according to the Sine-
Gordon equation.
#### 2.3.3 breather soliton
Discrete breathers (DB), also known as intrinsic localized modes, or nonlinear
localized excitations, are an important new phenomenon in physics, with
potential applications of sufficient significance to rival or surpass the
Soliton of integrable partial differential equations[6]. They occur in
networks (includes all crystalline lattices and also quasicrystal and
amorphous arrays) of oscillators (includes rotors and spins) rather than
spatially continuous media, and are time-periodic spatially localized
solutions.
Figure 6: breathers soliton
### 2.4 Classification of solitons in bases of nonlinear equations
Up to now we have considered two nonlinear equations which are used to
describe soliton solutions: the KdV equation and the sine-Gordon equation.
There is the third equation which exhibits true solitons it is called the
nonlinear Schr¨odinger (NLS) equation[2]. We now summarize soliton properties
on the basis of these three equations, namely, the Korteweg-de Vries equation:
$\displaystyle u_{t}=6uu_{x}-u_{xxx};$ (3)
the sine-Gordon equation:
$\displaystyle u_{tt}=u_{xx}-sinu;$ (4)
and the nonlinear Schrodinger equation:
$\displaystyle iu_{t}=-u_{xx}\pm|u|^{2}u;$ (5)
where $u_{z}$ means $\frac{\partial u}{\partial z}$. For simplicity, the
equations are written for the dimensionless function u depending on the
dimensionless time and space variables.
There are many other nonlinear equations (i.e. the Boussinesq equation) which
can be used for evaluating solitary waves, however, these three equations are
particularly important for physical applications. They exhibit the most famous
solitons: the KdV (pulse) solitons, the sine-Gordon (topological) solitons and
the envelope (or NLS) solitons. All the solitons are one-dimensional (or
quasi-one-dimensional). Figure .7 schematically shows these three types of
solitons. Let us summarize common features and individual differences of the
three most important solitons.
Figure 7: Schematic of the soliton solutions of: (a) the Korteweg-de Vries
equation; (b) the Sine-Gordon equation, and (c) the nonlinear schrodinger
equation.
A. The KdV solitons
The exact solution of the KdV equation is given by Eq. (7). The basic
properties of the KdV soliton, shown in Fig. 7a, can be summarized as follows
[7]:
i. Its amplitude increases with its velocity (and vice versa). Thus, they
cannot exist at rest.
ii. Its width is inversely proportional to the square root of its velocity.
iii. It is a unidirectional wave pulse, i.e. its velocity cannot be negative
for solutions of the KdV equation.
iv. The sign of the soliton solution depends on the sign of the nonlinear
coefficient in the KdV equation.
Figure 8: A schematic representation of a collision between two solitary
waves.
The KdV solitons are nontopological, and they exist in physical systems with
weakly nonlinear and with weakly dispersive waves. When a wave impulse breaks
up into several KdV solitons, they all move in the same direction (see, for
example, Fig. 3). The collision of two KdV solitons Fig .8, Under certain
conditions, the KdV solitons may be regarded as particles, obeying the
standard laws of Newton’s mechanics. In the presence of dissipative effects
(friction), the KdV solitons gradually decelerate and become smaller and
longer, thus, they are “mortal.”
B. The topological solitons
The basic properties of a topological (kink) soliton shown in Fig. 7b can be
summarized as follows [7]:
i. Its amplitude is independent of its velocity—it is constant and remains the
same for zero velocity, thus the kink may be static.
ii. Its width gets narrower as its velocity increases, owing to Lorentz
contraction.
iii. It has the properties of a relativistic particle.
iv. The topological kink which has a different screw sense is called an
antikink.
Topological solitons are extremely stable. Under the influence of friction,
these solitons only slow down and eventually stop and, at rest, they can live
“eternally.” In an infinite system, the topological soliton can only be
destroyed by moving a semi-infinite segment of the system above a potential
maximum. This would require an infinite energy. However, the topological
soliton can be annihilated in a collision between a soliton and an
antisoliton. In an integrable system having exact soliton solutions, solitons
and anti-solitons simply pass through each other with a phase shift, as all
solitons do, but in a real system like the pendulum chain which has some
dissipation of energy, the soliton-antisoliton equation may destroy the
nonlinear excitations. Figure .9 schematically shows a collision of a kink and
an antikink in an integrable system which has soliton solutions. In integrable
systems, the soliton-breather and breather-breather collisions are similar to
the kink-antikink collision shown in Fig. 9.
Figure 9: Sketch of a collision between a kink (K) and an antikink (AK).
The sine-Gordon equation has almost become ubiquitous in the theory of
condensed matter, since it is the simplest nonlinear wave equation in a
periodic medium.
C. The envelope solitons
The NLS equation is called the nonlinear Schr¨odinger equation because it is
formally similar to the Schr¨odinger equation of quantum mechanics
$\displaystyle(i\bar{h}\frac{\partial}{\partial
t}+\frac{\bar{h}^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}})\psi(x,t)=0$ (6)
where U is the potential, and $\psi(x,t)$ is the wave function. The NLS
equation describes self-focusing phenomena in nonlinear optics, one-
dimensional self-modulation of monochromatic waves, in nonlinear plasma etc.
In the NLS equation, the potential U is replaced by $|u|^{2}$ which brings
into the system self-interaction. The second term of the NLS equation is
responsible for the dispersion, and the third one for the nonlinearity. A
solution of the NLS equation is schematically shown in Fig. 7c. The shape of
the enveloping curve (the dashed line in Fig..7c) is given by
$\displaystyle u(x,t)=u_{0}\times sech((x-vt)/\ell)$ (7)
where $2\ell$ determines the width of the soliton. Its amplitude $u_{0}$
depends on $\ell$ , but the velocity $\nu$ is independent of the amplitude,
distinct from the KdV soliton. The shapes of the envelope and KdV solitons are
also different: the KdV soliton has a sech2 shape. Thus, the envelope soliton
has a slightly wider shape. However, other properties of the envelope solitons
are similar to the KdV solitons, thus, they are “mortal” and can be regarded
as particles. The interaction between two envelope solitons is similar to the
interactions between two KdV solitons (or two topological solitons).
In the envelope soliton, the stable groups have normally from 14 to 20 humps
under the envelope, the central one being the highest one. The groups with
more humps are unstable and break up into smaller ones. The waves inside the
envelope move with a velocity that differs from the velocity of the soliton,
thus, the envelope soliton has an internal dynamics. The relative motion of
the envelope and carrier wave is responsible for the internal dynamics of the
NLS soliton. The NLS equation is inseparable part of nonlinear optics where
the envelope solitons are usually called dark and bright solitons, and became
quasi-three-dimensional.
## References
* [1] J.S.Russell, Report on waves, in Rep. 14th Meet. British Assoc. Adv. Sci, 1844, P. 311.
* [2] A. Mourachkine, arXiv:cond-mat/0411452v1, 2004.
* [3] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, June 2007.
* [4] Nakahara and Mikio, Geometry, Topology and Physics, 2003.
* [5] Mermin, N. D. ”The topological theory of defects in ordered media”. Reviews of Modern Physics 51 (3), (1979).
* [6] N. N. Akhmediev; A. Ankiewicz, Solitons, non-linear pulses and beams. Springer(1997).
* [7] M. Remoissenet, waves called solitons(springer-verlag, Berlin, 1999).
|
arxiv-papers
| 2012-06-06T18:34:25 |
2024-09-04T02:49:31.562038
|
{
"license": "Public Domain",
"authors": "Yousef Yousefi and Khikmat Kh. Muminov",
"submitter": "Yousef Yousefi Dr",
"url": "https://arxiv.org/abs/1206.1294"
}
|
1206.1431
|
# Green functions and twist correlators for $N$ branes at angles.
Igor Pesando1
1Dipartimento di Fisica, Università di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
ipesando@to.infn.it
###### Abstract
We compute the Green functions and correlator functions for $N$ twist fields
for branes at angles on $T^{2}$ and we show that there are $N-2$ different
configurations labeled by an integer $M$ which is roughly associated with the
number of reflex angles of the configuration. In order to perform this
computation we use a $SL(2,\mathbb{R})$ invariant formulation and geometric
constraints instead of Pochammer contours. In particular the $M=1$ or $M=N-1$
amplitude can be expressed without using transcendental functions. We
determine the amplitudes normalization from $N\rightarrow N-1$ reduction
without using the factorization into the untwisted sector. Both the amplitudes
normalization and the OPE of two twist fields are unique (up to one constant)
when the $\epsilon\leftrightarrow 1-\epsilon$ symmetry is imposed. For
consistency we find also an infinite number of relations among Lauricella
hypergeometric functions.
keywords: D-branes, Conformal Field Theory
preprint: DFTT-6-2012
## 1 Introduction and conclusions
Since the beginning, D-branes have been very important in the formal
development of string theory as well as in attempts to apply string theory to
particle phenomenology and cosmology. However, the requirement of chirality in
any physically realistic model leads to a somewhat restricted number of
possible D-brane set-ups. An important class are intersecting brane models
where chiral fermions can arise at the intersection of two branes at angles.
An important issue for these models is the computation of Yukawa couplings and
flavour changing neutral currents.
Besides the previous computations many other computations often involve
correlators of twist fields and excited twist fields. It is therefore
important and interesting in its own to be able to compute these correlators.
As known in the literature [1] and explicitly shown in [2] for the case of
magnetized branes these computations boil down to the knowledge of the Green
function in presence of twist fields and of the correlators of the plain twist
fields.
In this technical paper we have analyzed the $N$ twist fields amplitudes at
tree level for open strings localized at $D$-branes intersections on $T^{2}$
using the classical path integral approach [1]. The subject has been explored
in many papers and in both the branes at angles setup and the magnetic branes
setup see for example ([12], [3], [4], [5], [6], [7], [8], [9]).
We have shown that there are different sectors with different amplitudes.
Sectors are labeled by an integer $M_{cw}$ ($1\leq M_{cw}\leq N-2$) and that
the number of sectors is equal to the number of reflex angles formed by the
brane configuration. This means that for example all the configurations in
fig. (1) have different amplitudes. In particular the quantum amplitudes with
$M_{cw}=1$ can be expressed using elementary functions only.
$a)$$b)$$c)$$d)$ Figure 1: The four different cases with $N=6$. $a)$
$M_{ccw}=2$ and $M_{cw}=4$ where $M_{ccw}$ is measured counterclockwise and
$M_{cw}$ clockwise. $b)$ $M_{ccw}=3$ and $M_{cw}=3$. $c)$ $M_{ccw}=4$ and
$M_{cw}=2$. $d)$ $M_{ccw}=5$ and $M_{cw}=1$.
This result generalizes the result previously obtained for both four point
amplitudes ([6], [7]) and for the $N$ point amplitudes [5] where only the
special case $M=N-2$ were considered. Since the $N=4$ $M=2$ amplitude has also
been obtained by a different approach in ([10], [11]). it would be interesting
to understand how this can come about in this different setup.
We have also obtained the normalizations (up to one constant) of both two
twist fields OPE and amplitudes. This result has been achieved using three
ingredients: the consistency of the $N$ twist fields correlator factorization
into $N-1$ twist fields one, the canonical normalization of the 2 twist
correlator
$\langle\sigma_{\epsilon}(x)\sigma_{1-\epsilon}(y)\rangle=1/(x-y)^{\epsilon(1-\epsilon)}$
and the assumption of the symmetry of under
$\sigma_{\epsilon}\leftrightarrow\sigma_{1-\epsilon}$.
Finally we have computed the Green functions in presence of $N$ twist fields
and we have shown that in order to do so there needs three different kinds of
derivatives instead of the usual two which are needed in the closed string
case.
This paper is organized as follows. In section 2 we review the geometrical
framework of branes at angles and we fix our conventions. In this section we
discuss carefully how to make use of the doubling trick in presence of
multiple cuts and the existence of local and global constraints. In section 3
we show the existence of $N-2$ different sectors and compute the corresponding
classical solutions. We show also explicitly the results for the $N=3$ and
$N=4$ cases. Moreover using the known relation between closed string and open
string amplitudes [13] we express the classical action as a sum of products of
holomorphic and antiholomorphic parts. Details on this computation are given
in appendix A. In section 4 we compute the Green functions for the different
sectors and give explicit expressions for $N=3$ and $N=4$ cases. In particular
we discuss the existence of infinite relations among polynomial of Lauricella
hypergeometric functions which must follow from the consistency of the
procedure. Finally in section 5 we compute the quantum correlators of $N$
twists and their normalization factors. In particular we show that the
$M_{cw}=1$ sector amplitudes can be expressed as a product of elementary
functions. Moreover we discuss how $N-1$ twist fields amplitudes can be
obtained from $N$ twist fields ones. A mathematically curious consequence is
that certain determinant of order $N-2$ involving Lauricella hypergeometric
functions of order $N-3$ are expressible as product of powers.
## 2 Review of branes at angles
The Euclidean action for the string configuration is given by
$S=\frac{1}{4\pi\alpha^{\prime}}\int
d\tau_{E}\int_{0}^{\pi}d\sigma~{}(\partial_{\alpha}X^{I})^{2}=\frac{1}{4\pi\alpha^{\prime}}\int_{H}d^{2}u~{}(\partial
X{\bar{\partial}}{\bar{X}}+{\bar{\partial}}X\partial{\bar{X}})$ (1)
where $u\in H$, the upper half plane,
$d^{2}u=e^{2\tau_{E}}d\tau_{E}d\sigma=\frac{du~{}d\bar{u}}{2i}$ and $I=1,2$ so
that $X=\frac{1}{\sqrt{2}}(X^{1}+iX^{2})$, ${\bar{X}}=X^{*}$. The complex
string coordinate is a map from the upper half plane to a closed polygon
$\Sigma$ in $\mathbb{C}$, i.e. $X:H\rightarrow\Sigma\subset\mathbb{C}$. For
example in fig. 2 we have pictured the interaction of $N=4$ branes at angles
$D_{i}$ with $i=1,\dots N$. The interaction between brane $D_{i}$ and
$D_{i+1}$ is at $f_{i}\in\mathbb{C}$ where we use the rule that index $i$ is
defined modulo $N$.
$D_{2}$$D_{1}$$D_{4}$$f_{4}$$f_{3}$$D_{3}$$f_{2}$$\Sigma$$f_{1}$$D_{1}$$D_{1}$$D_{4}$$D_{3}$$D_{2}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\tau_{4}$$\sigma=\pi$$\sigma=0$$X(\sigma,\tau)$
Figure 2: Map from the Minkowskian worldsheet to the target polygon $\Sigma$.
### 2.1 The local description
Locally at the interaction point $f_{i}$ the boundary conditions for the brane
$D_{i}$ are given by
$\displaystyle
Re(e^{-i\pi\alpha_{i}}X^{\prime}_{loc}|_{\sigma=0})=Im(e^{-i\pi\alpha_{i}}X_{loc}|_{\sigma=0})-g_{i}=0$
(2)
while those for the brane $D_{i+1}$ by
$\displaystyle
Re(e^{-i\pi\alpha_{i+1}}X^{\prime}_{loc}|_{\sigma=\pi})=Im(e^{-i\pi\alpha_{i+1}}X_{loc}|_{\sigma=\pi})-g_{i+1}=0$
(3)
with
$f_{i}=\frac{e^{i\pi\alpha_{i+1}}g_{i}-e^{i\pi\alpha_{i}}g_{i+1}}{\sin~{}\pi(\alpha_{i+1}-\alpha_{i})}$
(4)
When we write the Minkowskian string expansion as
$X(\sigma,\tau)=X_{L}(\tau+\sigma)+X_{R}(\tau-\sigma)$ the previous boundary
conditions imply (and not become since they are not completely equivalent
because of zero modes)
$\displaystyle
X^{\prime}_{L~{}loc}(\xi)=e^{i2\pi\alpha_{i}}X^{\prime}_{R~{}loc}(\xi),~{}~{}~{}~{}X^{\prime}_{L~{}loc}(\xi+\pi)=e^{i2\pi\alpha_{i+1}}X^{\prime}_{R~{}loc}(\xi-\pi)$
(5)
or in a more useful way in order to explicitly compute the mode expansion
$\displaystyle
X^{\prime}_{L~{}loc}(\xi+2\pi)=e^{i2\pi\epsilon_{i}}X^{\prime}_{L~{}loc}(\xi),~{}~{}~{}~{}X^{\prime}_{R~{}loc}(\xi+2\pi)=e^{-i2\pi\epsilon_{i}}X^{\prime}_{R~{}loc}(\xi)$
(6)
where we have defined
$\epsilon_{i}=\left\\{\begin{array}[]{c
c}(\alpha_{i+1}-\alpha_{i})&\alpha_{i+1}>\alpha_{i}\\\
1+(\alpha_{i+1}-\alpha_{i})&\alpha_{i+1}<\alpha_{i}\end{array}\right.$ (7)
so that $0<\epsilon_{i}<1$ and there is no ambiguity in the phase
$e^{i2\pi\epsilon_{i}}$ entering the boundary conditions. The quantity
$\pi\epsilon_{i}$ is the angle between the two branes $D_{i}$ and $D_{i+1}$
measured counterclockwise as shown in fig. 3.
$D_{i}$$D_{i+1}$$\pi\alpha_{i+1}$$\pi\alpha_{i}$$\pi\epsilon_{i}$$D_{i}$$D_{i+1}$$\pi\alpha_{i+1}$$\pi\alpha_{i}$$\pi\epsilon_{i}$
Figure 3: The connection between $\epsilon$ and the geometrical angles
$\alpha$s defining the branes.
A consequence of this definition is that $\epsilon$ becomes $1-\epsilon$ when
we flip the order of two branes. For example the angles in fig. 5 become those
in fig. 5 when we reverse the order we count the branes, i.e. when we follow
the boundary clockwise instead of counterclockwise the physics must obviously
not change.
$f_{4}$$f_{4}$$D_{4}$$D_{1}$$D_{2}$$f_{1}$$f_{2}$$D_{3}$$f_{3}$$\pi\epsilon_{1}$$\pi\epsilon_{2}$$\pi\epsilon_{3}$$\pi\epsilon_{4}$$\Sigma$
Figure 4: A polygon $\Sigma$ with an reflex angle and branes counted
counterclockwise with $N=4$ and $M_{ccw}=3$.
$f_{1}$$D_{2}$$D_{1}$$D_{4}$$f_{4}$$f_{3}$$D_{3}$$f_{2}$$\pi\epsilon_{4}$$\pi\epsilon_{3}$$\pi\epsilon_{2}$$\pi\epsilon_{1}$$\Sigma$
Figure 5: A polygon $\Sigma$ with an reflex angle and branes counted clockwise
with $N=4$ and $M_{cw}=1$.
We introduce as usual the Euclidean fields $X_{loc}(u,\bar{u})$,
$\bar{X}_{loc}(u,\bar{u})$ by a worldsheet Wick rotation in such a way they
are defined on the upper half plane by $u=e^{\tau_{E}+i\sigma}\in H$. The
previous choice of having brane $D_{i}$ at $\sigma=0$ (2) and brane $D_{i+1}$
at $\sigma=\pi$ (3) implies that in the local description where the
interaction point is at $x=0$ $D_{i}$ is mapped into $x>0$ and $D_{i+1}$ into
$x<0$. The boundary conditions (5) can then immediately be written as
$\displaystyle\partial X_{loc}(x+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{i}}\bar{\partial}\bar{X}_{loc}(x-i0^{+})~{}~{}0<x,$
$\displaystyle\partial X_{loc}(x+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{i+1}}\bar{\partial}\bar{X}_{loc}(x-i0^{+})~{}~{}~{}~{}x<0$
(8)
and similarly relations for ${\bar{X}}$ which can be obtained by complex
conjugation. When we add to the previous conditions the further constraints
$X(0,0)=f_{i},~{}~{}~{}~{}\bar{X}(0,0)=f_{i}^{*}$ (9)
we obtain a system of conditions which are equivalent to the original ones (2,
3).
In order to express the boundary conditions (6) in the Euclidean formulation
it is better to introduce the local fields defined on the whole complex plane
by the doubling trick as
$\displaystyle\partial{\cal X}_{loc}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial X_{loc}(u)&z=u\mbox{ with
}{Im~{}}z>0\mbox{ or }z\in\mathbb{R}^{+}\\\
e^{i2\pi\alpha_{i}}{\bar{\partial}}{\bar{X}}_{loc}(\bar{u})&z=\bar{u}\mbox{
with }{Im~{}}z<0\mbox{ or }z\in\mathbb{R}^{+}\end{array}\right.$ (12)
$\displaystyle\partial{\bar{\cal X}}_{loc}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial{\bar{X}}_{loc}(u)&z=u\mbox{
with }{Im~{}}z>0\mbox{ or }z\in\mathbb{R}^{+}\\\
e^{-i2\pi\alpha_{i}}{\bar{\partial}}X_{loc}(\bar{u})&z=\bar{u}\mbox{ with
}{Im~{}}z<0\mbox{ or }z\in\mathbb{R}^{+}\end{array}\right.$ (15)
In this way we can write eq.s (6) as
$\displaystyle\partial{\cal
X}_{loc}(e^{i2\pi}\delta)=e^{i2\pi\epsilon_{i}}\partial{\cal
X}_{loc}(\delta),~{}~{}~{}~{}\partial{\bar{\cal
X}}_{loc}(e^{i2\pi}\delta)=e^{-i2\pi\epsilon_{i}}\partial{\bar{\cal
X}}_{loc}(\delta)$ (16)
Notice that while the two Minkowskian boundary conditions (6) are one the
complex conjugate of the other the previous Euclidean ones are independent and
each is mapped into itself by complex conjugation therefore the Euclidean
classical solutions for ${\cal X}$ and ${\bar{\cal X}}$ are independent.
The quantization of the string with given boundary conditions yields
$\displaystyle X_{loc}(u,\bar{u})$ $\displaystyle=f_{i}$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{i}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(i)n}}{n+1-\epsilon_{i}}u^{-(n+1-\epsilon_{i})}-\frac{\alpha_{(i)n}^{\dagger}}{n+\epsilon_{i}}u^{n+\epsilon_{i}}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{i}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(i)n}^{\dagger}}{n+1-\epsilon_{i}}\bar{u}^{n+1-\epsilon_{i}}+\frac{\alpha_{(i)n}}{n+\epsilon_{i}}\bar{u}^{-(n+\epsilon_{i})}\right]$
$\displaystyle\bar{X}_{loc}(u,\bar{u})$ $\displaystyle=f_{i}^{*}$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{i}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(i)n}^{\dagger}}{n+1-\epsilon_{i}}u^{n+1-\epsilon_{i}}+\frac{\alpha_{(i)n}}{n+\epsilon_{i}}u^{-(n+\epsilon_{i})}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{i}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(i)n}}{n+1-\epsilon_{i}}\bar{u}^{-(n+1-\epsilon_{i})}-\frac{\alpha_{(i)n}^{\dagger}}{n+\epsilon_{i}}\bar{u}^{n+\epsilon_{i}}\right]$
(17)
with non trivial commutation relations ($n,m\geq 0$)
$[\alpha_{(i)n},\alpha_{(i)m}^{\dagger}]=(n+\epsilon_{i})\delta_{m,n},~{}~{}~{}~{}[\bar{\alpha}_{(i)n},\bar{\alpha}_{(i)m}^{\dagger}]=(n+1-\epsilon_{i})\delta_{m,n}$
(18)
and vacuum defined in the usual way by
$\alpha_{(i)n}|T_{i}\rangle=\bar{\alpha}_{(i)n}|T_{i}\rangle=0~{}~{}~{}~{}n\geq
0$ (19)
The vacuum is then generated from the twist operator
$\sigma_{\epsilon_{i},f_{i}}$ which depends both on the twist $\epsilon_{i}$
and on the position $f_{i}\in\mathbb{C}$. The dependence on the twist
$\epsilon_{i}$ can be read f.x. from the OPEs
$\displaystyle\partial X(u)\sigma_{\epsilon_{i},f_{i}}(x)$
$\displaystyle\sim(u-x)^{\epsilon_{i}-1}(\partial
X\sigma_{\epsilon_{i},f_{i}})(x)$
$\displaystyle\partial{\bar{X}}(u)\sigma_{\epsilon_{i},f_{i}}(x)$
$\displaystyle\sim(u-x)^{-\epsilon_{i}}(\partial{\bar{X}}\sigma_{\epsilon_{i},f_{i}})(x)$
(20)
which can be deduced from the local computations
$\displaystyle\partial X_{loc}(u)|T_{i}\rangle\sim
u^{\epsilon_{i}-1}~{}(-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{i-1}}\alpha_{(i)0}^{\dagger}|T_{i}\rangle),~{}~{}~{}~{}\partial{\bar{X}}_{loc}(u)|T_{i}\rangle\sim
u^{-\epsilon_{i}}~{}(-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{i-1}}\bar{\alpha}_{(i)0}^{\dagger}|T_{i}\rangle)$
(21)
On the other side the dependence on $f_{i}$ can be read from the OPE
$e^{ik\cdot
X(z,\bar{z})}\sigma_{\epsilon_{i},f_{i}}(x)\sim|z|^{-\alpha^{\prime}k^{2}}e^{-\frac{1}{2}R^{2}(\epsilon_{i})\alpha^{\prime}k^{2}}e^{ik\cdot
f_{i}}\sigma_{\epsilon_{i},f_{i}}(x)$ (22)
which can be deduced from the local computation
$|z|^{-\alpha^{\prime}k^{2}}e^{-\frac{1}{2}R^{2}(\epsilon_{i})\alpha^{\prime}k^{2}}:e^{ik\cdot
X_{loc}(z,\bar{z})}:|T_{i}\rangle\sim|z|^{-\alpha^{\prime}k^{2}}e^{ik\cdot
f_{i}}e^{-\frac{1}{2}R^{2}(\epsilon_{i})\alpha^{\prime}k^{2}}|T_{i}\rangle$
(23)
upon the identification [15] $e^{ik\cdot
X(z,\bar{z})}\leftrightarrow|z|^{-\alpha^{\prime}k^{2}}e^{-\frac{1}{2}R^{2}(\epsilon_{i})\alpha^{\prime}k^{2}}:e^{ik\cdot
X_{loc}(z,\bar{z})}:$ with
$R^{2}(\epsilon_{i})=2\psi(1)-\psi(\epsilon_{i})-\psi(1-\epsilon_{i})$,
$\psi(z)=\frac{d\ln\Gamma(z)}{dz}$ being the digamma function. Notice that
there is no obvious way of computing the angles $\alpha_{i}$ and
$\alpha_{i+1}$ from OPEs.
### 2.2 Global description
In the local description, where the interaction point is at $x=0$, $D_{i}$ is
mapped into $x>0$ and $D_{i+1}$ into $x<0$ this means that in the global
description the world sheet interaction points are mapped on the boundary of
the upper half plane so that $x_{i+1}<x_{i}$. The global equivalent of the
local boundary conditions eq.s (8) become
$\displaystyle\partial X_{L}(x+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{i}}\bar{\partial}\bar{X}_{R}(x-i0^{+})~{}~{}~{}~{}x_{i}<x<x_{i-1}$
$\displaystyle\partial{\bar{X}}_{L}(x+i0^{+})$
$\displaystyle=e^{-i2\pi\alpha_{i}}\bar{\partial}X_{R}(x-i0^{+})~{}~{}~{}~{}x_{i}<x<x_{i-1}$
(24)
To the previous constraints one must also add
$X_{loc}(x_{i},\bar{x}_{i})=f_{i},~{}~{}~{}~{}\bar{X}_{loc}(x_{i},\bar{x}_{i})=f_{i}^{*}$
(25)
in order to get a system of boundary conditions equivalent to the original
ones (2, 3). When we introduce the global fields defined on the whole complex
plane by the doubling trick as111 It is also possible to perform the doubling
trick by defining $\displaystyle\partial{\cal X}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial X(u)&z=u\mbox{ with
}{Im~{}}z>0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\\\
e^{i2\pi\alpha_{1}}{\bar{\partial}}{\bar{X}}(\bar{u})&z=\bar{u}\mbox{ with
}{Im~{}}z<0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\end{array}\right.$ and
similarly for $\partial{\bar{\cal X}}(z)$ but then all the formulae require a
cyclic permutation of the indexes as $2\rightarrow 1\rightarrow N\rightarrow
1$ so that the anharmonic ratio (43) becomes
$\omega_{z}=\frac{(z-x_{2})(x_{1}-x_{N})}{(z-x_{N})(x_{1}-x_{2})}$. This is
not a cyclic permutation for all indexes, i.e it is not $i\rightarrow i-1$ and
hence and all the $x_{j\neq 1,2,N}$ are mapped to $\omega_{j}<0$, nevertheless
$\sum_{j=2}^{N}\equiv\sum_{j\neq 1}\rightarrow\sum_{j\neq
N}\equiv\sum_{j=1}^{N-1}$ where in order to perform the change of indexes we
have rewritten $\sum_{j=2}^{N}$ as $\sum_{j\neq 1}$ and similarly for the
product $\prod_{j=2}^{N}\rightarrow\prod_{j=1}^{N-1}$. There is also a third
possibility and amounts to a cyclic permutation $2\rightarrow 1\rightarrow
N\rightarrow N-1\rightarrow...\rightarrow 2$.
$\displaystyle\partial{\cal X}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial X(u)&z=u\mbox{ with
}{Im~{}}z>0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{2}]-[x_{1},\infty]\\\
e^{i2\pi\alpha_{2}}{\bar{\partial}}{\bar{X}}(\bar{u})&z=\bar{u}\mbox{ with
}{Im~{}}z<0\mbox{ or
}z\in\mathbb{R}-[x_{N},x_{2}]-[x_{1},\infty]\end{array}\right.$ (28)
$\displaystyle\partial{\bar{\cal X}}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial{\bar{X}}(u)&z=u\mbox{ with
}{Im~{}}z>0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{2}]-[x_{1},\infty]\\\
e^{-i2\pi\alpha_{2}}{\bar{\partial}}X(\bar{u})&z=\bar{u}\mbox{ with
}{Im~{}}z<0\mbox{ or
}z\in\mathbb{R}-[x_{N},x_{2}]-[x_{1},\infty]\end{array}\right.$ (31)
the local boundary conditions (6) can be written in the global formulation as
$\displaystyle\partial{\cal X}(x_{i}+e^{i2\pi}\delta)$
$\displaystyle=e^{i2\pi\epsilon_{i}}\partial{\cal X}(x_{i}+\delta)$
$\displaystyle\partial{\bar{\cal X}}(x_{i}+e^{i2\pi}\delta)$
$\displaystyle=e^{-i2\pi\epsilon_{i}}\partial{\bar{\cal X}}(x_{i}+\delta).$
(32)
For the proper definition of the global constraints which follow from eq.s
(25), for example when dealing with the derivatives of the Green functions as
in section 4 it is worth noticing the behavior of the previously introduced
fields under complex conjugation when $z$ is restricted to
$z\in\mathbb{C}-[-\infty,x_{2}]-[x_{1},\infty]$
$\displaystyle[\partial{\cal X}(z)]^{*}$
$\displaystyle=e^{-i2\pi\alpha_{2}}\partial{\cal
X}(z\rightarrow\bar{z})={\bar{\partial}}{\bar{\cal
X}}(\bar{z})=\left\\{\begin{array}[]{c
c}{\bar{\partial}}{\bar{X}}({\bar{u}})&{\bar{z}}={\bar{u}}\\\
e^{-i2\pi\alpha_{2}}\partial X(u)&{\bar{z}}=u\end{array}\right.$ (35)
$\displaystyle[\partial{\bar{\cal X}}(z)]^{*}$
$\displaystyle=e^{-i2\pi\alpha_{2}}\partial{\bar{\cal
X}}(z\rightarrow\bar{z})={\bar{\partial}}{\cal
X}(\bar{z})=\left\\{\begin{array}[]{c
c}{\bar{\partial}}X({\bar{u}})&{\bar{z}}={\bar{u}}\\\
e^{i2\pi\alpha_{2}}\partial{\bar{X}}(u)&{\bar{z}}=u\end{array}\right.$ (38)
where $\partial{\cal X}(z\rightarrow\bar{z})$ means that the holomorphic
$\partial{\cal X}(z)$ is evaluated at ${\bar{z}}$. The previous expressions
also show that it is not necessary to introduce the antiholomorphic fields
${\bar{\partial}}{\cal X}(\bar{z})$ and ${\bar{\partial}}{\bar{\cal
X}}(\bar{z})$ which it is possible to construct applying the doubling trick on
${\bar{\partial}}X(\bar{u})$ and ${\bar{\partial}}{\bar{X}}(\bar{u})$
respectively.
## 3 The path integral approach
Following the by now classic method [1] we compute twists correlators by the
path integral
$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle=\int_{{\cal
M}(\\{x_{i},\epsilon_{i},f_{i}\\})}{\cal D}Xe^{-S_{E}}$ (39)
where ${\cal M}(\\{x_{i},\epsilon_{i},f_{i}\\})$ is the space of string
configurations satisfying the boundary conditions (24) and (25). Since the
integral is quadratic we can then efficiently separate the classical fields
from the quantum fluctuations as
$X(u,\bar{u})=X_{cl}(u,\bar{u})+X_{q}(u,\bar{u})$ (40)
where $X_{cl}$ satisfies the previous boundary conditions while $X_{q}$
satisfies the same boundary conditions but with all $f_{i}=0$. After this
splitting we obtain
$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle={\cal
N}(x_{i},\epsilon_{i})e^{-S_{E,cl}(x_{i},\epsilon_{i},f_{i})}$ (41)
The explicit expressions for $X_{cl}$ and $\bar{X}_{cl}$ given in eq.s (47,
48) show that they vanish when $f_{i}=0$ hence also the classical action
evaluated for $f_{i}=0$ $S_{E,cl}(x_{i},\epsilon_{i},f_{i}=0)$ is zero.
Actually because of translational invariance what said before works even when
all $f_{i}$ are equal, i.e. when $f_{i}=f$ and therefore we can identify
${\cal
N}(x_{i},\epsilon_{i})=\langle\sigma_{\epsilon_{1},f_{1}=f}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}=f}(x_{N})\rangle$
(42)
Our strategy is therefore first to compute the classical contribution in the
rest of this section and then compute the quantum contribution in section 5.
### 3.1 The classical solution
We want now to write the general solution for $\partial{\cal X}$ and
$\partial{\bar{\cal X}}$ in a way that the $SL(2,\mathbb{R})$ symmetry is
manifest. To this purpose we introduce the anharmonic ratio
$\omega_{z}=\frac{(z-x_{N})(x_{2}-x_{1})}{(z-x_{1})(x_{2}-x_{N})}$ (43)
and the corresponding ones $\omega_{j}$ where $z$ has been replaced with
$x_{j}$. In particular we get $\omega_{N}=0$, $\omega_{2}=1$ and
$\omega_{1}=-\infty$. The choice of $\omega_{1}=-\infty$ is dictated by the
request that powers are defined as
$(\omega-\omega_{i})^{\epsilon}=|\omega-\omega_{i}|^{\epsilon}e^{i\phi\epsilon}$
where $\phi=arg(\omega-\omega_{i})$ is counted from the real axis with range
$(-\pi,\pi)$ so that all cuts must be towards $-\infty$, as it is shown in
fig. (6)
$\omega_{2}$$\omega_{N}$$\omega_{3}$$\omega_{1}$ Figure 6: Cuts and
prevertexes positions in the $\omega$ plane.
We can now write the general solutions as
$\displaystyle\partial{\cal X}(z)$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}~{}\sum_{n=0}^{N-M-2}a_{n}(\omega_{j})\partial_{\omega}{\cal
X}^{(n)}(\omega_{z})$ $\displaystyle\partial{\bar{\cal X}}(z)$
$\displaystyle=e^{-i2\pi\alpha_{2}}\frac{\partial\omega_{z}}{\partial
z}~{}\sum_{r=0}^{M-2}b_{r}(\omega_{j})\partial_{\omega}{\bar{\cal
X}}^{(r)}(\omega_{z})$ (44)
where we have defined the basis
$\displaystyle\partial_{\omega}{\cal X}^{(n)}(\omega_{z})$
$\displaystyle=~{}\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-(1-\epsilon_{j})}~{}\omega_{z}^{n},~{}~{}~{}~{}0\leq
n\leq N-M-2$ $\displaystyle\partial_{\omega}{\bar{\cal X}}^{(r)}(\omega_{z})$
$\displaystyle=~{}\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-\epsilon_{j}}~{}\omega_{z}^{r},~{}~{}~{}~{}0\leq
r\leq M-2$ (45)
and we have also defined the integer
$M=\sum_{i=1}^{N}\epsilon_{i}$ (46)
When the target polygon $\Sigma$ is followed counterclockwise this integer $M$
is equal to the number of reflex angles plus 2 since every acute angle
internal the target polygon is $\pi-\pi\epsilon$ while every reflex one is
$2\pi-\pi\epsilon$ as shown in fig. (7). In a similar way when the target
polygon is followed clockwise $M$ is the number of acute angles minus 2.
$D_{i}$$D_{i+1}$$\pi\epsilon_{i}$$D_{i}$$D_{i+1}$$\pi\epsilon_{i}$ Figure 7:
When the target polygon $\Sigma$ (the shaded area) is followed
counterclockwise keeping the interior on the left side an internal acute angle
is equal to $\pi-\pi\epsilon$ while an reflex one to $2\pi-\pi\epsilon$.
Nevertheless it is important to notice how polygons having $M$ and
$M^{\prime}=N-M$ both measured counterclockwise or clockwise are not the same
polygons as it shown in fig. (1) in the case $N=6$. To distinguish between
these two cases it is necessary to compare expression (7) with the phases
$\alpha_{i}$ as derived from the geometrical relations $f_{i+1}-f_{i}=\pm
e^{i\pi\alpha_{i+1}}|f_{i+1}-f_{i}|$ (where the sign depends on the case) as
shown in fig. (8)
$D_{i+2}$$D_{i+1}$$\pi\alpha_{i+1}$$\pi\alpha_{i+2}$$f_{i+1}$$f_{i}$$D_{i}$
Figure 8: The connection between $f_{i+1}-f_{i}$ and the geometrical angle
$\alpha_{i+1}$ defining the brane.
Also when changing the $f$s while keeping fixed the $\epsilon$s the shape may
change as shown in fig. (9) for $N=4$ and $M_{ccw}=2$ and in fig. (10) for
$N=4$ and $M_{ccw}=3$. From now on we measure $M$ clockwise in not otherwise
stated. Since the number of reflex angles must be less or equal than $N-3$ we
deduce that $2\leq M_{ccw}\leq N-1$ or $1\leq M_{cw}\leq N-2$ and hence there
are $N-2$ different sectors222 The symmetry
$[X_{cl}(u,\bar{u};\\{1-\epsilon\\},\\{f^{*}\\})]^{*}=X_{cl}(u,\bar{u};\\{\epsilon\\},\\{f\\})$
maps $M_{ccw}$ into $M_{cw}=N-M_{ccw}$ because it is like the map
$X\rightarrow X^{*}$ which reverses the order in which a circuit is followed.
Hence it does not map $M_{ccw}$ into $M_{ccw}^{\prime}=N-M_{ccw}$. .
Figure 9: The four different cases with $N=4$ and $M_{ccw}=2$ and $M_{cw}=2$
which can be obtained moving the brane whose intersection points are the empty
circles. Figure 10: The four different cases with $N=4$ and $M_{ccw}=3$ and
$M_{cw}=1$ which can be obtained moving whose intersection points are the
empty circles.
In the previous expressions (44) $\frac{\partial\omega_{z}}{\partial z}$
ensures the proper transformation under $SL(2,\mathbb{R})$ and the product
$\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-(1-\epsilon_{j})}$ and the
corresponding one for $\partial{\bar{\cal X}}$ yields the proper monodromies
around all the point, $x_{1}$ included. The extrema of the summations, i.e.
the maximum allowed values of $n$ and $r$ are chosen in order to have a finite
action and in particular their values are determined by the analysis of the
behavior of the solutions around $z=x_{1}$ and not around $z=\infty$ as one
would naively expect. This happens because the solutions (44) behave as
$O\left(\frac{1}{z^{2}}\right)$ at $z=\infty$ because of the factor
$\frac{\partial\omega_{z}}{\partial z}$.
The powers of the products
$\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-(1-\epsilon_{j})}$ and
$\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-\epsilon_{j}}$ are chosen in order
to get a finite $X_{cl}(u,\bar{u})$ at the singular points, explicitly using
the definitions (31) and the expressions (44) we can write333 Because of way
we have chosen the cuts we have
$[(\omega_{z}-\omega_{j})^{\alpha}]^{*}=(\omega_{\bar{z}}-\omega_{j})^{\alpha}$
when $\omega_{j}$ is real.
$\displaystyle X_{cl}(u,\bar{u})$ $\displaystyle=f_{1}$
$\displaystyle+\sum_{n=0}^{N-M-2}a_{n}(\omega_{j})\int_{x_{1};z\in
H}^{u}dz~{}\frac{\partial\omega_{z}}{\partial
z}~{}\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-(1-\epsilon_{j})}\omega_{z}^{n}$
$\displaystyle+\sum_{r=0}^{M-2}b_{r}(\omega_{j})\int_{\bar{x}_{1};z\in
H^{-}}^{\bar{u}}dz~{}\frac{\partial\omega_{z}}{\partial
z}~{}\prod_{j=2}^{N}(\omega_{z}-\omega_{j})^{-\epsilon_{j}}\omega_{z}^{r}$
$\displaystyle=f_{1}$
$\displaystyle+\sum_{n=0}^{N-M-2}a_{n}(\omega_{j})\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}~{}\prod_{j=2}^{N}(\omega-\omega_{j})^{-(1-\epsilon_{j})}\omega^{n}$
$\displaystyle+\sum_{r=0}^{M-2}b_{r}(\omega_{j})\left[\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}~{}\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}\omega^{r}\right]^{*}$
(47)
where in the last step we have used the explicit definition of the power to
connect the integral performed in lower half plane with that performed in the
upper half plane. In a similar way we can write
$\displaystyle\bar{X}_{cl}(u,\bar{u})$ $\displaystyle=f_{1}^{*}$
$\displaystyle+e^{-i2\pi\alpha_{2}}\sum_{n=0}^{N-M-2}a_{n}(\omega_{j})\left[\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}~{}\prod_{j=2}^{N}(\omega-\omega_{j})^{-(1-\epsilon_{j})}\omega^{n}\right]^{*}$
$\displaystyle+e^{-i2\pi\alpha_{2}}\sum_{r=0}^{M-2}b_{r}(\omega_{j})\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}~{}\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}\omega^{r}$
(48)
where the coefficients are again $a$ and $b$ and not $a^{*}$ and $b^{*}$ as
naively expected because we computed both $X_{cl}$ and $\bar{X}_{cl}$ using
the definitions of $\partial{\cal X}$ and $\partial{\bar{\cal X}}$ (31) which
mix both $\partial X$ and ${\bar{\partial}}{\bar{X}}$ . On the other side
$\bar{X}_{cl}=(X_{cl})^{*}$ hence there are constraints on the coefficients
$a$ and $b$, i.e. $a^{*}=e^{-i2\pi\alpha_{2}}a$ and similarly for $b$ but
these constraints are precisely the ones needed to solve the equations (49)
when one takes into account the geometrical requirements that
$f_{i+1}-f_{i}=e^{i\pi\alpha_{i+1}}|f_{i+1}-f_{i}|$ as shown in fig. (8).
In order to determine the $N-M-1$ functions $a(\omega_{j})$ ($j\neq 1,2,N$)
and the $M-1$ $b(\omega_{j})$ we need simply to impose the $N-2$ geometrical
constraints
$X_{cl}(x_{i+1},\bar{x}_{i+1})-X_{cl}(x_{i},\bar{x}_{i})=f_{i+1}-f_{i}~{}~{}~{}~{}i=2,\dots
N-1$ (49)
There is actually one more equation one can obviously impose, the one with
$i=1$ but it turns out to be linearly dependent on the previous ones when the
geometrical constraints on $f$ and $\epsilon$ are imposed. It is worth
noticing that the previous constraints have an obvious geometrical meaning
differently from the the use of Pochammer path used in the literature. The
explicit solution of the previous constraints is given by solving the linear
system of $N-2$ equations444 The net effect of using the real valued integrals
$I^{(N)}_{i,n}(\epsilon)$ is simply the phase
$e^{+i\pi\sum_{j=2}^{i}\epsilon_{j}}$ which multiply the real integral. In
particular for $I^{(N)}_{i,n}(1-\epsilon)$ we get
$(-1)^{i-1}e^{-i\pi\sum_{j=2}^{i}\epsilon_{j}}$ and this explains the
alternating sign which appears weird at first sight.
$\displaystyle\sum_{n=0}^{N-M-2}(-)^{i-1}I^{(N)}_{i,n}(1-\epsilon_{j})a_{n}+\sum_{r=0}^{M-2}I^{(N)}_{i,r}(\epsilon_{j})b_{r}=e^{-i\pi\sum_{j=2}^{i}\epsilon_{j}}(f_{i}-f_{i+1})~{}~{}~{}~{}i=2,\dots
N-1$ (50)
where we have introduced the real valued integrals555 All these integrals can
be expressed using $I^{(N)}_{i,0}$ since $\omega_{N}=0$, explicitly
$I^{(N)}_{i,n}(\alpha_{j})=I^{(N)}_{i,0}(\alpha_{j}-\delta_{j,N}n)$ but we
have introduced this redundancy for notational simplicity. Moreover we have
used $\omega_{N+1}=\omega_{1}=-\infty$ because indexes are defined $mod~{}N$.
$\displaystyle I^{(N)}_{i,n}(\alpha_{j})$
$\displaystyle=\int_{\omega_{i+1}}^{\omega_{i}}d\omega~{}~{}\prod_{j=2}^{N}|\omega-\omega_{j}|^{-\alpha_{j}}\omega^{n}$
(51)
which are connected to type D Lauricella generalized hypergeometric function
when $i,i+1\neq 1$ by
$\displaystyle I^{(N)}_{i,0}(\alpha_{j})$ $\displaystyle=\prod_{j\neq
1,i,i+1}^{N}|\omega_{j}-\omega_{i+1}|^{-\alpha_{j}}~{}|\omega_{i}-\omega_{i+1}|^{1-\alpha_{i+1}}$
$\displaystyle~{}~{}~{}~{}\cdot\int_{0}^{1}dt~{}t^{-\alpha_{i+1}}~{}(1-t)^{-\alpha_{i}}\prod_{j=2}^{i-1}\left(1-\frac{\omega_{i}-\omega_{i+1}}{\omega_{j}-\omega_{i+1}}t\right)^{-\alpha_{j}}~{}\prod_{j=i+2}^{N}\left(1-\frac{\omega_{i}-\omega_{i+1}}{\omega_{i+1}-\omega_{j}}t\right)^{-\alpha_{j}}$
$\displaystyle=\prod_{j\neq
1,i,i+1}^{N}|\omega_{j}-\omega_{i+1}|^{-\alpha_{j}}~{}|\omega_{i}-\omega_{i+1}|^{1-\alpha_{i+1}-\alpha_{i}}\frac{1}{B(1-\alpha_{i},~{}1-\alpha_{i+1})}$
$\displaystyle~{}~{}~{}~{}\cdot
F^{(N-3)}_{D}(1-\alpha_{i+1};~{}\\{\alpha_{j\neq
1,i,i+1}\\};~{}2-\alpha_{i}-\alpha_{i+1};\xi_{a})$ (52)
where the parameters $\xi_{a}$ ($a=1,\dots N-3$) are given by
$\displaystyle\xi_{a}$ $\displaystyle=\left\\{\begin{array}[]{c
c}\frac{\omega_{i}-\omega_{i+1}}{\omega_{a+1}-\omega_{i+1}}&1\leq a\leq i-2\\\
\frac{\omega_{i}-\omega_{i+1}}{\omega_{a+3}-\omega_{i+1}}&i-1\leq a\leq
N-3\end{array}\right.$ (55)
In particular for $N=3$ $F^{(0)}_{D}$ is Euler Beta function $B$ and for $N=4$
$F^{(1)}_{D}$ is a plain hypergeometric ${}_{2}F_{1}$, explicitly the previous
expressions become
$\displaystyle I^{(3)}_{2,n}(1-\epsilon_{j})$
$\displaystyle=B(\epsilon_{2},\epsilon_{3}+n)$ $\displaystyle
I^{(4)}_{2,n}(1-\epsilon_{j})$
$\displaystyle=\omega_{3}^{\epsilon_{4}-1+n}~{}(1-\omega_{3})^{\epsilon_{2}+\epsilon_{3}-1}~{}\frac{1}{B(\epsilon_{3},\epsilon_{2})}~{}_{2}F_{1}(\epsilon_{3};1-n-\epsilon_{4};\epsilon_{2}+\epsilon_{3};\frac{\omega_{3}-1}{\omega_{3}})$
$\displaystyle I^{(4)}_{3,n}(1-\epsilon_{j})$
$\displaystyle=\omega_{3}^{\epsilon_{3}+\epsilon_{4}+n-1}~{}\frac{1}{B(\epsilon_{4}+n,\epsilon_{3})}~{}_{2}F_{1}(1-\epsilon_{2};n+\epsilon_{4};\epsilon_{3}+\epsilon_{4}+n;\omega_{3})$
(56)
We are now ready to compute the classical action for our solution. Using the
explicit expression for $\partial{\cal X}$ and ${\bar{\partial}}{\bar{\cal
X}}$ we can write 666$e^{-i\pi\alpha_{2}}a_{n}$ is real as discussed before
but it is by no means assured that it is positive.
$\displaystyle S_{cl}=$
$\displaystyle\frac{1}{8\pi\alpha^{\prime}}\Big{[}\sum_{n,m=0}^{N-M-2}(e^{-i\pi\alpha_{2}}a_{n})~{}(e^{-i\pi\alpha_{2}}a_{m})\int_{\mathbb{C}}d^{2}\omega~{}\prod_{j=2}^{N}|\omega-\omega_{j}|^{-2(1-\epsilon_{j})}~{}\omega^{n}\bar{\omega}^{m}$
$\displaystyle+\sum_{r,s=0}^{M-2}(e^{-i\pi\alpha_{2}}b_{r})(e^{-i\pi\alpha_{2}}b_{s})\int_{\mathbb{C}}d^{2}\omega~{}\prod_{j=2}^{N}|\omega-\omega_{j}|^{-2\epsilon_{j}}~{}\omega^{r}\bar{\omega}^{s}\Big{]}$
(57)
where an overall factor $\frac{1}{2}$ appears because we have extended the
integration domain from the upper half plane to the whole complex plane.
Notice that $e^{-i\pi\alpha_{2}}a,e^{-i\pi\alpha_{2}}b\in\mathbb{R}$ which is
however not enough to use their moduli $|a|$ and $|b|$ in the previous
expression. As explained in appendix A using the technique developed in [13]
the previous integrals can be expressed as a product of holomorphic and
antiholomorphic contour integrals as
$\displaystyle\int_{\mathbb{C}}d^{2}\omega~{}\prod_{j=2}^{N}|\omega-\omega_{j}|^{-2\epsilon_{j}}~{}\omega^{n}\bar{\omega}^{m}=$
$\displaystyle~{}~{}=\sum_{i=2}^{N-1}\sum_{l=i+1}^{N}\sin\left(\pi\sum_{j=i+1}^{l}\epsilon_{j}\right)I^{(N)}_{i,n}(\epsilon)I^{(N)}_{l,m}(\epsilon)$
(58)
### 3.2 The explicit $N=3$, $M_{cw}=1$ ($M_{ccw}=2$) case
Let us start examining the $M_{cw}=1$ computation. In this case we see
immediately that $\partial{\bar{\cal X}}$ is identically zero and that the
only unknown is $a_{0}$ which is not a function but simply a constant. The
eq.s (50) reduce simply to
$-a_{0}e^{i\pi\epsilon_{2}}B(\epsilon_{2},\epsilon_{3})=f_{2}-f_{3}$ (59)
because $I^{(3)}_{2,n}(\alpha_{j})=B(1-\alpha_{2},1+n-\alpha_{3})$ where
$B(\epsilon_{2},\epsilon_{3})$ is Euler beta function. The complete solution
is then
$X_{cl}^{(3,1_{cw})}(u,\bar{u})=f_{3}+\frac{e^{i\pi(1-\epsilon_{2})}(f_{2}-f_{3})}{B(1,\epsilon_{3})B(\epsilon_{2},\epsilon_{3})}~{}_{2}F_{1}(\epsilon_{3},~{}1-\epsilon_{2};~{}\epsilon_{3}+1;~{}\omega_{u})~{}\omega_{u}^{\epsilon_{3}}$
(60)
Consider now the $M_{ccw}=2$ case where only $b_{0}$ is different from zero
and therefore $\partial{\cal X}=0$. Proceeding as before we get
$b_{0}e^{i\pi\epsilon_{2}}B(1-\epsilon_{2},1-\epsilon_{3})=f_{2}-f_{3}$ (61)
from which follows
$X_{cl}^{(3,2_{ccw})}(u,\bar{u})=f_{3}+\frac{e^{-i\pi\epsilon_{2}}(f_{2}-f_{3})}{B(1,1-\epsilon_{3})B(1-\epsilon_{2},1-\epsilon_{3})}~{}\left[{}_{2}F_{1}(1-\epsilon_{3},~{}\epsilon_{2};~{}2-\epsilon_{3};~{}\omega_{u})~{}\omega_{u}^{1-\epsilon_{3}}\right]^{*}$
(62)
which explicitly shows the equivalence
$[X_{cl}^{(N,(N-M)_{cw})}(u,\bar{u};\\{1-\epsilon\\},\\{f^{*}\\})]^{*}=X_{cl}^{(N,M_{ccw})}(u,\bar{u};\\{\epsilon\\},\\{f\\})$.
### 3.3 The explicit $N=4$, $M_{cw}=1$ ($M_{ccw}=3$) case
In this case we again immediately realize that $\partial{\bar{\cal X}}$ is
identically zero and that the only unknowns are the two functions
$a_{0}(\omega_{3})$ and $a_{1}(\omega_{3})$. The eq.s (50) reduce simply to
$\displaystyle\sum_{n=0}^{1}(-)^{i-1}I^{(4)}_{i,n}(1-\epsilon_{3})~{}a_{n}(\omega_{3})=e^{-i\pi\sum_{j=2}^{i}\epsilon_{j}}(f_{i}-f_{i+1})~{}~{}~{}~{}i=2,3$
(63)
with $I^{(4)}$s given explicitly in eq.s (56) The classical solution then
reads
$\displaystyle X_{cl}^{(4,1_{cw})}(u,\bar{u})$ $\displaystyle=f_{1}$
$\displaystyle+\sum_{n=0}^{1}a_{n}(\omega_{3})\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}(\omega-1)^{\epsilon_{2}-1}(\omega-\omega_{3})^{\epsilon_{3}-1}\omega^{n+\epsilon_{4}-1}$
(64)
The $X_{cl}^{(4,3_{ccw})}(u,\bar{u})$ solution can then be obtained as
$X_{cl}^{(4,3)}(u,\bar{u};\\{\epsilon\\},\\{f\\})=[X_{cl}^{(4,1)}(u,\bar{u};\\{1-\epsilon\\},\\{f^{*}\\})]^{*}$.
### 3.4 The explicit $N=4$, $M=2$ case
In this case $M$ can be understood either as $M_{ccw}$ or as $M_{cw}$ which of
the two can be only decided looking at the phases $\\{\alpha_{i}\\}$. The
unknowns are the two functions $a_{0}(\omega_{3})$ and $b_{0}(\omega_{3})$ and
eq.s (50) reduce simply to
$\displaystyle(-)^{i-1}I^{(4)}_{i,0}(1-\epsilon_{3})~{}a_{0}(\omega_{3})+I^{(0)}_{i,0}(\epsilon_{3})~{}b_{r}(\omega_{3})=e^{-i\pi\sum_{j=2}^{i}\epsilon_{j}}(f_{i}-f_{i+1})~{}~{}~{}~{}i=2,3$
(65)
hence the classical solution reads
$\displaystyle X_{cl}^{(4,2)}(u,\bar{u})$ $\displaystyle=f_{1}$
$\displaystyle+a_{0}(\omega_{3})\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}(\omega-1)^{\epsilon_{2}-1}(\omega-\omega_{3})^{\epsilon_{3}-1}\omega^{\epsilon_{4}-1}$
$\displaystyle+b_{0}(\omega_{3})\left[\int_{-\infty;\omega\in
H}^{\omega_{u}}d\omega~{}(\omega-1)^{-\epsilon_{2}}(\omega-\omega_{3})^{-\epsilon_{3}}\omega^{-\epsilon_{4}}\right]^{*}$
(66)
### 3.5 Wrapping contributions
The wrapping contributions have been studied in [16] for the N=3 case and in
[5] for the case $M=N-2$ and there is not any difference among the different
$M$ values therefore the results obtained there are valid. Let us anyhow
quickly review them. Given a minimal $N$-polygon in $T^{2}$ with vertexes
$\\{f_{i}\\}$, i.e. with all vertexes in the fundamental cell, we can consider
non minimal polygons which wrap the $T^{2}$. These can be easierly described
as polygons which have vertexes $\\{\tilde{f}_{i}\\}$ in the covering
$\mathbb{R}^{2}$ where $T^{2}\equiv R^{2}/\Lambda$ with the lattice defined as
$\Lambda=\\{n_{1}e_{1}+n_{2}e_{2}|n_{1},n_{2}\in\mathbb{Z}\\}$. These
configurations give an additive contribution to the classical path integral as
$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle_{T^{2}}={\cal
N}(x_{i},\epsilon_{i})\sum_{\\{\tilde{f}_{i}\\}}e^{-S_{E,cl}(x_{i},\epsilon_{i},\tilde{f}_{i})}$
(67)
In order to determine the possible vertexes $\\{\tilde{f}_{i}\\}$ without
redundancy it is necessary to keep a vertex fixed and then expand the polygon.
For definiteness we keep fixed the vertex $\tilde{f}_{1}=f_{1}$ which lies at
the intersection between $D_{N}$ and $D_{1}$. We then move the next vertex
$f_{2}$ along the $D_{1}$ brane. Explicitly we write
$\tilde{f}_{2}=\tilde{f}_{1}+(f_{2}-f_{1})+n_{1}t_{1}=f_{2}+n_{1}t_{1}$ with
$n_{1}\in\mathbb{Z}$ and $t_{1}$ the shortest tangent vector to $D_{1}$ which
is in $\Lambda$. We can now continue for all the other vertexes for which we
have
$\tilde{f}_{i}=\tilde{f}_{i-1}+(f_{i}-f_{i-1})+n_{i-1}t_{i-1}=f_{i}+\sum_{k=1}^{i-1}n_{k}t_{k}$.
For consistency we need requiring $\tilde{f}_{N+1}\equiv\tilde{f}_{1}=f_{1}$,
therefore the possible wrapped polygons are obtained from the solution of the
Diophantine equation
$\displaystyle\sum_{i=1}^{N}n_{i}t_{i}=0$ (68)
which cannot be solved in general terms but only on a case by case basis as
discussed in [5].
## 4 Green functions for $N\geq 3$
Having determined the classical solution we now compute the Green functions in
presence of twist fields both as an intermediate step toward the computation
of the quantum part of correlators and as a key ingredient to the computation
of excited twist fields correlators.
Following partially the literature we define the following quantities for the
quantum fluctuations which are connected with the derivatives of the Green
functions as
$\displaystyle g_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\langle\partial{\cal X}_{q}(z)\partial{\bar{\cal
X}}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
$\displaystyle h_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\langle\partial{\bar{\cal X}}_{q}(z)\partial{\bar{\cal
X}}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
$\displaystyle l_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\langle\partial{\cal X}_{q}(z)\partial{\cal
X}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
(69)
We do not need to consider functions involving antiholomorphic quantities
because ${\bar{\partial}}{\cal X}$ and ${\bar{\partial}}{\bar{\cal X}}$ are
related to $\partial{\cal X}$ and $\partial{\bar{\cal X}}$ as in eq.s (38).
Quantum fluctuations are required to satisfy the boundary conditions
$\displaystyle
Re(e^{-i\pi\alpha_{i}}\partial_{y}X_{q}|_{y=0})=Im(e^{-i\pi\alpha_{i}}X_{q}|_{y=0})=0~{}~{}~{}~{}x_{i+1}<x<x_{i}.$
(70)
These conditions that can reformulated as a set of local constraints
$\displaystyle\partial{\cal X}_{q}(x_{i}+e^{i2\pi}\delta)$
$\displaystyle=e^{i2\pi\epsilon_{i}}\partial{\cal
X}_{q}(x_{i}+\delta),~{}~{}~{}~{}\partial{\bar{\cal
X}}_{q}(x_{i}+e^{i2\pi}\delta)=e^{-i2\pi\epsilon_{i}}\partial{\bar{\cal
X}}_{q}(x_{i}+\delta)$ (71)
and as a set of global constraints
$\displaystyle X_{q}(x_{i},\bar{x}_{i})$
$\displaystyle=X_{q}(x_{i+1},\bar{x}_{i+1}),~{}~{}~{}~{}\bar{X}_{q}(x_{i},\bar{x}_{i})=\bar{X}_{q}(x_{i+1},\bar{x}_{i+1}),$
(72)
In the spirit of what done in the previous section we use a $SL(2,\mathbb{R})$
invariant formulation and we write
$\displaystyle g_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{1}{(z-w)^{2}}\prod_{j\neq
1}\frac{(\omega_{z}-\omega_{j})^{\epsilon_{j}-1}}{(\omega_{w}-\omega_{j})^{\epsilon_{j}}}\sum_{n=0}^{N-M}\sum_{s=0}^{M}a_{ns}(\omega_{j})\omega_{z}^{n}\omega_{w}^{s}$
$\displaystyle h_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=e^{-i2\pi\alpha_{2}}\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\sum_{r,s=0}^{M-2}b_{rs}(\omega_{j})~{}\partial_{\omega}{\bar{\cal
X}}^{(r)}(\omega_{z})~{}\partial_{\omega}{\bar{\cal X}}^{(s)}(\omega_{w})$
$\displaystyle l_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=e^{i2\pi\alpha_{2}}\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\sum_{n,m=0}^{N-M-2}c_{nm}(\omega_{j})~{}\partial_{\omega}{\cal
X}^{(n)}(\omega_{z})~{}\partial_{\omega}{\cal X}^{(m)}(\omega_{w})$ (73)
where $a_{ns}(\omega_{j})$, $b_{rs}(\omega_{j})$ and $c_{nm}(\omega_{j})$ are
unknown functions of the anharmonic ratios $\omega_{j\neq 1,2,N}$.
Let us rapidly review the ingredients of the previous construction. The
factors $\frac{1}{(z-w)^{2}}$, $\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial w}$ and
$\frac{\partial\omega_{z}}{\partial z}\frac{\partial\omega_{w}}{\partial w}$
are there to ensure the proper $SL(2,\mathbb{R})$ transformations. The powers
of the singular parts have been chosen in order to reproduce the singularities
of OPEs
$\displaystyle\partial{\cal X}(z)\partial{\bar{\cal X}}(w)$
$\displaystyle\sim\frac{1}{(z-w)^{2}}+O(1)$ (74) $\displaystyle\partial{\cal
X}(z)\sigma_{\epsilon,f}(x)$
$\displaystyle\sim(z-x)^{\epsilon-1}(\partial{\cal X}\sigma_{\epsilon,f})(x)$
$\displaystyle\partial{\bar{\cal X}}(z)\sigma_{\epsilon,f}(x)$
$\displaystyle\sim(z-x)^{-\epsilon}(\partial{\bar{\cal
X}}\sigma_{\epsilon,f})(x)$ (75)
where $(\partial{\cal X}\sigma_{\epsilon,f})(x)$ and $(\partial{\bar{\cal
X}}\sigma_{\epsilon,f})(x)$ are excited twists. In particular eq.s (71) are
the same of eq.s (75) as it should be since twist operators have been
introduced to generate (71) constraints.
The upper bounds of the summation ranges are fixed by request that
singularities $z\rightarrow x_{1}$ and $w\rightarrow x_{1}$ are not worse than
those in eq.s (75) while the lower bound is fixed from the $z\rightarrow
x_{N}$ and $w\rightarrow x_{N}$ limits.
There is another consistency condition: when $x_{i}\rightarrow x_{j}$ we must
obtain the corresponding Green function with $N\rightarrow N-1$. It is worth
stressing that at first sight there is a further constraint. Usually the OPE
between two twists is written as
$\displaystyle\sigma_{\epsilon_{i},f}(x_{i})~{}\sigma_{\epsilon_{j},f}(x_{j})\sim\left\\{\begin{array}[]{c
c}(x_{i}-x_{j})^{-\epsilon_{i}\epsilon_{j}}~{}{\cal
M}(\epsilon_{i},\epsilon_{j})~{}\sigma_{\epsilon_{i}+\epsilon_{j},f}(x_{j})&\epsilon_{i}+\epsilon_{j}<1\\\
(x_{i}-x_{j})^{-(1-\epsilon_{i})(1-\epsilon_{j})}~{}{\cal
N}(\epsilon_{i},\epsilon_{j})~{}\sigma_{\epsilon_{i}+\epsilon_{j}-1,f}&\epsilon_{i}+\epsilon_{j}>1\end{array}\right.$
(78)
with ${\cal M}(\epsilon_{i},\epsilon_{j})={\cal
N}(\epsilon_{i},\epsilon_{j})=1$. We will discuss that it is not possible to
set both ${\cal M}$ and ${\cal N}$ to one in section 5.3. Now we would however
comment on the fact that the previous expression is written without higher
terms leading to the wrong impression that all the omitted terms are
descendants. If it were true that the OPE (78) has no other primaries in the
rhs this would imply that the derivatives of Green functions are analytic
functions of the variables $x_{i}$ too since the overall singularity in
$x_{i}-x_{j}$ due to the power factor would cancel between the numerator and
the denominator. This is not true as the explicit computations show but in the
$M_{cw}=1$, $M_{ccw}=N-1$ case and the reason is that the previous OPE
involves actually an infinite number of primary fields with powers of OPE
coefficients which do not differ by integers, explicitly for
$\epsilon_{i}+\epsilon_{j}<1$
$\displaystyle\sigma_{\epsilon_{i},f}(x_{i})\sigma_{\epsilon_{j},f}(x_{j})\sim$
$\displaystyle(x_{i}-x_{j})^{\epsilon_{i}\epsilon_{j}}~{}{\cal
M}(\epsilon_{i},\epsilon_{j})~{}\sigma_{\epsilon_{i}+\epsilon_{j},f}(x_{j})$
$\displaystyle+\sum_{k=1}c_{k}~{}(x_{i}-x_{j})^{\epsilon_{i}\epsilon_{j}+k(\epsilon_{i}+\epsilon_{j})}[(\partial
X)^{k}\sigma_{\epsilon_{i}+\epsilon_{j},f}](x_{j})+\dots$ (79)
where $c_{k}$ are certain numbers and $\dots$ stands for other primaries and
descendants. These primary fields have a simple interpretation as the states
associated to the Hilbert space of twisted string since all of them have
conformal dimensions which differ by multiples of $\pm\epsilon$.
To continue and write in a more compact way the following expressions we
define
$\displaystyle P(\omega_{z},\omega_{w})$
$\displaystyle=\prod_{j=2}^{N}\frac{(\omega_{z}-\omega_{j})^{\epsilon_{j}-1}}{(\omega_{w}-\omega_{j})^{\epsilon_{j}}}$
$\displaystyle S(\omega_{z},\omega_{w})$
$\displaystyle=\sum_{n=0}^{N-M}\sum_{s=0}^{M}a_{ns}(\omega_{j})\omega_{z}^{n}\omega_{w}^{s}$
(80)
When we impose the constraint from the $z\rightarrow w$ limit given in eq.
(74) we get
$\displaystyle S(\omega_{w},\omega_{w})$
$\displaystyle=\sum_{n=0}^{N-M}\sum_{s=0}^{M}a_{ns}(\omega_{j})\omega_{w}^{n+s}=\prod_{j=2}^{N}(\omega_{w}-\omega_{j})$
$\displaystyle\frac{\partial
S}{\partial\omega_{z}}\Big{|}_{\omega_{z}=\omega_{w}}$
$\displaystyle=\sum_{n=0}^{N-M}\sum_{s=0}^{M}na_{ns}(\omega_{j})\omega_{w}^{n+s-1}=\sum_{j=2}^{N}\frac{1-\epsilon_{j}}{\omega_{w}-\omega_{j}}\cdot\prod_{l=2}^{N}(\omega_{w}-\omega_{l})$
(81)
or the equivalent equations with $w\rightarrow z$ and $\epsilon\rightarrow
1-\epsilon$. These are $(N+1)+N$ equations for $a_{ns}$ but only $2N$ are
independent since both imply that $a_{N-M,~{}M}=0$. Generically, i.e. for
$M_{cw}\neq 1$ these equations are not sufficient to fix the $(N-M+1)(M+1)$
unknowns $a_{ns}$ and must be supplemented by the constraints which follow
from eq.s (72). These further constraints allow also to fix the remaining
$(M-1)^{2}+(N-M-1)^{2}$ unknowns functions $b_{rs}$ and $c_{nm}$. For example
from the first equation in (72) we get
$\displaystyle X_{q}(x_{i},\bar{x}_{i})-X_{q}(x_{i+1},\bar{x}_{i+1})$
$\displaystyle=\int_{x_{i+1}}^{x_{i}}dX_{q}=\int_{x_{i+1}}^{x_{i}}dx[\partial
X_{q}(x+i0^{+})+{\bar{\partial}}X_{q}(x-i0^{+})]$
$\displaystyle=\int_{x_{i+1}}^{x_{i}}dx[\partial{\cal
X}_{q}(x+i0^{+})+e^{i2\pi\alpha_{2}}{\bar{\partial}}{\bar{\cal
X}}_{q}(x-i0^{+})]=0$ (82)
which implies the constraints777 It is worth noticing that the segment
$[x_{i+1},x_{i}]$ is followed for one addend above and for the other below the
cut (it works also the other way round w.r.t. the main text). This ensures
that both addends have the same phase modulus $\pi$. Consistency among
possible formulations of the constraints is due to
$[g(z,w)]^{*}=g(\bar{z},\bar{w})$,
$[h(z,w)]^{*}=e^{i4\pi\alpha_{2}}h(\bar{z},\bar{w})$ and
$[l(z,w)]^{*}=e^{-i4\pi\alpha_{2}}l(\bar{z},\bar{w})$.
$\displaystyle\int_{x_{i+1}}^{x_{i}}dx~{}g_{(N,M)}(x+i0^{+},w)+e^{i2\pi\alpha_{2}}h_{(N,M)}(x-i0^{+},w)=0$
$\displaystyle\int_{x_{i+1}}^{x_{i}}dx~{}l_{(N,M)}(z,x-i0^{+})+e^{i2\pi\alpha_{2}}g_{(N,M)}(z,x+i0^{+})=0$
(83)
These can be explicitly written as
$\displaystyle\sum_{s=0}^{M}\omega_{w}^{s}$
$\displaystyle\sum_{n=0}^{N-M}a_{ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}\frac{d\omega}{(\omega-\omega_{w})^{2}}\prod_{j=2}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}~{}\omega^{n}$
$\displaystyle+\sum_{s=0}^{M-2}\omega_{w}^{s}\sum_{r=0}^{M-2}b_{rs}(\omega_{j})\int_{\omega_{i+1};\omega\in
H^{-}}^{\omega_{i}}{d\omega}\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}~{}\omega^{r}=0$
(84) $\displaystyle\sum_{n=0}^{N-M}\omega_{z}^{n}$
$\displaystyle\sum_{s=0}^{M}a_{ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H^{-}}^{\omega_{i}}\frac{d\omega}{(\omega_{z}-\omega)^{2}}\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}~{}\omega^{s}$
$\displaystyle+\sum_{n=0}^{N-M-2}\omega_{z}^{n}\sum_{m=0}^{N-M-2}c_{nm}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}{d\omega}\prod_{j=2}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}~{}\omega^{m}=0$
(85)
As in the case for the classical solution only $N-2$ of the previous intervals
give independent constraints, let us say $i=2,\dots N-1$. All these
constraints are then sufficient to fix completely and uniquely all the
coefficients. These constraints are actually much more than needed since they
equate a polynomial in $\omega_{w}$ or in $\omega_{z}$ to an analytic
function. If we expand around $\omega_{w}=\omega_{z}=\infty$ and consider only
the polynomial part we have enough equations to fix all the unknowns since to
the previous $2N$ constraints in eq.s (81) we add $M-1$ equations in
$\omega_{w}$ times $N-2$ intervals and $N-M-1$ equations for $\omega_{z}$
times $N-2$ intervals. Actually the previous equations are already
overdetermined in $a$ since the $2N$ eq.s (81) and the $(N-2)(M-1)$ ones in
$\omega_{w}$ are sufficient for fix both $a$ and $b$ and similarly for the
ones in $\omega_{z}$ therefore for consistency we suppose that this
overdetermined system is consistent as well as all the remaining equations
obtained from the polar part in $\omega_{w}$ and $\omega_{z}$. As far as the
consistency of the functions $a$ determined in the two ways we have checked it
in particular limits in the explicit cases treated afterward. Moreover all
constraints derived from the polar part are polynomials in the integrals
$I^{(N)}$ (51) since the functions $a$ and $b$ are solutions of a linear
system whose coefficients are precisely the $I^{(N)}$s. Nevertheless it is
easy to show that all constraints must be equivalent to a relation with
polynomial coefficients in $\omega_{j\neq 1,2,N}$ and $I^{(N)}$ and at most
linear in $\hat{I}^{(N+1)}$ (see eq. (97)) with one of the parameters equal to
$2$. This can be seen as follows. It is possible to split
$g_{(N,M)}(z,w;\\{x_{i}\\})$ in a singular part and a regular one as
$\displaystyle g_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=g_{s(N,M)}(z,w;\\{x_{i}\\})+g_{r(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle g_{s(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{1}{(z-w)^{2}}\prod_{j\neq
1}\frac{(\omega_{z}-\omega_{j})^{\epsilon_{j}-1}}{(\omega_{w}-\omega_{j})^{\epsilon_{j}}}\sum_{n=0}^{N-M}\sum_{s=0}^{M}a_{(0)ns}(\omega_{j})\omega_{z}^{n}\omega_{w}^{s}$
$\displaystyle g_{r(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\sum_{n=0}^{N-M-2}\sum_{r=0}^{M-2}\bar{a}_{ns}(\omega_{j})~{}\partial_{\omega}{\cal
X}^{(n)}(\omega_{z})~{}\partial_{\omega}{\bar{\cal X}}^{(r)}(\omega_{w})$ (86)
This splitting is completely arbitrary and therefore it is not uniquely
defined but it can be made unique imposing further conditions such for example
the request of setting to zero all the $a_{n,s=k-n}$ but the two888 Eq.s (81)
divide naturally the set of unknowns $\\{a_{ns}\\}$ into subsets
$\\{a_{ns}\\}_{s+n=k}$ and for each of these subsets there are two linear
equations. with lowest $n$ as for example in eq. (104) for the $(N=4,M=2)$
case or the request that the singular part $g_{s(N,M)}$ goes into
$g_{s(N-1,M^{\prime})}$ when $x_{i}\rightarrow x_{j}$ as shown in appendix B
and explicitly in eq. (182) for $(N=4,M=2)$ case. Once fixed by a “gauge
choice” the singular part the regular one is fixed by the global boundary
conditions.
Actually if we choose to split $g$ into a regular part and a singular part
(which is fixed not uniquely by the OPEs) as in eq.s (86) the equation (84)
can be written as
$\displaystyle\sum_{s=0}^{M}\omega_{w}^{s}$
$\displaystyle\sum_{n=0}^{N-M}a_{(0)ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}\frac{d\omega}{(\omega-\omega_{w})^{2}}\prod_{j=2}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}~{}\omega^{n}$
$\displaystyle+\sum_{s=0}^{M-2}\omega_{w}^{s}\sum_{n=0}^{N-M-2}\bar{a}_{ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}d\omega\partial_{\omega}{\cal X}^{(n)}(\omega)$
$\displaystyle+\sum_{s=0}^{M-2}\omega_{w}^{s}\sum_{r=0}^{M-2}b_{rs}(\omega_{j})\int_{\omega_{i+1};\omega\in
H^{-}}^{\omega_{i}}{d\omega}\partial_{\omega}{\bar{\cal X}}^{(r)}(\omega)=0$
(87)
which reveals that the singular part $g_{s(N,M)}$ contributes with a term
linear in $\hat{I}^{(N+1)}$ (with one parameter equal to $2$ because of the
term $(\omega-\omega_{w})^{-2}$) while the other terms have rational
coefficient in $\omega_{j\neq 1,2,N}$ and $I^{(N)}$ once we plug the solution
for the coefficients back. In a similar way we can write the equation
corresponding to (85) as
$\displaystyle\sum_{n=0}^{N-M}\omega_{z}^{n}$
$\displaystyle\sum_{s=0}^{M}a_{(0)ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}\frac{d\omega}{(\omega_{z}-\omega)^{2}}\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}~{}\omega^{n}$
$\displaystyle+\sum_{n=0}^{N-M-2}\omega_{z}^{n}\sum_{s=0}^{M-2}\bar{a}_{ns}(\omega_{j})\int_{\omega_{i+1};\omega\in
H}^{\omega_{i}}d\omega\partial_{\omega}{\bar{\cal X}}^{(s)}(\omega)$
$\displaystyle+\sum_{m=0}^{N-M-2}\omega_{z}^{m}\sum_{n=0}^{N-M-2}b_{nm}(\omega_{j})\int_{\omega_{i+1};\omega\in
H^{-}}^{\omega_{i}}{d\omega}\partial_{\omega}{\cal X}^{(n)}(\omega)=0$ (88)
Having determined the derivatives of the Green functions we can reconstruct
the actual Green functions as
$\displaystyle G^{X{\bar{X}}}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{i}\\})$
$\displaystyle=\int_{x_{i};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}(u^{\prime},v^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{2}}\int_{x_{i};u\in
H}^{u}du^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}l_{(N,M)}(u^{\prime},{\bar{v}}^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{i2\pi\alpha_{2}}\int_{x_{i};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}h_{(N,M)}({\bar{u}}^{\prime},v^{\prime};\\{x\\_i\\}))$
$\displaystyle+\int_{x_{i};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}({\bar{v}}^{\prime},{\bar{u}}^{\prime};\\{x_{i}\\}))$
(89)
and
$\displaystyle G^{XX}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{i}\\})$
$\displaystyle=\int_{x_{i};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}l_{(N,M)}(u^{\prime},v^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{i2\pi\alpha_{2}}\int_{x_{i};u\in
H}^{u}du^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}(u^{\prime},{\bar{v}}^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{i2\pi\alpha_{2}}\int_{x_{i};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}(v^{\prime},{\bar{u}}^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{i4\pi\alpha_{2}}\int_{x_{i};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}h_{(N,M)}({\bar{u}}^{\prime},{\bar{v}}^{\prime};\\{x_{i}\\}))$
(90)
and
$\displaystyle
G^{{\bar{X}}{\bar{X}}}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{i}\\})$
$\displaystyle=\int_{x_{i};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}h_{(N,M)}(u^{\prime},v^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{2}}\int_{x_{i};u\in
H}^{u}du^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}({\bar{v}}^{\prime},u^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{2}}\int_{x_{i};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}({\bar{u}}^{\prime},v^{\prime};\\{x_{i}\\}))$
$\displaystyle+e^{-i4\pi\alpha_{2}}\int_{x_{i};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{j};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}l_{(N,M)}({\bar{u}}^{\prime},{\bar{v}}^{\prime};\\{x_{i}\\}))$
(91)
where the arbitrariness of the lower integration limit is due to the
constraints (83) which allow to change $x_{i}\rightarrow x_{k}$.
### 4.1 The explicit $N=3$, $M_{cw}=1$ case
In this case $g_{(3,1)}$ is completely fixed by the local constraints to be
$\displaystyle g_{(3,1)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{1}{(z-w)^{2}}\frac{(\omega_{z}-1)^{\epsilon_{2}-1}}{(\omega_{w}-1)^{\epsilon_{2}}}\frac{\omega_{z}^{\epsilon_{3}-1}}{\omega_{w}^{\epsilon_{3}}}$
$\displaystyle\Big{[}(1-\epsilon_{2}-\epsilon_{3})\omega_{z}^{2}+(\epsilon_{2}+\epsilon_{3})\omega_{z}\omega_{w}-(1-\epsilon_{3})\omega_{z}-\epsilon_{3}\omega_{w}\Big{]}$
(92)
while $h_{(3,1)}=0$ since $M=1$. We get therefore a constraint from eq. (84)
or the equivalent form (87) which read
$\displaystyle\omega_{w}$
$\displaystyle\int_{0}^{1}d\omega~{}\frac{1}{(\omega-\omega_{w})^{2}}(\omega-1)^{\epsilon_{2}-1}~{}\omega^{\epsilon_{3}-1}[(\epsilon_{2}+\epsilon_{3})\omega-\epsilon_{3}]$
$\displaystyle+\int_{0}^{1}d\omega~{}\frac{1}{(\omega-\omega_{w})^{2}}(\omega-1)^{\epsilon_{2}-1}~{}\omega^{\epsilon_{3}-1}[(1-\epsilon_{2}-\epsilon_{3})\omega^{2}-(1-\epsilon_{3})\omega]=0$
(93)
which can be read either as a constraint on the hypergeometric functions
$\displaystyle(1-\epsilon_{2}-\epsilon_{3})~{}B(2+\epsilon_{3},\epsilon_{2})~{}_{2}F_{1}(2,2+\epsilon_{3};2+\epsilon_{2}+\epsilon_{3};\frac{1}{\omega_{w}})$
$\displaystyle-(1-\epsilon_{3})~{}B(1+\epsilon_{3},\epsilon_{2})~{}_{2}F_{1}(2,1+\epsilon_{3};1+\epsilon_{2}+\epsilon_{3};\frac{1}{\omega_{w}})$
$\displaystyle+\left[(\epsilon_{2}+\epsilon_{3})~{}B(1+\epsilon_{3},\epsilon_{2})~{}_{2}F_{1}(2,1+\epsilon_{3};1+\epsilon_{2}+\epsilon_{3};\frac{1}{\omega_{w}})-\epsilon_{3}~{}B(\epsilon_{3},\epsilon_{2})~{}_{2}F_{1}(2,1+\epsilon_{3};\epsilon_{2}+\epsilon_{3};\frac{1}{\omega_{w}})\right]\omega_{w}=0$
(94)
or as infinite constraints on the coefficients of the $\omega_{w}$ expansion
which relate different Beta functions. We are now left to determine
$l_{(3,1)}$ from eq. (85), explicitly
$\displaystyle\omega_{z}^{2}~{}\int_{0}^{1}d\omega~{}\frac{1}{(\omega_{z}-\omega)^{2}}(\omega-1)^{-\epsilon_{2}}~{}\omega^{-\epsilon_{3}}(1-\epsilon_{2}-\epsilon_{3})$
$\displaystyle+$
$\displaystyle\omega_{z}~{}\int_{0}^{1}d\omega~{}\frac{1}{(\omega_{z}-\omega)^{2}}(\omega-1)^{-\epsilon_{2}}~{}\omega^{-\epsilon_{3}}[(1-\epsilon_{2}-\epsilon_{3})\omega^{2}-(1-\epsilon_{3})\omega]$
$\displaystyle+$
$\displaystyle\int_{0}^{1}d\omega~{}\frac{1}{(\omega_{z}-\omega)^{2}}(\omega-1)^{-\epsilon_{2}}~{}\omega^{-\epsilon_{3}}(-\epsilon_{3}\omega)$
$\displaystyle+$ $\displaystyle
c_{00}~{}\int_{0}^{1}d\omega~{}(\omega-1)^{\epsilon_{2}-1}~{}\omega^{\epsilon_{3}-1}=0$
(95)
or
$\displaystyle\omega_{z}^{2}~{}(1-\epsilon_{2}-\epsilon_{3})~{}\hat{I}^{(4)}_{2,0}(\epsilon_{j};2)$
$\displaystyle+$
$\displaystyle\omega_{z}~{}[(1-\epsilon_{2}-\epsilon_{3})~{}\hat{I}^{(4)}_{2,2}(\epsilon_{j};2)-(1-\epsilon_{3})~{}\hat{I}^{(4)}_{2,1}(\epsilon_{j};2)]$
$\displaystyle-$
$\displaystyle\epsilon_{3}~{}\hat{I}^{(4)}_{2,1}(\epsilon_{j};2)+c_{00}~{}I^{(3)}_{2,0}(\epsilon_{j})=0$
(96)
where we have introduced the function
$\displaystyle\hat{I}^{(N)}_{i,n}(\alpha_{j};\beta)$
$\displaystyle=\int_{\omega_{i+1}}^{\omega_{i}}d\omega~{}~{}\prod_{j=2}^{N}|\omega-\omega_{j}|^{-\alpha_{j}}~{}(\omega-\omega_{w})^{-\beta}\omega^{n}$
(97)
which is a slight modification of our previous definition (51) and is still
connected to the Lauricella functions $F_{D}^{(n)}$. In the
$\omega_{z}\rightarrow\infty$ limit we can determine the unique unknown
coefficient and hence the $l_{(3,1)}$ normalization to be
$\displaystyle
c_{00}=-(1-\epsilon_{2}-\epsilon_{3})\frac{B(1-\epsilon_{2},1-\epsilon_{3})}{B(\epsilon_{2},\epsilon_{3})}$
(98)
We get also infinite constraints from the subleading orders in $\omega_{z}$ or
plugging the previous value for $c_{00}$ back into eq. (95) an equation of the
form $\sum B~{}_{2}F_{1}=0$ as eq. (94).
### 4.2 The explicit $N=4$, $M=1$ case
Again as the case before $g_{(4,1)}$ is completely fixed by the local
constraints only to be
$\displaystyle g_{(4,1)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{1}{(z-w)^{2}}\frac{(\omega_{z}-1)^{\epsilon_{2}-1}}{(\omega_{w}-1)^{\epsilon_{2}}}\frac{(\omega_{z}-\omega_{3})^{\epsilon_{3}-1}}{(\omega_{w}-\omega_{3})^{\epsilon_{3}}}\frac{\omega_{z}^{\epsilon_{4}-1}}{\omega_{w}^{\epsilon_{4}}}$
$\displaystyle\Big{[}\epsilon_{1}\omega_{z}^{3}+(1-\epsilon_{1})\omega_{z}^{2}\omega_{w}-[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]\omega_{z}^{2}$
$\displaystyle-[(\epsilon_{3}+\epsilon_{4})+(\epsilon_{2}+\epsilon_{4})\omega_{3}]\omega_{z}\omega_{w}+(1-\epsilon_{4})\omega_{3}\omega_{z}+\epsilon_{4}\omega_{3}\omega_{w}\Big{]}$
(99)
and $h_{(4,1)}=0$ since $M=1$. As in the $(3,1)$ case from eq. (84) or the
equivalent form (87) we get the constraints
$\displaystyle-\omega_{w}^{2}[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]~{}\hat{I}^{(5)}_{i,0}(1-\epsilon_{j};2)$
$\displaystyle+\omega_{w}\\{(1-\epsilon_{1})~{}\hat{I}^{(5)}_{i,0}(1-\epsilon_{j};2)-[(\epsilon_{3}+\epsilon_{4})+(\epsilon_{2}+\epsilon_{4})\omega_{3}]~{}\hat{I}^{(5)}_{i,1}(1-\epsilon_{j};2)+\epsilon_{4}\omega_{3}~{}\hat{I}^{(5)}_{i,0}(1-\epsilon_{j};2)\\}$
$\displaystyle+\epsilon_{1}~{}\hat{I}^{(4)}_{i,3}(1-\epsilon_{j})+(1-\epsilon_{4})~{}\omega_{3}~{}\hat{I}^{(4)}_{i,0}(1-\epsilon_{j})=0$
(100)
In particular notice that $\hat{I}^{(5)}_{i,n}\sim F_{D}^{(2)}$ is the Appell
function.
We can proceed to determine the $l_{(4,1)}$ function. This amounts to fixing
the four functions $c_{00},c_{01},c_{10},c_{11}$ from eq. (85) which reads
$\displaystyle(-1)^{i+1}\Big{\\{}\omega_{z}^{3}~{}\epsilon_{1}~{}\hat{I}^{(5)}_{i,0}(\epsilon_{j};2)+\omega_{z}^{2}~{}(1-\epsilon_{1})\hat{I}^{(5)}_{i,1}(\epsilon_{j};2)-\omega_{z}~{}[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]\hat{I}^{(5)}_{i,1}(1-\epsilon_{j};2)$
$\displaystyle-\omega_{z}~{}[(\epsilon_{3}+\epsilon_{4})+(\epsilon_{2}+\epsilon_{4})\omega_{3}]\hat{I}^{(5)}_{i,1}(\epsilon_{j};2)+\omega_{z}~{}(1-\epsilon_{4})~{}\omega_{3}~{}\hat{I}^{(5)}_{i,0}(\epsilon_{j};2)+\epsilon_{4}\omega_{3}\hat{I}^{(5)}_{i,1}(\epsilon_{j};2)\Big{\\}}$
$\displaystyle+c_{00}~{}I^{(4)}_{i,0}(1-\epsilon_{j})+c_{01}~{}I^{(4)}_{i,1}(1-\epsilon_{j})+\omega_{z}~{}c_{10}~{}I^{(4)}_{i,0}(1-\epsilon_{j})+\omega_{z}~{}c_{11}~{}I^{(4)}_{i,1}(1-\epsilon_{j})=0$
(101)
for $i=2,3$. When we consider the $\omega_{z}\rightarrow\infty$ limit we get
two sets of equations, the one from the coefficient of $\omega_{z}$
$\displaystyle(-1)^{i+1}\epsilon_{1}~{}I^{(4)}_{i,0}(\epsilon_{j})+c_{10}~{}I^{(4)}_{i,0}(1-\epsilon_{j})+c_{11}~{}I^{(4)}_{i,1}(1-\epsilon_{j})=0$
(102)
and the other from the coefficient of $\omega_{z}^{0}$
$\displaystyle(-1)^{i+1}(1+\epsilon_{1})~{}\hat{I}^{(4)}_{i,1}(\epsilon_{j})+c_{00}~{}I^{(4)}_{i,0}(1-\epsilon_{j})+c_{01}~{}I^{(4)}_{i,1}(1-\epsilon_{j})=0$
(103)
plus an infinite set of constraints from the coefficients of the polar
expansion in $\omega_{z}$ or, equivalently plugging the previous value back
into eq. (101) and equation of the form
$(_{2}F_{1})^{2}F^{(2)}_{D}+\sum(_{2}F_{1})^{3}=0$ analogously to eq. (94).
### 4.3 The explicit $N=4$, $M=2$ case
This is the first case where there are more unknown coefficients than
equations from the local constraints and therefore we must use the global
constraints to fix completely $g_{(4,2)}$ and determine both $h_{(4,2)}$ and
$l_{(4,2)}$ which are now both not vanishing. We can nevertheless fix the
singular part $g_{s(4,2)}$ by choosing $a_{20}=0$ so we can get
$\displaystyle g_{s(4,2)}$
$\displaystyle=\frac{1}{(z-w)^{2}}\frac{(\omega_{z}-1)^{\epsilon_{2}-1}}{(\omega_{w}-1)^{\epsilon_{2}}}\frac{(\omega_{z}-\omega_{3})^{\epsilon_{3}-1}}{(\omega_{w}-\omega_{3})^{\epsilon_{3}}}\frac{\omega_{z}^{\epsilon_{4}-1}}{\omega_{w}^{\epsilon_{4}}}$
$\displaystyle\Big{\\{}\epsilon_{1}\omega_{z}^{2}\omega_{w}+(1-\epsilon_{1})\omega_{z}\omega_{w}^{2}-[(2-\epsilon_{3}-\epsilon_{4})+(2-\epsilon_{2}-\epsilon_{4})\omega_{3}]\omega_{z}\omega_{w}$
$\displaystyle+[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]\omega_{w}^{2}+(1-\epsilon_{4})\omega_{3}\omega_{z}+\epsilon_{4}\omega_{3}\omega_{w}\Big{\\}}$
(104)
Using the global constraints for $g$ and $h$ as given in eq. (87) for $i=2,3$
it is then possible to determine $\bar{a}_{00}$ (which corresponds to $a_{20}$
after the split of $g_{(4,2)}$ into a regular and singular part) and $b_{00}$,
in particular taking the $\omega_{w}\rightarrow\infty$ limit we get
$\displaystyle\left\\{\begin{array}[]{c}\bar{a}_{00}~{}I^{(4)}_{2,0}(1-\epsilon)-b_{00}~{}I^{(4)}_{2,0}(\epsilon)=-(1-\epsilon_{1})~{}I^{(4)}_{2,1}(1-\epsilon)-[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]~{}I^{(4)}_{2,0}(1-\epsilon)\\\
\bar{a}_{00}~{}I^{(4)}_{3,0}(1-\epsilon)+b_{00}~{}I^{(4)}_{3,0}(\epsilon)=-(1-\epsilon_{1})~{}I^{(4)}_{3,1}(1-\epsilon)-[(1-\epsilon_{3}-\epsilon_{4})+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}]~{}I^{(4)}_{3,0}(1-\epsilon)\end{array}\right.$
(107)
where the minus sign in the lhs of the first line is due a careful treatment
of phases. In the limit $\omega_{w}\rightarrow\infty$ eq. (88) allows to fix
$c_{00}$ and again $\bar{a}_{00}$ as
$\displaystyle\left\\{\begin{array}[]{c}\bar{a}_{00}~{}I^{(4)}_{2,0}(\epsilon)-c_{00}~{}I^{(4)}_{2,0}(1-\epsilon)=-\epsilon_{1}~{}I^{(4)}_{2,1}(\epsilon)\\\
\bar{a}_{00}~{}I^{(4)}_{3,0}(\epsilon)+c_{00}~{}I^{(4)}_{3,0}(1-\epsilon)=-\epsilon_{1}~{}I^{(4)}_{3,1}(\epsilon)\end{array}\right.$
(110)
The two previous ways of fixing $\bar{a}_{00}$ must be compatible and this can
be easily verified at least in the $\omega_{3}\rightarrow 1^{-}$ limit.
## 5 The quantum twists correlators
In this section we want to compute the $N$ twists correlators in the $N-2$
different sectors determined by $M$. We can generically write the $N$ twists
correlators in the $M$ sector as
$\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}}(x_{i})\rangle=\frac{A_{(N,M)}(\omega_{j\neq
1,2,N})e^{-S_{E,cl}(x_{i},\epsilon_{i},f_{i})}}{\prod_{1\leq i<j\leq
N}(x_{i}-x_{j})^{\Delta_{ij}}}$ (111)
The powers $\Delta_{ij}$ can be completely fixed as follows. From the proper
behavior for $x_{i}\rightarrow\infty$ we get the constraints $\sum_{l\neq
i}\Delta_{il}=2\Delta(\sigma_{\epsilon_{i},f_{i}})=\epsilon_{i}(1-\epsilon_{i})$
where we have defined $\Delta_{ji}=\Delta_{ij}$ for $j>i$. Now redefining
$A\rightarrow A~{}\prod_{3\leq i<j\leq
N-1}(\omega_{i}-\omega_{j})^{\Delta_{ij}}~{}\prod_{3\leq i\leq
N-1}(1-\omega_{i})^{\Delta_{2i}}~{}\prod_{3\leq i\leq
N-1}\omega_{i}^{\Delta_{iN}}$ and remembering that $\omega_{2}=1$,
$\omega_{N}=0$ and $(\omega_{i}-\omega_{j})\propto(x_{i}-x_{j})$ we can set
all $\Delta$s to zero but $\Delta_{1i},\Delta_{12},\Delta_{1N}$ ($3\leq i\leq
N-1$) and $\Delta_{2N}$ which can now be fixed by the first set of
constraints. Therefore we can choose a “gauge” where the previous correlator
can be written
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}}(x_{i})\rangle=$
$\displaystyle\frac{1}{\prod_{3\leq i\leq
N-1}(x_{1}-x_{i})^{\epsilon_{i}(1-\epsilon_{i})}}$
$\displaystyle\cdot\frac{1}{(x_{1}-x_{2})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{2}(1-\epsilon_{2})-\epsilon_{N}(1-\epsilon_{N})]}}$
$\displaystyle\cdot\frac{1}{(x_{1}-x_{N})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{2}(1-\epsilon_{2})]}}$
$\displaystyle\cdot\frac{1}{(x_{2}-x_{N})^{\frac{1}{2}[\epsilon_{2}(1-\epsilon_{2})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{1}(1-\epsilon_{1})+\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})]}}$
$\displaystyle\cdot A_{(N,M)}(\omega_{j\neq
1,2,N})~{}e^{-S_{E,cl}(x_{i},\epsilon_{i},f_{i})}$ (112)
We can now proceed in the usual way. We first compute the expectation value of
the energy-momentum tensor as
$\displaystyle\langle\langle T(z)\rangle\rangle$
$\displaystyle=\frac{\langle\partial{\cal X}_{q}(z)\partial{\bar{\cal
X}}_{q}(z)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
$\displaystyle=\lim_{w\rightarrow z}g(z,w)-\frac{1}{(z-w)^{2}}$ (113)
then using the OPE
$T(z)\sigma_{\epsilon_{i},f_{i}}(x_{i})\sim\frac{\epsilon_{i}(1-\epsilon_{i})}{(z-x_{i})^{2}}+\frac{\partial_{x_{i}}\sigma_{\epsilon_{i},f_{i}}(x_{i})}{z-x_{i}}+O(1)$
(114)
we compute
$\displaystyle\partial_{x_{i}}\ln\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f}(x_{i})\rangle$
$\displaystyle=\lim_{z\rightarrow x_{i}}(z-x_{i})\left[\langle\langle
T(z)\rangle\rangle-\frac{\epsilon_{i}(1-\epsilon_{i})}{(z-x_{i})^{2}}\right]$
(115)
The function $A_{(N,M)}$ in the quantum case where $f_{i}=f$ can be determined
from eq. (111) ($j\neq 1,2,N$) as
$\displaystyle\partial_{x_{j}}\ln\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f}(x_{i})\rangle$
$\displaystyle=-\sum_{l\neq
j}\frac{\Delta_{lj}}{x_{j}-x_{l}}+\frac{\partial\omega_{j}}{\partial
x_{j}}\frac{\partial\ln A_{(N,M)}}{\partial\omega_{j}}$ (116)
### 5.1 The $M_{ccw}=N-1$, $M_{cw}=1$ cases
Using the expansion (80) for $S$ and the constraints (81) we can easily deduce
that
$\displaystyle\langle\langle T(z)\rangle\rangle$
$\displaystyle=\frac{1}{2}\left(\frac{\partial\omega_{z}}{\partial
z}\right)^{2}\left[\sum_{j=2}^{N}\frac{\epsilon_{j}}{(\omega_{z}-\omega_{j})^{2}}-\left(\sum_{j=2}^{N}\frac{\epsilon_{j}}{\omega_{z}-\omega_{j}}\right)^{2}+\prod_{j=2}^{N}\frac{1}{\omega_{z}-\omega_{j}}\frac{\partial^{2}S}{\partial\omega_{w}^{2}}\Big{|}_{\omega_{w}=\omega_{z}}\right]$
(117)
It then follows that ($j\neq 1,2,N$)
$\displaystyle\partial_{x_{j}}\log\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle$
$\displaystyle=-\frac{\epsilon_{j}}{x_{j}-x_{1}}$
$\displaystyle+\epsilon_{j}\left[-\sum_{l\neq
1,j}\frac{\epsilon_{l}}{x_{j}-x_{l}}+\frac{M-\epsilon_{1}}{x_{j}-x_{1}}\right]$
$\displaystyle+\frac{1}{2}\prod_{l\neq
1,j}\frac{1}{\omega_{j}-\omega_{l}}\frac{\partial\omega_{j}}{\partial
x_{j}}\frac{\partial^{2}S}{\partial\omega_{w}^{2}}\Big{|}_{\omega_{w}=\omega_{z}=\omega_{j}}$
(118)
from which we can obtain $A_{(N,M)}$ using eq. (116) to get
$\displaystyle\frac{\partial\ln A_{(N,M)}}{\partial\omega_{j}}$
$\displaystyle=\frac{1}{2}\prod_{l=2;l\neq
j}^{N}\frac{1}{\omega_{j}-\omega_{l}}\frac{\partial^{2}S}{\partial\omega_{w}^{2}}\Big{|}_{\omega_{w}=\omega_{z}=\omega_{j}}+\sum_{l=2;l\neq
j}^{N}\frac{\Delta_{jl}-\epsilon_{j}\epsilon_{l}}{\omega_{j}-\omega_{l}}$
(119)
The main issue is then to compute
$\frac{\partial^{2}S}{\partial\omega_{w}^{2}}\Big{|}_{\omega_{w}=\omega_{z}}$.
This can be done immediately in two cases, i.e. $M_{cw}=1$ for which
$\frac{\partial^{2}S}{\partial\omega_{w}^{2}}=0$ since the maximum
$\omega_{w}$ power is 1 and $M_{ccw}=N-1$ for which
$\frac{\partial^{2}S}{\partial\omega_{z}^{2}}=0$ since the maximum
$\omega_{z}$ power is 1 as it is obvious from eq. (80). In the former case we
get
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle=$
$\displaystyle C_{(N,M=1)}(\epsilon)~{}\frac{\prod_{3\leq j<l\leq
N-1}(\omega_{j}-\omega_{l})^{-\epsilon_{j}\epsilon_{l}}}{\prod_{3\leq i\leq
N-1}(x_{1}-x_{i})^{\epsilon_{i}(1-\epsilon_{i})}}$
$\displaystyle\cdot\frac{\prod_{3\leq l\leq
N-1}(1-\omega_{l})^{-\epsilon_{2}\epsilon_{l}}}{(x_{1}-x_{2})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{2}(1-\epsilon_{2})-\epsilon_{N}(1-\epsilon_{N})]}}$
$\displaystyle\cdot\frac{\prod_{3\leq j\leq
N-1}\omega_{j}^{-\epsilon_{j}\epsilon_{N}}}{(x_{1}-x_{N})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{2}(1-\epsilon_{2})]}}$
$\displaystyle\cdot\frac{1}{(x_{2}-x_{N})^{\frac{1}{2}[\epsilon_{2}(1-\epsilon_{2})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{1}(1-\epsilon_{1})+\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})]}}$
(120)
while in the latter we get the same result but with the substitution
$\epsilon\rightarrow 1-\epsilon$ by expanding $\omega_{z}$ around
$\omega_{w}$. The coefficients $C_{N,1}$ and $C_{N,N-1}$ will be fixed in
section 5.4 and are given in eq.s (160).
### 5.2 The $N\geq 4$ and $N-2\geq M\geq 2$ cases
For all the other cases it is enough to use a slight modification of the
technique used in [14] (see also [5]).
The main idea of this approach is to define a new basis for the classical
solutions (see eq.s (44)) and consequently for the non singular part of the
derivative of the Green function $g(z,w)$ (see eq. (86)) which are closed
under certain operations needed to compute the correlators.
We start therefore by defining a new basis for the classical solutions
$\displaystyle\partial_{\omega}{\cal X}^{(I)}(\omega)$
$\displaystyle=\prod_{j=2}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}\prod_{l\in
S_{I}}(\omega-\omega_{l})~{}~{}~{}~{}I\in S$
$\displaystyle\partial_{\omega}{\bar{\cal X}}^{(\bar{I})}(\omega)$
$\displaystyle=\prod_{j=2}^{N}(\omega-\omega_{j})^{-\epsilon_{j}}\prod_{l\in\bar{S}_{\bar{I}}}(\omega-\omega_{l})~{}~{}~{}~{}\bar{I}\in\bar{S}$
(121)
where we have defined two ordered sets
$S=\\{N-M-1~{}\mbox{arbitrary different indexes chosen among}~{}3,\dots
N-1\\}$ (122)
and
$\bar{S}=\\{M-1~{}\mbox{arbitrary different indexes chosen among}~{}3,\dots
N-1\\}$ (123)
and the subsets $S_{I}=S-\\{I\\}$ for any $I\in S$ and similarly for
$\bar{S}_{\bar{I}}$. In order to be able to define the previous basis as a
linear combination of the original one (45) we need that both $n\geq 0$ and
$r\geq 0$, i.e $N-2\geq M\geq 2$. In particular what follows works even if
either $S_{J}=\emptyset$ or ${\bar{S}}_{\bar{J}}=\emptyset$, i.e. $M=N-2$ or
$M=2$ for example when $N=4$ and $M=2$.
We can now expand the regular part of $g$ and $h$, $l$ as
$\displaystyle g_{r(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial w}\sum_{I\in
S}\sum_{{\bar{I}}\in{\bar{S}}}\bar{a}_{I{\bar{I}}}(\omega_{j})~{}\partial_{\omega}{\cal
X}^{(I)}(\omega_{z})~{}\partial_{\omega}{\bar{\cal
X}}^{({\bar{I}})}(\omega_{w})$ $\displaystyle h_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\sum_{{\bar{I}}\in{\bar{S}}}\sum_{{\bar{J}}\in{\bar{S}}}b_{{\bar{I}}{\bar{J}}}(\omega_{j})~{}\partial_{\omega}{\bar{\cal
X}}^{({\bar{I}})}(\omega_{z})~{}\partial_{\omega}{\bar{\cal
X}}^{({\bar{J}})}(\omega_{w})$ $\displaystyle l_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial w}\sum_{I\in S}\sum_{J\in
S}c_{IJ}(\omega_{j})~{}\partial_{\omega}{\cal
X}^{(I)}(\omega_{z})~{}\partial_{\omega}{\cal X}^{(J)}(\omega_{w})$ (124)
Then we can find a solution of the first of the constraints in eq. (83) as
$\displaystyle g_{(N,M)}(z,w;\\{x_{i}\\})$
$\displaystyle=g_{s(N,M)}(z,w;\\{x_{i}\\})-\frac{\partial\omega_{z}}{\partial
z}\sum_{i=1}^{N-2}\sum_{I\in S}(W^{-1})^{i}_{I}\partial_{\omega}{\cal
X}^{(I)}(\omega_{z})\int_{{\cal
I}_{i}}\frac{d\omega}{(\omega-\omega_{w})^{2}}PS_{0}(\omega,\omega_{w})$
$\displaystyle=g_{s(N,M)}(z,w;\\{x_{i}\\})-\frac{\partial\omega_{w}}{\partial
w}\sum_{i=1}^{N-2}\sum_{{\bar{I}}\in{\bar{S}}}(W^{-1})^{i}_{\bar{I}}\partial_{\omega}{\bar{\cal
X}}^{({\bar{I}})}(\omega_{w})\int_{{\cal
I}_{i}}\frac{d\omega}{(\omega_{z}-\omega)^{2}}PS_{0}(\omega_{z},\omega)$ (125)
where $S_{0}(\omega_{z},\omega_{w})$ is the same as in the second equation in
(80) but for the singular part of $g$, i.e. with coefficients $a_{(0)}$ as in
eq. (86). Moreover we have also defined the $(N-2)\times(N-2)$ matrix $W$
as999 Again as in eq.s (83) it is important that the integration is once above
and once below the cut as this ensures that both integrals have the same phase
modulus $\pi$.
$\displaystyle W_{i}^{~{}I}$
$\displaystyle=\int_{\omega_{i+2}}^{\omega_{i+1}}d\omega~{}\partial_{\omega}{\cal
X}^{(I)}(\omega+i0^{+})~{}~{}~{}~{}i=1,\dots N-2,~{}~{}~{}I\in S$
$\displaystyle W_{i}^{~{}{\bar{I}}}$
$\displaystyle=\int_{\omega_{i+2}}^{\omega_{i+1}}d\omega~{}\partial_{\omega}{\bar{\cal
X}}^{(I)}(\omega-i0^{+})~{}~{}~{}~{}i=1,\dots
N-2,~{}~{}~{}{\bar{I}}\in{\bar{S}}$ (126)
From this expression is immediate to compute the energy-momentum tensor
expectation value which can be split into a singular part as in eq. (117) but
with the substitution $S\rightarrow S_{0}$ and a regular part as
$\displaystyle\langle\langle T_{r}(w)\rangle\rangle$
$\displaystyle=-\frac{\partial\omega_{w}}{\partial
w}\sum_{i=1}^{N-2}\sum_{I\in S}(W^{-1})^{i}_{I}\int_{{\cal
I}_{i}}\frac{d\omega}{(\omega-\omega_{w})^{2}}\frac{\prod_{j=2}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}}{\prod_{j\not\in
S_{I}}(\omega_{w}-\omega_{j})}S_{0}(\omega,\omega_{w})$ (127)
where $j\not\in S_{I}$ means $j\in\\{2,\dots N\\}\setminus S_{I}$. If we
consider $J\in S$ we can then evaluate101010 There are two integrals which are
actually divergent but their sum is however convergent. These integrals
correspond to the intervals for which $\omega_{J}$ is a boundary point.
111111 We restrict to the case $J\in S$ because otherwise eq. (128) would
contain the sum over all $I\in S$ and eq. (129) would contain the sum over all
possible $\partial_{\omega_{j}}\partial_{\omega}{\cal X}^{(I)}(\omega)$ each
with a non trivial coefficient.
$\displaystyle\lim_{w\rightarrow x_{J}}$ $\displaystyle(w-x_{J})\langle\langle
T_{r}(w)\rangle\rangle$ $\displaystyle=-\frac{\partial\omega_{J}}{\partial
x_{J}}\sum_{i=1}^{N-2}(W^{-1})^{i}_{J}\int_{{\cal
I}_{i}}\frac{d\omega}{(\omega-\omega_{J})^{3-\epsilon_{J}}}\frac{\prod_{j=2;j\neq
J}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}}{\prod_{j\not\in
S}(\omega_{J}-\omega_{j})}S_{0}(\omega,\omega_{J})$ (128)
Now following [14] we rewrite the integrand as
$\displaystyle\frac{1}{(\omega-\omega_{J})^{3-\epsilon_{J}}}\frac{\prod_{j=2;j\neq
J}^{N}(\omega-\omega_{j})^{\epsilon_{j}-1}}{\prod_{j\not\in
S}(\omega_{J}-\omega_{j})}S_{0}(\omega,\omega_{J})$
$\displaystyle=\partial_{\omega_{J}}\partial_{\omega}{\cal
X}^{(J)}(\omega)+\sum_{L\in S}T^{J}_{L}\partial_{\omega}{\cal
X}^{(L)}(\omega)$ (129)
. The leading singularity is
$O\left((\omega-\omega_{J})^{-2+\epsilon_{J}}\right)$ because
$S_{0}(\omega_{J},\omega_{J})=0$ as follows from the first equation in (81)
when evaluated for $\omega_{w}=\omega_{J}$. Moreover when the left hand side
is subtracted the leading singularity and multiplied for
$\prod_{j=2}^{N}(\omega-\omega_{j})^{1-\epsilon_{j}}/\prod_{j\in
S}(\omega-\omega_{j})$ we are left with a rational function with poles at
$\omega_{I}$ ($I\in S$) which vanish at $\omega=\infty$ as the right hand
side. Because of the sum over $i$ in eq. (128) the only $T^{J}_{L}$ needed is
$\displaystyle T^{J}_{J}$ $\displaystyle=-(1-\epsilon_{J})\sum_{l\in
S_{J}}\frac{1}{\omega_{J}-\omega_{l}}+\frac{1}{2}\prod_{l\in
S_{J}}\frac{1}{\omega_{J}-\omega_{l}}\prod_{l\not\in
S}\frac{1}{\omega_{J}-\omega_{l}}\partial^{2}_{\omega_{z}}S_{0}(\omega_{J},\omega_{J})$
(130)
When we insert this value into eq. (128) and add the contribution from the
singular part which has the same expression as eq. (118) with $S\rightarrow
S_{0}$ and $\epsilon\rightarrow 1-\epsilon$ since we have here
$\partial^{2}_{\omega_{z}}S_{0}$ we get
$\displaystyle\partial_{x_{J}}\log\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle$
$\displaystyle=-\frac{(1-\epsilon_{J})(2-\epsilon_{J})}{x_{J}-x_{1}}$
$\displaystyle-2(1-\epsilon_{J})\left[\sum_{l\neq
1,J}\frac{1-\epsilon_{l}}{x_{J}-x_{l}}+\frac{M-N+1-\epsilon_{1}}{x_{J}-x_{1}}\right]$
$\displaystyle-\frac{\partial\omega_{J}}{\partial
x_{J}}\left[\sum_{i=1}^{N-2}(W^{-1})^{i}_{J}\partial_{\omega_{J}}W^{J}_{i}-(1-\epsilon_{J})\sum_{l\in
S_{J}}\frac{1}{\omega_{J}-\omega_{l}}\right]$ (131)
from which the dependence on $S_{0}$ has disappeared but we are left with a
dependence on $\partial_{\omega_{J}}W^{J}_{i}$. Differently from what done in
[14] we cannot rely on fact that twists have both an holomorphic and
antiholomorphic dependence in order to end the computation using
$\displaystyle\det W$ $\displaystyle=\sum_{i=1}^{N-2}\left[\sum_{I\in
S}(W^{-1})^{i}_{I}\partial_{\omega_{J}}W^{I}_{i}+\sum_{{\bar{I}}\in{\bar{S}}}(W^{-1})^{i}_{\bar{I}}\partial_{\omega_{J}}W^{\bar{I}}_{i}\right]$
(132)
and
$\partial_{\omega_{J}}W^{I\neq
J}_{i}=\frac{\epsilon_{I}}{\omega_{J}-\omega_{I}}\left(W^{J}_{i}-W^{I}_{i}\right)$
(133)
Instead we have to rely on the second of the (83) constraints (or better its
complex conjugate which is has the same expression with the substitution $\pm
i0^{+}\rightarrow\mp i0^{+}$). Analogously as before we require
${\bar{J}}\in{\bar{S}}$ and we get
$\displaystyle\partial_{x_{\bar{J}}}\log\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle$
$\displaystyle=-\frac{\epsilon_{\bar{J}}(1+\epsilon_{\bar{J}})}{x_{\bar{J}}-x_{1}}$
$\displaystyle-2\epsilon_{\bar{J}}\left[\sum_{l\neq
1,{\bar{J}}}\frac{\epsilon_{l}}{x_{\bar{J}}-x_{l}}+\frac{-M+\epsilon_{1}}{x_{\bar{J}}-x_{1}}\right]$
$\displaystyle-\frac{\partial\omega_{\bar{J}}}{\partial
x_{\bar{J}}}\left[\sum_{i=1}^{N-2}(W^{-1})^{i}_{\bar{J}}\partial_{\omega_{J}}W^{\bar{J}}_{i}-\epsilon_{\bar{J}}\sum_{l\in
S_{\bar{J}}}\frac{1}{\omega_{\bar{J}}-\omega_{l}}\right]$ (134)
then only if $J={\bar{J}}\in S\cap{\bar{S}}$ we can average the previous
expressions (131) and (134) Into this average we can use the analogous
expression of eq. (133)
$\partial_{\omega_{J}}W^{{\bar{I}}\neq{\bar{J}}}_{i}=\frac{1-\epsilon_{\bar{I}}}{\omega_{\bar{J}}-\omega_{\bar{I}}}\left(W^{{\bar{J}}}_{i}-W^{{\bar{I}}}_{i}\right)$
(135)
and
$\displaystyle\sum_{i=1}^{N-2}(W^{-1})^{i}_{J}\partial_{\omega_{J}}W^{J}_{i}$
$\displaystyle+\sum_{i=1}^{N-2}(W^{-1})^{i}_{\bar{J}}\partial_{\omega_{J}}W^{\bar{J}}_{i}$
$\displaystyle=\partial_{\omega_{J}}\det W-\sum_{I\in
S_{J}}\sum_{i=1}^{N-2}(W^{-1})^{i}_{I}\partial_{\omega_{J}}W^{I}_{i}-\sum_{{\bar{I}}\in{\bar{S}}_{J}}\sum_{i=1}^{N-2}(W^{-1})^{i}_{\bar{I}}\partial_{\omega_{J}}W^{\bar{I}}_{i}$
$\displaystyle=\partial_{\omega_{J}}\det W+\sum_{I\in
S_{J}}\frac{\epsilon_{l}}{\omega_{J}-\omega_{l}}+\sum_{{\bar{I}}\in{\bar{S}}_{J}}\frac{1-\epsilon_{l}}{\omega_{J}-\omega_{l}}$
(136)
to get finally
$\displaystyle\partial_{\omega_{J}}\log
A_{(N,M)}(\omega_{j})=\partial_{\omega_{J}}\log\Big{[}(\det W)^{-\frac{1}{2}}$
$\displaystyle\prod_{l\in
S_{J}}(\omega_{J}-\omega_{l})^{\frac{1}{2}}\prod_{l\in{\bar{S}}_{J}}(\omega_{J}-\omega_{l})^{\frac{1}{2}}$
$\displaystyle\prod_{l\neq
1,J}(\omega_{J}-\omega_{l})^{\Delta_{Jl}-\frac{1}{2}[(1-\epsilon_{J})(1-\epsilon_{l})+\epsilon_{J}\epsilon_{l}]}\Big{]}$
(137)
The previous equation is valid only for $J\in S\cap{\bar{S}}$ but if, in
either $S$ or in ${\bar{S}}$ there is at least one further element than those
contained in $S\cap{\bar{S}}$ or if $S\cap{\bar{S}}$ contains all the
independent $\omega_{j}$ as in the $N=4$ case, we can deduce that121212 In the
expression we have used $ord({\bar{I}})$ to indicate the order of $I$ in the
ordered set $S$.
$\displaystyle A_{(N,M)}(\omega_{j})$ $\displaystyle=\mbox{const}(\det
W)^{-\frac{1}{2}}\prod_{ord(I)<ord(J);I,J\in
S}(\omega_{I}-\omega_{J})^{\frac{1}{2}}\prod_{ord({\bar{I}})<ord({\bar{J}});{\bar{I}},{\bar{J}}\in{\bar{S}}}(\omega_{\bar{I}}-\omega_{\bar{J}})^{\frac{1}{2}}$
$\displaystyle~{}~{}\prod_{2\leq j<l\leq
N}(\omega_{j}-\omega_{l})^{\Delta_{jl}-\frac{1}{2}[(1-\epsilon_{j})(1-\epsilon_{l})+\epsilon_{j}\epsilon_{l}]}$
(138)
as a consequence of the independence of the result under a change of the
elements of $S$ and/or ${\bar{S}}$. Under the change $S\rightarrow
S^{\prime}=(S\setminus\\{I_{0}\\})\cup\\{I_{1}\\}$ the integrals
$W^{I}_{(S)i}$ in eq.s (126) transform as
$\displaystyle W^{L}_{(S^{\prime})i}$
$\displaystyle=\frac{\omega_{I_{1}}-\omega_{L}}{\omega_{I_{0}}-\omega_{L}}W^{L}_{(S)i}+\frac{\omega_{I_{1}}-\omega_{I_{0}}}{\omega_{L}-\omega_{I_{0}}}W^{I_{0}}_{(S)i}~{}~{}~{}~{}L\neq
I_{0}$ $\displaystyle W^{I_{1}}_{(S^{\prime})i}$
$\displaystyle=W^{I_{0}}_{(S)i}$ (139)
so that the transformation of the determinant
$\displaystyle\det W_{S^{\prime},{\bar{S}}}$ $\displaystyle=\det
W_{S,{\bar{S}}}\prod_{L\in
S_{I_{0}}}\frac{\omega_{I_{1}}-\omega_{L}}{\omega_{I_{0}}-\omega_{L}}$ (140)
is what is needed to compensate the change $\prod_{ord(I)<ord(J);I,J\in
S^{\prime}}(\omega_{I}-\omega_{J})^{\frac{1}{2}}\rightarrow\prod_{ord(I)<ord(J);I,J\in
S}(\omega_{I}-\omega_{J})^{\frac{1}{2}}$.
The final expression for the $N$ twists correlator in the $M$ sector is then
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle=$
$\displaystyle C_{(N,M)}(\epsilon)~{}\frac{\prod_{3\leq j<l\leq
N-1}(\omega_{j}-\omega_{l})^{-\frac{1}{2}[(1-\epsilon_{j})(1-\epsilon_{l})+\epsilon_{j}\epsilon_{l}]}}{\prod_{3\leq
i\leq N-1}(x_{1}-x_{i})^{\epsilon_{i}(1-\epsilon_{i})}}$
$\displaystyle\cdot\frac{\prod_{3\leq l\leq
N-1}(1-\omega_{l})^{-\frac{1}{2}[(1-\epsilon_{2})(1-\epsilon_{l})+\epsilon_{2}\epsilon_{l}]}}{(x_{1}-x_{2})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{2}(1-\epsilon_{2})-\epsilon_{N}(1-\epsilon_{N})]}}$
$\displaystyle\cdot\frac{\prod_{3\leq j\leq
N-1}\omega_{j}^{-\frac{1}{2}[(1-\epsilon_{j})(1-\epsilon_{N})+\epsilon_{j}\epsilon_{N}]}}{(x_{1}-x_{N})^{\frac{1}{2}[\epsilon_{1}(1-\epsilon_{1})-\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{2}(1-\epsilon_{2})]}}$
$\displaystyle\cdot\frac{1}{(x_{2}-x_{N})^{\frac{1}{2}[\epsilon_{2}(1-\epsilon_{2})+\epsilon_{N}(1-\epsilon_{N})-\epsilon_{1}(1-\epsilon_{1})+\sum_{i=3}^{N-1}\epsilon_{i}(1-\epsilon_{i})]}}$
$\displaystyle\cdot(\det
W_{S,{\bar{S}}})^{-\frac{1}{2}}\prod_{ord(I)<ord(J);I,J\in
S}(\omega_{I}-\omega_{J})^{\frac{1}{2}}\prod_{ord({\bar{I}})<ord({\bar{J}});{\bar{I}},{\bar{J}}\in{\bar{S}}}(\omega_{\bar{I}}-\omega_{\bar{J}})^{\frac{1}{2}}$
(141)
Notice that the previous expression is true even if there is only one element
in ${\bar{S}}$ in which case the product
$\prod(\omega_{\bar{I}}-\omega_{\bar{J}})^{\frac{1}{2}}$ is simply $1$.
Similarly for the $S$ case. In subsection 5.4 we will fix the constant
$C_{(N,M)}(\epsilon)$.
### 5.3 $N-1$ amplitudes from $N$ amplitudes
We want to check the consistency of the results of the previous section. We do
this by making $x_{j+1}$ coalesce with $x_{j}$ and so deducing the $N-1$
twists correlators from $N$ twists ones.
We start noticing that from the $(N,M)$ sector we can generically compute both
$(\tilde{N},\tilde{M})=(N-1,M)$ and $(\tilde{N},\tilde{M})=(N-1,M-1)$ sectors
depending whether $\epsilon_{j}+\epsilon_{j+1}<1$ or
$\epsilon_{j}+\epsilon_{j+1}>1$. Exceptions are the $M=1$ case where only
$\tilde{M}=1$ is possible and $M=N-1$ where only $\tilde{M}=\tilde{N}-1=N-2$
is possible.
#### 5.3.1 $(N,1)$ into $(N-1,1)$ case
Starting from eq. (120) we can very easily take the limit $x_{J+1}\rightarrow
x_{J}$. When we use
$\epsilon_{J}(1-\epsilon_{J})+\epsilon_{J+1}(1-\epsilon_{J+1})=\tilde{\epsilon}_{J}(1-\tilde{\epsilon}_{J})+2\epsilon_{J}\epsilon_{J+1}~{}~{}~{}~{}\tilde{\epsilon}_{J}=\epsilon_{J}+\epsilon_{J+1}$
(142)
and
$\omega_{J}-\omega_{J+1}=\frac{(x_{J+1}-x_{J})(x_{N}-x_{1})}{(x_{J}-x_{1})(x_{J+1}-x_{1})}\frac{x_{2}-x_{1}}{x_{2}-x_{N}}$
(143)
we find
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f}(x_{i})\rangle\sim_{x_{J+1}\rightarrow
x_{J}}(x_{J}-x_{J+1})^{-\epsilon_{J}\epsilon_{J+1}}~{}{\cal
M}(\epsilon_{J},\epsilon_{J+1})~{}\langle\prod_{\tilde{i}=1}^{N-1}\sigma_{\tilde{\epsilon}_{\tilde{i}},f}(x_{\tilde{i}})\rangle$
(144)
and the consistency relation for the normalizations
$C_{(N,1)}(\epsilon)=C_{(N-1,1)}(\tilde{\epsilon})~{}{\cal
M}(\epsilon_{J},\epsilon_{J+1})$ (145)
where $\tilde{\epsilon}$ are the twists of the $(N-1,1)$ theory defined by
$\tilde{\epsilon}_{j}=\epsilon_{j}$ for $j<J$,
$\tilde{\epsilon}_{J}=\epsilon_{J}+\epsilon_{J+1}$ for $j=J$ and
$\tilde{\epsilon}_{j}=\epsilon_{j+1}$ for $j>J$. Actually all the previous
equations work even when we consider the $\omega_{j}\rightarrow\infty$
($x_{j}\rightarrow x_{1}$) limit.
#### 5.3.2 $(N,N-1)$ into $(N-1,N-2)$ case
In a way completely analogous to that done in the previous subsection we get
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f}(x_{i})\rangle\sim_{x_{J+1}\rightarrow
x_{J}}(x_{J}-x_{J+1})^{-(1-\epsilon_{J})(1-\epsilon_{J+1})}~{}{\cal
N}(\epsilon_{J},\epsilon_{J+1})~{}\langle\prod_{\tilde{i}=1}^{N-1}\sigma_{\tilde{\epsilon}_{\tilde{i}},f}(x_{\tilde{i}})\rangle$
(146)
and the consistency relation the consistency relation for the normalization
coefficients
$C_{(N,1)}(\epsilon)=C_{(N-1,1)}(\tilde{\epsilon})~{}{\cal
N}(\epsilon_{J},\epsilon_{J+1})$ (147)
where $\tilde{\epsilon}$ are the twists of the $(N-1,1)$ theory defined by
$\tilde{\epsilon}_{j}=\epsilon_{j}$ for $j<J$,
$\tilde{\epsilon}_{J}=\epsilon_{J}+\epsilon_{J+1}-1$ for $j=J$ and
$\tilde{\epsilon}_{j}=\epsilon_{j+1}$ for $j>J$. Again all works in the
$\omega_{j}\rightarrow\infty$ ($x_{j}\rightarrow x_{1}$) limit.
#### 5.3.3 $(N,M)$ into $(N-1,M)$ with $2\leq M\leq N-2$ case
In this case we start from the general expression (141) and choose the sets
$S$ and ${\bar{S}}$ so that $J\in S$, $J\not\in{\bar{S}}$ and $J+1\not\in S$
then we now show that the new sets $\tilde{S}$ and $\tilde{\bar{S}}$ are given
by $\tilde{S}=S_{J}=S\setminus\\{J\\}$ and $\tilde{\bar{S}}={\bar{S}}$.
The previous choices are dictated by the need of having a simple and clean way
of computing the limit of $\det W_{S,{\bar{S}}}$. In particular while the
interval $[\omega_{J+1},\omega_{J}]$ vanishes $\partial{\cal X}^{(I\neq
J)}_{S}$ and $\partial{\bar{\cal X}}^{({\bar{I}})}_{\bar{S}}$ become the new
$\partial{\cal X}^{(I\neq J)}_{\tilde{S}}$ and $\partial{\bar{\cal
X}}^{({\bar{I}})}_{\tilde{\bar{S}}}$ and $\partial{\cal X}^{(J)}$ develops a
not integrable singularity at $\omega_{J}=\omega_{J+1}$ and gives the leading
singularity of $\det W_{S,{\bar{S}}}$, explicitly we find131313See appendix C
for an example of the computations involved in the special case $N=4$ $M=2$.
$\displaystyle\det W_{S,{\bar{S}}}\sim W^{(J)}_{(S,{\bar{S}})i=J-1}\det
W_{\tilde{S},\tilde{\bar{S}}}$ (148)
with
$\displaystyle
W^{(J)}_{(S,{\bar{S}})i=J-1}\sim(\omega_{J}-\omega_{J+1})^{1-\epsilon_{J}-\epsilon_{J+1}}~{}e^{-i\pi\epsilon_{J}}~{}B(\epsilon_{J},\epsilon_{J+1})~{}\prod_{l\neq
1,J,J+1}(\omega_{J}-\omega_{l})^{-\epsilon_{l}}\prod_{L\in
S_{J}}(\omega_{J}-\omega_{L})$ (149)
where $B(\cdot,\cdot)$ is Euler Beta function. Using these results into (141)
with a not so short computation we find the expected result
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle\sim_{x_{J+1}\rightarrow
x_{J}}(x_{J}-x_{J+1})^{-\epsilon_{J}\epsilon_{J+1}}~{}{\cal
M}(\epsilon_{J},\epsilon_{J+1})~{}\langle\prod_{i=1,i\neq
J}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle$ (150)
and a relation among the amplitude normalizations and the OPE normalization in
eq. (78) which up to a phase reads
$C_{(N,M)}(\epsilon)~{}[B(\epsilon_{J},\epsilon_{J+1})]^{-\frac{1}{2}}=C_{(N-1,M)}(\tilde{\epsilon})~{}{\cal
M}(\epsilon_{J},\epsilon_{J+1})$ (151)
where $\tilde{\epsilon}$ are the twists of the $(N-1,M)$ theory, i.e.
$\tilde{\epsilon}_{j}=\epsilon_{j}$ for $j<J$,
$\tilde{\epsilon}_{J}=\epsilon_{J}+\epsilon_{J+1}$ for $j=J$ and
$\tilde{\epsilon}_{j}=\epsilon_{j+1}$ for $j>J$.
It is worth noticing that the previous result (150) shows that eq. (141) is
valid even when $S$ has only one element. If we perform the reduction from
this case, i.e. with $(N+1,N-1)$ and we compare with the expression for the
$(N,N-1)$ amplitudes we deduce that
$\det
W_{(\emptyset,\bar{S})}\prod_{ord({\bar{I}})<ord({\bar{J}});{\bar{I}},{\bar{J}}\in\bar{S}}(\omega_{\bar{I}}-\omega_{\bar{J}})^{-1}\propto\prod_{2\leq
j<l\leq N}(\omega_{j}-\omega_{l})^{\epsilon_{j}+\epsilon_{l}-1}$ (152)
where $W_{(\emptyset,\bar{S})}$ is simply the matrix $\parallel
W_{i}^{\bar{I}}\parallel$. In other words certain determinants of order $N-2$
($card(\bar{S})=M-1=N-2$) of Lauricella hypergeometric functions of order
$N-3$ (since all $W_{i}^{\bar{I}}$ can be expressed using $I^{(N)}$) are a
product of powers. This could point to that also the general $\det
W_{S,{\bar{S}}}$ may be expressed as an elementary function.
For the special case where both $S$ and ${\bar{S}}$ have just one element,
i.e. for $N=4$, $M=2$ a direct and little different computation is needed but
the result is the same.
For checking the consistency of the approach and of the normalization
coefficients we determine in the next section it is worth considering the
$\omega_{J}\rightarrow\infty$ limit. The result for the normalization
coefficients in this case is based on the relation
$\displaystyle\det W_{S,{\bar{S}}}\sim$ $\displaystyle
W^{(J)}_{(S,{\bar{S}})i=J-1}~{}\omega_{J}^{\epsilon_{J}(N-2M-1)}~{}\det
W_{\tilde{S},\tilde{\bar{S}}}$ $\displaystyle\sim$ $\displaystyle
B(\epsilon_{J},1-\epsilon_{1}-\epsilon_{J})~{}\omega_{J}^{\epsilon_{J}(N-2M-1)-\epsilon_{1}}~{}\det
W_{\tilde{S},\tilde{\bar{S}}}$ (153)
and reads
$C_{(N,M)}(\epsilon)~{}[B(\epsilon_{J},1-\epsilon_{1}-\epsilon_{J})]^{-\frac{1}{2}}=C_{(N-1,M)}(\tilde{\epsilon})~{}{\cal
M}(\epsilon_{J},\epsilon_{1})$ (154)
with the new twists given by $\tilde{\epsilon}_{1}=\epsilon_{1}+\epsilon_{J}$,
$\tilde{\epsilon}_{j}=\epsilon_{j}$ for $1<j<J$ and
$\tilde{\epsilon}_{j}=\epsilon_{j+1}$ for $j>J$.
#### 5.3.4 $(N,M)$ into $(N-1,M-1)$ case
In this case we can choose the sets $S$ and ${\bar{S}}$ so that
${\bar{J}}\in{\bar{S}}$, ${\bar{J}}\not\in S$ and
${\bar{J}}+1\not\in{\bar{S}}$ then it is possible to show as in the previous
case that the new sets $\tilde{S}$ and $\tilde{\bar{S}}$ are given by
$\tilde{S}=S$ and $\tilde{\bar{S}}={\bar{S}}_{J}={\bar{S}}\setminus\\{J\\}$.
In particular it is possible to find analogously as before that the
determinant behaves in the $x_{\bar{J}}\rightarrow x_{{\bar{J}}+1}$ limit as
$\displaystyle\det W_{S,{\bar{S}}}\sim
W^{({\bar{J}})}_{(S,{\bar{S}})i={\bar{J}}-1}\det
W_{\tilde{S},\tilde{\bar{S}}}$ (155)
and the amplitude reduction gives
$\displaystyle\langle\prod_{i=1}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle\sim(x_{J}-x_{J+1})^{-(1-\epsilon_{J})(1-\epsilon_{J+1})}\langle\prod_{i=1,i\neq
J}^{N}\sigma_{\epsilon_{i},f_{i}=f}(x_{i})\rangle$ (156)
It follows a relation among the amplitude normalizations and the OPE
normalization in eq. (78) which up to a phase reads
$C_{(N,M)}(\epsilon)~{}[B(1-\epsilon_{J},1-\epsilon_{J+1})]^{-\frac{1}{2}}=C_{(N-1,M-1)}(\tilde{\epsilon})~{}{\cal
N}(\epsilon_{J},\epsilon_{J+1})$ (157)
where $\tilde{\epsilon}$ are the twists of the $(N-1,M-1)$ theory, i.e.
$\tilde{\epsilon}_{j}=\epsilon_{j}$ for $j<J$,
$\tilde{\epsilon}_{J}=\epsilon_{J}+\epsilon_{J+1}-1$ for $j=J$ and
$\tilde{\epsilon}_{j}=\epsilon_{j+1}$ for $j>J$.
As in the previous subsection starting from the $(N+1,2)$ amplitude and
reducing it to $(N,1)$ we deduce that
$\det W_{(S,\emptyset)}\prod_{ord(I)<ord(J);I,J\in
S}(\omega_{I}-\omega_{J})^{-1}\propto\prod_{2\leq j<l\leq
N}(\omega_{j}-\omega_{l})^{1-\epsilon_{j}-\epsilon_{l}}$ (158)
where $W_{(S,\emptyset)}$ is simply the matrix $\parallel W_{i}^{I}\parallel$.
### 5.4 Amplitudes and OPEs normalization
We normalize the 2-point amplitude as
$\langle\sigma_{\epsilon}(x)\sigma_{1-\epsilon}(y)\rangle=\frac{1}{(x-y)^{\epsilon(1-\epsilon)}}$
(159)
This normalization is not unique since any redefinition as
$\sigma_{\epsilon}\rightarrow{\cal R}(\epsilon)\sigma_{\epsilon}$ with ${\cal
R}(\epsilon)~{}{\cal R}(1-\epsilon)=1$ would work. In particular this kind of
redefinition can only be seen in amplitudes with at least three twist fields
since it leaves unchanged amplitudes involving two twist fields and an
arbitrary number of untwisted fields therefore it cannot be fixed factorizing
a 4 twists into an untwisted channel. If we require the normalizations to be
invariant under the symmetry $\epsilon\leftrightarrow 1-\epsilon$ then all the
normalizations are completely fixed (up one constant $k$ and phases) to be
$\displaystyle C_{(N,1)}$
$\displaystyle=k^{N-2}\left[\prod_{j=1}^{N}\frac{\Gamma(1-\epsilon_{j})}{\Gamma(\epsilon_{j})}\right]^{1/4}$
$\displaystyle C_{(N,M)}$
$\displaystyle=k^{N-2}\left[\frac{\prod_{j=2}^{N}\Gamma(\epsilon_{j})\Gamma(1-\epsilon_{j})}{\Gamma(\epsilon_{1})\Gamma(1-\epsilon_{1})}\right]^{1/4}~{}~{}~{}~{}2\leq
M\leq N-2$ $\displaystyle C_{(N,N-1)}$
$\displaystyle=k^{N-2}\left[\prod_{j=1}^{N}\frac{\Gamma(\epsilon_{j})}{\Gamma(1-\epsilon_{j})}\right]^{1/4}$
(160)
along with the OPE normalizations
$\displaystyle{\cal M}(\alpha,\beta)$
$\displaystyle=k\left[\frac{\Gamma(1-\alpha)}{\Gamma(\alpha)}\frac{\Gamma(1-\beta)}{\Gamma(\beta)}\frac{\Gamma(\alpha+\beta)}{\Gamma(1-\alpha-\beta)}\right]^{1/4}$
$\displaystyle=k\left[\frac{\Gamma(1-\alpha)}{\Gamma(\alpha)}\frac{\Gamma(1-\beta)}{\Gamma(\beta)}\frac{\Gamma(1-\gamma)}{\Gamma(\gamma)}\right]^{1/4}~{}~{}~{}~{}\alpha+\beta+\gamma=1$
$\displaystyle{\cal N}(\alpha,\beta)$
$\displaystyle=k\left[\frac{\Gamma(\alpha)}{\Gamma(1-\alpha)}\frac{\Gamma(\beta)}{\Gamma(1-\beta)}\frac{\Gamma(2-\alpha-\beta)}{\Gamma(\alpha+\beta-1)}\right]^{1/4}$
$\displaystyle=k\left[\frac{\Gamma(\alpha)}{\Gamma(1-\alpha)}\frac{\Gamma(\beta)}{\Gamma(1-\beta)}\frac{\Gamma(\delta)}{\Gamma(1-\delta)}\right]^{1/4}~{}~{}~{}~{}\alpha+\beta+\delta=2$
(161)
which also respect the symmetry $\epsilon\leftrightarrow 1-\epsilon$ as ${\cal
N}(\alpha,\beta)={\cal M}(1-\alpha,1-\beta)$. It is at first sight surprising
that there is not symmetry among the twist operators in the $M\neq 1,N-1$ case
but this is due to two reasons. The first is our choice of using a
$SL(2,\mathbb{R})$ invariant formalism which singles out some points and the
second is that not all twist operators are on the same footing since some
couples of twists sum to a quantity less than one while others to one bigger
than one. These normalization are the “square root” of the ones found in [17]
for the $N=4$ closed string case and matches those obtained for $N=3$ in the
magnetic brane case in [9] and for $N=4$ case in [10].
Let us see how we can get the previous results by exploiting the consequences
of equations of the previous subsections such as eq.s (151) and (157). First
we notice that we can always normalize the 2-points correlator as chosen
because the generic normalization factor $C_{(2,1)}(\epsilon,1-\epsilon)$ is
symmetric in the exchange $\epsilon\leftrightarrow 1-\epsilon$ hence we can
redefine the twist operators as
$\sigma_{\epsilon}=\tilde{\sigma}_{\epsilon}/\sqrt{C_{(2,1)}(\epsilon,1-\epsilon)}$.
From the reduction $(N=3,M=1)$ to $(\tilde{N}=2,\tilde{M}=1)$ with the help of
eq. (145) we find that ${\cal M}(\alpha,\beta)=C_{(3,1)}(\alpha,\beta,\gamma)$
with $\alpha+\beta+\gamma=1$ has the following basic symmetries
$\displaystyle{\cal M}(\alpha,\beta)$ $\displaystyle={\cal
M}(\beta,\alpha)={\cal M}(\alpha,1-\alpha-\beta)$ (162)
and all the others which follow from them.
In a similar way from the $(N=3,M=2)$ to $(\tilde{N}=2,\tilde{M}=1)$ reduction
and from eq. (147) we find
$\displaystyle{\cal N}(\alpha,\beta)$ $\displaystyle={\cal
N}(\beta,\alpha)={\cal N}(\alpha,2-\alpha-\beta)$ (163)
Now we can consider the $(N=4,M=2)$ to $(\tilde{N}=3,\tilde{M}=1)$ reduction
in two different ways. Either with
$(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})\rightarrow(\epsilon_{1},\epsilon_{2},\epsilon_{3}+\epsilon_{4}-1)$
which implies
$\displaystyle
C_{(4,2)}(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})[B(1-\epsilon_{3},1-\epsilon_{4})B(\epsilon_{2},\epsilon_{3}+\epsilon_{4}-1)]^{\frac{1}{2}}$
$\displaystyle={\cal
N}(\epsilon_{3},\epsilon_{4})C_{(3,1)}(\epsilon_{1},\epsilon_{2},\epsilon_{3}+\epsilon_{4}-1)$
(164)
or with
$(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})\rightarrow(\epsilon_{1},\epsilon_{2}+\epsilon_{3},\epsilon_{4}-1)$
which implies
$\displaystyle
C_{(4,2)}(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})[B(1-\epsilon_{3},1-\epsilon_{2})B(\epsilon_{4},\epsilon_{3}+\epsilon_{2}-1)]^{\frac{1}{2}}$
$\displaystyle={\cal
N}(\epsilon_{3},\epsilon_{2})C_{(3,1)}(\epsilon_{1},\epsilon_{2}+\epsilon_{3}-1,\epsilon_{4})$
(165)
Now taking the ratio of the two previous equations and using the symmetries of
${\cal M}$ and ${\cal N}$ we are led to the minimal ansatz
$\displaystyle{\cal M}(\alpha,\beta)$
$\displaystyle=k[\Gamma(\alpha)\Gamma(\beta)\Gamma(1-\alpha-\beta)]^{a}[\Gamma(1-\alpha)\Gamma(1-\beta)\Gamma(\alpha+\beta)]^{b}$
$\displaystyle{\cal N}(\alpha,\beta)$
$\displaystyle=k[\Gamma(\alpha)\Gamma(\beta)\Gamma(2-\alpha-\beta)]^{c}[\Gamma(1-\alpha)\Gamma(1-\beta)\Gamma(\alpha+\beta-1)]^{d}$
(166)
which gives an overconstrained system when plugged back into the ratio
constraint whose solution is $a=-b$ and $c=-d=\frac{1}{2}+a$. This solution
immediately yields both $C_{(3,2)}$ and $C_{(4,2)}$. Imposing the symmetry
$\epsilon\leftrightarrow 1-\epsilon$ then selects $a=-\frac{1}{4}$. It is then
easy to generalize to the full expressions. These can be checked in different
limits also when we consider $\omega_{j}\rightarrow\infty$ using to eq. (154).
Acknowledgments We thank M. Bianchi for pointing out a mistake in a figure.
## Appendix A Details on rewriting the classical action.
We want to give some details on the use of KLT technique for reducing the
integral
$J^{(N)}(\alpha+n,\bar{\alpha}+\bar{n})=\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}dy\prod_{j=2}^{N}(x+iy-\omega_{j})^{\alpha_{j}+n_{j}}(x-iy-\omega_{j})^{\bar{\alpha}_{j}+\bar{n}_{j}}$
(167)
with $n_{j},\bar{n}_{j}\in\mathbb{Z}$ to the sum of products of an holomorphic
and antiholomorphic integral. First we interpret the previous integral in $y$
as a line integral in the complex plane $Y=t+iy$. In the variable $Y$ the
integrand has cuts in $\pm(\omega_{j}-x)$, the main issue is then to properly
define the phase of
$(x+Y-\omega)^{\alpha}(x-Y-\omega)^{\bar{\alpha}}=|x+Y-\omega|^{\alpha}|x-Y-\omega|^{\bar{\alpha}}e^{i(\phi+\bar{\phi})}.$
(168)
The proper choice is shown in fig. (11) and is constrained by the request that
when $Y=iy$ and $\alpha=\bar{\alpha}$ then $\phi+\bar{\phi}=0$.
$Y=t+iy$$t$$y$$-(x-\omega)$$(x-\omega)$$\phi$$\bar{\phi}=-\phi$ Figure 11:
Proper definition of angles $\phi$ and $\bar{\phi}$ and therefore of phases
when $x-\omega>0$. When $x-\omega<0$ we substitute
$(x-\omega)\rightarrow-(x-\omega)$.
We can then rotate clockwise the path in $Y$ plane, change variables as
$\xi=x+t$, $\eta=x-t$ and then we can rewrite the $J^{(N)}$ integral as
$\displaystyle J^{(N)}(\alpha+n,\bar{\alpha}+\bar{n})=$
$\displaystyle-\frac{i}{2}\int_{-\infty}^{+\infty}d\xi\int_{-\infty}^{+\infty}d\eta\prod_{j=2}^{N}|\xi-\omega_{j}|^{\alpha_{j}}|\eta-\omega_{j}|^{\bar{\alpha}_{j}}$
$\displaystyle~{}~{}\times(\xi-\omega_{j})^{n_{j}}(\eta-\omega_{j})^{\bar{n}_{j}}$
$\displaystyle~{}~{}\times
e^{-i\pi\alpha_{j}~{}\theta(\omega_{j}-\xi)~{}\theta(\eta-\omega_{j})}e^{-i\pi\bar{\alpha}_{j}~{}\theta(\xi-\omega_{j})~{}\theta(\omega_{j}-\eta)}$
(169)
If $\bar{\alpha}=\alpha$ then we can proceed as in KLT. We fix $\xi$ and we
exam the $\eta$ integral. Each factor of the integrand can then be rewritten
as
$\displaystyle|\eta-\omega_{j}|^{\alpha_{j}}e^{-i\pi\alpha_{j}~{}\theta((\omega_{j}-\xi)(\eta-\omega_{j}))}$
$\displaystyle=\theta(\omega_{j}-\xi)~{}[\omega_{j}-(\eta+i0^{+})]^{\alpha_{j}}$
$\displaystyle+\theta(-\omega_{j}+\xi)~{}e^{+i\pi\alpha_{j}}[\omega_{j}-(\eta-i0^{+})]^{\alpha_{j}}$
(170)
when we choose the phase in the complex $\eta$ plane as in fig. (12),
obviously other choices would do the job as well. In words this means that
when $\xi<\omega_{j}$ we run above the cut from $-\infty$ to $\omega_{j}$ in
the complex $\eta$ plane while we run below the cut when $\omega_{j}<\xi$.
$Re\eta$$Im\eta$$\omega$ Figure 12: Definition of phase in $\eta$ plane in the
range $(-2\pi,0)$.
Hence the original integral can be written as
$\displaystyle J^{(N)}(\alpha+n,\alpha+\bar{n})=$
$\displaystyle-\frac{i}{2}\sum_{i=N-1}^{2}\int_{-\omega_{i+1}}^{\omega_{i}}d\xi\prod_{j=2}^{N}|\xi-\omega_{j}|^{\alpha_{j}}(\xi-\omega_{j})^{n_{j}}$
$\displaystyle\times
e^{i\sum_{l=i}^{2}\alpha_{l}}\int_{C_{i}}d\eta\prod_{j=2}^{N}(\omega_{j}-\eta)^{\alpha_{j}}(\eta-\omega_{j})^{\bar{n}_{j}}$
(171)
where the path $C_{i}$ is given in fig. (13). In particular the integrals
$\int_{\omega_{2}}^{\infty}d\xi$ and $\int^{\omega_{N}}_{-\infty}d\xi$ do not
contribute since the integrals over $d\eta$ runs either above or below the
cuts and are zero because of Jordan lemma.
$\omega_{i}$$\omega_{i+1}$$C_{i}$ Figure 13: The path $C_{i}$ in the complex
$\eta$ plane for $\omega_{i+1}<\xi<\omega_{i}$.
We can then rewrite the $C_{i}$ integral as an integral above (or below
depending the cases) the cuts plus a remainder. The final result is then
$\displaystyle J^{(N)}(\alpha+n,\alpha+\bar{n})=$
$\displaystyle-\sum_{i=2}^{N-1}\sum_{l=i+1}^{N}\sin\left(\pi\sum_{j=i+1}^{l}\alpha_{j}\right)$
$\displaystyle\times\int_{\omega_{i+1}}^{\omega_{i}}d\xi\prod_{j=2}^{N}|\xi-\omega_{j}|^{\alpha_{j}}(\xi-\omega_{j})^{n_{j}}$
$\displaystyle\times\int_{\omega_{l+1}}^{\omega_{l}}d\eta\prod_{j=2}^{N}|\omega_{j}-\eta|^{\alpha_{j}}(\eta-\omega_{j})^{\bar{n}_{j}}$
(172)
## Appendix B Fixing the singular part of $g(z,w)$ in a consistent way with
$N\rightarrow N-1$ reduction
Let us suppose that all coefficients $c_{ns}(\omega_{j})$ depend on
$\omega_{j}$ ($3\leq j\leq N-1$) in an analytic way. We want to show that it
is then possible to fix then in a recursive way starting from those of the
$N=3,M=1$ case. This can be done considering two limits $x_{j}\rightarrow
x_{N}$, i.e. $\omega_{j}\rightarrow 0$ and $x_{j}\rightarrow x_{1}$, i.e.
$\omega_{j}\rightarrow\infty$.
Combining the two cases when $\epsilon_{1}+\epsilon_{j}<1$ and
$\epsilon_{j}+\epsilon_{N}<1$ we get
$\displaystyle c^{(N,M)}_{n,s}(\omega,\epsilon)$
$\displaystyle=c^{(N-1,M)}_{n-1,s}(\check{\omega},\check{\epsilon})-c^{(N-1,M)}_{n,s}(\hat{\omega},\hat{\epsilon})\omega_{j}$
$\displaystyle c^{(N,M)}_{0,s}(\omega,\epsilon)$
$\displaystyle=-c^{(N-1,M)}_{n,s}(\hat{\omega},\hat{\epsilon})\omega_{j}$
$\displaystyle c^{(N,M)}_{N-M,s}(\omega,\epsilon)$
$\displaystyle=c^{(N-1,M)}_{N-M-1,s}(\check{\omega},\check{\epsilon})$ (173)
when $1\leq n\leq N-M-1,~{}~{}0\leq s\leq M$ and where we have defined
$\displaystyle\left\\{\begin{array}[]{c
r}\check{\epsilon}_{\check{\imath}}=\epsilon_{\check{\imath}}&\check{\imath}=1,\dots
j-1\\\
\check{\epsilon}_{\check{\imath}}=\epsilon_{\check{\imath}+1}&\check{\imath}=j,\dots
N-2\\\
\check{\epsilon}_{N-1}=\epsilon_{N}+\epsilon_{j}-\theta(\epsilon_{N}+\epsilon_{j}>1)\end{array}\right.$
(177)
and
$\displaystyle\left\\{\begin{array}[]{c
r}\hat{\epsilon}_{1}=\epsilon_{1}+\epsilon_{j}-\theta(\epsilon_{1}+\epsilon_{j}>1)\\\
\hat{\epsilon}_{\hat{\imath}}=\epsilon_{\hat{\imath}}&\hat{\imath}=2,\dots
j-1\\\
\hat{\epsilon}_{\hat{\imath}}=\epsilon_{\hat{\imath}+1}&\hat{\imath}=j,\dots
N-1\end{array}\right.$ (181)
and similar relations between $\check{\omega}$ with $\omega$ and
$\hat{\omega}$ with $\omega$. For example applying the previous formula to the
$N=4,M=2$ case we get
$\displaystyle g_{s}^{(4,2)}(z,w)$
$\displaystyle=\frac{1}{(z-w)^{2}}\prod_{2=2}^{4}\frac{(\omega_{z}-\omega_{j})^{\epsilon_{j}-1}}{(\omega_{w}-\omega_{j})^{\epsilon_{j}}}\Big{\\{}\epsilon_{1}\omega_{z}^{2}\omega_{w}+(1-\epsilon_{1})\omega_{z}\omega_{w}^{2}$
$\displaystyle+(1-\epsilon_{1}-\epsilon_{2})\omega_{z}^{2}-[\epsilon_{3}+\epsilon_{4}+(\epsilon_{1}+\epsilon_{3})\omega_{3}]\omega_{z}\omega_{w}+(1-\epsilon_{2}-\epsilon_{4})\omega_{3}\omega_{w}^{2}$
$\displaystyle+(1-\epsilon_{4})\omega_{3}\omega_{z}+\epsilon_{4}\omega_{3}\omega_{w}\Big{\\}}$
(182)
## Appendix C Some details on the $N=4$ reduction
As an example of the way we performed the $N\rightarrow N-1$ we give now some
details on the $N=4$ $M=2$ case, in particular we consider
$\omega_{3}\rightarrow 0$ when $\epsilon_{3}+\epsilon_{4}>1$. Under this
conditions we want to compute the behavior of
$\displaystyle\det W=\left|\begin{array}[]{c c}W^{3}_{1}&W^{\bar{3}}_{1}\\\
W^{3}_{2}&W^{\bar{3}}_{2}\end{array}\right|$ (185)
where we have chosen $S=\bar{S}=\\{3\\}$. The different entries of the
determinant have the following limits
$\displaystyle W^{3}_{1}$
$\displaystyle=\int^{1}_{\omega_{3}}d\omega~{}(\omega-1)^{\epsilon_{2}-1}~{}(\omega-\omega_{3})^{\epsilon_{3}-1}~{}\omega^{\epsilon_{4}-1}\sim\int^{1}_{0}d\omega~{}(\omega-1)^{\epsilon_{2}-1}~{}\omega^{(\epsilon_{3}+\epsilon_{4}-1)-1}$
(186)
$\displaystyle=e^{i\pi(\epsilon_{2}-1)}B(\epsilon_{2},\epsilon_{3}+\epsilon_{4}-1),$
(187)
$\displaystyle W^{3}_{2}$
$\displaystyle=\int^{\omega_{3}}_{0}d\omega~{}(\omega-1)^{\epsilon_{2}-1}~{}(\omega-\omega_{3})^{\epsilon_{3}-1}~{}\omega^{\epsilon_{4}-1}$
$\displaystyle=\omega_{3}^{\epsilon_{3}+\epsilon_{4}-1}\int^{1}_{0}dt~{}(\omega_{3}t-1)^{\epsilon_{2}-1}~{}(t-1)^{\epsilon_{3}-1}~{}t^{\epsilon_{4}-1}$
$\displaystyle\sim\omega_{3}^{\epsilon_{3}+\epsilon_{4}-1}~{}e^{i\pi(\epsilon_{2}+\epsilon_{3})}~{}B(\epsilon_{3},\epsilon_{4}),$
(188)
and
$\displaystyle W^{\bar{3}}_{1}$ $\displaystyle=\int^{1}_{\omega_{3};\omega\in
H^{-}}d\omega~{}(\omega-1)^{-\epsilon_{2}}~{}(\omega-\omega_{3})^{-\epsilon_{3}}~{}\omega^{-\epsilon_{4}}$
$\displaystyle=\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}\int^{1/\omega_{3}}_{1}dt~{}(\omega_{3}t-1)^{-\epsilon_{2}}~{}(t-1)^{-\epsilon_{3}}~{}t^{-\epsilon_{4}}$
$\displaystyle\sim~{}e^{+i\pi\epsilon_{2}}\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}\int^{+\infty}_{1}dt~{}(t-1)^{-\epsilon_{3}}~{}t^{-\epsilon_{4}}=~{}e^{+i\pi\epsilon_{2}}\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}~{}B(\epsilon_{3}+\epsilon_{4}-1,1-\epsilon_{3})$
(189)
finally the limit of the last entry can be obtained using again the
substitution $\omega=\omega_{3}t$ to be
$\displaystyle W^{\bar{3}}_{2}$ $\displaystyle=\int^{\omega_{3}}_{0;\omega\in
H^{-}}d\omega~{}(\omega-1)^{-\epsilon_{2}}~{}(\omega-\omega_{3})^{-\epsilon_{3}}~{}\omega^{-\epsilon_{4}}\sim~{}e^{+i\pi(\epsilon_{2}+\epsilon_{3})}\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}~{}B(1-\epsilon_{3},1-\epsilon_{4})$
(190)
Inserting all the previous asymptotic behaviors into the determinant we get
its $\omega_{3}\rightarrow 0$, $\epsilon_{3}+\epsilon_{4}>1$ limit to be
$\displaystyle\det W$ $\displaystyle=\left|\begin{array}[]{c
c}e^{i\pi(\epsilon_{2}-1)}B(\epsilon_{2},\epsilon_{3}+\epsilon_{4}-1)&e^{i\pi\epsilon_{2}}\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}~{}B(\epsilon_{3}+\epsilon_{4}-1,1-\epsilon_{3})\\\
e^{i\pi(\epsilon_{3}+\epsilon_{4})}\omega_{3}^{\epsilon_{3}+\epsilon_{4}-1}~{}B(\epsilon_{3},\epsilon_{4})&e^{i\pi(\epsilon_{2}+\epsilon_{3})}\omega_{3}^{1-\epsilon_{3}-\epsilon_{4}}~{}B(1-\epsilon_{3},1-\epsilon_{4})\end{array}\right|$
(193)
$\displaystyle\sim\omega_{3}^{(1-\epsilon_{3})+(1-\epsilon_{4})-1}~{}B(1-\epsilon_{3},1-\epsilon_{4})~{}B(\epsilon_{2},1-(1-\epsilon_{3})-(1-\epsilon_{4}))$
(194)
where it is worth noticing that we can drop the relative phases since only one
product is the leading one. This happens luckily also for all the other
computations which are needed to compute all the $N\rightarrow N-1$ reduction.
## References
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* [4] J. R. David, “Tachyon condensation in the D0 / D4 system,” JHEP 0010 (2000) 004 [hep-th/0007235].
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R. Russo, S. Sciuto, “The Twisted open string partition function and Yukawa
couplings,” JHEP 0704 (2007) 030. [hep-th/0701292].
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* [12] M. Bianchi, G. Pradisi and A. Sagnotti, “Planar duality in the discrete series,” Phys. Lett. B 273 (1991) 389.
* [13] H. Kawai, D. C. Lewellen and S. H. H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl. Phys. B 269 (1986) 1.
* [14] J. J. Atick, L. J. Dixon, P. A. Griffin, D. Nemeschansky, “Multiloop Twist Field Correlation Functions For Z(n) Orbifolds,” Nucl. Phys. B298 (1988) 1-35.
M. Bershadsky, A. Radul, “Conformal Field Theories with Additional Z(N)
Symmetry,” Int. J. Mod. Phys. A2 (1987) 165-178.
* [15] I. Pesando, “Strings in an arbitrary constant magnetic field with arbitrary constant metric and stringy form factors,” JHEP 1106 (2011) 138 [arXiv:1101.5898 [hep-th]].
I. Pesando, Phys. Lett. B 668 (2008) 324 [arXiv:0804.3931 [hep-th]].
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|
arxiv-papers
| 2012-06-07T09:48:42 |
2024-09-04T02:49:31.575776
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Igor Pesando",
"submitter": "Pesando Igor",
"url": "https://arxiv.org/abs/1206.1431"
}
|
1206.1566
|
# Non-Pauli Observables for CWS Codes
Douglas F.G. Santiago†, Renato Portugal‡, Nolmar Melo‡
${\dagger}$ Universidade Federal dos Vales do Jequitinhonha e Mucuri
Diamantina, MG 39100000, Brazil
douglassant@gmail.com
${\ddagger}$ Laboratório Nacional de Computação Científica
Petrópolis, RJ 25651-075, Brazil
portugal@lncc.br, nolmar@lncc.br
###### Abstract
It is known that nonadditive quantum codes are more optimal for error
correction when compared to stabilizer codes. The class of codeword stabilized
codes (CWS) provides tools to obtain new nonadditive quantum codes by reducing
the problem to finding nonlinear classical codes. In this work, we establish
some results on the kind of non-Pauli operators that can be used as decoding
observables for CWS codes and describe a procedure to obtain these
observables.
## 1 Introduction
It is known that quantum computers are able to solve hard problems in
polynomial time and to increase the speed of many algorithms [1, 2, 3, 4].
Decoherence problems are present in any practical implementation of quantum
devices, especially in large-scale quantum computer. Quantum error correcting
codes (QECCs) can be used to solve these problems by using extra qubits and
storing information using redundancy [5, 6, 7, 8].
The framework of stabilizer codes was used to obtain a large class of
important quantum codes [9, 10, 11]. A code is called a stabilizer code if it
is in the joint positive eigenspace of a commutative subgroup of Pauli group.
In certain cases, these codes are suboptimal, because there is larger class,
called nonadditive codes.
An important class of nonadditive codes, called CWS, has been studied recently
[12, 13, 14, 15, 16]. The framework of CWS codes generalizes the stabilizer
code formalism and has been used to build some good nonadditive codes, in some
cases enlarging the logical space of stabilizer codes of the same length. On
the one hand, many papers address codification procedures for CWS codes, and
on the other hand few papers address decodification procedures. Decoding
observables of specific codes are known, such as the ((9,12,3)) and
((10,24,3)) codes and associated families [13, 17]. A generic decoding
procedure for binary CWS codes was proposed in Ref. [18] and extended for
nonbinary CWS codes in Ref. [19].
In this work, we establish a condition to the existence of non-Pauli CWS
observables, that can be written in terms of the stabilizers associated to the
CWS code. We describe a procedure to find these observables, which is
specially useful for CWS codes that are close to stabilizer codes.
This paper is divided in the following parts. In Section 2, we review the
structure of CWS codes and introduce the notations that will used in this
work. In Section 3, we present the main results, in special Theorem 2, its
corollary and the procedure to find non-Pauli observables. In Section 4, we
give an example and in Section 5, we present the conclusions.
## 2 CWS codes
An $((n,K))$ CWS code in the Hilbert space $\mathcal{H}^{n}$ is described by
1. 1.
A stabilizer group $S=\langle s_{1},\ldots,s_{n}\rangle$, where $\\{s_{i}\\}$
is a generator set of independent and commutative Pauli operators (elements of
Pauli group ${\mathcal{G}}_{n}$). This group stabilizes a single codeword
$|\psi\rangle$;
2. 2.
A set of Pauli operators $W=\\{W_{1},\ldots,W_{K}\\}$. The set
$\\{W_{j}|\psi\rangle\\}$ spans the CWS code and each $W_{i}$ is called a
codeword operator.
Cross et al. [20] have showed that any binary CWS code is equivalent to a CWS
code in a standard form, which is characterized by: (1) a graph of $n$
vertices, (2) a set of Pauli operators $s_{i}=X_{i}Z^{r_{i}}$, where $r_{i}$
is the $i$-th line of the adjacency matrix $(M)$, and (3) codeword operators
$W_{j}=Z^{C_{j}}$, where $C_{1}=(0,\ldots,0)$, that is, $W_{1}=I$.
In the standard form, correctable Pauli errors can be expressed as binary
strings. A Pauli error $E=Z^{V}X^{U}$, where $V,U\in\mathbb{F}_{2}^{n}$, can
be mapped modulo a phase to an error $Z^{\mathrm{Cl}_{S}(E)}$ through function
${\mathrm{Cl}_{S}(Z^{V}X^{U})=V+MU\in\mathbb{F}_{2}^{n}.}$
The problem of finding good CWS quantum codes is reduced to the problem of
finding good classical codes. Theorem 3 of Cross et al. [20] states that a CWS
code in standard form with stabilizer $S$ spanned by
$\\{Z^{C_{i}}|\psi\rangle\\}$ detects errors in the set
${\mathcal{E}}=\\{E_{i}\\}$ if and only if the classical code $\\{C_{i}\\}$
detects errors in $\mathrm{Cl}_{S}({\mathcal{E}})$. This result is valid
because, for all $E_{i}$ satisfying $\mathrm{Cl}_{S}(E_{i})=0$, we disregard
all binary vectors $C$ such that $Z^{C}E_{i}=E_{i}Z^{C}$.
Our first goal is to analyze which Pauli operators can be used as observables
for CWS codes. If $W$ is the set of codeword operators and $g\in{N_{S}(W)}$,
where ${N_{S}(W)}$ is the normalizer of $W$ in $S$, $g$ can be used as a
decoding observable. This follows from the equalities
$\displaystyle
gE_{i}W_{j}|\psi\rangle=m_{i}E_{i}gW_{j}|\psi\rangle=m_{i}E_{i}W_{j}g|\psi\rangle=m_{i}E_{i}W_{j}|\psi\rangle,$
where $m_{i}=\pm 1$. It means that, for a fixed $E_{i}$ and for all $W_{j}$,
$E_{i}W_{j}|\psi\rangle$ lies entirely in the eigenspace associated with the
eigenvalue $m_{i}$ of $g$. So, there is no information leakage after the
measurement of observable $g$. When a CWS code is a stabilizer code, the
decoding procedure uses a generating set of ${N_{S}(W)}$ as observables. This
is not the only choice, because we can use non-Pauli observables.
Our second goal is to establish some results on the existence and form of non-
Pauli CWS observables on the group algebra $\mathbb{R}[S]$ over $\mathbb{R}$
spanned by $S$. An operator $A\in\mathbb{R}[S]$ can be written as
$\displaystyle A=\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}{\cal S}^{V},$
where we use the notation ${\cal S}^{V}$ as an element of $S=\langle
s_{1},\ldots,s_{n}\rangle$ given by
${\cal S}^{V}=s_{1}^{v_{1}}{\cdots}s_{n}^{v_{n}},$
where $V=(v_{i},\ldots,v_{n})$ is a binary vector. We will assign a type to
operator $A$ depending on the number of non-zeros coefficients $\alpha_{V}$.
This type notion is captured in the next definition.
* Definition
A type-$i$ observable is an operator $A\in\mathbb{R}[S]$ that satisfies
$A^{2}=I$ and is exactly a linear combination of $i$ different elements of
$S$.
Note that this definition makes sense because group $S$ is a subset of a basis
of the Hilbert space, and $\displaystyle
A=\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}{\cal S}^{V}$ is written in a unique
way.
Type-$1$ observables are Pauli operators. It is straightforward to show that
there are no type-2 or type-3 observables. In this work, we consider only
type-4 observables.
## 3 Main Results
If a unitary operator $A$ is an observable, then $A^{2}=I$. Since we are
dealing with observables in $\mathbb{R}[S]$, we have the following
proposition:
###### Proprosition 1.
Let $S$ be the stabilizer group of a CWS code in standard form and
$\displaystyle A=\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}{\cal S}^{V}$ an
element of $\mathbb{R}[S]$. Then, $A^{2}=I$ if and only if
$\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}^{2}=1\textrm{{ and
}}{\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}\alpha_{V+U}=0,}\;\forall
U\in\mathbb{F}_{2}^{n}\setminus\\{0\\}.$ (1)
###### Proof.
Take
$A^{2}=\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}^{2}I+\sum_{V\neq
V^{\prime}}\alpha_{V}\alpha_{V^{\prime}}{\cal S}^{V}{\cal S}^{V^{\prime}}.$
All terms ${\cal S}^{U}\in S\setminus\\{I\\}$ are present in the second sum,
each one as many times as $V+V^{\prime}=U$, that is, $2^{n}$. So, we can
rewrite this equation as
$\displaystyle A^{2}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}^{2}I+\displaystyle\sum_{U\in\mathbb{F}_{2}^{n}\setminus\\{0\\}}\sum_{V+V^{\prime}=U}\alpha_{V}\alpha_{V^{\prime}}{\cal
S}^{U}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}^{2}I+\displaystyle\sum_{U\in\mathbb{F}_{2}^{n}\setminus\\{0\\}}{\cal
S}^{U}\sum_{V\in\mathbb{F}_{2}^{n}}\alpha_{V}\alpha_{V+U}.$
Then, result (1) follows.
∎
Type-4 observables can be restricted by the following theorem:
###### Theorem 1.
A is a type-4 observable if and only if
$A=\pm\frac{{\cal S}^{V}}{2}\left(-I+{\cal S}^{V_{1}}+{\cal S}^{V_{2}}+{\cal
S}^{V_{1}+V_{2}}\right)$ (2)
with $V_{1}\neq V_{2}\in\mathbb{F}_{2}^{n}\setminus\\{0\\}$ and
$V\in\mathbb{F}_{2}^{n}$.
###### Proof.
If $A$ is given by Eq. (2), then it is straightforward to verify that
$A^{2}=I$. So, $A$ is a type-4 observable.
Reciprocally, take a type-4 observable $A=\alpha_{1}{\cal
S}^{U_{1}}+\alpha_{2}{\cal S}^{U_{2}}+\alpha_{3}{\cal
S}^{U_{3}}+\alpha_{4}{\cal S}^{U_{4}}$. We have
$\displaystyle A^{2}$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{4}\alpha_{i}^{2}\right)I+2\alpha_{1}\alpha_{2}{\cal
S}^{U_{1}+U_{2}}+2\alpha_{1}\alpha_{3}{\cal
S}^{U_{1}+U_{3}}+2\alpha_{1}\alpha_{4}{\cal S}^{U_{1}+U_{4}}+$ $\displaystyle
2\alpha_{2}\alpha_{3}{\cal S}^{U_{2}+U_{3}}+2\alpha_{2}\alpha_{4}{\cal
S}^{U_{2}+U_{4}}+2\alpha_{3}\alpha_{4}{\cal S}^{U_{3}+U_{4}}.$
The $\alpha$’s are not zero. So, $A^{2}=I$ implies that
$\sum_{i=1}^{4}\alpha_{i}^{2}=1$
and the sum of the remaining 6 terms is zero, which implies that
$U_{1}+U_{2}=U_{3}+U_{4}$, $U_{1}+U_{3}=U_{2}+U_{4}$ and
$U_{1}+U_{4}=U_{2}+U_{3}$.
We can rewrite $A$ by taking $V=U_{1}$, $V_{1}=U_{1}+U_{2}$ and
$V_{2}=U_{1}+U_{3}$, then $V_{1}+V_{2}=U_{1}+U_{4}$ and
$A=\frac{{\cal S}^{V}}{2}\left(\alpha_{1}I+\alpha_{2}{\cal
S}^{V_{1}}+\alpha_{3}{\cal S}^{V_{2}}+\alpha_{4}{\cal
S}^{V_{1}+V_{2}}\right).$
Note that $V_{1}\neq V_{2}$ and $V_{1}\neq 0\neq V_{2}$ because $U_{i}\neq
U_{j}$, if $i\neq j$. The solutions obeying constraints (1) belong to the set
$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})\in$
$\displaystyle\pm\frac{1}{2}\\{$
$\displaystyle(-1,1,1,1),(1,-1,1,1),(1,1,-1,1),(1,1,1,-1)\\}.$
The last three solutions can be obtained from the first one by collecting
${\cal S}^{V_{1}}$, ${\cal S}^{V_{2}}$ and ${\cal S}^{V_{1}+V_{2}}$,
respectively, and absorbing in ${\cal S}^{V}$.
∎
Let us introduce the following notation:
$\displaystyle{\cal S}^{(V_{1},V_{2})}=\frac{1}{2}\left(-I+{\cal
S}^{V_{1}}+{\cal S}^{V_{2}}+{\cal S}^{V_{1}+V_{2}}\right).$ (3)
Note that, for any $V_{1},V_{2}\in\mathbb{F}_{2}^{n}$, ${\cal
S}^{(V_{1},V_{2})}$ stabilizes $|\psi\rangle.$ In the next Lemma, we use
function $F:{\mathcal{G}_{n}}\mapsto\mathbb{F}_{2}^{n}$, which depends
implicitly on $V_{1}$ and $V_{2}$, and is defined by
$\displaystyle F(G)=\left\\{\begin{array}[]{cl}V_{1}+V_{2}&\textrm{if
}G\textrm{ anticommute with }{\cal S}^{V_{1}}\textrm{ and }{\cal
S}^{V_{2}};\\\ V_{1}&\textrm{if }G\textrm{ anticommute only with }{\cal
S}^{V_{2}};\\\ V_{2}&\textrm{if }G\textrm{ anticommute only with }{\cal
S}^{V_{1}};\\\ 0&\textrm{otherwise. }\end{array}\right.$ (8)
###### Lemma 1.
Let $G$ be a Pauli operator. If $G$ does not commute with ${\cal S}^{V_{1}}$
or ${\cal S}^{V_{2}}$, then ${\cal S}^{(V_{1},V_{2})}G=-{\cal
S}^{F(G)}G\,{\cal S}^{(V_{1},V_{2})}=-G\,{\cal S}^{F(G)}{\cal
S}^{(V_{1},V_{2})}=-G\,{\cal S}^{(V_{1},V0_{2})}{\cal S}^{F(G)}$.
###### Proof.
The verification is straightforward. ∎
The conditions to use an operator $A$ as a CWS observable is closely related
to the conditions that guarantees that $A$ stabilizes the code.
###### Proprosition 2.
Let $C_{i}=(c_{i}^{1},\ldots,c_{i}^{n})$, $i=1,\ldots,K$ be the classical
codewords of a CWS code in standard form. Let $V_{1}$, $V_{2}$,
$V\in\mathbb{F}_{2}^{n}$ and $p_{i}=\langle C_{i},V_{1}\rangle\vee\langle
C_{i},V_{2}\rangle$. Then, a type-4 observable $A$ stabilizes the code if and
only if $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ and $\langle
C_{i},V\rangle=p_{i}$ for all $i$.
###### Proof.
Suppose that $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ and $\langle
C_{i},V\rangle=p_{i}$ is true for all $i$. Then,
1. 1.
if $Z^{C_{i}}$ commutes with ${\cal S}^{V_{1}}$ and ${\cal S}^{V_{2}}$, then
$Z^{C_{i}}$ also commutes with ${\cal S}^{V}$ and ${\cal S}^{(V_{1},V_{2})}$,
that is,
${\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}Z^{C_{i}}|\psi\rangle=Z^{C_{i}}|\psi\rangle,$
2. 2.
if $Z^{C_{i}}$ does not commute with ${\cal S}^{V_{1}}$ or ${\cal S}^{V_{2}}$,
then $Z^{C_{i}}$ anticommutes with ${\cal S}^{V}$. Besides, Lemma 1 implies
that
$\displaystyle{\cal S}^{V}{\cal S}^{(V_{1},V_{2})}Z^{C_{i}}|\psi\rangle={\cal
S}^{V}(-{\cal S}^{F(Z^{C_{i}})})Z^{C_{i}}{\cal
S}^{(V_{1},V_{2})}|\psi\rangle=-{\cal S}^{V}Z^{C_{i}}{\cal
S}^{F(Z^{C_{i}})}|\psi\rangle=$ $\displaystyle-{\cal
S}^{V}Z^{C_{i}}|\psi\rangle=Z^{C_{i}}{\cal
S}^{V}|\psi\rangle=Z^{C_{i}}|\psi\rangle.$
In all cases, $A$ stabilizes the code.
Reciprocally, let $A$ be a type-4 observable. By Theorem 1, we have
$A=\pm{\cal{S}}^{V}{\cal{S}}^{\\{V_{1},V_{2}\\}}$. Then,
$A|\psi\rangle=\pm{\cal{S}}^{V}{\cal{S}}^{\\{V_{1},V_{2}\\}}|\psi\rangle=\pm|\psi\rangle$.
By supposition, $A$ stabilizes the code. Therefore,
$A={\cal{S}}^{V}{\cal{S}}^{\\{V_{1},V_{2}\\}}$. Besides, to stabilize the
code, we have:
1. 1.
If a codeword operator $W_{i}=Z^{C_{i}}$ commutes with ${\cal S}^{V_{1}}$ and
${\cal S}^{V_{2}}$, we have
${\cal S}^{V}{\cal S}^{(V_{1},V_{2})}Z^{C_{i}}|\psi\rangle={\cal
S}^{V}Z^{C_{i}}{\cal S}^{(V_{1},V_{2})}|\psi\rangle={\cal
S}^{V}Z^{C_{i}}|\psi\rangle=Z^{C_{i}}|\psi\rangle.$
The last equality implies that ${\cal S}^{V}$ commutes with $Z^{C_{i}}$. So,
$\langle C_{i},V\rangle=0$.
2. 2.
If $W_{i}=Z^{C_{i}}$ does not commute with ${\cal S}^{V_{1}}$ or ${\cal
S}^{V_{2}}$, the Lemma 1 implies that
$\displaystyle{\cal S}^{V}{\cal S}^{(V_{1},V_{2})}Z^{C_{i}}|\psi\rangle={\cal
S}^{V}(-{\cal S}^{F(Z^{C_{i}})})Z^{C_{i}}{\cal
S}^{(V_{1},V_{2})}|\psi\rangle=-{\cal S}^{V}Z^{C_{i}}{\cal
S}^{F(Z^{C_{i}})}|\psi\rangle$ $\displaystyle=-{\cal
S}^{V}Z^{C_{i}}|\psi\rangle=Z^{C_{i}}|\psi\rangle.$
The last equality implies that ${\cal S}^{V}$ anticommutes with $Z^{C_{i}}$.
So, $\langle C_{i},V\rangle=1$.
These results show that $\langle C_{i},V\rangle=p_{i}$ is true for all $i$. ∎
Taking $V=(v_{1},\ldots,v_{n})\in\mathbb{F}_{2}^{n}$ and $p_{i}=\langle
C_{i},V_{1}\rangle\vee\langle C_{i},V_{2}\rangle$, equations $\langle
C_{i},V\rangle=p_{i}$ can be put in matrix form
$\displaystyle C\left[\begin{array}[]{c}v_{1}\\\ \vdots\\\
v_{n}\end{array}\right]=\left[\begin{array}[]{c}p_{1}\\\ \vdots\\\
p_{k}\end{array}\right],$ (15)
where $C$ is the matrix of all classical codewords
$C=\left[\begin{array}[]{ccc}c_{1}^{1}&\ldots&c_{1}^{n}\\\
\vdots&\vdots&\vdots\\\ c_{K}^{1}&\ldots&c_{K}^{n}\end{array}\right].$ (16)
An operator $A$ can be used as a CWS observable in the decoding procedure, if
the encoded information is not lost after the measurement of $A$. We have to
guarantee that, for each $i$ and for all $j$, $E_{i}W_{j}|\psi\rangle$ belongs
to the eigenspace of $E_{i}W_{j}$ associated with the eigenvalues 1 or -1,
that is,
$AE_{i}W_{j}|\psi\rangle=E_{i}W_{j}|\psi\rangle,\;\forall j$
or
$AE_{i}W_{j}|\psi\rangle=-E_{i}W_{j}|\psi\rangle,\;\forall j.$
Those facts lead us to the following theorem:
###### Theorem 2.
Let $\mathcal{E}=\\{E_{i}\\}_{i=1}^{T}$ be a set of correctable Pauli errors
of a CWS code in standard form. Then, a type-4 observable $A={\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}$ can be used as a decoding observable if and only if for
all $i\in\\{1,\ldots,T\\}$ there is $V^{\prime}_{i}$ solution of Eq. (15) with
$V=V^{\prime}_{i}+F(E_{i})$.
###### Proof.
Suppose that $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ satisfies
$V=V_{i}+F(E_{i})$, for all $i$, where $V_{i}$ is a solution of Eq. (15). Let
${\cal S}^{V}E_{i}=m_{i}E_{i}{\cal S}^{V}$, where $m_{i}=\pm 1$. Then, by
Lemma 1 we have
1. 1.
if $E_{i}$ commutes with ${\cal S}^{V_{1}}$ and ${\cal S}^{V_{2}}$, then
$F(E_{i})=(0,\ldots,0)$ (8) and
$\displaystyle AE_{i}W_{j}|\psi\rangle$ $\displaystyle=$ $\displaystyle{\cal
S}^{V}{\cal S}^{(V_{1},V_{2})}E_{i}W_{j}|\psi\rangle={\cal S}^{V}E_{i}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle$ $\displaystyle=$ $\displaystyle
m_{i}E_{i}{\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle=m_{i}E_{i}{\cal S}^{V+F(E_{i})}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle$ $\displaystyle=$ $\displaystyle
m_{i}E_{i}{\cal S}^{V^{\prime}_{i}}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle=m_{i}E_{i}W_{j}|\psi\rangle.$
The last equality holds because ${\cal S}^{V^{\prime}_{i}}{\cal
S}^{(V_{1},V_{2})}$ stabilizes the code.
2. 2.
If $E_{i}$ does not commute with ${\cal S}^{V_{1}}$ or ${\cal S}^{V_{2}}$,
then
$\displaystyle AE_{i}W_{j}|\psi\rangle$ $\displaystyle=$ $\displaystyle{\cal
S}^{V}{\cal S}^{(V_{1},V_{2})}E_{i}W_{j}|\psi\rangle=-{\cal
S}^{V+F(E_{i})}E_{i}{\cal S}^{(V_{1},V_{2})}W_{j}|\psi\rangle$
$\displaystyle=$ $\displaystyle-m_{i}E_{i}{\cal S}^{V^{\prime}_{i}}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle=-m_{i}E_{i}W_{j}|\psi\rangle.$
Again, we have used that ${\cal S}^{V^{\prime}_{i}}{\cal S}^{(V_{1},V_{2})}$
stabilizes the code.
Reciprocally, suppose that $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ can be
used as a decoding CWS observable. Using ${\cal S}^{V}E_{i}=m_{i}E_{i}{\cal
S}^{V}$, where $m_{i}=\pm 1$, and repeating the commuting process, we have
1. 1.
if $E_{i}$ commutes with both ${\cal S}^{V_{1}}$ and ${\cal S}^{V_{2}}$, then
$\displaystyle AE_{i}W_{j}|\psi\rangle={\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}E_{i}W_{j}|\psi\rangle=m_{i}E_{i}{\cal S}^{V+F(E_{i})}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle$
where $F(E_{i})=(0,\ldots,0)$.
2. 2.
If $E_{i}$ does not commute with ${\cal S}^{V_{1}}$ or ${\cal S}^{V_{2}}$,
then
$\displaystyle AE_{i}W_{j}|\psi\rangle={\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}E_{i}W_{j}|\psi\rangle=-m_{i}E_{i}{\cal S}^{V+F(E_{i})}{\cal
S}^{(V_{1},V_{2})}W_{j}|\psi\rangle.$
We are assuming that $A$ can be used as a decoding CWS observable. In both
cases, we have
$\displaystyle AE_{i}W_{j}|\psi\rangle=E_{i}W_{j}|\psi\rangle,\;\forall j$
or
$\displaystyle AE_{i}W_{j}|\psi\rangle=-E_{i}W_{j}|\psi\rangle,\;\forall j.$
This implies that ${\cal S}^{V+F(E_{i})}{\cal S}^{(V_{1},V_{2})}$ stabilizes
the code for all $i$, and by Prop. 2 there is a solution $V^{\prime}_{i}$ of
Eq. (15) such that $V+F(E_{i})=V^{\prime}_{i}$ for all $i$.
∎
Theo. 2 allows us to make an exhaustive search for type-4 decoding observables
using expression $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$. We have to consider
all pairs $({\cal S}^{V_{1}},{\cal S}^{V_{2}})$ in $S$ such that $V_{1}\neq
V_{2}$ and look for solutions of Eq. (15) for each pair. This process can be
expensive. Next corollary addresses a more efficient way to search the
decoding observables by restricting the search space to
${N_{S}({\mathcal{E}})}$. In this case, some solutions may be lost.
###### Corollary 1.
Let $\mathcal{E}=\\{E_{i}\\}_{i=1}^{T}$ be a set of correctable errors of a
CWS code in standard form and ${N_{S}({\mathcal{E}})}$ the normalizer of
$\mathcal{E}$ in $S$. If $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ is a type-4
observable, where ${\cal S}^{V_{1}},{\cal S}^{V_{2}}\in{N_{S}({\mathcal{E}})}$
and $V$ is a solution of Eq. (15), then $A$ is a decoding observable for the
CWS code.
###### Proof.
If both ${\cal S}^{V_{1}}$ and ${\cal S}^{V_{2}}$ are in
${N_{S}({\mathcal{E}})}$, then $F(E_{i})=(0,\ldots,0)$ for all $i$, and Theo.
2 implies that $V=V_{i}$, where $V_{i}$ is a solution of Eq. (15).
∎
Corollary 1 helps us to build a procedure to find type-4 decoding observables,
which we describe now.
###### Procedure 1.
Let $\mathcal{E}=\\{E_{i}\\}$ be the set of correctable errors and
$W=\\{W_{j}\\}$ the set of codeword operators.
1. 1.
Find independent generators of ${N_{S}(W)}$.
2. 2.
Measure the generators. For each sequence of measurement results, there is set
${\mathcal{E}^{\prime}}$, subset of ${\mathcal{E}}$, of errors that were not
detected by the measurements.
3. 3.
For each ${\mathcal{E}^{\prime}}$ do
1. (a)
Find all elements in group ${N_{S}({\mathcal{E}^{\prime}})}$.
2. (b)
Take pairs $({\cal S}^{V_{1}},{\cal S}^{V_{2}})$ in
${N_{S}({\mathcal{E}^{\prime}})}$ such that ${V_{1}}\neq{V_{2}}$ until finding
a solution $V$ of Eq. (15) that distinguishes some errors in
${\mathcal{E^{\prime}}}$. This step may split ${\mathcal{E}^{\prime}}$ into
smaller subsets.
3. (c)
Repeat Step (a) and (b) with smaller subsets as many times as needed until
distinguishing Pauli errors in ${\mathcal{E}^{\prime}}$.
To find generators of ${N_{S}(W)}$ in Step 1, we employ the commuting
relations
$Z^{C_{i}}\mathcal{S}^{O_{j}}=(-1)^{\langle
C_{i},O_{j}\rangle}\mathcal{S}^{O_{j}}Z^{C_{i}}$ (17)
to show that $\mathcal{S}^{O_{j}}\in{N_{S}(W)}$ if and only if $\langle
C_{i},O_{j}\rangle=0$ for all $i$. This implies that $O_{j}$ must be in the
kernel of matrix $C$, described in Eq. (16). The independent generators for
${N_{S}(W)}$ are obtained from a basis of the kernel of $C$.
To find all elements in ${N_{S}(\mathcal{E}^{\prime})}$ in Step 3(a), we
convert the errors in $\mathcal{E}^{\prime}$ to classical words by using
function ClS and build a new matrix. The kernel of this matrix is in one-to-
one correspondence to the elements of ${N_{S}({\mathcal{E}^{\prime}})}$. Each
pair $({\cal S}^{V_{1}},{\cal S}^{V_{2}})$ and a solution $V$ of Eq. (15)
provides a non-Pauli observable for errors in ${\mathcal{E^{\prime}}}$. Step 3
can be improved by testing whether each non-Pauli observable can be used for
other sets ${\mathcal{E^{\prime}}}$ generated is Step 2.
## 4 Example
In this section, we employ Procedure 1 to find the decoding observables for
the $((10,20,3))$ code, described by Cross et al. [20]. This code is based on
the double ring graph, with the following generators:
$\begin{array}[]{cc}s_{1}=XZIIZZIIII&\,\,\,\,\,s_{6}=ZIIIIXZIIZ\\\
s_{2}=ZXZIIIZIII&\,\,\,\,\,s_{7}=IZIIIZXZII\\\
s_{3}=IZXZIIIZII&\,\,\,\,\,s_{8}=IIZIIIZXZI\\\
s_{4}=IIZXZIIIZI&\,\,\,\,\,s_{9}=IIIZIIIZXZ\\\
s_{5}=ZIIZXIIIIZ&\,\,\,\,\,s_{10}=IIIIZZIIZX\\\ \end{array}$
The associated classical codewords are
0000000000 | 1001100100 | 1001101111 | 0101100000
---|---|---|---
0000101001 | 1100101101 | 0111011011 | 0111010000
1011011111 | 1110010110 | 1100000100 | 1101111110
1111000101 | 0101101011 | 0001111010 | 0010010010
0010111011 | 1011010100 | 0011000001 | 1110111111
In Step 1 of Procedure 1, we have to find generators for ${N_{S}(W)}$. This is
accomplished by finding a basis $(O)$ for the kernel of matrix $C$, described
in Eq. (16). In this example, this basis is given in Table 1. Then, the
generators of ${N_{S}(W)}$ are Pauli observables ${\cal S}^{O_{1}},$ ${\cal
S}^{O_{2}}$, ${\cal S}^{O_{3}}$, ${\cal S}^{O_{4}}$. In Step 2, they are
measured one at a time. The results are displayed as signs $\pm$ on the top of
subtables in Fig. 1. For example, if the results of measuring these Pauli
observables are $+++-$, only two Pauli errors were not detected, namely,
$Y_{2}$ and $Z_{1}$. ${\mathcal{E}^{\prime}}$ is $\\{Y_{2},Z_{1}\\}$ in this
case.
Table 1: Decoding observables (Pauli type — ${\cal S}^{O_{i}}$) for the ((10,20,3)) code. $O_{1}$ | $O_{2}$ | $O_{3}$ | $O_{4}$
---|---|---|---
0001110011 | 0010011001 | 0100111110 | 1000000100
In Step 3 of Procedure 1, we obtain the first non-Pauli observable, $A_{1}$ in
Table 2, when ${\mathcal{E}^{\prime}}=\\{Y_{2},Z_{1}\\}$ . In this case, Step
3(a) is used only one time, because observable $A_{1}$ distinguishes all
errors in ${\mathcal{E}^{\prime}}$. Note that we can verify whether $A_{1}$
can be used for others ${\mathcal{E}^{\prime}}$. In this example, $A_{1}$ can
be used 4 times, as can be seen in Fig. 1. The next set will be
${\mathcal{E}^{\prime}}=\\{X_{4},Z_{3}\\}$.
Table 2: Decoding observables (non-Pauli) for the ((10,20,3)) code. They are type-4 observables described by $A={\cal S}^{V}{\cal S}^{(V_{1},V_{2})}$ (see Eq. (2)). | $V$ | $V_{1}$ | $V_{2}$
---|---|---|---
$A_{1}$ | 0000111001 | 0000100001 | 0001000011
$A_{2}$ | 0000111001 | 0000100010 | 0001000000
$A_{3}$ | 0000010001 | 0000000011 | 0000010010
$A_{4}$ | 0000110000 | 0000011000 | 0000100010
$A_{5}$ | 0000111001 | 0000011011 | 0000101011
$A_{6}$ | 0000111001 | 0000011000 | 0001000000
$A_{7}$ | 0000111001 | 0000110000 | 0010000010
At the end, we obtain seven type-4 decoding observables, which are listed in
Table 2. The form of those observables is given by ${\cal S}^{V}{\cal
S}^{(V_{1},V_{2})}$, which is described in Eq. (2). To decide which observable
must be measured, we have to analyze Fig. 1. Note that it is enough to measure
one non-Pauli observable for this code. We have not put the result ++++ in the
list of subtables, because it is trivial — only the identity operator appears
in this case.
Figure 1: Results of the measurements of the decoding observables. The signs
on the top of each subtable describe the results of measuring Pauli
observables of Table 1. The measurement of non-Pauli observables is
conditioned by the results of measuring Pauli observables.
$\begin{array}[]{|c|cc|}\lx@intercol\hfil+++-\hfil\lx@intercol\\\
\hline\cr&Y_{2}&Z_{1}\\\ \hline\cr A_{1}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil++-+\hfil\lx@intercol\\\
\hline\cr&Y_{10}&Z_{2}\\\ \hline\cr A_{1}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil++--\hfil\lx@intercol\\\
\hline\cr&X_{2}&Z_{8}\\\ \hline\cr A_{1}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil----\hfil\lx@intercol\\\
\hline\cr&X_{7}&Y_{5}\\\ \hline\cr A_{1}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil+-++\hfil\lx@intercol\\\
\hline\cr&X_{4}&Z_{3}\\\ \hline\cr A_{2}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil---+\hfil\lx@intercol\\\
\hline\cr&X_{10}&Z_{6}\\\ \hline\cr A_{2}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil-++-\hfil\lx@intercol\\\
\hline\cr&X_{3}&Y_{7}\\\ \hline\cr A_{3}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil--+-\hfil\lx@intercol\\\
\hline\cr&Y_{9}&Y_{3}\\\ \hline\cr A_{3}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil+-+-\hfil\lx@intercol\\\
\hline\cr&X_{5}&Y_{6}\\\ \hline\cr A_{4}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil--++\hfil\lx@intercol\\\
\hline\cr&Z_{10}&Y_{4}\\\ \hline\cr A_{4}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil-+++\hfil\lx@intercol\\\
\hline\cr&X_{8}&Z_{4}\\\ \hline\cr A_{5}&-&+\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil-+--\hfil\lx@intercol\\\
\hline\cr&X_{6}&Y_{8}\\\ \hline\cr A_{5}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil-+-+\hfil\lx@intercol\\\
\hline\cr&Z_{9}&Z_{5}\\\ \hline\cr A_{6}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil+--+\hfil\lx@intercol\\\
\hline\cr&X_{1}&Z_{7}\\\ \hline\cr A_{7}&+&-\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|cc|}\lx@intercol\hfil+---\hfil\lx@intercol\\\
\hline\cr&X_{9}&Y_{1}\\\ \hline\cr A_{7}&-&+\\\ \hline\cr\end{array}$
## 5 Conclusions
In this work, we have established two results on the existence and form of
type-4 decoding observables for CWS codes, namely, Theo. 2 and Corollary 1.
Those non-Pauli observables are necessary in non-stabilizer CWS codes. We have
described a procedure to obtain those observables, which has better chances to
succeed when the CWS code is close to a stabilizer code. The standard
procedure is to start measuring a list of Pauli observables that stabilizes
the code. In the next step, we search for type-4 decoding observables in the
search space described by Corollary 1.
The procedure does not succeed for all CWS codes, and it is interesting to
understand why it fails for some of them. In those cases, is it possible to
use type-$i$ observables, with $i>4$ as decoding observables? For example, the
$((10,18,3))$ code described in Ref. [20] cannot be decoded by type-4
observables.
It is also interesting to study methods, perhaps in family of codes, to obtain
the non-Pauli observables in a straightforward way, with less exhaustive
search by reducing the search space and to compare with the general method
proposed in Ref. [18].
## Acknowledgements
We acknowledge CNPq’s financial support
## References
* [1] P. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on, pp. 124 –134, nov 1994.
* [2] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, STOC ’96, (New York, NY, USA), pp. 212–219, ACM, 1996.
* [3] M. Mosca, “Quantum algorithms,” in Encyclopedia of Complexity and Systems Science, pp. 7088–7118, 2009.
* [4] A. M. Childs and W. van Dam, “Quantum algorithms for algebraic problems,” Rev. Mod. Phys., vol. 82, pp. 1–52, Jan 2010.
* [5] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A, vol. 54, pp. 1098–1105, Aug 1996.
* [6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A, vol. 54, pp. 3824–3851, Nov 1996.
* [7] A. M. Steane, “Simple quantum error-correcting codes,” Phys. Rev. A, vol. 54, pp. 4741–4751, Dec 1996.
* [8] E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A, vol. 55, pp. 900–911, Feb 1997.
* [9] D. Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys. Rev. A, vol. 54, pp. 1862–1868, Sep 1996.
* [10] D. Gottesman, Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997.
* [11] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett., vol. 78, pp. 405–408, Jan 1997.
* [12] J. A. Smolin, G. Smith, and S. Wehner, “Simple family of nonadditive quantum codes,” Phys. Rev. Lett., vol. 99, p. 130505, Sep 2007.
* [13] S. Yu, Q. Chen, C. H. Lai, and C. H. Oh, “Nonadditive quantum error-correcting code,” Phys. Rev. Lett., vol. 101, p. 090501, Aug 2008.
* [14] A. Cross, G. Smith, J. Smolin, and B. Zeng, “Codeword stabilized quantum codes,” in Information Theory, 2008. ISIT 2008. IEEE International Symposium on, pp. 364 –368, july 2008.
* [15] X. Chen, B. Zeng, and I. L. Chuang, “Nonbinary codeword-stabilized quantum codes,” Phys. Rev. A, vol. 78, p. 062315, Dec 2008.
* [16] I. Chuang, A. Cross, G. Smith, J. Smolin, and B. Zeng, “Codeword stabilized quantum codes: Algorithm and structure,” Journal of Mathematical Physics, vol. 50, no. 4, p. 042109, 2009.
* [17] S. Yu, Q. Chen, and C. H. Oh, “Two infinite families of nonadditive quantum error-correcting codes,” ArXiv e-prints, Jan. 2009.
* [18] Y. Li, I. Dumer, M. Grassl, and L. P. Pryadko, “Structured error recovery for code-word-stabilized quantum codes,” Phys. Rev. A, vol. 81, p. 052337, May 2010.
* [19] N. Melo, D. F. G. Santiago, and R. Portugal, “Decoder for Nonbinary CWS Quantum Codes,” ArXiv e-prints, Apr. 2012.
* [20] A. Cross, G. Smith, J. A. Smolin, and B. Zeng, “Codeword stabilized quantum codes,” IEEE Trans. Inf. Theor., vol. 55, pp. 433–438, Jan. 2009.
|
arxiv-papers
| 2012-06-07T18:16:00 |
2024-09-04T02:49:31.593183
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas F. G. Santiago, Renato Portugal, Nolmar Melo",
"submitter": "Nolmar Melo",
"url": "https://arxiv.org/abs/1206.1566"
}
|
1206.1619
|
# W/Z + JETS AND W/Z + HEAVY FLAVOR PRODUCTION AT THE LHC
A.A. PARAMONOV
The ATLAS and CMS experiments at the LHC conduct an extensive program to study
production of events with a $W^{\pm}$ or $Z^{0}$ boson and particle jets.
Dedicated studies focus on final states with the jets containing decays of
heavy-flavor hadrons ($b$-tagged jets). The results are obtained using data
from proton-proton collisions at $\sqrt{s}=7$ TeV from the LHC at CERN. The
set of measurements constitute a stringent test of the perturbative QCD
calculations.
## 1 Introduction
Production of jets in association with a massive vector boson ($W^{\pm}$ or
$Z^{0}$) is a well-understood process that provides tests of calculations
based on quantum chromodynamics (QCD). These events are also substantial
backgrounds to standard model (SM) measurements and searches for new physics.
The studies of the associated production constitute a foundation for
development of perturbative QCD (pQCD) calculations and Monte Carlo (MC)
simulations. The ATLAS $\\!{}^{{\bf?}}$ and CMS $\\!{}^{{\bf?}}$ experiments
at the LHC have reported their results using data from proton-proton
collisions at $\sqrt{s}=7$ TeV collisions in Refs.
$\\!{}^{{\bf?},{\bf?},{\bf?},{\bf?}}$. Previously, the associated production
of a massive vector boson and jets was studied at the Tevatron using
$p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV. The measurements at the LHC
offer wider reach in momenta of the jets than the previous studies.
Production of jets containing heavy-flavor hadrons in association with a
massive boson is of special interest. The results of these studies are
presented in Refs. $\\!{}^{{\bf?},{\bf?},{\bf?},{\bf?},{\bf?}}$.
Identification of jets with decays of heavy flavor hadrons, $b$-tagging, was
performed via reconstruction of a secondary vertex within a jets. In Ref.
$\\!{}^{{\bf?}}$ jets were not used but $B$-mesons were identified via
secondary vertices from $B\rightarrow D+X$ decays. The associated production
of heavy-flavor hadrons is less understood than of that of light particle
jets. Therefore, the experimental input is of key importance for development
of the MC simulations and pQCD calculations. Also, these measurements can
provide constraints on the parton density functions (PDF’s).
The measurements with a $W^{\pm}$ boson and a $Z^{0}$ boson are complementary.
Both final states are sensitive to similar physics processes but they are
different from the experimental point of view. The experimental signatures of
the two bosons are different. Identification of a $W^{\pm}$ boson requires a
well-identified lepton (an electron or a muon) and large imbalance of the
vector sum of transverse momenta of all reconstructed objects in event
(missing-$p_{\rm T}$). Identification of a $Z^{0}$ requires two oppositely-
charged leptons of the same flavor (two electrons or two muons).
All the experimental results have been corrected for all known instrumental
effects and are often quoted is a specific range of jet and lepton kinematics,
similar to the detector acceptance. That is done to avoid prediction-dependent
extrapolation and to facilitate comparisons with theoretical predictions.
Theoretical calculations at next-to-leading order (NLO) in pQCD are presented
for final states with a vector boson and up to four jets.
## 2 Backgrounds and Systematic Uncertainties
Reconstruction of the di-lepton invariant mass allows significant reduction of
backgrounds to events with a $Z^{0}$ boson. The majority of observed events
are from the associated production of a $Z^{0}$ and jets. The irreducible
backgrounds are the top quark pair production ($t\bar{t}$), dibosons, and
$Wt$. These are estimated using MC simulations normalized with the measured
luminosity and predicted cross sections. Background with one or two non-prompt
(“fake”) leptons are from events with a $W^{\pm}$ bosons and associated jets
and multi-jet events, correspondingly. Rates of events with “fake” leptons are
obtained using control regions in data. The requirement for a jet with decay
of a heavy-flavor hadron enhances the fraction of events from the $t\bar{t}$
production.
Events with a $W^{\pm}$ boson and jets are produced at a higher rate than with
a $Z^{0}$ boson. The major background with a non-prompt lepton is from the
multi-jet production. The background is evaluated using orthogonal control
regions in data. The contribution from multi-jet events is different for the
electron and muon decay modes of $W^{\pm}$ bosons. Therefore, comparison of
the measured cross section from the two decay modes can provide information of
biases related to the evaluation of the backgrounds. The backgrounds with a
prompt lepton are from $t\bar{t}$ production, dibosons, and events with a
$Z^{0}$ boson and jet. The top pair production becomes the dominant background
in final states with four or more jets (the jets are counted when $p_{\rm
T}>20$, 25, or 30 GeV). The top pair production is also substantial for events
with a $b$-tagged jet. The top pair production is the dominant background that
limits our ability to measure cross section for events with a $W^{\pm}$ and
two $b$-jets. The top background is less prominent for measurements involving
a $Z^{0}$ boson in the final state.
The major systematic uncertainties are from the jet energy scale (JES)
calibration and efficiency of $b$-tagging. The uncertainty on the JES grows
rapidly when the absolute value of jet rapidity is above two.
## 3 Results
The high cross section of the associated production of a massive boson and
jets allows detailed studies of the kinematic distributions using differential
and inclusive cross sections. Such studies have been performed by the CMS
$\\!{}^{{\bf?}}$ and ATLAS $\\!{}^{{\bf?},{\bf?},{\bf?}}$ collaborations.
Figs. 1 and 2 illustrate the cross sections measured as a function of
inclusive jet multiplicity and transverse momentum of the leading jet. The
studies have been conducted for a variety of kinematic observables such as
invariant mass of multiple jets, angular and rapidity separation between jets,
and so on. The measured ratios of cross sections allow cancellation of major
systematic uncertainties.
Figure 1: Measured cross sections as a function of jet multiplicity for events
with a $W^{\pm}$ boson 6 (left) and with a $Z^{0}$ boson 5 (right). The solid
bands correspond to the systematic uncertainties on the predicted cross
sections.
Figure 2: Measured cross sections as a function of $p_{\rm T}$ of the leading
jet for events with a $W^{\pm}$ boson 6 (left) and with a $Z^{0}$ boson 5
(right). The solid bands correspond to the systematic uncertainties on the
predicted cross sections.
The measured cross sections are compared to the NLO calculations from
BlackHat-Sherpa and MC simulations from Pythia, Sherpa and Alpgen matched to
Herwig. The NLO pQCD predictions are found in good agreement with data.
Leading-order (LO) matrix element calculations for final states with a vector
boson and up to five partons are matched to parton showering in Sherpa and
Alpgen+Herwig. These two generators are also in good agreement with data.
Production of a charm hadron in a jet and a $W^{\pm}$ boson is reported in
Ref. $\\!{}^{{\bf?}}$. The study has sensitivity to the strange quark PDF.
Ratios of cross sections were measured to be
$\sigma(W^{+}\bar{c}+X)/\sigma(W^{-}c+X)=0.92\pm$0.19(stat.)$\pm$0.04(syst.)
and $\sigma(Wc+X)/\sigma(W+jet+X)=0.143\pm 0.015$(stat.)$\pm$0.024(syst.). The
ratios are measured in the kinematic region $p^{\rm jet}_{\rm T}>$20 GeV,
$|\eta^{\rm jet}|<$2.1 for $W\rightarrow\mu\nu$ decays. The measured results
are in agreement with theoretical predictions at NLO based on available parton
distribution functions.
Studies of the associated production of jets with decays of $B$ mesons
($b$-jets) are described in Refs. $\\!{}^{{\bf?},{\bf?},{\bf?}}$. These final
state are backgrounds to the associated Higgs production; $pp\rightarrow HW$
and $pp\rightarrow HZ$, where $h\rightarrow b\bar{b}$. The results for
production of a $b$-jet and a $W^{\pm}$ boson are presented in Fig. 3. The
measured cross section slightly exceeds the predicted value for final states
with a single $b$-jet and another jet. Ref. $\\!{}^{{\bf?}}$ presents cross
sections for one and two $b$-jets with $p_{T}^{\rm jet}>25$ GeV and $\eta^{\rm
jet}<2.1$. The measured cross sections are $\sigma(Z^{0}+\>2\>b{\rm-
jets}+X)=0.37\pm 0.02$(stat.)$\pm 0.07$(syst.)$\pm 0.02$(theory) pb and
$\sigma(Z^{0}+\>b{\rm-jet}+X)=3.78\pm 0.05$(stat.)$\pm 0.31$(syst.)$\pm
0.11$(theory) pb. The cross section for two $b$-jets is in agreement with LO
pQCD predictions.
Figure 3: Exclusive cross sections for events with a $b$-jet and a $W^{\pm}$
(left) from ATLAS 8. Distribution in angular separation, $\Delta R$, between
$B$ meson candidates in events with a $Z^{0}$ (right) from CMS 10.
The study of the angular correlations between two $B$ hadrons produced in
association with a $Z^{0}$ boson is presented in Ref. $\\!{}^{{\bf?}}$.
Identification of $B$-hadron candidates utilizes displaced secondary vertices
without involving jets. That allows to analyze production of $B$ hadrons at
small angular separation. The normalized production cross section as function
of the angular separation is compared with QCD predictions at tree-level in
Fig 3. The measurement is performed in the kinematic region defined for $B$
hadrons with $p_{\rm T}>15$ GeV and $|\eta|<2$. This study gives further
insight into the properties of heavy quark pair-production in association with
a neutral vector bosons.
## References
## References
* [1] ATLAS Collaboration, JINST 3, S08003 (2008)
* [2] CMS Collaboration, JINST 3, S08004 (2008)
* [3] CMS Collaboration, JHEP 1201, 010 (2012).
* [4] ATLAS Collaboration, Phys. Lett. B 708, 221 (2012).
* [5] ATLAS Collaboration, Phys. Rev. D 85, 032009 (2012).
* [6] ATLAS Collaboration, Phys. Rev. D 85, 092002 (2012).
* [7] CMS Collaboration, CMS-PAS-SMP-12-003 (2012).
* [8] ATLAS Collaboration, Phys. Lett. B 707, 418 (2012).
* [9] ATLAS Collaboration, Phys. Lett. B 706, 295 (2012).
* [10] CMS Collaboration, CMS-PAS-EWK-11-015 (2012).
* [11] CMS Collaboration, CMS-PAS-EWK-11-013 (2011).
|
arxiv-papers
| 2012-06-07T21:08:10 |
2024-09-04T02:49:31.602055
|
{
"license": "Public Domain",
"authors": "A. A. Paramonov (for the ATLAS Collaboration and for the CMS\n Collaboration)",
"submitter": "Alexander Paramonov",
"url": "https://arxiv.org/abs/1206.1619"
}
|
1206.1692
|
# Invariant tensors related with natural connections for a class Riemannian
product manifolds
Dobrinka Gribacheva
###### Abstract.
Some invariant tensors in two Naveira classes of Riemannian product manifolds
are considered. These tensors are related with natural connections, i.e.
linear connections preserving the Riemannian metric and the product structure.
###### Key words and phrases:
Riemannian almost product manifold; Riemannian metric; product structure;
natural connection; curvature tensor; Riemannian P-tensor.
###### 2000 Mathematics Subject Classification:
53C15, 53C25.
## Introduction
A Riemannian almost product manifold $(M,P,g)$ is a differentiable manifold
$M$ for which almost product structure $P$ is compatible with the Riemannian
metric $g$ such that an isometry is induced in any tangent space of $M$.
The systematic development of the theory of Riemannian almost product
manifolds was started by K. Yano in [15].
In [11] A. M. Naveira gave a classification of Riemannian almost product
manifolds with respect to the covariant differentiation $\nabla P$, where
$\nabla$ is the Levi-Civita connection of $g$. This classification is very
similar to the Gray-Hervella classification in [1] of almost Hermitian
manifolds.
M. Staikova and K. Gribachev gave in [13] a classification of the Riemannian
almost product manifolds with ${\rm tr}P=0$. In this case the manifold $M$ is
even-dimensional.
For the class $\mathcal{W}_{1}$ of the Staikova-Gribachev classification is
valid
$\mathcal{W}_{1}=\overline{\mathcal{W}}_{3}\oplus\overline{\mathcal{W}}_{6}$,
where $\overline{\mathcal{W}}_{3}$ and $\overline{\mathcal{W}}_{6}$ are
classes of the Naveira classification. In some sense these manifolds have dual
geometries.
In [10], a connection $\nabla^{\prime}$ on a Riemannian almost product
manifold $(M,P,g)$ is called natural if $\nabla^{\prime}P=\nabla^{\prime}g=0$.
In [9], a tensor on such a manifold is called a Riemannian $P$-tensor if it
has properties similar to the properties of the Kähler tensor in Hermitian
geometry. In [4], a Riemannian $P$-tensor $K$ is defined on
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ by the
curvature tensor $R$ of $\nabla$ and the structure $P$.
In the present work111Partially supported by project NI11-FMI-004 of the
Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria,
we study manifolds $(M,P,g)$ from the class
$\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ for which the
curvature tensor of each natural connection is a Riemannian $P$-tensor.
We consider three tensors $B(L)$, $A(L)$ and $C(L)$ determined by arbitrary
Riemannian $P$-tensor $L$, where $B(L)$ is the Bochner tensor introduced in
[13]. We prove that $B(R^{\prime})=B(K)$ for arbitrary natural connection
$\nabla^{\prime}$ in Theorem 3.1. In Theorem 4.1 we prove that
$A(R^{\prime})=A(K)$ if $\nabla^{\prime}$ is the canonical connection
introduced in [10]. In Theorem 5.1 we prove that $C(R^{\prime})=C(K)$ if
$\nabla^{\prime}$ is a natural connection with parallel torsion. Moreover, we
consider a tensor $E(L)$ determined by a curvature-like tensor $L$. In Theorem
6.1 we prove that $E(R^{\prime})=E(R)$ for the natural connection
$\nabla^{\prime}=D$, considered in [3], in the case when $D$ has a parallel
torsion.
## 1\. Preliminaries
Let $(M,P,g)$ be a _Riemannian almost product manifold_ , i.e. a
differentiable manifold $M$ with a tensor field $P$ of type $(1,1)$ and a
Riemannian metric $g$ such that $P^{2}x=x$, $g(Px,Py)=g(x,y)$ for any $x$, $y$
of the algebra $\mathfrak{X}(M)$ of the smooth vector fields on $M$. Further
$x,y,z,w$ will stand for arbitrary elements of $\mathfrak{X}(M)$ or vectors in
the tangent space $T_{c}M$ at $c\in M$.
In [11] A.M. Naveira gives a classification of Riemannian almost product
manifolds with respect to the tensor $F$ of type (0,3), defined by
$F(x,y,z)=g\left(\left(\nabla_{x}P\right)y,z\right),$ where $\nabla$ is the
Levi-Civita connection of $g$.
In this work we consider manifolds $(M,P,g)$ with ${\rm tr}{P}=0$. In this
case $M$ is an even-dimensional manifold. We assume that $\dim{M}=2n$.
Using the Naveira classification, in [13] M. Staikova and K. Gribachev give a
classification of Riemannian almost product manifolds $(M,P,g)$ with ${\rm
tr}P=0$. The basic classes of this classification are $\mathcal{W}_{1}$,
$\mathcal{W}_{2}$ and $\mathcal{W}_{3}$. Their intersection is the class
$\mathcal{W}_{0}$ of the _Riemannian $P$-manifolds_ ([12]), determined by the
condition $F=0$. This class is an analogue of the class of Kähler manifolds in
the geometry of almost Hermitian manifolds.
The class $\mathcal{W}_{1}$ from the Staikova-Gribachev classification
consists of the Riemannian product manifolds which are locally conformal
equivalent to Riemannian $P$-manifolds. This class plays a similar role of the
role of the class of the conformal Kähler manifolds in almost Hermitian
geometry. We will say that a manifold from the class $\mathcal{W}_{1}$ is a
_$\mathcal{W}_{1}$ -manifold_.
The characteristic condition for the class $\mathcal{W}_{1}$ is the following
$\begin{array}[]{l}\mathcal{W}_{1}:F(x,y,z)=\frac{1}{2n}\big{\\{}g(x,y)\theta(z)-g(x,Py)\theta(Pz)\big{.}\\\\[4.0pt]
\phantom{\mathcal{W}_{1}:F(x,y,z)=\frac{1}{2n}}+g(x,z)\theta(y)-g(x,Pz)\theta(Py)\big{\\}},\end{array}$
where the associated 1-form $\theta$ is determined by
$\theta(x)=g^{ij}F(e_{i},e_{j},x).$ Here $g^{ij}$ will stand for the
components of the inverse matrix of $g$ with respect to a basis $\\{e_{i}\\}$
of $T_{c}M$ at $c\in M$. The 1-form $\theta$ is _closed_ , i.e. ${\rm
d}\theta=0$, if and only if
$\left(\nabla_{x}\theta\right)y=\left(\nabla_{y}\theta\right)x$. Moreover,
$\theta\circ P$ is a closed 1-form if and only if
$\left(\nabla_{x}\theta\right)Py=\left(\nabla_{y}\theta\right)Px$.
In [13] it is proved that
$\mathcal{W}_{1}=\overline{\mathcal{W}}_{3}\oplus\overline{\mathcal{W}}_{6}$,
where $\overline{\mathcal{W}}_{3}$ and $\overline{\mathcal{W}}_{6}$ are the
classes from the Naveira classification determined by the following
conditions:
$\begin{array}[]{rl}\overline{\mathcal{W}}_{3}:&\quad
F(A,B,\xi)=\frac{1}{n}g(A,B)\theta^{v}(\xi),\quad F(\xi,\eta,A)=0,\\\\[4.0pt]
\overline{\mathcal{W}}_{6}:&\quad
F(\xi,\eta,A)=\frac{1}{n}g(\xi,\eta)\theta^{h}(A),\quad
F(A,B,\xi)=0,\end{array}$
where $A,B,\xi,\eta\in\mathfrak{X}(M)$, $PA=A$, $PB=B$, $P\xi=-\xi$,
$P\eta=-\eta$, $\theta^{v}(x)=\frac{1}{2}\left(\theta(x)-\theta(Px)\right)$,
$\theta^{h}(x)=\frac{1}{2}\left(\theta(x)+\theta(Px)\right)$. In the case when
${\rm tr}P=0$, the above conditions for $\overline{\mathcal{W}}_{3}$ and
$\overline{\mathcal{W}}_{6}$ can be written for any $x,y,z$ in the following
form:
$\begin{array}[]{rl}\overline{\mathcal{W}}_{3}:&F(x,y,z)=\frac{1}{2n}\bigl{\\{}\left[g(x,y)+g(x,Py)\right]\theta(z)\\\\[4.0pt]
&+\left[g(x,z)+g(x,Pz)\right]\theta(y)\bigr{\\}},\quad\theta(Px)=-\theta(x),\\\\[4.0pt]
\overline{\mathcal{W}}_{6}:&F(x,y,z)=\frac{1}{2n}\bigl{\\{}\left[g(x,y)-g(x,Py)\right]\theta(z)\\\\[4.0pt]
&+\left[g(x,z)-g(x,Pz)\right]\theta(y)\bigr{\\}},\quad\theta(Px)=\theta(x).\end{array}$
In [13], a tensor $L$ of type (0,4) with properties
$L(x,y,z,w)=-L(y,x,z,w)=-L(x,y,w,z),$ $L(x,y,z,w)+L(y,z,x,w)+L(z,x,y,w)=0$
is called a _curvature-like tensor_. Such a tensor on a Riemannian almost
product manifold $(M,P,g)$ with the property
$L(x,y,Pz,Pw)=L(x,y,z,w)$
is called a _Riemannian $P$-tensor_ in [9]. This notion is an analogue of the
notion of a Kähler tensor in Hermitian geometry.
Let $S$ be a (0,2)-tensor on a Riemannian almost product manifold. In [13] it
is proved that
(1.1) $\begin{split}\psi_{1}(S)(x,y,z,w)&=g(y,z)S(x,w)-g(x,z)S(y,w)\\\\[4.0pt]
&+S(y,z)g(x,w)-S(x,z)g(y,w)\end{split}$
is a curvature-like tensor if and only if $S(x,y)=S(y,x)$, and the tensor
(1.2) $\psi_{2}(S)(x,y,z,w)=\psi_{1}(S)(x,y,Pz,Pw)$
is curvature-like if and only if $S(x,Py)=S(y,Px)$. Obviously
$\psi_{2}(S)(x,y,Pz,Pw)=\psi_{1}(S)(x,y,z,w).$
If $\psi_{1}(S)$ and $\psi_{2}(S)$ are curvature-like tensors, then
$\left(\psi_{1}+\psi_{2}\right)(S)$ is a Riemannian $P$-tensor. The tensors
(1.3)
$\pi_{1}=\frac{1}{2}\psi_{1}(g),\qquad\pi_{2}=\frac{1}{2}\psi_{2}(g),\qquad\pi_{3}=\psi_{1}(\widetilde{g})=\psi_{2}(\widetilde{g})$
are curvature-like, where $\widetilde{g}(x,y)=g(x,Py)$, and the tensors
$\pi_{1}+\pi_{2}$, $\pi_{3}$ are Riemannian $P$-tensors.
The curvature tensor $R$ of $\nabla$ is determined by
$R(x,y)z=\nabla_{x}\nabla_{y}z-\nabla_{y}\nabla_{x}z-\nabla_{[x,y]}z$ and the
corresponding tensor of type (0,4) is defined as follows
$R(x,y,z,w)=g(R(x,y)z,w)$. We denote the Ricci tensor and the scalar curvature
of $R$ by $\rho$ and $\tau$, respectively, i.e.
$\rho(y,z)=g^{ij}R(e_{i},y,z,e_{j})$ and $\tau=g^{ij}\rho(e_{i},e_{j})$. The
associated Ricci tensor $\rho^{*}$ and the associated scalar curvature
$\tau^{*}$ of $R$ are determined by $\rho^{*}(y,z)=g^{ij}R(e_{i},y,z,Pe_{j})$
and $\tau^{*}=g^{ij}\rho^{*}(e_{i},e_{j})$. In a similar way there are
determined the Ricci tensor $\rho(L)$ and the scalar curvature $\tau(L)$ for
any curvature-like tensor $L$ as well as the associated quantities
$\rho^{*}(L)$ and $\tau^{*}(L)$.
In [10], a linear connection $\nabla^{\prime}$ on a Riemannian almost product
manifold $(M,P,g)$ is called a _natural connection_ if
$\nabla^{\prime}P=\nabla^{\prime}g=0$.
In [2], it is established that the natural connections $\nabla^{\prime}$ on a
$\mathcal{W}_{1}$-manifold $(M,P,g)$ form a 2-parametric family, where the
torsion $T$ of $\nabla^{\prime}$ is determined by
(1.4)
$\begin{split}T(x,y,z)&=\frac{1}{2n}\left\\{g(y,z)\theta(Px)-g(x,z)\theta(Py)\right\\}\\\\[4.0pt]
&\phantom{=\
}+\lambda\left\\{g(y,z)\theta(x)-g(x,z)\theta(y)\right.\\\\[4.0pt]
&\phantom{=\
+\lambda\left\\{\right.}\left.+g(y,Pz)\theta(Px)-g(x,Pz)\theta(Py)\right\\}\\\\[4.0pt]
&\phantom{=\ }+\mu\left\\{g(y,Pz)\theta(x)-g(x,Pz)\theta(y)\right.\\\\[4.0pt]
&\phantom{=\
+\mu\left\\{\right.}\left.+g(y,z)\theta(Px)-g(x,z)\theta(Py)\right\\},\end{split}$
where $\lambda,\mu\in\mathbb{R}$.
Let $Q$ be the tensor determined by
(1.5) $\nabla^{\prime}_{x}y=\nabla_{x}y+Q(x,y).$
The corresponding tensor of type (0,3), according to [5], satisfies
(1.6) $Q(x,y,z)=T(z,x,y).$
Let us recall the following statement.
###### Theorem 1.1 ([5]).
Let $R^{\prime}$ is the curvature tensor of a natural connection
$\nabla^{\prime}$ on a $\mathcal{W}_{1}$-manifold $(M,P,g)$. Then the
following relation is valid:
(1.7)
$R=R^{\prime}-g(p,p)\pi_{1}-g(q,q)\pi_{2}-g(p,q)\pi_{3}-\psi_{1}(S^{\prime})-\psi_{2}(S^{\prime\prime}),$
where
$\begin{array}[]{l}p=\lambda\Omega+\left(\mu+\frac{1}{2n}\right)P\Omega,\quad
q=\lambda P\Omega+\mu\Omega,\quad g(\Omega,x)=\theta(x),\end{array}$
$\begin{array}[]{rl}&S^{\prime}(y,z)=\lambda\left(\nabla^{\prime}_{y}\theta\right)z+\left(\mu+\frac{1}{2n}\right)\left(\nabla^{\prime}_{y}\theta\right)Pz\\\\[4.0pt]
&\phantom{S^{\prime}(y,z)=}-\frac{1}{2n}\left\\{\lambda\theta(y)\theta(Pz)+\mu\theta(y)\theta(z)\right\\},\\\\[4.0pt]
&S^{\prime\prime}(y,z)=\lambda\left(\nabla^{\prime}_{y}\theta\right)z+\mu\left(\nabla^{\prime}_{y}\theta\right)Pz\\\\[4.0pt]
&\phantom{S^{\prime\prime}(y,z)=}+\frac{1}{2n}\left\\{\lambda\theta(Py)\theta(z)+\mu\theta(Py)\theta(Pz)\right\\}.\end{array}$
## 2\. Some properties of the natural connections on the manifolds of the
class $\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$
Let $(M,P,g)$ is a Riemannian product manifold of the class
$\overline{\mathcal{W}}_{3}$ or the class $\overline{\mathcal{W}}_{6}$, i.e.
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$. Then for
the 1-form $\theta$ and the vector $\Omega$ we have
(2.1) $\theta(Pz)=\varepsilon\theta(z),\qquad P\Omega=\varepsilon\Omega,$
where $\varepsilon=1$ for $(M,P,g)\in\overline{\mathcal{W}}_{3}$ and
$\varepsilon=-1$ for $(M,P,g)\in\overline{\mathcal{W}}_{6}$.
Let $\nabla^{\prime}$ be a natural connection on
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$.
Using (1.4), (1.6) and (2.1), we obtain for the tensor $Q$ determined by (1.5)
the following
$\begin{split}Q(x,y)&=\left[\lambda+\varepsilon\left(\mu+\frac{1}{2n}\right)\right]\left[g(x,y)-\theta(y)x\right]\\\\[4.0pt]
&\phantom{=\
}\left(\mu+\varepsilon\lambda\right)\left[g(x,Py)-\theta(y)Px\right].\end{split}$
Now, for the curvature tensors $R$ and $R^{\prime}$ of $\nabla$ and
$\nabla^{\prime}$, it is valid (1.7), where
(2.2) $\displaystyle
p=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{2n}\right)\Omega,\qquad
q=(\mu+\varepsilon\lambda)\Omega,$ (2.3) $\displaystyle
S^{\prime}(y,z)=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{2n}\right)\left(\nabla^{\prime}_{y}\theta\right)z-\frac{\mu+\varepsilon\lambda}{2n}\theta(y)\theta(z),$
(2.4) $\displaystyle
S^{\prime\prime}(y,z)=\left(\lambda+\varepsilon\mu\right)\left(\nabla^{\prime}_{y}\theta\right)z+\frac{\mu+\varepsilon\lambda}{2n}\theta(y)\theta(z).$
Further we consider manifolds
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with
closed 1-form $\theta$. In this case, the tensor $K$, determined by
(2.5) $K(x,y,z,w)=\frac{1}{2}\left[R(x,y,z,w)+R(x,y,Pz,Pw)\right],$
is a Riemannian $P$-tensor ([4]).
If $(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ has a
closed 1-form $\theta$, then the curvature tensor $R^{\prime}$ of a natural
connection $\nabla^{\prime}$ is also a Riemannian $P$-tensor. Indeed, from
(1.7) it is clear, that $R^{\prime}$ is a Riemannian $P$-tensor if and only if
$\psi_{1}(S^{\prime})$ and $\psi_{2}(S^{\prime\prime})$ are curvature-like
tensors, i.e. if and only if $S^{\prime}(y,z)=S^{\prime}(z,y)$ and
$S^{\prime\prime}(y,Pz)=S^{\prime\prime}(z,Py)$. According to (2.3) and (2.4),
the latter conditions are valid if and only if
(2.6)
$\left(\nabla^{\prime}_{y}\theta\right)z=\left(\nabla^{\prime}_{z}\theta\right)y.$
In [5], it is proved that for any $\mathcal{W}_{1}$-manifold the following
equality is valid:
$\begin{split}\left(\nabla^{\prime}_{y}\theta\right)z-\left(\nabla^{\prime}_{z}\theta\right)y&=\left(\nabla_{y}\theta\right)z-\left(\nabla_{z}\theta\right)y\\\\[4.0pt]
&-\frac{1}{2n}\left\\{\theta(Py)\theta(z)-\theta(y)\theta(Pz)\right\\}.\end{split}$
Bearing in mind (2.1), the latter equality implies that equality (2.6) is
valid on $(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$
if and only if
$\left(\nabla_{y}\theta\right)z=\left(\nabla_{z}\theta\right)y$, i.e. if and
only if the 1-form $\theta$ is closed.
###### Theorem 2.1.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. Then the following equality is valid
(2.7) $K=R^{\prime}-\left(\psi_{1}+\psi_{2}\right)(S),$
where
(2.8)
$\begin{split}S(y,z)&=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{4n}\right)\left(\nabla^{\prime}_{y}\theta\right)z\\\\[4.0pt]
&+\frac{g(p,p)+g(q,q)}{4}g(y,z)+\frac{g(p,q)}{2}g(y,Pz).\end{split}$
###### Proof.
According to Theorem 1.1, for $(M,P,g)$ it is valid the equality
(2.9)
$\begin{split}R(x,y,z,w)&=\left\\{R^{\prime}-g(p,p)\pi_{1}-g(q,q)\pi_{2}-g(p,q)\pi_{3}\right.\\\\[4.0pt]
&\phantom{=\left\\{\right.}\left.-\psi_{1}(S^{\prime})-\psi_{2}(S^{\prime\prime})\right\\}(x,y,z,w).\end{split}$
In (2.9), we substitute $Pz$ and $Pw$ for $z$ and $w$, respectively. We add
the obtained equality to (2.9). Then, taking into account (1.1), (1.2), (1.3),
(2.3), (2.4), (2.5), (2.8) and the properties of the curvature-like tensors
$\psi_{1}(S^{\prime})$ and $\psi_{2}(S^{\prime\prime})$, we get (2.7). ∎
In Section 3, Section 4 and Section 5, we find some Riemannian $P$-tensors
determined by $K$ on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$. We establish that the found tensors coincide with the
corresponding tensors determined by the curvature tensor $R^{\prime}$ of a
natural connection $\nabla^{\prime}$. In Section 6, we find a curvature-like
tensor determined by $R$ on such a manifold and establish that this tensor
coincides with the corresponding tensor determined by the curvature tensor
$R^{\prime}$ of the special natural connection $D$ investigated in [5], in the
case when $D$ has a parallel torsion.
## 3\. An arbitrary natural connection on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$
In [13], it is defined a Bochner tensor $B(L)$ for an arbitrary Riemannian
$P$-tensor $L$ on a $\mathcal{W}_{1}$-manifold $(M,P,g)$ ($\dim M\geq 6$) as
follows:
(3.1)
$\begin{split}B(L)&=L-\frac{1}{2(n-2)}\left\\{(\psi_{1}+\psi_{2})(\rho(L))\phantom{\frac{1}{2(n-1)}}\right.\\\\[4.0pt]
&\left.\phantom{=L}-\frac{1}{2(n-1)}\left[\tau(L)(\pi_{1}+\pi_{2})+\tau^{*}(L)\pi_{3}\right]\right\\}.\end{split}$
Let us remark that $B(L)$ is also a Riemannian $P$-tensor.
###### Theorem 3.1.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ ($\dim
M\geq 6$) be with a closed 1-form $\theta$. If $R^{\prime}$ is the curvature
tensor of a natural connection $\nabla^{\prime}$, then $B(R^{\prime})=B(K)$.
###### Proof.
Relation (2.7) implies the following equality for the Ricci tensors $\rho(K)$
and $\rho^{\prime}$ of $K$ and $R^{\prime}$, respectively:
(3.2) $\rho(K)=\rho^{\prime}-{\rm tr}S\ g-{\rm tr}\widetilde{S}\
\widetilde{g}-2(n-2)S,$
where $\widetilde{S}(y,z)=S(y,Pz)$. Then we get the following equalities for
the scalar curvatures:
(3.3) ${\rm tr}S=\frac{\tau^{\prime}-\tau(K)}{4(n-1)},\qquad{\rm
tr}\widetilde{S}=\frac{\tau^{\prime*}-\tau^{*}(K)}{4(n-1)}.$
Equalities (3.2) and (3.3) imply
(3.4)
$\begin{split}S=\frac{1}{2(n-2)}\left\\{\rho^{\prime}-\rho(K)-\frac{(\tau^{\prime}-\tau(K))g+(\tau^{\prime*}-\tau^{*}(K))\widetilde{g}}{4(n-1)}\right\\}.\end{split}$
From (1.1), (1.2), (1.3) and (3.4), we have
(3.5) $\begin{split}&(\psi_{1}+\psi_{2})(S)=\\\\[4.0pt]
&=\frac{1}{2(n-2)}\left\\{(\psi_{1}+\psi_{2})(\rho^{\prime})-(\psi_{1}+\psi_{2})(\rho(K))\phantom{\frac{1}{2(n-2)}\left\\{\right.=}\right.\\\\[4.0pt]
&\left.\phantom{\frac{1}{2(n-2)}\left\\{\right.=}-\frac{(\tau^{\prime}-\tau(K))(\pi_{1}+\pi_{2})+(\tau^{\prime*}-\tau^{*}(K))\pi_{3}}{2(n-1)}\right\\}.\end{split}$
Using (3.5), (2.7) and the definition (3.1) of the Bochner tensor, we obtain
$B(K)=B(R^{\prime})$. ∎
## 4\. The canonical connection on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$
The canonical connection on a Riemannian almost product manifold is a natural
connection introduced in [10] as an analogue of the Hermitian connection on
almost Hermitian manifold. A connection of such a type on almost contact
B-metric manifolds is considered in [7], [8].
We define the tensor $A(L)$ for an arbitrary Riemannian $P$-tensor $L$ by the
equality
(4.1) $A(L)=L-\frac{\tau(L)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$
Obviously, $A(L)$ is also a Riemannian $P$-tensor.
###### Theorem 4.1.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of the
canonical connection, then $A(R^{\prime})=A(K)$.
###### Proof.
In [5], it is shown the canonical connection on a $\mathcal{W}_{1}$-manifold
is determined by $\lambda=0$ and $\mu=-\frac{1}{4n}$. Then, (2.2) implies
$p=\frac{\varepsilon\Omega}{4n}$, $q=-\frac{\Omega}{4n}$ and therefore
(4.2) $g(p,p)=g(q,q)=-\varepsilon g(p,q)=\frac{\theta(\Omega)}{16n^{2}}.$
From (2.8) and (4.2) it is follows
$S=\frac{\theta(\Omega)}{32n^{2}}(g-\varepsilon\widetilde{g})$. Then, because
of (1.3), we have
$(\psi_{1}+\psi_{2})(S)=\frac{\theta(\Omega)}{16n^{2}}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})$.
Thus, (2.7) takes the form
(4.3)
$K=R^{\prime}-\frac{\theta(\Omega)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{16n^{2}}.$
By virtue of (4.3), we obtain the following equalities
(4.4)
$\begin{split}&\rho(K)=\rho^{\prime}-\frac{(n-1)\theta(\Omega)(g-\varepsilon\widetilde{g})}{8n^{2}},\\\\[4.0pt]
&\theta(\Omega)=\frac{4n(\tau^{\prime}-\tau(K))}{n-1}=-\frac{4n\varepsilon(\tau^{\prime*}-\tau^{*}(K))}{n-1}.\end{split}$
Bearing in mind (4.3) and (4.4), by suitable calculations we get
$R^{\prime}-\frac{\tau^{\prime}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}=K-\frac{\tau(K)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$
Then, according to (4.1), we have $A(R^{\prime})=A(K)$. ∎
In [12], a 2-plane $\alpha=(x,y)$ in $T_{c}M$ is called a totally real 2-plane
if $\alpha$ is orthogonal to $P\alpha$. Its sectional curvatures with respect
to $R^{\prime}$
$\nu^{\prime}=\frac{R^{\prime}(x,y,y,x)}{\pi_{1}(x,y,y,x)},\qquad\nu^{\prime*}=\frac{R^{\prime}(x,y,y,Px)}{\pi_{1}(x,y,y,x)}$
are called totally real sectional curvatures with respect to $R^{\prime}$.
###### Theorem 4.2.
A manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$ has point-wise constant totally real sectional
curvatures
$\nu^{\prime}=\frac{\tau^{\prime}}{4n(n-1)},\qquad\nu^{\prime*}=-\frac{\varepsilon\tau^{\prime}}{4n(n-1)}$
with respect to the curvature tensor $R^{\prime}$ of the canonical connection
if and only if $A(R^{\prime})=0$ (or equivalently $A(K)=0$).
###### Proof.
According to (4.1), the condition for annulment of $A(R^{\prime})$ is the
condition
$R^{\prime}=\frac{\tau^{\prime}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$
Then, bearing in mind [12], we establish the truthfulness of the statement. ∎
## 5\. An natural connection with parallel torsion on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$
We define the tensor $C(L)$ for an arbitrary Riemannian $P$-tensor $L$ by the
equality
(5.1) $C(L)=L-\frac{\tau(L)(\pi_{1}+\pi_{2})+\tau^{*}(L)\pi_{3}}{4n(n-1)}.$
Obviously, $C(L)$ is also a Riemannian $P$-tensor.
###### Theorem 5.1.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of a natural
connection with a parallel torsion, then $C(R^{\prime})=C(K)$.
###### Proof.
In [5], it is proved that a natural connection $\nabla^{\prime}$ on a
$\mathcal{W}_{1}$-manifold has a parallel torsion if and only if the 1-form
$\theta$ is also parallel, i.e. $\nabla^{\prime}\theta=0$. Then, (2.8) implies
$S=\frac{g(p,p)+g(q,q)}{4}g+\frac{g(p,q)}{2}\widetilde{g}.$
Then, because of (1.3), we have
$(\psi_{1}+\psi_{2})(S)=\frac{g(p,p)+g(q,q)}{2}(\pi_{1}+\pi_{2})+g(p,q)\pi_{3}.$
Thus, (2.7) takes the form
(5.2) $K=R^{\prime}-\frac{g(p,p)+g(q,q)}{2}(\pi_{1}+\pi_{2})+g(p,q)\pi_{3}.$
By virtue of (5.2), we obtain
$\rho(K)=\rho^{\prime}-(n-1)[g(p,p)+g(q,q)]g-2(n-1)g(p,q)\widetilde{g}),$
which implies
(5.3) $\begin{split}&\tau(K)=\tau^{\prime}-2n(n-1)[g(p,p)+g(q,q)],\\\\[4.0pt]
&\tau^{*}(K)=\tau^{\prime*}-4n(n-1)g(p,q).\end{split}$
Bearing in mind (5.2) and (5.3), by suitable calculations we get
$R^{\prime}-\frac{\tau^{\prime}(\pi_{1}+\pi_{2})+\tau^{\prime*}\pi_{3}}{4n(n-1)}=K-\frac{\tau(K)(\pi_{1}+\pi_{2})+\tau^{*}(K)\pi_{3}}{4n(n-1)}.$
Then, according to (5.1), we have $C(R^{\prime})=C(K)$. ∎
###### Theorem 5.2.
A manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$ has point-wise constant totally real sectional
curvatures
$\nu^{\prime}=\frac{\tau^{\prime}}{4n(n-1)},\qquad\nu^{\prime*}=\frac{\tau^{\prime*}}{4n(n-1)}$
with respect to the curvature tensor $R^{\prime}$ of an arbitrary natural
connection with parallel torsion if and only if $C(R^{\prime})=0$ (or
equivalently $C(K)=0$).
###### Proof.
According to (5.1), the condition for annulment of $C(R^{\prime})$ is the
condition
$R^{\prime}=\frac{\tau^{\prime}(\pi_{1}+\pi_{2})+\tau^{\prime*}\pi_{3}}{4n(n-1)}.$
Then, bearing in mind [12], we establish the truthfulness of the statement. ∎
## 6\. The natural connection $D$ ($\lambda=\mu=0$) with parallel torsion on
a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$
In [3], it is studied the natural connection $D$ determined by $\lambda=\mu=0$
on a $\mathcal{W}_{1}$-manifold $(M,P,g)$.
Now we consider the case when
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ is with a
closed 1-form $\theta$ and the connection $\nabla^{\prime}=D$ has a parallel
torsion. Then, from (2.2), (2.3) and (2.4) we have
$p=\frac{\varepsilon\Omega}{2n}$, $q=S^{\prime}=S^{\prime\prime}=0$ and
therefore (1.7) takes the form
(6.1) $R=R^{\prime}-\frac{\theta(\Omega)\pi_{1}}{4n^{2}}.$
The latter equality implies
$\rho=\rho^{\prime}-\frac{(2n-1)\theta(\Omega)}{4n^{2}}g,$ which gives us
(6.2)
$\tau=\tau^{\prime}-\frac{(2n-1)\theta(\Omega)}{2n},\qquad\tau^{*}=\tau^{\prime*}.$
We define the tensor $E(L)$ for an arbitrary curvature-like tensor $L$ by the
equality
(6.3) $E(L)=L-\frac{\tau(L)\pi_{1}}{2n(2n-1)}.$
Obviously, $E(L)$ is also a curvature-like tensor.
###### Theorem 6.1.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of the
connection $D$ with parallel torsion, then $E(R^{\prime})=E(R)$.
###### Proof.
Equalities (6.1) and (6.2) imply
$R-\frac{\tau\pi_{1}}{2n(2n-1)}=R^{\prime}-\frac{\tau^{\prime}\pi_{1}}{2n(2n-1)}.$
Then, according to (6.3), we have $E(R^{\prime})=E(R)$. ∎
###### Theorem 6.2.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$ and $D$ be with a parallel torsion. Then $D$ is flat if
and only if $E(R^{\prime})=0$ (or equivalently $E(R)=0$).
###### Proof.
Let $E(R^{\prime})=0$ be valid, i.e.
(6.4) $R^{\prime}(x,y,z,w)=\frac{\tau^{\prime}}{2n(2n-1)}\pi_{1}(x,y,z,w).$
In (6.4), we substitute $Pz$ and $Pw$ for $z$ and $w$, respectively. Taking
into account that $R^{\prime}$ is a Riemannian $P$-tensor and
$\pi_{1}(x,y,Pz,Pw)=\pi_{2}(x,y,z,w)$, we obtain
(6.5) $R^{\prime}=\frac{\tau^{\prime}}{2n(2n-1)}\pi_{2}.$
From (6.4) and (6.5) it is follows $\tau^{\prime}(\pi_{1}-\pi_{2})=0$ and
because of $\pi_{1}\neq\pi_{2}$ we have $\tau^{\prime}=0$. Then
$R^{\prime}=0$, according to (6.4), i.e. $D$ is a flat connection.
Vice versa, let $D$ be flat, i.e. $R^{\prime}=0$. Then $\tau^{\prime}=0$ and
bearing in mind the definition of $E(R^{\prime})$ we obtain $E(R^{\prime}=0$.
∎
###### Corollary 6.3.
Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$ and $D$ be flat with a parallel torsion. Then $(M,P,g)$
is a space form with a negative scalar curvature $\tau$.
###### Proof.
If $D$ is flat, then by Theorem 6.2 we have $E(R)=0$, i.e.
$R=\frac{\tau}{2n(2n-1)}\pi_{1}.$
This means that the manifold is a space form. Moreover, $\tau^{\prime}=0$ for
a flat connection $D$ and therefore $\tau=-\frac{2n-1}{2n}\theta(\Omega)$,
because of (6.2). Thus, since $\theta(\Omega)=g(\Omega,\Omega)>0$, we obtain
$\tau<0$. ∎
## References
* [1] A. Gray, L. Hervella, _The sixteen classes of almost Hermitian manifolds and their linear invariants._ Ann. Mat. Pura Appl. 123 (1980), 35–58.
* [2] D. Gribacheva, _Natural connections on Riemannian product manifolds._ Compt. rend. Acad. bulg. Sci. 64 (2011), no. 6, 799–806.
* [3] D. Gribacheva, _A natural connection on a basic class of Riemannian product manifolds._ Int. J. Geom. Methods Mod. Phys., 9 (2012), no. 7, 1250057 (14 pages).
* [4] D. Gribacheva, _Curvature properties of two Naveira classes of Riemannian product manifolds._ Plovdiv Univ. Sci. Works – Math. (In Press), arXiv:1204.5838.
* [5] D. Gribacheva, D. Mekerov, _Natural connections on conformal Riemannian $P$-manifolds._ Compt. rend. Acad. bulg. Sci. 65 (2012), no. 5, 581–590.
* [6] H. Hayden, _Subspaces of a space with torsion._ Proc. London Math. Soc. 34 (1934), 27–50.
* [7] M. Manev, M. Ivanova. _Canonical-type connection on almost contact manifolds with B-metric_ ; arXiv:1203.0137.
* [8] M. Manev, M. Ivanova. _Almost contact B-metric manifolds with curvature tensor of Kähler type_. Plovdiv Univ. Sci. Works – Math., vol. 39, no. 3 (2012) (In Press); arXiv:1203.3290.
* [9] D. Mekerov. _On Riemannian almost product manifolds with nonintegrable structure._ J. Geom. 89 (2008), no. 1-2, 119–129.
* [10] V. Mihova, _Canonical connections and the canonical conformal group on a Riemannian almost product manifold._ Serdica Math. P., 15 (1989), 351–358.
* [11] A. M. Naveira, _A classification of Riemannian almost product manifolds._ Rend. Math. 3 (1983), 577–592.
* [12] M. Staikova, _Curvature properties of Riemannian $P$-manifolds._ Plovdiv Univ. Sci. Works - Math. 32 (1987), no. 3, 241–251.
* [13] M. Staikova, K. Gribachev, _Canonical connections and their conformal invariants on Riemannian $P$-manifolds._ Serdica Math. P. 18 (1992), 150–161.
* [14] M. Staikova, K. Gribachev, D. Mekerov, _Riemannian $P$-manifolds of constant sectional curvatures._ Serdica Math. J. 17 (1991), 212–219.
* [15] K. Yano, _Differential geometry on complex and almost complex spaces._ Pure and Applied Math. 49, New York, Pergamon Press Book, 1965.
D. Gribacheva
Department of Algebra and Geometry
Faculty of Mathematics and Informatics
University of Plovdiv
236 Bulgaria Blvd
4003 Plovdiv, Bulgaria
dobrinka@uni-plovdiv.bg
|
arxiv-papers
| 2012-06-08T08:14:16 |
2024-09-04T02:49:31.609099
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dobrinka Gribacheva",
"submitter": "Dobrinka Gribacheva",
"url": "https://arxiv.org/abs/1206.1692"
}
|
1206.1846
|
# Warped Mixtures for Nonparametric Cluster Shapes
Tomoharu Iwata
University of Cambridge
ti242@cam.ac.uk &David Duvenaud
University of Cambridge
dkd23@cam.ac.uk &Zoubin Ghahramani
University of Cambridge
zoubin@eng.cam.ac.uk
###### Abstract
A mixture of Gaussians fit to a single curved or heavy-tailed cluster will
report that the data contains many clusters. To produce more appropriate
clusterings, we introduce a model which warps a latent mixture of Gaussians to
produce nonparametric cluster shapes. The possibly low-dimensional latent
mixture model allows us to summarize the properties of the high-dimensional
clusters (or density manifolds) describing the data. The number of manifolds,
as well as the shape and dimension of each manifold is automatically inferred.
We derive a simple inference scheme for this model which analytically
integrates out both the mixture parameters and the warping function. We show
that our model is effective for density estimation, performs better than
infinite Gaussian mixture models at recovering the true number of clusters,
and produces interpretable summaries of high-dimensional datasets.
## 1 Introduction
Probabilistic mixture models are often used for clustering. However, if the
mixture components are parametric (e.g. Gaussian), then the clustering
obtained can be heavily dependent on how well each actual cluster can be
modeled by a Gaussian. For example, a heavy tailed or curved cluster may need
many components to model it. Thus, although mixture models are widely used for
probabilistic clustering, their assumptions are generally inappropriate if the
primary goal is to discover clusters in data. Dirichlet process mixture models
can alleviate the problem of an unknown number of clusters, but this does not
address the problem that real clusters may not be well matched by any
parametric density.
| $\rightarrow$ |
---|---|---
Latent space | | Observed space
Figure 1: A sample from the iWMM prior. Left: In the latent space, a mixture
distribution is sampled from a Dirichlet process mixture of Gaussians. Right:
The latent mixture is smoothly warped to produce non-Gaussian manifolds in the
observed space.
In this paper, we propose a nonparametric Bayesian model that can find
nonlinearly separable clusters with complex shapes. The proposed model assumes
that each observation has coordinates in a latent space, and is generated by
warping the latent coordinates via a nonlinear function from the latent space
to the observed space. By this warping, complex shapes in the observed space
can be modeled by simpler shapes in the latent space. In the latent space, we
assume an infinite Gaussian mixture model rasmussen2000infinite , which allows
us to automatically infer the number of clusters. For the prior on the
nonlinear mapping function, we use Gaussian processes rasmussen38gaussian ,
which enable us to flexibly infer the nonlinear warping function from the
data. We call the proposed model the infinite warped mixture model (iWMM).
Figure 1 shows a set of manifolds and datapoints sampled from the prior
defined by this model.
To our knowledge this is the first probabilistic generative model for
clustering with flexible nonparametric component densities. Since the proposed
model is generative, it can be used for density estimation as well as
clustering. It can also be extended to handle missing data, integrate with
other probabilistic models, and use other families of distributions for the
latent components.
We derive an inference procedure for the iWMM based on Markov chain Monte
Carlo (MCMC). In particular, we sample the cluster assignments using Gibbs
sampling, sample the latent coordinates using hybrid Monte Carlo, and
analytically integrate out both the mixture parameters (weights, means and
covariance matrices), and the nonlinear warping function.
## 2 Gaussian Process Latent Variable Model
In this section, we give a brief introduction to the GPLVM, which can be
viewed as a special case of the iWMM. The GPLVM is a probabilistic model of
nonlinear manifolds. While not typically thought of as a density model, the
GPLVM does in fact define a posterior density over observations
nickisch2010gaussian . It does this by smoothly warping a single, isotropic
Gaussian density in the latent space into a more complicated distribution in
the observed space.
Suppose that we have a set of observations
$\mathbf{Y}=({\bf{y}}_{1},\cdots,{\bf{y}}_{N})^{\top}$, where
${\bf{y}}_{n}\in{\mathbb{R}}^{D}$, and they are associated with a set of
latent coordinates $\mathbf{X}=({\bf{x}}_{1},\cdots,{\bf{x}}_{N})^{\top}$,
where ${\bf{x}}_{n}\in{\mathbb{R}}^{Q}$. The GPLVM assumes that observations
are generated by mapping the latent coordinates through a set of smooth
functions, over which Gaussian process priors are placed. Under the GPLVM, the
probability of observations given the latent coordinates, integrating out the
mapping functions, is
$\displaystyle
p(\mathbf{Y}|\mathbf{X},\bm{\theta})=(2\pi)^{-\frac{DN}{2}}|\mathbf{K}|^{-\frac{D}{2}}\exp\left(-\frac{1}{2}{\rm
tr}(\mathbf{Y}^{\top}\mathbf{K}^{-1}\mathbf{Y})\right),$ (1)
where $\mathbf{K}$ is the $N\times N$ covariance matrix defined by the kernel
function $k({\bf{x}}_{n},{\bf{x}}_{m})$, and $\bm{\theta}$ is the kernel
hyperparameter vector. In this paper, we use an RBF kernel with an additive
noise term:
$\displaystyle k({\bf{x}}_{n},{\bf{x}}_{m})$
$\displaystyle=\alpha\exp\left(-\frac{1}{2\ell^{2}}({\bf{x}}_{n}-{\bf{x}}_{m})^{\top}({\bf{x}}_{n}-{\bf{x}}_{m})\right)$
$\displaystyle+\delta_{nm}\beta^{-1}.$ (2)
This likelihood is simply the product of $D$ independent Gaussian process
likelihoods, one for each output dimension.
Typically, the GPLVM is used for dimensionality reduction or visualization,
and the latent coordinates are determined by maximizing the posterior
probability of the latent coordinates, while integrating out the warping
function. In that setting, the Gaussian prior density on ${\bf{x}}$ is
essentially a regularizer which keeps the latent coordinates from spreading
arbitrarily far apart. In contrast, we instead integrate out the latent
coordinates as well as the warping function, and place a more flexible
parameterization on $p({\bf{x}})$ than a single isotropic Gaussian.
Just as the GPLVM can be viewed as a manifold learning algorithm, the iWMM can
be viewed as learning a set of manifolds, one for each cluster.
## 3 Infinite Warped Mixture Model
In this section, we define in detail the infinite warped mixture model (iWMM).
In the same way as the GPLVM, the iWMM assumes a set of latent coordinates and
a smooth, nonlinear mapping from the latent space to the observed space. In
addition, the iWMM assumes that the latent coordinates are generated from a
Dirichlet process mixture model. In particular, we use the following infinite
Gaussian mixture model,
$\displaystyle
p({\bf{x}}|\\{\lambda_{c},\bm{\mu}_{c},\mathbf{R}_{c}\\})=\sum_{c=1}^{\infty}\lambda_{c}{\cal
N}({\bf{x}}|\bm{\mu}_{c},\mathbf{R}_{c}^{-1}),$ (3)
where $\lambda_{c}$, $\bm{\mu}_{c}$ and $\mathbf{R}_{c}$ is the mixture
weight, mean, and precision matrix of the $c^{\text{th}}$ mixture component.
We place Gaussian-Wishart priors on the Gaussian parameters
$\\{\bm{\mu}_{c},\mathbf{R}_{c}\\}$,
$\displaystyle p(\bm{\mu}_{c},\mathbf{R}_{c})={\cal
N}(\bm{\mu}_{c}|{\bf{u}},(r\mathbf{R}_{c})^{-1}){\cal
W}(\mathbf{R}_{c}|\mathbf{S}^{-1},\nu),$ (4)
where ${\bf{u}}$ is the mean of $\bm{\mu}_{c}$, $r$ is the relative precision
of $\bm{\mu}_{c}$, $\mathbf{S}^{-1}$ is the scale matrix for $\mathbf{R}_{c}$,
and $\nu$ is the number of degrees of freedom for $\mathbf{R}_{c}$. The
Wishart distribution is defined as follows:
$\displaystyle{\cal
W}(\mathbf{R}|\mathbf{S}^{-1},\nu)=\frac{1}{G}|\mathbf{R}|^{\frac{\nu-Q-1}{2}}\exp\left(-\frac{1}{2}{\rm
tr}(\mathbf{S}\mathbf{R})\right),$ (5)
where $G$ is the normalizing constant. Because we use conjugate Gaussian-
Wishart priors for the parameters of the Gaussian mixture components, we can
analytically integrate out those parameters, given the assignments of points
to components. Let $z_{n}$ be the latent assignment of the $n^{\text{th}}$
point. The probability of latent coordinates $\mathbf{X}$ given latent
assignments $\mathbf{Z}=(z_{1},\cdots,z_{N})$ is obtained by integrating out
the Gaussian parameters $\\{\bm{\mu}_{c},\mathbf{R}_{c}\\}$ as follows:
$\displaystyle p(\mathbf{X}|\mathbf{Z},\mathbf{S},\nu,r)$
$\displaystyle=\prod_{c=1}^{\infty}\pi^{-\frac{N_{c}Q}{2}}\frac{r^{Q/2}|\mathbf{S}|^{\nu/2}}{r_{c}^{Q/2}|\mathbf{S}_{c}|^{\nu_{c}/2}}$
$\displaystyle\times\prod_{q=1}^{Q}\frac{\Gamma(\frac{\nu_{c}+1-q}{2})}{\Gamma(\frac{\nu+1-q}{2})},$
(6)
where $N_{c}$ is the number of data points assigned to the $c^{\text{th}}$
component, $\Gamma(\cdot)$ is Gamma function, and
$\displaystyle r_{c}=r+N_{c},\hskip 20.00003pt\nu_{c}=\nu+N_{c},$
$\displaystyle{\bf{u}}_{c}=\frac{r{\bf{u}}+\sum_{n:z_{n}=c}{\bf{x}}_{n}}{r+N_{c}},$
$\displaystyle\mathbf{S}_{c}=\mathbf{S}+\sum_{n:z_{n}=c}{\bf{x}}_{n}{\bf{x}}_{n}^{\top}+r{\bf{u}}{\bf{u}}^{\top}-r_{c}{\bf{u}}_{c}{\bf{u}}_{c}^{\top},$
(7)
are the posterior Gaussian-Wishart parameters of the $c^{\text{th}}$
component. We use a Dirichlet process with concentration parameter $\eta$ for
infinite mixture modeling maceachern1998estimating in the latent space. Then,
the probability of $\mathbf{Z}$ is given as follows:
$\displaystyle
p(\mathbf{Z}|\eta)=\frac{\eta^{C}\prod_{c=1}^{C}(N_{c}-1)!}{\eta(\eta+1)\cdots(\eta+N-1)},$
(8)
where $C$ is the number of components for which $N_{c}>0$. The joint
distribution is given by
$\displaystyle
p(\mathbf{Y},\mathbf{X},\mathbf{Z}|\bm{\theta},\bm{S},\nu,{\bf{u}},r,\eta)$
$\displaystyle=p(\mathbf{Y}|\mathbf{X},\bm{\theta})p(\mathbf{X}|\mathbf{Z},\bm{S},\nu,{\bf{u}},r)p(\mathbf{Z}|\eta),$
(9)
where factors in the right hand side can be calculated by (1), (6) and (8),
respectively.
In summary, the infinite warped mixture model generates observations
$\mathbf{Y}$ according to the following generative process:
1. 1.
Draw mixture weights $\bm{\lambda}\sim{\rm GEM}(\eta)$
2. 2.
For each component $c=1,\cdots,\infty$
1. (a)
Draw precision $\mathbf{R}_{c}\sim{\cal W}(\mathbf{S}^{-1},\nu)$
2. (b)
Draw mean $\bm{\mu}_{c}\sim{\cal N}({\bf{u}},(r\mathbf{R}_{c})^{-1})$
3. 3.
For each observed dimension $d=1,\cdots,D$
1. (a)
Draw function $f_{d}({\bf{x}})\sim{\rm
GP}(m({\bf{x}}),k({\bf{x}},{\bf{x}}^{\prime}))$
4. 4.
For each observation $n=1,\cdots,N$
1. (a)
Draw latent assignment $z_{n}\sim{\rm Mult}(\bm{\lambda})$
2. (b)
Draw latent coordinates ${\bf{x}}_{n}\sim{\cal
N}(\bm{\mu}_{z_{n}},\mathbf{R}_{z_{n}}^{-1})$
3. (c)
For each observed dimension $d=1,\cdots,D$
1. i.
Draw feature $y_{nd}\sim{\cal N}(f_{d}({\bf{x}}_{n}),\beta^{-1})$
Here, ${\rm GEM}(\eta)$ is the stick-breaking process sethuraman94 that
generates mixture weights for a Dirichlet process with parameter $\eta$, ${\rm
Mult}(\bm{\lambda})$ represents a multinomial distribution with parameter
$\bm{\lambda}$, $m({\bf{x}})$ is the mean function of the Gaussian process,
and ${\bf{x}},{\bf{x}}^{\prime}\in{\mathbb{R}}^{Q}$.
Figure 2: A graphical model representation of the infinite warped mixture
model, where the shaded and unshaded nodes indicate observed and latent
variables, respectively, and plates indicate repetition.
Figure 2 shows the graphical model representation of the proposed model. Here,
we assume a Gaussian for the mixture component, although we could in principle
use other distributions such as Student’s t-distribution or the Laplace
distribution.
The iWMM can be seen as a generalization of either the GPLVM or the infinite
Gaussian mixture model (iGMM). To be precise, the iWMM with a single fixed
spherical Gaussian density on the latent coordinates corresponds to the GPLVM,
while the iWMM with fixed direct mapping function $f_{d}({\bf{x}})=x_{d}$ and
$Q=D$ corresponds to the iGMM.
The iWMM offers attractive properties that do not exist in other probabilistic
models; principally, the ability to model clusters with nonparametric
densities, and to infer a seperate dimension for manifold.
## 4 Inference
We infer the posterior distribution of the latent coordinates $\mathbf{X}$ and
cluster assignments $\mathbf{Z}$ using Markov chain Monte Carlo (MCMC). In
particular, we alternate collapsed Gibbs sampling of $\mathbf{Z}$, and hybrid
Monte Carlo sampling of $\mathbf{X}$. Given $\mathbf{X}$, we can efficiently
sample $\mathbf{Z}$ using collapsed Gibbs sampling, integrating out the
mixture parameters. Given $\mathbf{Z}$, we can calculate the gradient of the
unnormalized posterior distribution of $\mathbf{X}$, integrating over warping
functions. This gradient allows us to sample $\mathbf{X}$ using hybrid Monte
Carlo.
First, we explain collapsed Gibbs sampling for $\mathbf{Z}$. Given a sample of
$\mathbf{X}$, $p(\mathbf{Z}|\mathbf{X},\mathbf{S},\nu,{\bf{u}},r,\eta)$ does
not depend on $\mathbf{Y}$. This lets resample cluster assignments,
integrating out the iGMM likelihood in close form. Given the current state of
all but one latent component $z_{n}$, a new value for $z_{n}$ is sampled from
the following probability:
$\displaystyle p(z_{n}=c|\mathbf{X},\mathbf{Z}_{\setminus
n},\bm{S},\nu,{\bf{u}},r,\eta)$
$\displaystyle\propto\\!\left\\{\begin{array}[]{ll}\\!\\!N_{c\setminus n}\cdot
p({\bf{x}}_{n}|\mathbf{X}_{c\setminus
n},\bm{S},\nu,{\bf{u}},r)&\text{{existing components}}\\\ \\!\\!\eta\cdot
p({\bf{x}}_{n}|\bm{S},\nu,{\bf{u}},r)&\text{{a new
component}}\end{array}\right.$ (12)
where $\mathbf{X}_{c}=\\{{\bf{x}}_{n}|z_{n}=c\\}$ is the set of latent
coordinates assigned to the $c^{\text{th}}$ component, and $\setminus n$
represents the value or set when excluding the $n^{\text{th}}$ data point. We
can analytically calculate $p({\bf{x}}_{n}|\mathbf{X}_{c\setminus
n},\bm{S},\nu,{\bf{u}},r)$ as follows:
$\displaystyle p({\bf{x}}_{n}|\mathbf{X}_{c\setminus
n},\bm{S},\nu,{\bf{u}},r)$ $\displaystyle=\pi^{-\frac{N_{c\setminus
n}Q}{2}}\frac{r_{c\setminus n}^{Q/2}|\mathbf{S}_{c\setminus
n}|^{\nu_{c\setminus n}/2}}{r_{c\setminus n}^{\prime
Q/2}|\mathbf{S}_{c\setminus n}^{\prime}|^{\nu_{c\setminus
n}^{\prime}/2}}\prod_{d=1}^{Q}\frac{\Gamma(\frac{\nu_{c\setminus
n}^{\prime}+1-d}{2})}{\Gamma(\frac{\nu_{c\setminus n}+1-d}{2})},$ (13)
where $r_{c}^{\prime}$, $\nu_{c}^{\prime}$, ${\bf{u}}_{c}^{\prime}$ and
$\mathbf{S}_{c}^{\prime}$ represent the posterior Gaussian-Wishart parameters
of the $c^{\text{th}}$ component when the $n^{\text{th}}$ data point is
assigned to the $c^{\text{th}}$ component. We can efficiently calculate the
determinant by using the rank one Cholesky update. In the same way, we can
analytically calculate the likelihood for a new component
$p({\bf{x}}_{n}|\bm{S},\nu,{\bf{u}},r)$.
Hybrid Monte Carlo (HMC) sampling of $\mathbf{X}$ from posterior
${p(\mathbf{X}|\mathbf{Z},\mathbf{Y},\bm{\theta},\bm{S},\nu,{\bf{u}},r)}$,
requires computing the gradient of the log of the unnormalized posterior
${\log p(\mathbf{Y}|\mathbf{X},\bm{\theta})+\log
p(\mathbf{X}|\mathbf{Z},\bm{S},\nu,{\bf{u}},r)}$. The first term of the
gradient can be calculated by
$\displaystyle\frac{\partial\log
p(\mathbf{Y}|\mathbf{X},\bm{\theta})}{\partial\mathbf{K}}=-\frac{1}{2}D\mathbf{K}^{-1}+\frac{1}{2}\mathbf{K}^{-1}\mathbf{Y}\mathbf{Y}^{T}\mathbf{K}^{-1},$
(14)
and
$\displaystyle\frac{\partial
k({\bf{x}}_{n},{\bf{x}}_{m})}{\partial{\bf{x}}_{n}}$
$\displaystyle=-\frac{\alpha}{\ell^{2}}\exp\left(-\frac{1}{2\ell^{2}}({\bf{x}}_{n}-{\bf{x}}_{m})^{\top}({\bf{x}}_{n}-{\bf{x}}_{m})\right)({\bf{x}}_{n}-{\bf{x}}_{m}),$
(15)
using the chain rule. The second term can be calculated as follows:
$\displaystyle\frac{\partial\log
p(\mathbf{X}|\mathbf{Z},\bm{S},\nu,{\bf{u}},r)}{\partial{\bf{x}}_{n}}=-\nu_{z_{n}}\bm{S}_{z_{n}}^{-1}({\bf{x}}_{n}-{\bf{u}}_{z_{n}}).$
(16)
We also infer kernel hyperparameters $\bm{\theta}=\\{\alpha,\beta,\ell\\}$ via
HMC, using the gradient of the log unnormalized posterior with respect to the
kernel hyperparameters. The complexity of each iteration of HMC is dominated
by the $\mathcal{O}(N^{3})$ computation of $\mathbf{K}^{{{-1}}}$ 111This
complexity could be improved by making use of an inducing point approximation
such as quinonero2005unifying ; snelson2006sparse .
In summary, we obtain samples from the posterior
$p(\mathbf{X},\mathbf{Z}|\mathbf{Y},\bm{\theta},\mathbf{S},\nu,{\bf{u}},r,\eta)$
by iterating the following procedures:
1. 1.
For each observation $n=1,\cdots,N$, sample the component assignment $z_{n}$
by collapsed Gibbs sampling (12).
2. 2.
Sample latent coordinates $\mathbf{X}$ and kernel parameters $\bm{\theta}$
using hybrid Monte Carlo.
### 4.1 Posterior Predictive Density
In the GP-LVM, the predictive density of at test point $y^{\star}$ is usually
computed by finding the point $x^{\star}$ which which is most likely to be
mapped to $y^{\star}$, then using the density of $p(x^{\star})$ and the
Jacobian of the warping at that point to approximately compute the density at
$y^{\star}$. When inference is done by simply optimizing the location of the
latent points, this estimation method simply requires solving a single
optimization for each $y^{\star}$.
For our model, we use approximate integration to estimate $p(y^{\star})$. This
is done for two reasons: First, multiple latent points (possibly from
different clusters) can map to the same observed point, meaning the standard
method can underestimate $p(y^{\star})$. Second, because we do not optimize
the latent coordinates but rather sample them, we would need to perform
optimizations for each $p(y^{\star})$ seperately for each sample. Our method
gives estimates for all $p({\bf{y}}^{\star})$ at once, but may not be accurate
in very high dimensions.
The posterior density in the observed space given the training data is simply:
$\displaystyle p({\bf{y}}_{*}|\mathbf{Y})$ $\displaystyle=\int\\!\\!\\!\int
p({\bf{y}}_{*},{\bf{x}}_{*},\mathbf{X}|\mathbf{Y})d{\bf{x}}_{*}d\mathbf{X}$
$\displaystyle=\int\\!\\!\\!\int
p({\bf{y}}_{*}|{\bf{x}}_{*},\mathbf{X},\mathbf{Y})p({\bf{x}}_{*}|\mathbf{X},\mathbf{Y})p(\mathbf{X}|\mathbf{Y})d{\bf{x}}_{*}d\mathbf{X}.$
(17)
We approximate $p(\mathbf{X}|\mathbf{Y})$ using the samples from the Gibbs and
hybrid Monte Carlo samplers. We approximate
$p({\bf{x}}_{*}|\mathbf{X},\mathbf{Y})$ by sampling points from the latent
mixture and warping them, using the following procedure:
1. 1.
Draw latent assignment
$z_{*}\sim{\rm
Mult}(\frac{N_{1}}{N+\eta},\cdots,\frac{N_{C}}{N+\eta},\frac{\eta}{N+\eta})$
2. 2.
Draw precision matrix
$\mathbf{R}_{*}\sim{\cal W}(\mathbf{S}^{-1}_{z_{*}},\nu_{z_{*}})$
3. 3.
Draw mean
$\bm{\mu}_{*}\sim{\cal N}({\bf{u}}_{z_{*}},(r_{z_{*}}\mathbf{R}_{*})^{-1})$
4. 4.
Draw latent coordinates
${\bf{x}}_{*}\sim{\cal N}(\bm{\mu}_{*},\mathbf{R}_{*}^{-1})$
When a new component $C+1$ is assigned to $z_{*}$, the prior Gaussian-Wishart
distribution is used for sampling in steps 2 and 3. The first factor of (17)
can be calculated by
$\displaystyle p({\bf{y}}_{*}|{\bf{x}}_{*},\mathbf{X},\mathbf{Y})$
$\displaystyle={\cal
N}({\bf{k}}_{*}^{\top}\mathbf{K}^{-1}\mathbf{Y},k({\bf{x}}_{*},{\bf{x}}_{*})-{\bf{k}}_{*}^{\top}\mathbf{K}^{-1}{\bf{k}}_{*}),$
(18)
where
${\bf{k}}_{*}=(k({\bf{x}}_{*},{\bf{x}}_{1}),\cdots,k({\bf{x}}_{*},{\bf{x}}_{N}))^{\top}$.
Each step of this procedure is exact, and since the observations
${\bf{y}}_{*}$ are conditionally normally distributed, each one adds a smooth
contribution to the empirical Monte Carlo estimate of the posterior density,
as opposed to a collection of point masses. This procedure was used to
generate the plots of posterior density in figures 1, 4, and 6.
## 5 Related work
The GPLVM is effective as a nonlinear latent variable model in a wide variety
of applications lawrence2004gaussian ; salzmann2008local ; lawrence2009non .
The latent positions $\mathbf{X}$ in the GPLVM are typically obtained by
maximum a posteriori estimation or variational Bayesian inference
titsias2010bayesian , placing a single fixed spherical Gaussian prior on
${\bf{x}}$. A prior which penalizes a high-dimensional latent space is
introduced by geiger2009rank , in which the latent variables and their
intrinsic dimensionality are simultaneously optimized. The iWMM can also infer
the intrinsic dimensionality of nonlinear manifolds: inferring the Gaussian
covariance for each latent cluster allows the variance of irrelevant
dimensions to become small. Because each latent cluster has a different set of
parameters, the effective dimension of each cluster can vary, allowing
manifolds of different dimension in the observed space. This ability is
demonstrated in figure 4b.
The iWMM can also be viewed as a generalization of the mixture of
probabilistic principle component analyzers tipping1999mixtures , or mixture
of factor analyzers ghahramani2000variational , where the linear mapping of
the mixtures is generalized to a nonlinear mapping by Gaussian processes, and
number of components is infinite.
There exist non-probabilistic clustering methods which can find clusters with
complex shapes, such as spectral clustering ng2002spectral and nonlinear
manifold clustering cao2006nonlinear ; elhamifar2011sparse . Spectral
clustering finds clusters by first forming a similarity graph, then finding a
low-dimensional latent representation using the graph, and finally, clustering
the latent coordinates via k-means. The performance of spectral clustering
depends on parameters which are usually set manually, such as the number of
clusters, the number of neighbors, and the variance parameter used for
constructing the similarity graph. In contrast, the iWMM infers such
parameters automatically. One of the main advantages of the iWMM over these
methods is that there is no need to construct a similarity graph.
The kernel Gaussian mixture model wang2003kernel can also find non-Gaussian
shaped clusters. This model estimates a GMM in the implicit high-dimensional
feature space defined by the kernel mapping of the observed space. However,
the kernel GMM uses a fixed nonlinear mapping function, with no guarantee that
the latent points will be well-modeled by a GMM. In contrast, the iWMM infers
the mapping function such that the latent co-ordinates will be well-modeled by
a mixture of Gaussians.
## 6 Experimental results
Figure 3: A sample from the 2-dimensional latent space when modeling a series
of 32x32 face images. Our model correctly discovers that the data consists of
two seperate manifolds, both approximately one-dimensional, which share the
same head-turning structure.
### 6.1 Clustering Faces
We first examined our model’s ability to model images without pre-processing.
We constructed a dataset consisting of 50 greyscale 32x32 pixel images of two
individuals from the UMIST faces dataset umistfaces . Both series of images
capture a person turning his head to the right. Figure 3 shows a sample from
the posterior over the latent coordinates and density model. The model has
recovered three relevant, interpretable features of the dataset. First, that
there are two distinct faces. Second, that each set of images lies
approximately along a smooth one-dimensional manifold. Third, that the two
manifolds share roughly the same structure: the front-facing images of both
individuals lie close to one another, as do the side-facing images.
Observed space
---
| | |
$\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$
| | |
Latent space
(a) 2-curve | (b) 3-semi | (c) 2-circle | (d) Pinwheel
Figure 4: Top row: The observed, unlabeled data points, and the clusters
inferred by the iWMM. Bottom row: Latent coordinates and Gaussian components,
shown for a single sample from the posterior. Each point in the latent space
corresponds to a point in the observed space. This figure is best viewed in
color.
### 6.2 Synthetic Datasets
Next, we demonstrate the proposed model on the four synthetic datasets shown
in Figure 4. None of these four datasets can be appropriately clustered by
Gaussian mixture models (GMM). For example, consider the 2-curve data shown in
Figure 4 (a), where 100 data points lie in one of two curved lines in a two-
dimensional observed space. A GMM with two components cannot separate the two
curved lines, while a GMM with many components could separate the two lines
only by breaking each line into many clusters. In contrast, with the iWMM, the
two non-Gaussian-shaped clusters in the observed space were represented by two
Gaussian-shaped clusters in the latent space, as shown at the bottom row of
Figure 4 (a). The iWMM separated the two curved lines by nonlinearly warping
two Gaussians from the latent space to the observed space.
Figure 4 (c) shows an interesting manifold learning challenge: a dataset
consisting of two circles. The outer circle is modeled in the latent space by
a Gaussian with effectively one degree of freedom. This linear topology fits
the outer circle in the observed space by bending the two ends until they
overlap. In contrast, the sampler fails to discover the 1D topology of the
inner circle, modeling it with a 2D manifold instead. This example
demonstrates that each cluster in the iWMM manifold can have a different
effective dimension.
| | |
---|---|---|---
(a) 1 | (b) 500 | (c) 1800 | (d) 3000
Figure 5: The inferred infinite GMMs over iterations in the two-dimensional
latent space with the iWMM using the 2-curve data. Labels indicate the number
of iterations of the sampler, and the color of each point represents its
ordering in the observed coordinates.
### 6.3 Mixing
An interesting side-effect of learning the number of latent clusters is that
this added flexibility can help the sampler escape local minima, helping the
sampler to mix properly. Figure 5 shows the samples of the latent coordinates
and clusters of the iWMM over time, when modeling the 2-curve data. 5(a) shows
the latent coordinates initialized at the observed coordinates, starting with
one latent component. At the 500th iteration 5(b), each curved line is modeled
by two components. At the 1800th iteration 5(c), the left curved line is
modeled by a single component. At the 3000th iteration 5(d), the right curved
line is also modeled by a single component, and the dataset is appropriately
clustered. This configuration was relatively stable, and a similar state was
found at the 5000th iteration.
|
---|---
(a) iWMM | (b) iWMM ($C=1$)
Figure 6: The posterior density in the observed space with the 2-curve data
inferred by the iWMM (a), and that inferred by the iWMM with one component
(b).
### 6.4 Density Estimation
Figure 6 (a) shows the posterior density in the observed space inferred by the
iWMM on the 2-curve data, computed using 1000 samples from the Markov chain.
The two separate manifolds of high density implied by the two curved lines was
recovered by the iWMM. Note also that the density along the manifold varies
with the density of data shown in Figure 4 (a). This result can be compared to
a special case of our model, which uses only a single Gaussian to model the
latent coordinates instead of an infinite GMM. Figure 6 (b) shows that the
result of the iWMM with $C=1$, where posterior is forced to place significant
density connecting the two clusters. Figure 6 (b) shows that the single-
cluster variant of the iWMM posterior is forced to place significant density
connecting the two clusters.
| | |
---|---|---|---
(a) iWMM | (b) iWMM ($C=1$) | (c) GPLVM | (d) BGPLVM
Figure 7: The estimated latent coordinates of the 2-curve data by (a) iWMM,
(b) iWMM ($C=1$), (c) GPLVM, and (d) Bayesian GPLVM.
### 6.5 Visualization
Next, we briefly investigate the potential of the iWMM for visualization.
Figure 7 (a) shows the latent coordinates obtained by averaging over 1000
samples from the posterior of the iWMM. Because rotating the latent
coordinates does not change their probability, averaging may not be an
adequate way to summarize the posterior. However, we show this result in order
to show the characteristics of latent coordinates obtained by the iWMM. The
estimated latent coordinates are clearly separated, and they form two straight
lines. This result indicates that in some cases, the iWMM can recover the
topology of the data before it has been warped into a manifold. For
comparison, Figure 7 (b) shows the latent coordinates estimated by the iWMM
when forced to use a single cluster: the latent coordinates lie in two
sections of a single straight line. Figure 7 (c) and (d) show the latent
coordinates estimated by the GPLVM when optimizing or integrating out the
latent coordinates, respectively. Recall that the iWMM ($C=1$) is a more
flexible model than the GPLVM, since the GPLVM enforces a spherical covariance
in the latent space. These methods did not unfold the two curved lines, since
the effective dimension of their latent representation is fixed beforehand. In
contrast, the iWMM effectively formed a low-dimensional representation in the
latent space.
Regardless of the dimension of the latent space, the iWMM will tend to model
each cluster with as low-dimensional a Gaussian as possible. This is because,
if the data in a cluster can be made to lie in a low-dimensional plane, a
narrowly-shaped Gaussian will assign the latent coordinates much higher
likelihood than a spherical Gaussian.
Table 1: The statistics of datasets used for evaluation. | 2-curve | 3-semi | 2-circle | Pinwheel | Iris | Glass | Wine | Vowel
---|---|---|---|---|---|---|---|---
number of samples: $N$ | 100 | 300 | 100 | 250 | 150 | 214 | 178 | 528
observed dimensionality: $D$ | 2 | 2 | 2 | 2 | 4 | 9 | 13 | 10
number of clusters: $C$ | 2 | 3 | 2 | 5 | 3 | 7 | 3 | 11
### 6.6 Clustering Performance
We more formally evaluated the density estimation and clustering performance
of the proposed model using four real datasets: iris, glass, wine and vowel,
obtained from LIBSVM multi-class datasets chang2011libsvm , in addition to the
four synthetic datasets shown above: 2-curve, 3-semi, 2-circle and Pinwheel
adams2009archipelago . The statistics of these datasets are summarized in
Table 1. In each experiment, we show the results of ten-fold cross-validation.
Results in bold are not significantly different from the best performing
method in each column according to a paired t-test.
Table 2: Average Rand index for evaluating clustering performance. | 2-curve | 3-semi | 2-circle | Pinwheel | Iris | Glass | Wine | Vowel
---|---|---|---|---|---|---|---|---
iGMM | $0.52$ | $0.79$ | $0.83$ | $0.81$ | $0.78$ | $0.60$ | $0.72$ | $\mathbf{0.76}$
iWMM(Q=2) | $\mathbf{0.86}$ | $\mathbf{0.99}$ | $\mathbf{0.89}$ | $\mathbf{0.94}$ | $\mathbf{0.81}$ | $\mathbf{0.65}$ | $0.65$ | $0.50$
iWMM(Q=D) | $\mathbf{0.86}$ | $\mathbf{0.99}$ | $\mathbf{0.89}$ | $\mathbf{0.94}$ | $0.77$ | $0.62$ | $\mathbf{0.77}$ | $\mathbf{0.76}$
Table 2 compares the clustering performance of the iWMM with the iGMM,
quantified by the Rand index rand1971objective , which measures the
correspondence between inferred clusters and true clusters. The iGMM is
another probabilistic generative model commonly used for clustering, which can
be seen as a special case of the iWMM in which the Gaussian clusters are not
warped. These experiments demonstrate the extent to which nonparametric
cluster shapes allow a mixture model to recover more meaningful clusters.
Table 3 lists average test log likelihood, comparing the proposed models with
kernel density estimation (KDE), and the infinite Gaussian mixture model
(iGMM). In KDE, the kernel width is estimated by maximizing the leave-one-out
log densities. Since the manifold on which the observed data lies can be at
most $D$-dimensional, we set the latent dimension $Q$ equal to the observed
dimension $D$ in iWMMs. We also include the $Q=2$ case in an attempt to
characterize how much modeling power is lost by forcing the latent
representation to be visualizable. The proposed models achieved high test log
likelihoods compared with the KDE and iGMM.
Table 3: Average test log likelihood for evaluating density estimation performance. | 2-curve | 3-semi | 2-circle | Pinwheel | Iris | Glass | Wine | Vowel
---|---|---|---|---|---|---|---|---
KDE | $-2.47$ | $-0.38$ | $-1.92$ | $-1.47$ | $\mathbf{-1.87}$ | $1.26$ | $-2.73$ | $\mathbf{6.06}$
iGMM | $-3.28$ | $-2.26$ | $-2.21$ | $-2.12$ | $-1.91$ | $3.00$ | $\mathbf{-1.87}$ | $-0.67$
iWMM(Q=2) | $\mathbf{-0.90}$ | $\mathbf{-0.18}$ | $\mathbf{-1.02}$ | $\mathbf{-0.79}$ | $\mathbf{-1.88}$ | $\mathbf{5.76}$ | $\mathbf{-1.96}$ | $\mathbf{5.91}$
iWMM(Q=D) | $\mathbf{-0.90}$ | $\mathbf{-0.18}$ | $\mathbf{-1.02}$ | $\mathbf{-0.79}$ | $\mathbf{-1.71}$ | $\mathbf{5.70}$ | $-3.14$ | $-0.35$
### 6.7 Source code
Code to reproduce all the above experiments is available at
http://github.com/duvenaud/warped-mixtures.
## 7 Future work
The Dirichlet process mixture of Gaussians in the latent space of our model
could easily be replaced by a more sophisticated density model, such as a
hierarchical Dirichlet process teh2006hierarchical , or a Dirichlet diffusion
tree neal2003density . Another straightforward extension of our model would be
making inference more scalable by using sparse Gaussian processes
quinonero2005unifying ; snelson2006sparse or more advanced hybrid Monte Carlo
methods zhang2011quasi . An interesting but more complex extension of the iWMM
would be a semi-supervised version of the model. The iWMM could allow label
propagation along regions of high density in the latent space, even if those
regions were stretched along low-dimensional manifolds in the observed space.
Another natural extension would be to allow a separate warping for each
cluster, which would also improve inference speed.
## 8 Conclusion
In this paper, we introduced a simple generative model of non-Gaussian density
manifolds which can infer nonlinearly separable clusters, low-dimensional
representations of varying dimension per cluster, and density estimates which
smoothly follow data contours. We then introduced an efficient sampler for
this model which integrates out both the cluster parameters and the warping
function exactly. We further demonstrated that allowing non-parametric cluster
shapes improves clustering performance over the Dirichlet process Mixture of
Gaussians.
Many methods have been proposed which can perform some combination of
clustering, manifold learning, density estimation and visualization. We
demonstrated that a simple but flexible probabilistic generative model can
perform well at all these tasks.
### Acknowledgements
The authors would like to thank Dominique Perrault-Joncas, Carl Edward
Rasmussen, and Ryan Prescott Adams for helpful discussions.
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|
arxiv-papers
| 2012-06-08T19:45:49 |
2024-09-04T02:49:31.619603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tomoharu Iwata, David Duvenaud, Zoubin Ghahramani",
"submitter": "David Duvenaud",
"url": "https://arxiv.org/abs/1206.1846"
}
|
1206.1927
|
# A topological set theory
implied by $\Th{ZF}$ and $\Th{GPK^{+}_{\infty}}$
Andreas Fackler
###### Abstract
We present a system of axioms motivated by a topological intuition: The set of
subsets of any set is a topology on that set. On the one hand, this system is
a common weakening of Zermelo-Fraenkel set theory $\Th{ZF}$, the positive set
theory $\Th{GPK}^{+}_{\infty}$ and the theory of hyperuniverses. On the other
hand, it retains most of the expressiveness of these theories and has the same
consistency strength as $\Th{ZF}$. We single out the additional axiom of the
universal set as the one that increases the consistency strength to that of
$\Th{GPK}^{+}_{\infty}$ and explore several other axioms and interrelations
between those theories. Our results are independent of whether the empty class
is a set and whether atoms exist.
This article is a revised version of the first part of the author’s doctoral
thesis [Fac12].
## Introduction
An axiomatic set theory can be thought of as an effort to make precise which
classes are sets. It simultaneously aims at providing enough freedom of
construction for all of classical mathematics and still remain consistent. It
therefore must imply that all “reasonable” class comprehensions
$\\{x\mid\phi(x)\\}$ produce sets and explain why the Russell class $\\{x\mid
x\notin x\\}$ does not. The answer given by e. g. Zermelo-Fraenkel set theory
($\Th{ZF}$) is the _Limitation of Size Principle_ : Only small classes are
sets. However, the totality of all mathematical objects, the _universe_
$\mathbb{V}=\\{x\mid x{=}x\\}$, is a proper class in $\Th{ZF}$.
Two very different ideas of sethood lead to a family of theories which do
allow $\mathbb{V}\in\mathbb{V}$:
Firstly, one might blame the negation in the formula $x\notin x$ for Russell’s
paradox. The collection of _generalized positive formula_ s is recursively
defined by several construction steps not including negation. If the existence
of $\\{x\mid\phi(x)\\}$ is stipulated for every generalized positive formula,
a beautiful “positive” set theory emerges.
Secondly, instead of demanding that every class is a set, one might settle for
the ability to approximate it by a least superset, a _closure_ in a
topological sense.
Surprisingly such “topological” set theories tend to prove the comprehension
principle for generalized positive formulas, and conversely, in positive set
theory, the universe is a topological space. More precisely, the sets are
closed with respect to intersections and finite unions, and the universe is a
set itself, so the sets represent the closed subclasses of a topology on
$\mathbb{V}$. A class is a set if and only if it is topologically closed.
The first model of such a theory was constructed by R. J. Malitz in [Mal76]
under the condition of the existence of certain large cardinal numbers. E.
Weydert, M. Forti and R. Hinnion were able to show in [Wey89, FH89] that in
fact a weakly compact cardinal suffices. In [Ess97] and [Ess99], O. Esser
exhaustively answered the question of consistency for a specific positive set
theory, $\Th{GPK}^{+}_{\infty}$ with a choice principle, and showed that it is
mutually interpretable with a variant of Kelley-Morse set theory.
### Atoms, sets, classes and topology
We incorporate proper classes into all the theories we consider. This enables
us to write down many arguments in a more concise yet formally correct way,
and helps separate the peculiarities of particular theories from the common
assumptions about atoms, sets and classes.
We use the language of set theory with atoms, whose non-logical symbols are
the binary relation symbol $\in$ and the constant symbol $\mathbb{A}$. We say
“$X$ is an element of $Y$” for $X\in Y$. We call $X$ an _atom_ if
$X\in\mathbb{A}$, and otherwise we call $X$ a _class_. If a class is an
element of any other class, it is a _set_ ; otherwise it is a _proper class_.
We denote the objects of our theories – all atoms, sets and classes – by
capital letters and use lowercase letters for sets and atoms only, so:
* •
${\forall}x\;\phi(x)$ means
${\forall}X.\;({\exists}Y\;X{\in}Y)\Rightarrow\phi(X)\;$ and
* •
${\exists}x\;\phi(x)$ means
${\exists}X.\;({\exists}Y\;X{\in}Y)\>\wedge\>\phi(X)$.
For each formula $\phi$ let $\phi^{C}$ be its relativization to the sets and
atoms, that is, every quantified variable in $\phi$ is replaced by a lowercase
variable in $\phi^{C}$.
Free variables in formulas that are supposed to be sentences are implicitly
universally quantified. For example, we usually omit the outer universal
quantifiers in axioms.
Using these definitions and conventions, we can now state the basic axioms
concerning atoms, sets and classes. Firstly, we assume that classes are
uniquely defined by their extension, that is, two classes are equal iff they
have the same elements. Secondly, atoms do not have any elements. Thirdly,
there are at least two distinct sets or atoms. And finally, any collection of
sets and atoms which can be defined in terms of sets, atoms and finitely many
fixed parameters, is a class. Formally:
Extensionality
$\displaystyle(X,Y{\notin}\mathbb{A}\;\wedge\;{\forall}Z.\;Z{\in}X\Leftrightarrow
Z{\in}Y)\;\Rightarrow\;X{=}Y$ Atoms $\displaystyle
X\in\mathbb{A}\quad\Rightarrow\quad Y\notin X$ Nontriviality
$\displaystyle{\exists}x{,}y\;\;x\neq y$
$\displaystyle\text{\emph{Comprehension}}(\psi)$
$\displaystyle{\exists}Z{\notin}\mathbb{A}.\;{\forall}x.\;x{\in}Z\Leftrightarrow\psi(x,\vec{P})\;\text{
for all formulas $\psi=\phi^{C}$}$
We will refer to these axioms as the _class axioms_ from now on. Note that the
object $\mathbb{A}$ may well be a proper class, or a set. The atoms axiom
implies however that $\mathbb{A}$ is not an atom.
We call the axiom scheme given in the fourth line the _weak comprehension_
scheme. It can be strengthened by removing the restriction on the formula
$\psi$, instead allowing $\psi$ to be any formula – even quantifying over all
classes. Let us call that variant the _strong comprehension_ scheme. The axiom
of extensionality implies the uniqueness of the class $Z$. We also write
$\\{w\mid\psi(w,\vec{P})\\}$ for $Z$, and generally use the customary notation
for comprehensions, e.g. $\\{x_{1},\ldots,x_{n}\\}=\\{y\mid
y{=}x_{1}\vee\ldots\vee y{=}x_{n}\\}$ for the class with finitely many
elements $x_{1},\ldots,x_{n}$, $\emptyset=\\{w\mid w{\neq}w\\}$ for the empty
class and $\mathbb{V}=\\{w\mid w{=}w\\}$ for the universal class. Also let
$\mathbb{T}=\\{x\mid\exists y.y{\in}x\\}$ be the class of nonempty sets. The
weak comprehension scheme allows us to define unions, intersections and
differences in the usual way.
Given the class axioms, we can now define several topological terms. They all
make sense in this weak theory, but one has to carefully avoid for now the
assumption that any class is a set. Also, the “right” definition of a topology
in our context is the collection of all nonempty closed sets instead of all
open sets.
For given classes $A$ and $T$, we call $A$ $T$-_closed_ if $A=\emptyset$ or
$A\in T$. A _topology_ on a class $X$ is a class $T$ of nonempty subsets of
$X$, such that:
* •
$X$ is $T$-closed.
* •
$\bigcap B$ is $T$-closed for every nonempty class $B{\subseteq}T$.
* •
$a\cup b$ is $T$-closed for all $T$-closed sets $a$ and $b$.
The class $X$, together with $T$, is then called a _topological space_. If $A$
is a $T$-closed class, then its complement $\complement A=X\setminus A$ is
$T$-_open_. A class which is both $T$-closed and $T$-open is $T$-_clopen_. The
intersection of all $T$-closed supersets of a class $A\subseteq X$ is the
least $T$-closed superset and is called the $T$-_closure_ $\mathrm{cl}_{T}(A)$
of $A$. Then $\mathrm{int}_{T}(A)=\complement\mathrm{cl}_{T}(\complement A)$
is the largest $T$-open subclass of $A$ and is called the $T$-_interior_ of
$A$. Every $A$ with $x\in\mathrm{int}_{T}(A)$ is a $T$-_neighborhood_ of the
point $x$. The explicit reference to $T$ is often omitted and $X$ itself is
considered a topological space, if the topology is clear from the context.
If $S\subset T$ and both are topologies, we call $S$ _coarser_ and $T$
_finer_. An intersection of several topologies on a set $X$ always is a
topology on $X$ itself. Thus for every class $B$ of subsets of $X$, if there
is a coarsest topology $T\supseteq B$, then that is the intersection of all
topologies $S$ with $B\subseteq S$. We say that $B$ is a _subbase_ for $T$ and
that $T$ is _generated_ by $B$.
If $A\subseteq X$, we call a subclass $B\subseteq A$ _relatively closed_ in
$A$ if there is a $T$-closed $C$ such that $B=A\cap C$, and similarly for
_relatively open_ and _relatively clopen_. If every subclass of $A$ is
relatively closed in $A$, we say that $A$ is _discrete_. Thus a $T$-closed set
$A$ is discrete iff all its nonempty subclasses are elements of $T$. Note that
there is an equivalent definition of the discreteness of a class $A\subseteq
X$ which can be expressed without quantifying over classes: $A$ is discrete
iff it contains none of its accumulation points, where an _accumulation point_
is a point $x\in X$ which is an element of every $T$-closed $B\supseteq
A{\setminus}\\{x\\}$. Formally, $A$ is discrete iff it has at most one point
or:
${\forall}x{\in}A\;{\exists}b{\in}T.\;\;A\subseteq
b{\cup}\\{x\\}\;\wedge\;x{\notin}b$
A topological space $X$ is $T_{1}$ if for all distinct $x,y\in X$ there exists
an open $U\subseteq X$ with $y\notin U\ni x$, or equivalently, if every
singleton $\\{x\\}\subseteq X$ is closed. $X$ is $T_{2}$ or _Hausdorff_ if for
all distinct $x,y\in X$ there exist disjoint open $U,V\subseteq X$ with $x\in
U$ and $y\in V$. It is _regular_ if for all closed $A\subseteq X$ and all
$x\in X\setminus A$ there exist disjoint open $U,V\subseteq X$ with
$A\subseteq U$ and $x\in V$. $X$ is $T_{3}$ if it is regular and $T_{1}$. It
is _normal_ if for all disjoint, closed $A,B\subseteq X$ there exist disjoint
open $U,V\subseteq X$ with $A\subseteq U$ and $B\subseteq V$. $X$ is $T_{4}$
if it is normal and $T_{1}$.
A map $f:X\rightarrow Y$ between topological spaces is _continuous_ if all
preimages $f^{-1}[A]$ of closed sets $A\subseteq Y$ are closed, and it is
_closed_ if all images $f[A]$ of closed sets $A\subseteq X$ are closed.
Let $\mathcal{K}$ be any class. We consider a class $A$ to be
$\mathcal{K}$-_small_ if it is empty or there is a surjection from a member of
$\mathcal{K}$ onto $A$, that is:
$A=\emptyset\quad\vee\quad{\exists}x{\in}\mathcal{K}\;{\exists}F{:}x{\rightarrow}A\;F[x]{=}A$
Otherwise, $A$ is $\mathcal{K}$-_large_. We say $\mathcal{K}$-_few_ for “a
$\mathcal{K}$-small collection of”, and $\mathcal{K}$-_many_ for “a
$\mathcal{K}$-large collection of”. Although we quantified over classes in
this definition, we will only use it in situations where there is an
equivalent first-order formulation.
If all unions of $\mathcal{K}$-small subclasses of a topology $T$ are
$T$-closed, then $T$ is called _$\mathcal{K}$ -additive_ or a _$\mathcal{K}$
-topology_. If $T$ is a subclass of every $\mathcal{K}$-topology $S\supseteq
B$ on $X$, then $T$ is $\mathcal{K}$-_generated_ by $B$ on $X$ and $B$ is a
$\mathcal{K}$-_subbase_ of $T$ on $X$. If every element of $T$ is an
intersection of elements of $B$, $B$ is a _base_ of $T$.
A topology $T$ on $X$ is $\mathcal{K}$-_compact_ if every $T$-cocover has a
$\mathcal{K}$-small $T$-subcocover, where a $T$-_cocover_ is a class
$B\subseteq T$ with $\bigcap B=\emptyset$. Dually, we use the more familiar
term _open cover_ for a collection of $T$-open classes whose union is $X$,
where applicable.
For all classes $A$ and $T$, let
$\square_{T}A=\\{b{\in}T\mid b{\subseteq}A\\}\quad\text{ and
}\quad\lozenge_{T}A=\\{b{\in}T\mid b\cap A\neq\emptyset\\}\text{.}$
If $T$ is a topology on $X$, and if for all $a,b\in T$ the classes
$\square_{T}a\cap\lozenge_{T}b$ are sets, then the set
$T=\square_{T}X=\lozenge_{T}X$, together with the topology $S$
$\mathcal{K}$-generated by $\\{\square_{T}a{\cap}\lozenge_{T}b\mid
a,b{\in}T\\}$ is called the $\mathcal{K}$-_hyperspace_ (or _exponential
space_) of $X$ and denoted by
$\mathrm{Exp}_{\mathcal{K}}(X,T)=\langle\square_{T}X,S\rangle$, or in the
short form: $\mathrm{Exp}_{\mathcal{K}}(X)$. Since
$\square_{T}a=\square_{T}a\cap\lozenge_{T}X$ and
$\lozenge_{T}a=\square_{T}X\cap\lozenge_{T}a$, the classes $\square_{T}a$ and
$\lozenge_{T}a$ are also sets and constitute another $\mathcal{K}$-subbase of
the exponential $\mathcal{K}$-topology. A notable subspace of
$\mathrm{Exp}_{\mathcal{K}}(X)$ is the space
$\mathrm{Exp}^{c}_{\mathcal{K}}(X)$ of $\mathcal{K}$-compact subsets. In fact,
this restriction suggests the canonical definition
$\mathrm{Exp}^{c}_{\mathcal{K}}(f)(a)=f[a]$ of a map
$\mathrm{Exp}^{c}_{\mathcal{K}}(f):\mathrm{Exp}^{c}_{\mathcal{K}}(X)\rightarrow\mathrm{Exp}^{c}_{\mathcal{K}}(Y)$
for every continuous $f:X\rightarrow Y$, because continuous images of
$\mathcal{K}$-compact sets are $\mathcal{K}$-compact. Moreover,
$\mathrm{Exp}^{c}_{\mathcal{K}}(f)$ is continuous itself.
$\mathcal{K}$ should be pictured as a cardinal number, but prior to stating
the axioms of essential set theory, the theory of ordinal and cardinal numbers
is not available. However, to obtain useful ordinal numbers, an axiom stating
that the additivity is greater than the cardinality of any discrete set is
needed. Fortunately, this can be expressed using the class $\mathcal{D}$ of
all discrete sets as the additivity.
## 1 Essential Set Theory
Consider the following, in addition to the class axioms:
1st Topology Axiom $\displaystyle\mathbb{V}\in\mathbb{V}$ 2nd Topology Axiom
If $A{\subseteq}\mathbb{T}$ is nonempty, then $\bigcap A$ is
$\mathbb{T}$-closed. 3rd Topology Axiom If $a$ and $b$ are
$\mathbb{T}$-closed, then $a{\cup}b$ is $\mathbb{T}$-closed. $\displaystyle
T_{1}$ $\\{a\\}$ is $\mathbb{T}$-closed. Exponential
$\square_{\mathbb{T}}a\cap\lozenge_{\mathbb{T}}b$ is $\mathbb{T}$-closed.
Discrete Additivity $\displaystyle\bigcup A\text{ is $\mathbb{T}$-closed for
every $\mathcal{D}$-small class $A$.}$
We call this system of axioms _topological set theory_ , or in short:
$\Th{TS}$, and the theory $\Th{TS}$ without the 1st topology axiom _essential
set theory_ or $\Th{ES}$. We will mostly work in $\Th{ES}$ and explicitly
single out the consequences of $\mathbb{V}\in\mathbb{V}$.
In $\Th{ES}$, the class
$\mathbb{T}=\square_{\mathbb{T}}\mathbb{V}=\lozenge_{\mathbb{T}}\mathbb{V}$ of
all nonempty sets satisfies all the axioms of a topology on $\mathbb{V}$,
except that it does not need to contain $\mathbb{V}$ itself. Although it is
not necessarily a class, we can therefore consider the collection of
$\mathbb{V}$ and all nonempty sets a topology on $\mathbb{V}$ and informally
attribute topological notions to it. We will call it the _universal topology_
and whenever no other topology is explicitly mentioned, we will refer to it.
Since no more than one element distinguishes the universal topology from
$\mathbb{T}$, any topological statement about it can easily be reformulated as
a statement about $\mathbb{T}$ and hence be expressed in our theory. Having
said this, we can interpret the third axiom as stating that the universe is a
$T_{1}$ space.
Alternatively one can understand the axioms without referring to collections
outside the theory’s scope as follows: Every set $a$ carries a topology
$\square a$, and a union of two sets is a set again. Then the $T_{1}$ axiom
says that all sets are $T_{1}$ spaces (and that all singletons are sets) and
the fourth says that every set’s hyperspace exists.
If $\mathbb{V}$ is not a set, we cannot interpret the exponential axiom as
saying that the universe’s hyperspace exists! Since $\square a=\square
a\cap\lozenge a$, it implies the power set axiom, but it does not imply the
sethood of $\lozenge a$ for every set $a$.
A very handy implication of the 2nd topology axiom and the exponential axiom
is that for all sets $b$, $c$ and every class $A$,
$\\{x{\in}c\mid A\subseteq x\subseteq
b\\}\quad=\quad\begin{cases}c\;\cap\;\bigcap_{y{\in}A}(\square
b\cap\lozenge\\{y\\})&\text{ if $A\neq\emptyset$.}\\\ (c\;\cap\;\square
b)\cup(c\cap\mathbb{A})&\text{ if $A=\emptyset$.}\end{cases}$
is closed, given that $c\cap\mathbb{A}$ is closed or $A$ is nonempty.
An important consequence of the $T_{1}$ axiom is that for each natural
number111Until we have defined them in essential set theory, we consider
natural numbers to be metamathematical objects. $n$, all classes with at most
$n$ elements are discrete sets. In particular, pairs are sets and we can
define ordered pairs as Kuratowski pairs $\langle
x,y\rangle=\\{\\{x\\},\\{x,y\\}\\}$. We adopt the convention that the
$n{+}1$-tuple $\langle x_{1},\ldots,x_{n+1}\rangle$ is $\langle\langle
x_{1},\ldots,x_{n}\rangle,x_{n+1}\rangle$ and that relations and functions are
classes of ordered pairs. With these definitions, all functional formulas
$\phi^{C}$ on sets correspond to actual functions, although these might be
proper classes. We denote the $\in$-relation for sets by
$\mathbf{E}=\\{\langle x,y\rangle\mid x{\in}y\\}$, and the equality relation
by $\Delta=\\{\langle x,y\rangle\mid x{=}y\\}$. Also, we write $\Delta_{A}$
for the equality $\Delta\cap A^{2}$ on a class $A$.
We have not yet made any stronger assumption than $T_{1}$ about the separation
properties of sets. However, many desirable set-theoretic properties,
particularly with respect to Cartesian products, apply only to Hausdorff sets,
that is, sets whose natural topology is $T_{2}$.
We denote by $\square_{<n}A$ the class of all $b\subseteq A$ with less than
$n$ elements. Given $t_{1},\ldots,t_{m}\in\\{1,\ldots,n\\}$. We define:
$F_{n,t_{1},\ldots,t_{m}}:\mathbb{V}^{n}\rightarrow\mathbb{V}^{m},F_{n,t_{1},\ldots,t_{m}}(x_{1},\ldots,x_{n})=\langle
x_{t_{1}},\ldots,x_{t_{m}}\rangle$
With the corresponding choice of $t_{1},\ldots,t_{m}$, all projections and
permutations can be expressed in this way.
For a set $a$, let $a^{\prime}$ be its _Cantor-Bendixson derivative_ , the set
of all its accumulation points, and let $a_{I}=a\setminus a^{\prime}$ be the
class of all its isolated points.
###### Proposition 1 ($\Th{ES}$).
Let $a$ and $b$ be Hausdorff sets and $a_{1},\ldots,a_{n}\subseteq a$.
1. 1.
$\square_{<n}a$ is a Hausdorff set.
2. 2.
The Cartesian product $a_{1}\times\ldots\times a_{n}$ is a Hausdorff set, too,
and its universal topology is at least as fine as the product topology.
3. 3.
Every continuous function $F:a_{1}\rightarrow a_{2}$ is a set.
4. 4.
For all $t_{1},\ldots,t_{m}\in\\{1,\ldots,n\\}$, the function
$F_{n,t_{1},\ldots,t_{m}}\upharpoonright a^{n}:a^{n}\rightarrow a^{m}$
is a Hausdorff set. It is even closed with respect to the product topology of
$a^{n+m}$.
5. 5.
For each $x\in a_{I}$, let $b_{x}\subseteq b$. Then for every map
$F:a_{I}\rightarrow b$, the class $F\cup(a^{\prime}{\times}b)$ is
$\mathbb{T}$-closed and we can define the product as follows:
$\prod_{x\in
a_{I}}b_{x}\quad=\quad\left\\{F\cup(a^{\prime}{\times}b)\;\mid\;F:a_{I}\rightarrow\mathbb{V},\;{\forall}x\;F(x)\in
b_{x}\right\\}$
It is $\mathbb{T}$-closed and its natural topology is at least as fine as its
product topology.
###### Proof.
(1): To show that it is a set it suffices to prove that it is a closed subset
of the set $\square a$, so assume $b\in\square a\setminus\square_{<n}a$. Then
there exist distinct $x_{1},\ldots,x_{n}\in b$, which by the Hausdorff axiom
can be separated by disjoint relatively open $U_{1},\ldots,U_{n}\subseteq a$.
Then $\lozenge U_{1}\cap\ldots\cap\lozenge U_{n}\cap\square a$ is a relatively
open neighborhood of $b$ disjoint from $\square_{<n}a$.
Now let $b,c\in\square_{<n}a$ be distinct sets. Wlog assume that there is a
point $x\in b\setminus c$. Since $c$ is finite and $a$ satisfies the Hausdorff
axiom, there is a relatively open superset $U$ of $c$ and a relatively open
$V\ni x$, such that $U\cap V=\emptyset$. Now $\lozenge V\cap\square_{<n}a$ is
a neighborhood of $b$ and $\square U\cap\square_{<n}a$ is a neighborhood of
$c$ in $\square_{<n}a$, and they are disjoint. Hence $\square_{<n}a$ is
Hausdorff.
(2): It suffices to prove that $a\times a$ is a set and carries at least the
product topology, because then it follows inductively that this is also true
for $a^{n}$ with $n\geq 2$. And from this in turn it follows that
$a_{1}\times\ldots\times a_{n}$ is closed in $a^{n}$ and carries the subset
topology, which implies the claim.
Since $a^{2}$ contains exactly the sets of the form $\\{\\{x\\},\\{x,y\\}\\}$
with $x,y\in a$, it is a subclass of the set $s=\square_{\leq 2}\square_{\leq
2}a\>\cap\>\lozenge\square_{\leq 1}a$ and we only have to prove that it is
closed in $s$. So let $c\in s\setminus a^{2}$. Then
$c=\\{\\{x\\},\\{y,z\\}\\}$ with $x\notin\\{y,z\\}$ and $x,y,z\in a$. Since
$a$ is Hausdorff, there are disjoint $U\ni x$ and $V\ni y,z$ which are
relatively open in $a$. Then $s\cap\lozenge\square_{\leq
1}U\cap\lozenge\square_{\leq 2}V$ is relatively open in $s$, and is a
neighborhood of $c$ disjoint from $a^{2}$.
It remains to prove the claim about the product topology, that is, that for
every subset $b\subseteq a$, $b\times a$ and $a\times b$ are closed, too. The
first one is easy, because $b\times a=a^{2}\cap\lozenge\square_{\leq 1}b$.
Similarly, $(b\times a)\cup(a\times b)=a^{2}\cap\lozenge\lozenge b$, so in
order to show that $a\times b$ is closed, let $c\in(b\times a)\cup(a\times
b)\setminus(a\times b)$, that is, $c=\\{\\{x\\},\\{x,y\\}\\}$ with $y\notin b$
and $x\in b$. Since $a$ is Hausdorff, there are relatively open disjoint
subsets $U\ni x$ and $V\ni y$ of $a$. Then $s\cap\lozenge\square_{\leq
1}U\cap\lozenge\lozenge(V\setminus b)$ is a relatively open neighborhood of
$c$ disjoint from $a\times b$.
(3): Let $F:a_{1}\rightarrow a_{2}$ be continuous and $\langle x,y\rangle\in
a_{1}{\times}a_{2}\setminus F$, that is, $F(x)\neq y$. Then $F(x)$ and $y$ can
be separated by relatively open subsets $U\ni F(x)$ and $V\ni y$ of $a_{2}$,
and since $F$ is continuous, $F^{-1}[U]$ is relatively open in $a_{1}$.
$F^{-1}[U]\times V$ is a neighborhood of $\langle x,y\rangle$ and disjoint
from $F$. This concludes the proof that $F$ is relatively closed in
$a_{1}{\times}a_{2}$ and hence a set.
(4): Let $F=F_{n,t_{1},\ldots,t_{m}}$. Then $F\subseteq a^{n}\times
a^{m}\in\mathbb{V}$, so we only have to find for every
$b=\langle\langle x_{1},\ldots,x_{n}\rangle,\langle
y_{1},\ldots,y_{m}\rangle\rangle\text{, such that $x_{t_{k}}\neq y_{k}$ for
some $k$,}$
a neighborhood disjoint from $F$. By the Hausdorff property, there are
disjoint relatively open $U\ni x_{t_{k}}$ and $V\ni y_{k}$. Then
$\left(a^{t_{k}-1}\times U\times a^{n-t_{k}}\right)\times\left(a^{k-1}\times
V\times a^{m-k}\right)$
is such a neighborhood.
(5): Firstly,
$F\cup(a^{\prime}{\times}b)\quad=\quad\bigcap_{x\in a_{I}}\left(\\{\langle
x,F(x)\rangle\\}\;\cup\;((a\setminus\\{x\\})\times b)\right)$
is a set for any such function $F$.
Secondly, the claim about the product topology follows as soon as we have
demonstrated the product to be $\mathbb{T}$-closed, because the product
topology is generated by classes of the form $\prod_{x\in a_{I}}c_{x}$, where
$c_{x}\subseteq b_{x}$ and only finitely many $c_{x}$ differ from $b_{x}$.
Since $a\times b$ is $\mathbb{T}$-closed and the product $P=\prod_{x\in
a_{I}}b_{x}$ is a subset of $\square(a\times b)$, it suffices to show that $P$
is relatively closed in $\square(a\times b)$, so let $r\in\square(a\times
b)\setminus P$. There are four cases:
* •
The domain of $r$ is not $a$. Then there is an $x\in a$ such that
$x\notin\mathrm{dom}(r)$. In that case, $\square(a\times
b)\cap\lozenge(\\{x\\}\times b)$ is a closed superset of $P$ omitting $r$.
* •
$a^{\prime}\times b\nsubseteq r$. Then some $\langle x,y\rangle\in
a^{\prime}\times b$ is missing and $\square(a\times b)\cap\lozenge\\{\langle
x,y\rangle\\}$ is a corresponding superset of $P$.
* •
$r\upharpoonright a_{I}$ is not a function. Then there is an $x\in a_{I}$,
such that there exist distinct $\langle x,y_{0}\rangle,\langle
x,y_{1}\rangle\in r$. Since $b$ is Hausdorff, there are closed
$u_{0},u_{1}\subseteq b$, such that $u_{0}\cup u_{1}=b$, $y_{0}\notin u_{0}$
and $y_{1}\notin u_{1}$. Then $P$ is a subclass of
$\square\left(\\{x\\}\times u_{0}\;\cup\;(a{\setminus}\\{x\\})\times
b\right)\quad\cup\quad\square\left(\\{x\\}\times
u_{1}\;\cup\;(a{\setminus}\\{x\\})\times b\right)\text{,}$
which does not contain $r$.
* •
$F=r\upharpoonright a_{I}$ is a function, but $F(x)\notin b_{x}$ for some
$x\in a_{I}$. Then
$\square\left(\\{x\\}\times b_{x}\;\cup\;(a{\setminus}\\{x\\})\times b\right)$
is a closed superclass of $P$ omitting $r$.
Thus for every $r\in\square(a\times b)\setminus P$, there is a closed
superclass of $P$ which does not contain $r$. Therefore $P$ is closed. ∎
The additivity axiom states that the universe is $\mathcal{D}$-additive, that
is, that the union of a discrete set’s image is $\mathbb{T}$-closed. In other
words: For every function $F$ whose domain is a discrete set, the union of the
range $\bigcup\mathrm{rng}(F)$ is a set or empty. Had we opted against proper
classes, the additivity axiom therefore could have been expressed as an axiom
scheme.
Even without a choice principle, we could equivalently have used injective
functions into discrete sets instead of surjective functions defined on
discrete sets: Point (2) in the following proposition is exactly the
additivity axiom.
###### Proposition 2.
In $\Th{ES}$ without the additivity axiom, the following are equivalent:
1. 1.
Images of discrete sets are sets, and unions of discrete sets are
$\mathbb{T}$-closed.
2. 2.
If $d$ is discrete and $F:d\rightarrow A$ surjective, then $\bigcup A$ is
$\mathbb{T}$-closed.
3. 3.
If $d$ is discrete and $F:A\hookrightarrow d$ injective, then $\bigcup A$ is
$\mathbb{T}$-closed.
###### Proof.
(1) $\Rightarrow$ (2): If images of discrete sets are sets, then they are
discrete, too, because all their subsets are images of subsets of a discrete
set. Thus $F[d]$ is discrete, and therefore its union $\bigcup F[d]$ is
closed.
(2) $\Rightarrow$ (1): If $d$ is discrete and $F$ is a function, consider the
function $G:\mathrm{dom}(F)\rightarrow\mathbb{V}$ defined by
$G(x)=\\{F(x)\\}$. Then $F[d]=\bigcup G[d]\in\mathbb{V}$. Applying (2) to the
identity proves that $\bigcup d=\bigcup\mathrm{id}[d]$ is closed.
(2) $\Rightarrow$ (3): If $F:A\hookrightarrow d$ is an injection, then
$F^{-1}:F[A]\rightarrow A$ is a surjection from the discrete subset
$F[A]\subseteq d$ onto $A$, so $\bigcup A$ is closed.
(3) $\Rightarrow$ (2): First we show that $\square d$ is discrete. We have to
show that any given $a\in\square d$ is not an accumulation point, i.e. that
$\square d\setminus\\{a\\}$ is closed. Since $a$ is a discrete set, every
$d\setminus\\{b\\}$ for $b\in a$ is closed, as well as $d\setminus a$. But
$\square d\setminus\\{a\\}=\square d\cap\left(\lozenge(d\setminus
a)\cup\bigcup_{b\in a}\square(d\setminus\\{b\\})\right)$
and this union can be seen to be closed by applying (3) to the map
$F:\\{\square(d\setminus\\{b\\})\mid b{\in}a\\}\hookrightarrow a,\quad
F(\square(d\setminus\\{b\\}))=b\text{.}$
Now we can prove (2):
Let $G:d\rightarrow\mathbb{V}$. Then $F:G[d]\rightarrow\square
d,F(x)=G^{-1}[\\{x\\}]$ is an injective function from $G[d]$ to the discrete
set $\square d$. Therefore, $\bigcup G[d]\in\mathbb{V}$. ∎
###### Proposition 3 ($\Th{ES}$).
$\square d$ is discrete for every discrete set $d$. Every $\mathcal{D}$-small
nonempty class is a discrete set and every nonempty union of $\mathcal{D}$-few
discrete sets is a discrete set.
###### Proof.
The first claim has already been shown in the proof of Proposition 2.
Let $A$ be $\mathcal{D}$-small and $B\subseteq A$. Then $B$ and
${\widetilde{B}}=\\{\\{b\\}\mid b{\in}B\\}$ are also $\mathcal{D}$-small.
Therefore $\bigcup{\widetilde{B}}=B$ is $\mathbb{T}$-closed by the additivity
axiom.
Finally, let $A$ be $\mathcal{D}$-small and let every $a\in A$ be a discrete
set. We have to show that every nonempty $B\subseteq\bigcup A$ is a set. But
if $A$ is $\mathcal{D}$-small, the class $C$ of all nonempty sets of the form
$B\cap a$ with $a\in A$ also is. Since $B\neq\emptyset$ and every $a\in A$ is
discrete, the union of $C$ is in fact $B$. ∎
## 2 Ordinal Numbers
We do not assume that the empty class is a set, so there may be no well-
founded sets at all, yet of course we want to define the natural numbers and
later we will even be looking for an interpretation of a well-founded theory.
To this end we need suitable variants of the concepts of well-foundedness and
von Neumann ordinal numbers.
Our starting point is finding a substitute for the empty set: A class or atom
$0$ is called a _zero_ if no element of $0$ is a superset of $0$. Zeros exist
in $\mathbb{V}$: By the nontriviality axiom, there are distinct
$x,y\in\mathbb{V}$, so we can set $0=\\{\\{x\\},\\{y\\}\\}$. But in many
interesting cases, there even is a definable zero: Let us set $0=\emptyset$ if
$\emptyset\in\mathbb{V}$, and if $\emptyset\notin\mathbb{V}$ but
$\mathbb{V}\in\mathbb{V}$, we set $0=\\{\\{\mathbb{V}\\}\\}$ (its element
$\\{\mathbb{V}\\}$ is not a superset of $0$, because by the nontriviality
axiom $\mathbb{V}$ is not a singleton). Note that all these examples are sets
with at most two elements.
Given a fixed zero $0$, we make the following definitions:
$\displaystyle A^{\oplus}$ $\displaystyle=$ $\displaystyle A\setminus 0$
$\displaystyle A\in_{0}B$ if $\displaystyle A\in B^{\oplus}\quad\text{ and
}\quad 0\subseteq B\text{.}$ $\displaystyle A\text{ is $0$-\emph{transitive}}$
if $\displaystyle c\in_{0}A\quad\text{ for all }\quad
c\in_{0}b\in_{0}A\text{.}$ $\displaystyle\text{A $0$-transitive }a\text{ is
$0$-\emph{pristine}}$ if $\displaystyle 0\subseteq
c\notin\mathbb{A}\quad\text{ for all }\quad c\in_{0}a\cup\\{a\\}\text{.}$
$\displaystyle\alpha\text{ is a $0$-\emph{ordinal number}}$ if
$\displaystyle\alpha\text{ is $0$-transitive, $0$-pristine and}$
$\displaystyle\alpha^{\oplus}\text{ is strictly well-ordered by $\in_{0}$,}$
where by a (strict) _well-order_ we mean a (strict) linear order such that
each nonempty sub _set_ has a minimal element. A (strict) order with the
property that every sub _class_ has a minimal element is called a (strict)
_strong well-order_ , and we will see shortly that in fact such
$\alpha^{\oplus}$ are strictly strongly well-ordered.
We denote the class of $0$-ordinals by $\mathit{On}_{0}$ and the $0$-ordinals
themselves by lowercase greek letters. If $\alpha$ and $\beta$ are
$0$-ordinals, we also write $\alpha\leq_{0}\beta$ for $\alpha\subseteq\beta$.
A $0$-ordinal $\alpha\neq 0$ is a $0$-_limit ordinal_ if it is not the
immediate $\leq_{0}$-successor of another $0$-ordinal, and it is a
$0$-_cardinal number_ if there is no surjective map from $\beta^{\oplus}$ onto
$\alpha^{\oplus}$ for any $\beta<_{0}\alpha$. If there is a least $0$-limit
ordinal distinct from $0$ itself, we call it $\omega_{0}$, otherwise we define
$\omega_{0}=\mathit{On}_{0}$. Its predecessors $n\in_{0}\omega_{0}$ are the
$0$-_natural numbers_. Obviously $0$ is the least $0$-ordinal, if
$0\in\mathbb{V}$.
For the remainder of this section, let us assume that our $0$ is an atom or a
finite set. Unless there is danger of confusion (as in the case of $\in_{0}$),
we omit the prefix and index $0$.
###### Proposition 4 ($\Th{ES}$).
Let $\alpha\in\mathit{On}$.
1. 1.
$\alpha\notin\alpha$, $\alpha$ is discrete and
$\alpha=0\;\cup\;\\{\beta\in\mathit{On}\mid\beta\in_{0}\alpha\\}$.
2. 2.
$\mathit{On}$ is strictly strongly well-ordered by $\in_{0}$ and $<$, and
these orders coincide.
3. 3.
$\alpha\cup\\{\alpha\\}$ is the unique immediate successor of $\alpha$.
4. 4.
If $A$ is a nonempty class of ordinals and $\bigcup A\in\mathbb{V}$, then
$\bigcup A$ is an ordinal and the least upper bound of $A$.
5. 5.
$\bigcup\mathit{On}=\mathit{On}\cup 0\notin\mathbb{V}$
###### Proof.
(1): Since $0\subseteq a$, the equality follows if we can prove that every
$x\in_{0}\alpha$ is an ordinal. Firstly, let $c\in_{0}b\in_{0}x$. Then
$b\in_{0}\alpha$ and $c\in_{0}\alpha$ by transitiviy of $\alpha$. Since
$\alpha^{\oplus}$ is strictly linearly ordered by $\in_{0}$, it follows that
$c\in_{0}x$, proving that $x$ is transitive. Again by the transitivity of
$\alpha$, we see that $x\subseteq\alpha$, and as a subset of a well-ordered
set, $x^{\oplus}$ is well-ordered itself. Also, every $c\in_{0}x\cup\\{x\\}$
is an element of $\alpha^{\oplus}$ and therefore a superset of $0$ not in
$\mathbb{A}$, so $x$ is pristine.
Since $\alpha$ is a superset of $0$, $\alpha\notin 0$. Thus if $\alpha$ were
an element of $\alpha$, it would be in $\alpha^{\oplus}$. But
$\alpha\in_{0}\alpha$ contradicts the condition that the elements of
$\alpha^{\oplus}$ are strictly well-ordered.
Because $0$ is a discrete set and $\alpha^{\oplus}=\\{x\in\alpha\mid
0\subseteq x\subseteq\alpha\\}$ is closed, it suffices to show that
$\alpha^{\oplus}$ is discrete. So let $\gamma\in_{0}\alpha$. Since the
elements of $\alpha^{\oplus}$ are strictly linearly ordered, every
$\delta\in\alpha^{\oplus}\setminus\\{\gamma\\}$ is either a predecessor or a
successor of $\gamma$. Hence
$\alpha^{\oplus}\setminus\\{\gamma\\}\quad=\quad\gamma^{\oplus}\;\cup\;\\{x\in\alpha^{\oplus}\mid\\{\gamma\\}\subseteq
x\subseteq\alpha\\}$
is closed.
(2): If $\alpha\in_{0}\beta$, then by transitivity of $\beta$, $\alpha$ is a
subset of $\beta$ and because $\alpha\notin_{0}\alpha$, it is a proper one.
For the converse assume $\alpha<\beta$, that is, $\alpha\subset\beta$.
$\beta^{\oplus}$ is discrete and well-ordered, so the nonempty subset
$\beta\setminus\alpha$ contains a minimal element $\delta$, which by (1) is an
ordinal number. For all $\gamma\in_{0}\delta$, it follows from the minimality
of $\delta$ that $\gamma\in_{0}\alpha$. Now let $\gamma\in_{0}\alpha$. Then
$\gamma\in_{0}\beta$ and since $\beta$ is linearly ordered, $\gamma$ is
comparable with $\delta$. But if $\delta\in_{0}\gamma$, then
$\delta\in_{0}\alpha$ by transitivity, which is false. Hence
$\gamma\in_{0}\delta$. We have shown that $\delta$ and $\alpha$ have the same
predecessors, so by (1), they are equal. Thus $\alpha=\delta\in_{0}\beta$ and
so the orders $\in_{0}$ and $<$ coincide on the ordinals.
Next we show that ordinals $\alpha,\beta\in\mathit{On}$ are always subsets of
each other and hence $\mathit{On}$ is linearly ordered, so assume they are
not. Let $\alpha_{0}$ be minimal in $\alpha\setminus\beta$ and $\beta_{0}$ in
$\beta\setminus\alpha$. Now all predecessors of $\alpha_{0}$ must be in
$\alpha\cap\beta$. And since $\alpha$ and $\beta$ are transitive,
$\alpha\cap\beta$ is an initial segment and therefore every element of
$\alpha\cap\beta$ is also in $\alpha_{0}$. The same argument applied to
$\beta_{0}$ shows that $\alpha_{0}=\alpha\cap\beta=\beta_{0}$, contradicting
our assumption.
Finally, given a nonempty subclass $A\subseteq\mathit{On}$, let $\alpha\in A$
be arbitrary. Then either $\alpha$ has no predecessor in $A$ and thus is
minimal itself, or $\alpha\cap A$ is nonempty and has a minimal element
$\delta$, because $\alpha^{\oplus}$ is well-ordered and discrete and
$\alpha\cap A\subseteq\alpha^{\oplus}$. For every $\gamma\in
A\setminus\alpha$, we then have $\delta<\alpha\leq\gamma$. Hence $\delta$ is
in fact minimal in $A$, concluding the proof that $\mathit{On}$ is strongly
well-ordered.
(3): First we verify that $\beta=\alpha\cup\\{\alpha\\}$ is an ordinal. Since
$\alpha$ is transitive, $\beta$ also is. Since $\alpha$ is pristine and
$0\subseteq\beta\notin\mathbb{A}$, $\beta$ is pristine itself. And
$\beta^{\oplus}$ is a set of ordinal numbers, which by (2) must be well-
ordered.
From $\alpha\notin\alpha$ it follows that in fact $\beta\neq\alpha$ and thus
$\beta>\alpha$. If $\gamma<\beta$, then $\gamma\in_{0}\beta$, so either
$\gamma\in_{0}\alpha$ or $\gamma=\alpha$, which shows that $\beta$ is an
immediate successor. Since the ordinals are linearly ordered, it is the only
one.
(4): As a union of transitive, pristine, well-founded sets, $\bigcup A$ is
transitive, pristine and well-founded itself. Since all its predecessors are
ordinals, they are strictly well-ordered by (2), so it is an ordinal itself.
For each $\beta\in A$, $\beta\subseteq\bigcup A$ and thus $\beta\leq\bigcup
A$, so it is an upper bound of $A$. If $\beta<\bigcup A$, there is an element
$\gamma\in A$ with $\beta<\gamma$, therefore it is the least upper bound.
(5): By (1), every element $x$ of an ordinal is in $0\cup\mathit{On}$.
Conversely, $0$ is an ordinal and by (3), every ordinal is an element of its
successor. Therefore, $0\cup\mathit{On}=\bigcup\mathit{On}$. If
$\bigcup\mathit{On}$ were a set, so would
$\bigcup\mathit{On}\cup\\{\bigcup\mathit{On}\\}$ be. But by (4), that would be
an ordinal strictly greater than all elements of $\mathit{On}$, which is a
contradiction. ∎
These features of $\mathit{On}$ are all quite desirable, and familiar from
Zermelo-Fraenkel set theory. Just as in $\Th{ZF}$, $\mathit{On}$ (or rather
$\mathit{On}\cup 0$) resembles an ordinal number itself, except that it is not
a set. But in $\Th{ZF}$, $\mathit{On}$ even has the properties of a regular
limit cardinal – a consequence of the replacement axiom. Also, our dependence
on the choice of a specific set $0$ is rather irritating. This is where the
additivity axiom comes in. In the context of ordinal numbers (and discrete
sets in general), it is the appropriate analog to the replacement axiom.
By the usual argument, all strongly well-ordered classes whose initial
segments are discrete sets are comparable with respect to their length: There
is always a unique isomorphism from one of them to an initial segment of the
other. In particular, for all finite zeros $0,{\widetilde{0}}\in\mathbb{V}$,
the well-orders of $\mathit{On}_{0}$ and $\mathit{On}_{{\widetilde{0}}}$ are
comparable. But if $A\subseteq\mathit{On}_{0}$ is an initial segment
isomorphic to $\mathit{On}_{{\widetilde{0}}}$, then in fact
$A=\mathit{On}_{0}$, because otherwise $A$ would be a discrete set and by the
additivity axiom, $\mathit{On}_{{\widetilde{0}}}\in\mathbb{V}$, a
contradiction. Hence $\mathit{On}_{0}$ and $\mathit{On}_{{\widetilde{0}}}$ are
in fact isomorphic and the choice of $0$ is not relevant to our theory of
ordinal numbers. Also, $\omega_{0}$ and $\omega_{{\widetilde{0}}}$ are equally
long and we can define a class $A$ to be _finite_ if there is a bijection from
$n^{\oplus}$ to $A$ for some natural number $n$. Otherwise it is _infinite_.
It is easy to prove that this definition is equivalent to $A$ being the image
of some $n^{\oplus}$ or embeddable into some $n^{\oplus}$. Also, it can be
stated without quantifying over classes, because such a bijection is defined
on a discrete set and therefore a discrete set itself.
Even if there is no limit ordinal, there might still be infinite sets – they
just cannot be discrete. So the proper axiom of infinity in the context of
essential set theory is the existence of a limit ordinal number:
Infinity $\displaystyle\omega\in\mathbb{V}$
We add the axiom of infinity to a theory by indexing it with the symbol
$\infty$.
Using induction on ordinal numbers, one easily proves that for each
$\alpha\in\mathit{On}$, the least ordinal $\kappa\in\mathit{On}$ such that
there is a surjection from $\kappa^{\oplus}$ to $\alpha^{\oplus}$ is a
cardinal, and there is a bijection from $\kappa^{\oplus}$ to
$\alpha^{\oplus}$.
###### Proposition 5 ($\Th{ES}$).
$\mathit{On}$ is a regular limit, that is:
1. 1.
Every function $F:\alpha^{\oplus}\rightarrow\mathit{On}$ is bounded.
2. 2.
The class of cardinal numbers is unbounded in $\mathit{On}$.
###### Proof.
(1): By the additivity axiom, $\bigcup F[\alpha^{\oplus}]$ is a discrete set,
so by Proposition 4, it is an ordinal number and an upper bound of
$F[\alpha^{\oplus}]$.
(2): Let us show that for each $\alpha$ there exists a cardinal $\nu>\alpha$.
This goes by the usual argument: Every well-order
$R\subseteq\alpha^{\oplus}\times\alpha^{\oplus}$ on a subset of
$\alpha^{\oplus}$ is a subclass of the discrete set
$\square\square\alpha^{\oplus}$, so it is a set itself and since
$\alpha^{\oplus}$ is discrete, it is even a strong well-order. Recursively,
isomorphisms from initial segments of $\alpha^{\oplus}$ with respect to $R$ to
initial segments of $\mathit{On}$ can be defined, and their union is a
function from $\alpha^{\oplus}$ onto some $\beta^{\oplus}$. We call $\beta$
the _order type_ of $R$. Now the class $A$ of all well-orders of $\alpha$ is a
subclass of $\square\square\square\alpha$ and hence also a discrete set.
Mapping every element of $A$ to its order type must therefore define a bounded
map $F:A\rightarrow\mathit{On}$. Let $\nu=\min(\mathit{On}\setminus\bigcup
F[A])$ be the least ordinal which is not an order type of any subset of
$\alpha^{\oplus}$. We show that $\nu$ is a cardinal above $\alpha$. Firstly,
$\alpha$ is the order type of a well-order of $\alpha^{\oplus}$, so
$\nu>\alpha$. Secondly, assume that $g:\gamma^{\oplus}\rightarrow\nu^{\oplus}$
is surjective and $\gamma<\nu$. Then this defines a well-order on
$\gamma^{\oplus}$ of order-type at least $\nu$, and since $\gamma$ is the
order type of a well-order on some subset of $\alpha^{\oplus}$ by definition,
$g$ would define a well-order on a subset of $\alpha^{\oplus}$ of order-type
$\nu$, a contradiction. ∎
If $\mathbb{V}\notin\mathbb{V}$, the closure of $\mathit{On}$ may well be all
of $\mathbb{V}$ and in particular does not have to be a set. But in the case
$\mathbb{V}\in\mathbb{V}$, the fact that all $\lozenge a$ are sets determines
the closure $\Omega$ of $0\cup\mathit{On}=\bigcup\mathit{On}$ much more
precisely. Moreover, $\mathit{On}$ then resembles a weakly compact cardinal,
which will in fact turn out to be crucial for the consistency strength of the
axiom $\mathbb{V}\in\mathbb{V}$.
###### Proposition 6 ($\Th{TS}$).
1. 1.
Every sequence $\langle x_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$ of
length $\mathit{On}$ has an accumulation point.
2. 2.
Every monotonously $\subseteq$-decreasing sequence $\langle
x_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$ of nonempty sets converges to
$\bigcap_{\alpha\in\mathit{On}}x_{\alpha}$. And every monotonously
$\subseteq$-increasing one to
$\mathrm{cl}\left(\bigcup_{\alpha\in\mathit{On}}x_{\alpha}\right)$.
3. 3.
$\Omega=0\cup\mathit{On}\cup\\{\Omega\\}$
4. 4.
$P=\\{x\mid 0\cup\\{\Omega\\}\subseteq x\subseteq\Omega\\}$ is a _perfect set_
, that is, $P^{\prime}=P\neq\emptyset$.
5. 5.
$\mathit{On}$ has the tree property, that is: If
$T\quad\subseteq\quad\\{f:\alpha^{\oplus}\rightarrow\mathbb{V}\mid\alpha\in\mathit{On}\\}$
such that $T_{\alpha}=\\{f{\upharpoonright}\alpha^{\oplus}\mid
f{\in}T,\>\alpha^{\oplus}{\subseteq}\mathrm{dom}(f)\\}$ is discrete and
nonempty for each ordinal $\alpha>0$, then there is a
$G:\mathit{On}\rightarrow\mathbb{V}$ such that:
$G\upharpoonright\alpha^{\oplus}\in T_{\alpha}\quad\text{ for every
}\quad\alpha\in\mathit{On}\text{.}$
###### Proof.
(1): Assume that there is no accumulation point. Then every point
$y\in\mathbb{V}$ has a neighborhood $U$ such that $\\{\alpha\mid
x_{\alpha}{\in}U\\}$ is bounded in $\mathit{On}$ and therefore discrete. Since
the class $\\{x_{\alpha}\mid x_{\alpha}{\in}U\\}$ of members in $U$ is the
image of $\\{\alpha\mid x_{\alpha}{\in}U\\}$, it is also a discrete set and
does not have $y$ as its accumulation point. It follows that firstly,
$\\{\alpha\mid x_{\alpha}{=}y\\}$ is discrete for each $y$, and secondly, the
image $\\{x_{\alpha}\mid\alpha{\in}\mathit{On}\\}$ of the sequence is also
discrete. But $\mathit{On}$ is the union of the sets $\\{\alpha\mid
x_{\alpha}{=}y\\}$ for $y\in\\{x_{\alpha}\mid\alpha{\in}\mathit{On}\\}$. Since
$\mathcal{D}$-small unions of discrete sets are discrete sets, this would
imply that $\mathit{On}$ is a discrete set, a contradiction.
(2): First let the sequence be decreasing. Then for every
$y\in\bigcap_{\alpha}x_{\alpha}$, every member of the sequence lies in the
closed set $\lozenge\\{y\\}$, so all its accumulation points do. Now let
$y\notin\bigcap_{\alpha}x_{\alpha}$. Then there is a $\beta\in\mathit{On}$
such that $y\notin x_{\beta}$, and hence from $x_{\beta}$ on, all members are
in $\square x_{\beta}$, so all accumulation points are. Thus the only
accumulation point is the intersection. (Note that the intersection therefore
is nonempty because $\lozenge\mathbb{V}$ is a closed set containing every
member of the sequence.)
Now assume that the sequence is ascending and let $A$ be its union. If $y\in
A$, then $y\in x_{\beta}$ for some $\beta\in\mathit{On}$. Then all members
from $x_{\beta}$ on are in $\lozenge\\{y\\}$, so each accumulation point also
is. Thus all accumulation points are supersets of $A$. But all members of the
sequence are in $\square\mathrm{cl}(A)$, so each accumulation point is a
subset of $\mathrm{cl}(A)$, and therefore equal to $\mathrm{cl}(A)$.
(3): It suffices to prove that $\Omega$ is the unique accumulation point of
$\mathit{On}$. Since $\mathit{On}$ is the image of an increasing sequence, its
accumulation point is indeed unique and is the closure of $\bigcup\mathit{On}$
by (2). But
$\mathrm{cl}\left(\bigcup\mathit{On}\right)=\mathrm{cl}(0\cup\mathit{On})=\Omega$.
(4): $P$ is closed, and it is nonempty because $\Omega\in P$. Given $x\in P$,
the sequences in $P$ given by
$y_{\alpha}\;=\;x\setminus(\mathit{On}\setminus\alpha^{\oplus})\quad\text{ and
}\quad z_{\alpha}\;=\;x\cup(\mathit{On}\setminus\alpha^{\oplus})$
both converge to $x$ by (2). If $x\cap\mathit{On}$ is unbounded, $x$ is not
among the $y_{\alpha}$, otherwise it is not among the $z_{\alpha}$, so in any
case, $x$ is the limit of a nontrivial sequence in $P$.
(5): Since for every $\alpha\in\mathit{On}$, $T_{\alpha}$ is nonempty, there
is for every $\alpha$ an $f\in T$ with
$\alpha^{\oplus}\subseteq\mathrm{dom}(f)$. Thus the map
$T\rightarrow\mathit{On},\quad f\mapsto 0\cup\mathrm{dom}(f)$
is unbounded in $\mathit{On}$ and therefore has a nondiscrete image. Hence $T$
is not discrete and has an accumulation point $g\in\mathbb{V}$. We set
$G=g\cap(\mathit{On}\times\mathbb{V})$.
For each $\alpha\in\mathit{On}$, the union $\bigcup_{\beta<\alpha}T_{\beta}$
is a discrete set, so $g$ is an accumulation point of the difference
$T\setminus\bigcup_{\beta<\alpha}T_{\beta}$, which is the class of all those
$f\in T$ whose domain is at least $\alpha^{\oplus}$. But every such $f$ is by
definition the extension of some $h\in T_{\alpha}$. Thus this difference is
the union of the classes $S_{h}=\\{f\in T\mid h\subseteq f\\}$ with $h\in
T_{\alpha}$. Since $T_{\alpha}$ is discrete, $\mathrm{cl}\left(\bigcup_{h\in
T_{\alpha}}S_{h}\right)=\bigcup_{h\in T_{\alpha}}\mathrm{cl}(S_{h})$, so $g$
must be in the closure of some $S_{h}$. But $S_{h}$ is a subclass of the
closed
$\left\\{x\mid h\subseteq x\subseteq
h\cup\left(\Omega{\setminus}\alpha\times\mathbb{V}\right)\right\\}\text{,}$
so $h\subseteq g\subseteq
h\cup\left(\Omega{\setminus}\alpha\times\mathbb{V}\right)$, too.
We have shown that for every $\alpha\in\mathit{On}$, the set
$g\cap\left(\alpha^{\oplus}\times\mathbb{V}\right)$ is an element of
$T_{\alpha}$. This implies that $G$ is a function defined on $\mathit{On}$,
and that $G\upharpoonright\alpha^{\oplus}=f\upharpoonright\alpha^{\oplus}$ for
some $f\in T$, concluding the proof. ∎
In fact, we have just shown that _every_ accumulation point $g$ of $T$ gives
rise to such a solution $G$. Hence firstly,
$T=\mathrm{cl}(T)\cap\square(\mathit{On}\times\mathbb{V})$, and secondly, $G$
can always be described as the intersection of a set $g$ with
$\mathit{On}\times\mathbb{V}$. In our formulation of the tree property, the
two quantifications over classes could thus be replaced by quantifications
over sets.
If $\Omega$ exists, the hierarchy of well-ordered sets extends well beyond the
realm of ordinal numbers. By _linearly ordered set_ we shall mean from now on
a set together with a linear order $\leq$ such that the set’s natural topology
is at least as fine as the order topology, that is, such that all
$\leq$-closed intervals are $\mathbb{T}$-closed. And by _well-ordered set_ we
mean a linearly ordered set whose order is a well-order (or a strong well-
order – which in this case is equivalent). Then all well-ordered sets are
comparable.
The significance of (4) is that even if $\mathbb{V}\in\mathbb{V}$, the
universe cannot be a well-ordered set, because well-ordered sets have no
perfect subset. Thus whenever $a$ is a well-ordered set, there is a $p\notin
a$ and the set $a\cup\\{p\\}$ can be well-ordered such that its order-type is
the successor of the order-type of $a$. We use the usual notation for
intervals in the context of linearly-ordered sets, and consider $\infty$
(resp. $-\infty$) as greater (resp. smaller) than all the elements of the set.
We will also sloppily write $a+b$ and $a\cdot b$ for order-theoretic sums and
products and say that an order-type _exists_ if there is a linearly ordered
set with that order-type.
Every linearly ordered set $a$ is a Hausdorff set and since its order is
closed with respect to the product topology, it is itself a set by Proposition
1. Moreover, the class
$\bigcap_{b{\subseteq}a\text{ initial segment}}\square b\;\cup\;\\{c\mid
b\subseteq c\subseteq a\\}$
of all its $\mathbb{T}$-closed initial segments is itself a linearly ordered
set in which $a$ can be embedded via $x\mapsto(-\infty,x]$. Thus we can limit
our investigations to well-ordered sets whose order is given by $\subseteq$
and whose union exists, which makes things considerably easier:
###### Lemma 7 ($\Th{ES}$).
If a class $A\subseteq\square a$ is linearly ordered by $\subseteq$, then
$\mathrm{cl}(A)$ is a linearly ordered set ordered by $\subseteq$. If $A$ is
well-ordered, then so is $\mathrm{cl}(A)$.
###### Proof.
First we prove that $\mathrm{cl}(A)$ is still linearly ordered. Let
$x,y\in\mathrm{cl}(A)$ and assume that $x\nsubseteq y$. Every $z\in A$ is
comparable to every other element of $A$, so $A$ is a subclass of the set
$\square z\cup\\{v\mid z\subseteq v\subseteq a\\}$ and thus $\mathrm{cl}(A)$
also is. Therefore both $x$ and $y$ are comparable to every element of $A$ and
$A$ is a subclass of $\square x\cup\\{v\mid x\subseteq v\subseteq a\\}$. Since
$y$ is not a superset of $x$, it must be in the closure of $A\cap\square x$
and thus a subset of $x$.
Since $\\{v{\in}\mathrm{cl}(A)\mid x\subseteq v\subseteq y\\}=[x,y]$ is
closed, $\mathrm{cl}(A)$ in fact carries at least the order topology.
Now assume that $A$ is well-ordered and let $B\subseteq\mathrm{cl}(A)$ be
nonempty. Wlog let $B$ be a final segment. If $B$ has only one element, then
that element is minimal, so assume it has at least two distinct elements.
Since $A$ is dense, it must then intersect $B$ and $A\cap B$ must have a
minimal element $x$. Assume that $x$ is not minimal in $B$. Then there is a
$y\subset x$ in $B\setminus A$, and this $y$ must be minimal, because if there
were a $z\subset y$ in $B$, then $(z,x)$ would be a nonempty open interval in
$\mathrm{cl}(A)$ disjoint from $A$. ∎
Thanks to this lemma, to prove that well-ordered sets of a certain length
exist, it suffices to give a corresponding subclass of some $\square a$ well-
ordered by $\subseteq$. As the next theorem shows, this enables us to do a
great deal of well-order arithmetic in essential set theory.
###### Proposition 8 ($\Th{ES}$).
If $a$ and $b$ are Hausdorff sets and $a_{x}\subseteq a$ is a well-ordered set
for every $x\in b_{I}$, then $\sup_{x\in b_{I}}a_{x}$ exists. If in addition,
$R$ is a well-order on $b_{I}$ (not necessarily a set), then $\sum_{x\in
b_{I}}a_{x}$ exists. In particular, the order-type of $R$ exists, and binary
sums and products of well-orders exist.
###### Proof.
Consider families $\langle r_{x}\>|\>x{\in}b_{I}\rangle$ of initial segments
$r_{x}\subseteq a_{x}$ with the following property: for all $x,y\in b_{I}$
such that $r_{x}\neq a_{x}$, the length of $r_{y}$ is the maximum of $r_{x}$
and $a_{y}$. Given such a family, the class
$b^{\prime}\times a\quad\cup\quad\bigcup_{x\in b_{I}}\\{x\\}\times r_{x}$
is a set. And the class of all such sets is a subclass of $\square(b\times a)$
well-ordered by $\subseteq$ and at least as long as every $a_{x}$, because
assigning to $y\in a_{x}$ the set
$b^{\prime}\times a\quad\cup\quad\bigcup_{z\in b_{I}}\\{z\\}\times
r_{z}\text{,}$
is an order-preserving map, where $r_{z}=a_{z}$ whenever $a_{z}$ is at most as
long as $a_{x}$, and $r_{z}=(-\infty,\tilde{y}]$ such that $r_{z}$ is oder-
isomorphic to $(-\infty,y]$ otherwise.
In the well-ordered case, consider for every $\langle x,y\rangle\in
b_{I}\times a$ with $y\in a_{x}$ the set
$b^{\prime}\times
a\quad\cup\quad\\{x\\}\times(-\infty,a_{x}]\quad\cup\quad(-\infty,x)_{R}\times
a\text{.}$
The class of these sets is again a subclass of $\square(b\times a)$ and well-
ordered by $\subseteq$. Its order-type is the sum of the orders $a_{x}$.
Setting $a_{x}=1^{\oplus}$ for each $x$ yields a well-ordered set of the
length of $R$. Using a two-point $b$ proves that binary sums exist. And if $b$
is a well-ordered set and $a_{x}=a$ for each $x\in b_{I}$, then
$(b+1^{\oplus})_{I}$ has at least the length of $b$ and $a\cdot b$ can be
embedded in $\sum_{x\in(b+1^{\oplus})_{I}}a_{x}$. ∎
## 3 Pristine Sets and Inner Models
Pristine sets are not only useful for obtaining ordinal numbers, but also
provide a rich class of inner models of essential set theory and prove several
relative consistency results. To this end, we need to generalize the notion of
a pristine set, such that it also applies to non-transitive sets.
But first we give a general criterion for interpretations of essential set
theory. The picture behind the following is this: The elements of the class
$Z$ are to be ignored, so $Z$ is interpreted as the empty class. We do this to
be able to interpret $\emptyset\in\mathbb{V}$ even if the empty class is
proper by choosing a nonempty set $Z\in\mathbb{V}$. Everything that is to be
interpreted as a class will be a superclass $X$ of $Z$, but only the elements
of $X\setminus Z$ correspond to actual objects of the interpretation. In
particular, $B\supseteq Z$ will be interpreted as the class of atoms and $W$
as the universe. So the extension of an element $x\in W\setminus B$ will be a
set $X$ with $Z\subseteq X\subseteq W$, which we denote by $\Phi(x)$. Theorem
9 details the requirements these objects must meet to define an interpretation
of $\Th{ES}$.
###### Theorem 9 ($\Th{ES}$).
Let $\mathcal{K}\subseteq\mathcal{D}$ and $Z\subseteq B\subseteq W$ be classes
and $\Phi:W\setminus B\rightarrow\mathbb{V}$ injective. We use the following
notation:
* •
$X$ is an _inner class_ if it is not an atom and $Z\subseteq X\subseteq W$. In
that case, let $X^{\oplus}=X\setminus Z$.
* •
$S=W\setminus B^{\oplus}$ and $T=\Phi[S^{\oplus}]$.
* •
$\overline{\Phi}=\Phi\cup\mathrm{id}_{B^{\oplus}}:W^{\oplus}\rightarrow\mathbb{V}$
Define an interpretation $\mathcal{I}$ as follows:
$\displaystyle X\text{ is in the domain of }\mathcal{I}$ if $\displaystyle
X\text{ is an inner class or }X\in B^{\oplus}\text{.}$ $\displaystyle
X\in^{\mathcal{I}}Y$ if $\displaystyle Y\text{ is an inner class and
}X\in\overline{\Phi}[Y^{\oplus}]\text{.}$
$\displaystyle\mathbb{A}^{\mathcal{I}}$ $\displaystyle=$ $\displaystyle B$
If the following conditions are satisfied, $\mathcal{I}$ interprets essential
set theory:
1. 1.
$W^{\oplus}$ has more than one element.
2. 2.
Every element of $T$ is an inner class, and no element of $B$ is an inner
class.
3. 3.
$Z\cup\\{x\\}\in T$ for every $x\in W^{\oplus}$.
4. 4.
Any intersection $\bigcap C$ of a nonempty $C\subseteq T$ is $Z$ or an element
of $T$.
5. 5.
$x\cup y\in T$ for all $x,y\in T$.
6. 6.
If $x\in T$ and $x\setminus\\{y\\}\in T$ for all $y\in x^{\oplus}$, then
$x^{\oplus}$ is $\mathcal{K}$-small.
7. 7.
Any union $\bigcup C$ of a nonempty $\mathcal{K}$-small $C\subseteq T$ is an
element of $T$.
8. 8.
For all $a,b\in T$, the class
$Z\cup\left\\{x{\in}S^{\oplus}\mid\Phi(x){\subseteq}a,\Phi(x){\cap}b{\neq}Z\right\\}$
is $Z$ or in $T$.
The length of $\mathit{On}^{\mathcal{I}}$ is the least $\mathcal{K}$-large
ordinal $\kappa$, or $\mathit{On}$ if no such $\kappa$ exists (for example in
the case $\mathcal{K}=\mathcal{D}$). In particular,
$(\omega\in\mathbb{V})^{\mathcal{I}}$ iff $\omega$ is $\mathcal{K}$-small.
###### Proof.
Let us first translate some $\mathcal{I}$-interpretations of formulas:
* •
$(X\notin\mathbb{A})^{\mathcal{I}}$ iff $X$ is an inner class, and
$(X\in\mathbb{A})^{\mathcal{I}}$ iff $X\in B^{\oplus}$.
* •
$(X\in\mathbb{V})^{\mathcal{I}}$ iff $X\in\overline{\Phi}[W^{\oplus}]$,
because $W$ is the union of all inner classes, so
$\mathbb{V}^{\mathcal{I}}=W$.
* •
If $(F:X_{1}\rightarrow X_{2})^{\mathcal{I}}$, then there is a function
$G:\overline{\Phi}[X_{1}^{\oplus}]\rightarrow\overline{\Phi}[X_{2}^{\oplus}]$,
defined by $G(Y_{1})=Y_{2}$ if $(F(Y_{1})=Y_{2})^{\mathcal{I}}$, and $G$ is
surjective resp. injective iff $(F\text{ is surjective})^{\mathcal{I}}$ resp.
$(F\text{ is injective})^{\mathcal{I}}$.
Now we verify the axioms of $\Th{ES}^{\mathcal{I}}$:
_Extensionality_ : Assume $(X_{1}\neq
X_{2}\;\wedge\;X_{1},X_{2}\notin\mathbb{A})^{\mathcal{I}}$. Then $X_{1}$ and
$X_{2}$ are inner classes. But $X_{1}\neq X_{2}$ implies that there exists an
element $y$ in $X_{1}\setminus X_{2}\subseteq W^{\oplus}$ or $X_{2}\setminus
X_{1}\subseteq W^{\oplus}$. $Y=\Phi(y)$ is either in $B^{\oplus}$ or an inner
class by (2). Since $\Phi$ is injective, this means by definition that
$(Y{\in}X_{1}\wedge Y{\notin}X_{2})^{\mathcal{I}}$ or vice versa.
The _atoms axiom_ follows directly from our definition of $\in^{\mathcal{I}}$,
because no element of $B^{\oplus}$ is an inner class, and we enforced
_Nontriviality_ by stating that $W^{\oplus}$ has more than one element.
_Comprehension( $\psi$):_ If
$Y=Z\cup\\{x{\in}W^{\oplus}\mid\psi^{\mathcal{I}}(\Phi(x),\vec{P})\\}$, then
$Y$ witnesses the comprehension axiom for the formula $\psi=\phi^{C}$ with the
parameters $\vec{P}$, because $X\in^{\mathcal{I}}Y$ iff
$X\in\overline{\Phi}[Y^{\oplus}]=\\{\Phi(x)\mid
x{\in}W^{\oplus}\wedge\psi^{\mathcal{I}}(\Phi(x),\vec{P})\\}\text{,}$
which translates to $X\in\overline{\Phi}[W^{\oplus}]$ and
$\psi^{\mathcal{I}}(X,\vec{P})$.
_$T_{1}$ :_ Let $(X\in\mathbb{V})^{\mathcal{I}}$. Then $X=\overline{\Phi}(x)$
for some $x\in W^{\oplus}$. By (3), $Y=Z\cup\\{x\\}\in T=\Phi[S^{\oplus}]$, so
in particular $(Y\in\mathbb{V})^{\mathcal{I}}$. But $X$ is the unique element
such that $X\in^{\mathcal{I}}Y$, so $(Y=\\{X\\})^{\mathcal{I}}$.
_2nd Topology Axiom:_ Assume $(D$ is a nonempty class of
sets$)^{\mathcal{I}}$, because if $(D$ contains an atom$)^{\mathcal{I}}$, the
intersection is empty in $\mathcal{I}$ anyway. Then $D$ is an inner class and
every $Y\in C=\overline{\Phi}[D^{\oplus}]$ is an inner class, which means
$Y\in\Phi[S^{\oplus}]$. So $C\subseteq\Phi[S^{\oplus}]$ and $C\neq\emptyset$.
We have $(X\in\bigcap D)^{\mathcal{I}}$ iff $X\in^{\mathcal{I}}Y$ for all
$Y\in^{\mathcal{I}}D$, that is:
$X\in\bigcap_{Y\in
C}\overline{\Phi}[Y^{\oplus}]=\overline{\Phi}\left[\left(\bigcap
C\right)^{\oplus}\right]\text{,}$
because $\overline{\Phi}$ is injective. Hence the inner class $\bigcap C$
equals $\left(\bigcap D\right)^{\mathcal{I}}$, and by (4), it is either in $T$
and therefore interpreted as a set, or it is $Z=\emptyset^{\mathcal{I}}$.
_Additivity_ : A similar argument shows that $\bigcup C$ equals $\left(\bigcup
D\right)^{\mathcal{I}}$. If $(D$ is a discrete set$)^{\mathcal{I}}$, then by
(6), $D^{\oplus}$ is $\mathcal{K}$-small and therefore the union of
$C=\overline{\Phi}[D^{\oplus}]$ is in $T$ by (7).
_3rd Topology Axiom:_ Let $(X_{1},X_{2}\in\mathbb{T})^{\mathcal{I}}$. Then
$X_{1},X_{2}\in T$ and $X_{1},X_{2}\neq Z$. By (5), $Y=X_{1}\cup X_{2}\in T$,
and $Y$ is interpreted as the union of $X_{1}$ and $X_{2}$.
The _Exponential_ axiom follows from (8), because
$Y=Z\cup\left\\{x{\in}S^{\oplus}\mid\Phi(x){\subseteq}a,\Phi(x){\cap}b{\neq}Z\right\\}$
equals $(\square a\cap\lozenge b)^{\mathcal{I}}$. In fact,
$X\in^{\mathcal{I}}Y$ iff $X\in T$, $X\subseteq a$ and $X\cap b\neq Z$, and
$X\subseteq a$ is equivalent to $(X\subseteq a)^{\mathcal{I}}$, while $X\cap
b\neq Z$ is equivalent to $(X\cap b\neq\emptyset)^{\mathcal{I}}$.
The statement about the length of $\mathit{On}^{\mathcal{I}}$ holds true
because the discrete sets are interpreted by the classes $X$ with
$\mathcal{K}$-small $X^{\oplus}$. ∎
All the conditions of the theorem only concern the image of $\Phi$ but not
$\Phi$ itself, so given such a model one can obtain different models by
permuting the images of $\Phi$. Also, if $\Phi[S^{\oplus}]$ is infinite and if
$Z\in\mathbb{V}$, one can toggle the truth of the statement
$(\emptyset\in\mathbb{V})^{\mathcal{I}}$ by including $Z$ in or removing $Z$
from $\Phi[S^{\oplus}]$.
###### Proposition 10 ($\Th{ES}$).
If $Z=\emptyset$, $T$ is a $\mathcal{K}$-compact Hausdorff
$\mathcal{K}$-topology on $W$, $W$ has at least two elements, $B\subseteq W$
is open and does not contain any subsets of $W$, and $\Phi:W\setminus
B\rightarrow\mathrm{Exp}_{\mathcal{K}}(W,T)$ is a homeomorphism, then all
conditions of Theorem 9 are met and therefore these objects define an
interpretation of $\Th{ES}$. In addition, they interpret the statements
$\mathbb{V}\in\mathbb{V}$ and that every set is $\mathcal{D}$-compact
Hausdorff.
###### Proof.
All conditions that we did not demand explicitly follow immediately from the
fact that $W$ is a $\mathcal{K}$-compact Hausdorff $\mathcal{K}$-topological
space and from the definition of the exponential $\mathcal{K}$-topology.
$(\mathbb{V}\in\mathbb{V})^{\mathcal{I}}$ holds true, because
$W\in\mathrm{Exp}_{\mathcal{K}}(W,T)$. And since the $\mathcal{K}$-small sets
are exactly those interpreted as discrete, the $\mathcal{K}$-compactness and
Hausdorff property of $W$ implies that $(\mathbb{V}$ is $\mathcal{D}$-compact
Hausdorff.$)^{\mathcal{I}}$. ∎
Such a topological space $W$, together with a homeomorphism $\Phi$ to its
hyperspace, is called a $\mathcal{K}$-_hyperuniverse_. These structures have
been extensively studied in [FHL96, FH96b, Ess03]. Here we will deal with a
different class of models given by pristine sets.
Let $Z\subseteq B$ be such that no element of $B$ is a super _set_ of $Z$
(they are allowed to be atoms). Again, write $X\in_{Z}Y$ for:
$X\in Y^{\oplus}\quad\text{ and }\quad Z\subseteq Y\text{.}$
And $X$ is $Z$-_transitive_ if $c\in_{Z}X$ whenever $c\in_{Z}b\in_{Z}X$. We
say that $X$ is $Z$-$B$-_pristine_ if:
* •
$X\in_{Z}B$ or:
* •
$Z\subseteq X\notin\mathbb{A}$, and there is a $Z$-transitive set $b\supseteq
X$, such that for every $c\in_{Z}b$ either $Z\subseteq c\notin\mathbb{A}$ or
$c\in_{Z}B$.
If $a$ has a $Z$-transitive superset $b$, then it has a least $Z$-transitive
superset $\mathrm{trcl}(a)=\bigcap\\{b{\supseteq}a\mid b\text{
$Z$-transitive}\\}$, the $Z$-_transitive closure_ of $a$. Obviously a set is
$Z$-transitive iff it equals its $Z$-transitive closure. Also, $a$ is
$Z$-$B$-pristine iff $\mathrm{trcl}(a)$ exists and is $Z$-$B$-pristine. A set
$a$ is $Z$-_well-founded_ iff for every $b\ni_{Z}a$, there exists an
$\in_{Z}$-minimal $c\in_{Z}b$.
###### Theorem 11 ($\Th{ES}$).
Let $Z\in\mathbb{V}$ and $B\supseteq Z$ such that no element of $B$ is a super
_set_ of $Z$, and $B^{\oplus}$ is $\mathbb{T}$-closed. Let $\Phi$ be the
identity on $W\setminus B$ and $\mathcal{K}=\mathcal{D}$. The following
classes $W_{i}^{\oplus}$ meet the requirements of Theorem 9 and therefore
define interpretations $\mathcal{I}_{i}$ of essential set theory:
* •
the class $W_{1}^{\oplus}$ of all $Z$-$B$-pristine $x$
* •
the class $W_{2}^{\oplus}$ of all $Z$-$B$-pristine $x$ with discrete
$\mathrm{trcl}(x)^{\oplus}$
* •
the class $W_{3}^{\oplus}$ of all $Z$-well-founded $Z$-$B$-pristine $x$ with
discrete $\mathrm{trcl}(x)^{\oplus}$
$Z$ is a member of all three classes and thus
$(\emptyset\in\mathbb{V})^{\mathcal{I}_{i}}$ holds true in all three cases. If
$i\in\\{2,3\\}$, then $(\text{every set is discrete})^{\mathcal{I}_{i}}$, and
in the third case, $(\text{every set is $\emptyset$-well-
founded})^{\mathcal{I}_{3}}$.
If $\mathbb{V}\in\mathbb{V}$, then:
1. 1.
$(\mathbb{V}\in\mathbb{V})^{\mathcal{I}_{1}}$
2. 2.
$(\text{$\mathit{On}$ has the tree property})^{\mathcal{I}}_{i}$ for all $i$.
3. 3.
If $B^{\oplus}$ is discrete, $\mathcal{I}_{3}$ satisfies the strong
comprehension principle.
###### Proof.
In this proof, we will omit the prefixes $Z$ and $B$: By “pristine” we always
mean $Z$-$B$-pristine, “transitive” means $Z$-transitive and “well-founded”
$Z$-well-founded.
Since $Z^{\oplus}$ is empty and $B$ is pristine and well-founded, $Z\in
W_{3}^{\oplus}\subseteq W_{2}^{\oplus}\subseteq W_{1}^{\oplus}$.
Before we go through the requirements of Theorem 9, let us prove that
$x^{\oplus}$ is closed for every $x\in S^{\oplus}$:
$x^{\oplus}\quad=\quad(x\cap B^{\oplus})\;\cup\;(\\{Z\\}\cap x)\;\cup\;\\{y\in
x\mid Z\subseteq y\notin\mathbb{A}\\}$
Since $x$ is pristine, there is a transitive pristine $c\supseteq x$, and we
can rewrite the class $\\{y\in x\mid Z\subseteq y\notin\mathbb{A}\\}$ as
$\\{y\in x\cap\square c\mid Z\subseteq y\subseteq c\\}$, which is closed.
Condition (1) of Theorem 9 is satisfied because $Z$ and $Z\cup\\{Z\\}$ are
distinct elements of $W_{3}^{\oplus}$.
(2): If $x\in B$, then $x$ is not a superset of $Z$ and therefore not an inner
class. Now let $x\in S_{1}^{\oplus}$. We have to show that $x=\Phi(x)$ is an
inner class. Since $x\notin B$ and $x$ is pristine, $x\notin\mathbb{A}$ and
$Z\subseteq x$, so it only remains to prove that $y\in W_{1}^{\oplus}$ for
every $y\in x^{\oplus}$. If $y\in_{Z}B$, $y$ is pristine. If $y\notin_{Z}B$,
then $Z\subseteq y$. Since every transitive superset of $x$ is also a superset
of $y$, $y$ is pristine in that case, too. If in addition,
$\mathrm{trcl}(x)^{\oplus}$ is discrete, $y$ also has that property, by the
same argument. And if $x$ is also well-founded, $y$ also is: For any
$b\ni_{Z}y$, $b^{\oplus}\cup\\{x\\}$ has a $\in_{Z}$-minimal element; since
$y\in_{Z}x$ and $y\in_{Z}b$, this cannot be $x$, so it must be in
$b^{\oplus}$. This concludes the proof that $y\in W_{i}^{\oplus}$ whenever
$x\in S_{i}^{\oplus}$.
(3): If $x\in W_{1}^{\oplus}$, then $Z\cup\\{x\\}$ is pristine, because if
$x\in B^{\oplus}$, it is already transitive itself, and otherwise if $c$ is a
transitive pristine superset of $x$, then $c\cup\\{x\\}$ is a transitive
pristine superset of $Z\cup\\{x\\}$. If moreover $c^{\oplus}$ is discrete,
then $c^{\oplus}\cup\\{x\\}$ also is, and if $x$ is well-founded,
$Z\cup\\{x\\}$ also is.
(4): Let $C\subseteq S_{i}^{\oplus}$ be nonempty. Then $\bigcap C\in
S_{i}^{\oplus}$, too, because every subset of a pristine set which is a
superset of $Z$ is pristine itself, every subset of a discrete set is
discrete, and every subset of a well-founded set is well-founded.
(6): Assume that for every $y\in x^{\oplus}$, we have $x\setminus\\{y\\}\in
S^{\oplus}$. Then $(x\setminus\\{y\\})^{\oplus}=x^{\oplus}\setminus\\{y\\}$ is
closed, and hence $x^{\oplus}$ is a discrete set.
(7) (and consequently (5)): Let $C\subseteq S_{1}^{\oplus}$ be a nonempty
discrete set. Then $\bigcup C\in W_{1}\setminus B$, because if $c_{b}$ is a
transitive pristine superset of $b$ for all $b\in C$, then $\bigcup c_{b}$ is
such a superset of the union. If all the $c_{b}$ are discrete, their union
also is, because they are only $\mathcal{D}$-few. And if every element of $C$
is well-founded, $\bigcup C$ also is.
(8): $Y=Z\cup\left\\{x{\in}S^{\oplus}\mid
x{\subseteq}a,x{\cap}b{\neq}Z\right\\}$ is pristine, because if $c$ is a
transitive pristine superset of $a$, then $z=Z\cup\\{Z\\}\cup\\{x{\in}\square
c\mid Z{\subseteq}x\\}$ is a transitive pristine superset of $Y$. And $Y$ is
in fact a set, because $b^{\oplus}$ is closed, so
$Y=Z\cup(z^{\oplus}\cap\square a\cap\lozenge b^{\oplus})$ also is. If
$c^{\oplus}$ is discrete, $\square c^{\oplus}$ is discrete, and so is
$z^{\oplus}\setminus\\{Z\\}=\\{y\cup Z\mid y\in\square c^{\oplus}\\}$. And if
$a$ is well-founded, any set of subsets of $a$ is well-founded, too.
The claims about discreteness and well-foundedness are immediate from the
definitions.
Now let us prove the remaining claims under the assumption that
$\mathbb{V}\in\mathbb{V}$:
(1): $\mathbb{V}\setminus\mathbb{A}$ is a set, namely
$\lozenge\mathbb{V}\cup\\{\emptyset\\}$ or $\lozenge\mathbb{V}$, depending on
whether $\emptyset\in\mathbb{V}$. Let:
$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\mathbb{V}$
$\displaystyle U_{n+1}$ $\displaystyle=$ $\displaystyle
B^{\oplus}\;\cup\;\\{x\in\mathbb{V}{\setminus}\mathbb{A}\mid Z\subseteq
x\subseteq Z\cup U_{n}\\}$ $\displaystyle U_{\omega}$ $\displaystyle=$
$\displaystyle\bigcap_{n\in\omega}U_{n}$
Then $U_{\omega}$ is a set. Since $W_{1}^{\oplus}\subseteq\mathbb{V}$ and
$W_{1}^{\oplus}\subseteq
B^{\oplus}\cup\\{x\in\mathbb{V}{\setminus}\mathbb{A}\mid Z\subseteq x\subseteq
Z\cup W_{1}^{\oplus}\\}$, it is a subset of $U_{\omega}$. It remains to show
that $U_{\omega}\subseteq W_{1}^{\oplus}$, that is, that every element of
$U_{\omega}$ is pristine, because then it follows that $W_{1}$ is a pristine
set itself and hence $W_{1}\in_{Z}W_{1}$. In fact, it suffices to prove that
$Z\cup U_{\omega}$ is a transitive pristine set, because then all
$x\in_{Z}U_{\omega}$ will be pristine, too. So assume $y\in_{Z}x\in_{Z}Z\cup
U_{\omega}$. If $x$ were in $B^{\oplus}$, then $y\notin_{Z}x$, so $x$ must be
in $\mathbb{V}\setminus\mathbb{A}$ and $Z\subseteq x\subseteq Z\cup U_{n}$ for
all $n$. Thus $x\subseteq Z\cup U_{\omega}$, which implies that
$y\in_{Z}U_{\omega}$.
(2) follows from Proposition 6.
(3): It suffices to show that $W_{3}^{\oplus}$ does not contain any of its
accumulation points, because that implies that every inner class corresponds
to a set – it’s closure –, so that the weak comprehension principle allows us
to quantify over all inner classes. Since $B^{\oplus}$ is discrete and
$S_{3}^{\oplus}\setminus\\{Z\\}\quad=\quad
W_{3}^{\oplus}\setminus(B\cup\\{Z\\})\quad\subseteq\quad\\{x\in\mathbb{V}{\setminus}\mathbb{A}\mid
Z\subseteq x\\}\quad\in\quad\mathbb{V}$
(recall that no element of $B$ is a superset of $Z$), $B$ certainly contains
no accumulation point of $W_{3}^{\oplus}$. So assume now that $x\in
W_{3}^{\oplus}$ is an accumulation point. Since it is well-founded and
$\mathrm{trcl}(x)^{\oplus}$ is a discrete set,
$\mathrm{trcl}(x)^{\oplus}\cup\\{x\\}$ has an $\in_{Z}$-minimal
$W_{3}^{\oplus}$-accumulation point $y$. Then $y\in S_{3}^{\oplus}$ and $y$ is
also an accumulation point of
$W_{3}^{\oplus}\setminus(B^{\oplus}\cup\\{Z\\})$. Since none of the
$\mathcal{D}$-few elements of $y^{\oplus}$ is an $W_{3}^{\oplus}$-accumulation
point, $W_{3}^{\oplus}\setminus(B^{\oplus}\cup\\{Z,y\\})$ is a subclass of
$\lozenge\mathrm{cl}(W_{3}^{\oplus}\setminus
y)\;\cup\;\bigcup_{z\in_{Z}y}\square\mathrm{cl}(W_{3}^{\oplus}\setminus\\{z\\})\text{,}$
which is closed and does not contain $y$, a contradiction. ∎
By the nontriviality axiom, there are distinct $x,y\in\mathbb{V}$. If we set
$Z=B=\\{\\{x\\},\\{y\\}\\}$, the requirements of Theorem 11 are satisfied, so
$\mathcal{I}_{i}$ interprets essential set theory with
$\emptyset\in\mathbb{V}$ in all three cases. Moreover, since $Z=B$, it
interprets $\mathbb{A}=\emptyset$. So $\mathbb{A}=\emptyset\in\mathbb{V}$ is
consistent relative to $\Th{ES}$. In the case $i=3$, moreover, $($every set is
$\emptyset$-well-founded and discrete$)^{\mathcal{I}_{3}}$! And if in addition
$\omega\in\mathbb{V}$, then $\omega$ is $\mathcal{D}$-small and thus
$(\omega\in\mathbb{V})^{\mathcal{I}_{3}}$ by Theorem 9.
But if in $\Th{ES}$ every set is discrete and $\emptyset$-well-founded, the
following statements are implied:
Pair, Union, Power, Empty Set $\displaystyle\\{a,b\\},\;\bigcup
a,\;\mathfrak{P}(a),\;\emptyset\;\in\;\mathbb{V}$ Replacement If $F$ is a
function and $a\in\mathbb{V}$, then $F[a]\in\mathbb{V}$. Foundation Every
$x\in\mathbb{T}$ has a member disjoint from itself.
And these are just the axioms of $\Th{ZF}$222With classes, of course. We avoid
the name $\Th{NBG}$, because that is usually associated with a strong axiom of
choice.! Conversely, all the axioms of $\Th{ES}$ hold true in $\Th{ZF}$, so
$\Th{ZF}$ could equivalently be axiomatized as follows333 We will soon
introduce a choice principle for $\Th{ES}$, the uniformization axiom, which
applies to all discrete sets. Since in $\Th{ZF}$ every set is discrete, that
axiom is equivalent to the axiom of choice.:
* •
$\Th{ES_{\infty}}$
* •
$\mathbb{A}=\emptyset\in\mathbb{V}$
* •
Every set is discrete and $\emptyset$-well-founded.
If in addition $\mathbb{V}\in\mathbb{V}$, then $\mathcal{I}_{3}$ even
interprets the strong comprehension axiom and therefore Kelley-Morse set
theory444The axiom of choice is not necessarily true in that interpretation,
but even the existence of a global choice function does not add to the
consistency strength, as was shown in [Ess04]. with $\mathit{On}$ having the
tree property. Conversely, O. Esser showed in [Ess97] and [Ess99] that this
theory is equiconsistent with $\Th{GPK}^{+}_{\infty}$, which in turn is an
extension of topological set theory that will be introduced in the next
section. In summary, we have the following results:
###### Corollary 12.
$\Th{ES}_{\infty}$ is equiconsistent with $\Th{ZF}$: The latter implies the
former and the former interprets the latter.
$\Th{TS}_{\infty}$ and $\Th{GPK}^{+}_{\infty}$ both are mutually interpretable
with:
Kelley-Morse set theory $\;+\;$ $\mathit{On}$ has the tree property.
$\mathcal{I}_{3}$ is a particularly intuitive interpretation if
$\emptyset,\mathbb{V}\in\mathbb{V}$, $\mathbb{A}=\emptyset$ and we set
$Z=B=\emptyset$. Then every set is ($\emptyset$-$\emptyset$-)pristine and
$\in_{N}$ is just $\in$. Also,
$\mathbb{V}\setminus\\{\emptyset\\}=\lozenge\mathbb{V}\in\mathbb{V}$, so
$\emptyset$ is an isolated point. If a set $x$ contains only isolated points,
it is discrete, and since $x=\bigcup_{y\in x}\\{y\\}$ and every $\\{y\\}$ is
open, $x$ is a clopen set. Moreover, $x$ is itself an isolated point, because
$\\{x\\}$ is open:
$\\{x\\}\quad=\quad\square x\cup\bigcup_{y\in x}\lozenge\\{y\\}$
Thus it follows that all ($\emptyset$-)well-founded sets are isolated. Define
the cumulative hierarchy as usual:
$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\emptyset$ $\displaystyle
U_{\alpha+1}$ $\displaystyle=$ $\displaystyle\square
U_{\alpha}\cup\\{\emptyset\\}$ $\displaystyle U_{\lambda}$ $\displaystyle=$
$\displaystyle\bigcup_{\alpha<\lambda}U_{\alpha}\;\text{ for limit ordinals
}\lambda$
Since images of discrete sets in $\mathit{On}$ are bounded and since every
nonempty class of well-founded sets has an $\in$-minimal element, the union
$\bigcup_{\alpha\in\mathit{On}}U_{\alpha}$ is exactly the class of all well-
founded sets, and in fact equals $W_{3}$.
## 4 Positive Specification
This section is a short digression from our study of essential set theory.
Again starting from only the class axioms we introduce specification schemes
for two classes of “positive” formulas as well as O. Esser’s theory
$\Th{GPK^{+}}$ (cf. [Ess97, Ess99, Ess00, Ess04]), and then turn our attention
to their relationship with topological set theory.
The idea of positive set theory is to weaken the inconsistent _naive
comprehension scheme_ – that every class $\\{x\mid\phi(x)\\}$ is a set – by
permitting only _bounded positive formula_ s (BPF), which are defined
recursively similarly to the set of all formulas, but omitting the negation
step, thus avoiding the Russell paradox. This family of formulas can
consistently be widened to include all _generalized positive formula_ s (GPF),
which even allow universal quantification over classes. But to obtain more
general results, we will investigate _specification_ schemes instead of
comprehension schemes, which only state the existence of subclasses
$\\{x{\in}c\mid\phi(x)\\}$ of sets $c$. If $\mathbb{V}$ is a set, this
restriction makes no difference.
We define recursively when a formula $\phi$ whose variables are among
$X_{1},X_{2},\ldots$ and $Y_{1},Y_{2},\ldots$ (where these variables are all
distinct) is a _generalized positive formula_ (GPF) _with parameters_
$Y_{1},Y_{2},\ldots$:
* •
The atomic formulas $X_{i}\in X_{j}$ and $X_{i}=X_{j}$ are GPF with parameters
$Y_{1},Y_{2},\ldots$.
* •
If $\phi$ and $\psi$ are GPF with parameters $Y_{1},Y_{2},\ldots$, then so are
$\phi\wedge\psi$ and $\phi\vee\psi$.
* •
If $i\neq j$ and $\phi$ is a GPF with parameters $Y_{1},Y_{2},\ldots$, then so
are ${\forall}X_{i}{\in}X_{j}\;\phi$ and ${\exists}X_{i}{\in}X_{j}\;\phi$.
* •
If $\phi$ is a GPF with parameters $Y_{1},Y_{2},\ldots$, then so is
${\forall}X_{i}{\in}Y_{j}\;\phi$.
A GPF with parameters $Y_{1},Y_{2},\ldots$ is a _bounded positive formula_
(BPF) if it does not use any variable $Y_{i}$, that is, if it can be
constructed without making use of the fourth rule. The _specification axiom_
for the GPF $\phi(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{n})$ with parameters
$Y_{1},Y_{2},\ldots$, whose free variables are among $X_{1},\ldots,X_{m}$, is:
$\displaystyle\\{x{\in}c\>|\>\phi(x,b_{2},\ldots,b_{m},B_{1},\ldots,B_{n})\\}\text{
is $\mathbb{T}$-closed}$ for all $c,b_{2},\ldots,b_{m}\in\mathbb{V}$ and all
classes $B_{1},\ldots,B_{n}$.
_GPF specification_ is the scheme consisting of the specification axioms for
all GPF $\phi$, and _BPF specification_ incorporates only those for BPF
$\phi$. Note that we did not include the formula $x\in\mathbb{A}$ or any other
formula involving the constant $\mathbb{A}$ in the definition, so
$x\in\mathbb{A}$ is not a GPF.
The following theorem shows that BPF specification is in fact finitely
axiomatizable, even without classes.555A similar axiomatization, but for
positive _comprehension_ , is given by M. Forti and R. Hinnion in [FH89]. On
the other hand, no finite axiomatization exists for _generalized_ positive
comprehension, as O. Esser has shown in [Ess04].
###### Theorem 13.
Assume only the class axioms and that for all $a,b\in\mathbb{V}$, the
following are $\mathbb{T}$-closed:
$\bigcup a,\quad\\{a,b\\},\quad a{\times}b$
Let $\Theta$ be the statement that for all sets $a,b\in\mathbb{V}$, the
following are $\mathbb{T}$-closed:
$\displaystyle\Delta{\cap}a,\quad\mathbf{E}{\cap}a,\quad\\{\langle
x,y\rangle{\in}b\>|\>{\forall}z{\in}y\;\langle x,y,z\rangle{\in}a\\},$
$\displaystyle\\{\langle y,x,z\rangle\>|\>\langle
x,y,z\rangle{\in}a\\},\quad\\{\langle z,x,y\rangle\>|\>\langle
x,y,z\rangle{\in}a\\}$
Then BPF specification is equivalent to $\Theta$. And GPF specification is
equivalent to $\Theta$ and the second topology axiom.
###### Proof.
Ordered pairs can be built from unordered ones, and the equality $\langle
x,y\rangle=z$ can be expressed as a BPF. Therefore the classes mentioned in
$\Theta$ can all be defined by applying BPF specification to a given set or
product of sets, so BPF specification implies $\Theta$.
GPF specification in addition implies the second topology axiom,
${\forall}B{\neq}\emptyset.\;\quad\emptyset{=}\bigcap B\quad\vee\quad\bigcap
B\in\mathbb{V}\text{,}$
because ${\forall}a{\in}B\;x{\in}a$ is clearly a GPF with parameter $B$, and
the intersection is a subclass of any $c\in B$.
To prove the converse, assume now that $\Theta$ holds. Since it is not yet
clear what we can do with sets, we have to be pedantic with respect to
Cartesian products. We define
$A\times_{2}B=\left\\{\langle a,b_{1},b_{2}\rangle\mid a{\in}A,\langle
b_{1},b_{2}\rangle{\in}B\right\\}\text{,}$
which is not the same as $A\times B$ for $B\subseteq\mathbb{V}^{2}$, because
$\langle a,b_{1},b_{2}\rangle=\langle\langle a,b_{1}\rangle,b_{2}\rangle$,
whereas the elements of $A\times B$ are of the form $\langle a,\langle
b_{1},b_{2}\rangle\rangle$. Yet we can construct this and several other set
theoretic operations from $\Theta$:
$\displaystyle a\times_{2}b$ $\displaystyle=$ $\displaystyle\\{\langle
z,x,y\rangle\>|\>\langle x,y,z\rangle\in b\times a\\}$ $\displaystyle a\cup b$
$\displaystyle=$ $\displaystyle\bigcup\\{a,b\\}$ $\displaystyle a\cap b$
$\displaystyle=$ $\displaystyle\bigcup\bigcup\\{\\{\\{x\\}\\}\mid x\in a\cap
b\\}=\bigcup\bigcup(\Delta\cap(a{\times}b))$ $\displaystyle
a\cap\mathbb{V}^{2}$ $\displaystyle=$ $\displaystyle
a\>\cap\>\left(\bigcup\bigcup a\right)^{2}$
$\displaystyle\\{\\{x\\}\mid\\{x\\}\in a\\}$ $\displaystyle=$ $\displaystyle
a\>\cap\>\bigcup\left(\Delta\cap\left(\bigcup a\right)^{2}\right)$
$\displaystyle\mathrm{dom}(a)$ $\displaystyle=$
$\displaystyle\bigcup\left\\{\\{x\\}\mid\\{x\\}\in\bigcup(a\cap\mathbb{V}^{2})\right\\}$
$\displaystyle a^{-1}$ $\displaystyle=$
$\displaystyle\mathrm{dom}\left(\\{\langle y,x,z\rangle\mid\langle
x,y,z\rangle\in a{\times}\\{a\\}\\}\right)$
We will prove by induction that for all GPF
$\phi(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{n})$ with parameters
$Y_{1},\ldots,Y_{n}$ and free variables $X_{1},\ldots,X_{m}$, and for all
classes $B_{1},\ldots,B_{n}$ and sets $a_{1},\ldots,a_{m}$,
$A^{\phi}_{a_{1},\ldots,a_{m}}=\\{\langle x_{1},\ldots,x_{m}\rangle\in
a_{1}{\times}\ldots{\times}a_{m}\>|\>\phi(x_{1},\ldots,x_{m},B_{1},\ldots,B_{n})\\}$
is $\mathbb{T}$-closed. This will prove the specification axiom for $\phi$,
because
$\\{x{\in}c\>|\>\phi(x,b_{2},\ldots,b_{m},B_{1},\ldots,B_{n})\\}\quad=\quad\mathrm{dom}\left(\ldots\mathrm{dom}\left(A^{\phi}_{c,\\{b_{2}\\},\ldots,\\{b_{m}\\}}\right)\ldots\right)\text{,}$
where the domain operation is applied $m-1$ times.
Each induction step will reduce the claim to a subformula or to a formula with
fewer quantifiers. Let us assume wlog that no bound variable is among the
$X_{1},\ldots$ or $Y_{1},\ldots$ and just always denote the bound variable in
question by $Z$.
Case 1: Assume $\phi$ is ${\forall}Z{\in}Y_{i}\;\psi$. Then
$A^{\phi}_{a_{1},\ldots,a_{m}}=\bigcap_{x\in
B_{i}}\mathrm{dom}\left(A^{\psi(Z/X_{m+1})}_{a_{1},\ldots,a_{m},\\{x\\}}\right)\text{,}$
where $\psi(Z/X_{m+1})$ is the formula $\psi$, with each free occurrence of
$Z$ substituted by $X_{m+1}$. This is the step which is only needed for GPF
formulas. Since it is the only point in the proof where we make use of the
closure axiom, we otherwise still obtain BPF specification as claimed in the
theorem.
Case 2: Assume $\phi$ is a bounded quantification. If $\phi$ is
${\exists}Z{\in}X_{i}\;\psi$, then
$A_{\phi,a_{1},\ldots,a_{m}}=\mathrm{dom}\left(A^{\psi(Z/X_{m+1})\;\wedge\;X_{m+1}{\in}X_{i}}_{a_{1},\ldots,a_{m},b}\right)\text{,}$
where $b=\bigcup a_{i}$. If $\phi$ is ${\forall}Z{\in}X_{i}\;\psi$, then
$A^{\phi}_{a_{1},\ldots,a_{m}}=\mathrm{dom}\left\\{\langle x,y\rangle\in
a_{1}{\times}{\ldots}{\times}a_{m}{\times}a_{i}\mid{\forall}z{\in}y\;\langle
x,y,z\rangle\in A^{\rho}_{a_{1},\ldots,a_{m},a_{i},b}\right\\}\text{,}$
where again $b=\bigcup a_{i}$, and $\rho$ is the formula
$\psi(Z/X_{m+2})\;\wedge\;X_{m+1}{=}X_{i}$. The class defined here is of the
form $\\{\langle x,y\rangle{\in}b\>|\>{\forall}z{\in}y\;\langle
x,y,z\rangle{\in}a\\}$ and therefore a set, by our assumption.
Case 3: Assume $\phi$ is a conjunction or disjunction. If $\phi$ is
$\psi\wedge\chi$ resp. $\psi\vee\chi$, then
$A^{\phi}_{a_{1},\ldots,a_{m}}=A^{\psi}_{a_{1},\ldots,a_{m}}\cap
A^{\chi}_{a_{1},\ldots,a_{m}}\quad\text{ resp. }\quad
A^{\phi}_{a_{1},\ldots,a_{m}}=A^{\psi}_{a_{1},\ldots,a_{m}}\cup
A^{\chi}_{a_{1},\ldots,a_{m}}\text{.}$
Case 4: Assume $\phi$ is atomic. If $X_{m}$ does not occur in $\phi$, then
$A^{\phi}_{a_{1},\ldots,a_{m}}=A^{\phi}_{a_{1},\ldots,a_{m-1}}\times a_{m}$.
If $\phi$ has more than one variable, but $X_{m-1}$ is not among them, then:
$A^{\phi}_{a_{1},\ldots,a_{m}}=\left\\{\langle
z,x_{m-1},x_{m}\rangle\mid\langle z,x_{m},x_{m-1}\rangle\in
A^{\phi(X_{m}/X_{m-1})}_{a_{1},\ldots,a_{m-2},a_{m}}\times a_{m-1}\right\\}$
Applying these two facts recursively reduces the problem to the case where
either $m=1$ or where $X_{m}$ and $X_{m-1}$ both occur in $\phi$:
$\displaystyle A^{X_{1}=X_{1}}_{a_{1}}$ $\displaystyle=$ $\displaystyle a_{1}$
$\displaystyle A^{X_{1}\in X_{1}}_{a_{1}}$ $\displaystyle=$
$\displaystyle\mathrm{dom}\left(\mathbf{E}\cap a_{1}^{2}\right)$
$\displaystyle A^{X_{m-1}=X_{m}}_{a_{1},\ldots,a_{m}}$ $\displaystyle=$
$\displaystyle a_{1}\times\ldots\times
a_{m-2}\times_{2}(\Delta\cap(a_{m-1}{\times}a_{m}))$ $\displaystyle
A^{X_{m}=X_{m-1}}_{a_{1},\ldots,a_{m}}$ $\displaystyle=$ $\displaystyle
a_{1}\times\ldots\times a_{m-2}\times_{2}(\Delta\cap(a_{m-1}{\times}a_{m}))$
$\displaystyle A^{X_{m-1}\in X_{m}}_{a_{1},\ldots,a_{m}}$ $\displaystyle=$
$\displaystyle a_{1}\times\ldots\times
a_{m-2}\times_{2}(\mathbf{E}\cap(a_{m-1}{\times}a_{m}))$ $\displaystyle
A^{X_{m}\in X_{m-1}}$ $\displaystyle=$ $\displaystyle a_{1}\times\ldots\times
a_{m-2}\times_{2}(\mathbf{E}^{-1}\cap(a_{m-1}{\times}a_{m}))$
∎
As we already indicated, the theory $\Th{GPK}^{+}$ uses GPF _comprehension_ ,
but if $\mathbb{V}\in\mathbb{V}$, specification entails comprehension.
$\Th{GPK}^{+}$ can be axiomatized as follows:
* •
$\mathbb{V}\in\mathbb{V}$
* •
$\mathbb{A}=\emptyset\in\mathbb{V}$
* •
GPF specification
###### Proposition 14.
$\Th{GPK}^{+}$ implies $\Th{TS}$ and that unions of sets are sets.
###### Proof.
If $B\subseteq\mathbb{T}$, then $\bigcap
B=\\{x\mid{\forall}y{\in}B\;x{\in}y\\}$ is $\mathbb{T}$-closed, and if
$a,b\in\mathbb{T}$, then $a\cup b=\\{x\mid x{\in}a\vee
x{\in}b\\}\in\mathbb{V}$, because these are defined by GPFs, proving the 2nd
and 3rd topology axioms. $\\{a\\}=\\{x\mid x{=}a\\}$ and $x{=}a$ is bounded
positive, so $T_{1}$ is also true.
$\square a\cap\lozenge
b=\\{c\mid{\exists}x{\in}b\;x{=}x\wedge{\forall}x{\in}c\;x{\in}a\wedge{\exists}x{\in}b\;x{\in}c\\}$
is defined by a positive formula as well, yielding the exponential axiom.
$\bigcup a=\\{c\mid{\exists}x{\in}a\;c{\in}x\\}$ is also $\mathbb{T}$-closed,
for the same reason.
The formula $z=\\{x,y\\}$ can be expressed as $x{\in}z\wedge
y{\in}z\wedge{\forall}w{\in}z\;(w{=}x\vee w{=}y)$, so it is bounded positive.
Using that, we see that ordered pairs, Cartesian products, domains and ranges
can all be defined by GPFs. This allows us to prove the additivity axiom:
Let $a\in\mathbb{T}$ be discrete and $F:a\rightarrow\mathbb{V}$. We first show
that $F\in\mathbb{V}$: Firstly, $F\subseteq a\times\mathbb{V}$ and
$a\times\mathbb{V}$ is $\mathbb{T}$-closed. Secondly, if $\langle
x,y\rangle\in(a\times\mathbb{V})\setminus F$, then $F(x)\neq y$, so $F$ is a
subclass of the $\mathbb{T}$-closed
$(a{\setminus}\\{x\\}\times\mathbb{V})\cup\\{\langle x,F(x)\rangle\\}$, which
does not contain $\langle x,y\rangle$. Thus $F$ is a set and hence
$\bigcup\mathrm{rng}(F)$ is $\mathbb{T}$-closed. ∎
## 5 Regularity and Union
After having seen that topological set theory is provable in $\Th{GPK}^{+}$,
we now aim for a result in the other direction. To this end we assume in
addition to $\Th{ES}$ the union axiom and that every set is a regular space:
Union $\displaystyle\bigcup a\text{ is $\mathbb{T}$-closed for every
$a\in\mathbb{V}$.}$ $T_{3}$ $\displaystyle x{\in}a\;\wedge\;b{\in}\square
a\quad\Rightarrow\quad{\exists}u,v.\;\;u{\cup}v{=}a\;\wedge\;x{\notin}u\;\wedge\;b{\cap}v{=}\emptyset$
These two axioms elegantly connect the topological and set-theoretic
properties of orders and products. Note that they, too, are theorems of
$\Th{ZF}$, because every discrete set is regular and its union is a set.
Recall that we use the term _ordered set_ only for sets with an order $\leq$,
whose order-topology is at least as fine as their natural topology. By
default, we consider the order itself to be the non-strict version.
###### Proposition 15 ($\Th{ES}+\text{Union}+T_{3}$).
1. 1.
Domains and ranges of sets are sets.
2. 2.
Every map in $\mathbb{V}$ is continuous and closed with respect to the natural
topology.
3. 3.
A linear order $\leq$ on a set $a$ is a set iff its order topology is at most
as fine as the natural topology of $a$.
4. 4.
The product topology of $a^{n}$ is equal to the natural topology.
5. 5.
If $\mathbb{A}$ is closed, GPF specification holds.
###### Proof.
(1): Let $a$ be a set. Then $c=\bigcup\bigcup a$ is a set, and in fact,
$c=\mathrm{dom}(a)\cup\mathrm{rng}(a)$. But
$\mathrm{dom}(a)=\bigcup(\square_{\leq 1}c\cap\bigcup a)$, which proves that
domains of sets are sets. Now $F_{2,2,1}\upharpoonright c^{2}:c^{2}\rightarrow
c^{2}$ is a set, and so is $(c^{2}\times a)\cap F_{2,2,1}$. But the domain of
this set is $a^{-1}$, and the domain of $a^{-1}$ is $\mathrm{rng}(a)$.
(2): Let $f\in\mathbb{V}$ be a map from $a$ to $b$, and let $c\subseteq b$ be
closed. Then $f\cap(a\times c)$ is a set, too, and so is
$f^{-1}[c]=\mathrm{dom}(f\cap(a\times c))$. Thus $f$ is continuous. Similarly,
if $c\subseteq a$ is closed, then $f[c]=\mathrm{rng}(f\cap(c\times b))$ is a
set and hence $f$ is closed.
(3): Now let $a$ be linearly ordered by $\leq$. If $x\in a$, then
$[x,\infty)=\mathrm{rng}((\\{x\\}\times a)\;\cap\leq)$ and
$(\infty,x]=\mathrm{dom}((a\times\\{x\\})\;\cap\leq)$. Conversely assume that
all intervals $[x,y]$ are sets. Then if $\langle x,y\rangle\in
a^{2}\setminus\leq$, that is, $x>y$. If there is a $z\in(y,x)$, then
$(z,\infty)\times(-\infty,z)$ is a relatively open neighborhood of $\langle
x,y\rangle$ disjoint from $\leq$. Otherwise, $(y,\infty)\times(-\infty,x)$ is
one.
(4): To show that the topologies on $a^{n}$ coincide, we only need to consider
the case $n=2$; the rest follows by induction, because products of regular
spaces are regular. Since $a$ is Hausdorff, we already know from Proposition 1
that the universal topology is at least as fine as the product topology, and
it remains to prove the converse.
Let $b\subseteq a^{2}$ be a set. We will show that it is closed with respect
to the product topology. Let $\langle x,y\rangle\in a^{2}\setminus b$. Then
$x\notin\mathrm{dom}(b\cap(a\times\\{y\\}))$, so by regularity, there is a
closed neighborhood $u\ni x$ disjoint from that set. Thus
$b\cap(a\times\\{y\\})\cap(u\times a)=\emptyset$, that is,
$y\notin\mathrm{rng}(b\cap(u\times a))$. Again by $T_{3}$, there is a closed
neighborhood $v\ni y$ disjoint from that. Hence $b\cap(u\times v)=\emptyset$
and $u\times v$ is a neighborhood of $\langle x,y\rangle$ with respect to the
product topology.
(5): We only have to prove $\Theta$ from Theorem 13: The statements about the
permutations of triples are true because the topologies on products coincide.
$\Delta\cap a$ is closed in $(\mathrm{dom}(a)\cup\mathrm{rng}(a))^{2}$, even
with respect to the product topology, because every set is Hausdorff.
$\mathbf{E}\cap a$ is a set by regularity: If $\langle x,y\rangle\in
a\setminus\mathbf{E}$, then $x\notin y$, so $x$ and $y$ can be separated by
disjoint $U\ni x$ and $V\supseteq y$ relatively open in
$\mathrm{dom}(a)\cup\mathrm{rng}(a)$. $a\cap(U\times V)$ is a neighborhood of
$\langle x,y\rangle$ disjoint from $\mathbf{E}$
It remains to show that $B=\\{\langle
x,y\rangle{\in}b\mid{\forall}z{\in}y\;\langle x,y,z\rangle{\in}a\\}$ is closed
for every $a\in\mathbb{V}$. Since
$B\quad=\quad b\;\cap\;\\{\langle
x,y\rangle{\in}c^{2}\mid{\forall}z{\in}y\;\langle x,y,z\rangle{\in}a\cap
c^{3}\\}\text{,}$
where $c=\mathrm{dom}(b)\cup\mathrm{rng}(b)\cup\bigcup\mathrm{rng}(b)$, we can
wlog assume that $b=c^{2}$ and $a\subseteq c^{3}$, and prove that $B$ is a
closed subset of $c^{2}$. Let $\langle x,y\rangle\in c^{2}\setminus B$, that
is, let ${\exists}z{\in}y\;\langle x,y,z\rangle{\notin}a$. By (4) there exist
relatively open neighborhoods $U$, $V$ and $W$ of $x$, $y$ and $z$ in $c$,
such that $U\times V\times W$ is disjoint from $a$. But then $c\cap\lozenge W$
equals $c\setminus(\mathbb{A}\cup\square(c\setminus W))$ or
$c\setminus(\mathbb{A}\cup\\{\emptyset\\}\cup\square(c\setminus W))$,
depending on whether $\emptyset\in\mathbb{V}$, so $c\cap\lozenge W$ is
relatively open and hence $U\times\left(V\cap\lozenge W\right)$ is an open
neighborhood of $\langle x,y\rangle$ in $c^{2}$ disjoint from $B$. ∎
Together with (5), Proposition 14 thus proves:
###### Corollary 16.
$\Th{GPK}^{+}_{(\infty)}+T_{3}\;$ is equivalent to
$\;\Th{TS}_{(\infty)}+(\mathbb{A}{=}\emptyset{\in}\mathbb{V})+\text{Union}+T_{3}$.
## 6 Uniformization
Choice principles in the presence of a universal set are problematic. By
Theorem 6, for example, $\mathbb{V}\in\mathbb{V}$ implies that there is a
perfect set and in particular that not every set is well-orderable. And in
[FH96a, FH98, Ess00], M. Forti, F. Honsell and O. Esser identified plenty of
choice principles as inconsistent with positive set theory. On the other hand,
many topological arguments rely on some kind of choice. The following
_uniformization axiom_ turns out to be consistent and yet have plenty of
convenient topological implications, in particular with regard to compactness.
A _uniformization_ of a relation $R\subseteq\mathbb{V}^{2}$ is a function
$F\subseteq R$ with $\mathrm{dom}(F)=\mathrm{dom}(R)$. The _uniformization
axiom_ states that we can simultaneously choose elements from a family of
classes as long as it is indexed by a discrete set:
Uniformization If $\mathrm{dom}(R)$ is a discrete set, $R$ has a
uniformization.
Unless the relation is empty, its uniformization will be a set by the
additivity axiom. Therefore the uniformization axiom can be expressed with at
most one universal and no existential quantification over classes, and thus
still be equivalently formulated in a first-order way, using axiom schemes.
Let us denote by $\Th{ESU}$ resp. $\Th{TSU}$ essential resp. topological set
theory with uniformization.
In these theories, at least all discrete sets are well-orderable. The
following proof goes back to S. Fujii and T. Nogura ([FN99]). We call
$f:\square a\rightarrow a$ a _choice function_ if $f(b)\in b$ for every
$b\in\square a$.
###### Proposition 17 ($\Th{ESU}$).
A set $a$ is well-orderable iff it is Hausdorff and there exists a continuous
choice function $f:\square a\rightarrow a$, such that $b\setminus\\{f(b)\\}$
is closed for all $b$.
In particular, every discrete set is well-orderable and in bijection to
$\kappa^{\oplus}$ for some cardinal $\kappa$.
###### Proof.
If $a$ is well-ordered, we only have to define $F(b)=\min(b)$. In a well-
order, the minimal element is always isolated, so $b\setminus F(b)$ is in fact
closed. To show that $F$ is a set, let $c\subseteq a$ be closed. Then the
preimage of $c$ consists of all nonempty subsets of $a$ whose minimal element
is in $c$. Assume $b\notin F^{-1}[c]$, that is $F(b)\notin c$. Then
$(\square a\;\cap\;\lozenge((-\infty,F(b)]\cap
c))\;\cup\;\square[F(b)+1,\infty)$
is a closed superset of $F^{-1}[c]$ omitting $b$, where by $F(b)+1$ we denote
the successor of $F(b)$, and if $F(b)$ is the maximal element, we consider the
right part of the union to be empty. Hence $F^{-1}[c]$ is in fact closed,
proving that $F$ is continuous and a set.
For the converse, assume now that $f$ is a continuous choice function. A set
$p\subseteq\square a$ is an _approximation_ if:
* •
$a\in p$
* •
$p$ is well-ordered by reverse inclusion $\supseteq$.
* •
For every nonempty proper initial segment $Q\subset p$, we have $\bigcap Q\in
p$.
* •
For every non-maximal $b\in p$, we have $b\setminus\\{f(b)\\}\in p$.
We show that two approximations $p$ and $q$ are always initial segments of one
another, so they are well-ordered by inclusion: Let $Q$ be the initial segment
they have in common. Since both contain $b=\bigcap Q$, that intersection must
be in $Q$ and hence the maximal element of $Q$. If $b$ is not the maximum of
either $p$ or $q$, both contain $b\setminus\\{f(b)\\}$, which is a
contradiction because that is not in $Q$.
Thus the union $P$ of all approximations is well-ordered. Assume $\bigcap P$
has more than one element. Then $P\cup\\{\bigcap P,\bigcap P\setminus
f(\bigcap P)\\}$ were an approximation strictly larger than $P$. Thus $\bigcap
P$ is empty or a singleton. Since there is no infinite descending chain, and
for every bounded ascending chain $Q\subseteq P$, we have $\bigcap Q\in P$,
$P$ is closed, so $P\in\mathbb{V}$. Also, $\supseteq$ is a set-well-order on
$P$. Thus $a$ is also set-well-orderable, because $f\upharpoonright P$ is a
continuous bijection onto $a$:
Firstly, it is injective, because after the first $b$ with $f(b)=x$, $x$ is
omitted. Secondly, it is surjective, because if $b\in P$ is the first element
not containing $x$, it cannot be the intersection of its predecessors and thus
has to be of the form $b=c\setminus\\{f(c)\\}$. Hence $x=f(c)$. If $x$ is a
member of every element of $P$, then $\bigcap P=\\{x\\}\in P$ and
$x=f(\\{x\\})$.
Now let $a$ be a discrete set. We only have to prove that a continuous choice
function $f:\square a\rightarrow a$ exists. In fact, any choice function will
do, since $\square a$ is discrete and hence every function on $\square a$ is
continuous. And the existence of such a function follows from the
uniformization axiom, applied to the relation $R\subseteq\square d\times d$
defined by: $xRy$ iff $y\in x$.
It follows that every discrete set $a$ is well-orderable. Therefore, it is
comparable in length to $\mathit{On}$. If an initial segment of $a$ were in
bijection to $\mathit{On}$, then as the image of a discrete set, $\mathit{On}$
would be a set. Hence $a$ must be in bijection to a proper initial segment
$\alpha^{\oplus}$ of $\mathit{On}$. If $\kappa$ is the cardinality of
$\alpha$, there is a bijection between $\kappa^{\oplus}$ and
$\alpha^{\oplus}$. Composing these bijections proves the claim. ∎
It follows that there exists an infinite discrete set iff
$\omega\in\mathit{On}$. The uniformization axiom also allows us to define for
every infinite cardinal $\kappa$ a cardinal $2^{\kappa}$, namely the least
ordinal in bijection to $\square\kappa^{\oplus}$. Proposition 17 then shows
that, just as in $\Th{ZFC}$, $\mathit{On}$ is not only a weak but even a
strong limit.
Like the axiom of choice, the uniformization axiom could be stated in terms of
products. Of course, it only speaks of products of $\mathcal{D}$-few factors
at first, but surprisingly it even has implications for larger products as
long as the factors are indexed by a $\mathcal{D}$-compact well-ordered set.
$\mathcal{D}$-compactness for a well-ordered set just means that no subclass
of cofinality $\geq\mathit{On}$ is closed.
###### Proposition 18 ($\Th{ESU}+T_{3}+\text{Union}$).
Let $w$ be a $\mathcal{D}$-compact well-ordered set, $a\in\mathbb{V}$ and
$a_{x}\subseteq a$ nonempty for every $x\in w_{I}$. Then the product
$\prod_{x\in w_{I}}a_{x}$ is nonempty.
###### Proof.
Recall that the product is defined as:
$\prod_{x\in
w_{I}}a_{x}\quad=\quad\left\\{F\cup(w^{\prime}{\times}a)\;\mid\;F:w_{I}\rightarrow\mathbb{V},\;{\forall}x\;F(x)\in
a_{x}\right\\}$
We do induction on the length of $w$ and we have to distinguish three cases:
Case 1: If $w$ has no greatest element, its cofinality must be
$\mathcal{D}$-small or else it would not be $\mathcal{D}$-compact Hausdorff.
So let $\langle y_{\alpha}\>|\>\alpha<\kappa\rangle$ be a cofinal strictly
increasing sequence. Using the induction hypothesis and the uniformization
axiom, choose for every $\alpha<\kappa$ an element
$f_{\alpha}\quad\in\quad\prod_{x\in]y_{\alpha},y_{\alpha+1}]_{I}}a_{x}$
Then the union of the $f_{\alpha}$ is an element of $\prod_{x\in w_{I}}a_{x}$.
Case 2: Assume that $w$ has a greatest element $p$ and that
$w\setminus\\{p\\}$ is a set. Then this is still $\mathcal{D}$-compact
Hausdorff and hence the induction hypothesis applies, so there is an element
$f:w\setminus\\{p\\}\rightarrow a$ of the product missing the last dimension.
For any $y\in a_{p}$, the set $f\cup\langle p,y\rangle$ is in $\prod_{x\in
w_{I}}a_{x}$.
Case 3: Finally assume that $w$ has a greatest element $p$ and that
$w\setminus\\{p\\}$ is not a set. By the induction hypothesis,
$P_{y}\;=\;\prod_{x\in[-\infty,y]_{I}}a_{x}$
is a nonempty set for every $y<p$. The union $Q=\bigcup_{y<p}P_{y}$ is not a
set, because otherwise its domain $\mathrm{dom}\left(\bigcup
Q\right)=w\setminus\\{p\\}$ would also be a set. But since
$Q\subseteq\square(w\times a)$, it does have a closure which is a set, and
this closure must have an element $g$ with $p\in\mathrm{dom}(g)$. We will show
that $f=g\cup(w^{\prime}\times a)$ witnesses the claim, that is,
$f\in\prod_{x\in w_{I}}a_{x}$.
If $z\in w_{I}$, then $g$ is not in the closure of $\bigcup_{y<z}P_{y}$,
because that is a subclass of the set $\square((-\infty,z]\times a)$. Thus $g$
is in the closure of $\bigcup_{z\leq y<p}P_{y}$, which is a subclass of:
$M_{z}\quad=\quad\square(w\times a)\;\cap\;\\{r\mid r\cap(\\{z\\}\times
a)\in\square_{\leq 1}a_{z}\\}$
If we can show that $M_{z}$ is closed, we can deduce that $g\in M_{z}$ for
every $z\in w_{I}$ and therefore $g\upharpoonright w_{I}=f\upharpoonright
w_{I}$ is a function from $w_{I}$ to $a$ with $f(x)\in a_{x}$ for all $x\in
w_{I}$. Thus $f$ is indeed an element of the product.
To prove that $M_{z}$ is closed in $\square(w\times
a)\cap\lozenge(\\{z\\}\times a_{z})$, assume $r$ is an element of the latter
but not of the former. Then there are distinct $x_{1},x_{2}\in a_{z}$, such
that $\langle z,x_{1}\rangle,\langle z,x_{2}\rangle\in z$. Since $a_{z}$ is
Hausdorff, there are $u_{1}$ and $u_{2}$, such that $x_{1}\notin u_{1}$,
$x_{2}\notin u_{2}$ and $u_{1}\cup u_{2}=a$, and
$\square\left((w\setminus\\{z\\})\times a\;\cup\;\\{z\\}\times
u_{1}\right)\quad\cup\quad\square\left((w\setminus\\{z\\})\times
a\;\cup\;\\{z\\}\times u_{2}\right)$
is a closed superset of $M_{z}$ omitting $r$. ∎
Some of the known models of topological set theory are ultrametrizable, which
in the presence of the uniformization axiom is a very strong topological
property. A set $a$ is _ultrametrizable_ if there is a decreasing sequence
$\langle\sim_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$ of equivalence
relations on $a$ such that $\bigcap_{\alpha}\sim_{\alpha}=\Delta_{a}$ and the
$\alpha$-_ball_ s $[x]_{\alpha}=\\{y\mid x\sim_{\alpha}y\\}$ for $x\in a$ and
$\alpha\in\mathit{On}$ are a base of the natural topology on $a$ in the sense
of open classes, that is, the relatively open classes $U\subseteq a$ are
exactly the unions of balls. If that is the case, the $\alpha$-balls partition
$a$ into clopen sets for every $\alpha$.
###### Proposition 19 ($\Th{ESU}$).
Every ultrametrizable set is a $\mathcal{D}$-compact linearly orderable set.
###### Proof.
For every $\alpha\in\mathit{On}$, the class $C_{\alpha}$ of all $\alpha$-balls
is a subclass of $\square a$. If $b\in\square a$ and $x\in b$, then
$\lozenge[x]_{\alpha}$ is a neighborhood of $b$ in $\square a$ which contains
only one element of $C_{\alpha}$, namely $[x]_{\alpha}$. Hence $C_{\alpha}$
has no accumulation points and is therefore a discrete set. That means there
are only $\mathcal{D}$-few $\alpha$-balls for every $\alpha\in\mathit{On}$.
Now let $A\subseteq\square a$ and $\bigcap A=\emptyset$. For each $\alpha$,
let $B_{\alpha}$ be the union of all $\alpha$-balls which intersect every
element of $A$. Then $\bigcap_{\alpha}B_{\alpha}=\emptyset$ and every
$B_{\alpha}$ is closed.
Assume that all $B_{\alpha}$ are nonempty. Then for every $\alpha$ all but
$\mathcal{D}$-few members of the sequence $\langle
B_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$ are elements of the closed set
$\square B_{\alpha}$, so every accumulation point must be in
$\bigcap_{\alpha\in\mathit{On}}\square B_{\alpha}$, which is empty. Thus
$\\{B_{\alpha}\mid\alpha{\in}\mathit{On}\\}$ has no accumulation point and is
a discrete subset of $\square B_{0}$. Hence it is $\mathcal{D}$-small, which
means that the sequence $\langle B_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$
is eventually constant, a contradiction.
Therefore there is a $B_{\alpha}$ which is empty, and by definition every
$\alpha$-ball is disjoint from some element of $A$. Since there are only
$\mathcal{D}$-few $\alpha$-balls, the uniformization axiom allows us to choose
for every $\alpha$-ball $[x]_{\alpha}$ an element $c_{[x]_{\alpha}}\in A$
disjoint from $[x]_{\alpha}$. The set of these $c_{[x]_{\alpha}}$ is discrete
and has an empty intersection. This concludes the proof of the
$\mathcal{D}$-compactness.
Since it is discrete, the set $C_{\alpha}$ can be linearly ordered and there
are only $\mathcal{D}$-few such linear orders for every $\alpha$. If $L$ is a
linear order on $C_{\alpha}$, let $R_{L}$ be the partial order relation on $a$
defined by $xR_{L}y$ iff $[x]_{\alpha}L[y]_{\alpha}$. $R_{L}$ is a set because
it is the union of $\mathcal{D}$-few sets of the form
$[x]_{\alpha}\times[y]_{\alpha}$. Let $S_{\alpha}$ be the set of all such
$R_{L}$. The sequence $\langle S_{\alpha}\>|\>\alpha{\in}\mathit{On}\rangle$
can only be eventually constant if $a$ is discrete, in which case it is
linearly orderable anyway. If $a$ is not discrete, however,
$S=\bigcup_{\alpha}S_{\alpha}$ must be $\mathcal{D}$-large and therefore have
an accumulation point $\leq$ in $\square a^{2}$. Because each $S_{\alpha}$ is
$\mathcal{D}$-small, $\leq$ is in the closure of every
$\bigcup_{\beta>\alpha}S_{\beta}$. For $x,y\in a$, let
$t_{\alpha,x,y}\quad=\quad\square(a^{2}\setminus([y]_{\alpha}\times[x]_{\alpha}))\;\cap\;\lozenge\\{\langle
x,y\rangle\\}\text{.}$
We will show that $\leq$ is a linear order on $a$:
Assume $x\neq y$. Then there is an $\alpha$ such that $x\nsim_{\alpha}y$.
Every element of $\bigcup_{\beta>\alpha}S_{\beta}$ assigns an order to
$[x]_{\alpha}$ and $[y]_{\alpha}$, so it is in exactly one of the disjoint
closed sets $t_{\alpha,x,y}$ and $t_{\alpha,y,x}$. Therefore the same must be
true of $\leq$, so we have $x\leq y$ iff not $y\leq x$. This proves
antisymmetry and totality.
If $x\leq y\leq z$ and $x,y,z$ are distinct, then there is an $\alpha$ such
that $x\nsim_{\alpha}y\nsim_{\alpha}z\nsim_{\alpha}x$. Then $\leq$ is in the
closure of neither $t_{\alpha,y,x}$ nor $t_{\alpha,z,y}$, and must therefore
be in the closure of
$\bigcup_{\beta>\alpha}S_{\beta}\;\cap\;t_{\alpha,x,y}\;\cap\;t_{\alpha,y,z}\text{,}$
which is a subset of $t_{\alpha,x,z}$, because every element of $S$ is
transitive. It follows that $\leq$ is also in $t_{\alpha,x,z}$ and thus $x\leq
z$, proving transitivity.
Finally, $\leq$ is reflexive because for every $x\in a$, all of $S$ lies in
the set $\square a^{2}\;\cap\;\lozenge\\{\langle x,x\rangle\\}$. ∎
Another consequence of the uniformization axiom is the following law of
distributivity:
###### Lemma 20 ($\Th{ESU}$).
If $d$ is discrete and for each $i\in d$, $J_{i}$ is a nonempty class, then
$\bigcup_{i\in d}\;\bigcap_{j\in J_{i}}\;j\quad=\quad\bigcap_{f\in\prod_{i\in
d}J_{i}\>}\;\bigcup_{i\in I}\;f(i)\text{.}$
###### Proof.
If $x$ is in the set on the left, there exists an $i\in d$ such that $x$ is an
element of every $j\in J_{i}$. Thus for every function $f$ in the product,
$x\in f(i)$. Hence $x$ is an element of the right hand side.
Conversely, assume that $x$ is not in the set on the left, that is, for every
$i\in d$, there is a $j\in J_{i}$ such that $x\notin j$. Let $f$ be a
uniformization of the relation $R=\\{\langle i,j\rangle\mid i\in d,\;x\notin
j\in J_{i}\\}$. Then $x\notin\bigcup_{i\in I}f(i)$. ∎
It implies that we can work with subbases in the familiar way. Let us call
$\mathcal{K}$ _regular_ if every union of $\mathcal{K}$-few
$\mathcal{K}$-small sets is $\mathcal{K}$-small again. Then in particular
$\mathcal{D}$ is regular.
###### Lemma 21 ($\Th{ESU}$).
Let $\mathcal{K}\subseteq\mathcal{D}$ and let $B$ be a $\mathcal{K}$-subbase
of a topology $T$ such that the union of $\mathcal{K}$-few elements of $B$
always is an intersection of elements of $B$. Then $B$ is a base of $T$.
###### Proof.
We only have to prove that the intersections of elements of $B$ are closed
with respect to $\mathcal{K}$-small unions and therefore constitute a
$\mathcal{K}$-topology. But if $I$ is $\mathcal{K}$-small, and each $\langle
b_{i,j}\>|\>j\in J_{i}\rangle$ is a family in $B$, we have
$\bigcup_{i\in I}\;\bigcap_{j\in
J_{i}}\;b_{i,j}\quad=\quad\bigcap_{f\in\prod_{i\in I}J_{i}\>}\;\bigcup_{i\in
I}\;b_{i,f(i)}$
by Lemma 20, and every $\mathcal{K}$-small union $\bigcup_{i\in I}b_{i,f(i)}$
is an element of $B$ again. ∎
Thus if $\mathcal{K}$ is regular and $S$ is a $\mathcal{K}$-subbase of $T$,
the class of all $\mathcal{K}$-small unions of elements of $S$ is a base of
$T$. Since $\bigcup_{i}\lozenge a_{i}=\lozenge\bigcup_{i}a_{i}$, the sets of
the following form constitute a base of the exponential
$\mathcal{K}$-topology:
$\lozenge_{T}a\;\cup\;\bigcup_{i{\in}I}\square_{T}b_{i}\text{,}$
where $I$ is $\mathcal{K}$-small and $a,b_{i}\in T$ for all $i\in I$. As that
is sometimes more intuitive, we also use open classes in our arguments instead
of closed sets. By setting $U=\complement a$ and $V_{i}=\complement b_{i}$, we
obtain that every open class is a union of classes of the following form:
$\square_{T}U\>\cap\>\bigcap_{i\in I}\lozenge_{T}V_{i}$
That is, these constitute a base in the sense of _open_ classes. Since
$\square U=\square U\cap\lozenge U$, the class $U$ can always be assumed to be
the union of the $V_{i}$.
Lemma 21 also implies that given a class $B$, the weak comprehension principle
suffices to prove the existence of the topology $\mathcal{K}$-generated by
$B$: A set $c$ is closed iff for every $x\in\complement a$, there is a
discrete family $(b_{i})_{i\in I}$ in $B$, such that
$c\subseteq\bigcup_{i}b_{i}$ and $x\notin\bigcup_{i}b_{i}$. In particular, the
$\mathcal{K}$-topology of $\mathrm{Exp}_{\mathcal{K}}(X)$ exists (as a class)
whenever the topology of $X$ is a set.
###### Lemma 22 ($\Th{ESU}$).
Let $\mathcal{K}$ be regular and $X$ a $\mathcal{K}$-topological
$T_{0}$-space.
1. 1.
If $X$ is $T_{1}$, then $\mathrm{Exp}_{\mathcal{K}}(X)$ is $T_{1}$ (but not
necessarily conversely).
2. 2.
$X$ is $T_{3}$ iff $\mathrm{Exp}_{\mathcal{K}}(X)$ is $T_{2}$.
3. 3.
$X$ is $T_{4}$ iff $\mathrm{Exp}_{\mathcal{K}}(X)$ is $T_{3}$.
###### Proof.
In this proof we use $\square$ and $\lozenge$ with respect to $X$, not the
universe, so if $T$ is the topology of $X$, we set $\square a=\square_{T}a$
and $\lozenge a=\lozenge_{T}a$.
(1): For $a\in\mathrm{Exp}_{\mathcal{K}}(X)$, the singleton $\\{a\\}=\square
a\cap\bigcap_{x\in a}\lozenge\\{x\\}$ is closed in
$\mathrm{Exp}_{\mathcal{K}}(X)$.
(As a counterexample to the converse consider the case where
$\mathcal{K}=\kappa$ is a regular cardinal number and $X=(\kappa+1)^{\oplus}$,
with the $\kappa$-topology generated by the singletons $\\{\alpha\\}$ for
$\alpha<\kappa$. This is not $T_{1}$, because $\\{\kappa\\}$ is not closed,
but it is clearly $T_{0}$. We show that its exponential $\kappa$-topology is
$T_{1}$: Let $a\in\mathrm{Exp}_{\mathcal{K}}(X)$. Then either
$a\subseteq\kappa$ is small or $a=X$.
In the first case, $\\{a\\}=\square a\cap\bigcap_{x\in a}\lozenge\\{x\\}$ is
closed. In the second case,
$\\{a\\}=\\{X\\}=\bigcap_{x\in\kappa}\lozenge\\{x\\}$ is also closed.)
(2): ($\Rightarrow$) Let $a,b\in\mathrm{Exp}_{\mathcal{K}}(X)$ be distinct,
wlog $x\in b\setminus a$. Then there are disjoint open $U,V\subseteq X$
separating $x$ from $a$. Hence $\lozenge U$ and $\square V$ separate $b$ from
$a$.
($\Leftarrow$) Firstly, we have to show that $X$ is $T_{1}$. Assume that
$\\{y\\}$ is not closed, so there exists some other
$x\in\mathrm{cl}(\\{y\\})$, and by $T_{0}$, $y$ is not in the closure of $x$,
so $\mathrm{cl}(\\{x\\})\subset\mathrm{cl}(\\{y\\})$. The two closures can be
separated by open base classes $\square U\cap\bigcap_{i}\lozenge U_{i}$ and
$\square V\cap\bigcap_{j}\lozenge V_{j}$ of $\mathrm{Exp}_{\mathcal{K}}(X)$,
whose intersection $\square(U\cap V)\cap\bigcap_{i}\lozenge
U_{i}\cap\bigcap_{j}\lozenge V_{j}$ is emtpy. Hence there either exists a
$U_{i}$ disjoint from $V$ – which is impossible because
$\mathrm{cl}(\\{x\\})\in\square V\cap\bigcap_{i}\lozenge U_{i}$ –, or there is
a $V_{j}$ disjoint from $U$: But since
$V_{j}\cap\mathrm{cl}(\\{y\\})\neq\emptyset$, we have $y\in V_{j}$. Hence
$y\notin U\ni x$, contradicting the assumption that $x$ is in the closure of
$y$.
Now let $x\notin a$. Then $a$ and $b=\\{x\\}\cup a$ can be separated by open
base classes $\square U\cap\bigcap_{i}\lozenge U_{i}$ and $\square
V\cap\bigcap_{j}\lozenge V_{j}$ of $\mathrm{Exp}_{\mathcal{K}}(X)$, whose
intersection $\square(U\cap V)\cap\bigcap_{i}\lozenge
U_{i}\cap\bigcap_{j}\lozenge V_{j}$ is emtpy. Hence there either exists a
$U_{i}$ disjoint from $V$ – which is impossible because $a\in\square
V\cap\bigcap_{i}\lozenge U_{i}$ –, or there is a $V_{j}$ disjoint from $U$:
Then $V_{j}$ and $U$ separate $x$ from $a$, because $b$ meets $V_{j}$ and $a$
does not, so $x\in V_{j}$.
(3): In both directions, the $T_{1}$ property follows from the previous
points.
($\Rightarrow$) Let $a\notin c$, $a\subseteq X$ closed and
$c\subseteq\mathrm{Exp}_{\mathcal{K}}(X)$ closed. Wlog666To verify that a
space $X$ is $T_{3}$ it suffices to separate each point $x$ from each subbase
set $b$ not containing $x$: Firstly, the $\mathcal{K}$-small unions of subbase
sets $b$ are a base, so if $x$ is not in a $\mathcal{K}$-small union
$\bigcup_{i}b_{i}$, it can be separated with $U_{i},V_{i}$ from every $b_{i}$,
and $\bigcap U_{i},\bigcup V_{i}$ separate $x$ from the union. This shows that
$x$ can then be separated from each base set. Secondly, every closed set is an
intersection $\bigcap_{i}b_{i}$ of base sets $b_{i}$, and if $x$ is not in
that intersection, there is an $i$ with $x\notin b_{i}$ and if $U_{i},V_{i}$
separate $x$ from $b_{i}$, they also separate $x$ from $\bigcap_{i}b_{i}$. let
$c$ be of the form $\square b$ or $\lozenge b$ with closed $b\subseteq X$. In
the first case, $a\nsubseteq b$, so let $U,V$ separate some $x\in a\setminus
b$ from $b$. Then $\lozenge U,\square V$ separate $\\{a\\},c$. In the second
case, $a\cap b=\emptyset$, so let $U,V$ separate them. Then $\square
U,\lozenge V$ separate $\\{a\\},c$.
($\Leftarrow$) Now let $\mathrm{Exp}_{\mathcal{K}}(X)$ be $T_{3}$ and let
$a,b\subseteq X$ be closed, nonempty and disjoint. Then $\\{a\\}$ and
$\lozenge b$ are disjoint and can be separated by disjoint open
$U,V\subseteq\mathrm{Exp}_{\mathcal{K}}(X)$. $U$ can be assumed to be an open
base class, so $U=\square W\cap\bigcap_{i}\lozenge W_{i}$. We claim that
$\mathrm{cl}(W)\cap b=\emptyset$, which proves the normality of $X$. So assume
that there exists $x\in\mathrm{cl}(W)\cap b$. Then $a\cup\\{x\\}\in\lozenge
b$, so one of the open base classes $\square Z\cap\bigcap_{j}\lozenge Z_{j}$
constituting $V$ must contain $a\cup\\{x\\}$. That means that either one of
the $Z_{j}$ must be disjoint from $W$ – which is impossible because $x\in
Z_{j}$ – or one of the $W_{i}$ must be disjoint from $Z$ – which also cannot
be the case, because all $W_{i}$ intersect $a$ and $a\subseteq Z$. ∎
## 7 Compactness
Hyperuniverses are $\mathcal{D}$-compact Hausdorff spaces, so
$\mathcal{D}$-compactness is another natural axiom to consider. In the case
$\mathbb{V}\notin\mathbb{V}$, the corresponding statement would be that every
set is $\mathcal{D}$-compact (note that this is another axiom provable in
$\Th{ZFC}$), but if $\mathbb{V}\in\mathbb{V}$, this is equivalent to
$\mathbb{V}$ being $\mathcal{D}$-compact. And in fact, $\Th{TSU}$ with a
$\mathcal{D}$-compact Hausdorff $\mathbb{V}$ implies most of the additional
axioms we have looked at so far, including the separation properties and the
union axiom:
Let $a\subseteq\mathbb{T}$ and $x\notin\bigcup a$. Then for every $y\in a$,
there is a $b$ such that $y\subseteq\mathrm{int}(b)$ and $x\notin b$. The sets
$\square\mathrm{int}(b)$ then cover $a$ and by $\mathcal{D}$-compact
Hausdorffness, a discrete subfamily also does. But then the union of these $b$
is a superset of $\bigcup a$ not containing $x$.
Another consequence of global $\mathcal{D}$-compactness is that most naturally
occurring topologies coincide: Point (2) of the following theorem not only
applies to hyperspaces $\square a$, but also to products, order topologies and
others. If the class of atoms is closed and unions of sets are sets, this even
characterizes compactness (note that these two assumptions are only used in
(3) $\Rightarrow$ (1)):
###### Theorem 23 ($\Th{ESU}+T_{2}+\text{Union}$).
If $\mathbb{A}$ is $\mathbb{T}$-closed, the following statements are
equivalent:
1. 1.
Every set is $\mathcal{D}$-compact, that is: If $\bigcap A{=}\emptyset$, there
is a discrete $d{\subseteq}A$ with $\bigcap d{=}\emptyset$.
2. 2.
Every Hausdorff $\mathcal{D}$-topology $T\in\mathbb{V}$ equals the natural
topology: $T=\square\bigcup T$
3. 3.
For every set $a$, the exponential $\mathcal{D}$-topology on $\square a$
equals the natural topology.
###### Proof.
(1) $\Rightarrow$ (2): Let $A=\bigcup T$. Since $A$ is $T$-closed in $A$,
$A\in T$ and thus $A\in\mathbb{V}$. By definition, $T\subseteq\square A$. For
the converse, we have to verify that each $b\in\square A$ is $T$-closed, so
let $y\in A\setminus b$. Consider the class $C$ of all $u\in T$, such that
there is a $v\in T$ with $u\cup v=A$ and $y\notin v$. By the Hausdorff axiom,
for every $x\in b$ there is a $u\in C$ omitting $x$, so $b\cap\bigcap
C=\emptyset$. By $\mathcal{D}$-compactness, there is a discrete $d\subseteq C$
with $b\cap\bigcap d=\emptyset$. By definition of $C$,
$y\in\mathrm{int}_{T}(u)$ for every $u$, and since $d$ is discrete, the
intersection $\bigcap_{u\in d}\mathrm{int}_{T}(u)$ is open. Therefore, every
$y\notin b$ has a $T$-open neighborhood disjoint from $b$.
(2) $\Rightarrow$ (3) is trivial, because as a $\mathcal{D}$-compact Hausdorff
$\mathcal{D}$-topological space, $a$ is $T_{3}$ and hence $\square a$ is
Hausdorff by Lemma 22.
(3) $\Rightarrow$ (1): Lemma 22 also implies that if $\square\square a$ is
$T_{2}$, then $\square a$ is $T_{3}$ and $a$ is $T_{4}$, so it follows from
the Hausdorff axiom that every set is normal.
Finally, we can prove $\mathcal{D}$-compactness. Let $A\subseteq\square a$,
$\bigcap A=\emptyset$ and let $c=\mathrm{cl}(A)$. Then $\bigcap c=\emptyset$.
Since every set is regular and $\mathbb{A}$ is closed, the positive
specification principle holds. Therefore
$B\quad=\quad\left\\{b{\in}\square c\mid\bigcap
b\neq\emptyset\right\\}\quad=\quad\\{b{\in}\square
c\mid{\exists}x\;{\forall}y{\in}b\;x{\in}y\\}$
is a closed subset of $\square c$ not containing $c$. In particular, there is
an open base class
$\square U\;\cap\;\bigcap_{i\in I}\lozenge V_{i}$
of the space $\square c$ containing $c$ which is disjoint from $B$. Every
$U\cap V_{i}$ is a relatively open subset of $c$, so there is an $x_{i}\in
A\cap U\cap V_{i}$, because $A$ is dense in $c$. The set $\\{x_{i}\mid
i{\in}I\\}$ – and here we used the uniformization axiom – then is a discrete
subcocover of $A$. ∎
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* [FHL96] M. Forti, F. Honsell, and M. Lenisa. Axiomatic characterizations of hyperuniverses and applications. In University of Southern, pages 140–163. Society Press, 1996.
* [FN99] S. Fujii and T. Nogura. Characterizations of compact ordinal spaces via continuous selections. Topology and its Applications, 91(1):65 – 69, 1999.
* [Mal76] R. J. Malitz. Set theory in which the axiom of foundation fails. PhD thesis, UCLA, 1976.
* [Wey89] E. Weydert. How to Approximate the Naive Comprehension Scheme inside of Classical Logic. PhD thesis, Friedrich-Wilhelms-Universität Bonn, 1989.
|
arxiv-papers
| 2012-06-09T10:45:23 |
2024-09-04T02:49:31.630750
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andreas Fackler",
"submitter": "Andreas Fackler",
"url": "https://arxiv.org/abs/1206.1927"
}
|
1206.2033
|
SEMISYMMETRIC GRAPHS OF ORDER $2p^{3}$
Li Wang and Shaofei Du
School of Mathematical Sciences
Capital Normal University
Beijing, 100048, P R China
###### Abstract
A simple undirected graph is said to be semisymmetric if it is regular and
edge-transitive but not vertex-transitive. Every semisymmetric graph is a
bipartite graph with two parts of equal size. It was proved in [J. Combin.
Theory Ser. B 3(1967), 215-232] that there exist no semisymmetric graphs of
order $2p$ and $2p^{2}$, where $p$ is a prime. The classification of
semisymmetric graphs of order $2pq$ was given in [Comm. in Algebra 28(2000),
2685-2715], for any distinct primes $p$ and $q$. Our long term goal is to
determine all the semisymmetric graphs of order $2p^{3}$, for any prime $p$.
All these graphs $\Gamma$ are divided into two subclasses: (I) $\hbox{\rm
Aut}(\Gamma)$ acts unfaithfully on at least one bipart; and (II) $\hbox{\rm
Aut}(\Gamma)$ acts faithfully on both biparts. This paper gives a group
theoretical characterization for Subclass (I) and based on this
characterization, we shall give a complete classification for this subclass in
our further research.
Keywords: permutation group, vertex-transitive graph, semisymmetric graph
## 1 Introduction
All graphs considered in this paper are finite, undirected and simple. For a
graph $\Gamma$ with the vertex set ${V}$ and edge set $E$, by $\\{u,v\\}$ and
$(u,v)$ we denote an edge and arc of $\Gamma$, respectively, by $\hbox{\rm
Aut}(\Gamma)$ to denote its full automorphism group. Set $A=\hbox{\rm
Aut}(\Gamma)$. If $\Gamma$ is bipartite with the bipartition $V=W\cup U$, then
we let $A^{+}$ be a subgroup of $A$ preserving both $W$ and $U$. Clearly if
$\Gamma$ is connected, then either $|A:A^{+}|=2$ or $A=A^{+}$, depending on
whether or not there exists an automorphism which interchanges the two
biparts. For $G\leq A^{+},$ the graph $\Gamma$ is said to be
$G$-semitransitive if $G$ acts transitively on both $W$ and $U$, while an
$A^{+}$-semitransitive graph is simply said to be semitransitive.
A graph is said to be semisymmetric if it is regular and edge-transitive but
not vertex-transitive. It is easy to see that every semisymmetric graph is a
semitransitive bipartite graph with two biparts of equal size.
The first person who studied semisymmetric graphs was Folkman. In 1967 he
constructed several infinite families of such graphs and proposed eight open
problems (see [14]). Afterwards, Bouwer, Titov, Klin, I.V. Ivanov, A.A. Ivanov
and others did much work on semisymmetric graphs (see [2, 3, 17, 18, 19, 25]).
They gave new constructions of such graphs and nearly solved all of Folkman’s
open problems. In particular, by using group-theoretical methods, Iofinova and
Ivanov [17] in 1985 classified cubic semisymmetric graphs whose automorphism
group acts primitively on both biparts, which was the first classification
theorem for semisymmetric graphs. More recently, following some deep results
in group theory which depend on the classification of finite simple groups and
some methods from graph coverings, some new results of semisymmetric graphs
have appeared. For instance, in [12] the second author and Xu classified
semisymmetric graphs of order $2pq$ for two distinct primes $p$ and $q$. For
more results on semisymmetric graphs, see ([4, 8, 9, 10, 11, 12, 13, 20, 21,
22, 23, 24, 26] and so on).
In [14], Folkman proved that there are no semisymmetric graphs of order $2p$
and $2p^{2}$ where $p$ is a prime. Then we are interested in determining
semisymmetric graphs of order $2p^{3}$, where $p$ is a prime. Since the
smallest semisymmetric graphs have order $20$ (see [14]), we let $p\geq 3$. It
was proved in [22] that the Gray graph of order $54$ is the only cubic
semisymmetric graph of order $2p^{3}$. To classify all the semisymmetric
graphs of order $2p^{3}$ is still one of attractive and difficult problems.
These graphs $\Gamma$ are naturally divided into two subclasses:
1. Subclass (I): $\hbox{\rm Aut}(\Gamma)$ acts unfaithfully on at least one bipart;
2. Subclass (II): $\hbox{\rm Aut}(\Gamma)$ acts faithfully on both biparts.
The aim of this paper is to give a group theoretical characterization for
Subclass (I). Based on this characterization, we shall give a complete
classification for this subclass in our further research.
In the following two paragraphs, we first introduce two definitions used
later.
Let $\cal P$ be a partition of the vertex set $V$. Then we let $\Gamma_{\cal
P}$ be the quotient graph of $\Gamma$ relative to $\cal P$, that is, the graph
with the vertex set $\cal P$, where two subsets $V_{1}$ and ${V_{2}}$ in $\cal
P$ are adjacent if there exist two vertices $v_{1}\in V_{1}$ and $v_{2}\in
V_{2}$ such that $v_{1}$ and $v_{2}$ are adjacent in $\Gamma$. In particular,
when $\cal P$ is the set of orbits of a subgroup $N$ of $\hbox{\rm
Aut}(\Gamma)$, we denote $\Gamma_{\cal P}$ by $\Gamma_{N}$.
Let $\Sigma=({\cal V},{\cal E})$ be a connected semitransitive and edge-
transitive graph with bipartition ${\cal V}={\cal W}\cup{\cal U}$, where
$|{\cal W}|=p^{3}$ and $|{\cal U}|=p^{2}$ for an odd prime $p$. Now we define
a bipartite graph $\Gamma=(V,E)$ with bipartition $V=W\cup U$, where
$\begin{array}[]{ll}&W={\cal W},\quad U={\cal U}\times Z_{p}=\\{({\bf
u},i)\bigm{|}{\bf u}\in{\cal U},i\in Z_{p}\\},\\\ &E=\\{\\{{\bf w},({\bf
u},i)\\}\bigm{|}\\{{\bf w},{\bf u}\\}\in{\cal E},i\in Z_{p}\\}.$$\end{array}$
Then we shall call that $\Gamma$ is the graph expanded from $\Sigma$. Clearly
$\Gamma$ is edge-transitive and regular. By the definion, we see that for any
${\bf u}\in{\cal U}$, the $p$ vertices $\\{({\bf u},i)\bigm{|}i\in Z_{p}\\}$
in $U$ have the same neighborhood in $\Gamma$. Therefore, $\Gamma$ is
semisymmetric, provided there exist no two vertices in ${\cal W}$ which have
the same neighborhood in $\Sigma$.
To state our main theorem, we first define four graphs: $\Sigma(3)$,
$\Sigma(9)$, $\Gamma(9)$ and $\Gamma(18)$.
###### Example 1.1
Let $\mathbb{V}=\mathbb{V}(3,3)$ be the 3-dimensional vector space over
$\hbox{\rm GF}(3)$. Take three 2-dimensional subspaces of $\mathbb{V}:$
$\begin{array}[]{lll}\mathbb{V}_{0}&=&\\{(0,b,c)\bigm{|}b,c\in\hbox{\rm
GF}(3)\\},\,\mathbb{V}_{1}=\\{(a,0,c)\bigm{|}a,c\in\hbox{\rm GF}(3)\\},\\\
\mathbb{V}_{2}&=&\\{(a,b,0)\bigm{|}a,b\in\hbox{\rm GF}(3)\\}.\end{array}$
Let ${\cal W}=\mathbb{V}$ and let ${\cal
U}=\\{\alpha+\mathbb{V}_{i}\bigm{|}\alpha\in\mathbb{V},i\in Z_{3}\\}$, the set
of nine 2-dimensional subspaces (not all) in the 3-dimensional affine geometry
$\hbox{\rm AG}(3,3)$ over $\hbox{\rm GF}(3)$.
Define a bipartite graph $\Sigma(3)$ with biparts ${\cal W}$ and ${\cal U}$,
whose edge-set is
$\\{\\{\alpha,\alpha+\mathbb{V}_{i}\\}\bigm{|}\alpha\in\mathbb{V},i\in
Z_{3}\\}.$
Define $\Sigma(6)$ to be the bi-complement of $\Sigma(3).$
Define $\Gamma(9)$ and $\Gamma(18)$ to be the graphs expanded from $\Sigma(3)$
and $\Sigma(6)$, respectively.
###### Lemma 1.2
Both $\Gamma(9)$ and $\Gamma(18)$ are semisymmetric graphs of order 54, with
valency 9 and 18, respectively.
Proof Let $N$ be the translation group of the affine group $\hbox{\rm
AGL}(3,3)$ and let $L$ be the subgroup of $\hbox{\rm GL}(3,3)$ consisting of
all those $3\times 3$ matrices with only one nonzero entry in each row and
column. Then it is easy to verify that $N\rtimes L$ preserves the edge-set of
both graphs $\Sigma(3)$ and $\Sigma(6)$ and acts edge-transitively on them.
Clearly, $\Gamma(9)$ and $\Gamma(18)$ are edge-transitive graphs of 54, with
valency 9 and 18, respectively. Since there exist no two vertices in ${\cal
W}$ having the same neighborhood in $\Sigma$ and since for each $i$, three
vertices $\\{(\alpha+\mathbb{V}_{i},j)\bigm{|}j\in Z_{3}\\}$ have the same
neighborhood in $\Gamma$, they are not vertex-transitive and then
semisymmetric.
The main results of this paper are the following Theorem 1.3 and Theorem 1.4.
###### Theorem 1.3
For any odd prime $p$, let $\Gamma=(V,E)$ be a semisymmetric graph of order
$2p^{3}$ with the partition $V=W\cup U$ and full automorphism group
$A=\hbox{\rm Aut}(\Gamma)$. Suppose that $A$ acts unfaithfully on at least one
bipart, say $W$, with the kernel $A_{(W)}$. Let ${\cal W}=W$ and $\cal U$ the
set of orbits of $A_{(W)}$ on $U$. Set $\Sigma=\Gamma_{A_{(W)}},$ the quotient
graph of $\Gamma$ induced by $A_{(W)}$, with the partition ${\cal W}\cup{\cal
U}$. Then the following hold.
1. (1)
Every orbit of $A_{(W)}$ on $U$ has length $p$, $A_{(W)}\cong(S_{p})^{p^{2}}$
and $\Gamma$ is expanded from $\Sigma$.
2. (2)
$A/A_{(W)}$ acts faithfully on ${\cal U}$ and so on ${\cal W}\cup{\cal U},$
and $A/A_{(W)}\cong\hbox{\rm Aut}(\Sigma)$.
3. (3)
$A$ acts faithfully on $U$ and there exist no two vertices in $W$ having the
same neighborhood in $\Gamma$.
By Theorem 1.3, we know that the graph $\Gamma$ is uniquely determined by its
quotient graph $\Sigma$. Now we turn to focus on the graph $\Sigma$, see the
following theorem.
###### Theorem 1.4
Adopting the notation in Theorem 1.3 and setting $F=\hbox{\rm Aut}(\Sigma)$,
we have
1. (1)
$F$ acts imprimitively on $\cal U$.
2. (2)
Suppose that $F$ acts primitively on $\cal W$. Then $p=3$, and
$\Sigma\cong\Sigma(3)$ or $\Sigma(6)$; $\Gamma\cong\Gamma(9)$ or $\Gamma(18)$,
see Example 1.1.
3. (3)
Suppose that $F$ acts imprimitively on $\cal W$, with a block (of length $p$)
system $\mathfrak{U}$ and the kernel $F_{(\mathfrak{U})}$. Then either
1. (3.1)
$F_{(\mathfrak{U})}$ is solvable and acts transitively on $\cal W$ and $F$ is
an affine group; or
2. (3.2)
$F_{(\mathfrak{U})}$ induces blocks of length $p^{2}$ on $\cal W$. Take any
$\bf w$ in $\cal W$. Then either
1. (3.2.1)
$\bf w$ is exactly adjacent to two blocks in $\mathfrak{U}$; or
2. (3.2.2)
$\bf w$ is adjacent to at least three blocks in $\mathfrak{U}$, $p\geq 5$, $F$
contains the nonabelian normal $p$-subgroup acting regularly on $\cal W$,
$F_{(\mathfrak{U})}$ is solvable, and $F/F_{(\mathfrak{U})}\cong Z_{p}\rtimes
Z_{r}$, where $r\bigm{|}(p-1)$ and $r\geq 3$.
###### Remark 1.5
To classify all the graphs $\Gamma$ in Subclass (I), it suffices to determine
the graphs $\Sigma$ in Theorem 1.4.(3.1), (3.2.1) and (3.2.2). However, the
determination of these graphs is still quite complicated. In this paper, we
just construct the respective examples, see Section 5, and by using the group
structures obtained in Theorem 1.4, we shall give a complete classification
for them in our further research.
After this introductory section, some preliminary results will be given in
Section 2; Theorem 1.3 and Theorem 1.4 will be proved in Section 3 and 4,
respectively. Finally, the related graphs will be constructed in Section 5.
## 2 Preliminaries
First we introduce some notation: by $K_{n}$ and $K_{m,n}$ we denote the
complete graph of order $n$ and the complete bipartite graph with two biparts
of size $m$ and $n$, respectively. For a graph $\Gamma$, by $d(v)$ we denote
the degree of a vertex $v\in V$. For a prime $p$, by $p^{i}\bigm{|}\bigm{|}n$
we mean $p^{i}\bigm{|}n$ but $p^{i+1}\nmid n$. By $Z_{n}$, $D_{2n}$ and
$S_{n}$, we denote the cyclic group of order $n$, the dihedral group of order
$2n$ and the symmetric group of degree $n$, respectively. By $\hbox{\rm
GF}(p)$, we denote the field of $p$ elements. For a ring $S,$ let $S^{*}$ be
the multiplicative group of all the units in $S.$ For a transitive group $G$
on $\Omega$ and a subset $\Omega_{1}$ of $\Omega$, by $G_{\Omega_{1}}$ and
$G_{(\Omega_{1})}$ we denote the setwise stabilizer and pointwise stabilizer
of $G$ relative to $\Omega_{1}$, respectively. A $m$-block of $G$ means a
block with length $m$.
For a group $G$ and a subgroup $H$ of $G$, use $Z(G),$ $C_{G}(H)$ and
$N_{G}(H)$ to denote the center of $G$, the centralizer and normalizer of $H$
in $G,$ respectively. A semidirect product of the group $N$ by the group $H$
is denoted by $N\rtimes H,$ where $N$ is normal. A wreath product of $N$ by
$H$ is denoted by $N\wr H$, that is $N^{n}\rtimes H$, where $H\leq S_{n}$. By
$[G:H]$ we denote the set of right cosets of $H$ in $G$. The action of $G$ on
$[G:H]$ is always assumed to be the right multiplication action.
For any $\alpha$ in the $n$-dimensional vector space
$\mathbb{V=}\mathbb{V}(n,p)$ over $\hbox{\rm GF}(P)$, we denote by
$t_{\alpha}$ the translation corresponding to $\alpha$ in the affine geometry
$\hbox{\rm AG}(\mathbb{V})$ and by $T$ the translation subgroup of the affine
group $\hbox{\rm AGL}(n,p)$. Then $\hbox{\rm AGL}(n,p)\cong T\rtimes\hbox{\rm
GL}(n,p).$ We adopt matrix notation for $\hbox{\rm GL}(n,p)$ and so we have
$g^{-1}t_{\alpha}g=(t_{\alpha})^{g}=t_{\alpha g}$ for any $t_{\alpha}\in
T\leq\hbox{\rm AGL}(n,p)$ and $g\in\hbox{\rm GL}(n,p).$
For group-theoretic concepts and notation not defined here the reader is
refereed to [5, 16].
To constructed graphs, we need to introduce the definition of bi-coset graphs
and two properties.
###### Definition 2.1
[12] Let $G$ be a group, $L$ and $R$ subgroups of $G$ and let $D=RdL$ be a
double coset of $R$ and $L$ in $G$. Let $[G:L]$ and $[G:R]$ denote the set of
right cosets of $G$ relative to $L$ and $R$ respectively. Define a bipartite
graph $\Gamma={\bf B}(G,L,R;D)$ with bipartition $V=[G:L]\cup[G:R]$ and edge
set $E=\\{\\{Lg,Rdg\\}\bigm{|}g\in G,d\in D\\}$. This graph is called the bi-
coset graph of $G$ with respect to $L$, $R$ and $D$.
###### Proposition 2.2
[12] The graph $\Gamma={\bf B}(G,L,R;D)$ is a well-defined bipartite graph.
Under the right multiplication action on $V$ of $G$, the graph $\Gamma$ is
$G$-semitransitive. The kernel of the action of $G$ on $V$ is $\hbox{\rm
Core}_{G}(L)\cap\hbox{\rm Core}_{G}(R)$, the intersection of the cores of the
subgroups $L$ and $R$ in $G$. Furthermore, we have
1. (i)
$\Gamma$ is $G$-edge-transitive;
2. (ii)
the degree of any vertex in $[G:L]$ (resp. $[G:R])$ is equal to the number of
right cosets of $R$ (resp. $L$) in $D$ (resp. $D^{-1})$, so $\Gamma$ is
regular if and only if $|L|=|R|;$
3. (iii)
$\Gamma$ is connected if and only if $G$ is generated by elements of
$D^{-1}D$.
###### Proposition 2.3
[12] Suppose $\Gamma^{\prime}$ is a $G$-semitransitive and edge-transitive
graph with bipartition $V=U$ $\cup\,W$. Take $u\in U$ and $w\in W$. Set
$D=\\{g\in G\bigm{|}w^{g}\in\Gamma^{\prime}_{1}(u)\\}.$ Then $D=G_{w}gG_{u}$
and $\Gamma^{\prime}\cong{\bf B}(G,G_{u},G_{w};D).$
Finally, several group theoretical results are given.
###### Proposition 2.4
[15] Let T be a nonabelian simple group with a subgroup $H<T$ satisfying
$|T:H|=p^{a},$ for $p$ a prime. Then one of the following holds:
1. (i)
$T=A_{n}$ and $H=A_{n-1}$ with $n=p^{a};$
2. (ii)
$T=\hbox{\rm PSL}(n,q)$, $H$ is the stabilizer of a projective point or a
hyperplane in $\hbox{\rm PG}(n-1,q)$ and $|T:H|=(q^{n}-1)/(q-1)=p^{a};$
3. (iii)
$T=\hbox{\rm PSL}(2,11)$ and $H=A_{5};$
4. (iv)
$T=M_{11}$ and $H=M_{10};$
5. (v)
$T=M_{23}$ and $H=M_{22};$
6. (vi)
$T=\hbox{\rm PSU}(4,2)$ and $H$ is a subgroup of index 27.
###### Proposition 2.5
[1] For an odd prime $p$, let $\overline{G}=\hbox{\rm PSL}(3,p)$ and
$\overline{H}$ a proper subgroup of $\overline{G}$. Then one of the following
holds:
1. (I)
If $\overline{H}$ has no nontrivial normal elementary abelian subgroup, then
$\overline{H}$ is conjugate in $\hbox{\rm GL}(3,p)/Z(\hbox{\rm SL}(3,p))$ to
one of the following groups:
1. (i)
$\hbox{\rm PSL}(2,7)$, with $p^{3}\equiv 1(\hbox{\rm mod }7);$
2. (ii)
$A_{6}$, with $p\equiv 1,19(\hbox{\rm mod }30);$
3. (iii)
$\hbox{\rm PSL}(2,5)$, with $p\equiv\pm 1(\hbox{\rm mod }10);$
4. (iv)
$\hbox{\rm PSL}(2,p)$ or $\hbox{\rm PGL}(2,p)$ for $p\geq 5$.
2. (II)
If $\overline{H}$ has a nontrivial normal elementary abelian subgroup, then
$\overline{H}$ is conjugate to a subgroup of one of the following subgroups:
1. (i)
$Z_{(p^{2}+p+1)/(3,p-1)}\rtimes Z_{3};$
2. (ii)
the subgroup $\overline{F}$ of all matrices with only one nonzero entry in
each row and column, and $\overline{F}$ contains the subgroup $\overline{D}$
of all diagonal matrices as a normal subgroup such that
$\overline{F}/\overline{D}\cong S_{3};$
3. (iii)
the point- or line-stabilizer of a given point $\langle(1,0,0)^{T}\rangle$ or
the line $\langle(0,$ $\alpha,\beta)^{T}\bigm{|}\alpha,\beta\in F_{p}\rangle$;
4. (iv)
the group $\overline{M}$ such that $\overline{M}$ contains a normal subgroup
$\overline{N}\cong Z_{3}^{2}$ and $\overline{M}/\overline{N}$ is isomorphic to
$\hbox{\rm SL}(2,3)$ if $p\equiv 1(\hbox{\rm mod }9)$ or to $Q_{8}$ if
$p\equiv 4,7(\hbox{\rm mod }9)$.
###### Proposition 2.6
[6] For an odd prime $p$, let $H$ be a maximal subgroup of $G=\hbox{\rm
GL}(2,p)$ and $H\neq\hbox{\rm SL}(2,p)$. Then up to conjugacy, $H$ is
isomorphic to one of the following subgroups:
1. (i)
$D\rtimes\langle b\rangle;$ where $D$ is the subgroup of diagonal matrices and
$b={\footnotesize\left(\begin{array}[]{ll}0&1\\\ 1&0\end{array}\right)};$
2. (ii)
$\langle a\rangle\rtimes\langle b\rangle$, where
$b={\footnotesize\left(\begin{array}[]{ll}1&0\\\ 0&-1\end{array}\right)}$ and
$\langle a\rangle$ is the Singer subgroup of $G$, defined by
$a$=$\left(\begin{array}[]{ll}\gamma&\delta\theta\\\
\delta&\gamma\end{array}\right)$$\in G,$ where
$F_{p}^{*}=\langle\theta\rangle$, $F_{p^{2}}=F_{p}({\bf t})$ for ${\bf
t}^{2}=\theta,$ and $F_{p^{2}}^{*}=\langle\gamma+\delta{\bf t}\rangle;$
3. (iii)
$\langle a\rangle\rtimes D$, where
$a={\footnotesize\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)};$
4. (iv)
$H/\langle z\rangle$ is isomorphic to $A_{4}\times Z_{\frac{p-1}{2}}$, for
$p\equiv 5(\hbox{\rm mod }8);$ $S_{4}\times Z_{\frac{p-1}{2}}$ for $p\equiv
1,3,7(\hbox{\rm mod }8);$ or $A_{5}\times Z_{\frac{p-1}{2}}$ for $p\equiv\pm
1(\hbox{\rm mod }10),$ where $z={\footnotesize\left(\begin{array}[]{ll}-1&0\\\
0&-1\end{array}\right)}$, $Z_{\frac{p-1}{2}}=Z(G)/\langle z\rangle;$
5. (v)
$H/\langle z\rangle=A_{4}\rtimes\langle s\rangle$, $\langle s^{2}\rangle\leq
Z(G)/\langle z\rangle$, if $p\equiv 1(\hbox{\rm mod }4).$
The following theorem can be extracted from [7].
###### Proposition 2.7
Let $G$ be a transitive permutation group of degree $p^{2}$, where $p\geq 5$ a
prime and let $P$ be a Sylow $p$-subgroup of $G$. Suppose that $G$ is
imprimitive and $|P|=p^{3}$. Then $P\lhd G$.
## 3 Proof of Theorem 1.3
From now on, we assume that $\Gamma$ is a semisymmetric graph of order
$2p^{3}$ with the bipartition $V=W\cup U$, where $p$ is a prime, and
$A=\hbox{\rm Aut}(\Gamma)$ acts unfaithfully on at least one part, say $W$.
For avoiding confusions, we need to emphasis the following notation:
$u:$ a vertex in $U$; ${\bf u}$: a block induced by $A_{(W)}$ on $U$;
${\cal U}$: the set of such blocks ${\bf u}$;
$\mathfrak{u}$: a block contained in ${\cal U}$; $\mathfrak{U}$: the set of
all such blocks $\mathfrak{u}$.
Symmetrically, for other bipart $W$, we let $w,$ ${\bf w},$ ${\cal W},$
$\mathfrak{w}$ and $\mathfrak{W}$ have the same meaning. Moreover, when
emphasizing on the set $\cal U$, we prefer to call ${\bf u}$ a vertex in
$\cal U$ but not a block in $U$.
Proof of Theorem 1.3: Now $A_{(W)}$ induces a complete $m$-block system
${\cal U}=\\{{\bf u_{0}},{\bf u_{1}},\cdots,{\bf u_{\frac{p^{3}}{m}-1}}\\},$
on $U,$ where $m\bigm{|}p^{3}$. Let $A_{(\cal U)}$ be the kernel of $A$ on
${\cal U}$. Then we divide the proof into the following six steps.
Step 1: Show that $p\geq 3$, $\Gamma\not\cong K_{p^{3},p^{3}}$ and $\Gamma$ is
connected.
By [14], there exists no semisymmetric graph of order less than 20. Hence
$p\geq 3$. Since the complete bipartite graph $K_{p^{3},p^{3}}$ is a symmetric
graph, $\Gamma\not\cong K_{p^{3},p^{3}}$.
Suppose that $\Gamma$ is disconnected. From the edge-transitivity of $\Gamma$,
we get that $\Gamma$ is isomorphic to either $p^{3}K_{2}$ or
$\frac{p^{3}}{m}\Phi_{2m}$ where $\Phi_{2m}$ (for $m\in\\{p,p^{2}\\})$ is a
regular edge-transitive bipartite graph of order $2m$. However, by [14], there
exist no semisymmetric graphs of order $p$ and $2p$, that is, $\Phi_{2m}$ is
vertex-transitive, which implies $\Gamma$ is vertex-transitive, a
contradiction. Therefore, $\Gamma$ is connected.
Step 2: Show that $m\neq 1,p^{3}$.
Since $A$ acts unfaithfully on $W$, we get $m\neq 1.$ Suppose that $m=p^{3}$.
Take $w\in W$. Since $A_{(W)}$ fixes $w$ and acts transitively on $U$, it
follows that $w$ is adjacent to all the vertices in $U$, which implies
$\Gamma\cong K_{p^{3},p^{3}},$ a contradiction.
Step 3: Show that $A_{(\cal U)}$ acts intransitively on $W$. Suppose that
$A_{(\cal U)}$ acts transitively on $W$. Then we shall show that for any $w\in
W$, we have $\Gamma_{1}(w)=U$, which implies $\Gamma\cong K_{p^{3},p^{3}}$, a
contradiction.
For any block ${\bf u}_{j}$ in ${\cal U}$, take an edge $\\{w_{1},u_{j}\\}$
where $w_{1}\in W$ and $u_{j}\in{\bf u}_{j}$. Since $A_{(\cal U)}$ fixes ${\bf
u}_{j}$ setwise and acts transitively on $W$, there exists a $g\in A_{(\cal
U)}$ sending $\\{w_{1},u_{j}\\}$ to $\\{w,u_{j}^{g}\\}$, which means that $w$
is adjacent to a vertex $u_{j}^{g}$ in ${\bf u}_{j}$. Moreover, since
$\\{w,u_{j}^{g}\\}\in E$ and since $A_{(W)}$ fixes $w$ and acts transitively
on ${\bf u}_{j}$, it follows that $w$ is adjacent to all the vertices in ${\bf
u}_{j}$. Therefore, $\Gamma_{1}(w)=U$.
Step 4: Show Theorem 1.3.(1). For the contrary, suppose $m=p^{2}.$ Then
$|{\cal U}|=p$ and $A/A_{(\cal U)}\lessapprox S_{p}$. Moreover, since
$A/A_{(W)}$ acts transitively on $W$, we get $p^{3}\bigm{|}|A/A_{(W)}|.$
Therefore, $A_{(W)}\lneqq A_{(\cal U)}$. Now $A_{(\cal U)}$ induces a complete
$n$-block system ${\cal W}=\\{{\bf w}_{1},\cdots,{\bf w}_{p^{3}/n}\\}$ on $W.$
Since $A/A_{(\cal U)}$ is transitive on $\cal W$ and $A/A_{(\cal U)}\leq
S_{p}$ , we get that $|{\cal W}|=p$. Then $n=p^{2}.$
Consider the quotient graph $\Gamma_{A_{(\cal U)}}$ induced by $A_{(\cal U)}$
with the bipartition ${\cal U}\bigcup{\cal W}.$ Take the edge $\\{{\bf
u}_{i},{\bf w}_{j}\\}$ in $\Gamma_{A_{(\cal U)}}$. A same argument as in the
proof of Step $1$ shows that the induced subgraph $\Gamma({\bf
u}_{i}\bigcup{\bf w}_{j})\cong K_{p^{2},p^{2}}.$ Therefore, for any ${\bf
u}_{i_{1}}\in{\cal U}$ and ${\bf w}_{j_{1}}\in{\cal W}$, the induced subgraph
$\Gamma({\bf u}_{i_{1}}\bigcup{\bf w}_{j_{1}})$ is either an empty graph or a
complete bipartite graph, which implies $\Gamma\cong\Gamma_{A_{(\cal
U)}}[p^{2}K_{1}].$ Since the graph $\Gamma_{A_{(\cal U)}}$ of order $2p$ is
symmetric by [14], we get $\Gamma$ is vertex-transitive, a contradiction
again.
Since $A_{(W)}$ fixes $W$ pointwise and acts transitively on each ${\bf
u}_{i}$ in ${\cal U}$, it follows that $p$ vertices in each ${\bf u}_{i}$ have
the same neighborhood in $\Gamma$. Therefore, $\Gamma$ is expanded from
$\Sigma$. Moreover, $A_{(W)}\cong S_{p}^{p^{2}}.$
Step 5: Show Theorem 1.3.(2).
From Step 4, we get $m=p$ and then $|{\cal U}|=p^{2}.$ Assume the contrary,
that is, $A/A_{(W)}$ acts unfaithfully on $\cal U$. Then $A_{(W)}\lneqq
A_{(\cal U)}$. As before, let ${\cal W}=\\{{\bf w}_{1},\cdots,{\bf
w}_{p^{3}/n}\\}$ be a complete $n-$block system of $A_{(\cal U)}$ on $W.$
If $n=p,$ then $|{\cal W}|=p^{2}$ and as in Step 4 again, one may easily see
$\Gamma\cong\Gamma_{A_{(\cal U)}}[pK_{1}],$ a contradiction.
Suppose that $n=p^{2}.$ Then $|{\cal W}|=p.$ Then $A/A_{(\cal W)}\lessapprox
S_{p}$ and $A_{(\cal U)}\leq A_{(\cal W)}$. Moreover, since $A/A_{(\cal U)}$
acts transitively on ${\cal U}$, it follows that $p^{2}\bigm{|}|A/A_{(\cal
U)}|.$ Then $A_{(\cal U)}\lneqq A_{(\cal W)}$. Naturally, we consider two
cases:
(i) $A_{(\cal W)}$ is transitive on ${\cal U}$.
On the one hand, since $A_{(\cal W)}$ fixes $\cal W$ pointwise and acts
transitively on $\cal U$ and since $|{\cal W}|=p\neq|{\cal U}|=p^{2}$, the
quotient graph of $\Gamma$ with partition ${\cal W}\cup{\cal U}$ is isomorphic
to $K_{p,p^{2}}$. On the other hand, for any block ${\bf u}_{i}\in\cal U$ and
${\bf w}_{j}\in\cal W$, by considering the actions of $A_{(W)}$ and $A_{(\cal
U)}$ we know that the induced subgraph $\Gamma({\bf u}_{i}\cup{\bf w}_{j})$ is
complete bipartite. Therefore, $\Gamma\cong K_{p^{3},p^{3}}$ a contradiction.
(ii) $A_{(\cal W)}$ has the blocks of length $p$ on ${\cal U}$.
Suppose that $A_{(\cal W)}$ has blocks of length $p$ on ${\cal U}$. Then
$A_{(\cal W)}$ has blocks of length $p^{2}$ on $U$. Then the quotient graph
$\Gamma_{A_{(\cal W)}}$ induced by $A_{(\cal W)}$ is an edge-transitive graph
of order $2p$ and then it is symmetric by [14] again.
Similarly, by considering the actions of $A_{(W)},$ $A_{(\cal U)}$ and
$A_{(\cal W)}$, we may show that the induced subgraph $\Gamma({\bf
u}_{i}\cup{\bf w}_{j})$ is either complete bipartite or empty. Therefore, the
graph $\Gamma$ is vertex-transitive, a contradiction.
This proves that $A/A_{(W)}$ acts faithfully on ${\cal U}.$
Finally we show that $\hbox{\rm Aut}(\Sigma)\cong A/A_{(W)}$. Since
$A/A_{(W)}$ acts faithfully on $\cal U$, it induces a faithful and edge-
transitive action on $\Sigma$, that is $A/A_{(W)}\lesssim\hbox{\rm
Aut}(\Sigma).$ Clearly, the graph $\Gamma$ is uniquely determined by its the
graph $\Sigma$. Then one may see that every automorphism of $\Sigma$ can be
extended to an automorphism of $\Gamma$ which preserves ${\cal W}$, that means
$|\hbox{\rm Aut}(\Sigma)|\leq|A/A_{(W)}|$. Therefore, $\hbox{\rm
Aut}(\Sigma)\cong A/A_{(W)}$.
Step 6: Show Theorem 1.3.(3).
Since $A/A_{(W)}$ acts faithfully on ${\cal U}$ by Step 5, it follows that
$A_{(\cal U)}=A_{(W)}$ and so $A_{(U)}\leq A_{(W)}$. Since $A_{(U)}\cap
A_{(W)}=1,$ we get $A_{(U)}=1$, equivalently, $A$ acts faithfully on $U$.
Suppose that there exist two vertices ${\bf w_{1}}$ and ${\bf w_{2}}$ in
${\cal W}$ having the same neighborhood in $\Sigma$. Then the permutation
$\tau$ exchanging ${\bf w_{1}}$ and ${\bf w_{2}}$ and fixing other vertices of
$\Sigma$ is clearly an automorphism of $\Sigma$, which forces that $A/A_{(W)}$
acts unfaithfully on ${\cal U}$, a contradiction. Therefore, there exist no
two vertices ${\bf w_{1}}$ and ${\bf w_{2}}$ in ${\cal W}$ having the same
neighborhood in $\Sigma$.
## 4 Proof of Theorem 1.4
By Theorem 1.3, from now on we focus on the quotient graph $\Sigma$ induced by
$A_{(W)}$ with biparts ${\cal W}\cup{\cal U}$, where $|{\cal W}|=p^{3}$ and
$|{\cal U}|=p^{2}$. Since ${\bf w}=\\{w\\}$ for some $w\in W$, we shall
identify ${\bf w}$ with $w$, and ${\cal W}$ with $W$ as well. Moreover,
$\Sigma$ is edge-transitive and there exist no two vertices in ${\cal W}$
having the same neighborhood in $\Sigma$.
To prove Theorem 1.4, we shall prove that $F=\hbox{\rm Aut}(\Sigma)$ acts
imprimitively on ${\cal U}$ in Subsection 4.1, that is Theorem 1.4.(1); and
deal with the cases when $\hbox{\rm Aut}(\Sigma)$ acts primitively on ${\cal
W}$ in Subsection 4.2, that is Theorem 1.4.(2), and imprimitively on ${\cal
W}$ in Subsection 4.3, that is Theorem 1.4.(3), respectively.
### 4.1 Proof of Theorem 1.4.(1)
First we prove a group theoretical result.
###### Lemma 4.1
For an odd prime $p$, let $G$ be a primitive group on $\Omega$, where
$|\Omega|=p^{2}$. Suppose that $G$ has a faithful transitive representation of
degree $p^{3}$. Then $G$ is isomorphic to one of the following groups:
1. (1)
${\rm P}{\rm\Gamma}{\rm L}(2,8)$, for $p=3$;
2. (2)
$Z_{3}^{2}\rtimes H$, where $H=\hbox{\rm SL}(2,3)$ or $\hbox{\rm GL}(2,3)$,
for $p=3$;
3. (3)
$Z_{5}^{2}\rtimes H$, where $H=\hbox{\rm SL}(2,5)$ or $\hbox{\rm GL}(2,5)$,
for $p=5$;
4. (4)
$Z_{7}^{2}\rtimes\hbox{\rm SL}(2,7)$, for $p=7$;
5. (5)
$Z_{11}^{2}\rtimes\hbox{\rm SL}(2,11)$, for $p=11$.
All these representations are imprimitive.
Proof By the well-known O’Nan-Scott Theorem [5], every primitive group $G$ of
degree $p^{2}$ is almost simple type, product type or affine type. Let
$T=\hbox{\rm soc}(G).$ Suppose $G$ has a faithful transitive representation on
$\Omega^{\prime}$, where $|\Omega^{\prime}|=p^{3}$. Then we divided the proof
into the following three cases.
Case 1: $G$ is almost simple type.
In this case, $T=\hbox{\rm soc}(G)$ is either $A_{p^{2}}$ or $\hbox{\rm
PSL}(n,q)$, where $\frac{q^{n}-1}{q-1}=p^{2}$, by checking Proposition 2.4.
First suppose that $G$ is primitive on $\Omega^{\prime}$. Then by checking
Proposition 2.4 again, the almost simple groups of degree $p^{3}$ are:
$A_{p^{3}}$, $\hbox{\rm PSU}(4,2)$ or $\hbox{\rm PSL}(n,q),$ where
$\frac{q^{n}-1}{q-1}=p^{3}$. Clearly, our group $G$ now cannot have any
faithful primitive representation of degree $p^{3}$.
In what follows, suppose that $G$ acts imprimitively on $\Omega^{\prime}$. Let
${\cal B}$ be an imprimitive complete $m-$block system. Then $T$ acts
transitively on ${\cal B}$ with the kernel $K.$ Since $T$ is the unique
minimal normal subgroup of $G$, it follows that either $T\leq K$ or $K=1.$ In
other words, if $T$ acts transitively on $\Omega^{\prime}$, then $K=1;$ if $T$
is intransitive on $\Omega^{\prime}$, then $T\leq K$ and $p||G:T|$.
(i) Firstly, suppose that $T=A_{p^{2}}.$ Since $|G:T|\leq 2,$ we can get that
$T$ is impritimitive and transitive on $\Omega^{\prime}.$ In this case, $K=1.$
Then $|{\cal B}|=p^{2}$ and $m=p.$ Take a block ${\bf b}$ in ${\cal B}.$ Then
$T_{\bf b}=A_{p^{2}-1},$ which should be transitive on $\bf b$. However,
$A_{p^{2}-1}$ has no subgroup of index $p$, a contradiction.
(ii) Secondly, suppose that $T=\hbox{\rm PSL}(n,q)$, where
$\frac{q^{n}-1}{q-1}=p^{2}$. Then $T\leq G\leq{\rm P}\Gamma{\rm L}(n,q)$,
where $q=p_{1}^{k}$ for a prime $p_{1}.$ It is known that $|{\rm P}\Gamma{\rm
L}(n,q):T|\bigm{|}(q-1)k$. If $n=2$, then $p^{2}-1=q=p_{1}^{k}$. From this
equation, we can get $p=3$ and $q=8,$ that is $T=\hbox{\rm PSL}(2,8)$ and
${\rm P}{\rm\Gamma}{\rm L}(2,8)=T\rtimes\langle f\rangle$, where $f$ is the
field automorphism of order 3 of $T$. This is a case in (1) of the lemma.
Suppose that $n\geq 3$. Then $p^{2}=\frac{q^{n}-1}{q-1}\geq q^{2}+q+1$ and so
we get $p>q=p_{1}^{k}$, which implies $p\nmid(q-1)$ and $p\nmid k$, and then
$p\nmid|G:T|$, that is, $T\nleq K$ and thus $T$ acts transitively on
$\Omega^{\prime}.$ However, in what follows we shall show that
$p^{3}\nmid|T|$.
Since
$|\hbox{\rm
PSL}(n,q)|=\frac{(q^{n}-1)(q^{n}-q)\cdots(q^{n}-q^{n-1})}{(q-1)(n,q-1)}=p^{2}\frac{(q^{n}-q)\cdots(q^{n}-q^{n-1})}{(n,q-1)}$
and since $p\nmid q$ and $p\nmid(q-1),$ it suffices to show
$p\nmid(q^{l}+q^{l-1}+\cdots+1)$ for any $1\leq l<n-1.$
Suppose that $k$ is the minimal positive integer such that
$p\mid(q^{k}+q^{k-1}+\cdots+1).$ Write
$\begin{array}[]{lll}&&p^{2}=q^{n-1}+q^{n-2}+\cdots+1\\\
&=&(1+q+q^{2}+\cdots+q^{k})+q^{k+1}(1+q+q^{2}+\cdots+q^{k})\\\
&&+\cdots+q^{n-i}(1+q+q^{2}+\cdots+q^{i}).\end{array}$
Then it follows $p\bigm{|}(1+q+q^{2}+\cdots+q^{i})$. From the minimality of
$k$, we get $i=k$ so that
$p^{2}=(1+q+q^{2}+\cdots+q^{k})(1+q^{k+1}+q^{2(k+1)}+\cdots+q^{n-k}),$
and so
$1+q+q^{2}+\cdots+q^{k}=1+q^{k+1}+q^{2(k+1)}+\cdots+q^{n-k},$
that is
$q(1+q+\cdots+q^{k-1}-q^{k}-q^{(2k-1)}-\cdots-q^{n-k-1})=1,$
a contradiction.
Case 2: $G$ is product type. In this case, $G=(M\times M)\rtimes Z_{2},$ where
$M$ is an irregular primitive group of degree $p.$ Clearly, $p^{3}\nmid|G|,$ a
contradiction.
Case 3: $G$ is affine type. Now $G=N\rtimes H,$ where $N\cong Z_{p}^{2}$ and
$H$ is an irreducible subgroup of $\hbox{\rm GL}(2,p).$ Clearly, $G$ acts
imprimitively on $\Omega^{\prime}.$ Since $|\hbox{\rm
GL}(2,p)|=p(p-1)^{2}(p+1),$ we know that $p^{3}\bigm{|}\bigm{|}|G|$ and then
$N$ induces a $p^{2}-$block system, say ${\cal B}$, on $\Omega^{\prime}$. Take
${\bf b}\in{\cal B}$ and $b\in{\bf b}$. Considering the action of $H$ on
${\cal B}$, we know that $|H:H_{\bf b}|=p$ and then $H_{\bf b}=H_{b}$, that
is, $H$ has a subgroup of index $p$. Checking Proposition 2.6, we get that
$H=\hbox{\rm SL}(2,p)$ for $p=3,5$, 7 and 11; or $H=\hbox{\rm GL}(2,p)$ for
$p=3,5$. This completes the proof of the lemma.
Proof of Theorem 1.4.(1): For the contrary, suppose that $F$ acts primitively
on ${\cal U}$. Then $F$ has a faithful primitive representation of degree
$p^{2}$. Since $|{\cal W}|=p^{3}$ , $F$ has a faithful transitive
representation of degree $p^{3}$ and so $p^{3}\bigm{|}|F|$. Then $F$ is one of
the groups in Lemma 4.1 and we divide the proof into two cases according to
$F={\rm P}{\rm\Gamma}{\rm L}(2,8)$ or $F$ is an affine group.
(i) $F={\rm P}{\rm\Gamma}{\rm L}(2,8)$
Let $F=T\rtimes\langle f\rangle$ and let $H\cong Z_{2}^{3}\rtimes Z_{7}$ be a
point stabilizer of $T$ on the projective line. Then $F_{\bf w}=H$ for some
${\bf w}\in{\cal W}$ and $F_{\bf u}=H\rtimes\langle f\rangle$ for some ${\bf
u}\in{\cal U}$. Since each of $F_{\bf w}$, $F_{\bf w}^{f}$ and $F_{\bf
w}^{f^{2}}$ fixes ${\bf u}$ and is transitive on other 8 vertices on ${\cal
U}$, three vertices ${\bf w}$, ${\bf w}^{f}$ and ${\bf w}^{f^{2}}$ have the
same neighborhood in the graph $\Sigma$, a contradiction (see Theorem
1.3.(3)).
(ii) $F$ is an affine group.
Now $F=N\rtimes H,$ where $N\cong Z_{p}^{2}$ and either $H=\hbox{\rm SL}(2,p)$
where $p=3,5,7,11$ or $\hbox{\rm GL}(2,p)$ where $p=3,5$. First let
$H=\hbox{\rm SL}(2,p)$. Let $Z$ be the center of $\hbox{\rm SL}(2,p)$.
Clearly, $Z\leq H_{\bf w}$. If $p=3$ then $\frac{H_{\bf w}}{Z}\cong
Z_{2}^{2};$ if $p=5$ then $\frac{H_{\bf w}}{Z}\cong A_{4}$; if $p=7$ then
$\frac{H_{\bf w}}{Z}\cong S_{4}$; and if $p=11$ then $\frac{H_{\bf w}}{Z}\cong
A_{5}$ where $H_{\bf w}\cong\hbox{\rm SL}(2,5)$. Let $P$ be a Sylow
$p$-subgroup of group $H$. Then $H=H_{\bf w}P$ where $H_{\bf w}\cap P=1$ and
$F=N\rtimes(PH_{\bf w})$. Now we may identify ${\cal U}$ with vector space
${\mathbb{V}}=\mathbb{V}(2,p)$. Let $\alpha=(1,0)$. Then
$H_{\alpha}=\\{\left(\begin{array}[]{cc}1&0\\\
c&1\end{array}\right)\bigm{|}c\in F_{p}\\}\cong Z_{p}$. Since for any $h\in
H$, we have
$((H_{\bf w})^{h})_{\alpha}=(H_{\bf w})^{h}\cap H_{\alpha}=1\,\quad{\rm
and}\,\quad|{\alpha}^{(H_{\bf w})^{h}}|=p^{2}-1=|(H_{\bf w})^{h}|.$
This implies that acting on ${\cal U}$, $(H_{\bf w})^{h}$ fixes 0 and is
transitive on $\mathbb{V}\setminus\\{0\\}$. Therefore, $p$ vertices $\\{{\bf
w}^{h}\bigm{|}h\in H\\}$ have the same neighborhood in $\Sigma$, a
contradiction.
For $H=\hbox{\rm GL}(2,p)$ where $p=3,5$, we have completely same argument as
last paragraph and get a contradiction again.
### 4.2 Proof of Theorem 1.4.(2)
The proof of Theorem 1.4.(2) consists of the following two lemmas.
###### Lemma 4.2
Suppose that $\hbox{\rm Aut}(\Sigma)$ acts primitively on ${\cal W}$. Then
$p=3$.
Proof. Again set $F=\hbox{\rm Aut}(\Sigma)$. By Theorem 1.3.(1), $F$ acts
imprimitively on ${\cal U}$. Let $\mathfrak{U}$ be a $p$-block system of $F$
on ${\cal U}$ with the kernel $F_{(\mathfrak{U})}$. Then
$F_{(\mathfrak{U})}\leq(S_{p})^{p}.$ Suppose that $F$ is primitive on ${\cal
W}$. Then $F_{(\mathfrak{U})}$ is transitive on ${\cal W}.$
Clearly, $F$ is neither a diagonal type or twisted wreath product type. So we
only need to deal with three cases separately: $F$ is almost simple type,
product type or affine type.
(i) $F$ is almost simple type.
Let $T=\hbox{\rm soc}(F).$ Then $T$ is transitive on ${\cal W}$. Since $T$ is
the unique minimal normal subgroup of $F$, it follows that $T\leq
F_{(\mathfrak{U})}$, which implies that $T$ is transitive on each block in
$\mathfrak{U}$. Thus $T$ has two faithful representations with respective
degree $p$ and $p^{3}$, which is impossible.
(ii) $F$ is product type.
In this case, $F=(M\times M\times M)\rtimes H$, where $M$ is a primitive and
irregular group of degree $p$, where $p\geq 5$. Since
$p^{3}\bigm{|}\bigm{|}|F|$ and $F_{(\mathfrak{U})}$ is transitive on ${\cal
W}$, we have $p^{3}\bigm{|}|F_{(\mathfrak{U})}|$ and so
$p\nmid|F/F_{(\mathfrak{U})}|$, a contradiction.
(iii) $F$ is affine type.
In this case, $F=N\rtimes H,$ where $N\cong Z_{p}^{3}$ and $H$ is an
irreducible subgroup of $\hbox{\rm GL}(3,p).$ Clearly, $N\leq
F_{(\mathfrak{U})}$ and thus $H$ must be transitive on $\mathfrak{U}$.
Therefore, $H$ has a subgroup $M$ of index $p$. Let $P$ be a Sylow
$p$-subgroup of $H$. Suppose that there exists an element $h$ of order $p$ in
$H\cap F_{(\mathfrak{U})}$. Since $N\langle h\rangle$ is a $p$-subgroup in
$F_{(\mathfrak{U})}$, it is abelian, and then $[h,N]=1$, a contradiction,
noting $F$ is an affine group. Therefore, $|P|=p$ and then $H=PM$. Set
$H_{1}=H\cap\hbox{\rm SL}(3,p)$. Noting $P\leq\hbox{\rm SL}(3,p)$, we get
$H_{1}=PM_{1}$, where $M_{1}=H_{1}\cap M.$ Set $Z=Z(\hbox{\rm SL}(3,p))$. Then
$Z\cong Z_{k}$ for $k=(3,p-1)$. Therefore, in $\hbox{\rm PSL}(3,p)$,
$|\overline{H_{1}}:\overline{M_{1}}|=|\overline{P}|=p$. Since
$\overline{H_{1}}$ is an irreducible subgroup which has a subgroup of index
$p$, by checking Proposition 2.5, the possible candidates are $\hbox{\rm
PSL}(2,5)$ or $\hbox{\rm PGL}(2,5)$ for $p=5$; $\hbox{\rm PSL}(2,7)$ for
$p=7$; $\hbox{\rm PSL}(2,11)$ for $p=11$; and $Z_{13}\rtimes Z_{3}$, $A_{4}$
or $S_{4}$ for $p=3$. Moreover, if $p=3,5,11$, then
$H_{1}\cong\overline{H_{1}}$; if $p=7$, the $H_{1}\cong\overline{H_{1}}$ or
$H_{1}\cong\overline{H_{1}}\times Z_{3}.$ In what follows, we shall show
$p\neq 5,7,11$ and then $p=3$, the lemma is proved.
For the contrary, suppose that $p\in\\{5,7,11\\}.$ Since $H=PM$, we get that
$H_{\mathfrak{u}}=M$ and $F_{\mathfrak{u}}=NM$, for some
$\mathfrak{u}\in\mathfrak{U}.$ Then
$\overline{F_{\mathfrak{u}}}=F_{\mathfrak{u}}/F_{(\mathfrak{u})}=\overline{N}\overline{M}\leq
S_{p}$. Since $\overline{F_{\mathfrak{u}}}$ contains a normal regular subgroup
$\overline{N}$, it is an affine group, which implies that $M/(M\cap
F_{(\mathfrak{u})})\cong\overline{M}\cong Z_{l}$ for $l\bigm{|}(p-1).$ In
particular, $M_{1}/(M_{1}\cap F_{(\mathfrak{u})})\cong Z_{l^{\prime}}$ for
$l^{\prime}\bigm{|}(p-1)$. Note that our group $M_{1}=A_{4}$ or $S_{4}$ for
$p=5$; $S_{4}$ or $S_{4}\times Z_{3}$ for $p=7$; and $A_{5}$ for $p=11$. In
all the cases, three exists a subgroup $M_{2}\cong A_{4}$ which is contained
in $F_{(\mathfrak{u})}$, that is, $M_{2}$ fixes $\mathfrak{u}$ pointwise. For
any ${\bf u}\in\mathfrak{u}$, we have that $N_{\bf u}\cong Z_{p}^{2}$ and
$N_{\bf u}M_{2}$ fixes $\mathfrak{u}$ pointwise.
Now let’s consider the subgroup $N_{\bf u}M_{2}$. Let $K_{0}$ be the kernel of
$M_{2}$ acting on $N_{\bf u}$ by conjugacy. Then $K_{0}$ fixes a 2-dimensional
subspace pointwise. It is easy to see that the subgroup of $\hbox{\rm
SL}(3,p)$ fixing a 2-dimensional subspace pointwise is isomorphic to
$Z_{p}^{2}\rtimes Z_{p-1}$. Since $p\nmid|M_{2}|$, we know that $K_{0}$ is
cyclic. But $A_{4}$ contains only one cyclic normal subgroup, that is 1, and
thus $K_{0}=1$ and then $M_{2}$ acts faithfully on $N_{\bf u}$, or
equivalently, $M_{2}\lesssim\hbox{\rm GL}(2,p).$ However, $\hbox{\rm GL}(2,p)$
does not contain any subgroup isomorphic to $A_{4}$, a contradiction.
###### Lemma 4.3
$\hbox{\rm Aut}(\Sigma)\cong S_{3}\wr S_{3}$, $\Sigma\cong\Sigma(3)$ or
$\Sigma(6)$; and $\Gamma\cong\Gamma(9)$ or $\Gamma(18)$ defined in Example
1.1.
Proof Suppose $p=3$. Continue the proof of (iii) in last paragraph. Then
$|{\cal U}|=9$, $F=N\rtimes H$, $H_{1}=H\cap\hbox{\rm SL}(3,p)$, and
$H_{1}=Z_{13}\rtimes Z_{3}$ $A_{4}$ or $S_{4}$. Clearly, $H_{1}\neq
Z_{13}\rtimes Z_{3}.$ Hence, we let $A_{4}\leq H_{1}\leq H\leq L$, where $L$
is the same group in Lemma 1.2, that is the subgroup of $\hbox{\rm GL}(3,3)$
consisting of all those $3\times 3$ matrices with only one nonzero entry in
each row and column and actually, $L\cong Z_{2}^{3}\rtimes S_{3}$, of order
$48$.
(i) First, suppose that $H=L$. Then $|F|=|N||H|=3^{4}2^{4}.$ Considering the
imprimitive action of $F$ on ${\cal U}$, we know that $F\lesssim S_{3}\wr
S_{3}$. Since $|S_{3}\wr S_{3}|=3^{4}2^{4}$, we get $F\cong S_{3}\wr S_{3}$.
In fact, $N\rtimes L$ is really isomorphic to $S_{3}\wr S_{3}.$
Since $F=N\times L$, an affine group, we may identify ${\cal W}$ with the
3-dimensional space $\mathbb{V}=\mathbb{V}(3,3)$. Let ${\bf w}$ be zero
vector. Then $F_{\bf w}=L.$ Take a vertex ${\bf u}\in{\cal U}$. Then $N_{\bf
u}=Z_{p}^{2}$, and $L_{\bf u}$ is a Sylow 2-subgroup of $L.$ Therefore,
$F_{\bf u}=N_{\bf u}\rtimes L_{\bf u}.$ Consider $N_{\bf u}$ as a
2-dimensional subspace, $L_{\bf u}$ must preserve it. As in Example 1.1, set
$\begin{array}[]{lll}{\mathbb{V}}_{0}&=&\\{(0,b,c)\bigm{|}b,c\in\hbox{\rm
GF}(3)\\},\,{\mathbb{V}}_{1}=\\{(a,0,c)\bigm{|}a,c\in\hbox{\rm GF}(3)\\},\\\
{\mathbb{V}}_{2}&=&\\{(a,b,0)\bigm{|}a,b\in\hbox{\rm GF}(3)\\}.\end{array}$
Without loss of generality, set $N_{\bf u}={\mathbb{V}}_{0}$. Take an element
$x={\footnotesize\left(\begin{array}[]{lll}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right)}\in L.$ Then $\langle x\rangle$ permutes
${\mathbb{V}}_{0}$, ${\mathbb{V}}_{1}$ and ${\mathbb{V}}_{2}$. Now, ${\cal U}$
may be identified with the set of nine lines:
${\cal U}=\\{\alpha+{\mathbb{V}}_{i}\bigm{|}\alpha\in{\mathbb{V}},i\in
Z_{3}\\}.$
It is easy to check that $F_{\bf w}(=L)$ has two orbits on ${\cal U}$ of
length 3 and 6, respectively. Therefore, we just get two graphs, which are
exactly $\Sigma(3)$ and $\Sigma(6)$, with $d({\bf w})=3$ and 6.
Set $\Sigma=\Sigma(3)$ or $\Sigma(6)$. From the argument of last section, we
know that there exist no graphs whose automorphism group acts primitively on
${\cal U}$ and so $\hbox{\rm Aut}(\Sigma)$ acts primitively on ${\cal W}$ and
imprimitively on ${\cal U}.$ Since $S_{3}\wr S_{3}$ is the maximal imprimitive
group of degree 9, $\hbox{\rm Aut}(\Sigma)\leq S_{3}\wr S_{3}\cong F$, and
then $\hbox{\rm Aut}(\Sigma)=S_{3}\wr S_{3}$.
Correspondingly, we get $\Gamma\cong\Gamma(9)$ or $\Gamma(18)$.
(ii) Secondly, suppose that $A_{4}\leq H\lvertneqq L$. Then $|\hbox{\rm
Aut}(\Sigma)|\lvertneqq 3^{4}2^{3}$ and $A_{4}\leq F_{\bf w}$. Consider the
action of $A_{4}$ on ${\cal U}$. Clearly, each subgroup $Z_{3}$ of $A_{4}$ is
transitive on ${\cal U}$ and the normal subgroup $Z_{2}^{2}$ of $A_{4}$ fixes
each block setwise. If $Z_{2}^{2}$ fixes pointwise in a block, then
$Z_{2}^{2}$ fixes ${\cal U}$ pointwise, which forces that $F$ acts
unfaithfully on ${\cal U}$. Therefore, acting in each block, $Z_{2}^{2}$ fixes
one vertex and exchange other two vertices, which implies that $F_{\bf w}$ has
two orbits on ${\cal U}$ with respective length 3 and 6. Hence, we get two
graphs as same as in (i), that is $\Sigma=\Sigma(3)$ and $\Sigma(6)$,
contradicting to $|\hbox{\rm Aut}(\Sigma)|=3^{4}2^{4}$.
### 4.3 Proof of Theorem 1.4.(3)
Before proving Theorem 1.4.(3), we first prove two group theoretical results.
###### Lemma 4.4
Let $\mathbb{V}=\mathbb{V}(3,p)$ and $G=\hbox{\rm GL}(3,p).$ Take
$x=\footnotesize{\left(\begin{array}[]{ccc}1&2&2\\\ 0&1&2\\\
0&0&1\end{array}\right)}\in G.$ Then
1. (1)
$x$ fixes setwise only one 1-dimensional subspace $\langle\alpha\rangle$ for
$\alpha=(0,0,1)\in\mathbb{V}$ and only one 2-dimensional subspace
$\mathbb{S}^{\prime}=\\{(0,a_{2},a_{3})\bigm{|}a_{2},a_{3}\in\hbox{\rm
GF}(p)\\}$.
2. (2)
For any 2-dimensional subspace $\mathbb{S}$ not including $\alpha$, we have
$\mathbb{S}^{x^{i}}\cap\mathbb{S}^{x^{j}}\cap\mathbb{S}^{x^{k}}=\\{0\\},$
where $i,j,k\in\hbox{\rm GF}(p)$ are distinct.
Proof (1) Checking directly.
(2) Let $\mathbb{S}$ be a 2-dimensional subspace and
$\alpha\not\in\mathbb{S}$. Suppose that
$0\neq\beta\in\mathbb{S}^{x^{i}}\cap\mathbb{S}^{x^{j}}\cap\mathbb{S}^{x^{k}},$
where $i,j,k$ are distinct. Then
$\beta^{x^{-i}},\beta^{x^{-j}},\beta^{x^{-k}}\in\mathbb{S}$. Since $x$ does
not fix $\langle\beta\rangle$, the subspace
$\langle\beta^{x^{-i}},\beta^{x^{-j}},$ $\beta^{x^{-k}}\rangle$ can not be
1-dimensional and so it is $\mathbb{S}$. Set $\beta=(a_{1},a_{2},a_{3})$. Note
that for any $l\in F_{p}$,
$x^{l}={\footnotesize\left(\begin{array}[]{ccc}1&2l&2l^{2}\\\ 0&1&2l\\\
0&0&1\end{array}\right)}.$
Let
$D=\left|\begin{array}[]{c}\beta^{x^{-i}}\\\ \beta^{x^{-j}}\\\
\beta^{x^{-k}}\end{array}\right|=\left|\begin{array}[]{ccc}a_{1}&-2ia_{1}+a_{2}&2i^{2}a_{1}-2ia_{2}+a_{3}\\\
a_{1}&-2ja_{1}+a_{2}&2j^{2}a_{1}-2ja_{2}+a_{3}\\\
a_{1}&-2ka_{1}+a_{2}&2k^{2}a_{1}-2ka_{2}+a_{3}\\\ \end{array}\right|.$
Since $\alpha\not\in\mathbb{S},$ we get $a_{1}\neq 0$. By computing we get
$D=4a_{1}^{3}(i-j)(k-i)(k-j)\neq 0,$
forcing dim($\mathbb{S}$)=3, a contradiction.
###### Lemma 4.5
Let $G$ be an imprimitive transitive group of degree $p^{2}$ on $\Omega$,
where $p\geq 3$ and $p^{3}\bigm{|}|G|$ and let $\mathcal{\mathcal{B}}$ be an
imprimitive $p$-block system of $G$. Let $P$ be a Sylow p-subgroup of $G$.
Then
1. (1)
$\hbox{\rm Exp }(P)\leq p^{2}$, $|Z(P)|=p$ and $P=(P\cap
G_{(\mathcal{B})})\langle t\rangle$, for some $t\in P$ such that $t^{p}\in
Z(P);$
2. (2)
Suppose that provided either $p=3$ or $p\geq 5$ and $|P\cap
G_{(\mathcal{B})}|\leq p^{p-1}.$ Then $G_{(\mathcal{B})}$ is solvable, $P\cap
G_{(\mathcal{B})}$ is a characteristic subgroup of $G_{(\mathcal{B})}$ and so
$P\cap G_{(\mathcal{B})}\lhd G$.
Proof (1) Check easily.
(2) If $p=3$, then the conclusion is clearly true.
Suppose $p\geq 5$ and $|P\cap G_{(\mathcal{B})}|\leq p^{p-1}.$ Set
$K=G_{(\mathcal{B})},\quad N=P\cap G_{(\mathcal{B})},\quad\mathcal{B}=\\{{\bf
b}_{0},{\bf b}_{1},\cdots,{\bf b}_{p-1}\\}.$
For any $g\in K\setminus\\{1\\}$, let $\ell(g)$ be the number of blocks ${\bf
b}_{i}$ in $\mathcal{B}$ such that the induced action $g^{{\bf b}_{i}}$ is
nontrivial and set
$\ell=min\\{\ell(g)\bigm{|}g\in K\setminus\\{1\\}\\}.$
Since $p^{3}\bigm{|}|G|$ and $|N|\leq p^{p-1}$, we get $\ell\neq p,1$. Hence,
$2\leq\ell\leq p-1$.
Take $g\in K$ such that $\ell(g)=\ell.$ Without loss of generality, say
$g^{{\bf b}_{i}}$ is nontrivial for $0\leq i\leq\ell-1$ and trivial for
$\ell\leq i\leq p-1$. Set $L=\cap_{\ell\leq i\leq{p-1}}K_{({\bf b}_{i})}.$
Then $L\lhd K$. Since $g\in L$, we get $L^{{\bf b}_{i}}$ is nontrivial for
$0\leq i\leq\ell-1$. Since $L\lhd K$, it follows that $L$ is transitive on
each such ${\bf b}_{i}.$ By the definition of $\ell$, we know that $L$ is
faithful on ${\bf b}_{i}$ and so $N\cap L\cong Z_{p}$.
Take an element $x\in P\setminus N$ such that ${\bf b}_{0}^{x}={\bf b}_{1}$.
Since $\langle x\rangle$ is transitive on ${\cal B}$, we have that $x$ cannot
fix setwise any proper subset of ${\cal B}.$ Therefore,
$1\leq|\\{{\bf b}_{0},{\bf b}_{1},\cdots{\bf b}_{\ell-1}\\}\cap\\{{\bf
b}_{0}^{x},{\bf b}_{1}^{x},\cdots,{\bf b}_{\ell-1}^{x}\\}|\leq{\ell-1}.$
For any $h=h_{0}h_{1}\cdots h_{\ell-1}\in L,$ where $1\neq h_{i}\in S^{{\bf
b}_{i}}$, we have that $h^{x}=h_{0}^{x}h_{1}^{x}\cdots h_{\ell-1}^{x}$. Noting
that $1\neq h_{0}^{x}\in S^{{\bf b}_{1}},$ we know that both $L$ and $L^{x}$
are nontrivial on ${\bf b}_{1}$. For any $h=h_{0}h_{1}\cdots h_{\ell-1}$ and
$h^{\prime}=h_{0}^{\prime}h_{1}^{\prime}\cdots h_{\ell-1}^{\prime}$ in $L,$ we
have $\ell([h,(h^{\prime})^{x}])\leq\ell-1.$ It follows the minimality of the
value $\ell$ that $[h,(h^{\prime})^{x}]=1$, equivalently, $[L,L^{x}]=1$. In
particular, $[L^{{\bf b}_{1}},(L^{x})^{{\bf b}_{1}}]=1$. Since $L^{{\bf
b}_{1}}\leq C_{S^{{\bf b}_{1}}}((L^{x})^{{\bf b}_{1}})$ and $(L^{x})^{{\bf
b}_{1}}$ is transitive on ${\bf b}_{1}$, it follows from a well-known theorem
in permutation group theory that $L^{{\bf b}_{1}}$ is regular, that is,
$L^{{\bf b}_{1}}\cong Z_{p}$. Moreover, since $L$ is faithful on each ${\bf
b}_{i}$ for $0\leq i\leq\ell-1$ by the arguments in last paragraph, we get
$L\cong Z_{p}$. It has been proved that $K^{{\bf b}_{1}}$ contains a regular
normal subgroup $L^{{\bf b}_{1}}$, and so $K^{{\bf b}_{1}}$ is solvable. This
in turn implies $K$ is solvable.
Let $N_{1}$ and $N_{2}$ be two Sylow $p$-subgroups of $K$. Since $K^{{\bf
b}_{i}}$ has the unique subgroup $Z_{p}$ for each block ${\bf b}_{i}$, we get
$[N_{1},N_{2}]=1.$ Now $N_{1}N_{2}$ is a $p$-subgroup of $K$, which forces
that $N_{1}=N_{2}$. Therefore, $N$char $K$ and then $N\lhd G$, as desired.
Proof of Theorem 1.4.(3): Suppose that $F=\hbox{\rm Aut}(\Sigma)$ acts
imprimitively on ${\cal W}$. By Theorem 1.4.(1), $F$ also acts imprimitively
on ${\cal U}$, with an imprimitive complete $p-$block system $\mathfrak{U}$.
Clearly, $F_{(\mathfrak{U})}\neq 1.$ Considering the imprimitive action of $F$
on ${\cal U}$, we find that $F\lesssim S_{p}\wr S_{p}=(S_{p})^{p}\rtimes
S_{p},$ and $F_{(\mathfrak{U})}\lesssim(S_{p})^{p}.$ Let $P\in\hbox{\rm
Syl}_{p}(F).$ Then $P\lesssim(Z_{p})^{p}\rtimes Z_{p}.$ Set $N=P\bigcap
F_{(\mathfrak{U})}$. Then $N\lesssim(Z_{p})^{p}$.
In what follows, we divide our proof into two cases depending on whether or
not $F_{(\mathfrak{U})}$ acts transitively on ${\cal W}$.
(1) $F_{(\mathfrak{U})}$ acts transitively on ${\cal W}$.
Suppose that $F_{(\mathfrak{U})}$ acts transitively on ${\cal W}$. Then $N$ is
also transitive on ${\cal W}$. Since $N$ is abelian, $N$ acts regularly on
${\cal W}$, that is $N\cong Z_{p}^{3}$ and then $|P|=p^{4}$. Take ${\bf
w}\in{\cal W}$. Then $P=N\rtimes P_{\bf w}\cong Z_{p}^{3}\rtimes Z_{p}.$
Considering the action of $P$ on ${\cal U}$, for ${\bf u}\in{\cal U}$ we have
that $P_{\bf u}=N_{\bf u}\cong Z_{p}^{2}.$
By Lemma 4.5, $F_{(\mathfrak{U})}$ is solvable and $N\lhd F$. Therefore, $F$
is an affine group, that is $F=N\rtimes F_{\bf w}$, where $N$ is identified
with the translation normal subgroup of $\hbox{\rm AGL}(3,p)$ and $F_{\bf w}$
with a reducible subgroup of $\hbox{\rm GL}(3,p)$. That is the case (3.1) in
Theorem 1.4.
(2) $F_{(\mathfrak{U})}$ acts intransitively on ${\cal W}$.
Suppose that $F_{(\mathfrak{U})}$ acts intransitively on ${\cal W}$. Since
$F/F_{(\mathfrak{U})}\leq S_{p}$, we get
$p\bigm{|}\bigm{|}|F/F_{(\mathfrak{U})}|$. Hence $|N|\geq p^{2}$ and so
$F_{(\mathfrak{U})}$ induces $p^{2}$-blocks on ${\cal W}$. Therefore, the
first conclusion of Theorem 1.4.(3.2) holds.
Let ${\bf w}$ be any vertex in ${\cal W}$. Then we deal with two cases
separately.
(2.1) Suppose that ${\bf w}$ is exactly adjacent to two blocks in
$\mathfrak{U}$. Then we are in case Theorem 1.4.(3.2.1).
(2.2) Suppose that ${\bf w}$ is adjacent to at least three blocks in
$\mathfrak{U}.$ Then in what follows we shall prove the conclusions of Theorem
1.4.(3.2.2).
Since $F/F_{(\mathfrak{U})}\lesssim S_{p}$, it acts faithfully on both
$\mathfrak{W}$ and $\mathfrak{U}$. Thus, we get $p\neq 3$ and so $p\geq 5$.
Since $|P|\geq p^{3}$ and $P$ acts faithfully on ${\cal U}$, it follows that
$P$ is nonabelian.
Now we show $|N|=p^{2}$. For the contrary, suppose that $|N|\geq p^{3}$. Let
$N_{1}$ be a normal subgroup of $P$ such that $|N_{1}|=p^{3}$ and $N_{1}\leq
N.$ Let $x_{0}\in P\setminus N$ and $P_{1}=N_{1}\langle x_{0}\rangle$. Then by
Lemma 4.5.(1), $|Z(P)|=p$, $x_{0}^{p}\in Z(P)$ and then $|P_{1}|=p^{4}$.
Clearly, for any ${\bf w}\in{\cal W}$, we have that $|(P_{1})_{\bf
w}|=|(N_{1})_{\bf w}|\geq p$; and for any ${\bf u}\in{\cal U},$ we have that
$(P_{1})_{\bf u}=(N_{1})_{\bf u}\cong Z_{p}^{2}$ and $N_{1}$ is transitive on
every block in ${\cal U}$. As the same reason as in (1), the conjugacy action
of $x_{0}$ on $N_{1}$ can be identified with the action of $x$ on
$\mathbb{V}(3,p)$, where $x$ is define as in Lemma 4.4. Suppose that ${\bf w}$
is adjacent to $p$ vertices in a block $\mathfrak{u}\in\mathfrak{U}$. Since
the edge-transitivity of $\Sigma$, we get that ${\bf w}$ is adjacent to $p$
vertices in any block such that one of whose vertex is adjacent to ${\bf w}$.
Considering the actions $N_{1}$ on ${\cal W}$ and ${\cal U}$, we know that the
vertices in ${\bf w}^{N_{1}}$ have the same neighborhood, a contradiction.
Therefore, if ${\bf w}$ is adjacent to a block $\mathfrak{u}\in\mathfrak{U}$,
then $(N_{1})_{\bf w}$ fixes pointwise $\mathfrak{u}$, equivalently
$(N_{1})_{\bf w}\leq(N_{1})_{\bf u}$, otherwise, ${\bf w}$ is adjacent to $p$
vertices in $\mathfrak{u}$. By the hypothesis, we assume that ${\bf w}$ is
adjacent to at least three blocks $\mathfrak{u}^{x^{i}},\mathfrak{u}^{x^{j}}$
and $\mathfrak{u}^{x^{k}}$, where $i,j,k$ are distinct in $Z_{p}$. Then
$(N_{1})_{\bf w}\leq((N_{1})_{\bf u})^{x^{i}}\cap((N_{1})_{\bf
u})^{x^{j}}\cap((N_{1})_{\bf u})^{x^{k}}$. Identifying $(N_{1})_{\bf
w},((N_{1})_{\bf u})^{x^{i}},((N_{1})_{\bf u})^{x^{j}}$ and $((N_{1})_{\bf
u})^{x^{k}}$ with the subspaces of $\mathbb{V}(3,p)$, we get from Lemma 4.4
that $(N_{1})_{\bf w}=1$, a contradiction.
Since $|N|=p^{2}$, we get $|P|=p^{3}$. Since $p\geq 5$, we get from
Proposition 2.7 that $P\lhd F,$ namely, $P$ is a normal subgroup of $F$ acting
regularly on ${\cal W}$. By Lemma 4.5, $F_{(\mathfrak{U})}$ is solvable.
Moreover, since $Z_{p}\cong PF_{(\mathfrak{U})}/F_{(\mathfrak{U})}\lhd
F/F_{(\mathfrak{U})}\leq S_{p}$, it follows that $F/F_{(\mathfrak{U})}$
contains a normal regular subgroup on $\mathfrak{U}$ and so it is affine
group. In other words, $F/F_{(\mathfrak{U})}\cong Z_{p}\rtimes Z_{r},$ where
$r\bigm{|}(p-1)$.
## 5 Examples of graphs
In this section, by defining three bi-coset graphs we show the existences of
the graphs in the three cases of Theorem 1.4.(3).
Graph $\Sigma_{1}(p):$ For $p\geq 5$, let $F=N\rtimes(\langle x\rangle\rtimes
H)\leq\hbox{\rm AGL}(3,p)$ where $N$ is the translation subgroup of $\hbox{\rm
AGL}(3,p)$, where
$x={\footnotesize\left(\begin{array}[]{ccc}1&2&2\\\ 0&1&2\\\
0&0&1\end{array}\right)},\quad
H=\langle{\footnotesize\left(\begin{array}[]{ccc}s^{2}t^{-1}&0&0\\\ 0&s&0\\\
0&0&t\end{array}\right)}\bigm{|}s,t\in\hbox{\rm GF}(p)^{*}\rangle\leq
N_{\hbox{\rm GL}(3,p)}(\langle x\rangle).$
Let $N_{0}=\langle t_{(1,0,0)},t_{(0,1,0)}\rangle\leq N.$ With the notation of
Definition 2.1, set
$L=\langle x\rangle H,\,\,R=N_{0}H,\,\,D=RL.$
Then $|F:L|=p^{3}$ and $|F:R|=p^{2}$. Since $R$ and $L$ are nonmaximal
subgroups of $F$, we get that $F$ acts imprimitively on both $[F:L]$ and
$[F:R]$.
Let $\Sigma_{1}(p)={\bf B}(F;L,R,D)$, a double coset graph. Since
$|D|/|R|=|L|/|L\cap R|=p,$ the degree of any vertex in $[F:L]$ is $p$.
Moreover, one may easily see that there exist no two vertices in $[F:L]$
having the same neighborhood. Now $F$ acts edge-transitively on $\Sigma$.
Since the proof of $\hbox{\rm Aut}(\Sigma)\cong F$ depends on several lemmas
and take a long argument, we do not try to write it in this paper but shall
put it in our further paper. This graph satisfies the condition of Theorem
1.4.(3.1). Finally, let $\Gamma_{1}(p)$ be the graph expanded from
$\Sigma_{1}(p)$.
Graph $\Sigma_{2}(p):$ For any prime $p\geq 5$, let
$\sigma=(0,1,\cdots,p-1),\quad\tau=(0)(1,-1)\cdots(\frac{p-1}{2},\frac{p+1}{2})\in
S_{p}.$
Then $\langle\sigma,\tau\rangle\cong D_{2p}.$ Let $M\cong S_{p}$, $H\leq M$
and $H\cong S_{p-1}$. Set
$\begin{array}[]{ll}&F=M\wr\langle\sigma,\tau\rangle=(\overbrace{M\times\cdots\times
M}^{p\,{\rm times}})\rtimes\langle\sigma,\tau\rangle,\\\
&L=(\overbrace{M\times\cdots\times M}^{\frac{p-1}{2}-1\,{\rm times}}\times
H\times H\times\overbrace{M\times\cdots\times M}^{\frac{p+1}{2}-1\,{\rm
times}})\rtimes\langle\tau\rangle,\\\
&R=(H\times\overbrace{M\times\cdots\times M}^{p-1\,{\rm
times}})\rtimes\langle\tau\rangle,\,\,D=R\sigma^{\frac{p-1}{2}}L.\end{array}$
Then $|F:L|=p^{3}$ and $|F:R|=p^{2}$. Clearly, $F$ acts imprimitively on both
$[F:L]$ and $[F:R]$.
Let $\Sigma_{2}(p)={\bf B}(F;L,R,D)$. Since $|D|/|R|=|L|/|L\cap R|=2,$ the
degree of any vertex in $[G:L]$ is $2$. Moreover, we shall show $\hbox{\rm
Aut}(\Sigma)\cong F$ in our further paper. Clearly, $M^{p}$ induces a
$p$-block system on $[F:R]$ and the vertex $L$ is exactly adjacent to two
blocks, corresponding to the double coset $R\sigma^{\frac{p-1}{2}}L$. This
graph satisfies the condition of Theorem 1.4.(3.2.1). Finally, let
$\Gamma_{2}(p)$ be the graph expanded from $\Sigma_{1}(p)$.
Graph $\Sigma_{3}(p):$ For any prime $p\geq 5$, suppose that
$P=\langle a,b\bigm{|}a^{p^{2}}=b^{p}=1,[b,a]=c,a^{p}=c,a^{b}=a^{1+p}\rangle.$
Pick up an element $s$ of order $p-1$ in $Z_{p^{2}}^{*}.$ Let
$F=P\rtimes\langle x\rangle,\,{\rm where}\,x^{p-1}=1,a^{x}=a^{s}$
Set
$L=\langle x\rangle,\,\,R=\langle b\rangle\langle x\rangle,\,\,D=RaL.$
Then $|F:L|=p^{3},|F:R|=p^{2}$. Clearly, $F$ acts imprimitively on both
$[F:L]$ and $[F:R]$.
Let $\Sigma_{3}(p)={\bf B}(F;L,R,D)$. Since $|D|/|R|=|L|/|L\cap R|=p-1,$ the
degree of any vertex in $[F:L]$ is $p-1$. Moreover, we shall show $\hbox{\rm
Aut}(\Sigma)\cong F$ in our further paper. Clearly, $\langle a^{p}\rangle$
induces a $p$-block system on $[F:R]$ and the vertex $L$ is adjacent to $p-1$
blocks. This graph satisfies the condition of Theorem 1.4.(3.2.2). Finally,
let $\Gamma_{3}(p)$ be the graph expanded from $\Sigma_{3}(p)$.
Acknowledgments: The authors thank the referee for the helpful comments and
suggestions. This work is partially supported by the National Natural Science
Foundation of China and Natural Science Foundation of Beijing.
## References
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* [16] B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967.
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|
arxiv-papers
| 2012-06-10T14:43:45 |
2024-09-04T02:49:31.652921
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li Wang and Shaofei Du",
"submitter": "Shaofei Du",
"url": "https://arxiv.org/abs/1206.2033"
}
|
1206.2280
|
# The Frobenius-Euler function and its applications
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Deyao Gao Yuren
Lab., No. 8 Tongsheng Road, Changsha, P. R. China 13607433711@163.com and
Mehmet Acikgoz University of Gaziantep, Faculty of Science and Arts,
Department of Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
In the present paper, we deal with Fourier-transformation of Frobenius-Euler
polynomials. We shall give its applications by using infinite series. Our
applications possess interesting properties which we state in this paper.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Frobenius-Euler numbers and polynomials, Fourier
transformation, infinite series.
## 1\. Introduction
The ordinary Frobenius-Euler numbers are defined by means of the following
generating function:
(1.1)
$\sum_{n=0}^{\infty}H_{n}\left(u\right)\frac{t^{n}}{n!}=e^{H\left(u\right)t}=\frac{1-u}{e^{t}-u}\text{.}$
where, in the umbral calculus, $H^{n}\left(u\right)$ is symbolically replaced
by $H_{n}\left(u\right)$ in the formal series expansion of
$e^{tH\left(u\right)}=\sum_{n=0}^{\infty}H_{n}\left(u\right)\frac{t^{n}}{n!}\text{.}$
From expression of this definition, we state the following
(1.2)
$\left(H\left(u\right)+1\right)^{n}-uH_{n}\left(u\right)=\left\\{\begin{array}[]{cc}1-u&\text{if
}n=0,\\\ 0&\text{if }n\in\mathbb{N},\end{array}\right.$
where $\mathbb{N}$ denotes the set of positive integers.
From (1.2), we note that
$H_{0}(u)=1,H_{1}\left(u\right)=-\frac{1}{1-u},H_{2}\left(u\right)=\frac{1+u}{\left(1-u\right)^{2}},\cdots.$
The Frobenius Euler polynomials are also introduced as
(1.3)
$e^{xt}\frac{1-u}{e^{t}-u}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}H_{n}\left(x,u\right)\text{.}$
By (1.1) and (1.3), we can find the following
$H_{n}\left(x,u\right)=\sum_{l=0}^{n}\binom{n}{l}x^{n-l}H_{l}\left(u\right)\text{.}$
By expression of (1.2), it is not difficult to show that the recurrence
relation for the Frobenius-Euler numbers as follows:
(1.4)
$\sum_{l=0}^{n}\binom{n}{l}H_{l}\left(u\right)-uH_{n}\left(u\right)=\left(1-u\right)\delta_{0,n}$
where $\delta_{m,n}$ is the Kronecker delta, is defined by
$\delta_{m,n}=\left\\{\begin{array}[]{cc}\text{ }1,&\text{if }m=n\\\ 0,&\text{
if }m\neq n.\end{array}\right.$
Thus, we easily procure the following:
(1.5) $H_{n}\left(1,u\right)-uH_{n}\left(u\right)=0\text{ }\left(\text{for
}n\in\mathbb{N}\right)\text{.}$
Thus, we arrive the following lemma.
###### Lemma 1.
For $n\in\mathbb{N}$, we have
$H_{n}\left(1,u\right)=uH_{n}\left(u\right)\text{.}$
Substituting $u=-1$ in the above lemma, it leads to
$H_{n}\left(-1,u\right)=E_{n}\left(1\right)=-E_{n}$
where $E_{n}$ is called Euler numbers, as is well-known, Euler numbers are
defined by the following generating function:
$\sum_{n=0}^{\infty}E_{n}\frac{t^{n}}{n!}=\frac{2}{e^{t}+1}$
(for more informations on this subjects, see[1-20]).
Recently, Fourier transformation of the special functions have been studied by
many mathematicians (cf., [1], [2], [4], [13], [14], [16]). In [16], Luo gave
Fourier expansions of Apostol-Bernoulli and Apostol-Euler polynomials and
derived some integral representations of Apostol-Bernoulli and Apostol-Euler
polynomials by using Fourier expansions. After, Bayad [1] introduced as
theoretical identities of the Fourier transformation of the Apostol-Bernoulli,
Apostol-Euler and Apostol-Genocchi polynomials. Next, T. Kim also defined the
Euler function which is Fourier transformation of Euler polynomials. We easily
see that Kim’s method is different from Bayad and Luo. Actually, Kim’s paper
[3, pp. 131-136] motivated us to write this paper. Thus, we also give Fourier
transformation of Frobenius-Euler function by using Kim’s method. In this
paper, we also show that this function is related to Lerch trancendent
$\Phi\left(z,s,a\right)$.
## 2\. On the Frobenius-Euler function
In this section, we consider Frobenius-Euler function by using infinite
series. For $m\in\mathbb{N}$, the Fourier transformation of Frobenius-Euler
function is introduced as
(2.1)
$H_{m}\left(x,u\right)=\sum_{n=-\infty}^{\infty}a_{n}^{\left(m\right)}\left(u\right)e^{\left(2n+1\right)\pi
ix}\text{, }\left(a_{n}^{\left(m\right)}\left(u\right)\in\mathbb{C}\right)$
where $\mathbb{C}$ denotes the set of complex numbers and
(2.2)
$a_{n}^{\left(m\right)}\left(u\right)=\int_{0}^{1}H_{m}\left(x,u\right)e^{-\left(2n+1\right)ix\pi}dx\text{.}$
By applying some technical method on (2.2), we procure the following
$\displaystyle a_{n}^{\left(m\right)}\left(u\right)$ $\displaystyle=$
$\displaystyle\left[\frac{H_{m+1}\left(x,u\right)}{m+1}e^{-\left(2n+1\right)ix\pi}\right]_{0}^{1}-\frac{\left(2n+1\right)\pi
i}{m+1}\int_{0}^{1}H_{m+1}\left(x,u\right)e^{-\left(2n+1\right)ix\pi}dx$
$\displaystyle=$
$\displaystyle-\frac{u+1}{m+1}H_{m+1}\left(u\right)-\frac{\left(2n+1\right)\pi
i}{m+1}a_{n}^{\left(m+1\right)}\left(u\right)\text{.}$
So from above, it leads to the following
$a_{n}^{\left(m+1\right)}\left(u\right)=\left[a_{n}^{\left(m\right)}+\left(u+1\right)\frac{H_{m+1}\left(u\right)}{m+1}\right]\frac{m+1}{\left(\left(2n+1\right)\pi
i\right)}\text{.}$
By continuing this process, becomes as follows:
(2.3a) $\displaystyle
a_{n}^{\left(m\right)}\left(u\right)=\left[a_{n}^{\left(1\right)}\left(u\right)+\left(u+1\right)\frac{H_{2}\left(u\right)}{2}\right]\frac{m!}{\left(\left(2n+1\right)\pi
i\right)^{m-1}}$
$\displaystyle+\left(u+1\right)\left[\frac{1}{\left(2n+1\right)\pi
i}H_{m}\left(u\right)+\frac{m}{\left(\left(2n+1\right)\pi
i\right)^{2}}H_{m-1}\left(u\right)+...+\frac{m!}{4!\left(\left(2n+1\right)\pi
i\right)^{m-3}}\right]\text{.}$
We want to note that
$\lim_{u\rightarrow-1}a_{n}^{\left(m\right)}\left(u\right)=a_{n}^{\left(m\right)}\left(-1\right):=a_{n}^{\left(m\right)}$
where $a_{n}^{\left(m\right)}\in\mathbb{C}$ is defined by Kim in [3] as
follows:
$a_{n}^{\left(m\right)}=\frac{m!}{\left(\left(2n+1\right)\pi
i\right)^{m-1}}a_{n}^{\left(1\right)}\left(u\right)\text{.}$
By using (2.1) and (2.3a), we readily derive the following
$H_{m}\left(x,u\right)=\sum_{n=-\infty}^{\infty}\left\\{\begin{array}[]{c}\left[a_{n}^{\left(1\right)}\left(u\right)+\left(u+1\right)\frac{H_{2}\left(u\right)}{2}\right]\frac{m!}{\left(\left(2n+1\right)\pi
i\right)^{m-1}}+\left(u+1\right)\\\ \times\left[\frac{1}{\left(2n+1\right)\pi
i}H_{m}\left(u\right)+\frac{m}{\left(\left(2n+1\right)\pi
i\right)^{2}}H_{m-1}\left(u\right)+...+\frac{m!}{4!\left(\left(2n+1\right)\pi
i\right)^{m-3}}\right]\end{array}\right\\}e^{\left(2n+1\right)\pi ix}\text{.}$
From this, we can state the following
$\displaystyle
H_{m}\left(x,u\right)=\sum_{n=-\infty}^{\infty}\left[a_{n}^{\left(1\right)}\left(u\right)+\left(u+1\right)\frac{H_{2}\left(u\right)}{2}\right]\frac{m!e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m-1}}$
$\displaystyle+\left(u+1\right)\sum_{n=-\infty}^{\infty}\left[\frac{1}{\left(2n+1\right)\pi
i}H_{m}\left(u\right)+\frac{m}{\left(\left(2n+1\right)\pi
i\right)^{2}}H_{m-1}\left(u\right)+...+\frac{m!}{4!\left(\left(2n+1\right)\pi
i\right)^{m-3}}H_{4}\left(u\right)\right]e^{\left(2n+1\right)\pi ix}$
After some calculations on the above equation, we have the following
$\displaystyle H_{m}\left(x,u\right)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\left[\frac{u+1}{u-1}\frac{1}{\left(2n+1\right)\pi
i}+\frac{2}{\left(\left(2n+1\right)\pi
i\right)^{2}}+\frac{1}{2}\left(\frac{u+1}{1-u}\right)^{2}\right]\frac{m!e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m-1}}$
$\displaystyle+\left(u+1\right)\sum_{n=-\infty}^{\infty}\left[\sum_{k=0}^{m-4}\frac{1}{\left(\left(2n+1\right)\pi
i\right)^{k+1}}\frac{d^{k}}{dx^{k}}H_{m}\left(x,u\right)\mid_{x=0}\right]e^{\left(2n+1\right)\pi
ix}$
As a result, we conclude the following theorem.
###### Theorem 1.
For $m\in\mathbb{N}$ and $0\leq x<1$, we have
$\displaystyle
H_{m}\left(x,u\right)=m!\sum_{n=-\infty}^{\infty}\left[\frac{u+1}{u-1}\frac{1}{\left(2n+1\right)\pi
i}+\frac{2}{\left(\left(2n+1\right)\pi
i\right)^{2}}+\frac{1}{2}\left(\frac{u+1}{1-u}\right)^{2}\right]\frac{e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m-1}}$
$\displaystyle+\left(u+1\right)\sum_{k=0}^{m-4}\frac{1}{\left(2\pi
i\right)^{k+1}}\frac{d^{k}}{dx^{k}}H_{m}\left(x,u\right)\mid_{x=0}\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(n+\frac{1}{2}\right)^{k+1}}\text{.}$
Considering generating functions of Euler and Frobenius-Euler polynomials, we
reach the following corollary.
###### Corollary 1.
Taking $u=-1$, we have Fourier transformation of Euler function, which is
defined by Kim in [3] as follows:
$H_{m}\left(x,-1\right)=E_{m}\left(x\right)=2m!\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m+1}}\text{ }\left(\text{for }0\leq
x<1\right)\text{.}$
The Lerch trancendent $\Phi\left(z,s,a\right)$ is the analytic continuation of
the series
(2.5)
$\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(n+a\right)^{s}}$
which converges for $a\in\mathbb{C}\backslash\mathbb{Z}_{0}^{-}$,
$s\in\mathbb{C}$ when $\left|z\right|<1$; $\Re\left(s\right)>1$ when
$\left|z\right|=1$ where
$\mathbb{Z}_{0}^{-}=\mathbb{Z}^{-}\cup\left\\{0\right\\}$,
$\mathbb{Z}^{-}=\left\\{-1,-2,-3,...\right\\}$. Lerch trancendent
$\Phi\left(z,s,a\right)$ is the proportional not only Riemann zeta funtion,
Hurwitz zeta function, the Dirichlet’s eta function but also Dirichlet beta
function, the Legendre chi function, the polylogarithm, the Lerch zeta
function (for details, see [15], [17]).
We now want to indicate that
$\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m}}$ is closely related to Lerch
trancendent $\Phi\left(z,s,a\right)$. So, we compute as follows:
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m}}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi
i\right)^{m}}\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(n+\frac{1}{2}\right)^{m}}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi
i\right)^{m}}\sum_{n=-\infty}^{-1}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(n+\frac{1}{2}\right)^{m}}+\frac{1}{\left(2\pi
i\right)^{m}}\sum_{n=0}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(n+\frac{1}{2}\right)^{m}}$
After some applications on the above equation, we procure the following
(2.6) $\sum_{n=-\infty}^{\infty}\frac{e^{\left(2n+1\right)\pi
ix}}{\left(\left(2n+1\right)\pi i\right)^{m}}=-\frac{e^{\pi ix}}{\left(\pi
i\right)^{m}}+\frac{\left(-1\right)^{m}e^{\pi ix}}{\left(2\pi
i\right)^{m}}\Phi\left(e^{-2\pi ix},m,-\frac{1}{2}\right)\text{.}$
By using Theorem 1 and (2.6), we give the following theorem.
###### Theorem 2.
The following equality holds true:
$\displaystyle H_{m}\left(x,u\right)=m!\frac{u+1}{u-1}\left(-\frac{e^{\pi
ix}}{\left(\pi i\right)^{m}}+\frac{\left(-1\right)^{m}e^{\pi ix}}{\left(2\pi
i\right)^{m}}+\Phi\left(e^{-2\pi ix},m,-\frac{1}{2}\right)\right)$
$\displaystyle+2m!\left(-\frac{e^{\pi ix}}{\left(\pi
i\right)^{m+1}}+\frac{\left(-1\right)^{m+1}e^{\pi ix}}{\left(2\pi
i\right)^{m+1}}+\Phi\left(e^{-2\pi ix},m+1,-\frac{1}{2}\right)\right)$
$\displaystyle+\frac{1}{2}\left(\frac{u+1}{u-1}\right)^{2}\left(-\frac{e^{\pi
ix}}{\left(\pi i\right)^{m-1}}+\frac{\left(-1\right)^{m-1}e^{\pi
ix}}{\left(2\pi i\right)^{m-1}}+\Phi\left(e^{-2\pi
ix},m-1,-\frac{1}{2}\right)\right)$
$\displaystyle+\left(u+1\right)\sum_{k=0}^{m-4}\frac{d^{k}}{dx^{k}}H_{m}\left(x,u\right)\mid_{x=0}\left(-\frac{e^{\pi
ix}}{\left(\pi i\right)^{k+1}}+\frac{\left(-1\right)^{k+1}e^{\pi
ix}}{\left(2\pi i\right)^{k+1}}+\Phi\left(e^{-2\pi
ix},k+1,-\frac{1}{2}\right)\right)\text{.}$
For $u=-1$ on the above theorem, we have the following corollary.
###### Corollary 2.
The following identity
$E_{m}\left(x\right)=2m!\left(-\frac{e^{\pi ix}}{\left(\pi
i\right)^{m+1}}+\frac{\left(-1\right)^{m+1}e^{\pi ix}}{\left(2\pi
i\right)^{m+1}}+\Phi\left(e^{-2\pi ix},m+1,-\frac{1}{2}\right)\right)$
is true.
Setting $x=1$ in Theorem 1, we obtain
(2.7) $\displaystyle
H_{m}\left(1,u\right)=-m!\sum_{n=-\infty}^{\infty}\left[\frac{u+1}{u-1}\frac{1}{\left(2n+1\right)\pi
i}+\frac{2}{\left(\left(2n+1\right)\pi
i\right)^{2}}+\frac{1}{2}\left(\frac{u+1}{1-u}\right)^{2}\right]\frac{1}{\left(\left(2n+1\right)\pi
i\right)^{m-1}}$
$\displaystyle-\left(u+1\right)\sum_{k=0}^{m-4}\frac{1}{\left(2\pi
i\right)^{k+1}}\frac{d^{k}}{dx^{k}}H_{m}\left(x,u\right)\mid_{x=0}\sum_{n=-\infty}^{\infty}\frac{1}{\left(n+\frac{1}{2}\right)^{k+1}}\text{.}$
By expressions of (2.7) and Lemma 1, we easily see the following corollary.
###### Corollary 3.
The following identity holds true:
$\displaystyle
uH_{m}\left(u\right)=-m!\sum_{n=-\infty}^{\infty}\left[\frac{u+1}{u-1}\frac{1}{\left(2n+1\right)\pi
i}+\frac{2}{\left(\left(2n+1\right)\pi
i\right)^{2}}+\frac{1}{2}\left(\frac{u+1}{1-u}\right)^{2}\right]\frac{1}{\left(\left(2n+1\right)\pi
i\right)^{m-1}}$
$\displaystyle-\left(u+1\right)\sum_{k=0}^{m-4}\frac{1}{\left(2\pi
i\right)^{k+1}}\frac{d^{k}}{dx^{k}}H_{m}\left(x,u\right)\mid_{x=0}\sum_{n=-\infty}^{\infty}\frac{1}{\left(n+\frac{1}{2}\right)^{k+1}}\text{.}$
Now, by using Kim’s method in [3], we discover the following
(2.8a)
$\displaystyle\frac{1}{1-ue^{-t}}=\sum_{n=0}^{\infty}u^{n}e^{-nt}=\sum_{n=0}^{\infty}u^{n}\left(e^{-t}\right)^{n}=\sum_{n=0}^{\infty}u^{n}\left(\sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{t^{k}}{k!}\right)^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\sum_{a_{1}+a_{2}+...+a_{n}=n}\frac{n!}{\left(a_{1}\right)!\left(a_{2}\right)!...}\frac{\left(-1\right)^{a_{1}+2a_{2}+...}}{\left(1!\right)^{a_{1}}\left(2!\right)^{a_{2}}...}\right)t^{a_{1}+2a_{2}+...}$
Let $p\left(i,j\right):a_{1}+2a_{2}+...=i,a_{1}+a_{2}+...=j$, from expression
of (2.8a), we compute as follows:
$\displaystyle\frac{1}{1-ue^{-t}}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}u^{n}\sum_{p\left(m,n\right)}\frac{n!}{\left(a_{1}\right)!\left(a_{2}\right)!...\left(a_{m}\right)!}\frac{\left(-1\right)^{a_{1}+2a_{2}+...+ma_{m}}}{\left(1!\right)^{a_{1}}\left(2!\right)^{a_{2}}...\left(m!\right)^{a_{m}}}\right)t^{a_{1}+2a_{2}+...+ma_{m}}$
$\displaystyle=\sum_{m=0}^{\infty}\left(-1\right)^{m}\left(\sum_{n=0}^{m}n!u^{n}\sum_{p\left(m,n\right)}\frac{m!}{\left(a_{1}\right)!\left(a_{2}\right)!...\left(a_{m}\right)!}\frac{\left(-1\right)^{m}}{\left(1!\right)^{a_{1}}\left(2!\right)^{a_{2}}...\left(m!\right)^{a_{m}}}\right)\frac{t^{m}}{m!}$
(2.9a)
$\displaystyle=\sum_{m=0}^{\infty}\left[\left(-1\right)^{m}\sum_{n=0}^{m}n!u^{n}s_{2}\left(m,n\right)\right]\frac{t^{m}}{m!}\text{.}$
where $s_{2}\left(m,n\right)$ is the second kind stirling number.
Via the definition of Frobenius-Euler numbers, we readily derive the following
$\displaystyle\frac{1}{1-ue^{-t}}$ $\displaystyle=$
$\displaystyle\frac{u^{-1}}{u^{-1}-1}\frac{1-u^{-1}}{e^{-t}-u^{-1}}$
$\displaystyle=$
$\displaystyle\frac{1}{1-u}\sum_{m=0}^{\infty}\left(-1\right)^{m}H_{m}\left(u^{-1}\right)\frac{t^{m}}{m!}\text{.}$
By comparing the coefficients of $\frac{t^{n}}{n!}$ on the both sides of
(2.9a) and (2), we reach the following theorem.
###### Theorem 3.
The following equality holds true:
$\frac{1}{1-u}H_{m}\left(u^{-1}\right)=\sum_{n=0}^{m}n!u^{n}s_{2}\left(m,n\right)\text{.}$
###### Corollary 4.
Substituting $u=-1$ in the above theorem, we get the following, which is
defined by Kim [3]
$E_{m}=2\sum_{n=0}^{m}n!\left(-1\right)^{n}s_{2}\left(m,n\right)\text{.}$
## References
* [1] A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol Genocchi polynomials, Mathematics of Computation, Volume 80, Number 276, October 2011, Pages 2219–2221.
* [2] J. Choi, D. S. Kim, T. Kim and Y. H. Kim, A note on Some identities of Frobeniu-Euler Numbers and Polynomials, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 861797, 9 pages.
* [3] T. Kim, Note on the Euler numbers and polynomials, Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.
* [4] T. Kim and B. Lee, Some Identities of the Frobenius-Euler polynomials, Abstract and Applied Analysis, Volume 2009, Article ID 639439, 7 pages.
* [5] T. Kim, Euler Numbers and Polynomials Associated with Zeta Functions, Abstract and Applied Analysis 2008 (2008), Article ID 581582, 11 pages.
* [6] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288-299.
* [7] T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 $\left(\text{2008}\right),$ 51-57.
* [8] T. Kim, $q$-extension of the Euler formula and trigonometric functions, Russian J. Math. Phys. 14 (2007), 275-278.
* [9] T. Kim, On Euler-Barnes multiple zeta functions, Russian J. Math. Phys. 10 (2003), 261-267.
* [10] T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27.
* [11] T. Kim, New approach to $q$-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225.
* [12] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4, 484–491.
* [13] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equatiaons, arXiv:1201.5088v1 [math.NT].
* [14] Y. Simsek, Y. Osman, V. Kurt, On interpolation functions of the twisted generalized Frobenius-Euler numbers, Adv. Stud. Contemp. Math. 15 (2007), 187-194.
* [15] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
* [16] Q-M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp., Volume 78 (2009), No. 268, 2193–2208.
* [17] M. Acikgoz and Y. Simsek, On multiple interpolation functions of the Nörlund-Type $q$-Euler polynomials, Abstract and Applied Analysis, Vol. 2009, Article ID 382574, 14 pages.
* [18] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, to appear in Bulletin of the Malaysian Mathematical Sciences and Society.
* [19] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [20] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, Bulletin of the Korean Mathematical Society (accepted for publication).
|
arxiv-papers
| 2012-06-11T16:40:27 |
2024-09-04T02:49:31.671172
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Deyao Gao and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1206.2280"
}
|
1206.2391
|
# PROPERTIES OF THE ACCELERATION REGIONS IN SEVERAL LOOP-STRUCTURED SOLAR
FLARES
Jingnan Guo11affiliation: Dipartimento di Matematica, Università di Genova,
via Dodecaneso 35, 16146 Genova, Italy; guo@dima.unige.it, piana@dima.unige.it
, A. Gordon Emslie22affiliation: Department of Physics and Astronomy, Western
Kentucky University, Bowling Green, KY 42101; emslieg@wku.edu , Anna Maria
Massone33affiliation: CNR - SPIN, via Dodecaneso 33, I-16146 Genova, Italy;
annamaria.massone@cnr.it , AND Michele Piana11affiliation: Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy;
guo@dima.unige.it, piana@dima.unige.it 33affiliation: CNR - SPIN, via
Dodecaneso 33, I-16146 Genova, Italy; annamaria.massone@cnr.it
###### Abstract
Using RHESSI hard X-ray imaging spectroscopy observations, we analyze electron
flux maps for a number of extended coronal loop flares. For each event, we fit
a collisional model with an extended acceleration region to the observed
variation of loop length with electron energy $E$, resulting in estimates of
the plasma density in, and longitudinal extent of, the acceleration region.
These quantities in turn allow inference of the number of particles within the
acceleration region and hence the filling factor $f$ – the ratio of the
emitting volume to the volume that encompasses the emitting region(s). We
obtain values of $f$ that lie mostly between $0.1$ and $1.0$; the (geometric)
mean value is $f=0.20\,\times\\!/\\!\div\,3.9$, somewhat less than, but
nevertheless consistent with, unity. Further, coupling information on the
number of particles in the acceleration region with information on the total
rate of acceleration of particles above a certain reference energy (obtained
from spatially-integrated hard X-ray data) also allows inference of the
specific acceleration rate (electron s-1 per ambient electron above the chosen
reference energy). We obtain a (geometric) mean value of the specific
acceleration rate $\eta(20$ keV) $=(6.0\,\times\\!/\\!\\!\div\,3.4)\times
10^{-3}$ electrons s-1 per ambient electron; this value has implications both
for the global electrodynamics associated with replenishment of the
acceleration region and for the nature of the particle acceleration process.
Acceleration of particles — Sun: flares — Sun: X-rays and gamma-rays
## 1 Introduction
An important diagnostic of high-energy electrons accelerated in solar flares
is the hard X-ray bremsstrahlung that they produce as they propagate through
the ambient solar atmosphere. The Ramaty High Energy Solar Spectroscopic
Imager (RHESSI ) has revealed a new class of flares in which the bulk of the
hard X-ray emission is produced predominantly not in dense chromospheric
footpoints, but rather in the coronal loop (Veronig & Brown, 2004; Sui et al.,
2004; Krucker et al., 2008). For such sources, the corona is not only the site
of particle acceleration, but also dense enough to act as a thick target,
stopping the accelerated electrons before they can penetrate to the
chromosphere.
For suprathermal electrons with energy substantially greater than the thermal
energy of the ambient electrons with which they interact, it is appropriate to
use a collisional cold-target energy loss rate (e.g., Emslie, 1978), for which
the penetration depth of electrons increases with energy. Xu et al. (2008)
analyzed a set of extended coronal flare loops located near the solar limb,
and were indeed able to account for the observed behavior of loop extent with
photon energy $\epsilon$ in terms of a cold-target collisional model with an
extended acceleration region. Guo et al. (2012) have extended this analysis
technique to a study of the variation of loop size with electron energy $E$,
in which the visibilities used to construct the electron flux images are
obtained by regularized spectral inversion of the visibility data in the count
domain (Piana et al., 2007).
Here we apply this new analysis technique to several simple coronal loop
events observed by RHESSI. In Section 2, we present basic data for the 22
events used in the study. In Section 3 we fit the variation of loop size with
electron energy $E$ to the parametric model of Guo et al. (2012) in order to
determine the acceleration region length $L_{0}$ and density $n$ for each
event. In Section 4 these values are used to determine estimates of two
important properties of the acceleration region – the filling factor $f$ (the
ratio of the volume that is actively involved in electron acceleration to the
overall volume that encompasses the acceleration region[s]) and the specific
acceleration rate $\eta(E_{0})$ (the rate of acceleration of electrons to
energies $\geq E_{0}$ per ambient electron), and we compare the values of
these quantities to the predictions of various acceleration models.
## 2 Events Studied
Table 1: Event List and Spectral Fit Parameters Event No. | Date | Time (UT) | EM ($10^{49}$ cm-3) | T (keV) | $\delta$ | $E_{t}$ (keV) | $d{\cal N}/dt$ ($10^{35}$ s-1)
---|---|---|---|---|---|---|---
1 | 2002-04-12 | 17:42:00-17:44:32 | $0.30$ | $1.53$ | $8.24$ | $15.5$ | $2.71$
2 | | 17:45:32-17:48:00 | $0.46$ | $1.54$ | $8.01$ | $15.5$ | $4.69$
3 | 2002-04-15 | 00:00:00-00:05:00 | $0.22$ | $1.75$ | $7.48$ | $15.5$ | $4.70$
4 | | 00:05:00-00:10:00 | $0.76$ | $1.61$ | $7.93$ | $15.5$ | $9.32$
5 | | 00:10:00-00:15:00 | $1.02$ | $1.60$ | $8.37$ | $15.5$ | $11.41$
6 | 2002-04-17 | 16:54:00-16:56:00 | $0.06$ | $1.51$ | $5.70$ | $15.5$ | $0.39$
7 | | 16:56:00-16:58:00 | $0.22$ | $1.43$ | $8.78$ | $14.8$ | $2.43$
8 | 2003-06-17 | 22:46:00-22:48:00 | $1.92$ | $1.71$ | $9.95$ | $16.5$ | $17.27$
9 | | 22:48:00-22:50:00 | $2.59$ | $1.67$ | $10.36$ | $16.5$ | $17.91$
10 | 2003-07-10 | 14:14:00-14:16:00 | $1.26$ | $1.45$ | $10.05$ | $15.5$ | $7.43$
11 | | 14:16:00-14:18:00 | $1.31$ | $1.34$ | $10.38$ | $14.8$ | $8.53$
12 | 2004-05-21 | 23:47:00-23:50:00 | $0.35$ | $1.85$ | $7.07$ | $18.5$ | $3.28$
13 | | 23:50:00-23:53:00 | $0.62$ | $1.75$ | $7.51$ | $18.5$ | $2.32$
14 | 2004-08-31 | 05:31:00-05:33:00 | $0.06$ | $1.61$ | $10.56$ | $15.5$ | $0.40$
15 | | 05:33:00-05:35:00 | $0.21$ | $1.57$ | $12.19$ | $18.5$ | $0.29$
16 | | 05:35:00-05:37:00 | $0.29$ | $1.48$ | $7.45$ | $18.5$ | $0.16$
17 | 2005-06-01 | 02:40:20-02:42:00 | $0.14$ | $1.81$ | $6.53$ | $17.5$ | $1.44$
18 | | 02:42:00-02:44:00 | $0.37$ | $1.70$ | $7.86$ | $17.5$ | $2.67$
19 | 2011-02-13 | 17:33:00-17:34:00 | $0.54$ | $1.39$ | $5.86$ | $10.5$ | $35.02$
20 | | 17:34:00-17:35:00 | $0.52$ | $1.68$ | $6.55$ | $14.5$ | $19.43$
21 | 2011-08-03 | 04:31:12-04:33:00 | $0.36$ | $1.61$ | $9.23$ | $15.5$ | $3.96$
22 | 2011-09-25 | 03:30:36-03:32:00 | $0.13$ | $1.44$ | $8.33$ | $14.5$ | $1.19$
|
---|---
Figure 1: Left panel: Light curves, in the energy intervals labeled at the
top right of the plot, for the flare on 2002 April 17. The vertical lines
delineate the time intervals for Events 6 and 7. Right panel: Spectral fit to
Event #7 (16:56:00 - 16:58:00 UT). The green histogram shows the thermal
component of the spectrum (EM = $0.216\times 10^{49}$ cm-3; $T=1.43$ keV) and
the yellow histogram shows the non-thermal thick-target component (transition
energy $E_{t}=14.8$ keV; spectral index $\delta=8.78)$. The red histogram
represents the sum of the thermal and nonthermal components and the lilac
histogram represents the background.
The list of events studied is shown in Table 1. In this context, an “event” is
a time interval during a flare for which spatial and spectral observations are
sufficiently good to permit both a determination of the source spatial
structure at a variety of energies and the overall spectrum of the hard X-ray
emission. Some flares provide multiple “events”; other flares only one (see
Table 1). For each event, we fit the spatially-integrated hard X-ray emission
with an isothermal-plus-power-law form, yielding values (Table 1) of the
emission measure EM (cm-3) and temperature $T$ (keV) of the thermal source,
the intensity and spectral index $\delta=\gamma+1$ of the injected nonthermal
electron spectrum (corresponding to the hard X-ray spectral index $\gamma$),
and $E_{t}$ (keV), the transition energy between the thermal and nonthermal
components. Straightforward thick-target modeling (Brown, 1971) then provides
$d{\cal N}/dt$ (s-1), the acceleration rate of electrons above (somewhat
arbitrary) reference energy $E_{0}=20$ keV.
Parenthetically, we note that all the injected electron spectra are rather
steep (the lowest value of $\delta$ is 5.70 [Event #6], and the typical value
is in the range 7 – 9). While this may indicate a property of the electron
acceleration process in a relatively dense (see below) medium, it may also
simply be an observational selection effect – the relative paucity of high-
energy electrons in such steep spectra is consistent with the absence of
footpoint emission that such high-energy electrons would produce.
Figure 1 shows the RHESSI count rate profiles for Events 6 and 7 (16:54:00 -
16:56:00 UT, and 16:56:00 - 16:58:00 UT, respectively, on 2002 April 17) in
five different energy channels. We have identified with vertical lines the
time intervals for each event. The right panel shows the spectrum for Event
#7, with the values of the spectral fit parameters provided in the caption
(and in Table 1).
---
Figure 2: Mean electron flux maps for each event, in the representative 18-20
keV energy bin. These maps were obtained by applying the uv-smooth procedure
(Massone et al., 2009) to the electron visibilities (Piana et al., 2007)
inferred from the RHESSI count visibility data.
Electron flux images of each event, in the representative 18-20 keV energy
channel, are shown in Figure 2. These images were produced by performing a
spectral inversion on the count visibility data to obtain the corresponding
electron visibilities and then using the uv-smooth algorithm (Massone et al.,
2009) to produce maps of the electron flux (weighted by the line-of-sight
column density; Piana et al., 2007). Two aspects of this technique are worthy
of note. First, the technique exploits the fact that for the bremsstrahlung
process counts of energy $q$ are produced by all electrons with energy $E\geq
q$, so that the method yields images at electron energies $E$ up to and beyond
the maximum count energy $q_{\rm max}$ observed. Second, the regularized
spectral inversion procedure that produces the electron visibilities results
in images that, by construction, vary more smoothly with electron energy $E$
than the “parent” count images vary with $q$. This smooth variation with $E$
greatly facilitates the analysis of the following section.
## 3 Analysis
Using the electron flux maps, we first calculate the principal longitudinal
and lateral directions $s$ and $t$ for the source, using the technique
described in Guo et al. (2012). The longitudinal extent of the source external
to the acceleration region can be found by considering the standard deviation
$\sigma(E)=\sqrt{\int_{0}^{\infty}s^{2}\,F(E,s)\,ds\over\int_{0}^{\infty}F(E,s)\,ds}\,\,\,,$
(1)
where $F(E,s)$ is the electron flux spectrum at longitudinal position $s$.
Various physical processes can in principle contribute to the behavior of
$F(E,s)$. Obviously, Coulomb collisions with ambient electrons (e.g., Emslie,
1978) must be considered. In addition, various authors (e.g., Knight &
Sturrock, 1977; Emslie, 1981; Zharkova & Gordovskyy, 2006) have stressed the
need to consider also the Ohmic energy losses associated with driving the
beam-neutralizing return current in a resistive medium. Such energy losses are
proportional to the decelerating voltage difference, which is in turn
proportional to the beam current. Return current losses are therefore likely
to be significant only in large events (e.g., Emslie, 1981) with electron
acceleration rates $d{\cal N}/dt$ substantially greater than those considered
here (Table 1). For similar reasons, we believe that collective plasma effects
(see, e.g., Hoyng & Melrose, 1977; Emslie & Smith, 1984) are also likely to be
unimportant. We therefore consider only Coulomb energy losses in the
computation of $F(E,s)$.
We therefore consider a cold-target collisional injection model and a target
of uniform density $n$ (cm-3). We also neglect pitch angle scattering and
dispersion around mean values, which typically affect $F(E,s)$ by factors of
order unity (Brown, 1972; Leach & Petrosian, 1981), and we shall address these
factors briefly in Section 4. For such a scenario, the form of $F(E,s)$ can be
deduced from the one-dimensional continuity and energy loss equations
$F(E)\,dE=F_{0}(E_{0})\,dE_{0}\,\,\,;\qquad{dE\over ds}=-{Kn\over E}\,\,\,.$
(2)
(Here $K=2\pi e^{4}\Lambda$, $e$ being the electronic charge and $\Lambda$
being the Coulomb logarithm.) The solution for $E(s)$ is
$E^{2}=E_{0}^{2}-2Kns$, so that $dE_{0}/dE=E/E_{0}$ and hence, for a power-law
injection spectrum $F_{0}(E_{0})\sim E_{0}^{-\delta}$,
$F(E,s)\sim{E\over(E^{2}+2Kns)^{(\delta+1)/2}}\,\,\,.$ (3)
Substituting the expression (3) into Equation (1), we obtain, after some
algebra,
$\sigma(E)=\sqrt{\frac{2}{(\delta-3)(\delta-5)}}\,{E^{2}\over Kn}\,\,\,.$ (4)
For a model111Xu et al. (2008) also discuss a more physically self-consistent
model which incorporates the finite density in the acceleration region in the
form for $L(E)$. The corresponding expression for $L(E)$ cannot be expressed
as a simple closed form and hence is more complicated to use in a best-fit
analysis. However, this more correct form nevertheless yields results for
$L_{0}$ and $n$ that are comparable to those obtained from the “tenuous
acceleration region” model used here. in which electrons are accelerated
within a region extending from [$-L_{0}/2$,$L_{0}/2$] and injected into an
external region with uniform density $n$, we therefore arrive at the
relationship between the observed longitudinal source extent $L$ and electron
energy $E$:
${L(E)\over 2}={L_{0}\over
2}+\frac{1}{Kn}\,\sqrt{\frac{2}{(\delta-3)(\delta-5)}}\,E^{2}.$ (5)
For each event, the values of the acceleration region length $L_{0}$ and the
loop density $n$ are obtained by best-fitting Equation (5) to the inferred
form of the variation of loop length $L(E)$ with electron energy $E$ in the
predominantly nonthermal domain $E\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$>$}}\,E_{t}$ (see Guo et al., 2012). The
resulting best-fit parameters are presented in Table 2. From the electron flux
images, we also straightforwardly obtain the width (lateral extent) $W$ of the
emitting region, which typically exhibits a much smaller variation with energy
$E$ than does $L$ (see Kontar et al., 2011) and can hence be taken as a
constant. From the inferred values of $L_{0}$, $W$, and $n$, we obtain the
volume of the acceleration region
$V_{0}={\pi W^{2}L_{0}\over 4}$ (6)
and the number of particles it contains
${\cal N}=n\,V_{0}\,\,\,.$ (7)
Values of $V_{0}$ and ${\cal N}$ are provided for each event in Table 2.
### 3.1 Specific Acceleration Rate
The specific acceleration rate (electrons s-1 per electron) is defined (Emslie
et al., 2008) as the ratio of two quantities: $d{\cal N}/dt(\geq E_{0})$, the
rate of acceleration of electrons beyond energy $E_{0}$, and ${\cal N}$, the
number of particles available for acceleration:
$\eta(E_{0})=\frac{1}{\cal{N}}\,\frac{d\cal{N}}{dt}(\geq E_{0})\,\,\,.$ (8)
The quantity $d{\cal N}/dt(\geq E_{0})$ is readily determined by spectral
fitting of the spatially-integrated hard X-ray emission – see Table 1. The
quantity ${\cal N}$ can be found from Equation (7) – see values in Table 2. We
can thus deduce the value of the specific acceleration rate $\eta(E_{0})$ in
each event; values are given in Table 2.
### 3.2 Filling Factor
The soft X-ray emission measure EM is related to the plasma density $n$ and
the emitting volume $V_{\rm emit}$ through EM $=n^{2}\,V_{\rm emit}$. Given
that an emitting region may be composed of a number of discrete emitting
subregion (e.g., “strands,” “kernels”), the emitting volume may be equal to or
smaller than the total flare volume $V$ estimated from observations of the
spatial extent of the source. The ratio of the emitting volume to the volume
$V$ of the observed region that encompasses the emitting region(s) is termed
the _filling factor_
$f={V_{\rm emit}\over V}=\frac{{\rm EM}}{n^{2}V}\,\,\,.$ (9)
For each event, we estimated $f$ by using the value of EM from the spectral
fit to the thermal portion of the hard X-ray spectrum (Table 1), the density
$n$ from the fit to Equation (5) (Table 2), and the observed volume $V$ of the
emitting portion of the loop, measured at the transition energy $E_{t}$ (Table
1), the maximum energy at which thermal emission is predominant:
$V=(\pi/4)\,W^{2}L(E_{t})$. Values of $f$ for each event are given in Table 2.
## 4 Results and Conclusions
Table 2: Acceleration Region Characteristics Event No. | $L_{0}$ (arcsec) | $W$ (arcsec) | $V_{0}$ (100 arcsec3) | $n$ ($10^{11}$ cm-3) | ${\cal N}$ ($10^{37}$) | $\eta$ (20 keV) ($10^{-3}$ s-1) | $f$
---|---|---|---|---|---|---|---
1 | $18.6$ | $7.0$ | $7.2$ | $1.5$ | $4.1$ | $6.5$ | $0.45$
2 | $16.3$ | $6.9$ | $6.2$ | $1.4$ | $3.2$ | $14.5$ | $0.83$
3 | $16.7$ | $7.3$ | $7.0$ | $4.4$ | $11.7$ | $4.0$ | $0.04$
4 | $16.6$ | $7.3$ | $7.0$ | $4.8$ | $12.8$ | $7.3$ | $0.11$
5 | $16.6$ | $8.2$ | $8.7$ | $10.5$ | $34.9$ | $3.3$ | $0.03$
6 | $11.9$ | $5.9$ | $3.3$ | $4.9$ | $6.0$ | $0.6$ | $0.02$
7 | $10.4$ | $6.0$ | $3.0$ | $1.8$ | $2.0$ | $12.1$ | $0.44$
8 | $17.8$ | $6.9$ | $6.4$ | $2.6$ | $7.1$ | $24.1$ | $0.90$
9 | $18.8$ | $6.6$ | $6.5$ | $2.9$ | $7.7$ | $23.1$ | $1.05$
10 | $15.1$ | $6.0$ | $4.2$ | $2.9$ | $5.4$ | $13.8$ | $0.72$
11 | $16.0$ | $5.7$ | $4.1$ | $1.9$ | $3.1$ | $27.8$ | $1.95$
12 | $10.3$ | $6.6$ | $3.5$ | $5.1$ | $6.7$ | $4.9$ | $0.08$
13 | $9.9$ | $6.5$ | $3.3$ | $4.6$ | $5.7$ | $4.1$ | $0.18$
14 | $21.5$ | $5.3$ | $4.8$ | $1.5$ | $2.8$ | $1.4$ | $0.13$
15 | $17.4$ | $6.3$ | $5.4$ | $0.8$ | $1.7$ | $1.7$ | $1.03$
16 | $17.8$ | $6.4$ | $5.8$ | $2.3$ | $5.1$ | $0.3$ | $0.18$
17 | $11.0$ | $6.2$ | $3.3$ | $3.9$ | $5.0$ | $2.9$ | $0.05$
18 | $9.9$ | $6.3$ | $3.1$ | $3.2$ | $3.8$ | $7.0$ | $0.22$
19 | $19.9$ | $6.2$ | $6.1$ | $11.1$ | $25.7$ | $13.6$ | $0.02$
20 | $14.5$ | $6.1$ | $4.2$ | $5.2$ | $8.3$ | $23.4$ | $0.10$
21 | $9.9$ | $6.1$ | $2.9$ | $2.2$ | $2.4$ | $16.5$ | $0.53$
22 | $12.4$ | $6.0$ | $3.6$ | $1.7$ | $2.3$ | $5.2$ | $0.26$
Geometric Mean | 14.5 | 6.4 | 4.7 | 2.9 | 5.4 | 6.0 | 0.20
$\times/\div$ | 1.3 | 1.1 | 1.4 | 1.9 | 2.2 | 3.4 | 3.9
The values of $L_{0}$, $W$, $V_{0}$, $n$, ${\cal N}$, $\eta(20$ keV) and $f$
for each event are presented in Table 2. While statistical uncertainties in
these values could readily be calculated through a Monte Carlo method in which
noise is added to the RHESSI count visibility data and the process repeated
(see Guo et al., 2012), we have intentionally refrained from doing so here,
since the approximations and assumptions used in the model doubtless entail
even larger uncertainties. Instead, we let the scatter of the inferred values
of the parameters across the 22 events determine the extent over which the
parameters range. We have calculated (Table 2) the value of the (geometric)
mean value of each quantity and the (multiplicative) uncertainty in this
value. In particular, we obtain $n=(2.9\times\\!\\!/\\!\div 1.9)\times
10^{11}$ cm-3, $f=0.20\,\times\\!/\\!\div 3.9$, and $\eta(20$ keV)
$=(6.0\times\\!/\\!\div 3.4)\times 10^{-3}$ electrons s-1 per ambient
electron.
Returning to the simplifying assumptions used in determining the form of the
electron flux $F(E,s)$ (Equations [2] and [3]), we note that inclusion of
electron trajectories that have a non-zero pitch angle to the guiding magnetic
field and/or a guiding magnetic field that is inclined to the longitudinal
axis (the direction defining the coordinate $s$) will add a factor
$\mu=\overline{\cos\theta}$, where $\theta$ is the angle between the electron
velocity vector and the longitudinal direction, to the energy-dependent term
in Equation (5). This will result in a decrease (by a multiplicative factor
$\mu$) in the inferred density $n$, which in turn, by Equations (7), (8) and
(9), will increase the values of $f$ and $\eta$ by factors of $1/\mu^{2}$ and
$1/\mu$, respectively. Inclusion of return current Ohmic energy losses and/or
energy losses to waves through collective plasma effects will also decrease
the electron penetration depth, leading to further decreases in the inferred
value of $n$ and so increases in $f$ and $\eta$. The values of $f$ and $\eta$
cited above are therefore in all likelihood lower limits.
The inferred values of $f$ are generally somewhat less than unity, with the
exception of three events (## 9, 11 and 15), for which $f=$ 1.05, 1.95 and
1.03, respectively. Given the uncertainties in the data, the approximations in
the analysis method, and the factor of four spread in the inferred values of
$f$, neither of these values exceeds unity by an alarming margin. The mean
value of the filling factor obtained is consistent, within a logarithmic
standard deviation or so, with unity. This result, while not entirely
surprising, is nevertheless still significant. It validates the assumption
used by many authors (e.g., Emslie et al., 2004) that most of the observed
flare volume contains bremsstrahlung-emitting electrons; the degree to which
the emission is fragmented (e.g., striated into “kernels” or “strands” of
emission situated within a relatively inert background medium) is quite small.
The inferred mean value of $\eta$(20 keV) $\simeq 5\times 10^{-3}$ electrons
s-1 per ambient electron is broadly consistent with the values reported for a
series of extended-loop-source events by Emslie et al. (2008). It should also
be noted that the value of the specific acceleration rate for Event #4 (the
“midnight flare” of 2002 April 15) has been determined independently by Torre
et al. (2012), who used a continuity equation analysis of the variation of the
electron flux spectrum throughout the source. The specific acceleration rate
$\eta(20$ keV)$=11\times 10^{-3}$ s-1 obtained by Torre et al. (2012) is
consistent with the value of $7.3\times 10^{-3}$ s-1 deduced here.
The observationally-deduced value $\eta\simeq 10^{-2}$ s-1 implies that all
available electrons would be energized and ejected towards the footpoints
within a few hundred seconds. This result has significant implications for
supply of electrons to the acceleration region, current closure, and the
global electrodynamic environment in which electron acceleration and
propagation occur (see, e.g., Emslie & Hénoux, 1995).
The values of the filling factor $f$ deduced herein are broadly consistent
with stochastic acceleration models (e.g., Petrosian & Liu, 2004; Bian et al.,
2011, Bian et al 2012, ApJ, in press) which generally involve a near-
homogeneous distribution of scattering centers. In addition, the deduced
values of the specific acceleration rates $\eta$ are also broadly consistent
with such models. For example, in their study of electron (and proton)
acceleration in a turbulent magnetohydrodynamic wave cascade, Miller et al.
(1996, their Figures 6, 7, 9, 10 and 12) derive values of the volumetric
electron acceleration rate $\sim(1.5-4)\times 10^{8}$ cm-3 s-1 above 20 keV,
with the exact value dependent on the assumptions of the various models
considered. In the Miller et al. (1996) model, the background number density
is $n=10^{10}$ cm-3, so that $\eta\sim(1.5-4)\times 10^{-2}$ s-1 above 20 keV.
On the other hand, values of the filling factor $f$ close to unity pose
significant challenges for particle acceleration models that involve highly
localized geometries, such as super-Dreicer acceleration in thin current
sheets (see, e.g., Litvinenko & Craig, 2000; Turkmani et al., 2006). Turning
to the specific acceleration rate in such an acceleration scenario,
Heerikhuisen et al. (2002) find (their equation [3.17]) a rate of proton
acceleration $d{\cal N}_{p}/dt\simeq 2\times 10^{37}\,\sqrt{\zeta}$ s-1, where
$\zeta$ ($\eta$ in the notation of Heerikhuisen et al., 2002) is the Lundquist
number, the ratio of the diffusive to advective terms in the magnetic
diffusion equation. Following Heerikhuisen et al. (2002), we take
$\zeta=10^{-8}$, giving a proton acceleration rate $d{\cal N}_{p}/dt\sim
2\times 10^{33}$ s-1, and we take the number of protons available for
acceleration as ${\cal N}_{p}\sim\varepsilon n\,L^{3}$, where
$\varepsilon\simeq 0.4\,\sqrt{\zeta}\simeq 4\times 10^{-5}$ is the angle at
the magnetic X-type neutral point at the origin of the acceleration region.
With an ambient density $n\simeq 10^{11}$ cm-3 and a longitudinal acceleration
region extent $L\sim 10^{9}$ cm, ${\cal N}_{p}\sim 4\times 10^{33}$ and so
$\eta\simeq 2$ s-1. Although this value is much greater than the values of
$\eta$ deduced here (it corresponds to the acceleration of all ambient
particles in less than a second), it must be again stressed that the
Heerikhuisen et al. (2002) model refers to highly efficient acceleration of a
relatively small number of protons in a very localized geometry. We encourage
calculations of specific acceleration rates for electrons in such a model, and
indeed in all theoretical particle acceleration scenarios.
JG, AMM and MP have been supported by the EU FP7 Collaborative grant HESPE,
grant No. 263086; AGE was supported by NASA Grant NNX10AT78J. The authors
thank the referee for helpful comments and Richard Schwartz, Gabriele Torre,
Eduard Kontar and Federico Benvenuto for useful discussions.
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|
arxiv-papers
| 2012-06-11T22:07:56 |
2024-09-04T02:49:31.679376
|
{
"license": "Public Domain",
"authors": "Jingnan Guo, A. Gordon Emslie, Anna Maria Massone, Michele Piana",
"submitter": "Jingnan Guo Dr.",
"url": "https://arxiv.org/abs/1206.2391"
}
|
1206.2554
|
RBC and UKQCD Collaborations
# Nonperturbative tuning of an improved relativistic heavy-quark action with
application to bottom spectroscopy
Y. Aoki RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY
11973, USA Kobayashi-Maskawa Institute for the Origin of Particles and the
Universe, Nagoya University, Nagoya 464-8602, Japan N. H. Christ Physics
Department, Columbia University, New York, NY 10027, USA J. M. Flynn School
of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
T. Izubuchi RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton,
NY 11973, USA Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA C. Lehner RIKEN-BNL Research Center, Brookhaven National
Laboratory, Upton, NY 11973, USA M. Li Physics Department, Columbia
University, New York, NY 10027, USA H. Peng Physics Department, Columbia
University, New York, NY 10027, USA A. Soni Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA R. S. Van de Water Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA O.
Witzel111Present address: Center for Computational Science, Boston University,
Boston, MA 02215, USA Physics Department, Brookhaven National Laboratory,
Upton, NY 11973, USA
###### Abstract
We calculate the masses of bottom mesons using an improved relativistic action
for the $b$-quarks and the RBC/UKQCD Iwasaki gauge configurations with 2+1
flavors of dynamical domain-wall light quarks. We analyze configurations with
two lattice spacings: $a^{-1}=1.729$ GeV ($a\approx 0.11$ fm) and
$a^{-1}=2.281$ GeV ($a\approx 0.086$ fm). We use an anisotropic, clover-
improved Wilson action for the $b$-quark, and tune the three parameters of the
action nonperturbatively such that they reproduce the experimental values of
the $B_{s}$ and $B_{s}^{*}$ heavy-light meson states. The masses and mass-
splittings of the low-lying bottomonium states (such as the $\eta_{b}$ and
$\Upsilon$) can then be computed with no additional inputs, and comparison
between these predictions and experiment provides a test of the validity of
our method. We obtain bottomonium masses with total uncertainties of $\sim
0.5$–$0.6\%$ and fine-structure splittings with uncertainties of $\sim
35$–$45\%$; for all cases we find good agreement with experiment. The
parameters of the relativistic heavy-quark action tuned for $b$-quarks
presented in this work can be used for precise calculations of weak matrix
elements such as $B$-meson decay constants and mixing parameters with lattice
discretization errors that are of the same size as in light pseudoscalar meson
quantities. This general method can also be used for charmed meson masses and
matrix elements if the parameters of the heavy-quark action are appropriately
tuned.
## I Introduction
Precise knowledge of mass spectrum, decay, and mixing properties of hadrons
containing one or more bottom or charm quarks is essential to advancing our
understanding of the parameters of the Standard Model. Lattice Quantum
Chromodynamics (QCD) provides methods to compute these quantities from first
principles. Conventional lattice calculations with heavy quarks are
challenging, however, because it is impractical to use a sufficiently small
lattice spacing to control the $O(ma)^{n}$ discretization errors directly.
One way to address this challenge is to adapt the lattice description of heavy
quarks to correctly describe heavy-quark physics in a carefully circumscribed
kinematic range. Such approaches include heavy-quark effective theory (HQET)
Eichten:1989zv in which the limit of infinite quark mass is considered and
the continuum limit of the lattice theory reproduces the continuum heavy quark
effective theory. A second method is non-relativistic QCD (NRQCD)
Thacker:1990bm ; Lepage:1992tx in which the mass of the heavy quark is
assumed to be much greater than the inverse lattice spacing but the momentum-
dependence of the heavy quark energy is included in the non-relativistic
limit. Each of these approaches has its own limitations. Specifically,
radiative corrections to the coefficients of the NRQCD Lagrangian contain
power-law divergences that blow up in the limit $ma\to 0$, while HQET cannot
deal with quarkonia.
The Fermilab or relativistic heavy quark (RHQ) method ElKhadra:1996mp ;
Aoki:2001ra ; Christ:2006us provides a more complete framework for heavy-
quark physics. It applies for all values of the heavy quark mass $m_{Q}$, for
both heavy-heavy and heavy-light systems, and allows a continuum limit. The
improved RHQ action accurately describes energies and amplitudes of on-shell
states containing heavy quarks whose spatial momentum $\vec{p}$ is small
compared to the lattice spacing. It can be shown Christ:2006us that all
errors of order $|\vec{p}a|$, $(m_{Q}a)^{n}$ and $|\vec{p}a|(m_{Q}a)^{n}$ for
all non-negative integers $n$ can be removed if an anisotropic, clover-
improved Wilson action is used for the heavy quark. This action depends on
three relevant parameters: the bare quark mass $m_{0}$, an anisotropy
parameter $\zeta(m_{0}a)$ and the coefficient $c_{P}(m_{0}a)$ of an isotropic
Sheikholeslami and Wohlert term.
In order to exploit this RHQ approach, values for these three parameters are
needed. The bare charm or bottom quark mass, $m_{0}$, must be determined from
experiment, usually by equating the known mass of a physical state containing
one or two heavy quarks with the mass determined from a lattice calculation
with the RHQ action. The remaining two parameters, $\zeta$ and $c_{P}$, may be
estimated from lattice perturbation theory or determined with a
nonperturbative technique. We cannot use the nonperturbative method of Rome-
Southampton Martinelli:1994ty to tune $c_{P}$ and $\zeta$ because the Rome-
Southampton approach depends on evaluating off-shell amplitudes, whereas the
3-parameter RHQ action only controls discretization errors for on-shell
states. On-shell step-scaling methods can be used, either via the Schrödinger
functional approach or a simple comparison of small volume spectra between
calculations with varying lattice scale but identical physical volumes
Lin:2006ur . Both of these step-scaling approaches, however, involve
substantial computational effort, requiring a series of carefully matched
finite volume simulations with varying lattice spacing.
In the work reported here, we use another approach and determine the two
remaining parameters $\zeta$ and $c_{P}$ nonperturbatively by imposing two
simple conditions. The first condition is the often-exploited requirement that
the energy of a specific heavy-heavy or heavy-light state depend on that
state’s spatial momentum in a fashion consistent with continuum relativity:
$E(\vec{p})^{2}=\vec{p}^{2}+M^{2}$. The second constraint is that a specific
mass splitting agree with its experimental value. For the case of bottom, a
natural choice is the $B_{s}^{*}-B_{s}$ mass splitting. Thus, using the bottom
system as an example and including the bare quark mass $m_{0}$, we determine
our three parameters $m_{0}$, $\zeta(m_{0}a)$ and $c_{P}(m_{0}a)$ by requiring
that experimental values are obtained for $m_{B_{s}}$ and $m_{B_{s}^{*}}$ and
that $E_{B_{s}}$ has the proper dependence on $\vec{p}_{B_{s}}$.
As is described below, these three conditions are straightforward to impose
and yield quite precise results for the three unknown parameters. This
approach has the disadvantage that a possible experimental prediction from
lattice QCD, a non-trivial spin-spin splitting, cannot be made. With this
approach, however, we can immediately determine the masses of a large number
of heavy-heavy and heavy-light states. These results can be viewed as tests of
QCD and can be used to explore the accuracy and limitations of the RHQ
approach. Finally, once the RHQ action has been determined by fixing these
three parameters, it can be used to compute phenomenologically-important charm
and bottom decay constants and mixing matrix elements, which are needed for
determinations of CKM matrix elements and constraints on the CKM unitarity
triangle
In this paper we present results for the bottom system. Our calculation is
performed on the 2+1 flavor, domain wall fermion (DWF) + Iwasaki gauge-field
ensembles generated by the LHP, RBC, and UKQCD collaborations with several
values of the light dynamical quark mass at two lattice spacings, $a\approx
0.11$ fm and $a\approx 0.086$ fm Allton:2008pn ; Aoki:2010dy . For the heavy-
light mesons, the heavy quark will typically carry a small spatial momentum,
$|\vec{p}|\approx\Lambda_{\rm QCD}$. Thus, for these systems the expected
$|\vec{p}a|^{2}$ errors are of the same size as those encountered in
calculations involving only light quarks. For heavy-heavy systems, however,
the spatial momenta will be larger: $|\vec{p}|\approx\alpha_{s}m_{Q}$, where
$m_{Q}$ is the heavy-quark mass and $\alpha_{s}$ the strong interaction
coupling constant evaluated at a scale appropriate for such a bound state.
While for charmonium $\alpha_{s}m_{Q}$ may be on the order of $\Lambda_{\rm
QCD}$, this is not the case for bottomonium where discretization errors are
expected to be three to four times larger due to the larger $b$-quark mass
($m_{b}/m_{c}\approx 3.3$). Thus we choose to tune the RHQ action for
$b$-quarks using bottom-light states in order to minimize systematic
uncertainties. In particular, we match to the experimentally-measured masses
of the bottom-strange states $B_{s}$ and $B_{s}^{*}$ in order to avoid the
need to perform a chiral extrapolation in the valence light-quark mass. Once
we have determined the values of the parameters $\\{m_{0},c_{P},\zeta\\}$ we
make predictions for the masses and mass-splittings of several low-lying
bottomonium states: $\eta_{B}$, $\Upsilon$, $\chi_{b0}$, $\chi_{b1}$, and
$h_{b}$. Our results agree with experiment within estimates of systematic
uncertainties, confirming the validity of the RHQ approach and bolstering
confidence in future computations of weak matrix elements with the RHQ action.
This work was begun by Li, who presented preliminary values for the RHQ
parameters and bottomonium masses on the coarser $a\approx 0.11$ fm ensemble
at Lattice 2008 Li:2008kb . We improve upon his results in several ways, most
notably in determining the RHQ parameters solely from quantities in the
bottom-strange system. (Li used a hybrid of bottom-strange and bottomonium
observables for the tuning.) This reduces the systematic errors in the
resulting parameters due to heavy-quark discretization effects, as discussed
above. We also significantly increased the statistics, more than quadrupling
the number of configurations analyzed, and optimized the spatial smearing
wavefunction used for the $b$-quarks in order to reduce excited-state
contamination in the bottom-strange 2-point correlators. More recently Peng
extended this work to the finer $a\approx 0.086$ fm ensembles and presented
preliminary values for the RHQ parameters and bottomonium masses at Lattice
2010 Peng:Lattice10 . Again, we polish this result with increased statistics
and improved $b$-quark smearings.
This paper is organized as follows. In Section II we first present the form of
the relativistic heavy-quark action used in this work. We then describe the
numerical strategy used to determine the three parameters $m_{0}$,
$\zeta(m_{0}a)$ and $c_{P}(m_{0}a)$. Next, in Section III we present the
tuning of the RHQ parameters for bottom. We give the actions and parameters
used in our numerical simulations, and then discuss the fits of heavy-light
meson 2-point correlators needed to extract the lattice values of the $B_{s}$
and $B_{s}^{*}$ meson masses. Using this data we tune the parameters of the
RHQ action such that it applies to $b$-quarks. In Section IV we use the
resulting RHQ parameters to predict the masses of several bottomonium states
and compare the results with experiment. Finally, we summarize our results and
discuss future plans in Section V.
## II Framework of the calculation
### II.1 Heavy-quark action
The relativistic heavy-quark method provides a consistent framework for
describing both light quarks ($am_{0}\ll 1$) and heavy quarks
($am_{0}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}1$) ElKhadra:1996mp ;
Christ:2006us ; Aoki:2001ra . This approach relies upon the fact that, in the
rest frame of bound states containing one or more heavy quarks, the spatial
momentum carried by each heavy quark is smaller than the mass of the heavy
quark: for heavy-heavy systems $|\vec{p}|\sim\alpha_{s}m_{0}$ and for heavy-
light systems $|\vec{p}|\sim\Lambda_{\textrm{QCD}}$. Then one can perform the
usual Symanzik expansion in powers of the spatial derivative $D_{i}$ (which
brings down powers of $a\vec{p})$. One must, however, include terms of all
orders in the mass $m_{0}a$ and the temporal derivative $D_{0}$. This suggests
that a suitable lattice formulation for heavy quarks should break the axis-
interchange symmetry between the spatial and temporal directions.
In this work we use the following anisotropic clover-improved Wilson action
for the $b$-quarks:
$\displaystyle S_{\textrm{lat}}$
$\displaystyle=a^{4}\sum_{x,x^{\prime}}\overline{\psi}(x^{\prime})\left(m_{0}+\gamma_{0}D_{0}+\zeta\vec{\gamma}\cdot\vec{D}-\frac{a}{2}(D^{0})^{2}-\frac{a}{2}\zeta(\vec{D})^{2}+\sum_{\mu,\nu}\frac{ia}{4}c_{P}\sigma_{\mu\nu}F_{\mu\nu}\right)_{x^{\prime}x}\psi(x)\,,$
(1) where $\displaystyle D_{\mu}\psi(x)$
$\displaystyle=\frac{1}{2a}\left[U_{\mu}(x)\psi(x+\hat{\mu})-U_{\mu}^{\dagger}(x-\hat{\mu})\psi(x-\hat{\mu})\right]\,,$
(2) $\displaystyle D^{2}_{\mu}\psi(x)$
$\displaystyle=\frac{1}{a^{2}}\left[U_{\mu}(x)\psi(x+\hat{\mu})+U_{\mu}^{\dagger}(x-\hat{\mu})\psi(x-\hat{\mu})-2\psi(x)\right]\,,$
(3) $\displaystyle F_{\mu\nu}\psi(x)$
$\displaystyle=\frac{1}{8a^{2}}\sum_{s,s^{\prime}=\pm
1}ss^{\prime}\left[U_{s\mu}(x)U_{s^{\prime}\nu}(x+s\hat{\mu})U_{s\mu}^{\dagger}(x+s^{\prime}\hat{\nu})U_{s^{\prime}\nu}^{\dagger}(x)-\textrm{h.c.}\right]\psi(x)\,,$
(4)
and $\gamma_{\mu}=\gamma_{\mu}^{\dagger}$ ,
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$ and
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. Christ, Li, and Lin
showed in Ref. Christ:2006us that one can absorb all positive powers of the
temporal derivative by allowing the coefficients $c_{P}(m_{0}a)$ and
$\zeta(m_{0}a)$ to be functions of the bare-quark mass $m_{0}a$. Thus, by
suitably tuning the three coefficients in the action – the bare-quark mass
$m_{0}a$, anisotropy parameter $\zeta$, and clover coefficient $c_{P}$ – one
can eliminate errors of ${\cal O}(|\vec{p}|a)$, ${\cal O}([m_{0}a]^{n})$, and
${\cal O}(|\vec{pa}|[m_{0}a]^{n})$ from on-shell Green’s functions. The
resulting action can be used to describe heavy quarks with
$m_{0}a\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}1$ with
discretization errors that are comparable to those for light-quark systems.
The relativistic heavy quark formulation based on Ref. Christ:2006us and used
in this work is one of several variations. This general approach was first
introduced by El-Khadra, Kronfeld, and Mackenzie in Ref. ElKhadra:1996mp , and
has been used recently by the Fermilab Lattice and MILC collaborations for
many phenomenolgical applications such as decay constant and form-factor
computations Bernard:2008dn ; Bailey:2008wp ; Evans:2009du ; Simone:2010zz .
In practice, however, Fermilab/MILC use a different approach to tune the
parameters in the action, Eq. (1), than our method described below in Sec.
II.2. They fix the anisotropy parameter $\zeta$ to unity and the clover
coefficient $c_{P}$ to its tree-level mean-field improved lattice perturbation
theory value $(1/u_{0}^{3})$, and then tune only the hopping parameter
$\kappa$ (which is equivalent to the bare-quark mass) nonperturbatively
Bernard:2010fr . The Tsukuba group uses a slightly different formulation of
the action in which they do not use field rotations to eliminate redundant
operators Aoki:2003dg ; hence their version of the action has four unknown
coefficients rather than three in the RHQ or Fermilab variants. For on-shell
Green’s functions the Tsukuba and RHQ/Fermilab actions are equivalent. In
practice, however, the inclusion of redundant couplings means that one cannot
nonperturbatively tune all four parameters simultaneously by only adjusting
the energies of heavy hadrons because one will run into flat directions in
parameter space, as was shown in Ref. Christ:2006us . Hence they rely upon
lattice perturbation theory for quark-quark scattering amplitudes to determine
at least one of the coefficients Aoki:2003dg .
Because the lattice action breaks Lorentz symmetry, mesons receive corrections
to their energy-momentum dispersion relation due to lattice artifacts:
$\displaystyle(aE)^{2}=(aM_{1})^{2}+\left(\frac{M_{1}}{M_{2}}\right)(a\vec{p})^{2}+{\cal
O}([a\vec{p}]^{4})\,.$ (5)
The quantities $M_{1}$ and $M_{2}$ are known as the rest mass and kinetic
mass, respectively,
$\displaystyle M_{1}=E(\vec{p}=0)\,,\qquad
M_{2}=M_{1}\times\left(\frac{\partial E^{2}}{\partial
p_{i}^{2}}\right)^{-1}_{\vec{p}=0}\,,$ (6)
and will generally be different for generic values of the parameters
$\\{m_{0}a,c_{P},\zeta\\}$. We will exploit this fact in the RHQ tuning
procedure described in the following subsection.
### II.2 Parameter tuning methodology
We tune the values for the RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$ to
describe bottom or charm quarks by requiring that calculations of specified
physical on-shell quantities with the action in Eq. (1) correctly reproduce
the experimentally-measured results. In particular, for $b$-quarks we
determine the RHQ action using the bottom-strange system because both
discretization errors and chiral extrapolation errors are expected to be
small. We match to the experimental values of the spin-averaged $B_{s}$ meson
mass:
$\displaystyle\overline{M}_{B_{s}}=\frac{1}{4}\left(M_{B_{s}}+3M_{B_{s}^{*}}\right)$
(7)
and hyperfine splitting:
$\displaystyle\Delta M_{B_{s}}=M_{B_{s}^{*}}-M_{B_{s}}\,.$ (8)
We also require that the $B_{s}$ meson rest and kinetic masses are equal:
$\displaystyle\frac{M_{1}^{B_{s}}}{M_{2}^{B_{s}}}=1\,,$ (9)
so that the $B_{s}$ meson satisfies the continuum energy-momentum dispersion
relation
$E^{2}_{B_{s}}(\vec{p})~{}=~{}\vec{p}^{2}_{B_{s}}~{}+~{}M^{2}_{B_{s}}$. (Note
that throughout this work we denote meson masses with a capital “$M$” and
quark masses with a lower-case “$m$” in order to avoid confusion in situations
where the context is insufficient.) We could in principle have used other
states (e.g. scalar or vector mesons), other mass-splittings (e.g. the spin-
orbit splitting), or even other systems (e.g. heavy-heavy mesons) to tune the
parameters of the RHQ action, since the same parameters should describe
$b$-quarks in all of these arenas. Instead, however, we can make predictions
for these quantities using the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$ and
use them to test the validity of our approach.
We determine the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$ nonperturbatively
using an iterative procedure. The bottom-strange meson masses in general will
have a nonlinear dependence on the RHQ parameters. We choose to work in a
region sufficiently close to the true parameters such that the following
linear approximation holds:
$\left[\begin{array}[]{c}\overline{M}_{B_{s}}\\\ \Delta M_{B_{s}}\\\
\frac{M_{1}^{B_{s}}}{M_{2}^{B_{s}}}\end{array}\right]=J\cdot\left[\begin{array}[]{c}m_{0}a\\\
c_{P}\\\ \zeta\end{array}\right]+A\,,$ (10)
where $J$ is a $3\times 3$ matrix containing the linear coefficients
(analogous to the slope in the $1\times 1$ case) and $A$ is a 3-element column
vector containing the constants (analogous to the intercept). For a single
step of the iteration procedure we compute the quantities
$\\{\overline{M}_{B_{s}},\Delta M_{B_{s}},M_{1}^{B_{s}}/M_{2}^{B_{s}}\\}$ at
seven sets of parameters (see Fig. 1) in which we vary one of the three
parameters $\\{m_{0}a,c_{P},\zeta\\}$ by a chosen uncertainty
$\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$ (not to be confused with the
statistical errors in the tuned parameters $\\{m_{0}a,c_{P},\zeta\\}$) while
holding the other two fixed:
$\displaystyle\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\\\
\end{array}\right],\left[\\!\\!\begin{array}[]{c}m_{0}a-\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a+\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}-\sigma_{c_{P}}\\\ \zeta\\\ \end{array}\\!\\!\right],\;$ (23)
$\displaystyle\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}+\sigma_{c_{P}}\\\
\zeta\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}\\\ \zeta-\sigma_{\zeta}\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta+\sigma_{\zeta}\\\ \end{array}\\!\\!\right]\,.$ (33)
This allows us to test whether or not the “box” of parameter space defined by
the seven parameter sets in Fig. 1 is in the linear region such that Eq. (10)
applies. If indeed we are in the linear region, we then compute the matrix $J$
and vector $A$ via a simple finite difference approximation of the
derivatives:
$\displaystyle J$
$\displaystyle=\left[\frac{Y_{3}-Y_{2}}{2\sigma_{m_{0}a}},\,\frac{Y_{5}-Y_{4}}{2\sigma_{c_{P}}},\,\frac{Y_{7}-Y_{6}}{2\sigma_{\zeta}}\right]\,,$
(34) $\displaystyle\rule[-10.00002pt]{0.0pt}{25.00003pt}A$
$\displaystyle=Y_{1}-J\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{T}\,,$ (35)
where $Y_{i}$ is the 3-element column vector containing the values of meson
masses and splittings measured on the $i^{\textrm{th}}$ parameter set listed
in Eq. (33):
$\displaystyle Y_{i}=\left[\overline{M}_{B_{s}},\Delta
M_{B_{s}},M_{1}^{B_{s}}/M_{2}^{B_{s}}\right]^{T}_{i}\,.$ (36)
Finally, the tuned RHQ parameters are given by:
$\displaystyle\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta\end{array}\right]^{\text{RHQ}}=J^{-1}\times\left(\left[\begin{array}[]{c}\overline{M}_{B_{s}}\\\
\Delta M_{B_{s}}\\\
\frac{M_{1}^{B_{s}}}{M_{2}^{B_{s}}}\end{array}\right]^{\text{PDG}}-A\right)\,.$
(43)
Figure 1: Location of the seven sets of parameters used to obtain the tuned
values of $\\{m_{0}a,c_{P},\zeta\\}$.
We consider the RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$ to be “tuned” when
all three of the values obtained via Eq. (43) are within the “box” defined by
the seven parameter sets in Fig. 1. This condition ensures that we are
interpolating, rather than extrapolating, to the tuned point. If the result
for any of the parameters $\\{m_{0}a,c_{P},\zeta\\}$ lies outside the box, we
re-center the box around the result of Eq. (43) and perform another iteration
step. We repeat this procedure until all three tuned RHQ parameters lie inside
the box.
Once the RHQ parameters have been tuned, we can use them to predict the masses
of other heavy-light and heavy-heavy meson states, and ultimately to compute
heavy-light meson weak matrix elements. We compute the desired quantities on
the same seven sets of parameters used for the final iteration of the tuning
procedure. We then propagate the statistical errors in the tuned RHQ
parameters to these quantities using the jackknife method; this accounts for
correlations between the parameters $m_{0}$, $c_{P}$, and $\zeta$.
## III Lattice calculation of RHQ parameters for bottom
### III.1 Lattice simulation parameters
The parameters of the RHQ action suitable for describing $b$-quarks depend
upon the choice of actions for the gauge fields and sea quarks. In this work
we perform our numerical lattice computations on the “2+1” flavor domain-wall
fermion ensembles generated by the LHP, RBC, and UKQCD Collaborations
Allton:2008pn ; Aoki:2010dy . These lattices include the effects of three
light dynamical quarks; the lighter two sea quarks are degenerate and we
denote their mass by $m_{l}$, while the heavier sea quark, whose mass we
denote by $m_{h}$, is a little heavier than the physical strange quark. The
RBC/UKQCD lattices combine the Iwasaki action for the gluons Iwasaki:1983ck
with the five-dimensional domain-wall action for the fermions Shamir:1993zy ;
Furman:1994ky . Use of the Iwasaki gauge action in combination with domain-
wall sea quarks allows for adequate tunneling between topological sectors
Antonio:2008zz , and in combination with domain-wall valence quarks reduces
chiral symmetry breaking and the size of the residual quark mass as compared
to the Wilson gauge action Wilson:1974sk .
We compute the RHQ $b$-quark parameters on several ensembles with different
light sea-quark masses; this allows us to study the sea-quark mass dependence,
which we find to be statistically insignificant. We also determine the
parameters at two lattice spacings; we refer to the coarser ensembles with
$a\approx 0.11$ fm as the “$24^{3}$” ensembles and the finer ensembles with
$a\approx 0.086$ fm as the “$32^{3}$” ensembles. Use of two lattice spacings
allows us to take a naïve continuum limit of physical quantities such as meson
masses and splittings, although we still include a conservative power-counting
estimate of the residual ${\cal O}(|\vec{p}a|^{2})$ discretization errors from
the RHQ action that may not be removed with this approach. Table 1 shows the
parameters of the ensembles used for the RHQ parameter tuning and bottomonium
spectroscopy presented in this work. On the finer lattice spacings we double
the statistics by performing two fermion inversions per gauge configuration
with the origins of the quark sources separated by half of the temporal
lattice extent.
Table 1: Lattice simulation parameters used in our determination of the RHQ parameters for $b$-quarks and in our predictions for the bottomonium masses and mass-splittings. The columns list the lattice volume, approximate lattice spacing, light ($m_{l}$) and strange ($m_{h}$) sea-quark masses, unitary pion mass, and number of configurations and time sources analyzed. | | | | | | # time
---|---|---|---|---|---|---
$\left(L/a\right)^{3}\times\left(T/a\right)$ | $\approx a$(fm) | $am_{l}$ | $am_{h}$ | $M_{\pi}$(MeV) | # configs. | sources
$24^{3}\times 64$ | 0.11 | 0.005 | 0.040 | 329 | 1636 | 1
$24^{3}\times 64$ | 0.11 | 0.010 | 0.040 | 422 | 1419 | 1
$32^{3}\times 64$ | 0.086 | 0.004 | 0.030 | 289 | 628 | 2
$32^{3}\times 64$ | 0.086 | 0.006 | 0.030 | 345 | 889 | 2
$32^{3}\times 64$ | 0.086 | 0.008 | 0.030 | 394 | 544 | 2
The ensembles listed in Table 1 have already been utilized to study the light
pseudoscalar meson sector; we can therefore take advantage of many results
from this earlier work. The amount of chiral symmetry breaking in the light-
quark sector can be parameterized in terms of an additive shift to the bare
domain-wall quark mass called the residual quark mass. At the values of
$M_{5}=1.8$ and $L_{s}=16$ used by RBC/UKQCD, the size of the residual quark
mass is quite small; $am_{\textrm{res}}=0.003152(43)$ on the $24^{3}$
ensembles and $am_{\textrm{res}}=0.0006664(76)$ on the $32^{3}$ ensembles
Aoki:2010dy . In order to compute the masses of $B_{s}$ and $B_{s}^{*}$ mesons
for the tuning procedure we also need the value of the physical strange-quark
mass on these ensembles. This was already determined in Ref. Aoki:2010dy ;
$am_{s}=0.0348(11)$ on the $24^{3}$ ensembles and $am_{s}=0.0273(7)$ on the
$32^{3}$ ensembles. (In practice we use slightly different values of the
strange-quark mass — $am_{s}=0.0343$ on the $24^{3}$ ensembles and
$am_{s}=0.0272$ on the $32^{3}$ ensembles — because we began this work before
the light pseudoscalar meson analysis in Ref. Aoki:2010dy was finalized.
These values, however, are within the stated statistical errors.) Finally, we
must convert lattice meson masses into physical units for the tuning procedure
and for comparison between predictions and experiment. The lattice scale was
determined from the $\Omega$ mass to be $a^{-1}=1.729(25)$ GeV on the $24^{3}$
ensembles and $a^{-1}=2.281(28)$ GeV on the $32^{3}$ ensembles Aoki:2010dy .
These values are consistent with an independent determination of the $24^{3}$
and $32^{3}$ lattice spacings using the $\Upsilon(2S)-\Upsilon(1S)$ mass-
splitting by Meinel Meinel:2010pv .
### III.2 Heavy-light meson correlator fits
We extract the $B_{s}$ and $B_{s}^{*}$ meson energies from the exponential
behavior of the following 2-point correlation functions:
$\displaystyle C_{B_{s}}(t,t_{0};\vec{p})$
$\displaystyle=\sum_{\vec{y}}e^{ip\cdot\vec{y}}\langle{\cal
O}^{\dagger}_{P}(\vec{y},t)\tilde{{\cal O}}_{P}(\vec{0},t_{0})\rangle\,,$ (44)
$\displaystyle C_{B_{s}^{*}}(t,t_{0};\vec{p})$
$\displaystyle=\frac{1}{3}\sum_{i}\sum_{\vec{y}}e^{ip\cdot\vec{y}}\langle{\cal
O}^{\dagger}_{V_{i}}(\vec{y},t)\tilde{{\cal
O}}_{V_{i}}(\vec{0},t_{0})\rangle\,,$ (45)
where ${\cal O}_{P}$ and ${\cal O}_{V_{i}}$ are the pseudoscalar and vector
heavy-strange meson interpolating operators, respectively:
$\displaystyle{\cal O}_{P}=\overline{b}\gamma_{5}s\,,\qquad{\cal
O}_{V_{i}}=\overline{b}\gamma_{i}s\,,$ (46)
and the index “$i$” denotes the three spatial directions. We will explain the
meaning of the tilde on some of the operators in Eqs. (44) and (45) later in
this section. At sufficiently large times, excited-state contributions to
these correlators will die away and the correlators will fall off as an
exponential function of the meson ground-state energy
exp[$-E(\vec{p})(t-t_{0})$]. We can therefore obtain the ground-state energy
from the following ratio of correlators:
$\displaystyle E_{\textrm{eff}}(\vec{p})=\lim_{t\gg
t_{0}}\textrm{cosh}^{-1}\left[\frac{C(t,t_{0};\vec{p})+C(t+2,t_{0};\vec{p})}{2C(t+1,t_{0};\vec{p})}\right]\,,$
(47)
which we refer to as the “effective energy”. In the above equation and
throughout the remainder of this work, meson masses and energies are given in
lattice units (where the factor of “$a$” is implied) unless other units (e.g.
GeV) are specified.
We use the Chroma lattice QCD software system to compute the heavy and
strange-quark propagators, as well as the 2-point correlation functions
Edwards:2004sx . In order to minimize autocorrelations between data on nearby
configurations, we translate the gauge field by a randomly chosen
4-dimensional vector before computing the strange-quark and $b$-quark
propagators. We generate the domain-wall light-quark propagators with a local
(point) source; this allows them to be re-used for a future computation of
$B$-meson decay constants and mixing matrix elements. In order to suppress
excited-state contamination we generate the $b$-quark propagators with a
gauge-invariant Gaussian source for the spatial wavefunction Alford:1995dm ;
Lichtl:2006dt :
$\displaystyle\tilde{b}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\sum_{\vec{y}}S(\vec{x},\vec{y};\sigma,N)b(\vec{y},t)\,,$ (48)
where the smearing function $S(\vec{x},\vec{y})$ depends upon the width
$\sigma$ and the number of smearing iterations $N$:
$\displaystyle
S(\vec{x},\vec{y};\sigma,N)=\left(1+\frac{\sigma^{2}}{4N}\nabla^{2}_{\vec{x},\vec{y}}\right)^{N}\,,$
(49)
$\displaystyle\nabla^{2}_{\vec{x},\vec{y}}=\sum_{k=1}^{3}\left(U_{k}(x)\delta_{\vec{x}+\hat{k},\vec{y}}+U^{\dagger}_{k}(\vec{x}-\hat{k})\delta_{\vec{x}-\hat{k},\vec{y}}-2\delta_{\vec{x},\vec{y}}\right)\,.$
(50)
As long as the parameters $(\sigma,N)$ satisfy the criteria $N>3\sigma^{2}/2$,
the source is spatially smooth and a good approximation to a Gaussian. For the
free-field case ($U=1$) with large $N$ and small $\sigma$, the root-mean-
squared (rms) radius $r_{\textrm{rms}}\approx\sqrt{3}\sigma/2$ independent of
$N$. Heavy-light meson interpolating operators with a Gaussian-smeared
$b$-quark are labeled with a tilde in Eqs. (44) and (45). We use a point sink,
however, for both the strange and $b$-quark in the sink meson interpolating
field because we find that this source-sink combination minimizes statistical
errors in the correlators.
Before beginning the iterative procedure to tune the RHQ parameters described
in Sec. II.2 we compute the zero-momentum heavy-light meson pseudoscalar
correlator [Eq. (44) with $B_{s}\to B_{l}$] for several values of the Gaussian
radius; these are given in Table 2. Because we expect both the light-quark and
$b$-quark mass-dependence of the optimal smearing choice to be mild, for each
lattice spacing we analyze data on a single sea-quark ensemble and with a
single light-quark mass and set of RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$.
For the smearing study on the $24^{3}$ ensembles we use the preliminary
results for the RHQ parameters in the chiral limit from Ref. MinPhDThesis ,
$\\{m_{0}a,c_{P},\zeta\\}=\\{7.38,3.89,4.19\\}$, which are similar to the
earlier values presented at Lattice 2008 Li:2008kb . We analyze the unitary
point on the $am_{l}=0.005$ ensemble. Figure 2 shows the heavy-light
pseudoscalar meson effective mass [$E_{\textrm{eff}}(\vec{p}=0)$] for several
choices of the Gaussian radius (including the limit of a point source). The
correlator generated with a $b$-quark spatial wavefunction with a root-mean-
squared (rms) radius of $r_{\textrm{rms}}=0.777$ fm clearly has the longest
plateau with the earliest onset; we therefore choose to use this spatial
wavefunction for the RHQ parameter tuning on the $24^{3}$ ensembles. One might
worry that the extremely long plateau in Fig. 2 is due to cancellations
between excited states with positive and negative amplitudes, and does not
correspond to the true ground-state mass. Figure 3 therefore shows a
comparison of the pseudoscalar meson effective mass in which the $b$-quark has
a smeared source and point sink and one in which the $b$-quark has both a
smeared source and sink. The two effective masses agree within statistical
errors, suggesting that we have obtained the true plateau.
Table 2: Root-mean-squared radii and corresponding Chroma Gaussian smearing parameters [defined in Eq. (49)] considered here. The parameters shown in bold are used to obtain the RHQ parameters in the following subsection. | $a\approx$ 0.086 fm | | $a\approx$ 0.11 fm
---|---|---|---
$r_{\textrm{rms}}$ (fm) | $\sigma$ | $N$ | | $\sigma$ | $N$
0.137 | 1.39 | 5 | | 1.83 | 5
0.275 | 2.78 | 15 | | 3.6 | 25
0.518 | 5.24 | 5 | | 6.92 | 80
0.777 | 7.86 | 100 | | 10.36 | 170
1.035 | 10.48 | 175 | | |
1.047 | | | | 13.98 | 310
Figure 2: Pseudoscalar meson effective mass for several choices for the
Gaussian radius of the $b$-quark in the heavy-light meson interpolating
operator. Results are shown for the unitary point on the $am_{l}=0.005$
$24^{3}$ ensemble with RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}=\\{7.38,3.89,4.19\\}$. Figure 3: Pseudoscalar meson
effective mass for the $b$-quark Gaussian radius
$r_{\textrm{rms}}={\color[rgb]{0,0,0}0.777}$ fm. The full symbols correspond
to correlators in which the $b$-quark is generated with a Gaussian spatial
wavefunction but has a point sink; the open points correspond to correlators
in which the $b$-quark has a Gaussian spatial wavefunction at both the source
and sink. The effective masses agree, but the smeared-point data has smaller
statistical errors.
For the smearing study on the $32^{3}$ ensemble we analyze the unitary point
on the $am_{l}=0.004$ sea-quark ensemble. We use the RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}=\\{3.70,3.60,2.20\\}$, which are close to the
preliminary results on the $am_{l}=0.004$ ensemble in Ref. Peng:Lattice10 . As
in the case of the $24^{3}$ ensembles, the Gaussian radius of
$r_{\textrm{rms}}=0.777$ fm leads to the best plateau, so we use it for the
RHQ parameter tuning procedure. This is consistent with expectations that the
size of the $B$-meson in physical units should be independent of the lattice
spacing.
We estimate the errors in the correlation functions and in the fitted meson
energies using a single-elimination jackknife procedure. This allows us to
propagate the statistical uncertainties including correlations between the
parameters $\\{m_{0}a,c_{p},\zeta\\}$ into subsequent steps of the RHQ
parameter tuning procedure. We find no evidence of residual autocorrelations
between subsequent trajectories, as measured by comparing the errors between
binned and un-binned data. We perform the $\chi$-squared minimization
including the full covariance matrix, and choose fit ranges that yield
acceptable correlated confidence levels ($p$-value222We adopt the PDG
convention that the $p$-value is the probability of finding a $\chi^{2}$ value
greater than that obtained in the fit; hence a larger $p$-value denotes a
stronger compatibility between the data and the fit hypothesis
Nakamura:2010zzi . $\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}$ 10%).
Because the $B_{s}$ and $B_{s}^{*}$ meson energies are largely insensitive to
the sea-quark mass, we expect the excited-state contamination to die off and
the onset of the ground-state plateau to occur at around the same location on
all sea-quark ensembles for a given lattice spacing. We therefore choose the
same fit range for all sea-quark ensembles on a given lattice spacing. The
requirement that we obtain a constant fit to the effective energy with a good
correlated confidence level using the same fitting range for all ensembles
helps to ensure that we obtain the true ground-state energy, and are not
misled by “wiggles” in the plateau that are due to fluctuations in the gauge
field, but are different on each ensemble. We do not, however, expect excited-
state contributions to be the same for all momenta, and, in fact, we observe
an earlier onset for the plateau in the zero momentum effective energy than
for the other momenta. Table 3 shows the fitting ranges used on the $24^{3}$
and $32^{3}$ ensembles. Figure 4 shows the $B_{s}$ and $B_{s}^{*}$ meson
effective energies for lattice momenta up to $(a\vec{p})^{2}=3$ on the
$am_{l}=0.005$ $24^{3}$ ensemble. Effective energy plots for the other
$24^{3}$ and $32^{3}$ ensembles look similar.
Figure 4: Heavy-strange pseudoscalar meson (blue circles) and vector meson (red triangles) effective energies on the $am_{l}=0.005$ $24^{3}$ ensemble with RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}=\\{8.40,5.80,3.20\\}$. From upper-left to lower-right the six plots show spatial momenta $(a\vec{p})^{2}=0$ through $(a\vec{p})^{2}=3$. For each plot the shaded horizontal band shows the fit range used and the fit result with jackknife statistical errors. Table 3: Time ranges used in plateau fits of the $B_{s}$ and $B_{s}^{*}$ effective energies. We use different ranges for zero and nonzero momenta, but use the same range for all sea-quark masses at a given lattice spacing. | fit range
---|---
| $\vec{p}=0$ | | $\vec{p}\neq 0$
$a\approx 0.11$ fm | [10,25] | | [10,25]
$a\approx 0.086$ fm | [11,21] | | [14,21]
### III.3 Determination of bottom-quark parameters
We begin our iterative tuning procedure using the preliminary values for
$\\{m_{0}a,c_{P},\zeta\\}$ determined in the pilot studies of Refs.
MinPhDThesis and Peng:Lattice10 . We compute the $B_{s}$ and $B_{s}^{*}$
meson energies for seven sets of parameters surrounding these values. We then
determine the ratio of the rest mass to the kinetic mass for these seven
parameter sets by fitting the nonzero momentum data for the $B_{s}$ meson to
the energy-momentum dispersion relation, Eq. (5). Finally, we determine the
predicted values of the RHQ parameters from Eq. (43) using the experimentally-
measured meson masses $M_{B_{s}}=5.366$ GeV and $M_{B_{s}^{*}}=5.415$ GeV
Nakamura:2010zzi . We find that the resulting values of
$\\{m_{0}a,c_{P},\zeta\\}$ lie outside the “box” determined by the seven
parameter sets. We therefore re-center the box around the newly-determined
values and repeat the procedure. We find that we need to iterate once or twice
before the values of $\\{m_{0}a,c_{P},\zeta\\}$ settle down and remain inside
the box. Here we only show results for the final iteration, since plots for
intermediate iterations look similar. The final sets of parameters used to
obtain the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$ on the $24^{3}$ and
$32^{3}$ ensembles are given in Table 4.
Table 4: Final “box” of parameters used to obtain the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$ (see Fig. 1). In each column the first number is the central value of the parameter and the second number is the variation. | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$a\approx 0.11$ fm | 8.40 $\pm$ 0.15 | 5.80 $\pm$ 0.45 | 3.20 $\pm$ 0.30
$a\approx 0.086$ fm | 3.98 $\pm$ 0.10 | 3.60 $\pm$ 0.30 | 1.97 $\pm$ 0.15
Figure 5 shows the energy-momentum dispersion relation fit for both the
$B_{s}$ and $B_{s}^{*}$ mesons on the $am_{l}=0.005$ $24^{3}$ ensemble.
Dispersion relation plots for the other sea-quark masses, RHQ parameter sets,
and lattice spacing look similar. The slopes ($M_{1}/M_{2}$) of the $B_{s}$
and $B_{s}^{*}$ energy-momentum dispersion relations agree with unity (and
hence with each other) within errors in the region of the parameter space near
the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$. We choose to use the
pseudoscalar meson data, however, for the parameter tuning because it has
smaller statistical errors. We perform a one parameter linear fit in which we
fix the intercept to go through the measured value of the rest mass
$E(\vec{p}=0)$ and allow the slope to vary. We include data with lattice
momenta through $(ap)^{2}=3$, and see no evidence for higher-order, e.g.
${\cal O}([ap]^{4})$, lattice discretization effects at these values of the
momenta. We account for correlations between data points by propagating the
jackknife values of the energies from the 2-point fits described in the
previous subsection. As a cross-check we compare the fit result with those of
a two-parameter fit in which we allow both the slope and intercept to vary; we
find that the results are consistent, and choose to use the one-parameter fit
because it leads to smaller statistical errors in
$M_{1}^{B_{s}}/M_{2}^{B_{s}}$.
Figure 5: $B_{s}$ (blue circles) and $B_{s}^{*}$ (red triangles) meson
squared-energy difference versus spatial momentum-squared on the
$am_{l}=0.005$ $24^{3}$ ensemble for the RHQ parameter values
$\\{m_{0}a,c_{P},\zeta\\}={\\{8.40,5.80,3.20\\}}$. The slope of the data gives
the ratio of the meson rest mass over the kinetic mass $(M_{1}/M_{2})$. Data
points shown with open symbol are not included in the fit.
In order to reliably determine the RHQ parameters via Eq. (43) we must be
interpolating in a regime in which the bottom-strange meson observables
$\\{\overline{M}_{B_{s}},\Delta M_{B_{s}},M_{1}^{B_{s}}/M_{2}^{B_{s}}\\}$
depend linearly upon the parameters in the action $\\{m_{0}a,c_{P},\zeta\\}$.
We test this assumption and look for signs of curvature by computing the
observables for three different boxes of seven parameters with sizes
$\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$, $\pm
2\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$, and $\pm
3\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$ (except for the parameter $m_{0}a$ on the
$24^{3}$ ensemble for which the largest box is $\pm 4\sigma_{m_{0}a}$). We
then determine the predicted values of the RHQ parameters for each of the
three boxes; we find that the difference is negligible within statistical
errors.
Figure 6 shows the dependence of the spin-averaged mass, hyperfine splitting,
and rest mass over kinetic mass on the parameters $m_{0}a$, $c_{P}$, and
$\zeta$, respectively, on the $am_{l}=0.005$ $24^{3}$ ensemble. We plot these
dependencies because these are the parameters to which each observable is most
sensitive. The bottom-strange observables $\\{\overline{M}_{B_{s}},\Delta
M_{B_{s}},M_{1}^{B_{s}}/M_{2}^{B_{s}}\\}$ depend linearly on the parameters
$\\{m_{0}a,c_{P},\zeta\\}$ throughout the range. The analogous plots for the
other sea-quark ensembles look similar.
Figure 6: Spin-averaged mass versus $m_{0}a$ (upper plot), hyperfine splitting
versus $c_{P}$ (center plot), and rest mass over kinetic mass versus $\zeta$
(lower plot) on the $am_{l}=0.005$ $24^{3}$ ensemble. The solid vertical lines
with shaded gray error bands denote the tuned values of the RHQ parameters
with jackknife statistical errors. For each quantity, the dashed line shows
the dependence on $m_{0}a$, $c_{P}$, or $\zeta$ calculated from Eqs.
(10)–(36).
Table 5 shows the nonperturbatively-tuned RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}$ obtained on the two $24^{3}$ ensembles. We do not
observe any statistically-significant sea-quark mass dependence. Hence,
instead of extrapolating the RHQ parameters to the physical light-quark
masses, we simply take an error-weighted average of the two values to obtain
our final preferred results. Similarly, Table 6 shows the nonperturbatively-
tuned RHQ parameters on the three $32^{3}$ ensembles and the corresponding
weighted average.
Table 5: Tuned RHQ parameter values on the $24^{3}$ ensembles determined using the parameter sets in Table 4. Because we do not observe any statistically-significant sea-quark mass dependence, we obtain the final preferred values from an error-weighted average of the two sets of results. | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$am_{l}=0.005$ | 8.43(7) | 5.7(2) | 3.11(9)
$am_{l}=0.01$ | 8.47(9) | 5.8(2) | 3.1(1)
average | 8.45(6) | 5.8(1) | 3.10(7)
Table 6: Tuned RHQ parameter values on the $32^{3}$ ensembles determined using the parameter sets in Table 4. Because we do not observe any statistically-significant sea-quark mass dependence, we obtain the final preferred values from an error-weighted average of the three sets of results. | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$am_{l}=0.004$ | 4.07(6) | 3.7(1) | 1.86(8)
$am_{l}=0.006$ | 3.97(5) | 3.5(1) | 1.94(6)
$am_{l}=0.008$ | 3.95(6) | 3.6(1) | 1.99(8)
average | 3.99(3) | 3.57(7) | 1.93(4)
### III.4 Comparison with perturbation theory
It is useful to compare the nonperturbatively-determined values of the RHQ
parameters with those computed in lattice perturbation theory. First, this
provides a consistency check of the nonperturbative tuning procedure. Second,
this allows us to see how well the perturbative estimates are working in a
case where we know the true nonperturbative value. Reasonable agreement
between the two approaches bolsters confidence in our ability to rely on
lattice perturbation theory in future situations where we do not have
nonperturbative matching factors available.
We calculate the RHQ parameters $c_{P}$ and $\zeta$ at 1-loop in mean-field
improved lattice perturbation theory Lepage:1992xa . The details of the
perturbative calculation will be given in a separate publication LehnerLPT .
The clover coefficient $c_{P}$ is obtained by matching the lattice quark-gluon
vertex to the continuum counterpart in the on-shell limit. At intermediate
steps of the calculation infrared divergences are regulated with a nonzero
gluon mass $\lambda$; the final results are obtained in the limit $\lambda\to
0$. Similarly, the anisotropy parameter $\zeta$ is obtained by requiring that
the lattice heavy-quark dispersion relation, extracted from the momentum
dependence of the pole in the heavy-quark propagator at one loop, agrees with
the continuum. We implement the mean-field improvement in two ways. First we
use the nonperturbative value of the fourth root of the plaquette,
$u_{0}=P^{1/4}$, to resum tadpole contributions as in Ref. Lepage:1992xa . We
also use the value of the spatial link field in Landau gauge to estimate
$u_{0}$. A comparison of these two approaches is useful for ascertaining the
systematic uncertainty due to the ambiguity in how to implement the tadpole
resummation. The lattice perturbation theory calculations of $c_{P}$ and
$\zeta$ also use the nonperturbatively-determined values of the bare-quark
mass $m_{0}a$ and the $2\times 1$ rectangle $R$ as inputs. The latter allows
for a refined resummation of tadpole contributions in improved gauge actions
Ali_Khan:2001tx .
Figure 7 compares results on the $24^{3}$ ensembles in both unimproved and
mean-field improved lattice perturbation theory with the nonperturbatively-
determined values. The results on the $32^{3}$ ensembles look qualitatively
similar. The use of mean-field improved lattice perturbation theory brings the
perturbative results into better agreement with the nonperturbative values. It
also reduces the size of the one-loop corrections, thereby appearing to
improve the convergence of the perturbative series, although one cannot be
entirely sure that this trend persists to higher orders. In the case of
$c_{P}$, the unimproved one-loop corrections are very large (approximately a
factor of 1.5) but are reduced to a more sensible level by resumming tadpole
contributions, whereas in the case of $\zeta$ the unimproved one-loop
corrections are already close to the naïve power-counting estimate of
$\alpha_{S}^{\overline{\rm MS}}(1/a_{24^{3}})\sim 23\%$ and the mean-field
improved one-loop correctons are even smaller.
Figure 7: Lattice perturbation theory calculations of $c_{P}$ (left plot) and
$\zeta$ (right plot) on the $24^{3}$ ensembles LehnerLPT . From left to right,
the perturbative calculations shown are (T) unimproved tree-level, (1)
unimproved 1-loop, $({\rm T}_{\rm P})$ mean-field improved tree-level using
the plaquette to estimate the tadpole factor $u_{0}$, ($1_{\rm PL}$), 1-loop
mean-field improved value using the lattice coupling and the plaquette,
($1_{\rm PM}$) 1-loop mean-field improved value using the $\overline{{\rm
MS}}$ coupling and the plaquette, (${\rm T_{U}}$), mean-field improved tree-
level using the spatial link in Landau gauge to estimate $u_{0}$, ($1_{\rm
UL}$), 1-loop mean-field improved value using the lattice coupling and the
spatial link, and ($1_{\rm UM}$) 1-loop mean-field improved value using the
$\overline{{\rm MS}}$ coupling and the spatial Landau link. In each plot, the
horizontal line indicates our choice of central value for $c_{P}$ or $\zeta$
while the solid horizontal band denotes our estimate of the uncertainty with
errors due to the truncation of the perturbative series and errors due to the
uncertainty in $m_{0}a$ added in quadrature. For comparison, the
nonperturbatively-determined values are shown at the far right with
statistical errors (solid inner error bar) and statistical and systematic
errors added in quadrature (dashed outer error bar).
We can use the results shown in Fig. 7 to estimate the uncertainties in the
values of $c_{P}$ and $\zeta$ calculated in lattice perturbation theory. We
consider two approaches for obtaining the error. A naïve power-counting
estimate of the size of the neglected 2-loop corrections would lead to a
predicted error of $\alpha_{S}^{2}\sim 5\%$. As mentioned earlier, however,
there is an ambiguity in how to estimate the tadpole factor $u_{0}$ used in
the resummation procedure. This is not strictly a measure of the size of
higher-order corrections, but taking the difference between the values of
$c_{P}$ and $\zeta$ computed at one-loop using $u_{0}$ from the plaquette and
from the spatial Landau link gives a larger estimate of the error in $c_{P}$
($\sim$10–12.5%) than the naïve power-counting approach. We therefore take
this difference to be the error in the perturbatively-calculated value of
$c_{P}$, but take $\alpha_{S}^{2}\sim 5\%$ to be the error in the
perturbatively-calculated value of $\zeta$. For the central values we quote
the average of the one-loop mean-field improved values expanded in the
$\overline{{\rm MS}}$ coupling at scale $a^{-1}$ and computed with $u_{0}$
obtained from the plaquette and from the spatial Landau link.
Our final perturbative estimates for $c_{P}$ and $\zeta$ on the $24^{3}$ and
$32^{3}$ ensembles are given in Table 7. They agree with the
nonperturbatively-determined values given in Table 8 in all cases. In order to
provide a fair comparison, we include an estimate of systematic errors for
both the perturbatively-calculated and nonperturbatively-computed values. The
largest source of uncertainty in the lattice perturbation theory
determinations is the error due to neglected terms in the coupling-constant
expansion of ${\cal O}(\alpha_{S}^{2})$ and higher. In contrast, the largest
source of uncertainty in the nonperturbative determinations of $c_{P}$ and
$\zeta$ is heavy-quark discretization errors from neglected operators in the
action of ${\cal O}(a^{2}p^{2})$ and higher (for $m_{0}a$ the uncertainty in
the lattice scale dominates). The good agreement between lattice perturbation
theory and the nonperturbative tuning procedure suggests that one-loop mean-
field improved lattice perturbation theory is sufficiently reliable that it
can be used in situations where the nonperturbative matching factors are not
available, such as in our future computations of decay constants and mixing
matrix elements.
Table 7: One-loop mean-field improved lattice perturbation theory predictions for the RHQ parameters $c_{P}$ and $\zeta$ (right panel) LehnerLPT . The nonperturbative inputs used in the calculation – the bare heavy-quark mass $m_{0}a$, the plaquette $P$, the $2\times 1$ rectangle $R$, and the spatial link in Landau gauge $L$ – are given in the center panel. The errors in $c_{P}$ and $\zeta$ are due to the truncation of lattice perturbation theory and the uncertainty in $m_{0}a$, respectively. The jackknife statistical errors in $P$, $R$, and $L$ are negligible. | nonperturbative | perturbative
---|---|---
| inputs | estimates
| $m_{0}a$ | $P$ | $R$ | $L$ | $c_{P}$ | $\zeta$
$a\approx 0.11$ fm | 8.45 | 0.588 | 0.344 | 0.844 | 4.8(6)(2) | 3.2(2)(1)
$a\approx 0.086$ fm | 3.99 | 0.616 | 0.380 | 0.861 | 3.04(28)(7) | 2.10(11)(5)
Table 8: Tuned values of the RHQ parameters on the $24^{3}$ and $32^{3}$ ensembles. The central values and statistical errors are from Tables 5 and 6. The systematic error estimates are obtained using the same approach as for the bottomonium masses and mass-splittings described in Sec. IV.3. The errors listed in $m_{0}a$, $c_{P}$, and $\zeta$ are from left to right: statistics, heavy-quark discretization errors, the lattice scale uncertainty, and the uncertainty in the experimental measurement of the $B_{s}$ meson hyperfine splitting, respectively. Errors that were considered but were found to be negligible are not shown. For the scale uncertainty we quote smaller errors on the $32^{3}$ ensembles because the lattice-spacing is determined more precisely than on the $24^{3}$ ensembles. | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$a\approx 0.11$ fm | 8. | 45(6)(13)(50)(7) | 5. | 8(1)(4)(4)(2) | 3. | 10(7)(11)(9)(0)
$a\approx 0.086$ fm | 3. | 99(3)(6)(18)(3) | 3. | 57(7)(22)(19)(14) | 1. | 93(4)(7)(3)(0)
## IV Bottomonium mass predictions
Given the determinations of the RHQ parameters described in the previous
section, we can now make predictions for other states involving $b$-quarks,
such as bottomonium masses and splittings. Comparison of the results with
experiment then provides a check of the relativistic heavy quark framework and
tuning methodology.
### IV.1 Heavy-heavy meson correlator fits
We extract the bottomonium meson masses from the following zero-momentum meson
2-point correlation functions:
$\displaystyle C_{\overline{b}b}(t,t_{0})$
$\displaystyle=\sum_{\vec{y}}\langle{{\cal
O}^{\Gamma}_{\overline{b}b}}^{\dagger}(\vec{y},t)\tilde{{\cal
O}}_{\overline{b}b}^{\Gamma}(\vec{0},t_{0})\rangle\,,$ (51)
where ${\cal O}^{\Gamma}_{\overline{b}b}$ is the $b\overline{b}$ meson
interpolating operator for the state with spin structure $\Gamma$:
$\displaystyle{\cal O}^{\Gamma}_{\overline{b}b}=\overline{b}\Gamma b\,.$ (52)
Table 9 shows the interpolating operators used in the computation of the
bottomonium 2-point functions. Again, the tilde over the interpolating
operator in Eq. (51) denotes that the $b$-quark in the operator was generated
with a Gaussian-smeared source.
Table 9: Interpolating operators used to compute the $\overline{b}b$ 2-point correlation functions. We average correlators over equivalent directions for the vector, axial-vector, and tensor states. meson | operator
---|---
$\eta_{b}$ | $\gamma_{5}$
$\Upsilon$ | $\gamma_{i}$
$\chi_{b0}$ | $1$
$\chi_{b1}$ | $\gamma_{i}\gamma_{5}$
$h_{b}$ | $\gamma_{i}\gamma_{j}$
Plots of the effective energy, Eq. (47), for the bottomonium correlators show
that excited-state contamination is significant for the choice of smearing
that we used to obtain the RHQ parameters. In fact, on the $32^{3}$ ensembles
excited-state contamination appears to persist over the entire time range up
to the temporal mid-point of the lattice, making a clean determination of the
ground-state mass difficult. We therefore choose to use a different smearing
for the $b$ quarks in the bottomonium correlators than for those in the
bottom-strange correlators. We perform a similar smearing study to that
described for bottom-strange states in Sec. III.2. Figure 8 shows the
$\Upsilon$ (vector) and $\chi_{b0}$ (scalar) meson effective masses on the
$am_{l}=0.005$ $24^{3}$ ensemble for several choices of the Gaussian radius
and values of the RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}=\\{8.40,5.80,3.20\\}$. The correlator generated with
a $b$-quark spatial wavefunction with $r_{\textrm{rms}}=0.137$ fm has the
longest plateau with the earliest onset; we therefore choose to use this
spatial wavefunction to compute the bottomonium masses and mass-splittings on
the $24^{3}$ ensembles. We perform an analogous smearing test on the
$am_{l}=0.004$ $32^{3}$ ensemble with RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}=\\{3.70,3.60,2.20\\}$. Again, we find that the
Gaussian spatial wavefunction with $r_{\textrm{rms}}=0.137$ fm is best.
Physically one expects a $\overline{b}b$ meson to have a narrower spatial
wavefunction than a $\overline{b}s$ meson, and this is consistent with our
observations. We find an optimal wavefunction that is approximately half as
wide as the bottomonium rms radius
$r_{\textrm{rms}}^{\textrm{Richardson}}=0.224(23)$ fm computed from the
Richardson potential model Menscher:2005kj .
Figure 8: $\Upsilon$ (upper plot) and $\chi_{b0}$ (lower plot) effective mass
for several choices for the Gaussian radius of the $b$-quark in the heavy-
light meson interpolating operator. Results are shown for the $am_{l}=0.005$
$24^{3}$ ensemble with RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}=\\{8.40,5.80,3.20\\}$.
Using the optimized $b$-quark smearing, we then compute the bottomonium
correlators, Eq. (51), on each ensemble for the final set of seven RHQ
parameters used in the iterative tuning procedure. This enables us to
propagate the statistical uncertainties in the RHQ parameters from the tuning
procedure into our determinations of the bottomonium masses and mass-
splittings. We determine the ground-state meson masses from constant fits to
the effective mass. We observe similar excited-state contamination in the
$\eta_{b}$ and $\Upsilon$ states, so we choose a fit range that yields a good
correlated confidence level for fits to both effective masses. Similarly, we
use the same fit range for the $\chi_{b0}$, $\chi_{b1}$, and $h_{b}$ states.
Finally, because we do not expect any significant sea-quark mass dependence,
we use the same fit range for all sea-quark ensembles with the same lattice
spacing. These constraints help to ensure that we are not fooled by false
plateaus due to fluctuations in the gauge field, which will differ among
uncorrelated ensembles. Table 10 gives the fit ranges to determine the various
meson masses on the two lattice spacings. Figure 9 shows sample bottomonium
effective masses and mass-splittings on the $am_{l}=0.005$ $24^{3}$ ensemble.
Plots for other sea-quark ensembles (including at the finer lattice spacing)
and other values of the RHQ parameters look similar.
Figure 9: Bottomonium masses and mass-splittings on the $am_{l}=0.005$ $24^{3}$ ensemble with RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}=\\{8.40,5.80,3.20\\}$. The meson states shown in each plot are specified in the legend. For each plot the shaded horizontal band shows the fit range used and the fit result with jackknife statistical errors. Table 10: Time ranges used in plateau fits of the bottomonium effective masses. We use different ranges for the $\eta_{b}$ and $\Upsilon$ states than for the $\chi$ and $h$ states, but use the same range for all sea-quark masses at a given lattice spacing. | fit range
---|---
| $\eta_{b}$ & $\Upsilon$ | | $\chi_{b0}$, $\chi_{b1}$, & $h_{b}$
$a\approx 0.11$ fm | [15,30] | | [4,12]
$a\approx 0.086$ fm | [13,30] | | [7,20]
### IV.2 Determination of bottomonium masses and fine-structure splittings
We determine the predicted values of the bottomonium masses at the tuned RHQ
parameters using equations similar to Eqs. (34)–(43):
$\displaystyle
M_{\overline{b}b}^{\text{RHQ}}=J_{M}\times\left[\begin{array}[]{c}m_{0}a\\\
c_{P}\\\ \zeta\end{array}\right]^{\text{RHQ}}+A_{M}\,,$ (56)
where the $1\times 3$ matrix $J_{M}$ and constant $A_{M}$ are determined from
a finite difference approximation of the derivatives:
$\displaystyle J_{M}$
$\displaystyle=\left[\frac{M_{3}-M_{2}}{2\sigma_{m_{0}a}},\,\frac{M_{5}-M_{4}}{2\sigma_{c_{P}}},\,\frac{M_{7}-M_{6}}{2\sigma_{\zeta}}\right]\,,$
(57) $\displaystyle A_{M}$
$\displaystyle=M_{1}-J_{M}\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{T}$ (58)
and $M_{i}$ is the $\overline{b}b$ meson mass measured on the
$i^{\textrm{th}}$ parameter set listed in Eq. (33). (Note that the values of
$M_{i}$, $J_{M}$, and $A_{M}$ are different for each bottomonium state.) For
each jackknife set we use the values of the tuned RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}^{\textrm{RHQ}}$ determined on that jackknife set,
thereby preserving correlations between the three parameters $m_{0}a$,
$c_{P}$, and $\zeta$. Hence the jackknife statistical errors in the
$\overline{b}b$ meson masses determined via Eq. (56) already include the
uncertainty due to the statistical errors in the tuned RHQ parameters.
The use of Eqs. (57) and (58) requires that we are in a regime in which the
bottomonium masses depend linearly on the RHQ parameters. We test this
assumption and look for signs of curvature by computing the bottomonium masses
for three different boxes of seven parameters with sizes
$\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$, $\pm
2\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$, and $\pm
3\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$. Figures 10 and 11 show the seven
bottomonium masses and splittings — $M_{\eta_{b}}$, $M_{\Upsilon}$,
$M_{\Upsilon}-M_{\eta_{b}}$, $M_{\chi_{b0}}$, $M_{\chi_{b1}}$,
$M_{\chi_{b1}}-M_{\chi_{b0}}$, and $M_{h_{b}}$ — versus $m_{0}a$, $c_{P}$, and
$\zeta$ on the $am_{l}=0.005$ $24^{3}$ ensemble; plots for the $am_{l}=0.004$
$32^{3}$ ensemble look similar. The statistical errors in the $\overline{b}b$
meson masses are approximately ten times smaller than those of the bottom-
strange meson masses, and we can resolve a nonlinear dependence of the
$\overline{b}b$ meson masses on the RHQ parameters within statistical errors.
This curvature is most pronounced in the hyperfine splitting
$M_{\Upsilon}-M_{\eta_{b}}$, and the dependence is strongest upon the
parameter $\zeta$. The nonlinear dependence is mild, however, within the
region of parameter space defined by the inner-most box of parameters. Hence
we expect that the use of Eq. (56) in this region will lead to only a small
error in the bottomonium mass predictions. Nevertheless, we will include a
systematic uncertainty in our predictions for the bottomonium meson masses due
to quadratic and higher-order corrections to Eq. (56).
Figure 10: Bottomonium masses versus $m_{0}a$ (upper plots), $c_{P}$ (center
plots), and $\zeta$ (lower plots) on the $am_{l}=0.005$ $24^{3}$ ensemble. The
meson states shown in each plot are specified in the legend. The solid
vertical lines with shaded gray error bands denote the tuned values of the RHQ
parameters with jackknife statistical errors. For each quantity, the dashed
line in the same color as the plotting symbol shows the dependence on the RHQ
parameters calculated from Eqs. (56)–(58).
Figure 11: Bottomonium mass-splittings versus $m_{0}a$ (upper plots), $c_{P}$
(center plots), and $\zeta$ (lower plots) on the $am_{l}=0.005$ $24^{3}$
ensemble. The hyperfine splitting $M_{\Upsilon}-M_{\eta_{b}}$ is shown on the
left and the splitting $M_{\chi_{b1}}-M_{\chi_{b0}}$ is shown on the right.
The solid vertical lines with shaded gray error bands denote the tuned values
of the RHQ parameters with jackknife statistical errors. For each quantity,
the dashed line shows the dependence on the RHQ parameters calculated from
Eqs. (56)–(58).
Once we have the results for the bottomonium masses and mass-splittings at
fixed sea-quark mass and lattice spacing, we must extrapolate to the physical
light-quark masses and the continuum limit. Because the $\overline{b}b$ states
contain no valence light quarks, we expect only a weak light-quark mass
dependence and a correspondingly mild chiral extrapolation. In practice, as
shown in Table 11, we do not observe any statistically significant dependence
of the observables on the light sea-quark masses at either lattice spacing. We
therefore compute the error-weighted average of each mass and mass-splitting
over the different sea-quark ensembles at the two lattice spacings.
Table 11: Bottomonium masses and mass-splittings on the five sea-quark ensembles and averaged for each lattice spacing. For the masses, we extrapolate the results on the two lattice spacings to the continuum limit linearly in $a^{2}$ as described in the text. Errors shown are statistical only, but include the uncertainty due to the statistical errors on the tuned RHQ parameters. | $a\approx 0.11$ fm | $a\approx 0.086$ fm | continuum
---|---|---|---
mass [MeV] | $am_{l}=0.005$ | $am_{l}=0.01$ | average | $am_{l}=0.004$ | $am_{l}=0.006$ | $am_{l}=0.008$ | average |
$M_{\eta_{b}}$ | 9328(14) | 9327(18) | 9328(11) | 9326(18) | 9341(15) | 9347(18) | 9338(10) | 9350(33)
$M_{\Upsilon}$ | 9367(14) | 9367(17) | 9367(11) | 9379(16) | 9388(13) | 9395(16) | 9388(9) | 9410(30)
$M_{\Upsilon}-M_{\eta_{b}}$ | 38.8(2.3) | 40.6(2.5) | 39.6(1.7) | 53.1(3.0) | 47.3(2.4) | 48.2(3.4) | 49.2(1.6) | —
$M_{\chi_{b0}}$ | 9853(15) | 9848(18) | 9851(12) | 9816(19) | 9836(15) | 9837(20) | 9831(10) | 9808(35)
$M_{\chi_{b1}}$ | 9884(15) | 9882(19) | 9883(12) | 9853(19) | 9873(15) | 9875(20) | 9868(10) | 9851(35)
$M_{\chi_{b1}}-M_{\chi_{b0}}$ | 31.2(1.8) | 33.5(2.0) | 32.3(1.3) | 37.8(2.7) | 36.6(2.2) | 38.8(2.6) | 37.5(1.4) | —
$M_{h_{b}}$ | 9895(16) | 9894(19) | 9895(12) | 9866(19) | 9884(16) | 9887(21) | 9879(10) | 9862(36)
Because the domain-wall fermion action is ${\cal O}(a)$-improved, the leading
lattice discretization effects from the light-quark and gluon sector are
proportional to $a^{2}$. With the relativistic heavy-quark formalism, heavy-
quark discretization errors depend on the lattice spacing as unknown functions
of $m_{0}a$ [with coefficients of ${\cal O}(1)$] whose behavior is only known
in the asymptotic limits of very large and very small $m_{0}a$; hence they do
not have to scale as $a^{2}$. As discussed in the following section, however,
we estimate that gluon discretization errors in the bottomonium masses are
larger than both light-quark and heavy-quark discretization errors, and
consequently dominate the scaling behavior of the masses. We therefore
extrapolate the bottomonium masses to the continuum linearly in $a^{2}$ in
order to remove gluon discretization errors. We estimate the remaining
systematic uncertainty from heavy-quark discretization errors using power-
counting, discussed below. Figure 12 shows the continuum extrapolation of the
five bottomonium masses along with the experimentally-measured values for
comparison.
In contrast, light-quark and gluon discretization errors largely cancel in the
fine-structure splittings, so the scaling behavior is dominated by the heavy-
quark discretization errors. With data at only two lattice spacings, however,
we cannot resolve quadratic or more complicated $m_{0}a$ dependence. We
therefore choose not to extrapolate the fine-structure splittings, and instead
quote the results obtained on the finer $32^{3}$ ensembles as our central
values. Again, we estimate the residual systematic uncertainty from heavy-
quark discretization errors using power-counting.
Figure 12: Continuum extrapolation of bottomonium masses and mass-splittings.
Upper left plot: $\Upsilon$ (filled blue symbols) and $\eta_{b}$ (open red
symbols) masses versus squared lattice spacing. Upper right plot: $\chi_{b1}$
(open green symbols) and $\chi_{b0}$ (filled pink symbols) masses versus
squared lattice spacing. Lower plot: $h_{b}$ mass versus squared lattice
spacing. On each plot the two lattice spacings $a\approx 0.086$ fm and
$a\approx 0.11$ fm are indicated by vertical black dash-dotted lines. Data
points at different light sea quark masses but the same lattice spacing are
shown with an offset for clarity. The average values at each lattice spacing
are given as shaded error bands in the same color as the symbols, and a linear
extrapolation in $a^{2}$ of the averaged values leads to the continuum limit
results denoted by circles. On the data points we show statistical errors
only. On the continuum-extrapolated values we denote the statistical errors
with solid error bars and the total statistical plus systematic errors with
additional dashed error bars. For comparison we show the experimentally-
measured values as stars.
### IV.3 Estimation of systematic errors
We now discuss the sources of systematic uncertainty in the bottomonium masses
and splittings. Table 12 presents the total statistical and systematic error
budget for each quantity.
#### IV.3.1 Statistics
We propagate the statistical errors through the entire multi-step analysis
procedure via a single-elimination jackknife procedure. Hence the statistical
errors include the uncertainty due to the statistical errors in the tuned RHQ
parameters, including correlations between $m_{0}a$, $c_{P}$, and $\zeta$.
#### IV.3.2 Heavy-quark discretization errors
The RHQ action gives rise to nontrivial lattice-spacing dependence in physical
quantities in the region $m_{0}a\sim 1$. Thus, instead of including additional
functions of $m_{0}a$ in the combined chiral-continuum extrapolation, we
estimate the size of discretization errors from the heavy-quark sector with
power-counting. We follow the method outlined by Oktay and Kronfeld in Ref.
Oktay:2008ex , in which they outline a general framework that applies to both
heavy-heavy and heavy-light systems.
We consider a nonrelativistic description of the heavy-quark action because
both the lattice and continuum theories can be described by effective
Lagrangians built from the same operators. Discretization errors arise due to
mismatches between the short-distance coefficients of higher-dimension
operators in the two theories. More precisely, for each operator ${\cal
O}_{i}$ in the heavy-quark effective Lagrangian, the associated discretization
error is given by
$\displaystyle\textrm{error}_{i}^{HQ}=\left({\cal C}_{i}^{\textrm{lat}}-{\cal
C}_{i}^{\textrm{cont}}\right)\langle{\cal O}_{i}^{HQ}\rangle\,.$ (59)
The “mismatch functions” $f_{i}\equiv{\cal C}_{i}^{\textrm{lat}}-{\cal
C}_{i}^{\textrm{cont}}$ are functions of the parameters of the lattice heavy-
quark action. They have been calculated at tree-level for the anisotropic
clover-improved Wilson action in Ref. Oktay:2008ex . The operators ${\cal
O}_{i}$ in Eq. (59) specify the ${\cal O}(a^{2})$ errors present in the heavy-
quark action and their expectation values $\langle{\cal O}_{i}\rangle$ depend
on the physical quantity of interest. When the sizes of operators in the
heavy-quark action are estimated with power-counting appropriate to heavy-
light meson systems, this framework leads to HQET. Similarly, when the sizes
of operators in the nonrelativistic heavy-quark action are estimated with
power-counting suitable for heavy-heavy meson systems, it leads to NRQCD.
We consider two sources of heavy-quark discretization errors in the
bottomonium system. The first is directly from operators that contribute to
bottomonium masses and fine-structure splittings. The second is indirect
contributions from discretization errors in the RHQ parameters; these are due
to heavy-quark discretization errors in the $B_{s}$ and $B_{s}^{*}$ energies
used in the tuning procedure. We discuss each source briefly in turn and
present the final error estimates here. Details are provided in the appendices
A–C.
To estimate the “direct” heavy-quark discretization errors, we compute the
values of the mismatch functions for our lattice simulation parameters and
estimate the sizes of the matrix elements of the higher-dimension operators
${\cal O}_{i}$ in Eq. (59) with power-counting appropriate to heavy-heavy
meson systems. We use $a^{-1}=2.281$ GeV Aoki:2010dy , which is the lattice
scale on our finer $32^{3}$ ensembles, and $m_{b}=4.2$ GeV Nakamura:2010zzi .
The RHQ parameters on the $32^{3}$ lattices are given by
$\\{m_{0}a,c_{P},\zeta\\}=\\{3.99,3.57,1.93\\}$. We also need an estimate for
the $b$-quark velocity $v$ in the $\overline{b}b$ mesons. Following Ref.
Thacker:1990bm , we expect that the mass difference between the $\Upsilon(1S)$
and $\Upsilon(2S)$ states, which is roughly 500 MeV, should be of the same
size as the average kinetic energy, $E\sim m_{b}v^{2}$. Taking the quark mass
to be half the meson mass gives an estimate for the $b$-quark velocity squared
of $v^{2}\sim 0.1$.
The numerical estimates of the relevant mismatch functions are given in
Appendix A. Because the $b$ quarks in the $\overline{b}b$ mesons are
nonrelativistic, we estimate the size of operators using the “NRQCD” power-
counting formulated in Ref. Lepage:1992tx :
$\vec{D}\sim m_{b}v\,,\quad g\vec{E}\sim m_{b}^{2}v^{3},\quad g\vec{B}\sim
m_{b}^{2}v^{4},\quad g^{2}\sim v\,,$ (60)
where the expansion parameter $v$ is the spatial velocity of the $b$ quarks.
Thus, in NRQCD, an operator’s numerical importance is determined by the order
in the heavy-quark velocity $v$, rather than the dimension. Within the NRQCD
power-counting framework, $b\overline{b}$ meson masses are approximately
$M\sim 2m_{b}$, generic mass splittings such as
$M_{\Upsilon}(2S)-M_{\Upsilon}(1S)$ are $\sim m_{b}v^{2}$ and fine-structure
splittings such as the hyperfine, spin-orbit, and tensor splittings are $\sim
m_{b}v^{4}$.
In the RHQ approach we tune the coefficients of the dimension five operators
in the Symanzik effective theory nonperturbatively; hence the leading heavy-
quark discretization errors come from operators of dimensions 6 and 7 in the
Symanzik effective theory (or alternatively the heavy-quark effective
Lagrangian) that are omitted from the lattice action. The dominant errors in
the $b\overline{b}$ meson masses come from operators that are of ${\cal
O}(v^{4})$ in the NRQCD power-counting. In Appendix B, we estimate the size of
their contributions to bottomonium masses to be $\sim 0.34\%$. Contributions
from operators of ${\cal O}(v^{4})$ cancel in the fine-structure splittings,
such that the dominant errors come from operators that are of ${\cal
O}(v^{6})$. In Appendix B, we estimate the size of their contributions to
hyperfine splittings to be $\sim 32\%$ and to $\chi$-state splittings to be
$\sim 43\%$. The errors in the hyperfine splittings are smaller because they
only come from operators containing the term $\vec{\sigma}\cdot\vec{B}$ (and
permutations thereof), where $\vec{B}$ is the chromomagnetic field.
To estimate the “indirect” heavy-quark discretization errors from the bottom-
strange mesons used in the RHQ tuning procedure, we use the same values of the
mismatch functions but estimate the sizes of operator matrix elements with
power-counting appropriate to heavy-light meson systems. We consider
separately heavy-quark discretization errors in the three input quantities:
the spin-averaged rest mass $\overline{M}_{B_{s}}$, the hyperfine splitting
$\Delta M_{B_{s}}$, and the ratio of rest-to-kinetic masses
$M_{1}^{B_{s}}/M_{2}^{B_{s}}$.
The $b$-quarks in $B$ hadrons typically carry a spatial momentum
$|\vec{p}|\approx\Lambda_{\rm QCD}$, the scale of the strong interactions.
Therefore we estimate the size of operators using HQET power-counting, which
in the continuum is an expansion in $|\vec{p}|/m_{b}$. The lattice introduces
an additional scale, $a$. Following Ref Oktay:2008ex , we therefore expand in
powers of $\lambda$, where $\lambda$ is either of the small parameters
$\lambda\sim a\Lambda_{\rm QCD},\Lambda_{\rm QCD}/m_{Q}\,.$ (61)
Within the HQET power-counting framework, $\overline{b}l$ meson masses are
approximately $M\sim m_{b}$ and hyperfine splittings are $\sim\Lambda_{\rm
QCD}^{2}/2m_{b}$.
As for the estimates above, we use the lattice-spacing and RHQ parameters on
the $32^{3}$ ensembles along with the experimentally-measured $b$-quark mass.
We also need an estimate for the $b$-quark momentum $\Lambda_{\rm QCD}$ in the
heavy-strange mesons. We choose $\Lambda_{\rm QCD}=500$ MeV because fits to
moments of inclusive $B$-decays using the heavy-quark expansion suggest that
the typical QCD scale that enters heavy-light quantities tends to be larger
than for light-light quantities Buchmuller:2005zv .
The dominant errors in the $B_{s}$ and $B_{s}^{*}$ meson rest masses come from
operators that are of ${\cal O}(\lambda^{2})$ in the HQET power-counting. In
Appendix C, we estimate the size of their contributions to
$M_{1}^{B_{s}^{(*)}}$ to be $\sim 0.05\%$. This is comparable to the size of
the statistical errors in the effective masses computed in our numerical
simulations (see the example fits in Fig. 4). As can be seen from Figs. 6,
such a small variation in the spin-averaged mass leads to a statistically-
negligible shift in the tuned value of $m_{0}a$ (i.e. well within the vertical
gray error band). Hence we neglect heavy-quark discretization effects in
$\overline{M}_{B_{s}}$ when determining the size of heavy-quark discretization
errors in the tuned RHQ parameters.
The dominant errors in the $B_{s}$ hyperfine splitting come from operators
that are of ${\cal O}(\lambda^{3})$ in the HQET power-counting. In Appendix C,
we estimate the size of their contributions to $\Delta M_{B_{s}}$ to be $\sim
4.4\%$. This is approximately twice as large as the statistical errors in the
hyperfine splittings computed in our numerical simulations. As can be seen
from Figs. 6, a variation of this size leads to a statistically-significant
shift in the tuned value of $c_{P}$, so we must propagate it to an uncertainty
in the tuned RHQ parameters. We estimate this error by varying the value of
$\Delta M_{B_{s}}$ used in the RHQ parameter-tuning procedure by $\pm 4.4\%$
and then re-computing the bottomonium masses and mass-splittings. For each
mass or mass-splitting we take the largest variation observed on any of the
sea-quark ensembles. We find that a $\sim 4.4\%$ error $\Delta M_{B_{s}}$
leads to a $\sim$ 0.0–0.1% error in the bottomonium masses, a $\sim 8.8\%$
error in the hyperfine splitting, and a $\sim 6.2\%$ error in the $\chi$-state
splittings.
Discretization errors in the $B_{s}$ kinetic meson mass arise from both the
constituent quarks’ kinetic energies and the binding energy. In Appendix C, we
estimate their size to be $\sim 2.6\%$ following the method of Ref.
Bernard:2010fr . This is comparable to the size of the statistical errors in
the $B_{s}$ meson kinetic masses computed in our numerical simulations (see
the example fits in Fig. 5). As can be seen from Figs. 6, a variation of this
size leads to a statistically-significant shift in the tuned value of $\zeta$,
so we must propagate it to an uncertainty in the tuned RHQ parameters. To
estimate the resulting error we follow the same procedure as described above
for the discretization errors in the hyperfine splitting. We find that a $\sim
2.6\%$ error $M_{1}^{B_{s}}/M_{2}^{B_{s}}$ leads to a $\sim$ 0.1–0.2% error in
the bottomonium masses, a $\sim 3.6\%$ error in the hyperfine splitting, and a
$\sim 1.0\%$ error in the $\chi$-state splittings.
To obtain the total heavy-quark discretization errors in the bottomonium
masses and mass-splittings, we add the direct errors and the indirect errors
in quadrature. The resulting estimates are given in Table 12. Numerically, the
indirect errors due to discretization errors in the RHQ parameters turn out to
be smaller than the direct errors for the $\overline{b}b$-meson masses, and
significantly smaller than the direct errors for the fine-structure
splittings.
#### IV.3.3 Light-quark and gluon discretization errors
We estimate the size of light-quark and gluon discretization errors following
the same approach as described for heavy-quark errors in the previous
subsection. In this case, the dimension 6 and higher-order light-quark and
gluon operators in the Symanzik effective Lagrangian have no counterpart in
the continuum QCD Lagrangian. (There are no dimension 5 operators because both
the light-quark and gluon actions are ${\cal O}(a)$-improved.) Thus the
coefficients of the continuum operators in the “mismatch functions” defined in
Eq. (59) are ${\cal C}_{i}^{\textrm{cont}}=0$. Further, the coefficients of
the lattice operators are not expected to be suppressed by any powers of the
heavy-quark mass $1/m_{Q}$. Thus we take them to be ${\cal
C}_{i}^{\textrm{lat}}=1$. The light-quark and gluon discretization errors are
then given by expectation values of light-quark and gluon operators between
heavy-heavy ($\overline{Q}Q$) meson states, i.e.:
$\displaystyle\textrm{error}_{i}^{LQ,g}=\langle{\cal O}_{i}^{LQ,g}\rangle\,,$
(62)
where we estimate their size using the NRQCD power-counting, Eq. (60).
The largest discretization errors in bottomonium masses from the light-quark
and gluon sector will arise from operators with only gluons. This is because
any operators containing light-quark fields must extract light quarks from the
sea, and their expectation values between $\overline{Q}Q$ meson states will be
suppressed by at least $\alpha_{s}^{2}$. A typical dimension 6 gluon operator
in the Symanzik effective Lagrangian is
${\cal O}_{\rm glue}=\rm{tr}[F_{\mu\nu}D^{2}F_{\mu\nu}]\,.$ (63)
Within the NRQCD power-counting we expect its size to be
$\langle{\cal O}_{\rm glue}\rangle^{\rm NRQCD}\sim a^{2}m^{3}v^{4}\,,$ (64)
where two powers of $mv$ come from the derivative operators, and we estimate
the size of $F^{2}$ to be the typical kinetic energy $mv^{2}$. On the $24^{3}$
($32^{3})$ ensembles the corresponding errors in the bottomonium masses are
$\textrm{error}_{\rm glue}\sim a^{2}m^{3}v^{4}/2m_{b}=3.0\%\,(1.7\%)\,,$ (65)
which are several times larger than the estimated sub-percent contributions of
heavy-quark discretization errors. Thus we conclude that, for bottomonium
masses, the ${\cal O}(a^{2})$ light-quark and gluon discretization errors will
dominate the scaling behavior, and we can remove them by extrapolating to the
continuum limit in $a^{2}$. The statistical errors in the continuum-limit
values reflect the uncertainty on the slope in $a^{2}$.
Contributions from light-quark and gluon operators will largely cancel in the
bottomonium fine-structure splittings, and we expect their contributions to
these quantities to be negligible as compared to the heavy-quark
discretization errors estimated previously.
#### IV.3.4 Input strange-quark mass
We tune the parameters of the RHQ action from the bottom-strange system using
the determination of the bare strange-quark mass on the two lattice-spacings
from RBC/UKQCD’s analysis of the light-pseudoscalar meson sector in Ref.
Aoki:2010dy . Hence the uncertainty in the bare strange-quark mass leads to a
systematic error in the RHQ parameters, and consequently in the bottomonium
masses and mass-splittings. We estimate this error by varying the valence
strange-quark mass in the $B_{s}$ and $B_{s}^{*}$ meson correlators used for
the tuning procedure, Eqs. (44) and (45), and then re-computing the
bottomonium masses and mass-splittings.
Figure 13 shows the dependence of the meson masses and mass-splittings on the
valence strange-quark mass used to tune the parameters of the RHQ action on
the $am_{l}=0.005$ $24^{3}$ ensemble. The results at the three strange-quark
mass values are consistent within statistical error, and analogous plots on
the $am_{l}=0.004$ $32^{3}$ ensemble look similar. Because the $\approx 1.2\%$
uncertainty in $m_{s}$ leads to a 0.1% or less change in the bottomonium
masses and a 0.3% or less change in the mass-splittings, we can safely neglect
its effect from our error budget.
Figure 13: Bottomonium masses and mass-splittings versus the valence strange-
quark mass in the bottom-strange meson correlators used to tune the parameters
of the RHQ action. Results are shown for the $am_{l}=0.005$ $24^{3}$ ensemble.
The meson states shown in each plot are specified in the legend. For each
quantity, the thicker line in the same color as the plotting symbol is an
uncorrelated linear fit used to obtain the slope $\Delta
M_{\overline{b}b}/\Delta m_{s}$. The vertical solid line with gray error band
denotes the value of the physical strange-quark mass obtained in Ref.
Aoki:2010dy . For each quantity, the two horizontal dashed lines show where
the linear fit crosses the edges of the error band, thereby indicating the
error due to the uncertainty in the strange-quark mass.
#### IV.3.5 Input scale uncertainty
At first glance, the value of the lattice spacing in physical units enters the
computation of the bottomonium masses and mass-splittings in two ways. It
first enters indirectly through the parameters of the RHQ action, which we
tune by matching the values of the $B_{s}$ and $B_{s}^{*}$ meson masses
obtained on the lattice to the experimentally-measured values from the PDG
Nakamura:2010zzi . It then enters directly when we convert the lattice values
of the bottomonium masses and mass splittings into GeV in order to compare
with experiment. In fact, however, the RHQ parameter tuning procedure allows
us to avoid this second source of scale uncertainty. This is because our
lattice calculation of the mass $M_{\overline{b}b}$ of a $\overline{b}b$ meson
gives directly the dimensionless ratio $M_{\overline{b}b}/M_{B_{s}}$ at the
tuned values of the RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$ without further
reference to the lattice scale. By construction, at the tuned point the
$B_{s}$-meson mass is fixed to the experimentally-measured value; hence we can
precisely obtain the bottomonium mass or mass-splitting in GeV by multiplying
the ratio by $M_{B_{s}}=5.3366$ GeV Nakamura:2010zzi . We therefore need only
consider the implicit dependence on the lattice spacing due to the RHQ
parameters when estimating the scale uncertainty in the
$\overline{b}{b}$-meson masses.
The absolute lattice scale ($a^{-1}$) has a quoted statistical error of $\sim
1\%$ ($1.5\%$) on the $32^{3}$ ($24^{3}$) lattices Aoki:2010dy , where the
errors on the two lattice spacings are highly correlated because they come
from a single fit to data at both lattice spacings. We estimate the
corresponding error in the bottomonium states by varying the lattice scale
$a^{-1}$ used in the RHQ parameter tuning procedure by plus and minus a
statistical sigma on each sea-quark ensemble. For each bottomonium mass or
mass-splitting, we then take the largest variation on any of the ensembles to
be the error due to the uncertainty in the lattice scale. We find that the
resulting uncertainty in the meson masses is 0.2% or less, and in the mass-
splittings is $\sim$1–3%; these errors are given in Table 12.
#### IV.3.6 Experimental inputs
We tune the parameters of the RHQ action by using the experimental
measurements of the spin-averaged $B_{s}$ meson mass and hyperfine splitting.
The $B_{s}$ and $B_{s}^{*}$ meson masses are both known to sub-percent
precision Nakamura:2010zzi , so the experimental error in
$\overline{M}_{B_{s}}$ contributes a negligible uncertainty to the tuned
values of the RHQ parameters. The experimental error in the hyperfine
splitting $\Delta M_{B_{s}}=49.0(1.5)$ MeV Nakamura:2010zzi , however, is
$\sim 3.1\%$ and cannot be neglected. We estimate the error in the bottomonium
masses and mass-splittings due to the experimental uncertainty in the $B_{s}$
meson hyperfine splitting by varying the value of $\Delta M_{B_{s}}$ used in
the RHQ tuning procedure by plus and minus 1.5 MeV. For each bottomonium mass
or mass-splitting, we then take the largest variation on any of the ensembles
to be the corresponding error. We find that the resulting uncertainty in the
meson masses is 0.1% or less, and in the mass-splittings is $\sim$4–6%; these
errors are given in Table 12.
#### IV.3.7 Linear approximation
We interpolate to the tuned values of the RHQ parameters assuming a linear
dependence upon $\\{m_{0}a,c_{P},\zeta\\}$. Hence any deviation from linearity
must be accounted for in the systematic error budget. In practice, as shown in
Figs. 6, we do not see any statistically significant deviation from linearity
for the heavy-strange states over a wide range of RHQ parameters. Nor do we
observe any statistically significant curvature for the $\chi$ states or the
$h_{b}$ (see the right-hand plots in Fig. 10). Thus the systematic uncertainty
in the $\chi$ states and the $h_{b}$ due to nonlinear dependence upon the RHQ
parameters is negligible. We can resolve nonlinear dependence of $\Upsilon$
and $\eta_{b}$ meson masses and the hyperfine splitting within the statistical
errors in the measured effective masses, as shown in Figs. 10 and 11. The
statistical errors in these data points, however, are almost two orders of
magnitude smaller than the statistical errors in the $\Upsilon$ and $\eta_{b}$
meson masses and the hyperfine splitting interpolated to the tuned RHQ
parameters given in Table 11; this is because the interpolated values include
the uncertainty due to the statistical errors in $\\{m_{0}a,c_{P},\zeta\\}$.
Hence we conclude that the systematic error due to deviations from linearity
is negligible for all bottomonium quantities considered here.
Table 12: Error budget for bottomonium masses and mass-splittings. The estimates of the size of each systematic uncertainty are given in the main text. Each error is given as a percentage, and we obtain the total systematic by adding the individual systematic uncertainties in quadrature. Errors that were considered but were found to be negligible (i.e. light-quark and gluon discretization errors, strange-quark mass uncertainty, and linear approximation) are not shown. | $M_{\eta_{b}}$ | $M_{\Upsilon}$ | $M_{\Upsilon}$-$M_{\eta_{b}}$ | $M_{\chi_{b0}}$ | $M_{\chi_{b1}}$ | $M_{\chi_{b1}}$-$M_{\chi_{b0}}$ | $M_{h_{b}}$
---|---|---|---|---|---|---|---
statistics | 0.4 | 0.3 | 3. | 3 | 0.4 | 0.4 | 3. | 7 | 0.4
heavy-quark discretization errors | 0.4 | 0.3 | 33. | 0 | 0.4 | 0.4 | 43. | 6 | 0.4
input scale uncertainty | 0.2 | 0.2 | 3. | 2 | 0.1 | 0.1 | 1. | 0 | 0.1
experimental inputs | 0.0 | 0.1 | 6. | 2 | 0.0 | 0.0 | 4. | 3 | 0.0
total systematic | 0.4 | 0.4 | 33. | 7 | 0.4 | 0.4 | 43. | 8 | 0.4
## V Results and conclusions
The relativistic heavy-quark formalism enables the description of systems
involving $b$-quarks, such as $B$-mesons and bottomonium states, on currently
available lattice spacings with lattice discretization errors from the heavy-
quark sector of the same size as those from the light-quark sector. We have
determined the $b$-quark parameters for the RHQ action on the RBC/UKQCD 2+1
flavor domain-wall lattices with lattice spacings $a\approx 0.11$ fm and
$a\approx 0.08$ fm. This is a continuation of and improvement upon the work of
Li and Peng, who each presented preliminary results for $B$-mesons and
bottomonium at conferences Li:2008kb ; Peng:Lattice10 .
In this work we tune the three parameters $\\{m_{0}a,c_{P},\zeta\\}$ using the
bottom-strange system, where discretization errors are expected to be of
${\cal O}([\vec{p}a]^{2})$ with $|\vec{p}|\approx\Lambda_{\textrm{QCD}}$. We
obtain the parameters nonperturbatively by imposing three simple conditions:
that the masses of the $B_{s}$ and $B_{s}^{*}$ mesons agree with the
experimental measurements, and that the $B_{s}$ meson on the lattice obey the
continuum relativistic dispersion relation $E^{2}=\vec{p}^{2}+M^{2}$. We then
test the reliability of the tuned parameters and the validity of the
relativistic heavy-quark approach by making predictions for the masses and
mass splittings of several bottomonium states.
As shown in Fig. 14 and Table 13, we obtain bottomonium masses with
$\sim$0.5–0.6% total uncertainties and mass-splittings with $\sim$35–45%
uncertainties, and find good agreement between our predicted values and
experiment for all the quantities that we study. In fact, the preliminary work
of Li successfully predicted the mass of the $h_{b}$ meson Li:2008kb before
it was first observed by the Belle collaboration Adachi:2011ji , thereby
lending further credence to the relativistic heavy-quark formalism. We also
find agreement with calculations of $M_{\eta_{b}}$, the hyperfine splitting
$M_{\Upsilon}-M_{\eta_{b}}$, and $M_{h_{b}}$ using the NRQCD formalism for the
$b$-quark Gray:2005ur ; Meinel:2010pv and with a calculation of the hyperfine
splitting using the Fermilab formalism Burch:2009az . Both the HPQCD and
Fermilab/MILC works use the MILC collaboration’s gauge configurations with 2+1
flavors of Asqtad-improved staggered sea quarks Aubin:2004fs ; our study of
$\overline{b}b$ meson spectroscopy using three flavors of dynamical domain-
wall light quarks provides a fully independent check of these results.
Although the calculation by Meinel Meinel:2010pv uses the same RBC/UKQCD
domain-wall + Iwasaki configurations as in this paper, our result is still
largely independent of his work because statistical errors (which are somewhat
correlated between the two results) are not the primary source of uncertainty.
Table 13: Comparison of predicted bottomonium masses and mass-splittings with experiment and, where possible, with other 2+1 flavor lattice calculations. The HPQCD and Meinel calculations use the NRQCD action for the $b$-quarks Lepage:1992tx , while the Fermilab/MILC calculation uses the Fermilab action ElKhadra:1996mp . For our results, the first error is statistical and the second is systematic; for the other results we add the errors in quadrature and quote the total. All results are given in MeV. | this work | Experiment | HPQCD Dowdall:2011wh | Fermilab/MILC Burch:2009az | Meinel Meinel:2010pv
---|---|---|---|---|---
$M_{\eta_{b}}$ | 9350 | (33)(37) | 9390. | 9(2.8) Nakamura:2010zzi | 9390 | (9) | | 9400. | 0(7.7)
$M_{\Upsilon}$ | 9410 | (30)(38) | 9460. | 30(26) Nakamura:2010zzi | | | | |
$M_{\Upsilon}$-$M_{\eta_{b}}$ | 49 | (02)(17) | 69. | 3(2.8) Nakamura:2010zzi | 70 | (9) | 54.0$\left({}^{+12.5}_{-12.4}\right)$ | 60. | 3(7.7)
$M_{\chi_{b0}}$ | 9808 | (35)(39) | 9859. | 44(52) Nakamura:2010zzi | | | | |
$M_{\chi_{b1}}$ | 9851 | (35)(39) | 9892. | 78(40) Nakamura:2010zzi | | | | |
$M_{\chi_{b1}}$-$M_{\chi_{b0}}$ | 38 | (01)(16) | 33. | 3(5) Brambilla:2004wf | | | | |
$M_{h_{b}}$ | 9862 | (36)(39) | 9899. | 1(1.1) Bellehb | 9905 | (7) | | 9899. | 8(1.0)
Given the successful predictions of the bottomonium states, we now plan to use
the nonperturbatively tuned parameters of the RHQ action to calculate
$B$-meson weak matrix elements of interest to flavor physics phenomenology. We
are currently computing the leptonic decay constants $f_{B}$ and $f_{B_{s}}$
and the neutral $B^{0}$-$\overline{B^{0}}$ mixing parameters VandeWater:2011gr
. These calculations are particularly timely given the observed approximately
$3\sigma$ tension in the CKM unitarity triangle Bona:2009cj ; Lenz:2010gu ;
Lunghi:2010gv ; Laiho:2012ss which currently favors the presence of new
physics in $B_{d}$-mixing or $B\to\tau\nu$ decays. Eventually we would also
like to use the RHQ framework to calculate more challenging quantities such as
$B\to\pi\ell\nu$ and $B\to D^{(*)}\ell\nu$ semileptonic form factors, which
are needed to extract the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$,
respectively, from exclusive channels. Like the Fermilab interpretation, our
relativistic heavy-quark formalism applies to any value of the quark mass, and
allows for a continuum limit. (This is in contrast to the NRQCD formalism, for
which errors increase away from the infinite heavy-quark limit.) Hence the
same framework can be used for charm quarks, which are neither particularly
heavy compared to $\Lambda_{\textrm{QCD}}$ nor light enough to be treated with
a standard lattice light-quark formulation with ${\cal O}(m_{c}a)^{2}$ errors
that are well-controlled. Treatment of both $b$\- and $c$-quarks within the
same framework allows for further tests of the methodology. We therefore also
plan to tune the parameters of the relativistic heavy-quark action for charm
quarks, such that we can compute the leptonic decay constants $f_{D}$ and
$f_{D_{s}}$, as well as other weak matrix elements such as the short-distance
contribution to $D^{0}$-$\overline{D^{0}}$ mixing.
This work demonstrates the validity of the relativistic heavy quark action on
bottom systems and opens a practical approach to obtain bottom and charm weak
matrix elements with high precision given the computer resources currently
available. Lattice QCD calculations of heavy-light weak matrix elements
provide critical inputs to the CKM unitarity triangle analysis. Hence
determinations with a variety of methods and independent sources of systematic
uncertainty will be essential to definitively uncovering new physics in the
flavor sector. Use of the relativistic heavy-quark formalism for $b$-quarks on
the RBC/UKQCD dynamical domain-wall lattices will provide phenomenologically-
important, independent determinations of key heavy-light weak-matrix elements
with comparable errors to other methods.
Figure 14: Comparison of predicted bottomonium masses (left panel) and mass-
splittings (right panel) with experiment. For the bottomonium masses we
extrapolate the results on the two lattice spacings to the continuum linearly
in $a^{2}$, whereas for the fine-structure splittings we take the results on
the finer $32^{3}$ ensembles as our central value. The solid error bars on the
data points show the statistical errors. For our preferred results, we also
show the systematic errors added in quadrature as dashed error bars.
## Acknowledgments
Computations for this work were carried out in part on facilities of the USQCD
Collaboration, which are funded by the Office of Science of the U.S.
Department of Energy. We thank BNL, Columbia University, Fermilab, RIKEN, and
the U.S. DOE for providing the facilities essential for the completion of this
work.
This work was supported in part by the U.S. Department of Energy under grant
No. DE-FG02-92ER40699 and by the Grant-in-Aid of the Ministry of Education,
Culture, Sports, Science and Technology, Japan (MEXT Grant), No.21540289, and
No.22224003, No, 2254030, and No. 23105715. JMF acknowledges support from STFC
grant ST/J0003961. This manuscript has been authored by employees of
Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with
the U.S. Department of Energy. CL acknowledges support from the RIKEN FPR
program.
## Appendix A Heavy-quark mismatch functions
In this section we collect the forms of the mismatch functions used to
estimate the size of heavy-quark discretization errors in heavy-heavy and
heavy-light systems for the RHQ action.
For each operator in the heavy-quark effective Lagrangian, the “mismatch
function” is defined as the difference between the short-distance coefficients
in the lattice and continuum theories. Hence the mismatch functions depend
upon the parameters of the lattice action. The mismatch functions have been
calculated at tree-level for the anisotropic clover-improved Wilson action in
Ref. Oktay:2008ex , but we present them here for completeness. Although Oktay
and Kronfeld derive general expressions for $c_{E}\neq c_{B}$ and $r_{s}\neq
1$ and include dimension 6 and higher order operators in the lattice action,
here we show the mismatch functions specific to the RHQ action. We obtain
these expressions from those in Ref. Oktay:2008ex by setting
$c_{E}=c_{B}=c_{P}/\zeta$ and $r_{s}=1$, and setting the coefficients of the
dimension 6 and higher-order operators to zero.
There are five relevant tree-level mismatch functions that enter our estimates
of heavy-quark discretization errors. The first is
$f_{E}(m_{0}a,c_{P},\zeta)=\frac{1}{8m_{E}^{2}a^{2}}-\frac{1}{8m_{2}^{2}a^{2}},$
(66)
where
$\displaystyle\frac{1}{m_{2}a}$ $\displaystyle=$
$\displaystyle\frac{2\zeta^{2}}{m_{0}a(2+m_{0}a)}+\frac{\zeta}{1+m_{0}a},$
(67) $\displaystyle\frac{1}{4m_{E}^{2}a^{2}}$ $\displaystyle=$
$\displaystyle\frac{\zeta^{2}}{[m_{0}a(2+m_{0}a)]^{2}}+\frac{\zeta
c_{P}}{m_{0}a(2+m_{0}a)}\,.$ (68)
The function $f_{E}$ vanishes when the “chromoelectric mass” $m_{E}$ equals
the $b$-quark’s kinetic mass $m_{2}$. The second tree-level mismatch function
is
$f_{w_{4}}(m_{0}a,c_{P},\zeta)=\frac{1}{6}w_{4}\,,$ (69)
where
$\displaystyle w_{4}$ $\displaystyle=$
$\displaystyle\frac{2\zeta^{2}}{m_{0}a(2+m_{0}a)}+\frac{r_{s}\zeta}{4(1+m_{0}a)}\,.$
(70)
The short-distance coefficient $w_{4}$ multiples the Lorentz-symmetry
violating $p_{i}^{4}$ term in the lattice $b$-quark’s energy-momentum
dispersion relation; hence the mismatch function $f_{w_{4}}$ vanishes when
$w_{4}=0$. The third tree-level mismatch function is
$\displaystyle
f_{m_{4}}(m_{0}a,c_{P},\zeta)=\frac{1}{8m_{4}^{3}a^{3}}-\frac{1}{8m_{2}^{3}a^{3}}\,,$
(71)
where
$\displaystyle\frac{1}{m_{4}^{3}a^{3}}$ $\displaystyle=$
$\displaystyle\frac{8\zeta^{4}}{[m_{0}a(2+m_{0}a)]^{3}}+\frac{4\zeta^{4}+8\zeta^{3}(1+m_{0}a)}{[m_{0}a(2+m_{0}a)]^{2}}$
(72) $\displaystyle+\frac{\zeta^{2}}{(1+m_{0}a)^{2}}\,.$
The short-distance coefficient $\frac{1}{m_{4}^{3}a^{3}}$ multiplies the
$(\vec{p}^{2})^{2}$ term in the $b$-quark’s energy-momentum dispersion
relation, so the mismatch function $f_{m_{4}}$ vanishes when $m_{4}=m_{2}$.
The fourth tree-level mismatch function is
$f_{w^{\prime}_{B}}(m_{0}a,c_{P},\zeta)=\frac{1}{12}w^{\prime}_{B}\,,$ (73)
where
$\displaystyle w^{\prime}_{B}$ $\displaystyle=$
$\displaystyle\frac{c_{P}}{1+m_{0}a}\,.$ (74)
The coefficient $w^{\prime}_{B}$ leads to a spin-dependent contribution to the
lattice quark-gluon vertex, so the mismatch function $f_{w^{\prime}_{B}}$
vanishes when $w^{\prime}_{B}=0$. The fifth tree-level mismatch function is
$f_{m_{B^{\prime}}}(m_{0}a,c_{P},\zeta)=\frac{1}{4m^{3}_{B^{\prime}}a^{3}}-\frac{1}{4m^{3}_{2}a^{3}}\,,$
(75)
where
$\displaystyle\frac{1}{m_{B^{\prime}}^{3}a^{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{4}^{3}a^{3}}-\frac{\zeta^{2}-\zeta
c_{P}}{(1+m_{0}a)^{2}}\,.$ (76)
The function $f_{m_{B^{\prime}}}$ vanishes when $m_{4}=m_{2}$ (as above) and
$c_{P}=\zeta$.
To estimate the size of heavy-quark discretization errors in our numerical
simulations, we evaluate the mismatch functions in Eqs. (66), (69), (71),
(73), and (75) at the tuned values of the RHQ parameters given in Tables 5 and
6. For the $24^{3}$ ensembles we use
$\\{m_{0}a,c_{P},\zeta\\}=\\{8.45,5.8,3.10\\}$ and for the $32^{3}$ ensembles
we use $\\{m_{0}a,c_{P},\zeta\\}=\\{3.99,3.57,1.93\\}$. The results are
presented in Table 14. Because the size of the heavy-quark discretization
errors is sensitive to the numerical values of the tree-level mismatch
functions, we have also tried evaluating Eqs. (66), (69), (71), (73), and (75)
at the tree-level values of the RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$. We
find that the results are similar to those in Table 14. We therefore conclude
that the mismatch functions given in Table 14 reflect the typical size of such
coefficients for our simulations, and use them for estimating the heavy-quark
discretization errors in the following appendices.
Table 14: Tree-level mismatch functions for the nonperturbatively-tuned parameters of the RHQ action on the $24^{3}$ and $32^{3}$ ensembles. | $f_{E}$ | $f_{w_{4}}$ | $f_{m_{4}}$ | $f_{w^{\prime}_{B}}$ | $f_{m_{B^{\prime}}}$
---|---|---|---|---|---
$a\approx$ 0.11 fm | 0.0640 | 0.0499 | 0.0353 | 0.0505 | 0.0934
$a\approx$ 0.086 fm | 0.0864 | 0.0681 | 0.0521 | 0.0596 | 0.1359
## Appendix B Discretization errors in heavy-heavy meson masses and fine-
structure splittings
In this section we estimate the size of heavy-quark discretization errors in
heavy-heavy mesons and fine-structure mass-splittings using the framework
described in Sec. IV.3.2. To estimate the numerical size of the operator
matrix elements, we use the NRQCD power-counting given in Eq. (60), and for
the size of the coefficients we use the mismatch functions on the $32^{3}$
ensembles given in Table 14.
### B.1 Masses
Here we consider operators of ${\cal O}(v^{4})$, which produce the dominant
discretization errors in bottomonium masses. Oktay and Kronfeld enumerate all
dimension 6 and 7 bilinear operators in the heavy-quark effective Lagrangian
consistent with symmetries in Table III of Ref. Oktay:2008ex . We do not need
to consider contributions from dimension 8 bilinears because they will be of
${\cal O}(v^{6})$ or higher.
#### B.1.1 ${\cal O}(a^{2})$ errors
There are two dimension six bilinears that are of ${\cal O}(v^{4})$ in the
NRQCD power-counting:
$\displaystyle\overline{h}\\{\bm{\gamma}\cdot\bm{D},\bm{\alpha}\cdot\bm{E}\\}h\,,$
(77)
$\displaystyle\overline{h}\gamma_{4}(\bm{D}\cdot\bm{E}-\bm{E}\cdot\bm{D})h\,.$
(78)
The expected size of these operators is
$\langle{\cal O}_{E}\rangle^{\rm NRQCD}\sim a^{2}m_{b}^{3}v^{4}\,.$ (79)
At tree level the coefficients of these operators are both equal to $f_{E}$,
Eq. (66). We therefore estimate the contribution to the error from each of
these operators to be
$\textrm{error}_{E}=f_{E}\langle{\cal O}_{E}\rangle^{\rm NRQCD}/2m_{b}\sim
0.15\%\,,$ (80)
where we obtain the relative error in the $b\overline{b}$ meson masses by
dividing by $2m_{b}$, the size of the meson masses in the NRQCD power
counting.
#### B.1.2 ${\cal O}(a^{3})$ errors
There are two dimension seven bilinears that are also of ${\cal O}(v^{4})$ in
the NRQCD power-counting:
$\displaystyle\overline{h}D_{i}^{4}h\,,$ (81)
$\displaystyle\overline{h}(\bm{D}^{2})^{2}h\,$ (82)
and the expected size of these operators is
$\langle{\cal O}_{4}\rangle^{\rm NRQCD}\sim a^{3}m_{b}^{4}v^{4}\,.$ (83)
At tree-level the mismatch function for the first operator is given by
$f_{w_{4}}$, Eq. (69), so we estimate its contribution to the error in
$\overline{b}b$ meson masses to be
$\textrm{error}_{w_{4}}=f_{w_{4}}\langle{\cal O}_{4}\rangle^{\rm
NRQCD}/2m_{b}\sim 0.21\%\,.$ (84)
The tree-level mismatch function for the second operator is given by
$f_{m_{4}}$, Eq. (71), so we estimate its contribution to the error in
$\overline{b}b$ meson masses to be
$\textrm{error}_{m_{4}}=f_{m_{4}}\langle{\cal O}_{4}\rangle^{\rm
NRQCD}/2m_{b}\sim 0.16\%\,.$ (85)
#### B.1.3 Total error
We obtain the total heavy-quark discretization error in the $\overline{b}b$
meson masses by adding the errors from the different operators in quadrature,
including ${\cal O}_{E}$ twice because there are two such operators:
$\displaystyle\textrm{error}^{M_{b\overline{b}}}_{\textrm{total}}$
$\displaystyle=$
$\displaystyle\left(2\times\textrm{error}_{E}^{2}+\textrm{error}_{m_{4}}^{2}+\textrm{error}_{w_{4}}^{2}\right)^{1/2}$
(86) $\displaystyle\sim$ $\displaystyle 0.34\%\,.$
### B.2 Hyperfine splittings
Only spin-dependent operators containing the term $\vec{\sigma}\cdot\vec{B}$
where $\vec{B}$ is the chromomagnetic field (and permutations thereof),
contribute to hyperfine splittings such as the mass difference
$M_{\Upsilon}-M_{\eta_{b}}$ Eichten:1980mw ; Peskin:1983up . There are five
dimension 7 bilinear operators of this form in the heavy-quark effective
action at ${\cal O}(v^{6})$:
$\displaystyle\sum_{i\neq j}\overline{h}\\{D_{j}^{2},i\Sigma_{i}B_{i}\\}h\,,$
(87) $\displaystyle\overline{h}\\{\bm{D}^{2},i\bm{\Sigma}\cdot\bm{B}\\}h\,,$
(88) $\displaystyle\sum_{i\neq j}\overline{h}i\Sigma_{i}D_{j}B_{i}D_{j}h\,,$
(89)
$\displaystyle\overline{h}\bm{\gamma}\cdot\bm{D}i\bm{\Sigma}\cdot\bm{B}\bm{\gamma}\cdot\bm{D}h\,,$
(90) $\displaystyle\overline{h}D_{i}i\bm{\Sigma}\cdot\bm{B}D_{i}h\,.$ (91)
Only the first two operators in Eqs. (87) and (88) have nonzero matching
coefficients at tree-level Oktay:2008ex . The matching coefficients of the
remaining three operators in Eqs. (89)–(91) are zero at tree-level
Oktay:2008ex , and have not been computed to one-loop. Higher-dimension
operators in the heavy-quark effective Lagrangian such as
$\overline{h}\\{\bm{D}^{2},\bm{\sigma}\cdot(\bm{D}\times\bm{E}-\bm{E}\times\bm{D})\\}h$
also contribute to hyperfine splittings at ${\cal O}(v^{6})$, but the full set
of dimension 8 heavy-heavy bilinears has not been worked-out in the
literature.
Given our incomplete knowledge of the ${\cal O}(v^{6})$ bilinear operators and
corresponding mismatch functions, we use a more naive error estimation
procedure for the bottomonium hyperfine splittings. The leading contribution
to the hyperfine splittings is $\sim mv^{4}$, so contributions of ${\cal
O}(v^{6})$ are suppressed by by a factor of $v^{2}\sim 0.1$. Hence we expect
that neglected ${\cal O}(v^{6})$ operators lead to 10% errors in hyperfine
splittings. We can check this estimate for the two cases in which the mismatch
functions are known, as shown below.
#### B.2.1 ${\cal O}(a^{3})$ errors
The expected size of the operators in Eqs. (87) and (88) is
$\langle{\cal O}_{\mathbf{\sigma}\cdot\mathbf{B}}\rangle^{\rm NRQCD}\sim
a^{3}m_{b}^{4}v^{6}\,.$ (92)
The tree-level mismatch function for the first operator is given by
$f_{w^{\prime}_{B}}$, Eq. (73), so we estimate its contribution to the error
to be
$\textrm{error}_{w^{\prime}_{B}}=f_{w^{\prime}_{B}}\langle{\cal
O}_{\mathbf{\sigma}\cdot\mathbf{B}}\rangle^{\rm NRQCD}/m_{b}v^{4}\sim
3.72\%\,,$ (93)
where we obtain the relative error in $\overline{b}b$ meson hyperfine
splittings by dividing by $m_{b}v^{4}$, the size of the hyperfine splittings
in the NRQCD power counting. The tree-level mismatch function for the second
operator is given by $f_{m_{B^{\prime}}}$, Eq. (75), so we estimate its
contribution to the error in bottomonium hyperfine splittings to be
$\textrm{error}_{m_{B^{\prime}}}=f_{m_{B^{\prime}}}\langle{\cal
O}_{\mathbf{\sigma}\cdot\mathbf{B}}\rangle^{\rm NRQCD}/m_{b}v^{4}\sim
8.48\%\,.$ (94)
Both of these estimates are consistent with the naive power-counting
expectation of 10% based on the order in the $b$-quark velocity $v$.
#### B.2.2 Total error
There are five dimension 7 and an unknown number of dimension 8 operators in
the heavy-quark effective action that contribute to the hyperfine splittings
at ${\cal O}(v^{6})$ in the NRQCD power-counting. If we assume that there are
the same number of ${\cal O}(v^{6})$ operators at dimensions 7 and 8, we
arrive at the estimate
$\textrm{error}^{\Delta
M_{HF}}_{\textrm{total}}=\Big{(}10\times({v^{2}})^{2}\Big{)}^{1/2}=31.62\%\,.$
(95)
### B.3 $\chi$-state splittings
The fine-structure splitting between $\chi$ mesons
$(M_{\chi_{b1}}-M_{\chi_{b0}})$ is a linear combination of the spin-orbit and
tensor splittings:
$\displaystyle\Delta_{M}^{\textrm{spin-orbit}}$
$\displaystyle=\frac{1}{9}\left(5M_{\chi_{b}2}-2M_{\chi_{b}0}-3M_{\chi_{b}1}\right),$
(96) $\displaystyle\Delta_{M}^{\textrm{tensor}}$
$\displaystyle=\frac{1}{9}\left(3M_{\chi_{b}1}-M_{\chi_{b}2}-2M_{\chi_{b}0}\right).$
(97)
Hence it receives contributions from both the spin-dependent operators
containing $\sigma\cdot\vec{B}$ considered above (which lead to the tensor
splitting Eichten:1980mw ) and from spin-dependent operators containing
$\vec{D}\times\vec{E}$ where $\vec{E}$ is the chromoelectric field (which lead
to the spin-orbit splitting Peskin:1983up ).
#### B.3.1 ${\cal O}(v^{4})$ errors
There is one relevant bilinear at dimension 6 which is of ${\cal O}(v^{4})$ in
the NRQCD power-counting:
$\overline{h}\\{\bm{\gamma}\cdot\bm{D},\bm{\alpha}\cdot\bm{E}\\}h\,.$ (98)
We estimate the size of its contribution to the error in the $\chi$-state
splittings to be
$\textrm{error}_{v^{4}}=f_{E}\langle{\cal O}_{E}\rangle^{\rm
NRQCD}/m_{b}v^{4}\sim 29.30\%\,.$ (99)
Note that the contribution of this operator to the $\chi$-state splittings is
not as large as the order in the $b$-quark velocity $v$ would suggest because
of the small numerical size of $f_{E}$.
#### B.3.2 ${\cal O}(v^{6})$ errors
The same ${\cal O}(v^{6})$ operators that contribute to the hyperfine
splittings also contribute to the splitting between the $\chi$ states. We
therefore estimate their contributions to be the same size as for the
hyperfine splittings:
$\textrm{error}_{v^{6}}=31.62\%\,.$ (100)
#### B.3.3 Total error
We obtain the total heavy-quark discretization error in the $\chi$ state
splittings by adding the ${\cal O}(v^{4})$ and ${\cal O}(v^{6})$ errors in
quadrature, yielding
$\textrm{error}^{\Delta
M_{\chi}}_{\textrm{total}}=\left(\textrm{error}_{v^{4}}^{2}+\textrm{error}_{v^{6}}^{2}\right)^{1/2}=43.11\%\,.$
(101)
## Appendix C Discretization errors in heavy-strange meson masses and
hyperfine splitting
In this section we estimate the size of heavy-quark discretization errors in
the heavy-strange meson quantities – the spin-averaged mass, hyperfine
splitting, and ratio of rest-to-kinetic masses – used in the RHQ parameter
tuning procedure. Again we use the framework described in Sec. IV.3.2. To
estimate the numerical size of the operators, we use the HQET power-counting
given in Eq. (61), and for the size of the coefficients we use the mismatch
functions on the $32^{3}$ ensembles given in Table 14.
### C.1 Rest mass
Because we tune the coefficients of the dimension 5 operators in the RHQ
action nonperturbatively, the leading discretization errors come from
operators of dimension 6 and higher in the effective theory. There are two
dimension 6 bilinears of ${\cal O}(\lambda^{2})$ in the HQET power-counting:
$\displaystyle\overline{h}\\{\bm{\gamma}\cdot\bm{D},\bm{\alpha}\cdot\bm{E}\\}h\,,$
(102)
$\displaystyle\overline{h}\gamma_{4}(\bm{D}\cdot\bm{E}-\bm{E}\cdot\bm{D})h\,.$
(103)
The estimated size of these operators is
$\langle{\cal O}_{E}\rangle^{\rm HQET}\sim a^{2}\Lambda_{\rm QCD}^{3}\,.$
(104)
We do not consider operators of dimension 7 and higher because they are all at
least of ${\cal O}(\lambda^{3})$. At tree-level the coefficients of the
operators in Eqs. (102) and (103) are both given by Eq. (66), so we estimate
their contributions to the error in the spin-averaged $B_{s}$ meson rest mass
to be
$\textrm{error}_{E}=f_{E}\langle{\cal O}_{E}\rangle^{\rm
HQET}/\overline{M}_{B_{s}}\sim 0.04\%\,.$ (105)
By construction, we tune the RHQ parameters such that the spin-averaged rest
mass equals the experimental value of $\frac{1}{4}(M_{B_{s}}+3M_{B_{s}}^{*})$,
so we obtain the relative error in $M_{1}$ by dividing by
$\overline{M}_{B_{s}}=5.4028$ GeV. We obtain the total heavy-quark
discretization error in the spin-averaged $B_{s}$ meson rest mass by adding
the contributions from the two operators in quadrature, which yields:
$\textrm{error}^{M_{1,B_{s}}}_{\textrm{total}}=\left(2\times\textrm{error}_{E}^{2}\right)^{1/2}=0.05\%\,,$
(106)
or $\sim$ 3 MeV.
### C.2 Kinetic mass
Discretization errors in the kinetic meson mass $M_{2}$ arise from both the
constituent quarks’ kinetic energies and from the binding energy. The Appendix
of Ref. Bernard:2010fr provides a semi-quantitative estimate of the
discretization error in $M_{2}$ (see also Ref. Kronfeld:1996uy ). Although
this estimate is made assuming that both quarks in the meson are
nonrelativistic, the result is interpreted a posteriori under the assumption
that the strange quark is light and relativistic. We follow the same approach
here.
The tree-level discretization error in $M_{2}$ through ${\cal O}(v^{4})$ in
the nonrelativistic expansion is given by Bernard:2010fr
$\delta
M_{2}=\frac{1}{3m_{2}}\frac{\langle\vec{p}^{2}\rangle}{2}\left[5\left(\frac{m_{2}^{3}}{m_{4}^{3}}-1\right)+4w_{4}(m_{2}a)^{3}\right]\,,$
(107)
where this result applies to $S$-wave states. Note that the $\delta M_{2}$ is
zero if the masses $m_{4}=m_{2}$ and the Lorentz-symmetry violating
coefficient $w_{4}=0$. To estimate the numerical size of the discretization
error in $M_{2}$ we replace $\langle\vec{p}^{2}\rangle$ with $\Lambda_{\rm
QCD}^{2}$ following the HQET power-counting prescription and use the
expressions for $m_{2}$, $m_{4}$, and $w_{4}$ given in Eqs. (67), (70), and
(72). By construction, we tune the RHQ parameters such that the kinetic meson
mass equals the experimental value of the $B_{s}$ meson mass, so we obtain the
relative error in $M_{2}$ by dividing by $M_{B_{s}}=5.366$ GeV. We obtain
$\textrm{error}^{M_{2,B_{s}}}_{\textrm{total}}=2.59\%\,,$ (108)
or $\sim$139 MeV.
### C.3 Hyperfine splitting
The bottom-strange hyperfine splitting receives contributions from spin-
dependent operators containing the term $\vec{\sigma}\cdot\vec{B}$ where
$\vec{B}$ is the chromomagnetic field (and permutations thereof)
Eichten:1980mw ; Peskin:1983up . The leading contribution is from the
dimension 5 operator $\overline{h}i\Sigma\cdot\bm{B}h$ and is of ${\cal
O}(\lambda)$ in the HQET power-counting. Because we tune the coefficient of
this operator nonperturbatively, there are no associated discretization
errors. Thus we consider discretization errors from operators of ${\cal
O}(\lambda^{2},\lambda^{3})$. There are five dimension 7 bilinear operators of
the type $\vec{\sigma}\cdot\vec{B}$ in the heavy-quark effective action at
${\cal O}(\lambda^{3})$; these are given in Eqs. (87)–(91). Operators of
dimension 8 and higher in the heavy-quark effective Lagrangian are all of
${\cal O}(\lambda^{4})$ or higher in the HQET power-counting.
#### C.3.1 ${\cal O}(a^{3})$ errors
The expected size of the operators in Eqs. (87) and (88) is
$\langle{\cal O}_{\sigma\cdot B}\rangle^{\rm HQET}\sim a^{3}\Lambda_{\rm
QCD}^{4}\,.$ (109)
By construction, we tune the RHQ parameters such that we reproduce the
experimental value of the bottom-strange hyperfine splitting
$M_{B_{s}}^{*}$-$M_{B_{s}}$. Hence we divide the contributions of these
operators by $\Delta M_{B_{s}}=49$ MeV to obtain the relative error in the
$B_{s}$ hyperfine splitting. The tree-level mismatch functions for the two
operators are $f_{w^{\prime}_{B}}$ [Eq. (73)] and $f_{m_{B^{\prime}}}$ [Eq.
(75)], so we estimate their contribution to the error in the bottom-strange
hyperfine splitting to be
$\displaystyle\textrm{error}_{w^{\prime}_{B}}$ $\displaystyle=$ $\displaystyle
f_{w^{\prime}_{B}}\langle{\cal O}_{\sigma\cdot B}\rangle^{\rm HQET}/\Delta
M_{B_{s}}$ (110) $\displaystyle\sim$ $\displaystyle 0.64\%\,,$
$\displaystyle\textrm{error}_{m_{B^{\prime}}}$ $\displaystyle=$ $\displaystyle
f_{m_{B^{\prime}}}\langle{\cal O}_{\sigma\cdot B}\rangle^{\rm HQET}/\Delta
M_{B_{s}}$ (111) $\displaystyle\sim$ $\displaystyle 1.46\%\,.$
#### C.3.2 ${\cal O}(\alpha_{s}a^{3})$ errors
The expected size of the operators in Eqs. (89)–(91) is also
$\langle{\cal O}_{\sigma\cdot B}\rangle^{\rm HQET}\sim a^{3}\Lambda_{\rm
QCD}^{4}\,.$ (112)
The mismatch functions of these operators, however, vanish at tree-level
Oktay:2008ex . Because they have not been computed to one-loop, we simply
estimate their size to be $\alpha_{s}^{\overline{{\rm
MS}}}(1/a_{32^{3}})=0.22$. Under this assumption, we estimate that the
contribution of each of these operators to the bottom-strange hyperfine
splitting is
$\displaystyle\textrm{error}_{\alpha_{s}}$ $\displaystyle=$
$\displaystyle\alpha_{s}\langle{\cal O}_{\sigma\cdot B}\rangle^{\rm
HQET}/\Delta M_{B_{s}}\sim 2.36\%\,.$ (113)
This estimate is likely conservative, given that we would naively expect
${\cal O}(\alpha_{s}a^{3})$ errors to be smaller than ${\cal O}(a^{3})$
errors, due to the fact that we have not considered any possible suppression
from the 1-loop mismatch functions.
#### C.3.3 Total error
We obtain the total heavy-quark discretization error in the bottom-strange
hyperfine splitting by adding the errors from the different operators in
quadrature, including $\textrm{error}_{\alpha_{s}}$ three times because there
are three 1-loop operators:
$\displaystyle\textrm{error}^{\Delta M_{B_{s}}}_{\textrm{total}}$
$\displaystyle=$
$\displaystyle\left(\textrm{error}_{w^{\prime}_{B}}^{2}+\textrm{error}_{m_{B^{\prime}}}^{2}+3\times\textrm{error}_{\alpha_{s}}^{2}\right)^{1/2}$
(114) $\displaystyle\sim$ $\displaystyle 4.40\%\,,$
or $\sim$ 2 MeV.
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|
arxiv-papers
| 2012-06-12T14:51:18 |
2024-09-04T02:49:31.691914
|
{
"license": "Public Domain",
"authors": "Yasumichi Aoki, Norman H. Christ, Jonathan M. Flynn, Taku Izubuchi,\n Christoph Lehner, Min Li, Hao Peng, Amarjit Soni, Ruth S. Van de Water,\n Oliver Witzel",
"submitter": "Ruth Van de Water",
"url": "https://arxiv.org/abs/1206.2554"
}
|
1206.2699
|
# Repulsive Casimir force between silicon dioxide and superconductor
Anh D. Phan Department of Physics, University of South Florida, Tampa,
Florida 33620, USA anhphan@mail.usf.edu N. A. Viet Institute of Physics, 10
Daotan, Badinh, Hanoi, Vietnam
###### Abstract
We have presented a detailed investigation of the Casimir interaction between
the superconductor $Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$ (BSCCO) and silicon
dioxide with the presence of bromobenzene in between. We found the dispersion
force is repulsive and the magnitude of the force can be changed by varying
the thickness of object and temperature. The repulsive force would provide a
method to deal with the stiction problems and bring much meaningful from
practical views.
###### pacs:
Valid PACS appear here
The Casimir force, one of the most important causes of stiction problems,
gives rise to critical impediments in fabrication and operation of nano/micro-
electromechanical (NEMS/MEMS) systems. In almost all cases in which the scale
is of hundreds of nanometers, the Casimir interactions produce such a
significant amount of friction as to draw the much attention of scientists
from a wide variety of research fields bib1 ; bib2 ; bib3 . In addition, the
theoretical understanding and measurements of the Casimir interactions have
grown substantially in the last ten years, allowing physicists to have a more
detailed understanding of fundamental physics, not only in nanophysics, but
particle physics and cosmology as well.
The simplest Casimir system was produced theoretically by Casimir in 1948 when
he developed a model describing the interaction between two parallel
conducting plates bib4 . Since then, the Casimir forces between real materials
such as metals bib5 , semiconductors bib6 , semimetals bib7 and high-Tc
superconductors bib8 have been extensively studied theoretically and
experimentally. It has been found that the presence of liquids between two
objects allow the sign of the Casimir force to switch bib1 ; bib2 ; bib3 .
These repulsive Casimir forces appear when the dielectric functions of object
1 and object 2 immersed in a medium 3 satisfy the relation
$\varepsilon_{1}(i\xi)<\varepsilon_{3}(i\xi)<\varepsilon_{2}(i\xi)$ over a
wide imaginary frequency range $\xi$. It is also possible to make the
repulsion using arrays of gold nanopillars on two plates bib9 . In addition,
metamaterials are promising candidates for creating these repulsive
interactions bib10 . The combination between experimental measurements and
theoretical calculations have provided essential information to help
researchers design nanoscale devices.
The well-known Lifshitz theory developed the generalization of the Casimir
force bib3 ; bib11 ; bib12 . In the theory, the force between uncharged
objects made of real materials was given by an analytical formula with the
frequency-dependent dielectric permittivity $\varepsilon(i\xi)$ . The
variation of the dielectric functions causes the change of the Casimir
interactions. The Casimir-Lifshitz force has been studied at for systems at
the thermal equilibrium between the atom-atom, plane-plane and atom-plane
configurations bib11 ; bib20 .
The cuprate superconductor, the high-Tc superconductor and the anisotropic
materials are widely used in various devices. It has been shown theoretically
that the Casimir force in the BSCCO-air-gold system is significantly affected
by the anisotropy in the dielectric functions. In the present paper, the
Casimir-Lifshitz force is calculated in the case of the perpendicular cleave
between the cuprate superconductor and silica with bromobenzene in between.
The force calculation take into account the thermal effect and the influence
of its thickness on the dispersion force. The force is repulsive and deals
with the sticking process in nano devices.
The general expression describing the Casimir interaction between two infinite
parallel plates is the Lifshitz formula. At a given separation $a$ and given
temperature $T$, the Casimir pressure between two plates is given bib11 ;
bib12
$\displaystyle
P(a)=-\dfrac{k_{B}T}{\pi}\sum_{n=0}^{\infty}\int_{0}^{\infty}qk_{\perp}dk_{\perp}\sum_{\alpha}\dfrac{r_{\alpha}^{(1)}r_{\alpha}^{(2)}}{e^{2qa}-r_{\alpha}^{(1)}r_{\alpha}^{(2)}},$
(1)
here $k_{B}$ is the Boltzmann constant, $k_{\perp}$ is the wave vector
component perpendicular to the plate, $\alpha=TM,TE$, $r_{TM}^{(1,2)}$ and
$r_{TE}^{(1,2)}$ denote the reflection coefficients of the transverse magnetic
(TM) and transverse electric (TE) field, respectively. The superscript (1) and
(2) correspond to the first body (silica) and the second body (BSCCO). In
addition, $q=\sqrt{k_{\perp}^{2}+\varepsilon_{3}\xi_{n}^{2}/c^{2}}$,
$\xi_{n}=2\pi nk_{B}T/\hbar$ are the Matsubara frequencies,$n$ is an integer,
and $\varepsilon_{3}\equiv\varepsilon_{3}(i\xi_{n})$ is the dielectric
function of medium in between two objects. In the calculation, bromobenzene is
medium. The dielectric function of the liquid can be described using the
oscillator model bib2
$\displaystyle\varepsilon_{3}(i\xi)=1+\sum_{i}\frac{C_{i}}{1+\xi^{2}/\omega_{i}^{2}},$
(2)
where parameters $C_{i}$ and $\omega_{i}$ were obtained by fitting with
experimental data in the large range of frequency bib2 .
It is important to note that for $n=0$, the prefactor of the integration is
$k_{B}T/(2\pi)$ instead of $k_{B}T/\pi$ for other values of $n$. In the case
of silicon dioxide, the reflection coefficients are presented bib2 ; bib3
$\displaystyle
r_{TM}^{(1)}=\frac{\varepsilon_{1}q-\varepsilon_{3}\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}}{\varepsilon_{1}q+\varepsilon_{3}\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}},$
(3) $\displaystyle
r_{TE}^{(1)}=\frac{q-\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}}{q+\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}},$
(4)
in which $\varepsilon_{1}\equiv\varepsilon_{1}(i\xi_{n})$ is the dielectric
function of silica. The dielectric fuction still has the form of an oscillator
model as in Eq.(2) and the parameters were generated in Ref.bib2 . Considering
the role of the thickness $D$ of the silica slab on the Casimir interaction,
the reflection coefficients $TM$ and $TE$ in Eq.(3) and Eq.(4) become bib13 ;
bib14
$\displaystyle r_{TM}^{(1)}\rightarrow
r_{TM}^{(1)}\frac{1-e^{-2\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}D}}{1-r_{TM}^{(1)2}e^{-2\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}D}},$
(5) $\displaystyle r_{TE}^{(1)}\rightarrow
r_{TE}^{(1)}\frac{1-e^{-2\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}D}}{1-r_{TE}^{(1)2}e^{-2\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}D}}.$
(6)
Because of the uniaxial property of BSCCO, its permittivity in the
perpendicular cleave is represented in the form of a tensor bib8
$\displaystyle\bar{\varepsilon_{2}}=\left[\begin{array}[]{ccc}\varepsilon_{2\perp}&0&0\\\
0&\varepsilon_{2\perp}&0\\\ 0&0&\varepsilon_{2||}\end{array}\right]$ (10)
where $\varepsilon_{2||}$ and $\varepsilon_{2\perp}$ are the dielectric
components along the optical axis and perpendicular to the optical axis.
Therefore, the expression of the reflection coefficients for BSCCO must be
modified. The TM and TE coefficients on the liquid-BSCCO interface are
performed bib8 ; bib15
$\displaystyle
r_{TM}^{(2)}=\frac{\varepsilon_{2\perp}q-\varepsilon_{3}\sqrt{\frac{\varepsilon_{2\perp}}{\varepsilon_{2||}}k_{\perp}^{2}+\varepsilon_{2\perp}\xi_{n}^{2}/c^{2}}}{\varepsilon_{2\perp}q+\varepsilon_{3}\sqrt{\frac{\varepsilon_{2\perp}}{\varepsilon_{2||}}k_{\perp}^{2}+\varepsilon_{2\perp}\xi_{n}^{2}/c^{2}}},,$
(11) $\displaystyle
r_{TE}^{(2)}=\frac{q-\sqrt{k_{\perp}^{2}+\varepsilon_{2\perp}\xi_{n}^{2}/c^{2}}}{q+\sqrt{k_{\perp}^{2}+\varepsilon_{2\perp}\xi_{n}^{2}/c^{2}}}.$
(12)
For BSCCO, the dielectric response $\varepsilon_{2||}$ and
$\varepsilon_{2\perp}$ are modeled based on the damped-multioscillator model.
Parameters corresponding to the resonance frequency, damping and oscillator
strength were given in Ref.bib8 .
As shown in Fig. 1, the Casimir forces are significantly influenced by the
thickness of the slab. The Casimir interactions in the real system are much
smaller than the force in the ideal case. It is clear that the presence of
bromodenzene makes the Casimir force in this case is repulsive. For $D>500$
$nm$, the effect of thickness nearly vanishes and the Casimir force in the
system can be modeled as an interactions between two plates. For metal such as
gold, when the thickness $D>30$ $nm$, a gold thin film can be treated as a
gold plate bib16 ; bib17 . The reason for the discrepancy between the two
cases is that the conductivity of metal is much larger than that of silicon
dioxide. When the thickness of a metal slab is reduced, it appears as if the
skin-depth effect occurs. This effect, however, is not present in the case of
silica.
Figure 1: (Color online) The relative Casimir pressures between a BSCCO plate
and a silica slab in the presence of bromobenzene, here $F_{0}(a)=\pi^{2}\hbar
c/(240a^{4})$ is the Casimir force between two ideal metal plates.
At small distances, there is not much change in the force with different
thicknesses. The reason is that at this range $D/a>>1$, so
$e^{-2\sqrt{k_{\perp}^{2}+\varepsilon_{1}\xi_{n}^{2}/c^{2}}D}<<1$. The
influence of thickness on the interaction disappears.
Figure 2: (Color online) The relative Casimir pressures are taken into account
the thermal effect with different values of thickness.
Bromobenzene molecules exist in liquid form over an important range of
temperature from $242$ $K$ to $429$ $K$. Fig. 2 shows the Casimir force in
this temperature range. It is evident that the interaction depends notably on
temperature. There are variations in the Casimir force at different
thicknesses. It is now possible to design a non-touching system because the
Casimir force is repulsive for entire range of distance. The Casimir force,
and the gravitational forces between the two bodies, with the earth lead to
the repulsive-attractive transition in our system and results in our system
reaching an equilibrium distance bib18 . Obviously, the stable position can be
varied by changing the sizes of bodies, the thickness and temperature. In
Ref.bib19 , authors presented the proposal for measuring the Casimir force
with the presence of a tiny spring in order to get a balanced position and the
oscillation frequency. However, in this case, the equilibrium positions exist
naturely. It is not necessary to attach a spring to the system to measure the
Casimir force. The force can be found via observation of the oscillation
frequencies. Because of the anisotropic property, the expressions of the
reflection coefficients TE and TM in the case of the parallel cleave
orientation are different from Eq.(11) and Eq.(12). This discrepancy is proof
that there is a difference between the Casimir forces in two orientations.
The thermal effect in the Casimir interaction plays an important role at long
distances bib12 . For short distances, this effect can be ignored. One can
used the double integration instead of summation and single integration as
Eq.(1) in calculations at short distances. The expression of the double
integration provides a good agreement with experiment.
This work presents a reliable anti-stiction method of NEMS/MEMS structures.
The presence of liquid can address the stiction issue causing catastrophic
failure in nanoscale devices. The temperature and thickness depedence of the
Casimir force allows control of the adhesion force between two surfaces. It is
currently difficult to measure properties in fluidic environments. However,
using liquid films in NEMS/MEMS devices with the range of the thickness of
liquid layer from 2 $nm$ to 70 $nm$ has been intensively investigated bib21 ;
bib22 . Moreover, authors in bib23 described the behavior of the tiny devices
in liquids. These research makes it possible to design nanostructures in the
microfluidic environment and our studies give an interesting view of what
happens physically in systems submerged in liquids.
###### Acknowledgements.
The work was partly funded by the Nafosted Grant No. 103.06-2011.51.
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|
arxiv-papers
| 2012-06-13T01:47:04 |
2024-09-04T02:49:31.710495
|
{
"license": "Public Domain",
"authors": "Anh D. Phan and N. A. Viet",
"submitter": "Anh Phan",
"url": "https://arxiv.org/abs/1206.2699"
}
|
1206.2794
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-161 LHCb-PAPER-2012-002 October 23, 2012
Measurement of $b$-hadron branching fractions for two-body decays into
charmless charged hadrons
The LHCb collaboration†††Authors are listed on the following pages.
Based on data corresponding to an integrated luminosity of 0.37
$\mathrm{fb}^{-1}$ collected by the LHCb experiment in 2011, the following
ratios of branching fractions are measured:
$\displaystyle\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$ $\displaystyle=$ $\displaystyle 0.262\pm 0.009\pm 0.017,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow K^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle 0.316\pm 0.009\pm 0.019,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$ $\displaystyle=$ $\displaystyle 0.074\pm 0.006\pm 0.006,$
$\displaystyle(f_{d}/f_{s})\cdot\mathcal{B}\left(B^{0}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle 0.018\,^{+\,0.008}_{-\,0.007}\pm 0.009,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle 0.050\,^{+\,0.011}_{-\,0.009}\pm 0.004,$
$\displaystyle\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow
p\pi^{-}\right)/\,\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow pK^{-}\right)$
$\displaystyle=$ $\displaystyle 0.86\pm 0.08\pm 0.05,$
where the first uncertainties are statistical and the second systematic. Using
the current world average of $\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$ and the ratio of the strange to light neutral $B$ meson
production $f_{s}/f_{d}$ measured by LHCb, we obtain:
$\displaystyle\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle(5.08\pm 0.17\pm 0.37)\times 10^{-6},$
$\displaystyle\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(23.0\pm 0.7\pm 2.3)\times 10^{-6},$
$\displaystyle\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(5.4\pm 0.4\pm 0.6)\times 10^{-6},$
$\displaystyle\mathcal{B}(B^{0}\rightarrow K^{+}K^{-})$ $\displaystyle=$
$\displaystyle(0.11\,^{+\,0.05}_{-\,0.04}\pm 0.06)\times 10^{-6},$
$\displaystyle\mathcal{B}(B^{0}_{s}\rightarrow\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(0.95\,^{+\,0.21}_{-\,0.17}\pm 0.13)\times
10^{-6}.$
The measurements of $\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$,
$\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)$ and
$\mathcal{B}(B^{0}\rightarrow K^{+}K^{-})$ are the most precise to date. The
decay mode $B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$ is observed for the first time
with a significance of more than $5\sigma$.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
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Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M.
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Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
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M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
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Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
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Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J.
Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H.
Voss10, R. Waldi55, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42,
A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches 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
## 1 Introduction
In the quest for physics beyond the Standard Model (SM) in the flavour sector,
the study of charmless $H_{b}\rightarrow h^{+}h^{\prime-}$ decays, where
$H_{b}$ is a $b$-flavoured meson or baryon, and $h^{(\prime)}$ stands for a
pion, kaon or proton, plays an important role. A simple interpretation of the
$C\\!P$-violating observables of the charmless two-body $b$-hadron decays in
terms of Cabibbo-Kobayashi-Maskawa (CKM) weak phases [1, *Kobayashi:1973fv] is
not possible. The presence of so-called penguin diagrams in addition to tree
diagrams gives non-negligible contributions to the decay amplitude and
introduces unknown hadronic factors. This then poses theoretical challenges
for an accurate determination of CKM phases. On the other hand, penguin
diagrams may have contributions from physics beyond the SM [3, 4, 5, 6, 7].
These questions have motivated an experimental programme aimed at the
measurement of the properties of these decays [8, 9, 10, 11, 12].
Using data corresponding to an integrated luminosity of $0.37$
$\mathrm{fb}^{-1}$ collected by the LHCb experiment in 2011, we report
measurements of the branching fractions $\mathcal{B}$ of the
$B^{0}\rightarrow\pi^{+}\pi^{-}$, $B_{s}^{0}\rightarrow K^{+}K^{-}$,
$B_{s}^{0}\rightarrow\pi^{+}K^{-}$, $B^{0}\rightarrow K^{+}K^{-}$ and
$B_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ decays. Furthermore, we also measure the
ratio of the $\Lambda^{0}_{b}\rightarrow p\pi^{-}$ and
$\Lambda^{0}_{b}\rightarrow pK^{-}$ branching fractions. The inclusion of
charge-conjugate decay modes is implied throughout the paper.
The ratio of branching fractions between any two of these decays can be
expressed as
$\frac{\mathcal{B}(H_{b}\rightarrow F)}{\mathcal{B}(H^{\prime}_{b}\rightarrow
F^{\prime})}=\frac{f_{H^{\prime}_{b}}}{f_{H_{b}}}\cdot\frac{N(H_{b}\rightarrow
F)}{N(H^{\prime}_{b}\rightarrow F^{\prime})}\cdot\frac{\varepsilon_{\rm
rec}(H^{\prime}_{b}\rightarrow F^{\prime})}{\varepsilon_{\rm
rec}(H_{b}\rightarrow F)}\cdot\frac{\varepsilon_{\rm
PID}(F^{\prime})}{\varepsilon_{\rm PID}(F)}$ (1)
where $f_{H_{b}^{(\prime)}}$ is the probability for a $b$ quark to hadronize
into a $H_{b}^{(\prime)}$ hadron, $N$ is the observed yield of the given decay
to the final state $F^{(\prime)}$, $\varepsilon_{\rm rec}$ is the overall
reconstruction efficiency, excluding particle identification (PID), and
$\varepsilon_{\rm PID}$ is the PID efficiency for the corresponding final
state hypothesis. We choose to measure ratios where a better cancellation of
systematic uncertainties can be achieved.
## 2 Detector, trigger and event selection
The LHCb detector [13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momenta. Charged hadrons are
identified using two ring-imaging Cherenkov (RICH) detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage which performs a full event
reconstruction.
The software trigger requires a two-, three- or four-track secondary vertex
with a high sum of the transverse momenta of the tracks, significant
displacement from the primary interaction, and at least one track with a
transverse momentum exceeding $1.7$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Furthermore, it exploits the impact parameter, defined as the smallest
distance between the reconstructed trajectory of the particle and the $pp$
collision vertex, requiring its $\chi^{2}$ to be greater than 16. A
multivariate algorithm is used for the identification of the secondary
vertices [14]. In addition, a dedicated two-body software trigger is used. To
discriminate between signal and background events, this trigger selection
imposes requirements on: the quality of the online-reconstructed tracks
($\chi^{2}$/ndf, where ndf is the number of degrees of freedom), their
transverse momenta ($p_{\mathrm{T}}$) and their impact parameters
($d_{\mathrm{IP}}$); the distance of closest approach of the daughter
particles ($d_{\mathrm{CA}}$); the transverse momentum of the $b$-hadron
candidate ($p_{\mathrm{T}}^{B}$), its impact parameter ($d_{\mathrm{IP}}^{B}$)
and its decay time ($t_{\pi\pi}$, calculated assuming decay into
$\pi^{+}\pi^{-}$). Only $b$-hadron candidates within the $\pi^{+}\pi^{-}$
invariant mass range 4.7–5.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are
accepted. The $\pi^{+}\pi^{-}$ mass hypothesis is chosen to ensure all
charmless two-body $b$-hadron decays are selected using the same criteria.
The events passing the trigger requirements are then filtered to further
reduce the size of the data sample. In addition to tighter requirements on the
kinematic variables already used in the software trigger, requirements on the
larger of the transverse momenta ($p_{\mathrm{T}}^{h}$) and of the impact
parameters ($d_{\mathrm{IP}}^{h}$) of the daughter particles are applied. As
the rates of the various signals under study span two orders of magnitude, for
efficient discrimination against combinatorial background three different sets
of kinematic requirements are used to select events for: (A) the measurements
of
$\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$, $\mathcal{B}\left(B^{0}_{s}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow K^{+}\pi^{-}\right)$ and
$\mathcal{B}(\Lambda^{0}_{b}\rightarrow
pK^{-})/\,\mathcal{B}(\Lambda^{0}_{b}\rightarrow p\pi^{-})$; (B) the
measurement of
$\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$; (C) the measurements of
$\mathcal{B}\left(B^{0}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$
and
$\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)$.
The kinematic requirements adopted in each selection are summarized in Table
1.
Table 1: Summary of criteria adopted in the event selections A, B and C defined in the text. Variable | Selection A | Selection B | Selection C
---|---|---|---
Track $p_{\mathrm{T}}\,[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>$ | $1.1$ | $>$ | $1.2$ | $>$ | $1.2$
Track $d_{\mathrm{IP}}\,[\mathrm{\mu m}]$ | $>$ | $150$ | $>$ | $200$ | $>$ | $200$
Track $\chi^{2}$/ndf | $<$ | $3$ | $<$ | $3$ | $<$ | $3$
$\mathrm{max}(p_{\mathrm{T}}^{h^{+}},\,p_{\mathrm{T}}^{h^{\prime-}})\,[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>$ | $2.8$ | $>$ | $3.0$ | $>$ | $3.0$
$\mathrm{max}(d_{\mathrm{IP}}^{h^{+}},\,d_{\mathrm{IP}}^{h^{\prime-}})\,[\mathrm{\mu m}]$ | $>$ | $300$ | $>$ | $400$ | $>$ | $400$
$d_{\rm CA}\,[\mathrm{\mu m}]$ | $<$ | $80$ | $<$ | $80$ | $<$ | $80$
$d_{\mathrm{IP}}^{B}\,[\mathrm{\mu m}]$ | $<$ | $60$ | $<$ | $60$ | $<$ | $60$
$p_{\mathrm{T}}^{B}\,[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>$ | $2.2$ | $>$ | $2.4$ | $>$ | $2.8$
$t_{\pi\pi}\,[\textrm{ps}]$ | $>$ | $0.9$ | $>$ | $1.5$ | $>$ | $2.0$
In order to evaluate the ratios of reconstruction efficiencies
$\varepsilon_{\rm rec}$, needed to calculate the relative branching fractions
of two $H_{b}\rightarrow h^{+}h^{\prime-}$ decays, we apply selection and
trigger requirements to fully simulated events. The results of this study are
summarized in Table 2, where the uncertainties are due to the finite size of
the simulated event samples. Other sources of systematic uncertainties are
negligible at the current level of precision. This is confirmed by studies on
samples of $D^{0}$ mesons decaying into pairs of charged hadrons, where
reconstruction efficiencies are determined from data using measured signal
yields and current world averages of the corresponding branching fractions.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [15] with a
specific LHCb configuration [16]. Decays of hadrons are described by EvtGen
[17] in which final state radiation is generated using Photos [18]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [19, *Agostinelli:2002hh] as described in
Ref. [21].
Table 2: Ratios of reconstruction efficiencies of the various channels, as determined from Monte Carlo simulation, corresponding to the three event selections of Table 1. PID efficiencies are not included here. The tight requirement on $t_{\pi\pi}$ used in selection C leads to a sizable difference from unity of the ratios in the last two rows, as the $B_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ and $B_{s}^{0}\rightarrow K^{+}K^{-}$ decays proceed mainly via the short lifetime component of the $B^{0}_{s}$ meson. Selection | Efficiency ratio | Value
---|---|---
A | $\varepsilon_{\rm rec}(B^{0}\rightarrow K^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm rec}(B^{0}\rightarrow\pi^{+}\pi^{-})$ | $0.98\pm 0.02$
$\varepsilon_{\rm rec}(B^{0}\rightarrow K^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm rec}(B_{s}^{0}\rightarrow K^{+}K^{-})$ | $1.00\pm 0.02$
$\varepsilon_{\rm rec}(\Lambda^{0}_{b}\rightarrow pK^{-})$ | $/$ | $\varepsilon_{\rm rec}(\Lambda^{0}_{b}\rightarrow p\pi^{-})$ | $1.00\pm 0.02$
B | $\varepsilon_{\rm rec}(B^{0}\rightarrow K^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm rec}(B_{s}^{0}\rightarrow\pi^{+}K^{-})$ | $0.98\pm 0.02$
C | $\varepsilon_{\rm rec}(B^{0}\rightarrow\pi^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm rec}(B_{s}^{0}\rightarrow\pi^{+}\pi^{-})$ | $1.10\pm 0.03$
$\varepsilon_{\rm rec}(B_{s}^{0}\rightarrow K^{+}K^{-})$ | $/$ | $\varepsilon_{\rm rec}(B^{0}\rightarrow K^{+}K^{-})$ | $0.92\pm 0.02$
## 3 Particle identification
In order to disentangle the various $H_{b}\rightarrow h^{+}h^{\prime-}$ decay
modes, the selected $b$-hadron candidates are divided into different final
states using the PID capabilities of the two RICH detectors. Different sets of
PID criteria are applied to the candidates passing the three selections, with
PID discrimination power increasing from selection A to selection C. These
criteria identify mutually exclusive sets of candidates. As discriminators we
employ the quantities $\Delta\ln\mathcal{L}_{K\pi}$ and
$\Delta\ln\mathcal{L}_{p\pi}$, or their difference $\Delta\ln\mathcal{L}_{Kp}$
when appropriate, where $\Delta\ln\mathcal{L}_{\alpha\beta}$ is the difference
between the natural logarithms of the likelihoods for a given daughter
particle under mass hypotheses $\alpha$ and $\beta$, respectively. In order to
determine the corresponding PID efficiency for each two-body final state, a
data-driven method is employed that uses $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $\Lambda\rightarrow p\pi^{-}$ decays as
control samples. In this analysis about 6.7 million $D^{*+}$ decays and 4.2
million $\Lambda$ decays are used.
The production and decay kinematics of the $D^{0}\rightarrow K^{-}\pi^{+}$ and
$\Lambda\rightarrow p\pi^{-}$ channels differ from those of the $b$-hadron
decays under study. Since the RICH PID information is momentum dependent, a
calibration procedure is performed by reweighting the
$\Delta\ln\mathcal{L}_{\alpha\beta}$ distributions of true pions, kaons and
protons obtained from the calibration samples, with the momentum distributions
of daughter particles resulting from $H_{b}\rightarrow h^{+}h^{\prime-}$
decays. The $\Delta\ln\mathcal{L}_{\alpha\beta}$ and momentum distributions of
the calibration samples and the momentum distributions of $H_{b}$ daughter
particles are determined from data. In order to obtain background-subtracted
distributions, extensive use of the _sPlot_ technique [22] is made. This
technique requires that extended maximum likelihood fits are performed, where
signal and background components are modelled. It is achieved by fitting
suitable models to the distribution of the variable $\delta
m=m_{K\pi\pi}-m_{K\pi}$ for $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$
decays, to the $p\pi^{-}$ mass for $\Lambda\rightarrow p\pi^{-}$ decays and,
for each of the three selections, to the invariant mass assuming the
$\pi^{+}\pi^{-}$ hypothesis for $H_{b}\rightarrow h^{+}h^{\prime-}$ decays.
The variables $m_{K\pi\pi}$ and $m_{K\pi}$ are the reconstructed $D^{*+}$ and
$D^{0}$ candidate masses, respectively.
In Fig. 1 the distributions of the variable $\delta m$ and of the invariant
mass of $\Lambda\rightarrow p\pi^{-}$ are shown. The superimposed curves are
the results of the maximum likelihood fits to the spectra.
Figure 1: Distributions of (a) $\delta m=m_{K\pi\pi}-m_{K\pi}$ for
$D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ candidates and (b) invariant
mass of $\Lambda\rightarrow p\pi^{-}$ candidates, used for the PID
calibration. The curves are the results of maximum likelihood fits. Figure 2:
Invariant $\pi^{+}\pi^{-}$ mass for candidates passing the selection A of
Table 1. The result of an unbinned maximum likelihood fit is overlaid. The
main contributions to the fit model are also shown.
Figure 3: Momentum distributions of (a) pions and (b) kaons from $D^{0}$
decays in the PID calibration sample (histograms). For comparison, the points
represent the inclusive momentum distribution of daughter particles in
$H_{b}\rightarrow h^{+}h^{\prime-}$ decays. The distributions are normalized
to the same area. This example corresponds to selection A.
The $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ signal $\delta m$ spectrum
has been modelled using the sum of three Gaussian functions ($G_{3}$) with a
common mean ($\mu$), convolved with an empirical function which describes the
asymmetric tail on the right-hand side of the spectrum:
$g(\delta m)=A\left[\Theta(\delta m^{\prime}-\mu)\cdot\left(\delta
m^{\prime}-\mu\right)^{s}\right]\otimes G_{3}(\delta m-\delta m^{\prime}),$
(2)
where $A$ is a normalization factor, $\Theta$ is the Heaviside (step)
function, $s$ is a free parameter determining the asymmetric shape of the
distribution, $\otimes$ stands for convolution and the convolution integral
runs over $\delta m^{\prime}$. In order to model the background shape we use
$h(\delta m)=B\left[1-\exp\left(-\frac{\delta m-\delta
m_{0}}{c}\right)\right],$ (3)
where $B$ is a normalization factor, and the free parameters $\delta m_{0}$
and $c$ govern the shape of the distribution. The fit to the
$\Lambda\rightarrow p\pi^{-}$ spectrum is made using a sum of three Gaussian
functions for the signal and a second order polynomial for the background.
Table 3: PID efficiencies (in %), for the various mass hypotheses, corresponding to the event samples passing the selections A, B and C of Table 1. Different sets of PID requirements are applied in the three cases. Selection A | $\pi^{+}\pi^{-}$ | $K^{+}K^{-}$ | $K^{+}\pi^{-}$ | $p\pi^{-}$ | $pK^{-}$
---|---|---|---|---|---
$B^{0}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | 43.1 | 0.33 | 28.6 | 1.53 | 0.13
$B^{0}_{s}$ | $\rightarrow$ | $K^{+}K^{-}$ | 0.05 | 55.0 | 15.4 | 0.05 | 1.63
$B^{0}_{(s)}$ | $\rightarrow$ | $K^{+}\pi^{-}$ | 1.40 | 4.17 | 67.9 | 0.72 | 0.06
$\bar{B}^{0}_{(s)}$ | $\rightarrow$ | $\pi^{+}K^{-}$ | 1.40 | 4.17 | 2.09 | 0.02 | 0.85
$\Lambda^{0}_{b}$ | $\rightarrow$ | $p\pi^{-}$ | 1.93 | 0.92 | 16.8 | 35.4 | 3.16
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $\pi^{+}\bar{p}$ | 1.93 | 0.92 | 0.95 | 0.03 | 0.18
$\Lambda^{0}_{b}$ | $\rightarrow$ | $pK^{-}$ | 0.06 | 12.2 | 1.92 | 1.18 | 40.2
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $K^{+}\bar{p}$ | 0.06 | 12.2 | 4.51 | 0.03 | 0.18
Selection B | $\pi^{+}\pi^{-}$ | $K^{+}K^{-}$ | $K^{+}\pi^{-}$ | $p\pi^{-}$ | $pK^{-}$
$B^{0}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | 42.8 | 0.33 | 2.06 | 1.51 | 0.13
$B^{0}_{s}$ | $\rightarrow$ | $K^{+}K^{-}$ | 0.05 | 54.5 | 1.09 | 0.05 | 1.63
$B^{0}_{(s)}$ | $\rightarrow$ | $K^{+}\pi^{-}$ | 1.38 | 4.12 | 35.7 | 0.72 | 0.06
$\bar{B}^{0}_{(s)}$ | $\rightarrow$ | $\pi^{+}K^{-}$ | 1.38 | 4.12 | 0.02 | 0.02 | 0.84
$\Lambda^{0}_{b}$ | $\rightarrow$ | $p\pi^{-}$ | 1.90 | 0.90 | 6.01 | 35.4 | 3.16
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $\pi^{+}\bar{p}$ | 1.90 | 0.90 | 0.03 | 0.03 | 0.17
$\Lambda^{0}_{b}$ | $\rightarrow$ | $pK^{-}$ | 0.06 | 11.8 | 0.09 | 1.19 | 40.2
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $K^{+}\bar{p}$ | 0.06 | 11.8 | 0.88 | 0.03 | 0.17
Selection C | $\pi^{+}\pi^{-}$ | $K^{+}K^{-}$ | $K^{+}\pi^{-}$ | $p\pi^{-}$ | $pK^{-}$
$B^{0}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | 40.5 | 0.00 | 1.64 | 1.51 | 0.00
$B^{0}_{s}$ | $\rightarrow$ | $K^{+}K^{-}$ | 0.04 | 21.4 | 0.98 | 0.04 | 1.01
$B^{0}_{(s)}$ | $\rightarrow$ | $K^{+}\pi^{-}$ | 1.27 | 0.11 | 32.4 | 0.70 | 0.00
$\bar{B}^{0}_{(s)}$ | $\rightarrow$ | $\pi^{+}K^{-}$ | 1.27 | 0.11 | 0.01 | 0.02 | 0.54
$\Lambda^{0}_{b}$ | $\rightarrow$ | $p\pi^{-}$ | 1.26 | 0.00 | 3.16 | 33.5 | 0.13
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $\pi^{+}\bar{p}$ | 1.26 | 0.00 | 0.02 | 0.02 | 0.03
$\Lambda^{0}_{b}$ | $\rightarrow$ | $pK^{-}$ | 0.04 | 1.35 | 0.05 | 1.08 | 23.9
$\bar{\Lambda}^{0}_{b}$ | $\rightarrow$ | $K^{+}\bar{p}$ | 0.04 | 1.35 | 0.65 | 0.02 | 0.03
Figure 2 shows the invariant mass assuming the $\pi^{+}\pi^{-}$ hypothesis for
selected $b$-hadron candidates, using the kinematic selection A of Table 1 and
without applying any PID requirement. The shapes describing the various signal
decay modes have been fixed by parameterizing the mass distributions obtained
from Monte Carlo simulation convolved with a Gaussian resolution function with
variable mean and width. The three-body and combinatorial backgrounds are
modelled using an ARGUS function [23], convolved with the same Gaussian
resolution function used for the signal distributions, and an exponential
function, respectively. The relative yields between the signal components have
been fixed according to the known values of branching fractions and
hadronization probabilities of $B^{0}$, $B^{0}_{s}$ and $\Lambda^{0}_{b}$
hadrons [24]. The fits corresponding to the kinematic selection criteria B and
C of Table 1 have also been made, although not shown, in order to take into
account possible differences in the momentum distributions due to different
selection criteria.
Table 4: Ratios of PID efficiencies used to compute the relevant ratios of branching fractions, corresponding to selection A. Efficiency ratio | Value
---|---
$\varepsilon_{\rm PID}(K^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm PID}(\pi^{+}\pi^{-})$ | $1.57\pm 0.09$
$\varepsilon_{\rm PID}(K^{+}\pi^{-})$ | $/$ | $\varepsilon_{\rm PID}(K^{+}K^{-})$ | $1.23\pm 0.06$
$\varepsilon_{\rm PID}(pK^{-})$ | $/$ | $\varepsilon_{\rm PID}(p\pi^{-})$ | $1.14\pm 0.05$
As mentioned above, the _sPlot_ procedure is used to determine the various
$\Delta\ln\mathcal{L}_{\alpha\beta}$ and momentum distributions, and these are
used to reweight the $D^{*+}$ and $\Lambda$ calibration samples. As an
example, the momentum distributions of pions and kaons from $D^{0}$ decays and
the inclusive momentum distribution of daughter particles in $H_{b}\rightarrow
h^{+}h^{\prime-}$ decays, the latter corresponding to selection A, are shown
in Fig. 3.
The PID efficiencies corresponding to the three selections are determined by
applying the PID selection criteria to the reweighted $D^{*+}$ and $\Lambda$
calibration samples. The results are reported in Table 3. Using these
efficiencies, the relevant PID efficiency ratios are determined and summarized
in Table 4. These ratios correspond to selection A only, since for the
measurements involved in B and C the final states are identical and the ratios
of PID efficiencies are equal to unity. It has been verified that the PID
efficiencies do not show any sizeable dependence on the flavour of the parent
hadron, as differences in the momentum distributions of the daughter particles
for different parent hadrons are found to be small. Owing to the large sizes
of the calibration samples, the uncertainties associated to the PID efficiency
ratios are dominated by systematic effects, intrinsically related to the
calibration procedure. They are estimated by means of a data-driven approach,
where several fits to the $B^{0}\rightarrow K^{+}\pi^{-}$ mass spectrum are
made. The mass distributions in each fit are obtained by varying the PID
selection criteria over a wide range, and then comparing the variation of the
$B^{0}\rightarrow K^{+}\pi^{-}$ signal yields determined by the fits to that
of the PID efficiencies predicted by the calibration procedure. The largest
deviation is then used to estimate the size of the systematic uncertainty.
## 4 Invariant mass fits to $H_{b}\rightarrow h^{+}h^{\prime-}$ spectra
Figure 4: Invariant mass spectra corresponding to selection A for the mass
hypotheses (a) $K^{+}\pi^{-}$, (b) $\pi^{+}\pi^{-}$, (c) $K^{+}K^{-}$, (d)
$pK^{-}$ and (e) $p\pi^{-}$, and to selection B for the mass hypothesis (f)
$K^{+}\pi^{-}$. The results of the unbinned maximum likelihood fits are
overlaid. The main components contributing to the fit model are also shown.
Unbinned maximum likelihood fits are performed to the mass spectra of events
passing the selections A, B and C with associated PID selection criteria. For
each selection we have five different spectra, corresponding to the final
state hypotheses $K^{+}\pi^{-}$, $\pi^{+}\pi^{-}$, $K^{+}K^{-}$, $pK^{-}$ and
$p\pi^{-}$, to which we perform a simultaneous fit. Since each signal channel
is also a background for all the other signal decay modes in case of
misidentification of the final state particles (cross-feed background), the
simultaneous fits to all the spectra allow a determination of the yields of
the signal components together with those of the cross-feed backgrounds, once
the appropriate PID efficiency factors are taken into account. The signal
component for each hypothesis is described by a single Gaussian distribution,
convolved with a function which describes the effect of the final state
radiation on the mass line shape [25]. The combinatorial background is
modelled by an exponential function and the shapes of the cross-feed
backgrounds are obtained from Monte Carlo simulation. The background due to
partially reconstructed three-body $B$ decays is parameterized by an ARGUS
function [23] convolved with a Gaussian resolution function that has the same
width as the signal distribution.
Figure 5: Invariant mass spectra corresponding to selection C for the mass hypotheses (a, b) $K^{+}K^{-}$ and (c, d) $\pi^{+}\pi^{-}$. Plots (b) and (d) are the same as (a) and (c) respectively, but magnified to focus on the rare $B^{0}\rightarrow K^{+}K^{-}$ and $B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$ signals. The results of the unbinned maximum likelihood fits are overlaid. The main components contributing to the fit model are also shown. Table 5: Signal yields determined by the unbinned maximum likelihood fits to the data samples surviving the event selections A, B and C of Table 1 with the associated PID criteria. Only statistical uncertainties are shown. Selection | Decay | Signal yield
---|---|---
A | $B^{0}$ | $\rightarrow$ | $K^{+}\pi^{-}$ | $9822$ | $\pm$ | $122$
$B^{0}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | $1667$ | $\pm$ | $51$
$B^{0}_{s}$ | $\rightarrow$ | $K^{+}K^{-}$ | $2523$ | $\pm$ | $59$
$\Lambda^{0}_{b}$ | $\rightarrow$ | $pK^{-}$ | $372$ | $\pm$ | $22$
$\Lambda^{0}_{b}$ | $\rightarrow$ | $p\pi^{-}$ | $279$ | $\pm$ | $22$
B | $B^{0}$ | $\rightarrow$ | $K^{+}\pi^{-}$ | $3295$ | $\pm$ | $59$
$B^{0}_{s}$ | $\rightarrow$ | $\pi^{+}K^{-}$ | $249$ | $\pm$ | $20$
C | $B^{0}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | $1076$ | $\pm$ | $36$
$B^{0}_{s}$ | $\rightarrow$ | $K^{+}K^{-}$ | $682$ | $\pm$ | $27$
$B^{0}$ | $\rightarrow$ | $K^{+}K^{-}$ | $13\,^{+\,6}_{-\,5}$
$B^{0}_{s}$ | $\rightarrow$ | $\pi^{+}\pi^{-}$ | $49\,^{+\,11}_{-\,9}$
Table 6: Ratios of signal yields needed for the measurement of the relative branching fractions. Only statistical uncertainties are shown. Selection | Ratio | Value
---|---|---
A | $\frac{N(B^{0}\rightarrow\pi^{+}\pi^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $0.170\pm 0.006$
$\frac{N(B_{s}^{0}\rightarrow K^{+}K^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $0.257\pm 0.007$
$\frac{N(\Lambda^{0}_{b}\rightarrow p\pi^{-})}{N(\Lambda^{0}_{b}\rightarrow pK^{-})}$ | $0.75\pm 0.07$
B | $\frac{N(B_{s}^{0}\rightarrow\pi^{+}K^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $0.076\pm 0.006$
C | $\frac{N(B^{0}\rightarrow K^{+}K^{-})}{N(B^{0}_{s}\rightarrow K^{+}K^{-})}$ | $0.019\,^{+\,0.009}_{-\,0.007}$
$\frac{N(B_{s}^{0}\rightarrow\pi^{+}\pi^{-})}{N(B^{0}\rightarrow\pi^{+}\pi^{-})}$ | $0.046\,^{+\,0.010}_{-\,0.009}$
The overall mass resolution determined from the fits is about 22
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Figure 4 shows the $K^{+}\pi^{-}$,
$\pi^{+}\pi^{-}$, $K^{+}K^{-}$, $pK^{-}$ and $p\pi^{-}$ invariant mass spectra
corresponding to selection A and the $K^{+}\pi^{-}$ spectrum corresponding to
selection B. Figure 5 shows the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ mass spectra
corresponding to selection C. As is apparent in the latter, while a
$B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$ mass peak is visible above the
combinatorial background, there are not yet sufficient data to observe a
significant $B^{0}\rightarrow K^{+}K^{-}$ signal. As an additional
complication, the mass peak of the $B^{0}\rightarrow K^{+}K^{-}$ decay is
expected in a region where various components give non-negligible
contributions, in particular the radiative tail of the $B^{0}_{s}\rightarrow
K^{+}K^{-}$ decay and the $B^{0}\rightarrow K^{+}\pi^{-}$ cross-feed
background. The relevant event yields for each of the three selections are
summarized in Table 5. Using the values listed in Table 5, we can calculate
the ratios of yields needed to compute the relative branching fractions. These
ratios are given in Table 6, with their statistical uncertainties.
## 5 Systematic uncertainties
Table 7: Systematic uncertainties on the ratios of signal yields. The total
systematic uncertainties are obtained by summing the individual contributions
in quadrature.
Syst. uncertainty | $\frac{N(B^{0}\rightarrow\pi^{+}\pi^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $\frac{N(B^{0}_{s}\rightarrow K^{+}K^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $\frac{N(\Lambda^{0}_{b}\rightarrow p\pi^{-})}{N(\Lambda^{0}_{b}\rightarrow pK^{-})}$ | $\frac{N(B_{s}^{0}\rightarrow\pi^{+}K^{-})}{N(B^{0}\rightarrow K^{+}\pi^{-})}$ | $\frac{N(B^{0}\rightarrow K^{+}K^{-})}{N(B^{0}_{s}\rightarrow K^{+}K^{-})}$ | $\frac{N(B^{0}_{s}\rightarrow\pi^{+}\pi^{-})}{N(B^{0}\rightarrow\pi^{+}\pi^{-})}$
---|---|---|---|---|---|---
PID calibration | $0.0002$ | $0.0012$ | $0.0075$ | $0.0013$ | $0.0005$ | $0.0002$
Final state rad. | $0.0019$ | $0.0043$ | $0.0140$ | $0.0012$ | $0.0093$ | $0.0013$
Signal model | negligible | $0.0001$ | $0.0013$ | $0.0052$ | $0.0010$ | $0.0031$
Comb. bkg model | $0.0013$ | $0.0006$ | $0.0086$ | negligible | $0.0012$ | $0.0004$
$K\pi$ 3-body bkg | $0.0018$ | $0.0048$ | $0.0239$ | $0.0011$ | negligible | negligible
Cross-feed bkg | $0.0023$ | $0.0045$ | $0.0042$ | $0.0008$ | $0.0008$ | $0.0002$
Total | $0.0038$ | $0.0080$ | $0.0304$ | $0.0056$ | $0.0095$ | $0.0034$
The systematic uncertainties on the ratios of signal yields are related to the
PID calibration and to the modelling of the signal and background components
in the maximum likelihood fits. Knowledge of PID efficiencies is necessary to
compute the number of cross-feed background events affecting the fit of any
$H_{b}$ mass spectrum. In order to estimate the impact of imperfect PID
calibration, we perform unbinned maximum likelihood fits after having altered
the number of cross-feed background events present in the relevant mass
spectra according to the systematic uncertainties affecting the PID
efficiencies. An estimate of the uncertainty due to possible imperfections in
the description of the final state radiation is determined by varying, over a
wide range, the amount of emitted radiation [25] in the signal line shape
parameterization. The possibility of an incorrect description of the core
distribution in the signal mass model is investigated by replacing the single
Gaussian with the sum of two Gaussian functions with a common mean. The impact
of additional three-body $B$ decays in the $K^{+}\pi^{-}$ spectrum, not
accounted for in the baseline fit — namely $B\rightarrow\pi\pi\pi$ where one
pion is missed in the reconstruction and another is misidentified as a kaon —
is investigated. The mass line shape of this background component is
determined from Monte Carlo simulation, and the fit is repeated after having
modified the baseline parameterization accordingly. For the modelling of the
combinatorial background component, the fit is repeated using a first-order
polynomial. Finally, for the cross-feed backgrounds, two distinct systematic
uncertainties are estimated: one due to a relative bias in the mass scale of
the simulated distributions with respect to the signal distributions in data,
and another accounting for the difference in mass resolution between
simulation and data. All the shifts from the relevant baseline values are
accounted for as systematic uncertainties. A summary of all systematic
uncertainties on the ratios of event yields is reported in Table 7. The total
uncertainties are obtained by summing the individual contributions in
quadrature. The uncertainties on the ratios of reconstruction and PID
efficiencies, reported in Tables 2 and 4, are also included in the computation
of the total systematic uncertainties on the ratios of branching fractions,
reported in the next section.
## 6 Results and conclusions
The following quantities are determined using Eq. (1) and the values reported
in Tables 2, 4, 6 and 7:
$\displaystyle\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$ $\displaystyle=$ $\displaystyle 0.262\pm 0.009\pm 0.017,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow K^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle 0.316\pm 0.009\pm 0.019,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow
K^{+}\pi^{-}\right)$ $\displaystyle=$ $\displaystyle 0.074\pm 0.006\pm 0.006,$
$\displaystyle(f_{d}/f_{s})\cdot\mathcal{B}\left(B^{0}\rightarrow
K^{+}K^{-}\right)/\,\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle 0.018\,^{+\,0.008}_{-\,0.007}\pm 0.009,$
$\displaystyle(f_{s}/f_{d})\cdot\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}\pi^{-}\right)/\,\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle 0.050\,^{+\,0.011}_{-\,0.009}\pm 0.004,$
$\displaystyle\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow
p\pi^{-}\right)/\,\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow pK^{-}\right)$
$\displaystyle=$ $\displaystyle 0.86\pm 0.08\pm 0.05,$
where the first uncertainties are statistical and the second systematic. Using
the current world average $\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})=(19.4\pm
0.6)\times 10^{-6}$ provided by the Heavy Flavor Averaging Group [24], and our
measurement of the ratio between the $b$-quark hadronization probabilities
$f_{s}/f_{d}=0.267\,^{+\,0.021}_{-\,0.020}$ [26], we obtain the following
branching fractions:
$\displaystyle\mathcal{B}\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle(5.08\pm 0.17\pm 0.37)\times 10^{-6},$
$\displaystyle\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(23.0\pm 0.7\pm 2.3)\times 10^{-6},$
$\displaystyle\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(5.4\pm 0.4\pm 0.6)\times 10^{-6},$
$\displaystyle\mathcal{B}(B^{0}\rightarrow K^{+}K^{-})$ $\displaystyle=$
$\displaystyle(0.11\,^{+\,0.05}_{-\,0.04}\pm 0.06)\times 10^{-6},$
$\displaystyle\mathcal{B}(B^{0}_{s}\rightarrow\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(0.95\,^{+\,0.21}_{-\,0.17}\pm 0.13)\times
10^{-6},$
where the systematic uncertainties include the uncertainties on
$\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})$ and $f_{s}/f_{d}$.
These results are compatible with the current experimental averages [24] and
with available theoretical predictions [27, *Lu:2005mx, *DescotesGenon:2006wc,
*Cheng:2009cn, *Williamson:2006hb, *Ali:2007ff, *Liu:2008rz, *Cheng:2009mu,
*Mohanta:2000nk, *Kaur:2006yr, *Lu:2009cm]. The measurements of
$\mathcal{B}\left(B^{0}_{s}\rightarrow K^{+}K^{-}\right)$,
$\mathcal{B}\left(B^{0}_{s}\rightarrow\pi^{+}K^{-}\right)$,
$\mathcal{B}(B^{0}\rightarrow K^{+}K^{-})$ and
$\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow
p\pi^{-}\right)/\,\mathcal{B}\left(\Lambda^{0}_{b}\rightarrow pK^{-}\right)$
are the most precise to date. Using a likelihood ratio test and including the
systematic uncertainties on the signal yield, we obtain for the
$B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$ signal a significance of 5.3$\sigma$.
This significance is estimated as $s_{\rm
stat}=\sqrt{-2\ln\frac{\mathcal{L}_{\rm B}}{\mathcal{L}_{\rm S+B}}}$, where
$\mathcal{L}_{\rm S+B}$ and $\mathcal{L}_{\rm B}$ are the values of the
likelihoods at the maximum in the two cases of signal-plus-background and
background-only hypotheses, respectively. The value of $s_{\rm
stat}=5.5\sigma$ is then corrected by taking into account the systematic
uncertainty as $s_{\rm tot}=s_{\rm stat}/\sqrt{1+\sigma_{\rm
syst}^{2}/\sigma_{\rm stat}^{2}}$, where $\sigma_{\rm stat}$ and $\sigma_{\rm
syst}$ are the statistical and systematic uncertainties. This is the first
observation at more than $5\sigma$ of the $B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$
decay.
## 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.
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|
arxiv-papers
| 2012-06-13T13:22:06 |
2024-09-04T02:49:31.718839
|
{
"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 S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer,\n Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, K. de Bruyn, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G.A. Cowan, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P.\n De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano,\n D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A.\n Falabella, 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, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, 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, H. Schindler, 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, N.A. Smith, E.\n Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B.\n Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica,\n S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. 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"submitter": "Vincenzo Maria Vagnoni",
"url": "https://arxiv.org/abs/1206.2794"
}
|
1206.2894
|
11institutetext: Max Planck Institute for Dynamics and Self-organization,
Göttingen, Germany
# Oscillation patterns in active emulsion networks
Shashi Thutupalli _Present address:_ Princeton University, Princeton, NJ, USA
Stephan Herminghaus shashi@princeton.edu
(Received: date / Revised version: date)
###### Abstract
We study water-in-oil emulsion droplets, running the Belousov-Zhabotinsky
reaction, that form a new type of active matter unit. These droplets,
stabilised by surfactants dispersed in the oil medium, are capable of internal
chemical oscillations and also self-propulsion due to dynamic interfacial
instabilities that result from the chemical reactions. The chemical
oscillations can couple via the exchange of activator and inhibitor type of
reaction intermediates across the droplets under precise conditions of
surfactant bilayer formation between the droplets. Here we present the
synchronization behaviour of networks of such chemical oscillators and show
that the resulting dynamics depend on the network topology. Further, we
demonstrate that the motion of droplets can be synchronized with the chemical
oscillations inside the droplets, leading to exciting possibilities in future
studies of active matter.
## 1 Introduction
There has been rapidly growing interest recently in so-called active matter,
which refers to open (soft) matter systems exhibiting complex dynamics and
collective behavior reminiscent of living organisms. Spontaneous oscillations
and self-sustained motion ramaswamy_mechanics_2010 ; kruse_oscillations_2005
represent the simplest examples of such complex dynamics and hence are ideally
suited for a theoretical analysis of the interactions and dynamic properties
of a complex active system. Systems such as those undergoing catalytic
reactions at interfaces, the autocatalytic Belousov-Zhabotinsky (BZ) reaction,
social amoeba under stress, populations of fireflies or the cells of heart
muscle tissue can all display spatio-temporal oscillatory behaviors following
similar patterns, which may be modelled by means of reaction-diffusion
dynamics or suitable mean-field approaches kuramoto ; Strogatz2000 . Self-
propelling entities such as motile bacteria, sperm, birds, and fish, or
analogous physical systems such as active emulsions have been found to exhibit
remarkable similarities in their collective behavior as well dos_santos_free-
running_1995 ; sumino_self-running_2005 ; linke_self-propelled_2006 ;
Howse2007 ; thutupalli_njp_2011 . There has been tremendous theoretical and
experimental progress towards the understanding of such systems in terms of
the dynamics of oscillators and motile particles. However, most of the
treatment of these phenomena has been in isolation ramaswamy_mechanics_2010 ;
kuramoto and in most of the studies of the collective behavior of active
matter, the individual unit has typically been abstracted to be either a point
like self propelled particle or a simple phase oscillator kuramoto ;
Strogatz2000 ; thutupalli_njp_2011 ; toner_hydrodynamics_2005 ;
bhattacharya_collective_2010 ; Schaller2010 ; guttal_social_2010 ;
romanczuk_collective_2009 .
While these studies have been very successful in understanding a range of
collective phenomena and synchronization, this is clearly not the complete
picture, as is obvious in many biological processes. For instance, during the
chemotactic self-organization of amoeba and in cellular organization during
embryogenesis, the internal dynamics are entwined with the macroscopic order
that emerges due to cell motility. These internal dynamics of an individual
unit, most often, manifest as biochemical oscillations, linked together with
the motility of the unit. Even in simple physical systems, it has been
observed that there is spontaneous symmetry breaking and emergence of
unexpected complex behavior when the internal degrees of freedom of coupled
non-equilibrium entities are taken into account danTanakaPRL ;
abramsChimeraPRL ; chimeraMetronomes ; erikChimerasChaos . Therefore, it may
be expected that a range of rich collective phenomena in active matter might
open up when the internal dynamics such as oscillations are studied together
with the resultant (or already existing) dynamics such as motility.
Here, we introduce a system that can potentially be a simple table-top
experiment to study the interplay between the internal dynamics of an
individual unit and its motion, and hence the collective behavior of the whole
system. Specifically, we enclose the Belousov-Zhabotinsky reaction in emulsion
droplets of a few tens to hundreds of microns in diameter to form chemical
oscillators. These chemical oscillators can also sustain self-propelled motion
with respect to the surrounding medium due to interfacial instabilities and
Marangoni stresses.
## 2 Experimental Techniques
Our chemical oscillators are made from incorporating the BZ reaction mixture
in aqueous droplets in an external oil phase of squalane contaning mono-olein
as surfactant. The droplets for these experiments were generated using a flow
focussing channel geometry in a PDMS microfluidic chip as shown in Fig. 1
RepProgPhys2012 . In order to prevent any pre-reaction and the formation of
unwanted gaseous bubbles of carbon-di-oxide, the BZ reaction mixture is
separated into two parts and they are combined on chip. The two parts are
created in stock with concentrations as follows: (i) 500 mM sulphuric acid
($\rm H_{2}SO_{4}$) and 280 mM sodium bromate ($\rm NaBrO_{3}$) (ii) 300 - 800
mM malonic acid ($\rm C_{3}H_{4}O_{4}$) and 3 mM ferroin
($C_{36}H_{24}FeN_{6}O_{4}S$). The concentration of the mono-olein in the
squalane ranges between 25 - 100 mM.
Figure 1: Production of monodisperse oscillator droplets. Left: The BZ
reaction consists of two loops (i) an autocatalytic and (ii) an inhibitory
cycle. The reaction state can be visualised by the colour of the ferroin
catalyst. In our setting, an additional reaction with the unsaturated mono-
olein surfactant occurs as shown. Right: The contents of the BZ reaction are
mixed on the microfluidic chip to prevent any pre-reaction. Seen through an
optical 480/20 nm notch filter, the transition from red colour to the blue
colour of the BZ reaction is seen by the change in the brightness of the
droplet.
As can be seen from the left panel of Fig. 1, the BZ reaction consists of an
autocatalytic cycle in which $\rm HBrO_{2}$ catalyses its own production via
the reduction of the ferroin catalyst, which changes its colour rapidly from
red to blue in response. This is when the inhibitory cycle proceeds, leading
to a slow production of bromine which quenches the autocatalysis. The effect
of this is a gradual change of the catalyst back from the blue colour to red.
In our case, an additional side-reaction occurs due to the addition of mono-
olein as a surfactant, which is used to stabilize the droplets against
coalescence. Since the surfactant has an unsaturated hydrocarbon chain as
shown, some of the bromine that is produced in the inhibitory cycle rapidly
reacts with the unsaturated bond. As we will discuss in more detail below,
this ’trapping’ of the bromine by the surfactant significantly affects the
coupling between droplet oscillators in our setup.
Figure 2: Storage of droplet oscillations in one and two dimensional
confinement for observation of their dynamics.
The droplet oscillators are stored as a monolayer in either a one-dimensional
(1d) or two-dimensional (2d) arrangement, as shown in Fig. 2. The 1d array is
created within a glass capillary with a square cross-section of inner width
100 $\rm\mu$m and outer width 135 $\rm\mu$m (Hilgenberg GmbH, Germany). The
inner walls of the capillary are hydrophobised using a coat of commerically
available hydrophobising agent ’Nano-protect’ (W5 Carcare). The 2d array is
created between two similarly hydrophobised glass slides with a PDMS spacer.
The reaction dynamics are recorded by video microscopy on an inverted
microscope (Olympus IX 81) through an appropriate optical filter. As described
earlier, the BZ reaction dynamics can be followed by the colour of the
catalyst as it changes from red to blue and vice versa. The optical filter we
chose is therefore a notch filter of 480/20 nm wavelength such that the red
colour of the catalyst has less transmittance through the filter. Droplets are
identified from the recorded images using Image-Pro Plus (Media Cybernetics)
as shown in Fig. 3 (left panel). The BZ oscillations within the droplet are
then identified by measuring the mean intensity value of the droplet. They are
recorded as traces similar to those of a relaxation oscillator as seen in the
right panel of Fig. 3. The sudden rise of intensity corresponds to the
autocatalytic cycle of the BZ reaction with the catalyst changing from red to
blue colour, and the gradual fall in intensity corresponds to the release of
bromine in the reaction, thus changing the catalyst colour back to red.
Further analysis on the obtained data as described in the results section is
done using MATLAB.
Figure 3: Image processing to identify the droplet oscillators and record
their dynamics.
## 3 Results and Discussion
### 3.1 Isolated BZ oscillators
The BZ oscillators described here are a closed reactor system i.e. there is no
cycling of reactants and final products during the course of the experiment.
Therefore all the reactants and the resultant products remain within the
droplet, except some outflux of promoter and inhibitor into the oil phase. As
a result, the nature of the oscillations gradually changes with time. As soon
as the experiment is started, the oscillation is set by the initial reaction
conditions and the amplitude of the oscillations and the frequency change as
time proceeds. This can be seen in Fig. 4. As time proceeds, the amplitude of
the oscillations reduces significantly, until they die out completely. The
frequency of the oscillation, shown in black, is gradually reduced as well. In
a sense, however, the BZ oscillators provide their own clock, and the
oscillators suggest themselves as a time normal of the experiment if
applicable. In any case, we did not observe any qualitative change of the
behaviour of our system as time proceeded and droplet oscillations slowed
down.
Figure 4: BZ oscillations in a closed reactor. The oscillation trace and its
corresponding frequency (black) are plotted as a function of time.
The BZ oscillators are suspended in an oil phase consisting of squalane, with
mono-olein at concentrations well above the critical micelle concentration
(CMC). The mono-olein serves two purposes. First, it forms dense surfactant
layers at the oil/water interface, which readily form bilayer membranes if
brought into close proximity. Second, the C=C double bond in the mono-olein
molecule acts as an efficient scavenger for bromine, since the latter rapidly
reacts with this site. The oil phase is thus expected to efficiently suppress
coupling between neighboring droplets, which is mediated by the exitatory and
the inhibitory species. That this is indeed the case can be seen in Fig. 5
which shows a two dimensional hexagonal packing of BZ oscillators. The
spherical shape of the droplets in the packing clearly shows that there is oil
between the droplets and that bilayer membranes have not yet formed
thutupalli_SM_2011 . The oscillation trace of a single oscillator is shown in
the lower panel of Fig. 5. We note that in our experiments we do not see a
systematic dependance of the oscillation frequency on the droplet size. When
such isolated droplets are close to each other, in spite of the fact that the
diffusion of the excitatory and inhibitory species can indeed cause coupling
between droplet, they are observed to be uncoupled. This is due to the fact
that, as we discussed before, they might be trapped via reaction with the
surfactant molecules.
Figure 5: Droplet oscillators in a hexagonal packing geometry within a PDMS
microchannel. Top: The different intensities of the droplets show the
different BZ reaction states within each droplet for two different droplet
sizes. Each droplet acts like an isolated individual oscillator without any
coupling with its neighbours. Image contrast is enhanced for better
visualization. Scale bar is 150 microns. Bottom: The intensity trace for a
single droplet is shown as a function of time. The constancy of the frequency
and amplitude of the oscillations can be clearly seen.
However, bilayer membranes form spontaneously between the droplets
thutupalli_SM_2011 , and this happens in the case of the droplet oscillators
too. As soon as bilayers form between droplets, their interfaces touch each
other very closely such that the droplets are not perfectly spherical anymore.
Consequently, the packing fraction increases as seen in Fig. 6, where the gaps
between the droplets due to oil that existed before bilayer formation are now
reduced significantly. Once the bilayers are formed, completely different
oscillatory dynamics are seen. Previously, we demonstrated that oscillator
coupling can be initiated by the formation of a bilayer membrane between the
oscillator droplets thutupalli_SM_2011 . Often, waves of synchronised activity
such as travelling waves are seen as in Fig. 6. Therefore we see a switch from
the individual to collective dynamics of the oscillators when bilayer networks
are formed. We discuss this aspect in greater detail in the next section.
Figure 6: The formation of travelling waves when bilayer membranes are formed
between oscillator droplets. Each image is 5 seconds apart. The droplet
diameter is 30 microns.
### 3.2 Synchronization patterns
Patterns such as pacemaker driven target waves, travelling waves and spirals
are most commonly seen in large assemblies of coupled oscillators. The BZ
droplet oscillators, connected by bilayer membranes, also give rise to a rich
variety of collective dynamics. In the present section, we discuss the various
patterns that emerge in connected networks of oscillators and the dependance
on the network topology of the type of behaviour that emerges. The
discreteness of the droplet oscillators allows us to clearly identify trigger
locations within the networks. All the following experiments are done with a
BZ reaction mixture as described in the experimental section, with a malonic
acid concentration of 500 mM.
First, we discuss the formation of target waves. These are characterised by a
pacemaker core which periodically tiggers excitatory waves that spread from
the core center outward. In our system, we observe that pacemakers
spontaneously emerge in the center of connected droplet ’islands’ or
’peninsulas’. An ’island’ is comprised of connected droplets as shown in the
left panel of Fig. 7 with the outer edges of the ’island’ open to the mono-
olein filled oil phase. A ’peninsula’ is a similar structure and is connected
by a narrow bridge of one or two droplets to a neighbouring ’island’. As we
discussed before, the inhibitory (bromine) and the excitatory ($\rm
BrO^{\cdot}_{2}$) components of the BZ reaction readily diffuse into the
external oil phase, where they are trapped by the surfactant. Therefore, at
the edges of the island, the oscillatory droplets lose their inhibitory and
excitatory components to the external oil phase. However, at the center of the
island, the concentration of the BZ coupling species increases since they come
in from all sides. Depending on the relative concentrations of the inbitory
and excitatory components, the center droplet can therefore either ’turn off’
(i.e. oscillations are inhibited) or trigger an oscillation. For a malonic
acid concentration of 500 mM together with the other concentrations as
described in the experimental section, we find that an oscillation is
triggered in the central droplet as can be seen in the right panel of Fig. 7.
This trigger from the central droplet then propagates outward as a target wave
throughout the ’island’. If it is a ’peninsula’ the connecting bridge can
couple the wave to the neighbouring ’island’ as well. This pattern then
repeats periodically. We find that droplets which are not connected via a
bilayer to the synchronously oscillating cluster are quite likely to not
oscillate at all. This represents an instance of quorum sensing, as reported
before in a similar system Showalter_QS_PRL .
Figure 7: Formation of a target pattern in a ’island’ or ’peninsula’ type of
droplet network. Top: A schematic of a hexagonal arrangement of droplets which
form an ’island’. At the outer edges of the structure, the excitatory and the
inhibitory components are lost to the oil phase (shown by the arrows), while
in the center, they are concentrated leading to the formation of pacemaker
center. Bottom: A target pattern develops within a ’peninsula’ of droplet
oscillators. The excitation and wave pattern are shown in blue for easy
visualization. The scale bar is 500 microns.
Next, we sought if we could induce the pacemaker patterns by confining the
oscillator droplets within a channel made of PDMS. In such a scenario as shown
in Fig. 8, the PDMS walls of the channel, in addition to the oil phase, act as
sinks for the BZ reaction species. Therefore, we expect that along the length
of the channel, multiple pacemaker centers form, each triggering target waves.
That this is indeed the case, can be seen in the right panel of Fig. 8. The
triggering of a wave from a core can be seen in the image sequence shown. In
addition, a wave can be seen coming in from the right of the images, clearly
triggered by a pacemaker upstream in the channel. Indeed there were also waves
coming in the from the left side, but are not shown here.
Figure 8: Formation of target waves with multiple pacemaker centers in a
oscillator network confined in a PDMS microchannel. Top: Schematic of the
oscillators in a PDMS channel. PDMS, in addition to the oil phase, absorbs the
excitatory and inhibitory components of the BZ at the edges (shown by the
arrows). Bottom: Two target wave patterns seen in the channel. One target wave
is forming at a pacemaker center clearly visible. The center of target pattern
coming in from the right is not visible.
Next, for the same concentrations, we looked at a very large network of
hexagonally packed droplet oscillators, such that the edges are too far away
from the cores to have a significant impact. This is shown in Fig. 9. In such
a case, we see the spontaneous emergence of travelling waves across the
network. Indeed, it may be expected that since the BZ concentrations are
rather uniform over the network, a random trigger in one of the oscillators
can set off a cascading wave of activity, which repeats periodically. However,
we were not able in this scenario to find the precise conditions and locations
at which the waves were triggered.
Figure 9: Travelling waves are formed in large densely connected oscillator
networks. Each image is spaced 5 seconds apart. The excitation is coloured
blue for easy visualization.
Spiral waves formed in our experiments, when the hexagonal packing was not
perfect such that not every oscillator is coupled to 6 nearest neighbours.
Yet, the networks were not so sparse as to form ’islands’ or peninsulas’,
where target waves were predominant. An instance of a spiral is seen in Fig.
10. A spiral wave can be clearly seen among the oscillator population. A
closer look into the network reveals that the local network of each oscillator
is not complete according to the 2-dimensional hexagonal packing. The lack of
local connections creates a refractory effect on the excitatory wave due to
the different speeds it travels at, in the different directions. This causes
the wave to turn and eventually forms a spiral or other rotary patterns
depending on the exact topology of the network. Such defects are well known to
provide cores for spiral waves also in other systems DictySpiral1974 ;
LutherHeartSpiral2011
Figure 10: Formation of spirals in oscillator networks. Top: A spiral can be
seen clearly in the oscillator population. Bottom: Image of the network at
higher magnification reavealing the lack of perfect local connectivity for
each droplet. Scale bar is 150 microns.
Finally, we note that the effect of the bilayer membrane is not just to easily
pass the various species of the BZ reaction from one oscillator to another,
such that a discrimination of the ’individual oscillator’ is simply lost. In
such a case, the behaviour of the oscillator network can be considered to be
the same as that of the BZ reaction in the bulk, in a 2d homogeneous planar
system. However, more complex relationships between neighbouring oscillators
connected by a bilayer are seen. As we mentioned before, both the excitatory
and inhibitory components of the BZ reaction can traverse the bilayer. The
formation of the target patterns as described before indicated that the BZ
mixture used in our experiments is excitatory i.e. the excitatory coupling
wins over the inhibitory coupling in determining the state of the coupled
oscillators, leading to wave like patterns as we have seen so far. However, it
has been reported in literature Toiya2008 ; Toiya2010 that due to inhibitory
coupling between BZ oscillators, it is possible to generate patterns that
strongly differ from wave-like patterns. We increased the concentration of
malonic acid to 700 mM compared to the 500 mM used for the previous
experiments. It is expected that increasing the concentration of the malonic
acid results in a greater production of the inhibitor, bromine, as shown in
the BZ reaction schematic in Fig. 1, thus possibly leading to an inhibitory
coupling effect. When the coupling is inhibitory i.e. non-excitatory, we
expect that wave like patterns will not result. As anticipated, the increase
in the concentration of Malonic acid resulted in a non-wave-like pattern as
shown in Fig. 11 where every oscillator droplet is found to be in strict anti-
phase with its neighbour. This can be understood to be a complicated interplay
between the inhibitory and excitatory coupling. In fact it has been shown that
the anti-phase state is an attractor for inhibitory coupling Toiya2008 ;
Toiya2010 . However, in such models of inhibitory coupling, the interdroplet
distance is quite large as compared to our experiments where the droplets are
separated only by a nanometric membrane. This illustrates that the membrane
between the oscillator droplets plays an important role in preserving the
individual properties of each oscillator. These studies, though only
demonstrative in the present work, must be performed in greater detail in
order to quantify the various synchronization patterns and their relation to
the network topology. In particular, a knowledge of the permeability of the
membrane to the various coupling intermediates is crucial to have predictive
control over the oscillator behaviour.
Figure 11: Antiphase pattern in a 1 dimensional oscillator network. Top:
Droplet oscillators in a glass microcapillary. Each droplet pair is connected
by bilayer membranes. The droplet diameter is 100 microns Bottom: Time trace
of the droplet oscillations shown for the three droplets in the center of the
top image. The red, blue and green traces correspond to the droplets marked as
shown.
## 4 Summary and Outlook
Active emulsions, consisting of chemical micro-oscillator droplets as
presented here may provide a crucial first step towards the realization of
active soft matter with complex dynamic functions. The Belousov-Zhabotinsky
reaction used here has been studied as a paradigm system for dynamical and
pattern forming systems for many years. In the present setting of using it
within microfluidic emulsion droplets, qualitatively new phenomena emerge due
to the interplay between the droplet network topologies and the type of
coupling between the oscillators. As we have shown, the bilayer membranes,
which form spontaneouly between adjacent droplets, play a crucial role in the
coupling and synchronization dynamics. In combination with our previous
results thutupalli_SM_2011 , this opens up the possibility to construct self-
organizing dynamic soft matter systems.
Figure 12: A self propelled droplet with oscillating BZ chemical reaction
taking place in the droplet. (a) Time snaps of a chemical wave within the
droplet (diameter $\rm\sim 600microns$). The droplet moves in the direction of
the wave propagation. (b) The speed of the above droplet shows roughly
periodic oscillations corresponding to the BZ waves inside the droplet. (c)
When the droplet size is reduced to $\rm\sim 80microns$ in diameter, the waves
inside the droplets are supressed. In this case, the oscillations in the
droplet speed (black trace) are perfectly synchronized with the optical
transmission of the droplet, plotted in red.
Also, we have shown previously thutupalli_njp_2011 that the BZ reaction
intermediates react with the surfactant in the oil phase and also at the
droplet interface creating artificial self propelled droplets. In Fig. 12, we
demonstrate that the internal oscillations of the BZ reaction affect the speed
of a self propelled droplet, similar to numerical predictions before
Kitahata2011 . Consequently, we can expect that the collective motion of
droplets thutupalli_njp_2011 will be strongly affected by their oscillatory
state and mutual coupling. On the other hand, as we have seen, their local
density affects the formation of bilayer membranes, and therefore acts back on
the mutual coupling of individual droplet oscillators. This provides an
exciting link to the rapidly evolving field of developmental evolution, which
considers the possible back-action mechanisms of the emerging phenotype (i.e.,
the collective arrangement of cells) onto the genotype, i.e., the microscopic
(genetic) state of the cell evoDevo2002 . It is an interesting option to use
emulsions containing chemical oscillator droplets such as BZ as a model for
systems with such mutual interaction of different levels of integration.
When mechanisms such as chemotaxis can be engineered into such systems, we
envisage that many exciting possibilities might be opened in the field of
active matter. For example, our results may provide a useful step to
addressing some of the challenges in the design of artificial self organizing
assemblies capable of achieving complex tasks Kolmakov2010 . Finally, we may
develop well controlled experiments to understand how the internal degrees of
freedom of collections of similar nonequilibrium units couple with their self-
emergent mesoscopic order danTanakaPRL ; thutupalliThesis .
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|
arxiv-papers
| 2012-06-13T18:49:56 |
2024-09-04T02:49:31.730899
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shashi Thutupalli and Stephan Herminghaus",
"submitter": "Shashi Thutupalli",
"url": "https://arxiv.org/abs/1206.2894"
}
|
1206.2897
|
# Conjugates, Correlation and Quantum Mechanics
Alexander Wilce Department of Mathematics, Susquehanna University,
Selinsgrove, PA
###### Abstract
The Jordan structure of finite-dimensional quantum theory is derived, in a
conspicuously easy way, from a few very simple postulates concerning abstract
probabilistic models (defined by a set of basic measurements and a convex set
of states). The key assumption is that each system $A$ can be paired with an
isomorphic conjugate system, $\overline{A}$, by means of a non-signaling
bipartite state $\eta_{A}$ perfectly and uniformly correlating each basic
measurement on $A$ with its counterpart on $\overline{A}$. In the case of a
quantum-mechanical system associated with a complex Hilbert space
$\boldsymbol{\mathcal{H}}$, the conjugate system is that associated with the
conjugate Hilbert space $\overline{\boldsymbol{\mathcal{H}}}$, and $\eta_{A}$
corresponds to the maximally entangled state on
$\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$.
## I Introduction and Background
This paper derives the Jordan structure of finite-dimensional quantum theory
from a very lean set of postulates, and in a conspicuously easy way — easy, at
any rate, if one knows the Koecher-Vinberg Theorem, which relates euclidean
Jordan algebras to homogeneous, self-dual cones 111See section 2 for a brief
explanation of these terms, and FK for an accessible proof of the Koecher-
Vinberg Theorem. Given the use of such a powerful off-the-shelf mathematical
tool, I am perhaps “using a canon to shoot a canary”. But this is the sport of
kings; and anyway, it’s a big canary.. This brings us within hailing distance
of orthodox finite-dimensional quantum theory: every euclidean Jordan algebra
is a direct sum of self-adjoint parts of real, complex or quaternionic matrix
algebras, spin factors, and the exceptional jordan algebra of self-adjoint
$3\times 3$ matrices over the octonions FK .
In fact, I present two slightly different results, both resting on the idea of
a conjugate system. Here is a sketch. As is well known, any mixed state on a
quantum-mechanical system is the marginal of a pure bipartite state on a
composite of two copies of that system. This latter state perfectly correlates
some pair of basic observables on these two copies. Using not two copies of
the same system but rather, a system and its conjugate system (associated with
the conjugate of the first system’s Hilbert space), the maximally mixed state
arises as the marginal of a bipartite state — essentially, the maximally
entangled state — perfectly correlating every observable with its conjugate.
These correlational features can be abstracted. A finite-dimensional
probabilistic model $A$ is characterized by a set of basic measurements and a
finite-dimensional convex set of states. From this data, one can construct, in
a canonical way, a pair of ordered real vector space ${\mathbf{V}}(A)$ and
${\mathbf{E}}(A)\leq{\mathbf{V}}(A)^{\ast}$, the former generated by $A$’s
states, and the latter by its basic measurement outcomes. Suppose that all
basic measurements of $A$ have a common number $n$ of outcomes. Define a
conjugate for $A$ to be a model $\overline{A}$, plus a fixed isomorphism
$\gamma_{A}:A\simeq\overline{A}$ and a fixed bipartite, non-signaling state
$\eta_{A}$ between $A$ and $\overline{A}$, such that, for every basic
measurement outcome $x$ of $A$, $\eta_{A}(x,\gamma_{A}(x))=1/n$. Assume now
that (i) $A$ has a conjugate system, and, further, that (ii) every state of
$A$ dilates to a non-signaling bipartite state between $A$ and $\overline{A}$
that perfectly correlates some pair of basic measurements. Further, suppose
(iii) every basic measurement outcome has probability one in a unique state,
and (iv) every non-singular state (every state strictly positive on basic
measurement outcomes) can be prepared, up to normalization, from the maximally
mixed state by means of a reversible process.
Theorem 1 Subject to conditions (i)-(iii), ${\mathbf{E}}(A)$ is self-dual, and
isomorphic to ${\mathbf{V}}(A)$. Subject to condition (iv), ${\mathbf{V}}(A)$
is homogeneous. Thus, subject to conditions (i)-(iv), ${\mathbf{E}}(A)$ is
homogeneous and self-dual, and hence, by the Koecher-Vinberg Theorem, has the
structure of a euclidean Jordan algebra.
In the presence of a conjugate, a slightly stronger preparability hypothesis
than (iv) yields both the homogeneity and the self-duality of
${\mathbf{E}}(A)_{+}$, without the need for assumptions (ii) and (iii) above.
A filter is a process $\Phi$ (that is, a positive mapping
${\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$) that independently attenuates the
reliability of each outcome of some basic measurement, in the sense that for
each outcome $x$, there is a constant $t_{x}\in(0,1]$ with
$\Phi(\alpha)(x)=t_{x}\alpha(x)$ for all states $\alpha$. Suppose $A$ has a
conjugate, $\overline{A}$: starting in the canonical state $\eta_{A}$, we can
apply $\Phi$ to the first component of the composite system $A\overline{A}$ to
obtain a new bipartite state $(\Phi\otimes 1_{\overline{A}})(\eta_{A})$ (where
$1_{\overline{A}}$ is the identity transformation on $\overline{A}$’s state
space). We can also apply the counterpart of $\overline{\Phi}$ to
$\overline{A}$, obtaining $(1\otimes\overline{\Phi})(\eta_{A})$. Call $\Phi$
symmetric iff these states are the same.
Theorem 2: Let $A$ have a conjugate system $\overline{A}$. If every interior
state of $A$ arises from the maximally mixed state by a symmetric reversible
filter, then $A$ satisfies (ii) and (iii); hence, ${\mathbf{E}}(A)_{+}$ is
homogeneous and self-dual.
The proofs of both of these results are quite short and straightforward.
Several of the ideas in this paper were earlier explored, and somewhat similar
results derived, in 4.5 and SSD . However, the approach taken here is much
simpler and more direct.
### I.1 General Probabilistic Theories
The general framework for this paper is that of BW12 ; 4.5 , which I’ll now
quickly review. In a few places (set off in numbered definitions), my usage
differs slightly from that of the cited works. See Alfsen-Shultz ; FK for
further information on ordered vector spaces and on Jordan algebras.
Probabilistic Models A probabilistic model is characterized by a set
${\mathcal{M}}(A)$ of basic measurements or tests, and a set $\Omega(A)$ of
states. Identifying each test with its outcome-sest, ${\mathcal{M}}(A)$ is
simply a collection of non-empty sets (a test space, in the language of BW12
). I’ll write $X(A)$ for the union of this collection, i.e., the space of all
outcomes of all basic measurements. I understand a state to be an assignment
of probabilities to measurement-outcomes, that is, a function
$\alpha:X(A)\rightarrow[0,1]$ such that $\sum_{x\in E}\alpha(x)=1$ for all
tests $E\in{\mathcal{M}}(A)$. To reflect the possibility of forming
statistical mixtures, I also assume that $\Omega(A)$ is convex. 222It is
usually also assumed that $\Omega(A)$ is closed (hence, compact) in the
product topology on ${\mathbb{R}}^{X}$. This assumption isn’t needed here,
however.
By the dimension of a model $A$, I mean the affine dimension of $\Omega(A)$.
In the simplest finite-dimensional classical model, ${\mathcal{M}}(A)$
consists of a single, finite test, and $\Omega(A)$ is the simplex of all
probability weights on that test. Of more immediate interest to us is the
quantum model
$A(\boldsymbol{\mathcal{H}})=({\mathcal{M}}(\boldsymbol{\mathcal{H}}),\Omega(\boldsymbol{\mathcal{H}}))$
associated with a complex Hilbert space $\boldsymbol{\mathcal{H}}$. The test
space ${\mathcal{M}}(\boldsymbol{\mathcal{H}})$ is the set of orthonormal
bases of $\boldsymbol{\mathcal{H}}$; thus, the outcome-space
$X(\boldsymbol{\mathcal{H}})$ is the set of unit vectors of
$\boldsymbol{\mathcal{H}}$. The state space $\Omega(\boldsymbol{\mathcal{H}})$
consists of the quadratic forms associated with density operators on
$\boldsymbol{\mathcal{H}}$, so that a state
$\alpha\in\Omega(\boldsymbol{\mathcal{H}})$ has the form $\alpha(x)=\langle
W_{\alpha}x,x\rangle$ for some density operator $W_{\alpha}$, and all unit
vectors $x\in X(\boldsymbol{\mathcal{H}})$. Real and quaternionic quantum
models, corresponding to real or quaternionic Hilbert spaces, are defined in
the same way.
Remark: Not every physically accessible observable on a finite-dimensional
quantum system is represented by an orthonormal basis. Rather, the general
observable corresponds to a positive-operator-valued measurement. Similarly,
for an arbitrary probabilistic model $A$, the test space ${\mathcal{M}}(A)$
may, but need not, represent a complete catalogue of all possible measurements
one might make on the system represented by $A$: rather, it is some convenient
catalogue of such measurements, sufficiently large to determine the system’s
states.
The spaces ${\mathbf{E}}(A)$ and ${\mathbf{V}}(A)$ An ordered vector space is
a real vector space ${\mathbf{E}}$ equipped with a convex cone
${\mathbf{E}}_{+}$ with ${\mathbf{E}}_{+}\cap-{\mathbf{E}}_{+}=\\{0\\}$ and
${\mathbf{E}}={\mathbf{E}}_{+}-{\mathbf{E}}_{+}$ — that is, ${\mathbf{E}}$ is
spanned by ${\mathbf{E}}_{+}$. The cone induces a partial order, invariant
under translation and multiplication by non-negative scalars, given by $a\leq
b$ iff $b-a\in{\mathbf{E}}_{+}$. An example is the space
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$ of hermitian operators on a
real, complex or quaternionic Hilbert space, ordered by the cone of positive
semi-definite operators. A linear mapping
$T:{\mathbf{E}}\rightarrow{\mathbf{F}}$ between ordered vector spaces
${\mathbf{E}}$ and ${\mathbf{F}}$ is positive iff
$T({\mathbf{E}}_{+})\subseteq{\mathbf{F}}_{+}$. The dual cone,
${\mathbf{E}}^{\ast}_{+}\subseteq{\mathbf{E}}^{\ast}$, consists of positive
linear functionals $f\in{\mathbf{E}}^{\ast}$.
Any probabilistic model $A$ gives rise in a canonical way to a pair ordered
vector spaces ${\mathbf{E}}(A)$ and ${\mathbf{V}}(A)$. The latter is simply
the span of the state space $\Omega(A)$ in the space ${\mathbb{R}}^{X(A)}$,
ordered by the cone ${\mathbf{V}}(A)_{+}$ of non-negative multiples of states.
Every outcome $x\in X(A)$ corresponds to a linear evaluation functional
$\widehat{x}:{\mathbf{V}}(A)\rightarrow{\mathbb{R}}$, given by
$\widehat{x}(\alpha)=\alpha(x)$ for all $\alpha\in{\mathbf{V}}(A)$. The space
${\mathbf{E}}(A)$ is the span of these functionals in
${\mathbf{V}}(A)^{\ast}$, ordered by the cone ${\mathbf{E}}(A)_{+}$ consisting
of all finite linear combinations $\sum_{i}t_{i}\widehat{x}_{i}$ having non-
negative coefficients $t_{i}$. Note that $a\in{\mathbf{E}}(A)_{+}$ implies
$a(\alpha)\geq 0$ for all $\alpha\in{\mathbf{V}}(A)_{+}$. The converse is
generally false. Note, too, that there is a distinguished vector
$u_{A}\in{\mathbf{E}}(A)_{+}$ given by $u_{A}=\sum_{x\in E}\widehat{x}$; this
is independent of the choice of $E\in{\mathcal{M}}(A)$. Any state
$\alpha\in\Omega(A)$ satisfies $u_{A}(\alpha)=1$; again, the converse is
generally false.333A model is said to be state-complete iff every
$\alpha\in{\mathbf{V}}(A)_{+}$ with $u_{A}(\alpha)=1$ belongs to $\Omega(A)$.
State-completeness is frequently, though often tacitly, assumed in much of the
recent literature on generalized probabilistic theories, including many of my
earlier papers, but is not assumed here.
Processes A physical process on a system $A$ is naturally represented by an
affine (convex-linear) mapping $T:\Omega(A)\rightarrow{\mathbf{V}}(A)$ such
that, for every $\alpha\in\Omega(A)$, $T(\alpha)=p\beta$ for some
$\beta\in\Omega(A)$ and some constant $0\leq p\leq 1$ (depending on $\alpha$),
which we can regard as the probability that the process occurs, given that the
initial state is $\alpha$. Such a mapping extends uniquely to a positive
linear mapping $T:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$ with
$T(\alpha)(u_{A})\leq 1$ for all $\alpha\in\Omega(A)$. I will say that a
process $T:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(B)$ prepares a state
$\alpha$ of $A$ from another state, $\beta$, if $\alpha$ is a multiple of
$T(\beta)$, i.e., $T(\beta)$ coincides with $\alpha$ up to normalization.
A process $T:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(B)$ has a dual action on
$V(A)^{\ast}$, given by $T^{\ast}(f)=f\circ T$ for all
$f\in{\mathbf{V}}(A)^{\ast}$, with $T{\ast}(u)\leq u$. In our finite-
dimensional setting, we can identify ${\mathbf{V}}(A)^{\ast}$ with
${\mathbf{E}}(A)$ as vector spaces, but not, generally, as ordered vector
spaces. While $\Phi^{\ast}$ will preserve the dual cone
${\mathbf{V}}(A)^{\ast}_{+}$, it is not required that $T^{\ast}$ preserve the
cone ${\mathbf{E}}(A)\leq{\mathbf{V}}(A)^{\ast}$. This reflects the idea that
not every physically accessible measurement need appear among the tests in
${\mathcal{M}}(A)$, as discussed above.
A process $T:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$ reversible iff there
is another process, $S$, such that, for every state $\alpha$, there exists a
constant $p\in(0,1]$ with $S(T(\alpha))=p\alpha$. In other words, $S$ allows
us to recover $\alpha$ from $T(\alpha)$, up to normalization. It is not hard
to see that $p$ must be independent of $\alpha$, so that $S=pT^{-1}$. In
particular, $T$ is an order-isomorphism of ${\mathbf{V}}(A)$.
Self-Duality and Jordan Algebras For both classical and quantum models, the
ordered spaces ${\mathbf{E}}(A)$ and ${\mathbf{V}}(A)$ are isomorphic. In the
former case, where ${\mathcal{M}}(A)$ consists of a single test $E$ and
$\Omega(A)$ is the simplex of all probability weights on $E$, we have
${\mathbf{V}}(A)\simeq{\mathbb{R}}^{E}$ and
${\mathbf{E}}(A)\simeq({\mathbb{R}}^{E})^{\ast}$. The standard inner product
on ${\mathbb{R}}^{E}$ provides an order-isomorphism between these spaces, that
is, a linear bijection taking the positive cone of one space onto that of the
other. If $\boldsymbol{\mathcal{H}}$ is a finite-dimensional real or complex
Hilbert space, we have an affine isomorphism between the state space of
$\Omega(\boldsymbol{\mathcal{H}})$ and the set of density operators on
$\boldsymbol{\mathcal{H}}$, allowing us to identify
${\mathbf{V}}(A(\boldsymbol{\mathcal{H}}))$ with
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$. For any $x\in
X(\boldsymbol{\mathcal{H}})$, the evaluation functional
$\widehat{x}\in{\mathbf{V}}(A)$ is then given by $W\mapsto\langle
Wx,x\rangle=\mbox{Tr}(WP_{x})$. It follows that
${\mathbf{E}}(A(\boldsymbol{\mathcal{H}}))\simeq{\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})^{\ast}\simeq{\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$,
with the latter isomorphism implemented by the tracial inner product $\langle
a,b\rangle=\mbox{Tr}(ab)$.
More generally, any inner product $\langle,\rangle$ on an ordered vector space
${\mathbf{E}}$ defines a positive linear mapping
${\mathbf{E}}\rightarrow{\mathbf{E}}^{\ast}$. If this is an order-isomorphism,
one says that ${\mathbf{E}}$ is self-dual with respect to this inner product.
This is equivalent to the condition $a\in{\mathbf{E}}_{+}$ iff $\langle
a,b\rangle\geq 0$ for all $b\in{\mathbf{E}}_{+}$. In this language, the
standard inner product on ${\mathbb{R}}^{E}$ and the tracial inner product on
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$ are self-dualizing, where $E$ is
a finite set and $\boldsymbol{\mathcal{H}}$, a finite-dimensional Hilbert
space.
In fact, any euclidean Jordan algebra, ordered by its cone of squares, is
self-dual with respect to its canonical tracial inner product. Another
property shared by all euclidean Jordan algebras is homogeneity: the group of
order-automorphisms of ${\mathbf{E}}$ acts transitively on the interior of the
positive cone ${\mathbf{E}}_{+}$. The Koecher-Vinberg Theorem FK states that,
conversely, any self-dual, homogenous order-unit space ${\mathbf{E}}$ can be
equipped with the structure of a euclidean Jordan algebra. This structure is
unique, up to the choice of an element of the interior of the cone
${\mathbf{E}}_{+}$ to serve as a unit for the Jordan algebra.
Definition 1: Let us say that a model $A$ is self-dual iff ${\mathbf{E}}(A)$
is self-dual and ${\mathbf{E}}(A)_{+}\simeq{\mathbf{V}}(A)_{+}$. Call $A$
homogeneous iff ${\mathbf{V}}(A)$ is homogeneous.
If the model $A$ is both homogeneous and self-dual, then ${\mathbf{E}}(A)$ is
homogeneous and self-dual, and hence, ${\mathbf{E}}(A)_{+}$ is the cone of
squares of a euclidean Jordan algebra. Of these two properties, homogeneity is
the easier to interpret in physical terms: it amounts to assumption (iv) in
the introduction, namely, that every state in the interior of
${\mathbf{V}}(A)_{+}$ can be obtained, up to normalization, from some
particular such state by means of a reversible process. Self-duality is less
easily motivated, but will emerge from the other assumptions sketched in the
introduction.
Composite Systems A composite of two probabilistic models $A$ and $B$ is a
model $AB$, together with a mapping $X(A)\times X(B)\rightarrow X(AB)$
allowing us to interpret a pair $(x,y)$ of outcomes belonging to the two
systems separately as a single product outcome $xy\in X(AB)$, in such a way
that, for any tests $E\in{\mathcal{M}}(A)$ and $F\in{\mathcal{M}}(B)$, the set
$EF=\\{xy|x\in E,y\in F\\}$ is a test in $AB$. It follows that any state
$\omega\in\Omega(AB)$ pulls back to a function — which I’ll also denote by
$\omega$ — on $X(A)\times X(B)$, given by $\omega(x,y)=\omega(xy)$, satisfying
$\sum_{x\in E,y\in F}\omega(xy)=1$ for all tests $E\in{\mathcal{M}}(A)$,
$F\in{\mathcal{M}}(B)$. One understands $\omega(x,y)$ as the joint probability
of the outcomes $x$ and $y$ in the bipartite state $\omega$.
Remark: This definition is weaker than that used in, e.g., BW12 , where it is
also assumed that, for any pair of states $\alpha\in\Omega(A)$ and
$\beta\in\Omega(B)$, there exists a unique state $\alpha\otimes\beta$ on $AB$
such that $(\alpha\otimes\beta)(x,y)=\alpha(x)\beta(y)$ for all outcomes $x\in
X(A)$, $y\in X(B)$. This assumption is not needed in what follows.
Non-Signaling Composites A state $\omega$ on a composite $AB$ is non-signaling
iff the marginal states $\omega_{1}(x)=\sum_{y\in F}$ and
$\omega_{2}(y)=\sum_{x\in E}\omega(xy)$ are well-defined, i.e., independent of
the choice of tests $E$ and $F$. One can then also define conditional states
$\omega_{2|x}(y):=\omega(x,y)/\omega_{1}(x)$ (with, say, $\omega_{2|x}$
identically zero if $\omega_{1}(x)=0$), and similarly for $\omega_{1|y}$. This
gives us the following bipartite version of the law of total probability FR :
$\omega_{2}=\sum_{x\in E}\omega_{1}(x)\omega_{2|x}\ \ \mbox{and}\ \
\omega_{1}=\sum_{y\in F}\omega_{2}(y)\omega_{1|y}$ (1)
for any choice of $E\in{\mathcal{M}}(A)$ or $F\in{\mathcal{M}}(B)$.
Classical and quantum bipartite states are clearly non-signaling.
Definition 2: A non-signaling composite of $A$ and $B$, I mean a composite
$AB$ such that every state $\omega$ of $AB$ is non-signaling, and the
conditional states $\omega_{1|y}$ and $\omega_{2|x}$ are valid states of $A$
and $B$, respectively, for all outcomes $y\in X(A)$ and $x\in X(A)$.
As an example, if $\boldsymbol{\mathcal{H}}$ and $\boldsymbol{\mathcal{K}}$
are real or complex Hilbert spaces, there is a natural mapping
$X(\boldsymbol{\mathcal{H}})\times X(\boldsymbol{\mathcal{K}})\rightarrow
X(\boldsymbol{\mathcal{H}}\otimes\boldsymbol{\mathcal{K}})$, namely
$(x,y)\mapsto x\otimes y$. It is straightforward to check that this makes
$A(\boldsymbol{\mathcal{H}}\otimes\boldsymbol{\mathcal{K}})$ a non-signaling
composite of $A(\boldsymbol{\mathcal{H}})$ and $A(\boldsymbol{\mathcal{K}})$,
in the above sense.
It follows from (1) that if $\omega\in\Omega(AB)$ is non-signaling, the
marginal states $\omega_{1}$ and $\omega_{2}$ also belong to $\Omega(A)$ and
$\Omega(B)$, respectively. It is not hard to show that a state $\omega$ on a
non-signaling composite $AB$ gives rise to a bilinear form
$\omega:{\mathbf{E}}(A)\times{\mathbf{E}}(B)\rightarrow{\mathbb{R}}$, uniquely
defined by $\omega(\widehat{x},\widehat{y})=\omega(xy)$ for all outcomes $x\in
X(A),y\in X(B)$, and hence also to a positive linear conditioning map
$\widehat{\omega}:{\mathbf{E}}(A)\rightarrow{\mathbf{V}}(B)$
given by $\widehat{\omega}(a)(b)=\omega(a,b)$. Note that for any $x\in X(A)$,
$\widehat{\omega}(x)=\omega_{1}(x)\omega_{2|x}$, whence the terminology. If
$\widehat{\omega}$ is an order-isomorphism, then $\omega$ is called an
isomorphism state.
Probabilistic Theories As a rule, one wants to think of physical theories, not
as loosely structured classes, but rather, as categories of systems Abramsky-
Coecke ; BW11 . In what follows, a probabistic theory is understood to be a
category, of probabilistic models, with morphisms corresponding to processes —
that is, if $A$ and $B$ are models belonging to the theory, then morphism
$A\rightarrow B$ are positive linear mappings
$T:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(B)$, subject to the condition that
$T(\alpha)(u_{B})\leq 1$ for all $\alpha\in\Omega(A)$, as discussed above. A
monoidal probabilistic theory is one closed under some definite operation
$A,B\mapsto AB$ of forming non-signaling composites, and subject also to the
further condition that, for all models $A_{1},A_{2},B_{1}$ and
$B_{2}\in{\mathcal{C}}$ and all processes
$T_{1}:{\mathbf{V}}(A_{1})\rightarrow{\mathbf{V}}(B_{1})$ and
$T_{2}:{\mathbf{V}}(A_{2})\rightarrow{\mathbf{V}}(B_{2})$, there is process
$T_{1}\otimes
T_{2}:{\mathbf{V}}(A_{1}A_{2})\rightarrow{\mathbf{V}}(B_{1}B_{2})$ such that
$(T_{1}\otimes T_{2})(\omega)(x,y)=\omega(T_{1}^{\ast}x,T_{2}^{\ast}y)$ (2)
for all $x\in X(A_{1})$ and $y\in X(A_{2})$.
The category of quantum models and completely positive mappings is a monoidal
probabilistic theory in this sense, as is the category of real quantum models
and completely positive mappings. It will simplify the discussion to assume,
in the balance of this paper, that we are working in some monoidal non-
signaling probabilistic theory, so that we can take advantage of (2).
## II Correlational properties of quantum composites
In this section, $\boldsymbol{\mathcal{H}}$, $\boldsymbol{\mathcal{K}}$ are
finite-dimensional real or complex Hilbert spaces (with inner products
conjugate-linear in the second argument, in the complex case). The space of
linear operators on $\boldsymbol{\mathcal{H}}$ is
${\mathcal{L}}(\boldsymbol{\mathcal{H}})$; the space of Hermitian operators,
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$. As discussed above, if
$A(\boldsymbol{\mathcal{H}})$ is the corresponding quantum probabilistic model
(with $X(\boldsymbol{\mathcal{H}})$ the set of unit vectors of
$\boldsymbol{\mathcal{H}}$, ${\mathcal{M}}(\boldsymbol{\mathcal{H}})$, the set
of orthonormal bases of $\boldsymbol{\mathcal{H}}$, and
$\Omega(\boldsymbol{\mathcal{H}})$, the set of states associated with density
operators on $\boldsymbol{\mathcal{H}}$), then
${\mathbf{E}}(\boldsymbol{\mathcal{H}}):={\mathbf{E}}(A(\boldsymbol{\mathcal{H}}))$
is canonically isomorphic to ${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$.
Let $\overline{\boldsymbol{\mathcal{H}}}$ denote $\boldsymbol{\mathcal{H}}$’s
conjugate space (with
$\overline{\boldsymbol{\mathcal{H}}}=\boldsymbol{\mathcal{H}}$ if
$\boldsymbol{\mathcal{H}}$ is real). If $x\in\boldsymbol{\mathcal{H}}$, write
$\overline{x}$ for the same vector in $\overline{\boldsymbol{\mathcal{H}}}$,
so that, for any scalar $c$, $c\overline{x}=\overline{\overline{c}x}$, and
$\overline{cx}=\overline{c}\,\overline{x}$. Note that inner product on
$\overline{H}$ is given by
$\langle\overline{x},\overline{y}\rangle=\overline{\langle x,y\rangle}=\langle
y,x\rangle$. If $T\in{\mathcal{L}}(\boldsymbol{\mathcal{H}})$, write
$\overline{T}$ for the operator $\overline{T}\overline{x}=\overline{Tx}$.
While $T\mapsto\overline{T}$ is conjugate linear as a mapping from
${\mathcal{L}}(\boldsymbol{\mathcal{H}})$ to
${\mathcal{L}}(\boldsymbol{\mathcal{H}})$, it is linear as a mapping between
the real vector spaces
${\mathbf{E}}(\boldsymbol{\mathcal{H}})={\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$
and
${\mathbf{E}}(\overline{\boldsymbol{\mathcal{H}}})={\mathcal{L}}_{h}(\overline{\boldsymbol{\mathcal{H}}})$.
Indeed, $T\mapsto\overline{T}$ defines an order-isomorphism between these
spaces.
For any vectors $x,y\in\boldsymbol{\mathcal{H}}$, let $x\odot y$ denote the
rank-one operator on $\boldsymbol{\mathcal{H}}$ given by $(x\odot y)z=\langle
z,y\rangle x$. (In Dirac notation, this is $|x\rangle\langle y|$.) If $x$ is a
unit vector, then $x\odot x=P_{x}$, the orthogonal projection operator
associated with $x$. The mapping $x,y\mapsto x\odot y$ is sesquilinear, that
is, linear in its first, and conjugate linear in its second, argument; it
therefore extends uniquely to a linear mapping
$\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}\rightarrow{\mathcal{L}}(\boldsymbol{\mathcal{H}})$.
It is easy to check that this is inective, and hence, on dimensional grounds,
an isomorphism.
Suppose now that $\alpha$ is a state on $A(\boldsymbol{\mathcal{H}})$,
represented by a density operator $W$ on $\boldsymbol{\mathcal{H}}$, so that
$\alpha(x)=\langle Wx,x\rangle$ for all unit vectors $x\in
X(\boldsymbol{\mathcal{H}})$. Let $W$ have spectral resolution
$W=\sum_{x\in E}\lambda_{x}P_{x}=\sum_{x\in E}\lambda_{x}x\odot x$
where $E$ is an orthonormal basis for $\boldsymbol{\mathcal{H}}$ (so that
$\lambda_{x}=\alpha(x)$ for all $x\in E$). Functional calculus gives us
$W^{1/2}=\sum_{x\in E}\lambda^{1/2}_{x}x\odot x$. Using the isomorphism
${\mathcal{L}}(\boldsymbol{\mathcal{H}})\simeq\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$,
we can interpret $W^{1/2}$ as a vector
$v_{W}\in\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$,
namely
$v_{W}:=\sum_{x\in E}\lambda^{1/2}_{x}x\otimes\overline{x}.$ (3)
This is a unit vector, and so, in turn, defines a pure bipartite state
$\omega$ on the composite quantum system
$A\overline{A}:=A(\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}})$,
with $\omega(y,\overline{z})=|\langle y\otimes\overline{z},v_{W}\rangle|^{2}$.
for all $y,z\in X(\boldsymbol{\mathcal{H}})$. The marginal, or reduced, state
$\omega_{1}$ on first component system assigns to an effect $a$ a probability
$\displaystyle\ \omega_{1}(a)$ $\displaystyle=$ $\displaystyle\langle(a\otimes
1_{\overline{\boldsymbol{\mathcal{H}}}})v_{W},v_{W}\rangle$ $\displaystyle=$
$\displaystyle\sum_{x\in E}\sum_{y\in
E}\lambda_{x}^{1/2}\lambda_{y}^{1/2}\langle
ax,y\rangle\langle\overline{x},\overline{y}\rangle$ $\displaystyle=$
$\displaystyle\sum_{x\in E}\lambda_{x}\langle ax,x\rangle$ $\displaystyle=$
$\displaystyle\sum_{x}\lambda_{x}\mbox{Tr}(P_{x}a)=\mbox{Tr}(Wa).$
In other words, $\omega_{1}=\alpha$. A similar computation gives us
$\omega_{2}=\overline{\alpha}$, the state on
$A(\overline{\boldsymbol{\mathcal{H}}})$ corresponding to $\overline{W}$.
Indeed, the state $\omega$ is symmetric, in the sense that
$\omega(x,\overline{y})=\omega(y,\overline{x})$ for all $x,y\in
X(\boldsymbol{\mathcal{H}})$.
Notice that, by (3), we have $|\langle
x\otimes\overline{x},v_{W}\rangle|^{2}=\alpha(x)$ for every $x\in E$.
Evidently, the pure state $\omega$ corresponding to $v_{W}$ sets up a perfect
correlation between $E\in{\mathcal{M}}(\boldsymbol{\mathcal{H}})$ and the
corresponding test
$\overline{E}=\\{\overline{x}|x\in{\mathbf{E}}\\}\in{\mathcal{M}}(\overline{\boldsymbol{\mathcal{H}}})$.
This works for any basis $E$ diagonalizing $W$, i.e., $\omega$ correlates
every such basis with the corresponding basis $\overline{E}$ for
$\overline{\boldsymbol{\mathcal{H}}}$. Indeed, if $U$ is a unitary operator on
$\boldsymbol{\mathcal{H}}$ with $UW=WU$, then
$\omega(Ux,\overline{U}\overline{y})=\omega(x,\overline{y})$ for all $x,y\in
X(\boldsymbol{\mathcal{H}})$.
Definition 3: A non-signaling state $\omega$ on a composite $AB$ is
correlating iff there exists some pair of tests $E\in{\mathcal{M}}(A)$ and
$F\in{\mathcal{M}}(B)$ and a bijection $f:E\rightarrow F$ such that
$\omega(x,y)=0$ for $(x,y)\in E\times F$ with $y\not=f(x)$ — equivalently,
$\sum_{x\in E}\omega(x,f(x))=1$.
Using this language, we can summarize the foregoing discussion as follows:
every state $\alpha$ on a quantum model $A(\boldsymbol{\mathcal{H}})$ is the
marginal of a pure, correlating state on
$A(\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}})$,
invariant under the group of unitaries of the form $U\otimes\overline{U}$ with
$U$ stabilizing $\alpha$.444It is also true, by the Schmidt decomposition,
that every pure state on
$\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$ is
correlating. In the special case where $\alpha$ is the maximally mixed state,
i.e., where $W=1/n$, where $n=\dim(\boldsymbol{\mathcal{H}}))$ and $1$ is the
identity operator on $\boldsymbol{\mathcal{H}}$, we the corresponding vector
from (2) is $v_{W}=\frac{1}{n}\sum_{x\in E}x\otimes\overline{x}=:\Psi$, the
maximally entangled stateSince $W=1/n$ is diagonalized by every orthonormal
basis, the expansion above is valid for all
$E\in{\mathcal{M}}(\boldsymbol{\mathcal{H}})$. Let’s denote the corresponding
non-signaling state on
${\mathbf{E}}(\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}})$
by $\eta_{A}$, so that
$\eta_{A}(a,\overline{b})=|\langle\Psi,a\otimes\overline{b}\rangle|^{2}$. This
state perfectly correlates every test
$E\in{\mathcal{M}}(\boldsymbol{\mathcal{H}})$ with its counterpart in
${\mathcal{M}}(\overline{\boldsymbol{\mathcal{H}}})$, and the correlation is
uniform, in that $\eta_{A}(x,\overline{x})=1/n$ for all correlated pairs of
outcomes $x$ and $\overline{x}$. If fact, It is not hard to see that, for
arbitrary effects $a,b\in{\mathbf{E}}(A)$, we have
$\eta_{A}(a,\overline{b})=\mbox{Tr}(ab)$. In other words, the maximally
entangled state on
$\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$ (rather
than on $\boldsymbol{\mathcal{H}}\otimes\boldsymbol{\mathcal{H}}$) provides a
direct operational interpretation of the tracial inner product on
${\mathbf{E}}(\boldsymbol{\mathcal{H}})={\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$.
Suppose now that $\omega$ is any non-signaling state on the composite quantum
system associated with any two Hilbert spaces $\boldsymbol{\mathcal{H}}_{A}$
and $\boldsymbol{\mathcal{H}}_{B}$. Since the spaces ${\mathbf{E}}(A)$ and
${\mathbf{V}}(B)$ are canonically isomorphic to the spaces
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}}_{A})$ and
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}}_{B})$ of Hermitian operators on
$\boldsymbol{\mathcal{H}}_{A}$ and $\boldsymbol{\mathcal{H}}_{B}$, we can
regard the conditioning map
$\widehat{\omega}:{\mathbf{E}}(A)\rightarrow{\mathbf{E}}(B)$ associated with
$\omega$ as a mapping
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}}_{A})\rightarrow{\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}}_{B})$.
The following is well known, and straightforward to verify:
Lemma 1: Let $W$ be any density operator on $\boldsymbol{\mathcal{H}}$, and
let $\omega$ be the pure state on
$A(\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}})$
corresponding to $v_{W}$, as given by (2) and (3) above. Then the conditioning
map corresponding to $\omega$ is given by
$\widehat{\omega}(a)=\overline{W^{1/2}aW^{1/2}}$ for all
$a\in{\mathbf{E}}(\boldsymbol{\mathcal{H}})$.
Corollary 1: Let $W$ and $\omega$ be as in Lemma 1. If $W$ is non-singular,
then $\omega$ is an isomorphism state.
Proof: If $W$ is non-singular, so is $W^{1/2}$, with inverse given by
$W^{-1/2}$ – again, a positive operator. So
$\widehat{\omega}:{\mathcal{L}}(\boldsymbol{\mathcal{H}})\rightarrow{\mathcal{L}}(\overline{\boldsymbol{\mathcal{H}}})$
given by
$\widehat{\omega}(a)=\overline{W}^{1/2}\overline{a}\overline{W}^{1/2}$ is
invertible, with inverse $\widehat{\omega}^{-1}:a\mapsto W^{-1/2}aW^{-1/2}$ —
again, a positive mapping. $\Box$
Remark: Quaternionic Hilbert spaces present special problems, owing to the
difficulty of defining a tensor product over a non-commutative division ring.
However Baez , one can view a quaternionic Hilbet space as a pair
$(\boldsymbol{\mathcal{H}},J)$ where $\boldsymbol{\mathcal{H}}$ is a complex
Hilbert space and $J$ is an anti-unitary operator on
$\boldsymbol{\mathcal{H}}$ satisfying $J^{2}=-1$. Likewise, a real Hilbert
spaces can be identified with complex Hilbert spaces equipped with anti-
unitary operator $J$ satisfying $J^{2}=1$. Given two quaternionic Hilbert
spaces $(\boldsymbol{\mathcal{H}}_{1},J_{1})$, a natural candidate for the
tensor product of two pairs $(\boldsymbol{\mathcal{H}}_{i},J_{i})$ with
$J_{i}$ anti-unitary and satisfying $J^{2}=-1$, is
$(\boldsymbol{\mathcal{H}}_{1}\otimes\boldsymbol{\mathcal{H}}_{2},J_{1}\otimes
J_{2})$. Since $J_{i}^{2}=-1$, we have $(J_{1}\otimes J_{2})^{2}=1$, i.e., the
tensor product should be thought of as a real Hilbert space. Understanding
composites of quaternionic quantum systems in this way, with a little care one
can show that QM over $\boldsymbol{\mathcal{H}}$ still enjoys the
correlational features just disucussed. In other words, these features are
equally consistent with real, complex and quaternionic quantum mechanics.
## III Conjugate Systems
We’ve seen that the composite quantum system
$A(\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}})$ has
some very strong correlational properties. As I’ll now show, one can derive
the Jordan structure of finite-dimensional QM from these properties, with a
minimum of fuss. As a first step, we need to generalize the relationship
between $A(\boldsymbol{\mathcal{H}})$ and
$A(\overline{\boldsymbol{\mathcal{H}}})$. Throughout this section, let $A$ is
a model of uniform rank $n$, meaning that all tests have cardinality $n$. By
an isomorphism between two models $A$ and $B$, I mean a bijection
$\phi:X(A)\rightarrow X(B)$ such that $\phi(E)\in{\mathcal{M}}(B)$ iff
$E\in{\mathcal{M}}(A)$, and $\beta\circ\phi\in\Omega(A)$ iff
$\beta\in\Omega(B)$.
Definition 4: A conjugate for $A$ is a triple
$(\overline{A},\gamma_{A},\eta_{A})$ where $\gamma_{A}:A\simeq\overline{A}$ is
an isomorphism and $\eta_{A}$ is a non-signaling state on (some composite of)
$A$ and $\overline{A}$ such that (a)
$\eta(x,\gamma_{A}(y))=\eta(y,\gamma_{A}(x))$ and (b)
$\eta_{A}(x,\gamma_{A}(x))=1/n$ for every $x,y\in X(A)$.555The requirement
that $\eta_{A}$ be symmetric is mainly a convenience: given any uniformly
perfectly correlating bipartite state $\eta$ on $A\overline{A}$, i.e., any
state satisfying $\eta(x,\gamma_{A}(x))=1/n$ for all $x\in X(A)$, the state
$\eta^{t}$ defined by $\eta^{t}(x,\gamma(y))=\eta(y,\gamma(x))$ is also
perfectly uniformly correlating, whence, so is the symmetic state
$(\eta+\eta^{t})/2$.
In the context of finite-dimensional quantum mechanics, where
$A=A(\boldsymbol{\mathcal{H}})$ for a Hilbert space
$\boldsymbol{\mathcal{H}}$, we can take
$\overline{A}=A(\overline{\boldsymbol{\mathcal{H}}})$, with
$\gamma_{A}(x)=\overline{x}$; for the state $\eta_{A}$, we can use the
standard maximally entangled state on
$\boldsymbol{\mathcal{H}}\otimes\overline{\boldsymbol{\mathcal{H}}}$. Thus, we
can think of $\eta_{A}$ for an arbitrary probabilistic model $A$, as a
generalized maximally entangled state.
Remark: One might wonder whether one can use the isomorphism $\gamma_{A}$ to
simply identify $A$ with its conjugate. Certainly we can use $\gamma_{A}$ to
pull the correlator $\eta_{A}$ on $A\overline{A}$ back to a positive bilinear
form on ${\mathbf{E}}(A)\times{\mathbf{E}}(A)$, namely
$\eta_{A}^{\prime}(a,b)=\eta_{A}(a,\gamma_{A}(a))$. However, whether this
corresponds to a legitimate bipartite state on a legitimate composite $AA$ of
$A$ with itself, depends on the particular probabilistic theory at hand. For
example, if $A=A(\boldsymbol{\mathcal{H}})$ is the quantum model associated
with a Hilbert space $\boldsymbol{\mathcal{H}}$, and $\overline{A}$ is the
model associated with $\overline{\boldsymbol{\mathcal{H}}}$ in the usual way,
then the state $\eta_{A}$ pulls back along the isomorphism
$a\mapsto\overline{a}$ to a bilinear form on
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})\times{\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$
— but one associated with the non-completely positive partial-transpose
operation, which corresponds to no state on
$A(\boldsymbol{\mathcal{H}}\otimes\boldsymbol{\mathcal{H}})$. Thus, the notion
of a conjugate system is best understood as applying to an entire
probabilistic theory, rather than to a single probabilistic model.
Four Axioms We can now restate the four conditions discussed in the
introduction in more precise terms:
Axiom 1 (Conjugates): $A$ has a conjugate, $\overline{A}$.
Axiom 2 (Correlation): Every state of $A$ is the marginal of a corelating
state on $A\overline{A}$.
Axiom 3 (Sharpness): For every outcome $x\in X(A)$, there exists a unique
state $\delta_{x}\in\Omega(A)$ such that $\delta_{x}(x)=1$.
Axiom 4 (State Preparation): Every state $\alpha$ with $\alpha(x)>0$ for all
outcomes $x$, can be prepared by a reversible process from the maximally mixed
state $\rho(x)\equiv 1/n$.
Axioms 1-3 are trivially satisfied in discrete classical probability theory.
As observed in Section 2 they are also satisfied in finite-dimensional quantum
theory. Axiom 4 is equivalent to the homogeneity of the cone
${\mathbf{V}}(A)_{+}$, and hence, also satisfied by classical and quantum
probability theory.
We are now ready to prove Theorem 1. The main order of business is to show
that a model satisfying Axioms 1, 2 and 3 is self-dual. The proof is not
difficult. I’ll break it up into a sequence of even easier lemmas. In the
interest of readability, in what follows I conflate outcomes $x\in X(A)$ with
the corresponding effects $\widehat{x}\in{\mathbf{E}}(A)$, and write
$\overline{x}$ for $\gamma_{A}(x)$.
Lemma 2: For every $\alpha\in{\mathbf{V}}_{+}(A)$, there exists a test $E$
such that
$\alpha\ =\ \sum_{x\in E}\alpha(x)\delta_{x}$ (4)
Proof: We can assue that $\alpha$ is a normalized state. Then, by Axiom 2,
$\alpha=\omega_{1}$ where $\omega$ correlates some pair of tests
$E\in{\mathcal{M}}(A)$ with $\overline{F}\in{\mathcal{M}}(\overline{A})$ along
a bijection $f:E\rightarrow\overline{F}$. By the law of total probability (1)
for non-signaling states,
$\alpha=\sum_{\overline{y}\in\overline{F}}\omega_{2}(\overline{y})\omega_{1|\overline{y}}$.
Since $\omega$ is correlating along $f$, if $x\in E$ and $y=f(x)$, we have
$\omega_{1|\overline{y}}(x)=1$. Thus, by sharpness (Axiom 3),
$\omega_{1|\overline{y}}=\delta_{x}$. Hence, $\alpha=\sum_{x\in
E}\omega_{2}(f(x))\delta_{x}$. It follows that $\omega_{2}(f(x))=\alpha(x)$,
giving us (4). $\Box$
Lemma 3: $\widehat{\eta_{A}}$ is an isomorphism state.
Proof: We need to show that
$\widehat{\eta_{A}}:{\mathbf{E}}(A)\rightarrow{\mathbf{V}}(A)$ is a linear
isomorphism with a positive inverse. It is enough to show that
$\widehat{\eta_{A}}$ maps the positive cone of ${\mathbf{E}}(A)$ onto that of
${\mathbf{V}}(\overline{A})$. Since $x\mapsto\overline{x}$ is an isomorphism
between $A$ and $\overline{A}$, we can apply Lemma 2 to $\overline{A}$: if
$\alpha\in{\mathbf{V}}_{+}(\overline{A})$, we have $\alpha=\sum_{x\in
E}\alpha(\overline{x})\delta_{\overline{x}}$. Since
$\eta_{A}(x,\overline{x})=1/n$, we have
$\widehat{\eta_{A}}(x)=\frac{1}{n}\delta_{\overline{x}}$ for every $x\in
X(A)$. Hence, $\widehat{\eta_{A}}(\sum_{x\in E}n\alpha({x})x)=\alpha$. $\Box$
Lemma 4: Every $a\in{\mathbf{E}}(A)$ has a representation $a=\sum_{x\in
E}t_{x}x$ for some test $E\in{\mathcal{M}}(A)$ and some coefficients $t_{x}$.
Proof:If $a\in{\mathbf{E}}(A)_{+}$, then by Lemma 2,
$\widehat{\eta_{A}}(a)=\sum_{x\in F}t_{x}\delta_{\overline{x}}$ for some
$E\in{\mathcal{M}}(A)$. By Lemma 3, $\eta_{A}$ is an order-isomorphism.
Applying $\eta_{A}^{-1}$ to this expansion gives the desired result. Now for
an arbitrary $a\in{\mathbf{E}}(A)_{+}$, we can find some $N$ such that $a\leq
Nu$. If $a=a_{1}-a_{2}$ with $a_{1},a_{2}\in{\mathbf{E}}(A)_{+}$, we can find
$N\geq 0$ with $a_{2}\leq Nu$. Thus, $b:=a+Nu=a_{1}+(Nu-a_{2})\geq 0$. Then
$b:=\sum_{x\in E}t_{x}x$ for some $E\in{\mathcal{M}}$, so that
$a=b-Nu=\sum_{x\in E}t_{x}x-N(\sum_{x\in E}x)=\sum_{x\in E}(t_{x}-N)x$. $\Box$
Lemma 5: The bilinear form $\langle a,b\rangle=\eta_{A}(a,\gamma_{A}(b))$ is
positive-definite, i.e., an inner product.
Proof: By assumption, it’s symmetric. It’s also bilinear, because $\eta_{A}$
is non-signaling and $\gamma_{A}$ is linear. We need to show that
$\langle~{},~{}\rangle$ is positive semi-definite. Let $a\in{\mathbf{E}}(A)$.
From Lemma 4, we have $a=\sum_{x\in E}t_{x}x$ for some test $E$ and some
coeffcients $t_{x}$. Now
$\displaystyle\langle a,a\rangle=\left\langle\sum_{x\in E}t_{x}x,\sum_{y\in
E}t_{y}y\right\rangle$ $\displaystyle=$ $\displaystyle\sum_{x,y\in E\times
E}t_{x}t_{y}\langle x,y\rangle$ $\displaystyle=$ $\displaystyle\sum_{x,y\in
E\times E}t_{x}t_{y}\eta_{A}(x,\overline{y})$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{x\in E}{t_{x}}^{2}\geq 0.$
This is zero only where all coefficients $t_{x}$ are zero, i.e., only for
$a=0$. $\Box$
Lemma 2 tells us that ${\mathbf{E}}(A)\simeq{\mathbf{V}}(A)$, so it remains
only to show that the inner product $\langle\ ,\ \rangle$ is self-dualizing.
Clearly ${\mathbf{E}}(A)_{+}\subseteq{\mathbf{E}}(A)^{+}$, since
$\eta(a,\overline{b})\geq 0$ for all $a,b\in{\mathbf{E}}(A)_{+}$. For the
reverse inclusion, suppose $\langle a,b\rangle\geq 0$ for all
$b\in{\mathbf{E}}(A)_{+}$. Then $\langle a,y\rangle\geq 0$ for all $y\in X$.
By Lemma 4, $a=\sum_{x\in E}t_{x}x$ for some test $E$. Thus, for all $y\in E$
we have $\langle a,y\rangle=t_{y}\geq 0$, whence, $a\in{\mathbf{E}}(A)_{+}$.
$\Box$
Axiom 4 now guarantees then ${\mathbf{V}}(A)$ and hence, ${\mathbf{E}}(A)$, is
homogeneous. Indeed, as discussed above, a reversible process is an order-
automorphism $\phi:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$; to say that
this prepares $\alpha$, up to normalization, from $\rho$, is simply to say
that $\alpha=t\phi(\rho)$ for some appropriate constant $t>0$ (namely,
$t=u_{A}(\phi(\rho))^{-1}$). Since $t\phi$ is again an order-automorphism, it
follows that the group of order-automorphisms of ${\mathbf{V}}(A)$ act
transitively on the interior of ${\mathbf{V}}(A)_{+}$, i.e., ${\mathbf{V}}(A)$
is homogeneous.
We now have the advertised result: Subject to axioms 1-4, ${\mathbf{E}}(A)$ is
homogeneous and self-dual, whence, by the Koecher-Vinberg Theorem, can be
equipped with a euclidean Jordan structure making ${\mathbf{E}}(A)_{+}$ the
cone of squares. Indeed, this Jordan structure is unique, subject to the
condition that $u_{A}$ act as the unit for this Jordan structure. (One can
further show that the outcome-set $X(A)$ is precisely the set of primitive
idempotents in ${\mathbf{E}}(A)$ with respect to this Jordan structure, and
that basic measurements in ${\mathcal{M}}(A)$ correspond to Jordan frames,
i.e., maximal pairwise orthogonal sets of idempotents. See Lemma 5 in LTHSD .)
Remarks (1) Note that the only point at which Axioms 2 and 3 were invoked was
in the proof of Lemma 2. Thus, any other assumptions leading to the
representation (4) could be used instead. Moreover, since $\widehat{\eta}_{A}$
is a linear isomorphism, and hence (in finite dimensions) an homeomorphism, it
is enough to obtain this representation for states in the interior of
$\Omega$. From this we have, as in the proof of Lemma 3, that the interior of
the cone ${\mathbf{V}}_{+}$ is in the range of $\widehat{\eta}_{A}$, from
which it follows that $\widehat{\eta}$ is a linear isomorphism, and hence,
that $K:=\widehat{\eta}^{-1}({\mathbf{V}}_{+}^{\circ})$ is an open sub-cone of
${\mathbf{E}}_{+}$, spanning the latter. Moreover, every point in the interior
of $K$ has a corresponding “spectral” decomposition of the form $\sum_{x\in
E}t_{x}x$ with $t_{x}\geq 0$, where $E$ is some test in ${\mathcal{M}}(A)$.
Arguing exactly as in the proof of Lemma 4, we can extend this to arbitrary
points $a\in{\mathbf{E}}$ by decomposing $a$ as $a_{1}-a_{2}$, where
$a_{1},a_{2}\in K$. The rest of the proof of Theorem 1 then proceeds just as
before.
(2) Since axioms 1 and 2 have such a similar character, it is natural to look
for a single principle that encompasses them both. Suppose $G$ is a group
acting transitively on the outcome-space $X(A)$ of the model $A$, and leaving
the state-space $\Omega(A)$ invariant. If $G$ is compact, there will exist an
invariant state, $\rho$, obtained by group averging; by the transitivity of
$G$ on outcomes, this state must be constant, i.e., $\rho$ is the maximally
mixed state $\rho(x)\equiv 1/n$. Now consider the following variant of Axiom 2
(here $G_{\alpha}$ denotes the stabilizer in $G$ of the state $\alpha$):
Axiom 2′ There exists a system $\overline{A}$ and an isomorphism
$\gamma_{A}:A\simeq\overline{A}$, such that every state $\alpha$ is the
marginal, $\omega_{1}$, of some correlating bipartite state $\omega$ on $AB$
with $\omega(gx,\gamma_{A}(gy))=\omega(x,\gamma_{A}(y))$ for every $g\in
G_{\alpha}$.
As observed in Section 2, this is satisfied by finite-dimensinal quantum
models. Applied to the maximally mixed state $\rho$, this produces a
correlator $\eta_{A}$ turning $\overline{A}$ into a conjugate in the sense of
Definition 3. Thus, we have
Corollary 2: Let $A$ carry a compact, transitive-on-outcomes group action, as
described above, and satisfy Axioms $2^{\prime}$ and 3. Then ${\mathbf{E}}(A)$
is self-dual.
(3) Given Axioms 1-3, any condition guaranteeing the homogeneity of
${\mathbf{V}}(A)$ will also secure that of ${\mathbf{E}}(A)$. As observed in
BGW , homogeneity follows from the requirement that every interior state of
$A$ be the marginal of an isomorphism state on a composite of two isomorphic
copies of $A$. Thus, in place of Axiom 4, we could simply strengthen Axiom 2
to
Axiom 2′′ Every interior state of $A$ is the marginal of an isomorphism state
on $A\overline{A}$, correlating some pair of tests.
Corollary 3: For any model $A$ satisfying Axioms 1, $2^{\prime\prime}$ and 3,
${\mathbf{E}}(A)$ is homogeneous and self-dual.
For irreducible systems, isomorphism states are pure BGW , so Axiom
$2^{\prime\prime}$ is related to the purification postulate of CDP . The
latter asserts that every system $A$ has a “conjugate system” (in their usage)
$B$, such that every state on $A$ arises as the marginal of a pure state of
$AB$, unique up to symmetries of $B$.
## IV Filters
To this point, I’ve been leaning heavily on the assumption of sharpness (Axiom
3). It would surely be preferable to define $\delta_{x}$ to be the conditional
state $\eta_{1|\overline{x}}$, for each $x\in X(A)$, and to prove that this is
the unique state making $x$ certain. In fact, this can be done if we replace
Axiom 2 with a slightly stronger, but very plausible, axiom concerning the
existence of certain processes called filters 4.5 . In many kinds of
laboratory experiments, the distinct outcomes of an experiment correspond to
physical detectors, the efficiency of which can independently be attenuated,
if desired, by the experimenter.
In fact, this can be done reversibly. Let $A$ be a finite-dimensional quantum
system, with corresponding Hilbert space $\boldsymbol{\mathcal{H}}$, and
identify ${\mathbf{E}}(A)$ with
${\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}}_{A})$. If $E$ is an orthonormal
basis representing a basic measurement on this system, define a positive self-
adjoint operator
$V:\boldsymbol{\mathcal{H}}\rightarrow\boldsymbol{\mathcal{H}}$ by setting
$Vx=t_{x}^{1/2}x$ for every $x\in E$, where $0<t_{x}\leq 1$. This gives us a
completely positive linear mapping
$\phi:{\mathbf{E}}(A)\rightarrow{\mathbf{E}}(A)$, namely $\phi(a)=VaV$. This
has a completely positive inverse $\phi^{-1}(a)=V^{-1}aV^{-1}$, hence, is an
order automorphism. For each $x\in E$, the corresponding effect
$\widehat{x}\in{\mathbf{E}}(A)\simeq{\mathcal{L}}_{h}(\boldsymbol{\mathcal{H}})$
is the projection operator $P_{x}$. It is easy to check that
$VP_{x}V=t_{x}P_{x}$, i.e., that $\phi(\widehat{x})=t_{x}\widehat{x}$ for
every $x\in E$.
Definition 5: A filter on a probabilistic model $A$ is a positive linear
mapping $\Phi:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$ such that, for some
test $E\in{\mathcal{M}}(A)$ and coefficients $0\leq t_{x}\leq 1$,
$\Phi(\alpha)(x)=t_{x}\alpha(x)$ for all $x\in E$ and all states $\alpha$.
Let us say that $\Phi$ prepares a state $\alpha$ if $\alpha$ equals
$\phi(\rho)$ up to normalization, where where $\rho$ is the maximally mixed
state $\rho(x)\equiv 1/n$. If every interior state is preparable by a
reversible filter, then ${\mathbf{V}}(A)_{+}$ is homogeneous. Suppose that $A$
has a conjugate system, $\overline{A}$, and that $\phi$ is a filter for a test
$E\in{\mathcal{M}}(A)$. By applying $\phi$ to one wing of the composite system
$A\overline{A}$, we can convert the correlator $\eta_{A}$ into a new non-
signaling, sub-normalized joint state $\omega$, given by
$\omega(x,y)=\eta_{A}(\phi^{\ast}x,y)$ for all $x\in X(A),y\in X(B)$. Noticing
that $\Phi^{\ast}(x)=t_{x}x$ for every $x\in E$, we see that $\omega$
correlates $E$ with $\overline{E}$: if $x,y\in E$ with $x\not=y$, we have
$\omega(x,\overline{y})=\eta_{A}(t_{x}x,\overline{y})=t_{x}\eta_{A}(x,\overline{y})=0$.
Since $\omega_{1}=\rho\circ\phi$, it follows that any state preparable from
$\rho$ by a filter, is the marginal of a correlating state. So, in the
presence of sharpness we can replace Axiom 2 by the axiom that every state be
preparable by a filter. In fact, we can do a bit better.
The isomorphism $\gamma_{A}:A\simeq\overline{A}$ extends to an order-
automorphism ${\mathbf{V}}(A)\simeq{\mathbf{V}}(\overline{A})$, given by
$\alpha\mapsto\overline{\alpha}$, with
$\overline{\alpha}(\overline{x})=\alpha(x)$ for all $x\in X(A)$. Hence, a
positive linear mapping $\Phi:{\mathbf{V}}(A)\rightarrow{\mathbf{V}}(A)$ has a
countepart
$\overline{\Phi}:{\mathbf{V}}(\overline{A})\rightarrow{\mathbf{V}}(\overline{A})$,
given by $\overline{\Phi}(\overline{\alpha})=\overline{\Phi(\alpha)}$. Let us
say that $\Phi$ is symmetric with respect to a conjugate $\overline{A}$ iff
$\eta_{A}(\Phi^{\ast}(x),\overline{y})=\eta_{A}(x,\overline{\Phi^{\ast}}(y))$
for all $x,y\in X(A)$, i.e., iff
$\eta_{A}\circ(\Phi\otimes\overline{1})=\eta_{A}\circ(1\otimes\overline{\Phi})$.
Lemma 6: Let $A$ have a conjugate $\overline{A}$. Suppose that every state of
$A$ is preparable by a symmetric filter. Then $\langle
a,b\rangle:=\eta_{A}(a,\gamma_{A}(b))$ is a self-dualizing inner product on
${\mathbf{E}}(A)$.
Proof: Let $\alpha=\Phi(\rho)$, where $\Phi$ is a symmetric filter on some
test $E$. Consider the bipartite state
$\omega:=(\Phi\otimes 1)(\eta_{A})=(1\otimes\overline{\Phi})(\eta_{A}).$
For each outcome $x\in X(A)$, let $\delta_{x}$ denote the conditional state
$(\eta_{A})_{1|\overline{x}}$. For all $x\in E$, we have
$\displaystyle\omega_{1|\overline{x}}(y)=\frac{\eta_{A}(\Phi^{\ast}(y),\overline{x})}{\eta_{A}(\Phi^{\ast}(u_{A}),\overline{x})}$
$\displaystyle=$
$\displaystyle\frac{\eta_{A}(y,\overline{\Phi^{\ast}}(\overline{x}))}{\eta_{A}(u_{A},\overline{\Phi^{\ast}}(\overline{x}))}$
$\displaystyle=$
$\displaystyle\frac{\eta_{A}(y,t_{x}\overline{x})}{\eta_{A}(u_{A},t_{x}\overline{x})}$
$\displaystyle=$
$\displaystyle\frac{\eta_{A}(y,\overline{x})}{\eta_{A}(u_{A},\overline{x})}=(\eta_{A})_{1|\overline{x}}=\delta_{x}.$
Now $\omega_{1}=\Phi((\eta_{A})_{1})=\Phi(\rho)=\alpha$, and also, by the law
of total probability (1), $\omega_{1}=\sum_{x\in
E}\omega_{2}(\overline{x})\omega_{1|\overline{x}}=\sum_{x\in
E}t_{x}\delta_{x}$, where $t_{x}=\omega_{2}(\overline{x})$. Thus, every state
on $A$ is a convex combination of the states $\delta_{x}$. Hence, the cone
generated by these states coincides with ${\mathbf{V}}(A)_{+}$. We now have
that $\widehat{\eta}$ maps ${\mathbf{E}}(A)_{+}$ onto ${\mathbf{V}}(A)_{+}$,
as in the proof of Lemma 3. The proof that $\langle
a,b\rangle:=\eta(a,\overline{b})$ defines an inner product on
${\mathbf{E}}(A)$ now proceeds as in the proof of Lemmas 4 and 5. $\Box$
This suggests another axiom, combining Axiom 1 with a strengthened form of
Axiom 4:
Axiom 4b $A$ has a conjugate system, and every interior state is preparable by
a reversible symmetric filter.
This clearly implies the homogeneity of ${\mathbf{V}}(A)$. In fact, it is
strong enough to allow us to do without Axioms 2 and 3. Indeed, as noted in
Remark (1) following the proof of Theorem 1, it is sufficient to obtain the
decomposition (4) for points in the interior of ${\mathbf{V}}_{+}$. By Lemma
6, for any system satisfying Axiom 5, all states in the interior of $\Omega$
can be decomposed as in equation (4); as noted in Remark (1) following the
proof of Theorem 1, this is enough to secure the self-duality of
${\mathbf{E}}(A)$, and its isomorphism with ${\mathbf{V}}(A)$. This proves
Theorem 2.
## V Conclusion
We’ve seen that any of several related sets of assumptions, e.g., Axioms 1-4,
or Axioms 2’,3 and 4, pr Axioms 1, 2’ and 3, or the two-part Axiom 4b, lead in
a very simple way the homogeneity and self-duality of the cone
${\mathbf{E}}(A)_{+}$ associated with a probabilistic model $A$. Hence, by the
Koecher-Vinberg Theorem, the space ${\mathbf{E}}(A)$ carries a canonical
Jordan struture. While this is not the only route one can take to deriving
this structure (see, e.g, MU and SSD for approaches stressing symmetry
principles), it does seem especially straightforward.
Among Jordan-algebraic probabilistic theories, finite-dimensional quantum
mechanics over ${\mathcal{C}}$ can be singled out as follows. A non-signaling
composite system $AB$ locally tomographic iff every state $\omega\in AB$ is
uniquely determined by the joint probability function $\omega(x,y)$ that it
induces. It is well known, and easy to see on dimensional grounds, that among
finite-dimensional real, complex and quaternionic quantum mechanics, only the
complex version is locally tomographic. Call a probabilistic theory monoidal
iff it is a symmetric monoidal category, in which the monoidal product is a
non-signaling composite in the sense of Definition 2 above. By exploiting a
result of Hanche-Olsen Hanche-Olsen , one can show BW12 that a Jordan-
algebraic theory in which all composites are locally tomographic, and which
contains at least one system having the structure of a qubit, must be a direct
sum of finite-dimensional complex matrix algebras — that is, finite-
dimensional complex QM with superselection rules. One should perhaps not rush
to embrace local tomography as a universal principle, however. Indeed, the
very fact that it excludes real and quaternionic quantum theory suggests that
it is too strong: see Baez for some cogent reasons not to exclude these
cases.
Several other recent papers (e.g, CDP ; Dakic-Brukner ; Hardy ; Masanes-
Mueller ; Rau ) have derived standard finite-dimensional quantum mechanics,
over $\mathbb{C}$, from operational axioms. Besides the fact that the
mathematical development here is much quicker and easier (modulo invocation of
the KV theorem), the axiomatic basis is different, and arguably leaner. The
papers cited in the introduction tend to impose strong constraints on
“subspaces”, along the lines of assuming that every face of the state space
corresponds to the state space of a system satisfying the remaining axioms. A
related assumption, also used in several of the cited papers, is that all
systems characterized by the same “information-carrying capacity” are
isomorphic. The present approach entirely avoids such assumptions. I also
avoid the assumption, used in Masanes-Mueller ; SSD that every element of
${\mathbf{V}}(A)^{\ast}$ corresponds to a physically accessible measurement
result. Finally, all of the cited papers assume some form of local tomography.
In view of the comments above, it seems valuable to be able to delineate
clearly what does and what does not depend on this assumption (particularly if
we are interested in the possibilities for a “post-quantum” theory).
This brings us to the interesting question of whether all euclidean Jordan
algebras actually satisfy the axioms discussed here. Let $X$ denote the set of
primitive (that is, minimal) idempotents in ${\mathbf{E}}$, let
${\mathcal{M}}$ denote the set of Jordan frames (i.e., maximal sets of
pairwise orthogonal idempotents), and let $\Omega$ denote the set of states on
${\mathbf{E}}$ (i.e., positive linear functionals
$\alpha\in{\mathbf{E}}^{\ast}$ with $\alpha(u)=1$, where $u$ is the unit
element of ${\mathbf{E}}$). Then $A=({\mathcal{M}},\Omega)$ is a probabilistic
model with ${\mathbf{E}}(A)={\mathbf{E}}$. In particular, $A$ is self-dual and
sharp. Taking $\eta_{A}(a,b)=\frac{1}{r}\mbox{Tr}(ab)$, where $r$ is the rank
of ${\mathbf{E}}$, we have a perfectly correlating, non-signaling bipartite
state. Using the spectral theorem for Jordan algebras, plus the quadratic
representation FK , one can show that every state of $A$ can be prepared by a
filter. Hence, every state is the marginal of a correlating bipartite state,
as discussed in Section 4.
What isn’t obvious is how to interpret the state $\eta$ just defined. In
orthodox QM, in fact, it isn’t a state at all: rather, one needs to invoke the
complex conjugate Hilbert space. What meaning one should attach to $\eta_{A}$
must depend on the choice of the conjugate system $\overline{A}$, and on that
of the composite system $A\overline{A}$ —- and this, in turn, depends, not on
the individual model $A$, but on the entire probabilistic theory at hand. So
the question remains: can we embed arbitrary euclidean Jordan algebras in a
probabilistic theory in which axioms 1, 2’ and 3 are satisfied?
Even assuming that this is possible mathematically, there is still a question
of how to interpret the conjugate system $\overline{A}$. In quantum theory,
one can regard the conjugate Hilbert space
$\overline{\boldsymbol{\mathcal{H}}}$ as representing a time-reversed version
of the system represented by $\boldsymbol{\mathcal{H}}$. Whether some such
interpretation can be maintained more generally remains to be addressed.
Alternatively, one might view the existence of a conjugate as a way of
formulating von Neumann’s “projection postulate”: if we take $A$ to represent
the system at one moment, and $\overline{A}$, the system “immediately
afterwards”, then $\eta_{A}$ represents a state in which we expect that,
whatever measurement is made, and whatever result is secured, if that same
measurement were immediately repeated, the result would be the same. In any
case, the meaning of the conjugate, and of the correlator $\eta_{A}$, must
certainly depend on the entire probabilistic theory. These matters will be
taken up elsewhere.
Acknowledgements I wish to thank Giulio Chiribella, Chris Heunen, Matt Leifer
and Markus Müller for helpful comments on earlier drafts of this paper.
## References
* (1) S. Abramsky and B. Coecke, Abstract Physical Traces, Theory and Applications of Categories, vol 14, pages 111–124, 2005 (arXiv: arXiv:0910.3144, 2009)
* (2) E. Alfsen and F. Shultz, Geometry of State Spaces of Operator Algebras, Birhauser, 2003
* (3) J. Baez, Division algebras and quantum theory, Found. Phys. 42 (2012), 819-855 (arXiv:1101.5690)
* (4) H. Barnum, C.P. Gaebbler and A. Wilce, Steering etc., Ensemble steering, weak self-duality, and the structure of probabilistic theories, arXiv:0912.5532, 2009
* (5) H. Barnum and A. Wilce, Information processing in convex operational theories, Electronic Notes in Theoretical Computer Science 270 (2011), 3-15.
* (6) H. Barnum and A. Wilce, Post-classical probability theory, to appear in G. Chiribella and R. Spekkens (eds.), Quantum Theory: Informational Foundations and Foils, Springer (arXiv:1205.3833)
* (7) H. Barnum and A. Wilce, Local tomography and the Jordan structure of quantum theory, arXiv:1202.4513, 2012
* (8) G. Chiribella, M. D’Ariano and P. Perinotti, Informational derivation of quantum theory, Physical Review A 84 (2011), 012311.
* (9) B. Dakic and C. Brukner, Quantum theory and beyond: is entanglement special? (arXiv:0911.0695, 2009)
* (10) J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford, 1994
* (11) D. J. Foulis and C. H. Randall, Empirical logic and tensor products, in H. Neunmann (ed.), Foundations of Interpretations and Foundations of Quantum Mechanics, B.I.-Wissenshaftsverlag, 1981
* (12) H. Hanche-Olsen, JB algebras with tensor products are $C^{\ast}$ algebras, in H. Araki et al. (eds.), Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics 1132 (1985), 223-229.
* (13) L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012, 2001.
* (14) L. Massanes and M. Müller, A derivation of quantum theory from physical requirements, New J. Phys. 13 (2011) (arXiv:1004.1483, 2010)
* (15) M. Müller and C. Ududec, The structure of reversible computation determines the self-duality of quantum theory Phys. Rev. Lett. 108 (2012), 130401- (arXiv:1110.3516, 2011)
* (16) J. Rau, On quantum vs. classical probability, Annals of Physics 324 (2009) 2622–2637 (arXiv:0710.2119, 2007)
* (17) A. Wilce, Four and a half axioms for finite-dimensional quantum theory in Y. Ben-Menahem and M. Hemmo (eds.), Probability in Physics, Springer, 2012 (arXiv:0912.5530, 2009)
* (18) A. Wilce, Symmetry, self-duality and the Jordan structure of finite-dimensional quantum theory, arXiv:1110.6607 (2011)
|
arxiv-papers
| 2012-06-13T19:05:35 |
2024-09-04T02:49:31.739004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Alexander Wilce",
"submitter": "Alexander Wilce",
"url": "https://arxiv.org/abs/1206.2897"
}
|
1206.2975
|
Enumerating the total number of subtrees of trees***Financially supported by
the National Natural Science Foundation of China (Grant No. 11071096) and the
Special Fund for Basic Scientific Research of Central Colleges (CCNU11A02015).
Shuchao Li†††E-mail: lscmath@mail.ccnu.edu.cn (S.C. Li), wang06021@126.com
(S.J. Wang), Shujing Wang
Faculty of Mathematics and Statistics, Central China Normal University, Wuhan
430079, P.R. China
Abstract: Over some types of trees with a given number of vertices, which
trees minimize or maximize the total number of subtrees or leaf containing
subtrees are studied. Here are some of the main results: (1) Sharp upper bound
on the total number of subtrees (resp. leaf containing subtrees) among
$n$-vertex trees with a given matching number is determined; as a consequence,
the $n$-vertex tree with domination number $\gamma$ maximizing the total
number of subtrees (resp. leaf containing subtrees) is characterized. (2)
Sharp lower bound on the total number of leaf containing subtrees among
$n$-vertex trees with maximum degree at least $\Delta$ is determined; as a
consequence the $n$-vertex tree with maximum degree at least $\Delta$ having a
perfect matching minimizing the total number of subtrees (resp. leaf
containing subtrees) is characterized. (3) Sharp upper (resp. lower) bound on
the total number of leaf containing subtrees among the set of all $n$-vertex
trees with $k$ leaves (resp. the set of all $n$-vertex trees of diameter $d$)
is determined.
Keywords: Subtrees; Leaves; Matching number; Domination number; Diameter;
AMS subject classification: 05C05, 05C10
## 1 Introduction
We consider only simple connected graphs (i.e. finite, undirected graphs
without loops or multiple edges). Let $G=(V_{G},E_{G})$ be a graph with
$u,v\in V_{G}$, $d_{G}(u)$ (or $d(u)$ for short) denotes the degree of $u$;
the distance $d_{G}(u,v)$ is defined as the length of the shortest path
between $u$ and $v$ in $G$; $D_{G}(v)$ (or $D(v)$ for short) denotes the sum
of all distances from $v$. The eccentricity $\varepsilon(v)$ of a vertex $v$
is the maximum distance from $v$ to any other vertex. Vertices of minimum
eccentricity form the center (see [1]). A tree $T$ has exactly one or two
adjacent center vertices. In what follows, if a tree has a bicenter, then our
considerations apply to any of its center vertices.
A subset $S$ of $V_{G}$ is called a dominating set of $G$ if for every vertex
$v\in V_{G}\setminus S$, there exists a vertex $u\in S$ such that $v$ is
adjacent to $u$. A vertex in the dominating set is called a dominating vertex.
For a dominating set $S$ of graph $G$ with $v\in S$ and $u\in V_{G}\setminus
S$, if $vu\in E_{G}$, then $u$ is said to be dominated by $v$. The domination
number of graph $G$, denoted by $\gamma(G)$, is defined as the minimum
cardinality of dominating sets of $G$. For a connected graph $G$ of order $n$,
Ore [9] obtained that $\gamma(G)\leqslant\frac{n}{2}$. And the equality case
was characterized independently in [3, 20]. Given a graph $G$, the matching
number of $G$ is the cardinality of one of its maximum matchings.
Throughout the text we denote by $P_{n},\,K_{1,n-1}$ the path and star on $n$
vertices, respectively. $G-v,\,G-uv$ denote the graph obtained from $G$ by
deleting vertex $v\in V_{G}$, or edge $uv\in E_{G}$, respectively (this
notation is naturally extended if more than one vertex or edge is deleted).
Similarly, $G+uv$ is obtained from $G$ by adding edge $uv\not\in E_{G}$. For
$v\in V_{G},$ let $N_{G}(v)$ (or $N(v)$ for short) denote the set of all the
adjacent vertices of $v$ in $G.$ The diameter diam$(G)$ of graph $G$ is the
maximum eccentricity of any vertex in $G$. We refer to vertices of degree 1 of
a tree $T$ as leaves (or pendants), and the edges incident to leaves are
called pendant edges. The unique path connecting two vertices $v,u$ in $T$
will be denoted by $P_{T}(v,u)$.
Let
$W(T)=\frac{1}{2}\sum_{v\in V_{T}}D_{T}(v)$
denote the Wiener index of $T,$ which is the sum of distances of all unordered
pairs of vertices. This topological index was introduced by Wiener [19], which
has been one of the most widely used descriptors in quantitative structure-
activity relationships. Since the majority of the chemical applications of the
Wiener index deals with chemical compounds with acyclic molecular graphs, the
Wiener index of trees has been extensively studied over the past years; see
[1, 4, 5, 6, 10] and the references there for details.
Given a tree $T$, a subtree of $T$ is just a connected induced subgraph of
$T$. The number of subtrees as well the related subjects has been studied. Let
$T$ denote an $n$-vertex tree each of whose non-pendant vertices has degree at
least three, Andrew and Wang [16] showed that the average number of vertices
in the subtrees of $T$ is at least $\frac{n}{2}$ and strictly less than
$\frac{3n}{4}$. Székely and Wang [12] characterized the binary tree with $n$
leaves that has the greatest number of subtrees. Kirk and Wang [7] identified
the tree, given a size and maximum vertex degree, which has the greatest
number of subtrees. Székely and Wang [15] gave a formula for the maximal
number of subtrees a binary tree can possess over a given number of vertices.
They also showed that caterpillar trees (trees containing a path such that
each vertex not belonging to the path is adjacent to a vertex on the path)
have the smallest number of subtrees among binary trees. Yan and Ye [22]
characterized the tree with the diameter at least $d$, which has the maximum
number of subtrees, and they characterized the tree with the maximum degree at
least $\Delta$, which has the minimum number of subtrees. Consider the
collection of rooted labeled trees with $n$ vertices, Song [11] derived a
closed formula for the number of these trees in which the child of the root
with the smallest label has a total of $p$ descendants. He also derived a
recurrence relation for the number of these trees with the property that for
each non-terminal vertex $v$, the child of $v$ with the smallest label has no
descendants. The authors [8] here determined the maximum (resp. minimum) value
of the total number of subtrees of trees among the set of all $n$-vertex trees
with given number of leaves and characterize the extremal graphs. As well we
determined the maximum (resp. minimum) value of the total number of subtrees
of trees with a given bipartition, the corresponding extremal graphs are
characterized. For some related results on the enumeration of subtrees of
trees, the reader is referred to Székely and Wang [13, 14] and Wang [18].
It is well known that the Wiener index is maximized by the path and minimized
by the star among general trees with the same number of vertices. It is
interesting that the Wiener index and the total number of subtrees of a tree
share exactly the same extremal structure (i.e. the tree that
maximizes/minimizes the corresponding index) among trees with a given number
of vertices and maximum degree, although the values of the indices are in no
general functional correspondence. On the other hand, an acyclic molecule can
be expressed by a tree in quantum chemistry (see [4]). Obviously, the number
of subtrees of a tree can be regarded as a topological index. Hence, Yan and
Ye [22] pointed out that to explore the role of the total number of subtrees
in quantum chemistry is an interesting topic. As a continuance of those works
in [7, 8, 12, 15, 16, 22] which studied the correlations between the Wiener
index and the number of subtrees of trees, in this paper we continue to
characterize the extremal tree among some types of trees which minimizes or
maximizes the total number of subtrees. Through a similar approach, we also
identify the extremal trees that maximize (minimize) the number of leaf
containing subtrees.
## 2 Preliminaries
Given a tree $T$ on $n$ vertices. Let $\mathscr{S}(T)$ denote the set of
subtrees of $T$. For two fixed vertices $u,v$ in $V_{T}$, denote by
$\mathscr{S}(T;u)$ (resp. $\mathscr{S}(T;u,v)$) the set of all subtrees of
$T$, each of which contains $u$ (resp. $u$ and $v$). Let $\mathscr{S}^{*}(T)$
denote the set of all subtrees of $T$ each of which contains at least one leaf
in $T$. Given a vertex $w$ in $V_{T}$, denote by $\mathscr{S}^{*}(T;w)$ the
set of all subtrees of $T$ each of which contains $w$ and at least one leaf of
$T$ different from $w$. For convenience, we call the subtree that contains at
least one leaf of $T$ leaf containing subtree. Set
$F(T)=|\mathscr{S}(T)|,f_{T}(v)=|\mathscr{S}(T;v)|,f_{T}(v_{i}*v_{j})=|\mathscr{S}(T;v_{i},v_{j})|,F^{*}(T)=|\mathscr{S}^{*}(T)|,f_{T}^{*}(v)=|\mathscr{S}^{*}(T;v)|.$
Let $PV(T)$ be the set of leaves of $T$; it is routine to check the following
fact.
###### Fact 1.
Given a tree $T$, then $H(T):=T-PV(T)$ is a tree and $F^{*}(T)=F(T)-F(H)$.
###### Lemma 2.1 ([15]).
Among trees on $n\geqslant 3$ vertices, the path $P_{n}$ minimizes $F^{*}$
with $F^{*}(P_{n})=2n-1$; while the star $K_{1,n-1}$ maximizes $F^{*}$ with
$F^{*}(K_{1,n-1})=2^{n-1}+n-2$.
Figure 1: Path $P_{W}(x,y)$ connecting vertices $x$ and $y$.
Consider the tree $W$ in Fig. 1 with $x,\,y\in PV(W)$, and
$P_{W}(x,y)=xx_{1}\ldots x_{n}zy_{n}\ldots y_{1}y(xx_{1}\ldots
x_{n}y_{n}\ldots y_{1}y)$
if $d_{W}(x,y)$ is even (odd) for any $n\geqslant 0$. After the deletion of
all the edges of $P_{W}(x,y)$ from $W$, some connected components will remain.
Let $X_{i}$ denote the component that contains $x_{i}$, let $Y_{i}$ denote the
component that contains $y_{i}$, for $i=1,2,\ldots,n$, and let $Z$ denote the
component that contains $z$. (Note that $z$ and $Z$ exist if and only if
$d_{W}(x,y)$ is even.)
###### Lemma 2.2 ([18]).
In the above situation, if $f_{X_{i}}(x_{i})\geqslant f_{Y_{i}}(y_{i})$ and
$f_{X_{i}}^{*}(x_{i})\geqslant f_{Y_{i}}^{*}(y_{i})$ for $i=1,\ldots,n$, then
$f_{W}(x)\geqslant f_{W}(y)$ (2.1)
and
$f_{W}^{*}(x)\geqslant f_{W}^{*}(y).$ (2.2)
Furthermore, if a strict inequality $f_{X_{i}}(x_{i})\geqslant
f_{Y_{i}}(y_{i})$ holds for any $i,i\in\\{1,2\ldots,n\\}$, then we have the
strict inequalities in (2.1) and (2.2).
###### Lemma 2.3.
Let $T^{\prime}$ be a graph obtained from a tree $T$ by deleting one leaf.
Then $F(T^{\prime})<F(T)$ and $F^{*}(T^{\prime})<F^{*}(T).$ Furthermore, we
have $f_{T^{\prime}}(v)<f_{T}(v)$ and $f_{T^{\prime}}^{*}(v)\leqslant
f_{T}^{*}(v)$ for any vertex $v$ in $V_{T^{\prime}}$, with equality if and
only if $T$ is a path and $v$ is the other leaf of $T$.
###### Proof.
It is straightforward to check that this result is true. We omit the procedure
here. ∎
If we have a tree $T$ with $x$ and $y$ in $V_{T}$, and a rooted tree $X$ that
is not a single vertex, then we can build two new trees, first $T^{\prime}$,
by identifying the root of $X,u$ with $x$, second $T^{\prime\prime}$, by
identifying the root of $X$, $u$ with $y$ (as depicted in Fig. 2).
Figure 2: Trees $T^{\prime}$ and $T^{\prime\prime}$.
###### Lemma 2.4.
In the above situation, if $f_{T}(x)>f_{T}(y)$, then we have
$F(T^{\prime})>F(T^{\prime\prime})$. Further more, if $x$ is not a leaf of $T$
and $f_{T}^{*}(x)\geqslant f_{T}^{*}(y)$, we have
$F^{*}(T^{\prime})>F^{*}(T^{\prime\prime})$. And if both $x$ and $y$ are
leaves of $T$ and $f_{T}^{*}(x)\geqslant f_{T}^{*}(y)$, we also have
$F^{*}(T^{\prime})\geqslant F^{*}(T^{\prime\prime})$, with equality if and
only if both $x$ and $y$ are leaves of $T$, $f_{T}^{*}(x)=f_{T}^{*}(y)$ and
$X$ is a path with $d_{X}(u)=1$.
###### Proof.
Note that
$\begin{split}F(T^{\prime})&=f_{T^{\prime}}(x)+F(T^{\prime}-x)=f_{T^{\prime}}(x)+F(T-x)+F(X-u)\\\
&=f_{T}(x)f_{X}(u)+(F(T)-f_{T}(x))+(F(X)-f_{X}(u)),\\\
F(T^{\prime\prime})&=f_{T^{\prime\prime}}(y)+F(T^{\prime\prime}-y)=f_{T^{\prime\prime}}(y)+F(T-y)+F(X-u)\\\
&=f_{T}(y)f_{X}(u)+(F(T)-f_{T}(y))+(F(X)-f_{X}(u)).\end{split}$
Hence,
$F(T^{\prime})-F(T^{\prime\prime})=(f_{T}(x)-f_{T}(y))(f_{X}(u)-1)>0,$
i.e., $F(T^{\prime})>F(T^{\prime\prime})$.
We partition the set $\mathscr{S}^{*}(T^{\prime})$ of leaf containing subtrees
of $T^{\prime}$ as follows:
$\mathscr{S}^{*}(T^{\prime})=\mathscr{S}^{*}_{1}(T^{\prime})\cup\mathscr{S}^{*}_{2}(T^{\prime})\cup\mathscr{S}^{*}_{3}(T^{\prime}),$
where
* •
$\mathscr{S}^{*}_{1}(T^{\prime})=\\{\hat{T}:\ \hat{T}$ is a subtree of
$T^{\prime}$ with $x\in V_{\hat{T}}$ and
$V_{T^{\prime}}\cap(PV(T)\setminus\\{x\\})\not=\emptyset$. }.
* •
$\mathscr{S}^{*}_{2}(T^{\prime})=\\{\hat{T}:\ \hat{T}$ is a subtree of
$T^{\prime}$ with $x\in V_{\hat{T}}$,
$V_{T^{\prime}}\cap(PV(T)\setminus\\{x\\})=\emptyset$,
$V_{T^{\prime}}\cap(PV(X)\setminus\\{u\\})\not=\emptyset$}.
* •
$\mathscr{S}^{*}_{3}(T^{\prime})=\\{\hat{T}:\ \hat{T}$ is a subtree of
$T^{\prime}$ with $x\notin V_{\hat{T}}$, $V_{T^{\prime}}\cap
PV(T^{\prime})\not=\emptyset$ }.
Then we have
$|\mathscr{S}^{*}_{1}(T^{\prime})|=f_{X}(u)f_{T}^{*}(x),\ \ \
|\mathscr{S}^{*}_{2}(T^{\prime})|=f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x))$
and
$|\mathscr{S}^{*}_{3}(T^{\prime})|=F^{*}(T-x)+F^{*}(X-u)=\left\\{\begin{aligned}
F^{*}(T)-f_{T}(x)+F^{*}(X-u),&\ \ \ \ x\in PV(T),\\\
F^{*}(T)-f_{T}^{*}(x)+F^{*}(X-u),&\ \ \ \ x\not\in PV(T),\end{aligned}\right.$
Hence,
$F^{*}(T^{\prime})=f_{X}(u)f_{T}^{*}(x)+f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x))+\left\\{\begin{aligned}
F^{*}(T)-f_{T}(x)+F^{*}(X-u),&\ \ \ \ x\in PV(T),\\\
F^{*}(T)-f_{T}^{*}(x)+F^{*}(X-u),&\ \ \ \ x\not\in PV(T),\end{aligned}\right.$
(2.3)
Similarly,
$F^{*}(T^{\prime\prime})=f_{X}(u)f_{T}^{*}(y)+f_{X}^{*}(u)(f_{T}(y)-f_{T}^{*}(y))+\left\\{\begin{aligned}
F^{*}(T)-f_{T}(y)+F^{*}(X-u),&\ \ \ \ y\in PV(T),\\\
F^{*}(T)-f_{T}^{*}(y)+F^{*}(X-u),&\ \ \ \ y\not\in PV(T),\end{aligned}\right.$
(2.4)
First consider that neither $x$ nor $y$ is a leaf of $T$, then (2.3) and (2.4)
give
$\begin{split}F^{*}(T^{\prime})-F^{*}(T^{\prime\prime})&=(f_{T}^{*}(x)-f_{T}^{*}(y))f_{X}(u)+f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x)-f_{T}(y)+f_{T}^{*}(y))-f_{T}^{*}(x)+f_{T}^{*}(y)\\\
&=(f_{T}^{*}(x)-f_{T}^{*}(y))(f_{X}(u)-f_{X}^{*}(u)-1)+f_{X}^{*}(u)(f_{T}(x)-f_{T}(y))>0.\end{split}$
Next consider $y$ is a leaf while $x$ is not a leaf of $T$, then in view of
(2.3) and (2.4) we have
$\begin{split}F^{*}(T^{\prime})-F^{*}(T^{\prime\prime})&=(f_{T}^{*}(x)-f_{T}^{*}(y))f_{X}(u)+f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x)-f_{T}(y)+f_{T}^{*}(y))-f_{T}^{*}(x)+f_{T}(y)\\\
&>(f_{T}^{*}(x)-f_{T}^{*}(y))f_{X}(u)+f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x)-f_{T}(y)+f_{T}^{*}(y))-f_{T}^{*}(x)+f_{T}^{*}(y)\\\
&=(f_{T}^{*}(x)-f_{T}^{*}(y))(f_{X}(u)-f_{X}^{*}(u)-1)+f_{X}^{*}(u)(f_{T}(x)-f_{T}(y))>0.\end{split}$
Now consider that both $x$ and $y$ are leaves of $T$, then (2.3) and (2.4)
give
$\displaystyle F^{*}(T^{\prime})-F^{*}(T^{\prime\prime})$
$\displaystyle=(f_{T}^{*}(x)-f_{T}^{*}(y))f_{X}(u)+f_{X}^{*}(u)(f_{T}(x)-f_{T}^{*}(x)-f_{T}(y)+f_{T}^{*}(y))-f_{T}(x)+f_{T}(y)$
$\displaystyle=(f_{T}^{*}(x)-f_{T}^{*}(y))(f_{X}(u)-f_{X}^{*}(u))+(f_{X}^{*}(u)-1)(f_{T}(x)-f_{T}(y))$
$\displaystyle\geqslant 0.$ (2.5)
Note that $f_{X}(u)>f_{X}^{*}(u)$ and $f_{T}(x)>f_{T}(y)$, hence the equality
holds in (2.5) if and only if $f_{T}^{*}(x)=f_{T}^{*}(y)$ and
$f_{X}^{*}(u)=1$. Thus, $F^{*}(T^{\prime})=F^{*}(T^{\prime\prime})$ if and
only if both $x$ and $y$ are leaves of $T,\,f_{T}^{*}(x)=f_{T}^{*}(y)$ and
$f_{X}^{*}(u)=1$ with $X$ is a path and $u$ is a pendant vertex of $X$. ∎
###### Lemma 2.5.
Given an $n$-vertex path $P_{n}=v_{1}v_{2}\ldots v_{n}$, one has
* (i)
([8]) $f_{P_{n}}(v_{k})=f_{P_{n}}(v_{n-k+1})=k(n-k+1),k\in\\{1,2,\ldots,n\\}$
and
$f_{P_{n}}(v_{1})<f_{P_{n}}(v_{2})<\cdots<f_{P_{n}}(v_{i})<f_{P_{n}}(v_{i+1})<\cdots<f_{P_{n}}(v_{\lfloor\frac{n+1}{2}\rfloor})=f_{P_{n}}(v_{\lceil\frac{n+1}{2}\rceil}).$
* (ii)
$f_{P_{n}}^{*}(v_{1})=f_{P_{n}}^{*}(v_{n})=1,f_{P_{n}}^{*}(v_{k})=n$ for
$k\in\\{2,\ldots,n-1\\}$.
###### Proof.
(ii) follows directly by the the definition of $f_{T}^{*}(v)$. ∎
By Lemmas 2.4 and 2.5, the following lemma follows immediately.
###### Lemma 2.6.
Given a tree $T$ with at least two vertices and a path $P_{k}=v_{1}v_{2}\ldots
v_{k}$, let $T_{i}$ be a tree obtained from $T$ and $P_{k}$ by identifying one
vertex of $T$ with $v_{i}$ of $P_{k}$. Then
$F(T_{i})=F(T_{k-i+1}),F^{*}(T_{i})=F^{*}(T_{k-i+1}),$
$F(T_{1})<F(T_{2})<\cdots<F(T_{i})<\cdots<F(T_{\lfloor\frac{k+1}{2}\rfloor}),$
and
$F^{*}(T_{1})<F^{*}(T_{2})<\cdots<F^{*}(T_{i})<\cdots<F^{*}(T_{\lfloor\frac{k+1}{2}\rfloor}).$
###### Lemma 2.7.
Given a tree $T$ with $uv\in E_{T}$ and $d_{T}(u)=1$, one has
* (i)
([8]) $f_{T}(u)\leqslant f_{T}(v)$ with equality if and only if $T\cong
K_{2}$.
* (ii)
$f_{T}^{*}(u)\leqslant f_{T}^{*}(v)$ with equality if and only if $T\cong
K_{2}$.
###### Proof.
If $T\cong K_{2},$ it’s routine to check that $f_{T}^{*}(u)=f_{T}^{*}(v)=1$.
In what follows, we consider that $T\not\cong K_{2}.$ Note that $uv$ is a
pendant edge, the map
$f:\,\mathscr{S}^{*}(T,u)\rightarrow\mathscr{S}^{*}(T-u,v)$ that sends each
$T$ to $T-u$ is a bijection. On the other hand, by Lemma 2.3, we have
$|\mathscr{S}^{*}(T-u,v)|<|\mathscr{S}^{*}(T,v)|$, i.e.,
$f_{T-u}^{*}(v)<f_{T}^{*}(v)$, hence our results follows immediately. ∎
## 3 Three transformations on trees
In this section, we introduce three transformations on trees, which will be
used to prove our main results.
###### Definition 1.
Let $T^{\prime}$ (resp. $T^{\prime\prime}$) be a tree with $u\in
V_{T^{\prime}}$ (resp. $v\in V_{T^{\prime\prime}}$), where
$|V_{T^{\prime\prime}}|=r+1\geqslant 2$. Let $T_{1}$ be a tree obtained from
$T^{\prime}$ and $T^{\prime\prime}$ by identifying vertices $u$ with $v$; see
Fig 3. In particular, if $T^{\prime\prime}\cong P_{r+1}$ with $v$ being an
endvertex, we may denote the resultant graph by $T_{2}$; see Fig 3. We say
that $T_{2}$ is an $A$-transformation of $T_{1}$ on $T^{\prime\prime}$.
Figure 3: Trees $T_{1}$ and $T_{2}$.
###### Lemma 3.1.
Let $T_{1}$ and $T_{2}$ be the trees defined as above. Then
* (i)
([22]) $F(T_{1})\geqslant F(T_{2})$ with equality if and only if
$T^{\prime\prime}=P_{r+1}$ with $d_{T^{\prime\prime}}(v)=1.$
* (ii)
$F^{*}(T_{1})\geqslant F^{*}(T_{2})$ with equality if and only if
$T^{\prime\prime}=P_{r+1}$ with $d_{T^{\prime\prime}}(v)=1.$
###### Proof.
In view of the proof of Lemma 2.4, let $T^{\prime}$ be $X$ and
$T^{\prime\prime}$ (resp. $P_{r+1}$) be $T$ in Lemma 2.4. Then we have
$\displaystyle F^{*}(T_{2})=$ $\displaystyle
f_{T^{\prime}}(u)f_{P_{r+1}}^{*}(v)-f_{T^{\prime}}^{*}(u)(f_{P_{r+1}}(v)-f_{P_{r+1}}^{*}(v))+F^{*}(P_{r+1})-f_{P_{r+1}}(v)+F^{*}(T^{\prime}-u)$
$\displaystyle=$
$\displaystyle(f_{T^{\prime}}(u)-f_{T^{\prime}}^{*}(u))+(f_{T^{\prime}}^{*}(u)-1)(r+1)+F^{*}(P_{r+1})+F^{*}(T^{\prime}-u),$
(3.1) $\displaystyle F^{*}(T_{1})=$ $\displaystyle
f_{T^{\prime}}(u)f_{T^{\prime\prime}}^{*}(v)+f_{T^{\prime}}^{*}(u)(f_{T^{\prime\prime}}(v)-f_{T^{\prime\prime}}^{*}(v))+\left\\{\begin{aligned}
F^{*}(T^{\prime\prime})-f_{T^{\prime\prime}}(v)+F^{*}(T^{\prime}-u)&\ \ \ \
v\in PV(T^{\prime\prime}),\\\
F^{*}(T^{\prime\prime})-f_{T^{\prime\prime}}^{*}(v)+F^{*}(T^{\prime}-u)&\ \ \
\ v\not\in PV(T^{\prime\prime})\end{aligned}\right.$ $\displaystyle=$
$\displaystyle(f_{T^{\prime}}(u)-f_{T^{\prime}}^{*}(u))f_{T^{\prime\prime}}^{*}(v)+f_{T^{\prime}}^{*}(u)f_{T^{\prime\prime}}(v)+\left\\{\begin{aligned}
F^{*}(T^{\prime\prime})-f_{T^{\prime\prime}}(v)+F^{*}(T^{\prime}-u)&\ \ \ \
v\in PV(T^{\prime\prime}),\\\
F^{*}(T^{\prime\prime})-f_{T^{\prime\prime}}^{*}(v)+F^{*}(T^{\prime}-u)&\ \ \
\ v\not\in PV(T^{\prime\prime})\end{aligned}\right.$ $\displaystyle\geqslant$
$\displaystyle(f_{T^{\prime}}(u)-f_{T^{\prime}}^{*}(u))f_{T^{\prime\prime}}^{*}(v)+(f_{T^{\prime}}^{*}(u)-1)f_{T^{\prime\prime}}(v)+F^{*}(T^{\prime\prime})+F^{*}(T^{\prime}-u)$
(3.2) $\displaystyle\geqslant$
$\displaystyle(f_{T^{\prime}}(u)-f_{T^{\prime}}^{*}(u))+(f_{T^{\prime}}^{*}(u)-1)f_{T^{\prime\prime}}(v)+F^{*}(T^{\prime\prime})+F^{*}(T^{\prime}-u)$
(3.3) $\displaystyle\geqslant$
$\displaystyle(f_{T^{\prime}}(u)-f_{T^{\prime}}^{*}(u))+(f_{T^{\prime}}^{*}(u)-1)(r+1)+F^{*}(P_{r+1})+F^{*}(T^{\prime}-u)$
(3.4) $\displaystyle=$ $\displaystyle F^{*}(T_{2}).$
Equality holds in (3.2) if and only if $v$ is a leaf of $T^{\prime\prime}.$
Note that $f_{T^{\prime}}(u)>f_{T^{\prime}}^{*}(u)$, hence equality holds in
(3.3) if and only if $f^{*}_{T^{\prime\prime}}(v)=1$, i.e., $v$ is a leaf of a
path; Equality holds in (3.4) if and only if $T^{\prime\prime}\cong P_{r+1}$
and $v$ is a leaf. The last equality follows by (3.1). Hence,
$F^{*}(T_{1})=F^{*}(T_{2})$ if and only if $T^{\prime\prime}=P_{r+1}$ and $v$
is one of its endvertices, as desired. ∎
###### Definition 2.
Let $T_{1}$ be the graph as depicted in Fig. 4, where $T^{\prime}$ (resp.
$T^{\prime\prime}$) is a tree with at least two vertices. Let $\hat{T}_{2}$ be
a tree obtained from $T_{1}$ by deleting the edge $uv$ and identifying its
endvertices. Let $T_{2}$ be the tree obtained from $\hat{T}_{2}$ by attaching
an pendant edge to $u;$ see Fig. 4. We call the procedure constructing $T_{2}$
from $T_{1}$ the $B$-transformation of $T_{1}$.
Figure 4: Trees $T_{1},\hat{T}_{2}$ and $T_{2}$.
By Lemmas 2.4 and 2.7, the following lemma follows immediately.
###### Lemma 3.2.
Let $T_{1}$ and $T_{2}$ be the trees defined as above, we have
$F(T_{1})<F(T_{2})$ and $F^{*}(T_{1})<F^{*}(T_{2})$.
Figure 5: $C$-transformation on $v$.
###### Definition 3.
Let $T$ be an arbitrary tree, rooted at a center vertex $u$ and let $v$ be a
vertex with degree $m+1$. Suppose that $wv\in E_{T}$ and $P_{T}(u,w)\subset
P_{T}(u,v)$(we call $w$ the parent of $v$ in $T$) and that
$T_{1},T_{2},\dots,T_{m}$ are subtrees under $v$ with root vertices
$v_{1},v_{2},\dots v_{m}$, such that the tree $T_{m}$ is actually a path. We
form a tree $T_{0}$ by removing the edges $vv_{1},vv_{2},\dots vv_{m-1}$ from
$T$ and adding new edges $wv_{1},wv_{2},\dots wv_{m-1}$; see Fig. 5. If $v$ is
not a center vertex, we say that $T_{0}$ is a $C$-transformation of $T$. And
if $w$ and $v$ are both center of $T$ with $d_{T}(w)>2$, we say that $T_{0}$
is a $C^{\prime}$-transformation of $T$.
This transformation preserves the number of pendant vertices in a tree $T$,
and does not increase its diameter.
###### Lemma 3.3.
Let $T$ and $T_{0}$ be the trees defined as above, we have $F(T)<F(T_{0})$ and
$F^{*}(T)<F^{*}(T_{0})$.
###### Proof.
Let $W$ be the component that contains $v$ in
$T-\\{vv_{1},vv_{2},\ldots,vv_{m-1}\\}$ and $X$ be the component that contains
$v$ in $T-\\{w,v_{m}\\}$. Now we consider $f_{W}(w),f_{W}(v),f_{W}^{*}(w)$ and
$f_{W}^{*}(v)$. It is routine to check that
$f_{W}(w)=f_{W-v}(w)+f_{W}(w*v),f_{W}(v)=f_{W-w}(v)+f_{W}(w*v).$
Hence,
$f_{W}(w)-f_{W}(v)=f_{W-v}(w)-f_{W-w}(v)=f_{W-v}(w)-|V_{T_{m}}|-1.$
If $v$ is not a center of $T$ and its parent is $w$, then there is a proper
subtree of the component that contains $w$ in $W-v$, say $T^{\prime}$, with
$T^{\prime}\cong P_{|V_{T_{m}}|+1}$. Hence, we have
$f_{W-v}(w)>f_{T^{\prime}}(w)\geqslant|V_{T_{m}}|+1,$ i.e.,
$f_{W}(w)>f_{W}(v)$. By Lemma 2.4 we have $F(T_{0})>F(T).$
Note that neither $w$ nor $v$ is a leaf of $W$, hence by a similar discussion
as above we also have
$f_{W}^{*}(w)-f_{W}^{*}(v)=f_{W-v}^{*}(w)-f_{W-w}^{*}(v)=f_{W-v}^{*}(w)-1\geqslant
0,$
i.e., $f_{W}^{*}(w)\geqslant f_{W}^{*}(v)$. By Lemma 2.4 we have
$F^{*}(T_{0})>F^{*}(T)$. If $w=u$ is the center of $T$ with $d_{T}(w)>2$, then
we can also have a proper subtrees of the component that contains $w$ in $W-v$
say $T^{\prime\prime}$ with $T^{\prime\prime}\cong P_{|V_{T_{m}}|+1}$. By a
similar discussion as above, we can also have
$F(T_{0})>F(T),F^{*}(T_{0})>F^{*}(T).$ This completes the proof. ∎
## 4 Enumeration of subtrees of some types of trees
In this section, we determine sharp upper (or lower) bound on the total number
of subtrees (or leaf containing subtrees) of some type of trees.
The matching number of a graph $G$ is the maximum size of an independent
(pair-wise nonincident) set of edges of $G$ and will be denoted by $q(G)$. Let
$\mathscr{M}_{n,q}$ be the set of all $n$-vertex trees with matching number
$q$. Let $A(n,q)$ be the tree that is obtained by attaching $q-1$ pendant
edges to $q-1$ pendant vertices of the star $K_{1,n-q}$. It is routine to
check that $A(n,q)\in\mathscr{M}_{n,q}$. Given a vertex $w$ in $G$, call $w$ a
perfectly matched vertex if it is matched in any maximum matching of $G$.
###### Theorem 4.1.
Among $\mathscr{M}_{n,q}$ precisely the graph $A(n,q)$, which has
$2^{n-2q+1}\cdot 3^{q-1}+n+q-2$ subtrees, maximizes the total number of
subtrees and has $2^{n-2q+1}\cdot 3^{q-1}-2^{q-1}+n-1$ leaf containing
subtrees, maximizes the total number of leaf containing subtrees.
###### Proof.
Choose $T$ in $\mathscr{M}_{n,q}$ such that its total number of subtrees
(resp. leaf containing subtrees) is as large as possible. If $T$ contains a
pendant path of length $p>2$, say $v_{1}v_{2}v_{3}\ldots v_{p}v$ with
$v_{1}\in PV(T)$ and $d_{T}(v)\geqslant 3$, then
$f_{T-v_{2}-v_{1}}(v_{3})<f_{T-v_{2}-v_{1}}(v_{4}),\,f^{*}_{T-v_{2}-v_{1}}(v_{3})<f^{*}_{T-v_{2}-v_{1}}(v_{4})$
by Lemma 2.7. Let
$T_{0}=T-v_{2}v_{3}+v_{2}v.$
It is routine to check that $T_{0}$ is in $\mathscr{M}_{n,q}$. By Lemma 2.4 we
get $F(T)<F(T_{0}),F^{*}(T)<F^{*}(T_{0})$, a contradiction. Hence, any pendant
path contained in $T$ is of length at most 2.
If there exists a non-center vertex $v\in V_{T}$ such that $T$ contains $r$
pendant edges and $s$ pendant paths of length 2 attached to $v$, then assume
that $w$ is the parent of $v$. Consider the following possible cases.
$\bullet$ $s=0$ and $w$ is perfectly matched. Apply $C$-transformation at $v$
once, and get $r-1$ pendant edges and one pendant path $P_{3}$ attached at $w$
in the resultant graph, say $\hat{T}$. It is routine to check that $\hat{T}$
is in $\mathscr{M}_{n,q}$. By Lemma 3.3 we get
$F(T)<F(\hat{T}),F^{*}(T)<F^{*}(\hat{T})$, a contradiction.
$\bullet$ $s=0$ and $w$ is not perfectly matched. Applying $B$-transformation
at the edge $wv$, we get $r+1$ pendant edges at $w$ in the resultant graph,
say $\hat{T}$. It is routine to check that $\hat{T}$ is in
$\mathscr{M}_{n,q}$. By Lemma 3.2 we get
$F(T)<F(\hat{T}),F^{*}(T)<F^{*}(\hat{T})$, a contradiction.
$\bullet$ $r=0$. Applying $C$-transformation at $v$, we get $s-1$ pendant
paths $P_{3}$’s and one pendant path $P_{4}$ attached at $w$ in the resultant
graph, say $\hat{T}$. It is routine to check that $\hat{T}$ is in
$\mathscr{M}_{n,q}$. By Lemma 3.3 we get
$F(T)<F(\hat{T}),F^{*}(T)<F^{*}(\hat{T})$, a contradiction.
$\bullet$ $r>0,s>0$ and $w$ is perfectly matched. Applying $C$-transformation
at $v$, we get $r-1$ pendant edges and $s+1$ pendant paths $P_{3}$’s attached
at $w$ in the resultant graph, say $\hat{T}$. It is routine to check that
$\hat{T}$ is in $\mathscr{M}_{n,q}$. By Lemma 3.3 we get
$F(T)<F(\hat{T}),F^{*}(T)<F^{*}(\hat{T})$, a contradiction.
$\bullet$ $r>0,s>0$ and $w$ is not perfectly matched. Applying
$B$-transformation at the edge $wv$, we get $r+1$ pendant edges and $s$
pendant paths $P_{3}$’s attached at $w$ in the resultant graph, say $\hat{T}$.
It is routine to check that $\hat{T}$ is in $\mathscr{M}_{n,q}$. By Lemma 3.2
we get $F(T)<F(\hat{T}),F^{*}(T)<F^{*}(\hat{T})$, a contradiction.
Hence, all the pendant paths of length at most 2 are attached only to the
centers of $T$. In order to characterize the structure of $T$, it suffices to
show that $T$ contains just one center whose degree is larger than $2$.
Otherwise, assume that $T$ contains two centers, say $c_{1},c_{2}$, with
$d_{T}(c_{1})>2$ and $d_{T}(c_{2})>2$. Apply$C^{\prime}$-transformation on
$c_{1}$ in $T$ to get a new tree, say $T^{\prime}$. It’s routine to check that
$T^{\prime}$ is in $\mathscr{M}_{n,q}$. By Lemma 3.3, we get
$F(T^{\prime})>F(T)$ and $F^{*}(T^{\prime})>F(T)$, a contradiction.
Therefore, we get that $T\cong A(n,q)$ with the center $c.$ Note that in
$A(n,q)$ there exist $n-2q+1$ pendant edges and $q-1$ pendant paths of length
2 attached to $c$, hence
$\displaystyle f_{A(n,q)}(c)$ $\displaystyle=$ $\displaystyle 2^{n-2q+1}\cdot
3^{q-1},$ $\displaystyle F(A(n,q)-c)$ $\displaystyle=$ $\displaystyle
F((n-2q+1)P_{1}\cup(q-1)P_{2})=n-2q+1+3(q-1)=n+q-2,$
and
$F^{*}(A(n,q))=F(A(n,q))-F(A(n,q)-PV(A(n,q)))=F(A(n,q))-F(K_{1,q-1}).$
This gives
$F(A(n,q))=f_{A(n,q)}(c)+F(A(n,q)-c)=2^{n-2q+1}\cdot 3^{q-1}+n+q-2$
and
$F^{*}(A(n,q))=2^{n-2q+1}\cdot 3^{q-1}+n+q-2-(2^{q-1}+q-1)=2^{n-2q+1}\cdot
3^{q-1}-2^{q-1}+n-1,$
as desired. ∎
Let $\mathscr{S}(n,\gamma)$ be the set of $n$-vertex trees with domination
number $\gamma$.
###### Theorem 4.2.
Among $\mathscr{S}(n,\gamma),$ the tree $A(n,\gamma)$ maximizes the total
number of subtrees (resp. leaf containing subtrees).
###### Proof.
It is known from [21] that $\gamma(G)\leqslant q(G),$ where $q(G)$ is the
matching number of $G$. In order to complete the proof, it suffices to show
the following claim.
###### Claim 1.
If $T_{0}\in\mathscr{S}(n,\gamma)$ maximizes the total number of subtrees
(resp. leaf containing subtrees), then we have $\gamma(T_{0})=q(T_{0})$.
###### Proof.
It suffices to show that $\gamma(T_{0})\geqslant q(T_{0})$. Otherwise, by the
definition of the set $\mathscr{T}(n,\gamma)$, we have
$q(T_{0})>\gamma(T_{0})=\gamma$. Assume that
$S=\\{v_{1},v_{2},\ldots,v_{\gamma}\\}$ is a dominating set with cardinality
$\gamma$. Then there exist $\gamma$ independent edges
$v_{1}v_{1}^{\prime},v_{2}v_{2}^{\prime},\ldots,v_{\gamma}v_{\gamma}^{\prime}$
in $T_{0}$. Note that $q(T_{0})>\gamma(T_{0})=\gamma$, there must exist
another edge, say $w_{1}w_{2}$, which is independent of each of edges
$v_{i}v_{i}^{\prime}$, $i=1,2,\ldots,\gamma$.
Figure 6: The structures of $T_{0}$ and $T_{0}^{\prime}$ in Claim 1
If the two vertices $w_{1},w_{2}$ is dominated by the same vertex $v_{i}\in
S$, then a triangle $C_{3}=w_{1}w_{2}v_{i}$ occurs. This is impossible because
of the fact that $T_{0}$ is a tree. Therefore $w_{1},w_{2}$ are dominated by
two different vertices from $S$. Without loss of generality, assume that
$w_{i}$ is dominated by the vertex $v_{i}$ for $i=1,2$ (see Fig. 6). Now we
construct a new tree $T_{0}^{\prime}\in\mathscr{T}(n,\gamma)$ by
$B$-transformation of $T_{0}$ on the edges $v_{1}w_{1}$ and $v_{2}w_{2}$,
respectively. By Lemma 3.2, we have
$F(T_{0})<F(T_{0}^{\prime}),F^{*}(T_{0})<F^{*}(T_{0}^{\prime})$, a
contradiction. This completes the proof of Claim 1. ∎
Theorem 4.2 follows immediately from Theorem 4.1 and Claim 1. ∎
Let $H\circ K_{1}$ be the graph obtained by attaching a leaf to each of the
vertices of the graph $H$.
###### Theorem 4.3.
Let $T\in\mathscr{S}(n,\frac{n}{2})(n\geqslant 4)$, then
$F(T)\geqslant 2^{\frac{n}{2}+2}-\frac{n}{2}-4,\ \ \ \ \ F^{*}(T)\geqslant
2^{\frac{n}{2}+2}-\frac{n}{2}-4-{{\frac{n}{2}+1}\choose{2}}.$ (4.1)
Each of the equalities in (4.1) holds if and only if $T\cong
P_{\frac{n}{2}}\circ K_{1}.$
###### Proof.
It is known [3, 20] that if $n=2\gamma$, then a tree $T$ belongs to
$\mathscr{S}(n,\gamma)$ if and only if there exists a tree $H$ of order
$\gamma=\frac{n}{2}$ such that $T=H\circ K_{1}$. Hence it suffices to show the
following fact.
###### Fact 2.
For any tree $T,$ one has
$F(T\circ K_{1})\geqslant F(P_{|V_{T}|}\circ K_{1}),\ \ \ F^{*}(T\circ
K_{1})\geqslant F^{*}(P_{|V_{T}|}\circ K_{1}).$ (4.2)
Each of the equalities in (4.2) holds if and only if $T\cong P_{|V_{T}|}.$
###### Proof.
For any $u$ in $V_{T}$ and $1\leqslant m\leqslant|V_{T}|$, let
$\mathscr{S}^{m}(T;u)$ denote the set of all $m$-vertex subtrees of a tree $T$
each of which contains $u.$ It is routine to check that
$\displaystyle F(T\circ K_{1})$ $\displaystyle=$
$\displaystyle\sum_{T_{1}\in\mathscr{S}(T)}2^{|V_{T_{1}}|}+|V_{T}|$
$\displaystyle=$
$\displaystyle\sum_{T_{1}\in\mathscr{S}(T-u)}2^{|V_{T_{1}}|}+\sum_{T_{1}\in\mathscr{S}(T;u)}2^{|V_{T_{1}}|}+|V_{T}|$
$\displaystyle=$
$\displaystyle\sum_{T_{1}\in\mathscr{S}(T-u)}2^{|V_{T_{1}}|}+\sum_{m=1}^{|V_{T}|}|\mathscr{S}^{m}(T;u)|2^{m}+|V_{T}|$
(4.4)
and
$\displaystyle F^{*}(T\circ K_{1})$ $\displaystyle=$ $\displaystyle F(T\circ
K_{1})-F(T)$ (4.5) $\displaystyle=$
$\displaystyle\sum_{T_{1}\in\mathscr{S}(T)}(2^{|V_{T_{1}}|}-1)+|V_{T}|$
$\displaystyle=$
$\displaystyle\sum_{T_{1}\in\mathscr{S}(T-u)}(2^{|V_{T_{1}}|}-1)+\sum_{m=1}^{|V_{T}|}|\mathscr{S}^{m}(T;u)|(2^{m}-1)+|V_{T}|.$
(4.6)
Assume that $T\not\cong P_{|V_{T}|}$. If $|V_{T}|=2$ or $3$, our result is
clearly true. If $|V_{T}|=4$, there exist only two trees, i.e., $P_{4}$ and
$K_{1,3}$, hence $T=K_{1,3}$. In this case, for any $u\in PV(T)$ we have
$|\mathscr{S}^{1}(T;u)|=|\mathscr{S}^{2}(T;u)|=|\mathscr{S}^{4}(T;u)|=1,|\mathscr{S}^{3}(T;u)|=2$
(4.7)
And for any $v\in PV(P_{4})$, we have
$|\mathscr{S}^{1}(P_{4};v)|=|\mathscr{S}^{2}(P_{4};v)|=|\mathscr{S}^{3}(P_{4};v)|=|\mathscr{S}^{4}(P_{4};u)|=1.$
(4.8)
Note that $P_{4}-u=K_{1,3}-v$, hence by (4.4), (4.6)-(4.8) we have
$F(K_{1,3}\circ K_{1})>F(P_{4}\circ K_{1}),F^{*}(K_{1,3}\circ
K_{1})>F^{*}(P_{4}\circ K_{1}).$
In what follows we assume that the inequalities hold in (4.2) for all trees of
order less than $|V_{T}|$. On the one hand, for any $u\in PV(T)$ and $v\in
PV(P_{|V_{T}|})$, we have
$F((T-u)\circ K_{1})\geqslant F((P_{|V_{T}|}-v)\circ K_{1}),F^{*}((T-u)\circ
K_{1})\geqslant F^{*}((P_{|V_{T}|}-v)\circ K_{1}),$ (4.9)
Each of the equalities in (4.9) holds if and only if $T-u\cong P_{|V_{T}|}-v$.
Hence by (4) and (4.5), we have
$\sum_{T_{1}\in\mathscr{S}(T-u)}2^{|V_{T_{1}}|}\geqslant\sum_{T_{1}\in\mathscr{S}(P_{|V_{T}|}-v)}2^{|V_{T_{1}}|},\sum_{T_{1}\in\mathscr{S}(T-u)}(2^{|V_{T_{1}}|}-1)\geqslant\sum_{T_{1}\in\mathscr{S}(P_{|V_{T}|}-v)}(2^{|V_{T_{1}}|}-1).$
(4.10)
On the other hand, it is easy to see that for any $w\in
PV(T)\setminus\\{u\\}$, $T-w\in\mathscr{S}^{|V_{T}|-1}(T;u)$, so we have
$|\mathscr{S}^{|V_{T}|-1}(T;u)|>1=|\mathscr{S}^{|V_{T}|-1}(P_{|V_{T}|};v)|.$
(4.11)
Furthermore, for $m=1,2,\ldots,|V_{T}|-2,|V_{T}|$,
$|\mathscr{S}^{m}(T;u)|\geqslant 1=|\mathscr{S}^{m}(P_{|V_{T}|};v)|.$ (4.12)
Hence, (4.2) follows by (4.4),(4.6),(4.10)-(4.12). This completes the proof of
Fact 1. ∎
Note that $|\mathscr{S}^{m}(P_{n};v)|=1$ for $m=1,2\ldots,n$, hence by (4) and
(4.4) we get
$F(P_{n}\circ
K_{1})=\sum_{T_{1}\in\mathscr{S}(P_{n-1})}2^{|V_{T_{1}}|}+\sum_{m=1}^{n}2^{m}+n=F(P_{n-1}\circ
K_{1})+2^{n+1}-1.$
Hence we have
$F(P_{n}\circ K_{1})=F(P_{1}\circ
K_{1})+\sum_{i=3}^{n+1}2^{i}-(n-1)=3+\sum_{i=3}^{n+1}2^{i}-(n-1)=2^{n+2}-n-4.(n\geqslant
2)$
and
$F^{*}(P_{n}\circ K_{1})=F(P_{n}\circ
K_{1})-F(P_{n})=2^{n+2}-n-4-{{n+1}\choose{2}}(n\geqslant 2).$
This completes the proof. ∎
Let $P_{k}(1^{a},1^{b})$ be a tree obtained by attaching $a$ and $b$ pendant
vertices to the two endvertices of $P_{k}$, respectively.
###### Theorem 4.4.
Let $T\in\mathscr{S}(n,2)$ with $n\geqslant 6$, then
$\displaystyle F(T)$ $\displaystyle\geqslant$ $\displaystyle
3\cdot\left(2^{\lfloor\frac{n-4}{2}\rfloor}+2^{\lceil\frac{n-4}{2}\rceil}\right)+2^{n-4}+n-1,$
(4.13) $\displaystyle F^{*}(T)$ $\displaystyle\geqslant$ $\displaystyle
3\cdot\left(2^{\lfloor\frac{n-4}{2}\rfloor}+2^{\lceil\frac{n-4}{2}\rceil}\right)+2^{n-4}+n-11.$
(4.14)
The equality in (4.13) (resp. (4.14)) holds if and only if $T\cong
P_{4}(1^{\lfloor\frac{n-4}{2}\rfloor},1^{\lceil\frac{n-4}{2}\rceil})$.
###### Proof.
When $n=6$, this theorem holds as $P_{6}\in\mathscr{S}(n,2)$. So we only
consider the case when $n\geqslant 7$. Choose $T\in\mathscr{S}(n,2)$ such that
its total number of subtrees (resp. leaf containing subtrees) is as small as
possible. Let $S=\\{w_{1},w_{2}\\}$ be a dominating set of $T$.
If $d_{T}(w_{1},w_{2})=1$, $T$ must be the form $P_{2}(1^{a},1^{b})$ with
$a+b=n-2$. Without loss of generality, assume that $a\leqslant b$. Note that
$T^{\prime}=P_{3}(1^{a},1^{b-1})\in\mathscr{S}(n,2)$, and by Lemma 3.2 we have
$F(T^{\prime})<F(T),F^{*}(T^{\prime})<F^{*}(T)$, a contradiction. By a similar
discussion we can show that $d_{T_{1}}(w_{1},w_{2})\not=2$. We omit the
procedure here.
If $d_{T}(w_{1},w_{2})\geqslant 4$, then there exists at least one vertex $x$
on $P_{T}(w_{1},w_{2})$ $x$ can not be dominated by $w_{1}$ or $w_{2}$, which
implies that $T\not\in\mathscr{S}(n,2)$. Hence we get $d_{T}(w_{1},w_{2})=3$.
That is to say, $T\cong P_{4}(1^{a},1^{b})$ with $a+b=n-4,1\leqslant
a\leqslant b$. (Note that $P_{4}(1^{0},1^{n-4})=P_{3}(1^{1},1^{n-5})$). Hence,
by direct computing we have
$\displaystyle F(P_{4}(1^{a},1^{b}))$ $\displaystyle=$ $\displaystyle
3\cdot(2^{a}+2^{b})+2^{n-4}+n-4+{3\choose{2}}=3\cdot(2^{a}+2^{b})+2^{n-4}+n-1,$
$\displaystyle F^{*}(P_{4}(1^{a},1^{b}))$ $\displaystyle=$ $\displaystyle
F(P_{4}(a,b))-F(P_{4})=3\cdot(2^{a}+2^{b})+2^{n-4}+n-11.$
Note that when $a=b-1,b$, our results hold immediately. Hence, we consider
$a\leqslant b-2$ in what follows. It is routine to check that
$2^{a}+2^{b}>2^{a+1}+2^{b-1}>\cdots>2^{\lfloor\frac{n-4}{2}\rfloor}+2^{\lceil\frac{n-4}{2}\rceil}.$
Hence we have
$\displaystyle
F(P_{4}(1^{a},1^{b}))>F(P_{4}(1^{a+1},1^{b-1}))>\cdots>F(P_{4}(1^{\lfloor\frac{n-4}{2}\rfloor},1^{\lceil\frac{n-4}{2}\rceil})),$
$\displaystyle
F^{*}(P_{4}(1^{a},1^{b}))>F^{*}(P_{4}(1^{a+1},1^{b-1}))>\cdots>F^{*}(P_{4}(1^{\lfloor\frac{n-4}{2}\rfloor},1^{\lceil\frac{n-4}{2}\rceil})).$
This completes the proof. ∎
###### Theorem 4.5.
Let $\Delta$ be a positive integer more than two, and let $T$ be an $n$-vertex
tree with maximum degree at least $\Delta$. Then
$F^{*}(T)\geqslant(n-\Delta+1)\cdot 2^{\Delta-1}+\Delta-1.$ The equality holds
if and only if $T\cong T_{n,\Delta}$, where $T_{n,\Delta}$ is obtained from
$P_{n-\Delta+1}$ by attaching $\Delta-1$ pendant vertices to one endvertex of
$P_{n-\Delta+1}$; see Fig. 7(a).
###### Proof.
Choose an $n$-vertex tree $T$ with maximum degree at least $\Delta$ such that
its total number of leaf containing subtrees is as small as possible. Then
there exists a vertex $u$ in $V_{T}$ such that $d_{T}(u)\geqslant\Delta$.
Without loss of generality, we assume that
$\\{v_{1},v_{2},\ldots,v_{\Delta-1}\\}\subseteq N_{T}(u)$. Obviously, the
graph $T-\\{uv_{1},uv_{2},\ldots,uv_{\Delta-1}\\}$ contains $\Delta$
components $T_{1},T_{2},\ldots,T_{\Delta-1},T_{\Delta}$, where $T_{i}$
contains vertex $v_{i}$ for $i=1,2,\ldots,\Delta-1$ and $T_{\Delta}$ contains
at least two vertices with $u\in V_{T_{\Delta}};$ see Fig. 7(b).
Figure 7: Trees $T,T^{*}$ and $T_{n,\Delta}$ in the proof of Theorem 4.5.
Next we are to show that each $T_{i}$ is a path for $i=1,2\ldots,\Delta$. In
fact, if there exists an $i\in\\{1,2\ldots,\Delta\\}$ such that $T_{i}$ is not
a path. Applying $A$-transformations of $T$ on $T_{i}$ to get a tree, say
$\hat{T}$. By Lemma 3.1 we have $F^{*}(\hat{T})<F^{*}(T)$, a contradiction.
Hence $T\cong T^{*}$, where $T^{*}$ is depicted in Fig. 7(c).
Now we show that for any $u_{i},u_{j}\in PV(T)$, $u_{i}u\in E_{T}$ or
$u_{j}u\in E_{T}$. In fact, if there exists two pendant vertices, say
$u_{1},u_{2}$, such that $u_{1}u,u_{2}u\not\in E_{T}$. Let
$P_{T}(u_{1},u_{2})=u_{1}w_{1}w_{2}\ldots w_{r}u_{2}$ with
$u,v_{1},v_{2}\in\\{w_{1},w_{2},\ldots,w_{r}\\}$ and $u\not=w_{1},w_{r}$. Let
$T^{**}=T-\\{uv_{3},uv_{4},\ldots,uv_{\Delta}\\}+\\{w_{1}v_{3},w_{1}v_{4},\ldots,w_{1}v_{\Delta}\\}$.
By Lemma 2.6, $F^{*}(T^{**})<F^{*}(T)$, a contradiction. Hence, $T\cong
T_{n,\Delta}$.
Note that $f_{T_{n,\Delta}}(u)=(n-\Delta+1)\cdot
2^{\Delta-1},F(T_{n,\Delta}-u)=F((\Delta-1)P_{1}\cup P_{n-\Delta})$ and
$F(H(T_{n,\Delta}))=F(T_{n,\Delta}-PV(T_{n,\Delta}))=F(P_{n-\Delta}),$ hence
we have
$F(T_{n,\Delta})=f_{T_{n,\Delta}}(u)+F(T_{n,\Delta}-u)=(n-\Delta+1)\cdot
2^{\Delta-1}+\Delta-1+{n-\Delta+1\choose{2}}.$
So we have
$\displaystyle F^{*}(T_{n,\Delta})$ $\displaystyle=$ $\displaystyle
F(T_{n,\Delta})-F(H(T_{n,\Delta}))=(n-\Delta+1)\cdot 2^{\Delta-1}+\Delta-1,$
as desired. ∎
###### Theorem 4.6.
Let $\Delta$ be a positive integer more than two, and let $T$ be an $n$-vertex
tree with maximum degree at least $\Delta$ having a perfect matching. Then
$\displaystyle F(T)$ $\displaystyle\geqslant$ $\displaystyle
2(n-2\Delta+3)\cdot 3^{\Delta-2}+3\cdot\Delta-5+{n-2\Delta+3\choose{2}},$
(4.15) $\displaystyle F^{*}(T)$ $\displaystyle\geqslant$ $\displaystyle
2(n-2\Delta+3)\cdot 3^{\Delta-2}-(n-2\Delta+2)\cdot 2^{\Delta-2}+n-1.$ (4.16)
Equality holds in (4.15) (resp. (4.16)) if and only if $T\cong
T_{n,\Delta}^{\prime}$, where $T_{n,\Delta}^{\prime}$ is the tree obtained
from $P_{n-2\Delta+1}$ by attaching $(\Delta-2)\,\,P_{3}$’s and one $P_{2}$
to one endvertex of $P_{n-2\Delta+3};$ see Fig. 8.
Figure 8: Tree $T_{n,\Delta}^{\prime}$.
###### Proof.
Choose an $n$-vertex tree $T$ with maximum degree at least $\Delta$ having a
perfect matching such that its total number of subtree (resp. leaf containing
subtrees) is as small as possible. By a similar discussion as in the proof of
Theorem 4.5, we can obtain that $T$ is the graph depicted in Fig. 7(b).
Note that for any two $n$-vertex tree $T_{1}$ and $T_{2}$, if $T_{2}$ is an
$A$-transformation of $T_{1}$, then the maximum matching number of $T_{2}$ is
no less than that of $T_{1}$. Hence, by a similar discussion as in Theorem
4.5, we have $T\cong T^{*}$ as depicted in Fig. 7(c) and $T^{*}$ contains a
perfect matching, say $M$. Note that for the vertex $u$ in $T^{*}$, $u$ is
saturated by $M$, hence without loss of generality we assume that $uv_{1}\in
M$. Then we have $|V_{T_{1}}|,|V_{T_{\Delta}}|$ are odd and
$|V_{T_{2}}|,\ldots|V_{T_{\Delta-1}}|$ are even.
Next we show that $v_{1}\in PV(T)=\\{u_{1},u_{2}\ldots u_{\Delta}\\}$. Suppose
that $P_{T}(u_{1},u_{\Delta})=u_{1}w_{1}w_{2}\ldots w_{r}u_{\Delta}$ with
$u,v_{1},v_{\Delta}\in\\{w_{1},w_{2}\ldots,w_{r}\\}$ and $u\not=w_{1},w_{r}$.
Let
$\hat{T}=T-\\{uv_{2},uv_{3},\ldots,uv_{\Delta-1}\\}+\\{w_{1}v_{2},w_{1}v_{3},\ldots,w_{1}v_{\Delta-1}\\}$,
$M$ is also a perfect matching of $\hat{T}$. By Lemma 2.6, we have
$F(T)>F(\hat{T}),F^{*}(T)>F^{*}(\hat{T})$, a contradiction. Hence we have
$v_{1}\in PV(T)$. For convenience, let $u_{1}:=v_{1}.$
Now we show that for any $u_{i},u_{j}\in PV(T)\setminus\\{u_{1}\\}$,
$d_{T}(u_{i},u)=2$ or $d_{T}(u_{j},u)=2$. Note that $T$ contains a perfect
matching, hence $d_{T}(u_{i},u)\geqslant 2$ for $u_{i}\in
PV(T)\setminus\\{u_{1}\\}$. If there exist two vertices in
$PV(T)\setminus\\{u_{1}\\}$, say $u_{2},u_{3}$, such that
$d_{T}(u_{2},u)>2,\,d_{T}(u_{3},u)>2$. Denote the unique path connecting
$u_{2},u_{3}$ by $P_{T}(u_{2},u_{3})=u_{2}s_{1}s_{2}\ldots s_{t-1}s_{t}u_{3}$
with $u=s_{k}$, where $k\not=1,2,t-1,t$. Let $T^{**}=T-\\{uv_{1},uv_{4},\ldots
u_{\Delta}\\}+\\{s_{2}v_{1},s_{2}v_{4},\ldots s_{2}v_{\Delta}\\}$. Note that
$M-uv_{1}+s_{2}v_{1}$ is a perfect matching of $T^{**}$. By Lemma 2.6, we have
$F(T)>F(T^{**}),F^{*}(T)>F^{*}(T^{**})$, a contradiction. So we have $T\cong
T_{n,\Delta}^{\prime}$; see Fig. 8.
Note that $f_{T_{n,\Delta}^{\prime}}(u)=2(n-2\Delta+3)\cdot
3^{\Delta-2},F(T_{n,\Delta}^{\prime}-u)=F((\Delta-2)P_{2}\cup P_{1}\cup
P_{n-2\Delta+2}),$ hence
$F(T_{n,\Delta}^{\prime})=2(n-2\Delta+3)\cdot
3^{\Delta-2}+3(\Delta-2)+1+{n-2\Delta+3\choose{2}}.$ (4.17)
Note that
$H(T_{n,\Delta}^{\prime})=T_{n,\Delta}^{\prime}-PV(T_{n,\Delta}^{\prime})=T_{n-\Delta,\Delta-1}.$
Hence,
$F(H(T_{n,\Delta}^{\prime}))=(n-2\Delta+2)\cdot
2^{\Delta-2}+\Delta-2+{n-2\Delta+2\choose{2}}.$
Together with (4.17) and Fact 1, we have
$F^{*}(T_{n,\Delta}^{\prime})=F(T_{n,\Delta}^{\prime})-F(H(T_{n,\Delta}^{\prime}))=2(n-2\Delta+3)\cdot
3^{\Delta-2}-(n-2\Delta+2)\cdot 2^{\Delta-2}+n-1,$
as desired. ∎
Let $\mathscr{S}_{n}^{k}$ be the set of all $n$-vertex trees with $k$ leaves
($2\leqslant k\leqslant n-1$). A spider is a tree with at most one vertex of
degree more than 2, called the hub of the spider (if no vertex of degree more
than two, then any vertex can be the hub). A leg of a spider is a path from
the hub to a leaf. Let $T_{n}^{k}$ be an $n$-vertex tree with $k$ legs
satisfying all the lengths of $k$ legs, say $l_{1},l_{2},\ldots,l_{k}$, are
almost equal lengths, i.e., $|l_{i}-l_{j}|\leqslant 1$ for $1\leqslant
i,j\leqslant k.$ It is easy to see that $T_{n}^{k}\in\mathscr{S}_{n}^{k}$ and
$l_{i}+l_{j}\in\\{2\lfloor\frac{n-1}{k}\rfloor,\lfloor\frac{n-1}{k}\rfloor+\lceil\frac{n-1}{k}\rceil,2\lceil\frac{n-1}{k}\rceil\\}$,
where $1\leqslant i,j\leqslant k.$
###### Theorem 4.7.
Among ${\mathscr{S}}_{n}^{k}$ with $n\geqslant 2$, precisely the graph
$T_{n}^{k}$, has
$\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}-\left(\left\lfloor\frac{n-1}{k}\right\rfloor\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil\right)^{j}+i\left\lfloor\frac{n-1}{k}\right\rfloor+j\left\lceil\frac{n-1}{k}\right\rceil$
leaf containing subtrees, maximizes the total number of leaf containing
subtrees, where $i+j=k$ and $n-1\equiv j\pmod{k}$.
###### Proof.
Choose $T\in\mathscr{S}_{n}^{k}$ such that its total number of leaf containing
subtrees is as large as possible. If $k=2$ or, $n-1$, it is easy to see that
$\mathscr{S}_{n}^{k}=\\{T_{n}^{k}\\}$, our result follows immediately. Hence,
in what follows we consider $2<k<n-1$. For convenience, let $W$ be the set of
vertex of degree larger than 2 in $T$.
First we show that for any $v\in W$, $v$ is a center of $T$. Otherwise, apply
a $C$ transformation to $v$ of $T$ to get a new tree $T^{\prime}$. It’s
straightforward to check that $T^{\prime}\in\mathscr{S}_{n}^{k}$. By Lemma
3.3, we have $F^{*}(T)<F^{*}(T^{\prime})$, a contradiction to the choice of
$T$. Hence, for any vertex $w\in V_{T}$ that is not the center of $T$, we have
$d_{T}(w)\leqslant 2$. If there are two center vertices $c_{1}$ and $c_{2}$ in
$W$, apply a $C^{\prime}$-transformation to $c_{1}$ of $T$ to get a new tree
$T^{\prime}$. Then $T^{\prime}$ is a spider and by Lemma 3.3 we have
$F^{*}(T^{\prime})>F^{*}(T)$, a contradiction.
Now suppose $c$ is the only vertex in $W$. We are to show that for any
$u_{i},u_{j}\in PV(T)$, one has $|d_{T}(c,u_{i})-d_{T}(c,u_{j})|\leqslant 1.$
Assume to the contrary that there exist two pendant vertices, say
$u_{t},u_{l}$, in $PV(T)$ such that
$|d_{T}(c,u_{t})-d_{T}(c,u_{l})|\geqslant 2.$ (4.18)
Denote the unique path connecting $u_{t}$ and $u_{l}$ by
$P_{s}=w_{1}w_{2}\ldots w_{i-1}w_{i}w_{i+1}\ldots w_{s},$ where
$w_{1}=u_{t},w_{s}=u_{l}$ and $w_{i}=c,1\leqslant i\leqslant s$. In view of
(4.18), we have
$\text{$c=w_{i}\neq w_{\lfloor\frac{s+1}{2}\rfloor}$\ \ \ and\ \ \
$c=w_{i}\neq w_{\lceil\frac{s+1}{2}\rceil}$}.$
Hence, by Lemma 2.6 there exists an $n$-vertex tree
$T^{\prime}\in\mathscr{S}_{n}^{k}$ such that $F^{*}(T)<F^{*}(T^{\prime})$, a
contradiction to the choice of $T$. So we have $T\cong T_{n}^{k}$.
Furthermore, we know from ([8]) that
$F(T_{n}^{k})=\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}+i{\lfloor\frac{n-1}{k}\rfloor+1\choose{2}}+j{\lceil\frac{n-1}{k}\rceil+1\choose{2}},$
(4.19)
where $i+j=k$ and $n-1\equiv j\pmod{k}$.
By Fact 1,
$F^{*}(T_{n}^{k})=F(T_{n}^{k})-F(T_{n}^{k}-PV(T_{n}^{k}))=F(T_{n}^{k})-F(T_{n-k}^{k}).$
Hence in view of (4.19),
$\begin{split}F^{*}(T_{n}^{k})=&\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}+i{\lfloor\frac{n-1}{k}\rfloor+1\choose{2}}+j{\lceil\frac{n-1}{k}\rceil+1\choose{2}}\\\
&-\left[\left(\left\lfloor\frac{n-1}{k}\right\rfloor\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil\right)^{j}+i{\lfloor\frac{n-1}{k}\rfloor\choose{2}}+j{\lceil\frac{n-1}{k}\rceil\choose{2}}\right]\\\
=&\left(\left\lfloor\frac{n-1}{k}\right\rfloor+1\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil+1\right)^{j}-\left(\left\lfloor\frac{n-1}{k}\right\rfloor\right)^{i}\left(\left\lceil\frac{n-1}{k}\right\rceil\right)^{j}+i\left\lfloor\frac{n-1}{k}\right\rfloor+j\left\lceil\frac{n-1}{k}\right\rceil,\end{split}$
where $i+j=k$ and $n-1\equiv j\pmod{k}$.
This completes the proof. ∎
Let $\mathscr{S}_{n,d}$ denote the set of all $n$-vertex trees of diameter
$d$. Let $\hat{T}_{n,k}^{d}$ be the $n$-vertex tree obtained from
$P_{d+1}=v_{1}v_{2}\ldots v_{d}v_{d+1}$ by attaching $n-d-1$ pendant edges to
$v_{k}$; see Fig. 9.
Figure 9: Tree $\hat{T}_{n,k}^{d}.$
###### Theorem 4.8.
For any $n\geqslant 2$, precisely the graph $\hat{T}_{n,i}^{d}$, which has
$2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)+{\lfloor\frac{d}{2}\rfloor+1\choose{2}}{\lceil\frac{d}{2}\rceil+1\choose{2}}+n-d-1-{d\choose{2}}$
leaf containing subtrees, minimizes the total number of leaf containing
subtrees among $\mathscr{S}_{n,d}$, where $i=\lfloor\frac{d}{2}\rfloor+1$ or
$i=\lceil\frac{d}{2}\rceil+1$.
###### Proof.
If $T\in\mathscr{S}_{n,d}$, it is easy to see that $|V_{H(T)}|\geqslant d-1$.
By Lemma 2.1 we have $F(H)\geqslant F(P_{d-1})$ with equality if and only if
$H\cong P_{d-1}$, which is equivalent to that $T$ is a caterpillar tree of
diameter $d$. On the one hand, it is known ([22]) that $F(T)\leqslant
F(\hat{T}_{n,i}^{d})$ with equality if and only if $T\cong\hat{T}_{n,i}^{d}$,
where $i=\lfloor\frac{d}{2}\rfloor+1$ or $i=\lceil\frac{d}{2}\rceil+1$.
Together with Fact 1, for any $T\in\mathscr{S}_{n,d}$, we have
$F^{*}(T)=F(T)-F(H(T))\leqslant
F(\hat{T}_{n,i}^{d})-F(P_{d-1})=F^{*}(\hat{T}_{n,i}^{d})$ (4.20)
with equality if and only if $T\cong\hat{T}_{n,i}^{d}$ for
$i=\lfloor\frac{d}{2}\rfloor+1$ or $i=\lceil\frac{d}{2}\rceil+1$. On the other
hand, it is known ([8]) that
$F(\hat{T}_{n,i}^{d})=2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)+{\lfloor\frac{d}{2}\rfloor+1\choose{2}}{\lceil\frac{d}{2}\rceil+1\choose{2}}+n-d-1.$
Together with (4.20), we have
$F^{*}(\hat{T}_{n,i}^{d})=2^{n-d-1}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right)+{\lfloor\frac{d}{2}\rfloor+1\choose{2}}{\lceil\frac{d}{2}\rceil+1\choose{2}}+n-d-1-{d\choose{2}}.$
This completes the proof. ∎
## 5 Concluding remarks
Du and Zhou [2] characterized the extremal trees with matching number $q$ that
minimize the Wiener index; in this paper we show the counterparts of these
results for the total number of subtrees of $n$-vertex trees with matching
number $q$. In view of Theorem 4.2, we conjecture that there exist the
counterparts of these results for the Wiener index among the $n$-vertex trees
with domination number $\gamma$. Furthermore, for the Wiener index, sharp
upper and lower bounds of trees with given degree sequence are determined; see
[17, 23, 24]. It is natural for us to determine sharp upper and lower bounds
on the total number of subtrees of trees with given degree sequence. It is
difficult but interesting and it is still open. We leave these problems for
future study.
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* [16] A. Vince, H. Wang, The average order of a subtree of a tree, J. Combin. Theory Ser. B 100 (2) (2010) 161-170.
* [17] H. Wang, The extremal values of the Wiener index of a tree with given degree sequence, Discrete Appl. Math. 156 (14) (2008) 2647-2654.
* [18] H. Wang, Subtrees of Trees, Wiener Index and Related Problems, PhD Thesis, Department of Mathematics, University of South Carolina, 2005\.
* [19] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17-20.
* [20] B. Xu, E.J. Cockayne, T.W. Haynes, S.T. Hedetniemi, S.C. Zhou, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216 (2000) 1-10.
* [21] K.X. Xu, L.H. Feng, Extremal energies of trees with a given domination number, Linear Algebra Appl. 435 (2011) 2382-2393.
* [22] W.G. Yan, Y.N. Ye, Enumeration of subtrees of trees, Theoret. Comput. Sci. 369 (2006) 256 - 268.
* [23] X.D. Zhang, Y. Liu, M.X. Han, Maximum Wiener index of trees with given degree sequence, MATCH Commun. Math. Comput. Chem. 64 (3) (2010) 661-682.
* [24] X.D. Zhang, Q.Y. Xiang, L.Q. Xu, R.Y. Pan, The Wiener index of trees with given degree sequences, MATCH Commun. Math. Comput. Chem. 60 (2) (2008) 623-644.
|
arxiv-papers
| 2012-06-14T00:38:25 |
2024-09-04T02:49:31.751481
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuchao Li, Shujing Wang",
"submitter": "Shuchao Li",
"url": "https://arxiv.org/abs/1206.2975"
}
|
1206.3027
|
# Social Networks, Functional Differentiation of Society, and Data Protection
Jörg Pohle Humboldt University at Berlin, Institute of Computer Science,
Computer Science in Education and Society, pohle@informatik.hu-berlin.de.
###### Abstract
Abstract. Most scholars, politicians, and activists are following
individualistic theories of privacy and data protection. In contrast, some of
the pioneers of the data protection legislation in Germany like Adalbert
Podlech, Paul J. Müller, and Ulrich Dammann used a systems theory approach.
Following Niklas Luhmann, the aim of data protection is (1) maintaining the
functional differentiation of society against the threats posed by the
possibilities of modern information processing, and (2) countering undue
information power by organized social players. It could be, therefore, no
surprise that the first data protection law in the German state of Hesse
contained rules to protect the individual as well as the balance of power
between the legislative and the executive body of the state.
Social networks like Facebook or Google+ do not only endanger their users by
exposing them to other users or the public. They constitute, first and
foremost, a threat to society as a whole by collecting information about
individuals, groups, and organizations from different social systems and
combining them in a centralized data bank. They transgress the boundaries
between social systems that act as a shield against total visibility and
transparency of the individual and protect the freedom and the autonomy of the
people. Without enforcing structural limitations on the organizational use of
collected data by the social network itself or the company behind it, social
networks pose the worst totalitarian peril for western societies since the
fall of the Soviet Union.
###### Abstract
Keywords: privacy, data protection, right of personality, functional
differentiation, separation of duties, balance of power, social networks
## 1 Introduction
General social networking services111Specialized networks like LinkedIn and
XING will not be surveyed. are used by more than a billion users worldwide.
They produce a large volume of data about their users, either collected
directly from the individuals themselves or derived from the social graph
around each of them. The information contained in the data relates to all
areas of life, all social systems, and the past, the present, and the future
alike. Additionally, they often also relate to individuals unaffiliated with
said social networks. Social networks are often harshly criticized from
privacy activists, data protection commissioners, and politicians alike for
exposing their users to the public, for their low privacy and security
standards, and their incomprehensible privacy policies.
The aim of this paper is to show that the danger of social networks to
individuals and society alike are better understood from a structuralist point
of view than from the individualist one used by most scholars, politicians,
and activists.
In Chapter 2, I begin with a brief survey of the history and the three main
lines of the modern information privacy and data protection discourse. Even
though the history of privacy regulation can be traced back to the Hippocratic
oath and the secrecy of confession, modern theories of privacy appear in the
second half of the nineteenth century. A hundred years later, the right to
privacy and the general right of personality are accepted legal concepts in
the US and in Germany, respectively. Against this background the advent of the
computer and the electronic data processing ignites the public debate on
information privacy. Most scholars follow one of two traditional privacy
theories: (1) privacy as secrecy and confidentiality, or (2) privacy as right
of personality and individual self-determination. In contrast, some of the
pioneers of the data protection legislation in German use an approach based on
concepts borrowed from Luhmann’s sociological systems theory (3).
In Chapter 3, I give a short overview of the concept of functional
differentiation used to characterize social systems in modern western
societies. I show how society, social systems, and the individual benefit from
functional differentiated social systems, and how modern information
processing threatens these achievements. I then present the aim of data
protection as envisioned by some of the pioneers of the German data protection
legislation.
Finally, in Chapter 4, I examine the effects of general social networking
services on the functional differentiation of society, the balance of power
between social systems, and the individual’s chance for self-determined role-
playing. I then demonstrate how social networks and the companies behind them
must be constrained in their abilities to process, use and disseminate
individual-related information. The same limitations must be applied to
private organizations and public authorities trying to access the information
in social networks to use it for their own purposes.
## 2 Theories of Privacy and Data Protection
The history of privacy regulation can be traced back to ancient times. The
Hippocratic oath is one of the oldest examples, protecting the confidentiality
of all private information about the patient becoming known to her physician.
Since at least the Fourth Council of the Lateran 1215 the clergy must protect
the secrecy of confession. The bank secrecy is accepted since the seventeenth
century, at least in Germany. All of these are based on a contractual or
quasi-contractual protection of information entrusted to the care of specific
professions. In addition the Roman law knew with the _actio iniuriarum_ a
protection against specific indiscretions also outside of special trusted
relationships [24].
In the end of the nineteenth century the modern privacy debate started almost
simultaneously in the US and in Europe. Based on Roman legal tradition, French
and British precedents, and the history of the copyright Josef Kohler
formulated the right of the author to decide whether to publish or not her
sensations and feelings in writing as a special case of a general
_Individualrecht_ (right of the individual) envisioned by him [13]. The very
same argumentation was used by Samuel D. Warren and Louis D. Brandeis in their
seminal paper on the right of privacy [44]. Their _right to be let alone_ was
just an update of Kohler’s _noli me tangere_ in light of the development of
the instantaneous photography and the emergence of the yellow press. Otto
Gierke then generalized and reformulated Kohler’s _Individualrecht_ as the
_allgemeines Persönlichkeitsrecht_ (general right of personality) [8].
The right to privacy was quite successful used as a term and a concept in the
American common law albeit there was no consensus of what it means and how it
must be treated. After many extremely varying court decisions William L.
Prosser argued that behind the right to privacy there are in fact four
different interests against four different intrusions: (1) the intrusion upon
the individual’s seclusion or solitude, or into her private affairs, (2)
public disclosure of embarrassing private facts about the individual, (3)
publicity which places the individual in a false light in the public eye, and
(4) appropriation, for the intruder’s advantage, of the individual’s name or
likeness [32]. Prosser was heavily criticized by Edward J. Bloustein for his
distorting citations, improper classifications, and untenable conclusions [4].
While many scholars followed Bloustein in his holistic approach to regard
privacy as an aspect of human dignity, the American legislator and most courts
followed Prosser’s more pragmatic approach. Prosser also heavily influenced
the upcoming privacy debate in the computer science field and most of their
technical approaches to implement privacy in electronic data processing
systems.
In Germany, the legislator and the jurisprudence for a long time did not
recognize a general right of personality as a legal concept. Instead they
protected specific expressions of this right as independent rights, like the
_Recht am eigenen Bild_ (right to one’s own picture), or the _Recht am
gesprochenen Wort_ (right to the spoken word). After the _Bonner Grundgesetz_
(German Basic Law) came into effect the _Bundesgerichtshof_ (Federal Court of
Justice) acknowledged the general right of personality as constitutionally
protected (Article 2 (1) in conjunction with Article 1 (1) Basic Law). Since
the middle of the 1950s this is settled case law of the
_Bundesverfassungsgericht_ (Federal Constitutional Court), too. The general
right of personality is therefore equally protected in constitutional law as
in common law while its specific expressions might be protected in either the
constitutional law, or the common law, or both. For more information about
similarities and differences between the German right of personality and the
American right of privacy at the end of the pre-digital era see [14], [42],
and [11].
The public dispute about the National Data Center in the mid-60s can be
considered as the starting point of the modern information privacy and data
protection debate [6]. Besides a strong focus on data quality and safety
requirements the debate centered around guaranteeing secrecy and
confidentiality for collected information, and some restrictions on how to
collect information. Privacy was first and foremost used in its meaning of a
private sphere, not as a right of personality [26], [38]. The individual’s
need for such a private sphere was often justified with a psychological
rationale. The main distinction was between information being private or being
public. Public information should not need any protection because it is not
private. While this concept of privacy lost ground in the legal field after
the beginning of the 1970s the computer science scholars still based their
research on the distinction of the private and the public. For a few years the
public discourse is based again on this outdated concept, most notably with
respect to social networks.
The second main individualistic concept of information privacy and data
protection takes the more holistic approach based on the general right of
personality, and an equal understanding of the right to privacy. The
predominant functions of information privacy according to this view are
individual self-determination, personal autonomy, and limited and protected
communication [45]. Privacy is not viewed as limited to secrecy [37], [12].
Instead it means the control over information about oneself [45] or
informational self-determination [1]. Most modern (European) data protection
laws are based on this concept: The individual must know who knows what about
her, and should retain some control over the collection, the processing, and
the dissemination of information about her even while the information itself
is owned by the data processor [7], [43].
While most scholars only examined the consequences of electronic data
processing on the individual, others also considered the effects on structural
aspects of society. Jeffrey A. Meldman noticed that Americans have always
rather tolerated inefficiency than permitted the occurrence of unchecked
power, particularly if centralized [25]. Malcolm Warner and M. G. Stone also
warned of possible bureaucratic omnipotence stemming from a broad employment
of computers to process data [41]. Even legislators paid attention to the
broader problem. The first data protection law in the German state of Hesse
contained rules to protect the balance of power between the legislative and
the executive body of the state [9]. Adalbert Podlech noticed first, as far as
I can see, that for the formulation of a holistic data protection law a theory
is needed encompassing the consequences both on the individual and the
societal institutions [30]. Concepts borrowed from both Niklas Luhmann’s works
[18], [19], [21], [22], [20], and [23] about his sociological systems theory
and state organization law theories were used to describe the social function
of data protection [31], [40], [3], [34]. While in general Luhmann’s systems
theory lost the scientific discourse against more modern sociological
theories, it nevertheless provided the basis for some important new directions
in data protection like _Systemdatenschutz_ (system data protection) [29],
data protection conformity in system design [5], or identity management [33].
## 3 Functional Differentiation and the Social Function of Data Protection
Most important is the concept of functional differentiation [22], [17], and
its consequences on social systems and individuals alike. Functional
differentiation means that a social system differentiates itself from an
environment through the function it performs for the overall system [16]. That
means different systems are distinguishable from each other by the different
functions they perform. This separation of duties is often being used as a
protection measurement, i.e. the separation of church and state, or the
separation of powers in modern, democratic states to balance their powers and
protect society. The functional differentiation thus parallels the division of
labor, with its reasons as well as with its consequences. Losing one of them
modern societies cannot be sustained.
On the societal level Luhmann distinguishes general social systems like
Science, Politics, Economics, Law, or Religion. Each of these systems is using
its own defining binary code: Science uses true vs. false, Politics uses power
vs. no power, Economics uses payment vs. nonpayment, etc. That means for
example that in Science only scientific truth matters, neither money nor
power. If you can buy scientific truth with money or enforce it with power,
you have no Science.
The correspondent to the functional differentiation on the state level in
western democracies is the separation of power. This separation is applied
horizontally as well as vertically. Horizontally, the _trias_ consists in the
legislature, the executive, and the judiciary. They serve different functions:
The legislature makes the law, the executive acts under the law, and the
judiciary controls the executive actions with respect to the law. Vertically,
modern states are differentiated in a municipal level, a state or intermediate
level, and the national level. Using both these differentiations we are able
to balance the powers even as states as a hole are much more powerful entities
than in the past [35].
The executive body itself became historically more and more differentiated,
too. The many authorities perform different duties using different means. One
example is the separation of the police and the social welfare administration,
previously being part of the _Polizey_. Much later, the same happened with the
separation of the police and the registry office, or the separation of the
police and the intelligence service [15]. In a constitutional state the police
acts primarily repressive while the intelligence service may also act
preventative, nevertheless it is not allowed to arrest people. There is no
_Einheit der Verwaltung_ (unity of authority) as in an absolutistic state—the
different authorities are structurally and informationally separated, and
therefore limited in their power.
Before the emergence of digital computers, data processing and information
processing was done by humans, and therefore slow, inefficient and error-
prone. The computer tends to revoke all limitations of manual data processing,
eases the processing of information, and therefore undermines the protective
character of inefficiency [39]. Because information serve the production of
decisions whoever controls the collection and processing of information
controls the decisions based thereon. External entities like the parliament,
the data protection commissioner, or even more the data subject are
structurally unable to control the data processor. The control of the
processing of information becomes more and more centralized even if the
computing itself becomes decentralized or distributed. Local authorities and
local democratic institutions therefore tend to lose power to centralized
ones.
The functional differentiation not only has consequences for social systems or
the society but also for the individual. In modern differentiated societies
the individual plays different roles in different contexts. Although there
exist expectations on how to behave in specific contexts the individual
autonomously decides how to play her roles as sister, friend, neighbor,
colleague, principal, patient, business partner, client, voter, tax payer,
etc. Every human being and every social system she interacts with only sees
the individual in her current role. The roles are generally separated, the
individual—the totality of her roles—only being known to the individual
itself. Her role-playing is basis for and product of her right to personality
[27]. Social players being able to consolidate information from different
roles based on the abilities of modern data processing technology, ubiquitous
data collection, and sophisticated data mining methods would threaten the
individual with total visibility and transparency. The data processor would be
able to predict the future role-playing of the individual and to base its
decisions on this information advantage. This information power threatens the
autonomy of the individual [28].
The aim of data protection is therefore (1) to maintain the functional
differentiation of society against the threats posed by modern information
processing, and (2) to counter undue information power by organized social
players. Data protection guarantees the balance of power between different
social systems, societal institutions, and other social groups, and protects
the role-playing of the individual and therefore her autonomy. This is done by
controlling the flow of information between individuals, institutions, and
different sectors of society. Therefore data protection is the controlled
assignment or retention of information to prevent socially undesirable
information processing and to limit organizational power over individuals,
groups, social systems, and society.
## 4 Social Networks and Data Protection
Social networks like Facebook or Google+ often get criticized for exposing
people to the public gaze against their will, or for helping criminals spying
or stalking by making “private” data publicly available. So called privacy
critics counter with pointing on the consent of the individuals, or by
claiming that information given to the social networks are public, and
therefore not deserving protection. But as shown before these are not the
problems data protection tries to prevent. They are either IT security
problems, or based on outdated privacy theories.
Instead, the main problem of social networks is their ability to collect
information on different roles of the individual and merge them into one
holistic and exhaustive picture. Because this modeling of the individual is
not only based on information provided by the individual itself but also based
on information provided by other people or intrinsic properties of the social
graph around each one, the ability of the individual to control what parts of
her personality is known to the owner of the social networking service is
seriously limited. Most people are not even aware of how much information may
be deduced from the social graph, see for example [10]. The roles being made
transparent cover all areas of life: family, education, work life, hobbies,
and even politics. With their members made transparent, informal groups of
people, formal associations, companies, and even institutions become
transparent, too. In addition more and more groups, associations, and
institutions use social networks for their own internal or external
communications.
Second, there are usually no limitations for the company behind the social
network to process and use all information, either collected or derived. There
are also almost no limitations on handing over information from or about the
individual to government authorities, or private organizations. For example,
Facebook’s privacy policy reads: “We may also share information when we have a
good faith belief it is necessary to prevent fraud or other illegal activity,
to prevent imminent bodily harm, or to protect ourselves and you from people
violating our Statement of Rights and Responsibilities. This may include
sharing information with other companies, lawyers, courts or other government
entities,” (cited after [36]). What the social network knows, the state knows,
too.
With individuals, groups, and social systems made visible and transparent
alike, and widespread sharing of information between social networks and
primarily government authorities, the balance of power is shifted in favor of
centralized bureaucracies, either private or public. Legislation loses its
ability to control the executive if the latter acts _in arcanum_ , and the
former is being made transparent and therefore predictable. Individuals,
groups, and associations alike lose their autonomy—and with it their
freedom—against businesses and public authorities. The functional
differentiation as a limitation of power of social systems in modern western
societies may collapse—at least in some areas—if there exist social players
being able to transgress informational boundaries between social systems. The
modern state as the most powerful member of society may become total again.
From a data protection point of view there would be two fundamental claims:
(1) General social networking services must be information sinks concerning
information about individuals and groups. They should not be allowed to
disseminate individual-related information to other organizations, either
private or public. In the same way as the inviolability of the home (Article
13 (1) Basic Law) or the _Grundrecht auf Gewährleistung der Vertraulichkeit
und Integrität informationstechnischer Systeme_ (right to the provision of
confidentiality and integrity of information technology systems) [2] protect
the individual and her personality through the protection of structures (the
home and the computer, respectively), information stored in social networks
must be protected as a whole due to their ability to provide a total picture
of the individual. There also must be a general prohibition of retrieving and
using such information by public authorities, especially police and
intelligence services. (2) For structurally limiting the power of companies
behind social networks, they should be treated and regulated like monopolies.
First of all, it must be prevented that they use their informational power
over their users to gain a hold in other areas, especially politics or in
collaboration with government agencies.
## 5 Conclusion
General social networks should not only be criticized for exposing their users
to the public, for their low privacy and security standards, and their
incomprehensible privacy policies. Based on sociological theories of Niklas
Luhmann and the pioneering works of Adalbert Podlech, Paul J. Müller, and
others concerning the foundations of data protection I have shown in this
paper that general social networks and their ability to collect, store, and
process vast amounts of information about individuals, groups, and
organizations from different social systems and to combine them in centralized
data banks constitute a threat (1) to the individual’s ability to control her
own role-playing—and therefore her autonomy and freedom—in modern,
functionally differentiated societies, (2) to groups, associations, and social
systems being dependent on functional differentiation as protection against
overly powerful public or private entities, and (3) to the functionally
differentiated society as a whole with its dependency on a balance of power to
guarantee freedom, democracy, and a state of law.
### Acknowledgements.
The author would like to thank Martin Rost for providing the idea to study the
consequences of social networking services on data protection from a
structuralistic point of view. He also wishes to thank Martin Warnke, Wolfgang
Coy, and especially Jochen Koubek for enlightening and fruitful discussions.
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* [40] Wilhelm Steinmüller “Stellenwert der EDV in der Öffentlichen Verwaltung und Prinzipien des Datenschutzrechts” In _Öffentliche Verwaltung und Datenverarbeitung_ 2.11, 1972, pp. 453–462
* [41] M. G. Stone and Malcolm Warner “Politics, Privacy, and Computers” In _The Political Quarterly_ 40.3, 1969, pp. 256–267 DOI: 10.1111/j.1467-923X.1969.tb00022.x
* [42] Stig Strömholm “Right of Privacy and Rights of the Personality” Working Paper prepared for the Nordic Conference on Privacy organized by the International Commission of Jurists, Stockholm May 1967 VIII, Acta Instituti Upsaliensis Iurisprudentiae Comparativae Stockholm: P. A. Norstedt & Söners Förlag, 1967
* [43] Marie-Theres Tinnefeld, Eugen Ehmann and Rainer W. Gerling “Einführung in das Datenschutzrecht” München: Oldenbourg Verlag, 2005
* [44] Samuel D. Warren and Louis D. Brandeis “The Right to Privacy” In _Harvard Law Review_ , 1890, pp. 193–220
* [45] Alan F. Westin “Privacy and Freedom” New York: Atheneum, 1967
|
arxiv-papers
| 2012-06-14T07:59:05 |
2024-09-04T02:49:31.762362
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J\\\"org Pohle",
"submitter": "J\\\"org Pohle",
"url": "https://arxiv.org/abs/1206.3027"
}
|
1206.3040
|
# Quasifission at extreme sub-barrier energies
V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, W. Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, M.
Manougian 1, 0025, Yerevan, Armenia
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
With the quantum diffusion approach the behavior of the capture cross-section
is investigated in the reactions 92,94Mo + 92,94Mo, 100Ru + 100Ru, 104Pd +
104Pd, and 78Kr + 112Sn at deep sub-barrier energies which are lower than the
ground state energies of the compound nuclei. Because the capture cross
section is the sum of the complete fusion and quasifission cross sections, and
the complete fusion cross section is zero at these sub-barrier energies, one
can study experimentally the unique quasifission process in these reactions
after the capture.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, quantum diffusion approach, quasifission
The first evidences of hindrance for compound nucleus formation in the
reactions with massive nuclei ($Z_{1}\times Z_{2}>1600$) at energies near the
Coulomb barrier were observed at GSI already long time ago GSI ; GSI2 ; GSI3 .
The theoretical investigations showed that the probability of complete fusion
depends on the competition between the complete fusion and quasifission after
the capture stage Volkov ; nasha ; Avaz . As known, this competition can
strongly reduce the value of the fusion cross section and, respectively, the
value of the evaporation residue cross section in reactions producing heavy
and superheavy nuclei. The quasifission is related to the binary decay of the
nuclear system after the capture, but before a compound nucleus is formed
which could exist at angular momenta treated Volkov ; Schroder ; nasha ; Avaz
. The quasifission process was originally ascribed only to reactions with
massive nuclei. But it is the general phenomenon which takes place in
reactions with the massive and medium-mass nuclei at energies above and below
the Coulomb barrier EPJSub1 ; EPJSub2 . The mass and angular distributions of
the quasifission products depend on the entrance channel and the bombarding
energy Schroder .
For systems with negative $Q$-value, the complete fusion cross section
$\sigma_{fus}$ is equal to zero at bombarding energies $E_{\rm c.m.}<E_{\rm
c.m.}^{0}=-Q$:
$\sigma_{fus}(E_{\rm c.m.}<E_{\rm c.m.}^{0})=0.$
This expression implies that the fusion cross section or the fusion
probability $P_{fus}$ must go to zero when the center-of-mass energy $E_{\rm
c.m.}$ approaches the ground state energy, -$Q$, of the compound nucleus.
Since the sum of the complete fusion cross section $\sigma_{fus}$ and the
quasifission cross section $\sigma_{qf}$ gives the capture cross section
$\sigma_{cap}=\sigma_{fus}+\sigma_{qf},$
at $E_{\rm c.m.}<E_{\rm c.m.}^{0}=-Q$ we have
$\sigma_{cap}(E_{\rm c.m.}<E_{\rm c.m.}^{0})=\sigma_{qf}.$
So, at these deep sub-barrier energies the quasifission is only contribution
to the capture cross section and there is no the overlapping between the
fusion-fission and quasifission processes as at higher bombarding energies. At
deep sub-barrier energies, the quasifission event corresponds to the formation
of a nuclear-molecular state or dinuclear system with small excitation energy
that separates by quantum tunneling through the Coulomb barrier in a binary
event with mass and charge close to the entrance channel.
Although many measurements do not reach such deep sub-barrier energies $E_{\rm
c.m.}<E_{\rm c.m.}^{0}=-Q$, it is still possible to find systems with
relatively small values of $V_{b}-E_{\rm c.m.}^{0}=V_{b}+Q$ ($V_{b}=V(R_{b})$
is the height of the Coulomb barrier for the spherical nuclei, $R_{b}$ is the
position of this barrier) for the experimental study of the quasifission
process. The purpose of the present article is to find such type of systems
and to estimate the capture cross sections at $E_{\rm c.m.}<E_{\rm
c.m.}^{0}=-Q$. The quantum diffusion approach EPJSub1 ; EPJSub2 ; EPJSub ;
EPJSub3 is applied to study the capture process more thoroughly.
In our quantum diffusion approach EPJSub1 ; EPJSub2 ; EPJSub ; EPJSub3 the
collisions of nuclei are treated in terms of a single collective variable: the
relative distance between the colliding nuclei. The nuclear deformations are
taken into account through the dependence of the nucleus-nucleus potential on
the quadrupole deformations and mutual orientations of the colliding nuclei.
Our approach regards the fluctuation and dissipation effects in the collision
of heavy ions and models the coupling with various channels (for example,
coupling of the relative motion with low-lying collective modes such as
dynamical quadrupole and octupole modes of the target and projectile Ayik333
). We have to mention that many quantum-mechanical and non-Markovian effects
accompanying the passage through the potential barrier are considered in our
formalism EPJSub ; our through the friction and diffusion. To calculate the
nucleus-nucleus interaction potential $V(R)$, we use the procedure presented
in Refs. EPJSub ; EPJSub1 ; EPJSub2 . For the nuclear part of the nucleus-
nucleus potential, the double-folding formalism with a Skyrme-type density-
dependent effective nucleon-nucleon interaction is used. The absolute values
of the quadrupole deformation parameters $\beta_{2}$ of deformed nuclei were
taken from Ref. Ram .
The calculated results for all reactions are obtained with the same set of
parameters as in Refs. EPJSub ; EPJSub2 and are rather insensitive to a
reasonable variation of them. One should stress that diffusion models, which
also include quantum statistical effects, were proposed in Refs. Hofman ; Ayik
; Hupin too.
Symmetric and near symmetric dinuclear systems with neutron-deficient stable
nuclei have the smallest values of $(V_{b}+Q)$. For example, the sub-barrier
energies with respect to the Coulomb barrier are $V_{b}-E_{\rm
c.m.}^{0}=V_{b}+Q=13,14.8,18,19.4,21.8$ MeV for the systems 92Mo + 92Mo, 104Pd
+ 104Pd, 94Mo + 94Mo, 100Ru + 100Ru, 78Kr + 112Sn, respectively. Here
predictions of unknown mass-excesses of the compound nuclei are taken from
Ref. MN . In Figs. 1–3 the calculated capture cross sections for these
reactions are presented.
Figure 1: The calculated capture cross sections vs $E_{\rm c.m.}$ for the
reactions 92,94Mo + 92,94Mo. The dashed and solid arrows show $E_{\rm
c.m.}=E_{\rm c.m.}^{0}=-Q$ and $E_{\rm c.m.}=V_{b}$, respectively. Figure 2:
The same as in Fig. 1, but for the reactions 100Ru + 100Ru and 104Pd + 104Pd.
Figure 3: The same as in Fig. 1, but for the 78Kr + 112Sn reaction.
All systems show a steady decrease of the sub-barrier fusion cross sections
with a pronounced change of slope. With $E_{\rm c.m.}$ decreasing below the
Coulomb barrier the interaction changes because at the external turning point
the colliding nuclei do no more reach the region of the nuclear interaction
where the friction plays a role. As result, at smaller $E_{\rm c.m.}$ the
cross sections fall with a smaller rate. For sub-barrier energies, the results
of calculations are very sensitive to the quadrupole deformation parameters
$\beta_{2}$ of the interacting nuclei. The influence of nuclear deformation is
straightforward. If the target and projectile nuclei are prolate in their
ground states, the Coulomb field on its tips is lower than on its sides. This
increases the capture probability at energies below the barrier corresponding
to the spherical nuclei. The enhancement of sub-barrier capture for the
reactions 104Pd + 104Pd, 100Ru + 100Ru, and 78Kr + 112Sn in the contrast to
the reactions 92,94Mo + 92,94Mo is explained by the deformation effect: the
deformations in the former systems are larger the ones in the later systems.
In Figs. 1–3 the calculated capture cross sections at $E_{\rm c.m.}=E_{\rm
c.m.}^{0}=-Q$ are $\sigma_{cap}=$0.2 nb, 5.1 nb, 2.3 $\mu$b, 24.4 $\mu$b, and
0.7 mb for the reactions 92Mo + 92Mo, 94Mo + 94Mo, 78Kr + 112Sn, 100Ru +
100Ru, and 104Pd + 104Pd, respectively. So, 104Pd + 104Pd, 100Ru + 100Ru, and
78Kr + 112Sn are the optimal reactions for studying capture and quasifission
at deep sub-barrier energies $E_{\rm c.m.}<E_{\rm c.m.}^{0}=-Q$ where the
complete fusion channel is closed ($\sigma_{fus}=0$). At these sub-barrier
energies the quasifission process can be studied in future experiments: from
the measurement of the mass (charge) distribution in collisions with total
momentum transfer one can show the distinct components which are due to
quasifission (with respect to the quasielastic components). Because the
angular momentum is $J<10$ at these energies, the angular distribution would
have a small anisotropy. The low-energy experimental quasifission data would
probably provide straight information since the high-energy data may be shaded
by competing the fusion-fission processes. The lifetime of nuclear molecule
formed seems to be long enough to separate it mass from other reaction
products. Then one can observe the decay of this molecule into two fragments.
In conclusion, the quantum diffusion approach was applied to calculate the
capture cross sections for the reactions 92Mo + 92Mo, 104Pd + 104Pd, 94Mo +
94Mo, 100Ru + 100Ru, and 78Kr + 112Sn at extreme sub-barrier energies which
are too low for complete fusion. The quasifission near the entrance channel is
the unique binary decay process after the capture. The reactions 104Pd +
104Pd, 100Ru + 100Ru, and 78Kr + 112Sn seem to be optimal systems for a
experimental study of the true quasifission at extreme sub-barrier energies.
This work was supported by DFG, NSFC, and RFBR. The IN2P3(France) -
JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully
acknowledged.
## References
* (1) C.-C. Sahm, H.-G. Clerc, K.-H. Schmidt, W. Reisdorf, P. Armbruster, F.P. Hessberger, J.G. Keller, G. Miinzenberg, and D. Vermeulen, Z. Phys. A 319, 113 (1984).
* (2) J.G. Keller, K.-H. Schmidt, F.P. Hessberger, G. Miinzenberg, W. Reisdorf, H.-G. Clerc, and C.-C. Sahm, Nucl. Phys. A452, 173 (1986).
* (3) A.B. Quint, W. Reisdorf, K.-H. Schmidt, P. Armbruster, F.P. Hessberger, S. Hofmann, J. Keller, G. Miinzenberg, H. Stelzer, H.-G. Clerc, W. Morawek, and C.-C. Sahm, Z. Phys. A 346, 119 (1993).
* (4) V.V. Volkov, Particles and Nuclei, 35, 797 (2004).
* (5) G.G. Adamian, N.V. Antonenko, and W.Scheid, Phys. Rev. C 68, 034601 (2003); Lecture Notes in Physics 848, Clusters in Nuclei, Vol. 2, ed. by C. Beck (Springer-Verlag, Berlin, 2012) p. 165.
* (6) G. Giardina et al., Nucl. Phys. A671, 165 (2000); A. Nasirov et al., Nucl. Phys. A759, 342 (2005); Z.-Q. Feng, G.-M. Jin, J.-Q. Li, and W. Scheid, Phys. Rev. C 76, 044606 (2007); H.Q. Zhang, C.L. Zhang, C.J. Lin, Z.H. Liu, F. Yang, A.K. Nasirov, G. Mandaglio, M. Manganaro, and G. Giardina, Phys. Rev. C 81, 034611 (2010).
* (7) W.-U. Schröder and J.R. Huizenga, in Treatise on Heavy-Ion Science, edited by D.A. Bromley, Vol. 2 (Plenum Press, New York, 1984) p.115.
* (8) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Eur. Phys. J. A 47, 38 (2011); J. of Phys.: Conf. Ser. 282, 012001 (2011); EPJ Web Conf. 17, 04003 (2011).
* (9) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Phys. C 84, 064614 (2011); Phys. Rev. C 85, 024616 (2012).
* (10) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 45, 125 (2010).
* (11) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, C.J. Lin, and H.Q. Zhang, Phys. Phys. C 85, 017603 (2012); Phys. Phys. C 85, 037602 (2012).
* (12) S. Ayik, B. Yilmaz, and D. Lacroix, Phys. Rev. C 81, 034605 (2010).
* (13) V.V. Sargsyan, Z. Kanokov, G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. C 80, 034606 (2009); Phys. Rev. C 80, 047603 (2009); V.V. Sargsyan, Z. Kanokov, G.G. Adamian, and N.V. Antonenko, Part. Nucl. 41, 175 (2010).
* (14) H. Hofmann, Phys. Rep. 284, 137 (1997); J.D. Bao and Y.-Z. Zhuo, Phys. Rev. C 67, 064606 (2003).
* (15) N. Takigawa, S. Ayik, K. Washiyama, and S. Kimura, Phys. Rev. C 69, 054605 (2004); S. Ayik, B. Yilmaz, A. Gokalp, O. Yilmaz, and N. Takigawa, Phys. Rev. C 71, 054611 (2005).
* (16) G. Hupin and D. Lacroix, Phys. Rev. C 81, 014609 (2010).
* (17) S. Raman, C.W. Nestor, Jr, and P. Tikkanen, At. Data Nucl. Data Tables 78, 1 (2001).
* (18) P. Möller, J.R. Nix, W.D. Myers, and W.J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995).
|
arxiv-papers
| 2012-06-14T09:10:40 |
2024-09-04T02:49:31.770318
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid, and H. Q.\n Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1206.3040"
}
|
1206.3065
|
# Stability analysis and controller design for a linear system with Duhem
hysteresis nonlinearity
Ruiyue Ouyang, Bayu Jayawardhana B. Jayawardhana and Ruiyue Ouyang are with
the Dept. Discrete Technology and Production Automation, University of
Groningen, Groningen 9747AG, The Netherlands e-mail: bayujw@ieee.org,
r.ouyang@rug.nl
###### Abstract
In this paper, we investigate the stability of a feedback interconnection
between a linear system and a Duhem hysteresis operator, where the linear
system and the Duhem hysteresis operator satisfy either the counter-clockwise
(CCW) or clockwise (CW) input-output dynamics. More precisely, we present
sufficient conditions for the stability of the interconnected system that
depend on the CW or CCW properties of the linear system and the Duhem
operator. Based on these results we introduce a control design methodology for
stabilizing a linear plant with a hysteretic actuator or sensor without
requiring precise information on the hysteresis operator.
## I Introduction
Hysteresis is a common phenomenon that is present in diverse systems, such as
piezo-actuator, ferromagnetic material and mechanical systems. For describing
hysteresis phenomena, several hysteresis models have been proposed in the
literature, see, for example, [4, 18, 16]. These include backlash model [27]
which is used to describe gear trains, Preisach model for modeling the
ferromagnetic systems and elastic-plastic model which is used to study
mechanical friction [4, 18]. From the perspective of input-output behavior,
the hysteresis phenomena can exhibit counterclockwise (CCW) input-output (I/O)
dynamics [1], clockwise (CW) I/O dynamics [20], or even more complex I/O map
(such as, butterfly map [3]). For example, backlash model generates CCW
hysteresis loops, elastic-plastic model generates CW hysteresis loops and
Preisach model can generate CCW, CW or butterfly hysteresis loops depending on
the weight of the hysterons which are used in the Preisach model.
The CCW and CW I/O dynamics of a system can also be related to certain
dissipation inequalities [1, 23, 26]. Denoting $AC$ as the class of absolutely
continuous functions, we show in [11] that for a class of Duhem hysteresis
operator $\Phi:AC({\mathbb{R}}_{+})\times{\mathbb{R}}\rightarrow
AC({\mathbb{R}}_{+})$, we have that for every $u_{\Phi}\in
AC({\mathbb{R}}_{+})$ there exists a function
$H_{\circlearrowleft}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ which
satisfies
$\frac{{\rm d}\hbox{\hskip
0.5pt}H_{\circlearrowleft}(y_{\Phi}(t),u_{\Phi}(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\dot{y}_{\Phi}(t)u_{\Phi}(t)$ (1)
for almost every $t$, where $y_{\Phi}=\Phi(u_{\Phi},y_{\Phi_{0}})$ and
$y_{\Phi_{0}}\in{\mathbb{R}}$ is the initial condition. The inequality (1)
characterizes the CCW I/O property of the operator $\Phi$. We will discuss
this property in detail in Section II. Here, we use the symbol
$\circlearrowleft$ in $H_{\circlearrowleft}$ to indicate the counterclockwise
behavior of $\Phi$.
As a dual result to [11], in [24] we give sufficient conditions on the Duhem
hysteresis operator such that it exhibits CW input-output dynamics. In
particular for a class of Duhem operator $\Phi$, we construct a function
$H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ which
satisfies
$\frac{{\rm d}\hbox{\hskip
0.5pt}H_{\circlearrowright}(y_{\Phi}(t),u_{\Phi}(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\dot{u}_{\Phi}(t)y_{\Phi}(t)$ (2)
for almost every $t$. Correspondingly, the symbol $\circlearrowright$ in
$H_{\circlearrowright}$ indicates the clockwise behavior of $\Phi$.
Figure 1: Feedback interconnection between a linear plant $\mathbf{P}$ and a
Duhem operator ${\bf\Phi}$.
In this paper, we exploit our knowledge on $H_{\circlearrowleft}$ and
$H_{\circlearrowright}$ to study the stability of an interconnected system as
shown in Figure 1, where $\mathbf{P}$ is a linear system and ${\bf\Phi}$ is
the hysteresis operator. We consider four cases of interconnections where the
plant $\mathbf{P}$ and the hysteresis operator ${\bf\Phi}$ can assume either
CCW or CW I/O dynamics. These four cases are summarized in Table I
Table I: Four Possible cases of interconnection ${\bf\Phi}$ $\mathbf{P}$ | CCW | CW
---|---|---
CCW | ⓐ | ⓑ
CW | ⓒ | ⓓ
In Theorem IV.1 of this paper, the interconnection case $ⓐ$ in Table I is
considered, where both the linear system $\mathbf{P}$ and the hysteresis
operator ${\bf\Phi}$ have CCW I/O dynamics. In particular, we give sufficient
conditions on $\mathbf{P}$ which are dependent on the underlying anhysteresis
function of ${\bf\Phi}$ that ensure the stability of the closed-loop system
with a positive-feedback interconnection. This result is motivated by recent
results on the positive-feedback interconnection of negative imaginary system
[23] and of CCW systems [26]. A motivating example for the interconnection
case $ⓐ$ is the piezo-actuated stages which are commonly used in the high-
precision positioning mechanisms, see, for example [15]. The piezo-actuated
stage contains two parts: a piezo-actuator and a positioning mechanism, which
can be described by
$\left.\begin{array}[]{rr}\mathbf{P}:&\begin{array}[]{rl}m\ddot{x}+b\dot{x}+kx&=F_{{\rm
piezo}},\\\ V&=cx,\end{array}\\\\[14.22636pt] {\bf\Phi}:&F_{{\rm
piezo}}=\Phi(V),\end{array}\right\\}$ (3)
where $m$ is the mass, $b$ is the damping constant, $k$ is the spring
constant, $c$ is the proportional gain, $V$ is the input voltage of the piezo-
actuator, $F_{{\rm piezo}}$ denotes the force generated by the piezo-actuator
and $x$ denotes the displacement of the stage. The piezoelectric actuator has
been shown to have CCW hysteresis loops from the input voltage to the output
generated force (see, for example [7]). It can be checked that the linear
mass-damper-spring system $\mathbf{P}$ is also CCW from $F_{{\rm piezo}}$ to
$x$ or, equivalently, $\mathbf{P}$ is a negative-imaginary system [23].
In Theorem IV.3, we consider the interconnection case $ⓑ$ in Table I, where
the linear system $\mathbf{P}$ has CW I/O dynamics and the hysteresis operator
${\bf\Phi}$ has CCW I/O dynamics. In this case Theorem IV.3 provides
sufficient conditions on $\mathbf{P}$ which are independent of ${\bf\Phi}$
such that the closed-loop system with a negative feedback interconnection is
stable. An example for this case is the active vibration mechanism using
piezo-actuator, which has been used for vibration control in mechanical
structures [13]. The mechanism can be described by
$\left.\begin{array}[]{rr}\mathbf{P}:&\begin{array}[]{rl}m\ddot{x}+b\dot{x}+kx&=F_{{\rm
piezo}},\\\ V&=-c\ddot{x},\end{array}\\\\[14.22636pt] {\bf\Phi}:&F_{{\rm
piezo}}=\Phi(V).\end{array}\right\\}$ (4)
As described before, the piezoelectric actuator has CCW I/O dynamics and it
can be checked that the mass-damper-spring system $\mathbf{P}$ is CW from
$F_{{\rm piezo}}$ to $\ddot{x}$.
Theorem V.1 deals with the interconnection case $ⓒ$, where $\mathbf{P}$ has
CCW I/O dynamics and ${\bf\Phi}$ has CW I/O dynamics. A motivating example for
this case is the mechanical systems with friction [21], which is given by
$\left.\begin{array}[]{rrl}\mathbf{P}:&m\ddot{x}+kx&=-F_{{\rm
friction}},\\\\[14.22636pt] {\bf\Phi}:&F_{{\rm
friction}}&=\Phi(x),\end{array}\right\\}$ (5)
where $F_{{\rm friction}}$ is the friction force. As discussed in [21], the
friction force has CW I/O dynamics where the input is the displacement. On the
other hand, the mechanical system is CCW from the friction force $-F_{{\rm
friction}}$ to the displacement $x$.
As a completion to the Table I, we present the analysis of the interconnection
case $ⓓ$ in Theorem V.3. Based on these results, we present in Section VI a
control design methodology for a linear plant with a hysteretic
actuator/sensor ${\bf\Phi}$ and we provide two numerical examples in Section
VII.
## II preliminaries
In this section we give the definitions of the CCW and CW dynamics based on
the work by Angeli [1] and Padthe [20]. Figure 2 illustrates the CCW and CW
input-output dynamics of a (nonlinear) operator $G:u\mapsto G(u)=:y$. We
denote $AC({\mathbb{R}}_{+},{\mathbb{R}}^{n})$ the space of absolutely
continuous function $f:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}^{n}$.
Figure 2: A graphical illustration of counter-clockwise (CCW) and clockwise
(CW) I/O dynamics of an operator $G:u\longmapsto y$. $(a)$ CCW I/O dynamics;
$(b)$ CW I/O dynamics.
### II-A Counterclockwise dynamics
###### Definition II.1
[1, 20] A (nonlinear) map $G:AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})\rightarrow
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ is counterclockwise (CCW) if for every
$u\in AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ with the corresponding output map
$y:=Gu$, the following inequality holds
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{\langle\dot{y}(t),u(t)\rangle
dt>-\infty}.$ (6)
For an operator $G$, inequality (6) holds if there exists a function
$V:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ such that for every input
signal $u$, the inequality
$\frac{{\rm d}\hbox{\hskip 0.5pt}V(y(t),u(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\langle\dot{y}(t),u(t)\rangle,$ (7)
holds for almost every $t$ where the output signal $y:=Gu$.
###### Definition II.2
A (nonlinear) map $G:AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})\rightarrow
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ is strictly counterclockwise (S-CCW)
(see also [1]), if for every input $u\in
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$, there exists a constant
$\varepsilon>0$ such that the inequality
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{\langle\dot{y}(t),\
u(t)\rangle-\varepsilon\|\dot{y}(t)\|^{2}{\rm d}\hbox{\hskip
0.5pt}t>-\infty},$ (8)
holds where $y:=Gu$.
Note that for systems described by the state space representation as follows:
$\Sigma:\left.\begin{array}[]{rl}\dot{x}&=f(x,u),\qquad x(0)=x_{0}\\\
y&=h(x),\end{array}\right\\}$ (9)
where $x(t)\in{\mathbb{R}}^{n}$ is the state, $u(t)\in{\mathbb{R}}^{m}$ is the
input, $y(t)\in{\mathbb{R}}^{m}$ is the output and $f$, $h$ are sufficiently
smooth functions, the following lemma provides sufficient conditions for
$\Sigma$ to be CCW (and S-CCW).
###### Lemma II.3
Consider the state space system $\Sigma$ as in (9). If there exists
$V:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{+}$ and $\varepsilon\geq 0$, such
that
$\frac{\partial V(x)}{\partial x}f(x,u)\leq\left\langle\frac{\partial
h(x)}{\partial x}f(x,u),u\right\rangle-\varepsilon\left\|\frac{\partial
h(x)}{\partial x}f(x,u)\right\|^{2},$
holds for all $x\in{\mathbb{R}}^{n}$ and $u\in{\mathbb{R}}^{m}$, then $\Sigma$
is CCW. Moreover if $\varepsilon>0$, it is S-CCW.
### II-B Clockwise dynamics
Dual to the concept of counterclockwise I/O dynamics, the notion of clockwise
I/O dynamics can be defined as follows.
###### Definition II.4
[20] A (nonlinear) map $G:AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})\rightarrow
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ is clockwise (CW) if for every input
$u\in AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ with the corresponding output map
$y:=Gu$, the following inequality holds:
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{y(t)^{T}\dot{u}(t){\rm
d}\hbox{\hskip 0.5pt}t>-\infty}.$ (10)
For a nonlinear operator $G$, inequality (10) holds if there exists a function
$V:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ such that for every input
signal $u\in AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$, the inequality
$\frac{{\rm d}\hbox{\hskip 0.5pt}V(y(t),u(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\langle y(t),\dot{u}(t)\rangle,$ (11)
holds for a.e. $t$ where the output signal $y:=Gu$.
###### Lemma II.5
Consider the state space system $\Sigma$ as in (9). If there exist
$\alpha,V:{\mathbb{R}}^{m+n}\rightarrow{\mathbb{R}}_{+}$, such that $V$ is
positive definite and proper, and
$\left[\begin{array}[]{cc}\frac{\partial V(w,x)}{\partial w}&\frac{\partial
V(w,x)}{\partial x}\end{array}\right]\left[\begin{array}[]{c}q\\\
f(x,w)\end{array}\right]\leq\langle h(x),w\rangle-\alpha(w,x),$ (12)
holds for all $x\in{\mathbb{R}}^{n}$, $w\in{\mathbb{R}}^{m}$ and
$q\in{\mathbb{R}}^{m}$, then $\Sigma$ is CW.
Proof: Define the extended state space system (9) as follows
$\left.\begin{array}[]{rl}\dot{w}&=q,\\\ \dot{x}&=f(x,w),\\\
y&=h(x).\end{array}\right.$ (13)
Note that $w$ defines the input in (9). It follows from (12) and (13) that
$\displaystyle\dot{V}$ $\displaystyle\leq\langle h(x),q\rangle-\alpha(x,w),$
$\displaystyle=\langle y,\dot{w}\rangle-\alpha(x,w),$
which completes our proof by taking $w=u$. $\Box$
## III Duhem Hysteresis operator
The Duhem operator $\Phi:AC({\mathbb{R}}_{+})\times\mathbb{R}\to
AC({\mathbb{R}}_{+}),(u_{\Phi},y_{\Phi_{0}})\mapsto\Phi(u_{\Phi},y_{\Phi_{0}})=:y_{\Phi}$
is described by
$\dot{y}_{\Phi}(t)=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\
y_{\Phi}(0)=y_{\Phi_{0}},$ (14)
where $\dot{u}_{\Phi+}(t):=\max\\{0,\dot{u}_{\Phi}(t)\\}$,
$\dot{u}_{\Phi-}(t):=\min\\{0,\dot{u}_{\Phi}(t)\\}$ and
$f_{1}:\mathbb{R}^{2}\to\mathbb{R}$, $f_{2}:\mathbb{R}^{2}\to\mathbb{R}$ are
$C^{1}$. We refer to [18, 19, 28] for standard properties of the Duhem
operator, such as causality, monotonicity and rate-independency.
The existence of solutions to (14) has been reviewed in [18]. In particular,
if for every $v\in{\mathbb{R}}$, the functions $f_{1}$ and $f_{2}$ satisfy
$\displaystyle(\gamma_{1}-\gamma_{2})[f_{1}(\gamma_{1},v)-f_{1}(\gamma_{2},v)]$
$\displaystyle\leq\lambda_{1}(v)(\gamma_{1}-\gamma_{2})^{2},$ (15)
$\displaystyle(\gamma_{1}-\gamma_{2})[f_{2}(\gamma_{1},v)-f_{2}(\gamma_{2},v)]$
$\displaystyle\geq-\lambda_{2}(v)(\gamma_{1}-\gamma_{2})^{2},$
for all $\gamma_{1}$, $\gamma_{2}\in{\mathbb{R}}$, where $\lambda_{1}$ and
$\lambda_{2}$ are nonnegative, then (14) has a unique global solution and
$\Phi$ maps $AC({\mathbb{R}}_{+})\times{\mathbb{R}}\rightarrow
AC({\mathbb{R}}_{+})$.
### III-A Duhem operator with CCW characterization
To show the CCW properties of the Duhem operator, we review our previous
results in [11]. In [11], we define a function
$H_{\circlearrowleft}:\mathbb{R}^{2}\to{\mathbb{R}}_{+}$ for the Duhem
operator $\Phi$ such that (1) holds (under certain conditions on $f_{1}$ and
$f_{2}$). Before we can define the function $H_{\circlearrowleft}$ for $\Phi$,
we need to define three functions which depend on $f_{1}$ and $f_{2}$.
Firstly, we define a traversing function $\omega_{\Phi}$ which describes the
possible trajectory of $\Phi$ when a monotone increasing $u_{\Phi}$ and a
monotone decreasing $u_{\Phi}$ is applied to $\Phi$ from an initial condition.
For every pair $(y_{\Phi_{0}},u_{\Phi_{0}})\in{\mathbb{R}}^{2}$, let
$\omega_{\Phi,1}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}}):[u_{\Phi_{0}},\infty)\to{\mathbb{R}}$
be the solution of
$z(v)-y_{\Phi_{0}}=\int^{v}_{u_{\Phi_{0}}}{f_{1}(z(\sigma),\sigma)\ {\rm
d}\hbox{\hskip 0.5pt}\sigma},\quad\forall v\in[u_{\Phi_{0}},\infty),$
and let
$\omega_{\Phi,2}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}}):(-\infty,u_{\Phi_{0}}]\to{\mathbb{R}}$
be the solution of
$z(v)-y_{\Phi_{0}}=\int_{u_{\Phi_{0}}}^{v}{f_{2}(z(\sigma),\sigma)\ {\rm
d}\hbox{\hskip 0.5pt}\sigma},\quad\forall v\in(-\infty,u_{\Phi_{0}}].$
Using the above definitions, for every pair
$(y_{\Phi_{0}},u_{\Phi_{0}})\in{\mathbb{R}}^{2}$, the traversing function
$\omega_{\Phi}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}}):{\mathbb{R}}\to{\mathbb{R}}$
is defined by the concatenation of
$\omega_{\Phi,2}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}})$ and
$\omega_{\Phi,1}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}})$:
$\omega_{\Phi}(v,y_{\Phi_{0}},u_{\Phi_{0}})=\left\\{\begin{array}[]{ll}\omega_{\Phi,2}(v,y_{\Phi_{0}},u_{\Phi_{0}})&\forall
v\in(-\infty,u_{\Phi_{0}})\\\
\omega_{\Phi,1}(v,y_{\Phi_{0}},u_{\Phi_{0}})&\forall
v\in[u_{\Phi_{0}},\infty).\end{array}\right.$ (16)
Again, we remark that the curve
$\omega_{\Phi}(\cdot,y_{\Phi_{0}},u_{\Phi_{0}})$ is the (unique) hysteresis
curve where the curve defined in $(-\infty,u_{\Phi_{0}}]$ is obtained by
applying a monotone decreasing $u_{\Phi}\in
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ to $\Phi(u_{\Phi},y_{\Phi_{0}})$ with
$u_{\Phi}(0)=u_{\Phi_{0}}$ and $\lim_{t\to\infty}u_{\Phi}(t)=-\infty$ and,
similarly, the curve defined in $[u_{\Phi_{0}},\infty)$ is produced by
introducing a monotone increasing $u_{\Phi}\in
AC({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ to $\Phi(u_{\Phi},y_{\Phi_{0}})$ with
$u_{\Phi}(0)=u_{\Phi_{0}}$ and $\lim_{t\to\infty}u_{\Phi}(t)=\infty$.
The second function we need to define is the anhysteresis function $f_{an}$,
which represents the curve where $f_{1}(f_{an}(v),v)=f_{2}(f_{an}(v),v)$.
Another function that is needed for defining $H_{\circlearrowleft}$ is the
intersecting function between the anhysteresis function $f_{an}$ and the
function $\omega_{\Phi}$ as defined above. The function
$\Omega:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ is the CCW intersecting
function if
$\omega_{\Phi}(\Omega(\gamma,v),\gamma,v)=f_{an}(\Omega(\gamma,v))$ for all
$(\gamma,v)\in{\mathbb{R}}^{2}$ and $\Omega(\gamma,v)\geq v$ whenever
$\gamma\geq f_{an}(v)$ and $\Omega(\gamma,v)<v$ otherwise. For simplicity, we
assume that $\Omega$ is differentiable. In [11, Lemma 3.1] sufficient
conditions on $f_{1}$ and $f_{2}$ which guarantee the existence of such
$\Omega$ are $f_{an}$ be monotone increasing and
$\displaystyle f_{1}(\gamma,v)<\frac{{\rm d}\hbox{\hskip 0.5pt}f_{an}(v)}{{\rm
d}\hbox{\hskip 0.5pt}v}-\epsilon$ whenever $\displaystyle\gamma>f_{an}(v)\ $
(17) $\displaystyle f_{2}(\gamma,v)<\frac{{\rm d}\hbox{\hskip
0.5pt}f_{an}(v)}{{\rm d}\hbox{\hskip 0.5pt}v}-\epsilon$ whenever
$\displaystyle\gamma<f_{an}(v)\ $ (18)
hold with $\epsilon>0$.
###### Theorem III.1
Consider the Duhem hysteresis operator $\Phi$ defined in (14) with $C^{1}$
functions $f_{1},f_{2}:{\mathbb{R}}^{2}\to{\mathbb{R}}_{+}$. Let $f_{an}$ be
the corresponding anhysteresis function which is monotone increasing and
satisfies (17) and (18). Denote by $\Omega$ the corresponding CCW intersecting
function. Suppose that for all $(\gamma,v)$ in ${\mathbb{R}}^{2}$,
$f_{1}(\gamma,v)\geq f_{2}(\gamma,v)$ whenever $\gamma\leq f_{an}(v)$ and
$f_{1}(\gamma,v)<f_{2}(\gamma,v)$ otherwise. Then $\Phi$ is CCW with the
function $H_{\circlearrowleft}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ be
given by
$H_{\circlearrowleft}(\gamma,v)=\gamma
v-\int_{0}^{v}{\omega_{\Phi}(\sigma,\gamma,v)\ {\rm d}\hbox{\hskip
0.5pt}\sigma}+\int_{0}^{\Omega(\gamma,v)}{\omega_{\Phi}(\sigma,\gamma,v)-f_{an}(\sigma)\
{\rm d}\hbox{\hskip 0.5pt}\sigma}.$ (19)
Proof: The proof follows from Lemma 3.1 and Theorem 3.3 in [11]. In
particular, it is shown in [11] that
$\frac{{\rm d}\hbox{\hskip
0.5pt}H_{\circlearrowleft}(y_{\Phi}(t),u_{\Phi}(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\langle\dot{y}_{\Phi}(t),u_{\Phi}(t)\rangle,$ (20)
where $y_{\Phi}:=\Phi(u_{\Phi},y_{\Phi_{0}})$ and $H_{\circlearrowleft}$ is
non-negative. By integrating (20) from $0$ to $T$ we have
$H_{\circlearrowleft}\big{(}y_{\Phi}(T),u_{\Phi}(T)\big{)}-H_{\circlearrowleft}\big{(}y_{\Phi}(0),u_{\Phi}(0)\big{)}\\\
=\int_{0}^{T}{\dot{y}_{\Phi}(\tau)u_{\Phi}(\tau){\rm d}\hbox{\hskip
0.5pt}\tau}.$
Since $H_{\circlearrowleft}$ is nonnegative then
$\int_{0}^{T}{\dot{y}_{\Phi}(\tau)u_{\Phi}(\tau){\rm d}\hbox{\hskip
0.5pt}\tau}\geq-H_{\circlearrowleft}(y_{\Phi}(0),u_{\Phi}(0))>-\infty.$
$\Box$
An example of the CCW hysteresis phenomenon is the magnetic hysteresis in
ferromagnetic material, which has CCW behavior from the input (an applied
electrical field) to the output (the magnetization). The magnetic hysteresis
can be modeled by the Coleman-Hodgdon model [5] given by
$\dot{y}_{\Phi}(t)=C_{\alpha}|\dot{u}_{\Phi}(t)|[f(u_{\Phi}(t))-y_{\Phi}(t)]+\dot{u}_{\Phi}(t)g(u_{\Phi}(t)),$
(21)
where $C_{\alpha}$ is a positive constant,
$f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a monotone increasing $C^{1}$
function, such that $f(0)=0$ and $g$ is locally Lipschitz. The Coleman-Hodgdon
model in (21) can be rewritten into the form of (14) where:
$f_{1}(y_{\Phi},u_{\Phi})=C_{\alpha}[f(u_{\Phi})-y_{\Phi}]+g(u_{\Phi}),\
f_{2}(y_{\Phi},u_{\Phi})=-C_{\alpha}[f(u_{\Phi})-y_{\Phi}]+g(u_{\Phi})$ (22)
In this case, it has the same structure as in (14) with $f_{an}=f$. Figure 3
shows the behaviour of the Coleman-Hodgdon model using the functions $f$ and
$g$ given by
$f(u_{\Phi})=bu_{\Phi},\quad g(u_{\Phi})=a,$ (23)
where $b>0$ and $a>0$. It can be easily checked that for every
$u_{\Phi}(t)\in{\mathbb{R}}$, $f_{1}$ and $f_{2}$ satisfy (15), i.e., for
every $u_{\Phi}\in AC({\mathbb{R}}_{+})$ and for every
$y_{\Phi}(0)\in{\mathbb{R}}$, the solution of (22) exists for all
$t\in{\mathbb{R}}_{+}$.
Figure 3: Behaviour of the Coleman-Hodgdon model using $f$ and $g$ as in (23)
with $b=5\times 10^{-3}$, $C_{\alpha}=1\times 10^{-2}$, $a=2.5\times 10^{-3}$
and $y_{\Phi_{0}}=0$.
Calculating the curve $\omega_{\Phi}$, we have
$\omega_{\Phi}(\sigma,y_{\Phi}(t),u_{\Phi}(t))=\left\\{\begin{array}[]{ll}b\sigma+\frac{a-b}{C_{\alpha}}+(y_{\Phi}(t)-bu_{\Phi}(t)+\frac{b-a}{C_{\alpha}})\mathop{e}^{-C_{\alpha}(\sigma-
u_{\Phi})}\ \ \sigma\in[u_{\Phi}(t),\ \infty),\\\
b\sigma+\frac{b-a}{C_{\alpha}}+(y_{\Phi}(t)-bu_{\Phi}(t)+\frac{a-b}{C_{\alpha}})\mathop{e}^{C_{\alpha}(\sigma-
u_{\Phi})}\ \ \sigma\in(-\infty,\ u_{\Phi}(t)].\end{array}\right.$ (24)
The CCW intersecting function $\Omega(y_{\Phi}(t),u_{\Phi}(t))$ is given by
$\Omega(y_{\Phi}(t),u_{\Phi}(t))=\left\\{\begin{array}[]{ll}u_{\Phi}(t)-\frac{1}{C_{\alpha}}{\rm
ln}\left[\frac{\frac{b-a}{C_{\alpha}}}{y_{\Phi}(t)-bu_{\Phi}(t)+\frac{b-a}{C_{\alpha}}}\right]\
y_{\Phi}(t)\geq f_{an}(u_{\Phi}(t)),\\\ u_{\Phi}(t)+\frac{1}{C_{\alpha}}{\rm
ln}\left[\frac{\frac{a-b}{C_{\alpha}}}{y_{\Phi}(t)-bu_{\Phi}(t)+\frac{a-b}{C_{\alpha}}}\right]\
y_{\Phi}(t)<f_{an}(u_{\Phi}(t)).\end{array}\right.$ (25)
Since $f_{1}$ and $f_{2}$ satisfy the hypotheses in Theorem III.1, $\Phi$ is
CCW. Denoting $u_{\Phi}^{*}(t)=\Omega(y_{\Phi}(t),u_{\Phi}(t))$, we can
compute explicitly $H_{\circlearrowleft}$ in (19) as follows
$H_{\circlearrowleft}(y_{\Phi}(t),u_{\Phi}(t))\\\
=\left\\{\begin{array}[]{ll}u_{\Phi}(t)y_{\Phi}(t)-\frac{1}{2}bu_{\Phi}(t)^{2}+\frac{a-b}{C_{\alpha}}(u_{\Phi}^{*}(t)-u_{\Phi}(t))\\\
+\frac{1}{C_{\alpha}}(y_{\Phi}(t)-bu_{\Phi}(t)+\frac{b-a}{C_{\alpha}})(1-\mathop{e}^{C_{\alpha}(u_{\Phi}(t)-u_{\Phi}^{*}(t))})\quad
y_{\Phi}(t)\geq f_{an}(u_{\Phi}(t)),\\\
u_{\Phi}(t)y_{\Phi}(t)-\frac{1}{2}bu_{\Phi}(t)^{2}+\frac{b-a}{C_{\alpha}}(u_{\Phi}^{*}(t)-u_{\Phi}(t))\\\
+\frac{1}{C_{\alpha}}(y_{\Phi}(t)-bu_{\Phi}(t)+\frac{a-b}{C_{\alpha}})(\mathop{e}^{C_{\alpha}(u_{\Phi}^{*}(t)-u_{\Phi}(t))}-1)\quad
y_{\Phi}(t)\leq f_{an}(u_{\Phi}(t)).\end{array}\right.$ (26)
The graphical interpretation of $H_{\circlearrowleft}$ is shown in Figure 4,
where the value of $H_{\circlearrowleft}$ at a given time $t$ is given by the
area in grey.
Figure 4: Graphical interpretation of the function
$H_{\circlearrowleft}(y_{\Phi}(t),u_{\Phi}(t))$ of the Coleman-Hodgdon model
using $f$ and $g$ as in (23) with $b=5\times 10^{-3}$, $C_{\alpha}=1\times
10^{-2}$, $a=2.5\times 10^{-3}$ and $y_{\Phi_{0}}=0$.
###### Proposition III.2
Consider the Duhem operator $\Phi$ satisfying the hypotheses in Theorem III.1.
Suppose that $f_{an}(0)=0$. Then the function $H_{\circlearrowleft}(\cdot,v)$
(where $H_{\circlearrowleft}$ is as in (19)) is radially unbounded for every
$v$.
Proof: Let us consider $v>0$. To show the properness of
$H_{\circlearrowleft}(\cdot,v)$, let us first consider the case where
$\gamma\geq f_{an}(v)$. In this case, we rewrite the function
$H_{\circlearrowleft}$, as follows
$H_{\circlearrowleft}(\gamma,v)=\int_{0}^{v}{\gamma-f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int_{v}^{\Omega(\gamma,v)}{\omega_{\Phi}(\sigma,\gamma,v)-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
Due to the property of the CCW intersecting function $\Omega$, $\gamma\geq
f_{an}(v)$ implies that $\Omega(\gamma,v)\geq v$. Hence the last term on the
RHS of the above equation is non-negative, i.e.,
$\int_{v}^{\Omega(\gamma,v)}{\omega_{\Phi}(\sigma,\gamma,v)-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}\geq 0$. Then,
$H_{\circlearrowleft}(\gamma,v)\geq\int_{0}^{v}{\gamma-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}\geq\int_{0}^{v}{\gamma-f_{an}(v){\rm
d}\hbox{\hskip 0.5pt}\sigma}=(\gamma-c)v,$ (27)
where $c:=f_{an}(v)$. Equation (27) indicates that for every $v>0$,
$H_{\circlearrowleft}(\gamma,v)\rightarrow\infty$ as
$\gamma\rightarrow\infty$.
To evaluate the other limit when $\gamma\rightarrow-\infty$, let us consider
the case when $\gamma<0$. Note that in this case $\gamma<f_{an}(v)$ due to the
monotonicity assumption on $f_{an}$ and $f_{an}(0)=0$. Rewriting
$H_{\circlearrowleft}$, we have
$\displaystyle H_{\circlearrowleft}(\gamma,v)$
$\displaystyle=\int_{0}^{\Omega(\gamma,v)}{\gamma-f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int_{\Omega(\gamma,v)}^{v}{\gamma-\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle\geq\int_{0}^{\Omega(\gamma,v)}{\gamma-f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}=\int^{0}_{\Omega(\gamma,v)}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}.$
The last inequality is obtained due to the property of the CCW intersecting
function $\Omega$, where $\Omega(\gamma,v)<v$ whenever $\gamma<f_{an}(v)$.
Since $\omega_{\Phi}$ is monotone and non-decreasing (due to the positivity of
$f_{1}$ and $f_{2}$) and using the fact that $f_{an}$ is monotone increasing
and $f_{an}(0)=0$, it can be checked that $\gamma<0$ implies that
$\Omega(\gamma,v)<0$.
Now let us fix $\bar{\gamma}$ such that $0>\bar{\gamma}>\gamma$. Using the
fact that
$\omega_{\Phi}(\sigma,\bar{\gamma},v)\geq\omega_{\Phi}(\sigma,\gamma,v)$ for
all $\sigma<v$ and using monotonicity of $f_{an}$, it follows that
$0>\bar{\Omega}>\Omega(\gamma,v)$ where the constant
$\bar{\Omega}:=\Omega(\bar{\gamma},v)$. Thus
$\displaystyle H_{\circlearrowleft}(\gamma,v)$
$\displaystyle\geq\int^{0}_{\Omega(\gamma,v)}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle=\int_{\Omega(\gamma,v)}^{\bar{\Omega}}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}+\int^{0}_{\bar{\Omega}}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle\geq\int^{0}_{\bar{\Omega}}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip
0.5pt}\sigma}\geq\int^{0}_{\bar{\Omega}}{f_{an}(\bar{\Omega})-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}$ $\displaystyle=(\gamma-
f_{an}(\bar{\Omega}))\bar{\Omega}.$
The last equality shows that as $\gamma\rightarrow-\infty$,
$H_{\circlearrowleft}\rightarrow\infty$ since $\bar{\Omega}<0$. Therefore, we
can conclude that for the case $v>0$, the function
$H_{\circlearrowleft}(\cdot,v)$ is radially unbounded.
Using similar arguments we can get the same conclusion for the case when
$v\leq 0$. $\Box$
### III-B Duhem operator with CW characterization
The CW property of the Duhem operator has been discussed in our previous
results in [24], where we also constructed a function
$H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ for the
Duhem operator such that (2) holds. Following a similar procedure as before,
the construction of the function $H_{\circlearrowright}$ requires three
functions: the traversing function $\omega_{\Phi}$, the anhysteresis function
$f_{an}$ and the intersecting function $\Lambda$. The definitions of the
functions $\omega_{\Phi}$ and $f_{an}$ are the same as those given in Section
III-A. However the CW intersecting function $\Lambda$ has a different
definition than that of the function $\Omega$.
The function $\Lambda:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is a CW intersecting
function if
$\omega_{\Phi}(\Lambda(\gamma,v),\gamma,v)=f_{an}(\Lambda(\gamma,v))$ for all
$(\gamma,v)\in{\mathbb{R}}^{2}$ and $\Lambda(\gamma,v)\leq v$ whenever
$\gamma\geq f_{an}(v)$ and $\Lambda(\gamma,v)>v$ otherwise. Here we assume
$\Lambda$ is differentiable. In [24, Lemma 1] sufficient conditions on $f_{1}$
and $f_{2}$ which ensure that such $\Lambda$ exists are $f_{an}$ be monotone
increasing and
$\displaystyle f_{1}(\gamma,v)>\frac{{\rm d}\hbox{\hskip 0.5pt}f_{an}(v)}{{\rm
d}\hbox{\hskip 0.5pt}v}+\epsilon$ whenever $\displaystyle\gamma>f_{an}(v)\ $
(28) $\displaystyle f_{2}(\gamma,v)>\frac{{\rm d}\hbox{\hskip
0.5pt}f_{an}(v)}{{\rm d}\hbox{\hskip 0.5pt}v}+\epsilon$ whenever
$\displaystyle\gamma<f_{an}(v)\ $ (29)
hold with $\epsilon>0$.
We recall our main results in [24] in the following theorem, which gives the
sufficient conditions on $\Phi$ such that it is CW.
###### Theorem III.3
[24, Theorem $1$] Consider the Duhem hysteresis operator $\Phi$ defined in
(14) with $C^{1}$ functions $f_{1},f_{2}:{\mathbb{R}}^{2}\to{\mathbb{R}}_{+}$.
Let $f_{an}$ be the corresponding anhysteresis function which satisfies (28)
and (29). Denote by $\Lambda$ the corresponding CW intersecting function.
Suppose that for all $(\gamma,v)$ in ${\mathbb{R}}^{2}$, $f_{1}(\gamma,v)\geq
f_{2}(\gamma,v)$ whenever $\gamma\leq f_{an}(v)$ and
$f_{1}(\gamma,v)<f_{2}(\gamma,v)$ otherwise. Let the anhysteresis function
$f_{an}$ satisfies $f_{an}(0)=0$. Then $\Phi$ is CW with the storage function
$H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ be given
by
$H_{\circlearrowright}(\gamma,v)=\int_{0}^{\Lambda(\gamma,v)}{f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}-\int_{v}^{\Lambda(\gamma,v)}{\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip 0.5pt}\sigma},$ (30)
The proof is similar to that of Theorem III.1 where we use also the result in
[24, Theorem $1$] which shows that $H_{\circlearrowright}$ satisfies (2).
An example of the CW hysteresis phenomenon is the friction-induced hysteresis
in mechanical system, which has CW behavior from the input (i.e., the relative
displacement) to the output (i.e., the friction force). One of the standard
model to describe friction-induced hysteresis is the Dahl model [6, 21], which
is given by
$\dot{y}_{\Phi}(t)=\rho\left|1-\frac{y_{\Phi}(t)}{F_{c}}\textrm{sgn}(\dot{u}_{\Phi}(t))\right|^{r}\\\
\textrm{sgn}\left(1-\frac{y_{\Phi}(t)}{F_{c}}\textrm{sgn}(\dot{u}_{\Phi}(t))\right)\dot{u}_{\Phi}(t),$
(31)
where $y_{\Phi}$ denotes the friction force, $u_{\Phi}$ denotes the relative
displacement, $F_{c}>0$ denotes the Coulomb friction force, $\rho>0$ denotes
the rest stiffness and $r\geq 0$ is a parameter that determines the shape of
the hysteresis loops.
The Dahl model can be described by the Duhem hysteresis operator (14) with
$f_{1}(y_{\Phi},u_{\Phi})=\rho\left|1-\frac{y_{\Phi}}{F_{c}}\right|^{r}\textrm{sgn}\left(1-\frac{y_{\Phi}}{F_{c}}\right),\
f_{2}(y_{\Phi},u_{\Phi})=\rho\left|1+\frac{y_{\Phi}}{F_{c}}\right|^{r}\textrm{sgn}\left(1+\frac{y_{\Phi}}{F_{c}}\right).$
(32)
In Figure 5, we illustrate the behavior of the Dahl model where $F_{c}=0.75$,
$\rho=1.5$ and $r=1$.
Figure 5: The input-output dynamics of the Dahl model with $F_{c}=0.75$,
$\rho=1.5$ and $r=1$.
It is immediate to check that $f_{1}$ and $f_{2}$ satisfy the hypotheses in
(15), which means that for all $u_{\Phi}\in AC({\mathbb{R}}_{+})$ and
$y_{\Phi}(0)\in{\mathbb{R}}$ the solution of (31) exists for all
$t\in{\mathbb{R}}_{+}$. The anhysteresis function of the Dahl model is
$f_{an}(u_{\Phi}(t))=0$.
Calculating the curve $\omega_{\Phi}$, we have
$\omega_{\Phi}(\sigma,y_{\Phi}(t),u_{\Phi}(t))=\left\\{\begin{array}[]{ll}F_{c}+(y_{\Phi}(t)-F_{c})\mathop{e}^{\frac{\rho}{F_{c}}(u_{\Phi}(t)-\sigma)}\quad\sigma\in[u_{\Phi}(t),\
\infty),\\\ -F_{c}+(y_{\Phi}(t)+F_{c})\mathop{e}^{\frac{\rho}{F_{c}}(\sigma-
u_{\Phi}(t))}\quad\sigma\in(-\infty,\ u_{\Phi}(t)].\end{array}\right.$ (33)
The CW intersecting function $\Lambda(y_{\Phi}(t),u_{\Phi}(t))$ is given by
$\Lambda(y_{\Phi}(t),u_{\Phi}(t))=\left\\{\begin{array}[]{ll}u_{\Phi}(t)+\frac{F_{c}}{\rho}{\rm
ln}{\frac{F_{c}}{y_{\Phi}(t)+F_{c}}}\quad y_{\Phi}(t)\geq 0,\\\
u_{\Phi}(t)-\frac{F_{c}}{\rho}{\rm ln}{\frac{-F_{c}}{y_{\Phi}(t)-F_{c}}}\quad
y_{\Phi}(t)<0,\end{array}\right.$ (34)
Figure 6: Graphical interpretation of the function
$H_{\circlearrowright}(y_{\Phi}(t),u_{\Phi}(t))$ of the Dahl model using
$f_{1}$ and $f_{2}$ as in (32) with $\sigma=1$, $F_{c}=0.5$ and
$y_{\Phi_{0}}=0$.
Denoting $u_{\Phi}^{*}(t)=\Lambda(y_{\Phi}(t),u_{\Phi}(t))$, we can compute
explicitly the function $H_{\circlearrowright}$ in (30) as follows
$H_{\circlearrowright}(y_{\Phi}(t),u_{\Phi}(t))=\left\\{\begin{array}[]{ll}-F_{c}(u_{\Phi}(t)-u_{\Phi}^{*}(t))+\frac{F_{c}}{\rho}(y_{\Phi}(t)+F_{c})(1-\mathop{e}^{\frac{\rho}{F_{c}}(u_{\Phi}^{*}(t)-u_{\Phi}(t))})\quad
y_{\Phi}(t)\geq 0,\\\
F_{c}(u_{\Phi}(t)-u_{\Phi}^{*}(t))+\frac{F_{c}}{\rho}(y_{\Phi}(t)-F_{c})(\mathop{e}^{\frac{\rho}{F_{c}}(u_{\Phi}(t)-u_{\Phi}^{*}(t))}-1)\quad
y_{\Phi}(t)<0.\end{array}\right.$
The graphical interpretation of $H_{\circlearrowright}$ is shown in Figure 6,
where the value of $H_{\circlearrowright}$ at a given time $t$ is given by the
area in grey.
###### Proposition III.4
Consider a Duhem operator $\Phi$ satisfying the hypotheses in Theorem III.3.
Suppose that $f_{an}$ is monotone increasing and $f_{an}(0)=0$. Then the
function $H_{\circlearrowright}(\cdot,v)$ (where $H_{\circlearrowright}$ is as
in (30)) is radially unbounded for every $v$.
Proof: To show the properness of $H_{\circlearrowright}(\cdot,v)$ for any
given $v$, we first consider the case $\gamma\geq f_{an}(v)$. Since the Duhem
operator $\Phi$ satisfies the hypotheses in Theorem III.3, the function
$H_{\circlearrowright}$ is nonnegative. Thus, using (30) and since
$\Lambda(f_{an}(v),v)=v$ we have
$\displaystyle H_{\circlearrowright}(\gamma,v)$ $\displaystyle\geq
H_{\circlearrowright}(\gamma,v)-H_{\circlearrowright}(f_{an}(v),v)$
$\displaystyle=\int_{\Lambda(f_{an}(v),v)}^{\Lambda(\gamma,v)}{f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}-\int_{v}^{\Lambda(\gamma,v)}{\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int_{v}^{\Lambda(f_{an}(v),v)}{\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle=\int_{v}^{\Lambda(\gamma,v)}{f_{an}(\sigma)-\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip 0.5pt}\sigma}.$
By the definition of the CW intersecting function $\Lambda$, $\gamma\geq
f_{an}(v)$ implies that $\Lambda(\gamma,v)<v$. Using the monotonicity of
$\omega_{\Phi}$, $\omega_{\Phi}(\sigma,\gamma,v)\leq\gamma$ for all
$\sigma<v$, and thus, it follows from the above inequality that
$\displaystyle H_{\circlearrowright}(\gamma,v)$
$\displaystyle\geq\int_{v}^{\Lambda(\gamma,v)}{f_{an}(\sigma)-\omega_{\Phi}(\sigma,\gamma,v){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle\geq\int_{v}^{\Lambda(\gamma,v)}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma},$
Now let us fix $\bar{\gamma}$ s.t. $f_{an}(v)<\bar{\gamma}<\gamma$. Since
$\omega_{\Phi}(\sigma,\bar{\gamma},v)<\omega_{\Phi}(\sigma,\gamma,v)$ for all
$\sigma<v$ and using the monotonicity of $f_{an}$, we have that
$\Lambda(\gamma,v)<\bar{\Lambda}$, where
$\bar{\Lambda}=\Lambda(\bar{\gamma},v)$. Therefore,
$\displaystyle H_{\circlearrowright}(\gamma,v)$
$\displaystyle\geq\int_{v}^{\Lambda(\gamma,v)}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip
0.5pt}\sigma}\geq\int_{v}^{\bar{\Lambda}}{f_{an}(\sigma)-\gamma{\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle\geq\int_{v}^{\bar{\Lambda}}{f_{an}(v)-\gamma{\rm d}\hbox{\hskip
0.5pt}\sigma}=(\gamma-c)(v-\bar{\Lambda})>0$
where $c:=f_{an}(v)$ and $\bar{\Lambda}=\Lambda(\bar{\gamma},v)$ for any given
$v$. Hence, it implies that for every $v$,
$H_{\circlearrowright}(\gamma,v)\rightarrow\infty$ as
$\gamma\rightarrow\infty$.
We can apply similar arguments to show that for every $v$,
$H_{\circlearrowright}(\gamma,v)\rightarrow\infty$ as
$\gamma\rightarrow-\infty$ by evaluating the case when $\gamma<f_{an}(v)$.
$\Box$
## IV Linear system with CCW Duhem hysteresis
In this section we analyze the stability of a feedback interconnection of a
linear system and a CCW Duhem hysteresis operator. The stability of the
closed-loop system is analyzed by exploiting the CCW or CW properties of each
subsystem.
###### Theorem IV.1
Consider a positive feedback interconnection of a minimal single-input single-
output linear system and a Duhem operator $\Phi$ as shown in Figure 1
satisfying the hypotheses in Theorem III.1 as follows
$\left.\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=Ax+Bu,\\\
y&=Cx,\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=y_{\Phi},\ u_{\Phi}=y,\end{array}\right\\}$ (35)
where $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times 1}$ and
$C\in{\mathbb{R}}^{1\times n}$. Let $\varepsilon:=(CB)^{-1}$ where we assume
that $CB>0$. Suppose that there exist $\xi>0$ and $Q=Q^{T}>0$ such that
$\displaystyle\frac{1}{2}(A^{T}Q+QA)+\varepsilon A^{T}C^{T}CA$
$\displaystyle\leq 0,$ (36) $\displaystyle QB+A^{T}C^{T}$ $\displaystyle=0,$
(37) $\displaystyle Q-\xi C^{T}C$ $\displaystyle>0,$ (38)
hold and the anhysteresis function $f_{an}$ satisfies $(f_{an}(v)-\xi v)v\leq
0$ for all $v\in{\mathbb{R}}$ (i.e. $f_{an}$ belongs to the sector $[0,\xi]$).
Then for every initial condition $(x(0),y_{\Phi}(0))$, the state trajectory of
the closed-loop system (35) is bounded and converges to the largest invariant
set in $\\{(x,y_{\Phi})|CAx+CBy_{\Phi}=0\\}$.
Proof: Using $V(x)=\frac{1}{2}x^{T}Qx$ and (36) and (37), it can be checked
that
$\displaystyle\dot{V}$ $\displaystyle=\frac{1}{2}x^{T}(A^{T}Q+QA)x+x^{T}QBu$
$\displaystyle\leq-\varepsilon x^{T}A^{T}C^{T}CAx-x^{T}A^{T}C^{T}u$
$\displaystyle=(u^{T}B^{T}C^{T}+x^{T}A^{T}C^{T})u-\varepsilon(CAx+CBu)^{T}(CAx+CBu)$
$\displaystyle=\langle\dot{y},u\rangle-\varepsilon\dot{y}^{2}.$
It follows from Lemma II.3 that the linear system is S-CCW.
By the assumptions of the theorem, the Duhem operator $\Phi$ is also CCW with
the storage function
$H_{\circlearrowleft}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ as given in
(19).
Now let $H_{cl}(x,y_{\Phi})=V(x)+H_{\circlearrowleft}(y_{\Phi},Cx)-Cxy_{\Phi}$
be the Lyapunov function of the interconnected system (35). We show first that
$H_{cl}$ is lower bounded. Substituting the representation of $V$ and
$H_{\circlearrowleft}$, we have
$\displaystyle H_{cl}$
$\displaystyle=\frac{1}{2}x^{T}Qx+zCx-\int_{0}^{Cx}{\omega_{\Phi}(\sigma,y_{\Phi},Cx)\
{\rm d}\hbox{\hskip
0.5pt}\sigma}+\int_{0}^{\Omega(y_{\Phi},Cx)}{\omega_{\Phi}(\sigma,y_{\Phi},Cx){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle-\int_{0}^{\Omega(y_{\Phi},Cx)}{f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}-Cxy_{\Phi}$
$\displaystyle=\frac{1}{2}x^{T}Qx-\int_{0}^{Cx}{f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int_{Cx}^{\Omega(y_{\Phi},Cx)}{\omega_{\Phi}(\sigma,y_{\Phi},Cx)-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle\geq\frac{1}{2}x^{T}Qx-\int_{0}^{Cx}{(f_{an}(\sigma)-\xi\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}-\int_{0}^{Cx}{\xi\sigma{\rm d}\hbox{\hskip
0.5pt}\sigma}+\int_{Cx}^{\Omega(y_{\Phi},Cx)}{\omega_{\Phi}(\sigma,y_{\Phi},Cx)-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ $\displaystyle\geq\frac{1}{2}x^{T}(Q-\xi
C^{T}C)x+\int_{Cx}^{\Omega(y_{\Phi},Cx)}{\omega_{\Phi}(\sigma,y_{\Phi},Cx)-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}.$ (39)
where the last inequality is due to the sector condition on $f_{an}$. In the
following, we will prove that the last term on the RHS of (39) is lower
bounded. Notice that since $f_{1}\geq 0$, $f_{2}\geq 0$, (17) and (18) imply
that $\frac{{\rm d}\hbox{\hskip 0.5pt}f_{an}(v)}{{\rm d}\hbox{\hskip
0.5pt}v}>\epsilon$ for some $\epsilon>0$. Hence $f_{an}$ is strictly
increasing and invertible.
Consider the case when $y_{\Phi}\geq f_{an}(Cx)$ which implies also that
$\Omega(y_{\Phi},Cx)\geq f_{an}^{-1}(y_{\Phi})\geq Cx$ by the definition of
$\Omega$. Using the monotonicity of $\omega_{\Phi}$ we have
$\int_{Cx}^{\Omega(y_{\Phi},Cx)}{\omega_{\Phi}(\sigma,y_{\Phi},Cx)-f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}\geq\int_{Cx}^{\Omega(y_{\Phi},Cx)}{y_{\Phi}-f_{an}(\sigma){\rm
d}\hbox{\hskip
0.5pt}\sigma}\geq\int_{Cx}^{f_{an}^{-1}(y_{\Phi})}{y_{\Phi}-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}.$
Define
$V(y_{\Phi},Cx):=\int_{Cx}^{f_{an}^{-1}(y_{\Phi})}{y_{\Phi}-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ and let
$c:=\frac{y_{\Phi}-f_{an}(Cx)}{2}+f_{an}(Cx)$. It follows that
$f_{an}^{-1}(y_{\Phi})\geq f_{an}^{-1}(c)$ and $f_{an}(\sigma)\leq c$ for all
$\sigma\in[Cx,f_{an}^{-1}(c)]$. Therefore
$\displaystyle V(y_{\Phi},Cx)$
$\displaystyle\geq\int_{Cx}^{f_{an}^{-1}(c)}{y_{\Phi}-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}\geq\int_{Cx}^{f_{an}^{-1}(c)}{y_{\Phi}-c\ {\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle=\frac{1}{2}(y_{\Phi}-f_{an}(Cx))(f_{an}^{-1}(c)-Cx)\geq 0.$
Thus, $\int_{Cx}^{\Omega(y_{\Phi},Cx)}{y_{\Phi}-f_{an}(\sigma){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ is lower bounded by $V(y_{\Phi},Cx)$ which is
positive definite (it is equal to zero only if $y_{\Phi}=f_{an}(Cx)$) and
$V(y_{\Phi},Cx)\rightarrow\infty$ as $y_{\Phi}\rightarrow\infty$.
When $y_{\Phi}<f_{an}(Cx)$, we can obtain the same result where
$\int_{Cx}^{\Omega(y_{\Phi},Cx)}{y_{\Phi}-f_{an}(\sigma){\rm d}\hbox{\hskip
0.5pt}\sigma}$ is lower bounded by $V(y_{\Phi},Cx)$ which is positive definite
and $V(y_{\Phi},Cx)\rightarrow\infty$ as $y_{\Phi}\rightarrow-\infty$.
Therefore, using (39), we have
$H_{cl}\geq\frac{1}{2}x^{T}(Q-\xi C^{T}C)x+V(y_{\Phi},Cx),$
which is radially unbounded.
Now computing the time derivative of $H_{cl}$, we obtain
$\dot{H}_{cl}=\dot{V}+\dot{H}_{\circlearrowleft}-C\dot{x}y_{\Phi}-Cx\dot{y}_{\Phi}\leq-\varepsilon\dot{y}^{2}.$
This inequality together with the radially unboundedness of $H_{cl}$ imply
that the trajectory $(x,y_{\Phi})$ is bounded. Using the Lasalle’s invariance
principle, we conclude that the trajectory $(x,y_{\Phi})$ of (35) converges to
the largest invariant set contained in
$M:=\\{(x,y_{\Phi})\in{\mathbb{R}}^{n}\times{\mathbb{R}}|CAx+CBy_{\Phi}=0\\}$.
$\Box$
We illustrate Theorem IV.1 in the following simple example.
###### Example IV.2
Consider
$\begin{array}[]{rl}\mathbf{P}:&\dot{x}=-x+u,\ y=x,\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=y_{\Phi},\ u_{\Phi}=y,\end{array}$
where $x(t)\in\mathbb{R}$ and the functions $f_{1}$, $f_{2}$ satisfy the
hypotheses in Theorem III.1. Using $Q=1$, it can be checked that (36) $-$ (38)
hold. Using $H_{cl}$ as in the proof of Theorem IV.1, let us define
$H_{cl}(x,y_{\Phi})=\frac{1}{2}x^{2}+H_{\circlearrowleft}(y_{\Phi},y)-yy_{\Phi}$
and a routine computation shows that
$\displaystyle\dot{H}_{cl}$
$\displaystyle\leq\dot{y}y_{\Phi}+\dot{\overbrace{\Phi(y)}}y-\dot{y}y_{\Phi}-y\dot{y}_{\Phi}-\dot{y}^{2}$
$\displaystyle=-(-x+y_{\Phi})^{2}.$
Note that $Q=1$, $C=1$, so that (55) holds for $\xi<1$. This means that the
result in Theorem IV.1 holds if the anhysteresis function $f_{an}$ satisfies
$(f_{an}(v)-\xi v)v\leq 0$, for all $v\in{\mathbb{R}}$ and $\xi<1$. In other
words, $f_{an}$ should belong to the sector $[0,\xi]$ for the stability of the
closed-loop system.
$\hfill\triangle$
The result in Theorem IV.1 deals with a positive feedback interconnection of a
linear system and a Duhem hysteresis operator. This is motivated by the study
of an interconnection between counterclockwise systems as studied in [1] for
the general case and in [23] for the linear case. In the following result, we
consider the other case where a negative feedback is used instead.
###### Theorem IV.3
Consider a negative feedback interconnection of a minimal single-input single-
output linear system and a Duhem operator $\Phi$ as shown in Figure 1
satisfying the hypotheses in Theorem III.1 as follows
$\left.\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=Ax+Bu,\\\
y&=Cx+Du,\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=-y_{\Phi},\ u_{\Phi}=y,\end{array}\right\\}$ (40)
where $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times 1}$,
$C\in{\mathbb{R}}^{1\times n}$ and $D\in{\mathbb{R}}$. Assume that there exist
$P=P^{T}>0$, $L$ and $\delta>0$ such that the following linear matrix
inequalities (LMI)
$P\left[\begin{smallmatrix}1\\\ 0^{n\times
1}\end{smallmatrix}\right]=\left[\begin{smallmatrix}D\\\
C^{T}\end{smallmatrix}\right],\\\ $ (41)
$\frac{1}{2}\left(P\left[\begin{smallmatrix}0&0^{n\times n}\\\
B&A\end{smallmatrix}\right]+\left[\begin{smallmatrix}0&B^{T}\\\ 0^{n\times
n}&A^{T}\end{smallmatrix}\right]P\right)+\delta L^{T}L\leq 0,$ (42)
hold. Then for every initial condition $(x(0),y_{\Phi}(0))$, the state
trajectory of the closed-loop system (40) is bounded and converges to the
largest invariant set in
$\\{(x,y_{\Phi})|L\left[\begin{smallmatrix}-y_{\Phi}\\\
x\end{smallmatrix}\right]=0\\}$.
Proof: By the assumptions of the theorem, the Duhem operator $\Phi$ is CCW
with the function
$H_{\circlearrowleft}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ as given in
(19).
Define the extended state space of the linear system in (40) by
$\left.\begin{array}[]{rl}\mathbf{P}_{ext}:&\begin{array}[]{rl}\dot{w}&=q,\\\
\dot{x}&=Ax+Bw,\\\ y&=Cx+Dw,\end{array}\end{array}\right\\}$ (43)
where $w=u$.
Using $V=\frac{1}{2}[w\ x^{T}]^{T}P\left[\begin{array}[]{c}w\\\
x\end{array}\right]$, a routine computation shows that
$\dot{V}=\frac{1}{2}[\begin{array}[]{cc}w&x^{T}\end{array}]\left(\left[\begin{array}[]{cc}0&B^{T}\\\
0^{n\times
n}&A^{T}\end{array}\right]P\right.+P\left.\left[\begin{array}[]{cc}0&0^{n\times
n}\\\ B&A\end{array}\right]\right)\left[\begin{array}[]{c}w\\\
x\end{array}\right]+[\begin{array}[]{cc}w&x^{T}\end{array}]P\left[\begin{array}[]{c}1\\\
0^{n\times 1}\end{array}\right]q.$
Using (41) and (42),
$\dot{V}\leq\langle
y,q\rangle-\delta\left\|L\left[\begin{array}[]{c}-y_{\Phi}\\\
x\end{array}\right]\right\|^{2}.$ (44)
This inequality (44) with $q=\dot{u}$ (by the relation in (43)) implies that
the linear system defined in (40) is CW.
Now take $H_{cl}(x,y_{\Phi})=H_{\circlearrowleft}(y_{\Phi},Cx-
Dy_{\Phi})+V(x,y_{\Phi})$ as the Lyapunov function of the interconnected
system (40), where $H_{cl}$ is radially unbounded by the non-negativity of
$H_{\circlearrowleft}$ and the properness of $V$. It is straightforward to see
that
$\displaystyle\dot{H}_{cl}$
$\displaystyle=\dot{H}_{\circlearrowleft}+\dot{V},$ $\displaystyle\leq\langle
y,\dot{u}\rangle+\langle\dot{y}_{\Phi},u_{\Phi}\rangle-\delta\left\|L\left[\begin{array}[]{c}-y_{\Phi}\\\
x\end{array}\right]\right\|^{2},$ (47)
$\displaystyle=-\delta\left\|L\left[\begin{array}[]{c}-y_{\Phi}\\\
x\end{array}\right]\right\|^{2},$ (50)
where the last equation is due to the interconnection conditions $u=-y_{\Phi}$
and $y=u_{\Phi}$. It follows from (50) and from the radial unboundedness (or
properness) of $H_{cl}$, the signals $x$ and $y_{\Phi}$ are bounded.
Based on the Lasalle’s invariance principle [17], the semiflow $(x,y_{\Phi})$
of (40) converges to the largest invariant set contained in
$M:=\\{(x,y_{\Phi})\in{\mathbb{R}}^{n}\times{\mathbb{R}}|L\left[\begin{smallmatrix}-y_{\Phi}\\\
x\end{smallmatrix}\right]=0\\}$. $\Box$
To illustrate Theorem IV.3, let us consider the following simple example.
###### Example IV.4
Let
$\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=-3x+u,\\\
y&=-2x+u,\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=-y_{\Phi},\ u_{\Phi}=y,\end{array}$
where $x(t)\in\mathbb{R}$ and the functions $f_{1}$, $f_{2}$ satisfy the
hypotheses in Theorem III.1. By using $P=\left[\begin{smallmatrix}1&-2\\\
-2&6\end{smallmatrix}\right]$, it can be checked that (41) $-$ (42) hold.
Following the same construction as in the proof of Theorem IV.3, we define
$H_{cl}(x,y_{\Phi})=\frac{1}{2}x^{T}Px+H_{\circlearrowleft}(-y_{\Phi},-2x+y_{\Phi})$
and a routine computation shows that
$\displaystyle\dot{H}_{cl}$
$\displaystyle\leq-2(-3x+y_{\Phi})^{2}+y\dot{y}_{\Phi}-\dot{\overbrace{\Phi(y)}}y$
$\displaystyle=-2(-3x+y_{\Phi})^{2}.$
Thus, we can conclude that $(x,y_{\Phi})$ converges to the invariant set where
$x=\frac{1}{3}y_{\Phi}$.
$\hfill\triangle$
## V Linear system with CW Duhem hysteresis
Dual to the result that we present in the previous section, the feedback
interconnection of a linear system and a CW Duhem hysteresis is considered in
this section.
###### Theorem V.1
Consider a negative feedback interconnection of a minimal single-input single-
output linear system and a Duhem operator $\Phi$ as shown in Figure 1
satisfying the hypotheses in Theorem III.3 as follows
$\left.\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=Ax+Bu,\\\
y&=Cx,\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=-y_{\Phi},\ u_{\Phi}=y,\end{array}\right\\}$ (51)
where $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times 1}$ and
$C\in{\mathbb{R}}^{1\times n}$. Let $\varepsilon:=(CB)^{-1}$ where we assume
$CB>0$ and assume that there exist $Q=Q^{T}>0$ such that
$\displaystyle\frac{1}{2}(A^{T}Q+QA)+\varepsilon A^{T}C^{T}CA$
$\displaystyle\leq 0,$ (52) $\displaystyle QB+A^{T}C^{T}$ $\displaystyle=0,$
(53)
hold. Then for every initial condition $(x(0),y_{\Phi}(0))$, the state
trajectory of the closed-loop system (51) is bounded and converges to the
largest invariant set in $\\{(x,y_{\Phi})|CAx-CBy_{\Phi}=0\\}$.
Proof: Let $V(x)=\frac{1}{2}x^{T}Qx$, and using (52)$-$(53), it can be checked
that
$\displaystyle\dot{V}$ $\displaystyle=\frac{1}{2}x^{T}(A^{T}Q+QA)x+x^{T}QBu$
$\displaystyle\leq\langle\dot{y},u\rangle-\varepsilon\dot{y}^{2}.$
It follows from Lemma II.3 that the linear system is S-CCW.
By the assumptions of the theorem, the Duhem operator $\Phi$ is CW with the
function $H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$
as given in (30).
Now let $H_{cl}(x,y_{\Phi})=V(x)+H_{\circlearrowright}(y_{\Phi},Cx)$ as the
Lyapunov function of the system (51). According to Proposition III.4,
$H_{\circlearrowright}(y_{\Phi},Cx)$ is radially unbounded for every $x$,
which implies that $H_{cl}(x,y_{\Phi})$ is radially unbounded.
Computing the time derivative of $H_{cl}$, we obtain
$\dot{H}_{cl}=\dot{V}+\dot{H}_{\circlearrowleft}\leq-\varepsilon\dot{y}^{2}.$
This inequality together with the radially unboundedness of $H_{cl}$ imply
that the trajectory $(x,y_{\Phi})$ is bounded. Using the Lasalle’s invariance
principle, we conclude that the trajectory $(x,y_{\Phi})$ of (51) converges to
the largest invariant set contained in
$M:=\\{(x,y_{\Phi})\in{\mathbb{R}}^{n}\times{\mathbb{R}}|CAx-CBy_{\Phi}=0\\}$.
$\Box$
To illustrate Theorem V.1 we could use the same linear system as given in the
Example IV.2.
###### Example V.2
$\begin{array}[]{rl}\mathbf{P}:&\dot{x}=-x+u,\ y=x,\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=-y_{\Phi},\ u_{\Phi}=y,\end{array}$
where $x(t)\in\mathbb{R}$ and the functions $f_{1}$, $f_{2}$ satisfy the
hypotheses in Theorem III.3. Using $Q=1$, it can be checked that (52) and (53)
hold. Define
$H_{cl}(x,y_{\Phi})=\frac{1}{2}x^{2}+H_{\circlearrowright}(y_{\Phi},y)$, a
routine computation shows that
$\displaystyle\dot{H}_{cl}$
$\displaystyle\leq\dot{y}y_{\Phi}-\dot{y}y_{\Phi}-\dot{y}^{2}$
$\displaystyle=-(-x+y_{\Phi})^{2},$
which implies that $(x,y_{\Phi})$ converges to the invariant set where
$x=y_{\Phi}$.
$\hfill\triangle$
###### Theorem V.3
Consider a positive feedback interconnection of a minimal single-input single-
output linear system and a Duhem operator $\Phi$ as shown in Figure 1
satisfying the hypotheses in Theorem III.3 as follows
$\left.\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=Ax+Bu,\\\
y&=Cx+Du,\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t),\\\
&u=y_{\Phi},\ u_{\Phi}=y,\end{array}\right\\}$ (54)
where $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times 1}$,
$C\in{\mathbb{R}}^{1\times n}$ and $D\in{\mathbb{R}}$. Assume that there exist
$P$, $L$, $\delta$ and $\eta>0$ such that
$P=P^{T}>\eta\left[\begin{smallmatrix}D^{2}&DC\\\
C^{T}D&C^{T}C\end{smallmatrix}\right]\geq 0$ and the following linear matrix
inequalities (LMI)
$P\left[\begin{smallmatrix}1\\\ 0^{n\times
1}\end{smallmatrix}\right]=\left[\begin{smallmatrix}D\\\
C^{T}\end{smallmatrix}\right],\\\ $ (55)
$\frac{1}{2}\left(P\left[\begin{smallmatrix}0&0^{n\times n}\\\
B&A\end{smallmatrix}\right]+\left[\begin{smallmatrix}0&B^{T}\\\ 0^{n\times
n}&A^{T}\end{smallmatrix}\right]P\right)+\delta L^{T}L\leq 0,$ (56)
hold. Assume further that $f_{1}(\gamma,v)\leq\frac{\eta}{2}$ and
$f_{2}(\gamma,v)\leq\frac{\eta}{2}$ for all $(\gamma,v)\in{\mathbb{R}}^{2}$.
Then for every initial condition $(x(0),y_{\Phi}(0))$, the state trajectory of
the closed-loop system (54) is bounded and converges to the largest invariant
set in $\\{(x,y_{\Phi})|L\left[\begin{smallmatrix}y_{\Phi}\\\
x\end{smallmatrix}\right]=0\\}$.
Proof: Define an extended system $\mathbf{P}_{ext}$ as in (43) and let
$V(w,x)=\frac{1}{2}\left[\begin{smallmatrix}w&\
x^{T}\end{smallmatrix}\right]P\left[\begin{smallmatrix}w\\\
x\end{smallmatrix}\right]$. Using (55), (56) and (43), we have
$\dot{V}\leq\langle
y,\dot{u}\rangle-\delta\left\|L\left[\begin{array}[]{c}y_{\Phi}\\\
x\end{array}\right]\right\|^{2}.$ (57)
Equation (57) indicates that the linear system is CW.
Next we take
$H_{cl}(x,y_{\Phi})=H_{\circlearrowright}(y_{\Phi},y)+V(y_{\Phi},x)-yy_{\Phi}$
as the Lyapunov function of the interconnected system. We will show first that
$H_{cl}$ is lower bounded. Using the definition of $V$ as above and
$H_{\circlearrowright}$ as in (30), we have
$H_{cl}=\frac{1}{2}\left[\begin{matrix}w&\
x^{T}\end{matrix}\right]P\left[\begin{matrix}w\\\
x\end{matrix}\right]+\int^{\Lambda(y_{\Phi},u_{\Phi})}_{0}{f_{an}(\sigma)-\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi}){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int^{u_{\Phi}}_{0}{\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi}){\rm
d}\hbox{\hskip 0.5pt}\sigma}-u_{\Phi}y_{\Phi}.$
Since $u_{\Phi}=y$ (by the interconnection),
$u_{\Phi}^{2}=\left[\begin{smallmatrix}w&\
x^{T}\end{smallmatrix}\right]\left[\begin{smallmatrix}D^{2}&DC\\\
C^{T}D&C^{T}C\end{smallmatrix}\right]\left[\begin{smallmatrix}w\\\
x\end{smallmatrix}\right]$. By the assumption on $P$, there exists
$\eta,\varepsilon>0$ such that $P-\eta\left[\begin{smallmatrix}D^{2}&DC\\\
C^{T}D&C^{T}C\end{smallmatrix}\right]>\varepsilon I$. Then
$\displaystyle H_{cl}$ $\displaystyle=\frac{1}{2}\left[\begin{matrix}w&\
x^{T}\end{matrix}\right]\left(P-\eta\left[\begin{matrix}D^{2}&DC\\\
C^{T}D&C^{T}C\end{matrix}\right]\right)\left[\begin{matrix}w\\\
x\end{matrix}\right]+\frac{\eta}{2}u_{\Phi}^{2}+\int^{\Lambda(y_{\Phi},u_{\Phi})}_{0}{f_{an}(\sigma)-\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi}){\rm
d}\hbox{\hskip 0.5pt}\sigma}$
$\displaystyle+\int^{u_{\Phi}}_{0}{\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi}){\rm
d}\hbox{\hskip 0.5pt}\sigma}-u_{\Phi}y_{\Phi}$
$\displaystyle\geq\frac{\varepsilon}{2}\left\|\left[\begin{matrix}w\\\
x\end{matrix}\right]\right\|^{2}+\int^{\Lambda(y_{\Phi},u_{\Phi})}_{0}{f_{an}(\sigma)-\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi}){\rm
d}\hbox{\hskip
0.5pt}\sigma}+\int^{u_{\Phi}}_{0}{\left(\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi})-y_{\Phi}+\frac{\eta}{2}u_{\Phi}\right){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ (58)
It can be checked that the second term of (58) is nonnegative. Indeed, it
follows from the property of the CW intersecting function $\Lambda$ that if
$\Lambda(y_{\Phi},u_{\Phi})\geq 0$ we have that
$f_{an}(\sigma)\geq\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi})$ for all
$\sigma\in[0,\Lambda(y_{\Phi},u_{\Phi})]$ and if
$\Lambda(y_{\Phi},u_{\Phi})<0$ then
$f_{an}(\sigma)\leq\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi})$ for all
$\sigma\in[\Lambda(y_{\Phi},u_{\Phi}),0]$.
To check whether the last term of (58) is lower bounded, we use the definition
of $\omega_{\Phi}$ given in the Section III-A. Consider the case $u_{\Phi}\geq
0$. Using the definition of $\omega_{\Phi}$ in (16), the last term of (58) can
be written by
$\displaystyle\int^{u_{\Phi}}_{0}{\left(\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi})-y_{\Phi}+\frac{\eta}{2}u_{\Phi}\right){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ $\displaystyle\
=\int^{u_{\Phi}}_{0}{\left(y_{\Phi}+\int^{\sigma}_{u_{\Phi}}{f_{2}(\omega_{\Phi}(s,y_{\Phi},u_{\Phi}),s){\rm
d}\hbox{\hskip 0.5pt}s}\right){\rm d}\hbox{\hskip
0.5pt}\sigma}+\int_{0}^{u_{\Phi}}{\frac{\eta}{2}u_{\Phi}-y_{\Phi}{\rm
d}\hbox{\hskip 0.5pt}\sigma}$ $\displaystyle\
=\int^{u_{\Phi}}_{0}{\int^{u_{\Phi}}_{\sigma}{\frac{\eta}{2}-f_{2}(\omega_{\Phi}(s,y_{\Phi},u_{\Phi}),s){\rm
d}\hbox{\hskip 0.5pt}s}{\rm d}\hbox{\hskip
0.5pt}\sigma}+\frac{\eta}{4}u_{\Phi}^{2}\geq 0,$
where the last inequality is due to fact that
$f_{2}(\gamma,v)\leq\frac{\eta}{2}$ for all $(\gamma,v)\in{\mathbb{R}}^{2}$.
In a similar way, we can obtain the non-negativity of
$\int^{u_{\Phi}}_{0}{(\omega_{\Phi}(\sigma,y_{\Phi},u_{\Phi})-y_{\Phi}+\frac{\eta}{2}u_{\Phi}){\rm
d}\hbox{\hskip 0.5pt}\sigma}$ for the case $u_{\Phi}<0$. Therefore, (58)
implies that $H_{cl}$ is lower bounded and radially unbounded.
It can be computed that
$\dot{H}_{cl}=\dot{V}+\dot{H}_{\circlearrowright}-\dot{y}y_{\Phi}-y\dot{y}_{\Phi}\leq-\delta\left\|L\left[\begin{matrix}y_{\Phi}\\\
x\end{matrix}\right]\right\|^{2}.$ (59)
Hence, by the radially unboundedness of $H_{cl}$, (59) implies that
$(x,y_{\Phi})$ is bounded. Using the Lasalle’s invariance principle, we can
conclude that the trajectory $(x,y_{\Phi})$ of (54) converges to the largest
invariant set contained in
$M:=\\{(x,y_{\Phi})\in{\mathbb{R}}^{n}\times{\mathbb{R}}|L\left[\begin{smallmatrix}y_{\Phi}\\\
x\end{smallmatrix}\right]=0\\}$. $\Box$
To illustrate Theorem V.3, let us consider the Example IV.4, where we replace
the negative feedback interconnection by a positive one.
###### Example V.4
$\begin{array}[]{rl}\mathbf{P}:&\begin{array}[]{rl}\dot{x}&=-3x+y_{\Phi},\\\
y&=-2x+y_{\Phi},\end{array}\\\
{\bf\Phi}:&\dot{y}_{\Phi}=f_{1}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi+}(t)+f_{2}(y_{\Phi}(t),u_{\Phi}(t))\dot{u}_{\Phi-}(t)\\\
&u=y_{\Phi},\ u_{\Phi}=y,\end{array}$
where $x(t)\in\mathbb{R}$ and the functions $f_{1}$, $f_{2}$ satisfy the
hypotheses in Theorem III.3. By using $P=\left[\begin{smallmatrix}1&-2\\\
-2&6\end{smallmatrix}\right]$, the conditions in (55) and (56) hold with
$L=\left[\begin{smallmatrix}1&-3\end{smallmatrix}\right]$ and $\delta=2$. Also
$P=P^{T}>\frac{1}{2}\left[\begin{smallmatrix}1&-2\\\
-2&4\end{smallmatrix}\right]$, i.e., $\eta=\frac{1}{2}$. Using
$H_{cl}(x,y_{\Phi})=\frac{1}{2}\left[\begin{smallmatrix}y_{\Phi}&x^{T}\end{smallmatrix}\right]P\left[\begin{smallmatrix}y_{\Phi}\\\
x\end{smallmatrix}\right]+H_{\circlearrowright}(y_{\Phi},-2x+y_{\Phi})-yy_{\Phi}$,
routine computation shows that
$\dot{H}_{cl}\leq-2(-3x+y_{\Phi})^{2}.$
Hence, if $f_{1}(\gamma,v)\leq\frac{1}{4}$ and
$f_{2}(\gamma,v)\leq\frac{1}{4}$ for all $(\gamma,v)\in{\mathbb{R}}^{2}$, then
$(x,y_{\Phi})$ converges to the invariant set where $x=-\frac{1}{3}y_{\Phi}$
following Theorem V.3.
$\hfill\triangle$
Figure 7: Feedback interconnection of a linear plant $\mathbf{G}$, controller
$\mathbf{C}$ and hysteresis operator ${\bf\Phi}$. (a) An interconnection
example where the plant $\mathbf{G}$ is driven by a hysteretic actuator
$\mathbf{\Phi}$; (b) An interconnection example where the dynamics of
$\mathbf{G}$ is measured by a hysteretic sensor $\mathbf{\Phi}$.
## VI Controller design
The stability analysis given in the previous sections can be used to design a
controller for a linear plant with hysteretic sensor/actuator. Consider the
closed-loop system as shown in Figure 7, where $\mathbf{G}$ and $\mathbf{C}$
are the linear plant and controller, respectively, and they are given by
$\mathbf{G}:\left\\{\begin{array}[]{rl}\dot{x}_{G}&=A_{G}x_{G}+B_{G}u_{G},\\\
y_{G}&=C_{G}x_{G}+D_{G}u_{G},\end{array}\right.\
\mathbf{C}:\left\\{\begin{array}[]{rl}\dot{x}_{C}&=A_{C}x_{C}+B_{C}u_{C},\\\
y_{C}&=C_{C}x_{C}+D_{C}u_{C}.\end{array}\right.$ (60)
Thus depending on the location of the hysteretic element, the cascaded linear
systems can be compactly written into
$\left.\begin{array}[]{rl}\dot{x}&=Ax+Bu\\\ y&=Cx+Du,\end{array}\right.$ (61)
where $x=\left[\begin{smallmatrix}x_{G}\\\ x_{C}\end{smallmatrix}\right]$ and
for the case of hysteretic actuator as shown in Figure 7(a),
$A=\left[\begin{smallmatrix}A_{G}&0\\\
B_{C}C_{G}&A_{C}\end{smallmatrix}\right]$,
$B=\left[\begin{smallmatrix}B_{G}\\\ B_{C}D_{G}\end{smallmatrix}\right]$,
$C=\left[\begin{smallmatrix}D_{C}C_{G}&C_{C}\end{smallmatrix}\right]$,
$D=D_{C}D_{G}$, or for the case of hysteretic sensor as shown in Figure 7(b),
$A=\left[\begin{smallmatrix}A_{G}&B_{G}C_{C}\\\
0&A_{C}\end{smallmatrix}\right]$, $B=\left[\begin{smallmatrix}B_{G}D_{C}\\\
B_{C}\end{smallmatrix}\right]$,
$C=\left[\begin{smallmatrix}C_{G}&D_{G}C_{C}\end{smallmatrix}\right]$,
$D=D_{G}D_{C}$. The controller design can then be carried out as follows.
* •
Control design algorithm for the case of CCW $\Phi$:
1. 1.
Determine the anhysteresis function $f_{an}$ of the Duhem operator $\Phi$ and
possibly, the desired $L$.
2. 2.
Find $\mathbf{C}$ such that either (36)-(38) or (41)-(42) holds.
3. 3.
If (36)-(38) is solvable, then $\mathbf{C}$ stabilizes the closed-loop system
with a negative feedback interconnection; otherwise
4. 4.
If (41)-(42) is solvable, then $\mathbf{C}$ stabilizes the closed-loop system
with a positive feedback interconnection.
* •
Control design algorithm for the case of CW $\Phi$:
1. 1.
Determine the functions $f_{1}$ and $f_{2}$ of the Duhem operator $\Phi$ and
possibly, the desired $L$.
2. 2.
Find $\mathbf{C}$ such that either (52)-(53) or (55)-(56) holds.
3. 3.
If (52)-(53) is solvable, then $\mathbf{C}$ stabilizes the closed-loop system
with a negative feedback interconnection; otherwise
4. 4.
If (55)-(56) is solvable, then $\mathbf{C}$ stabilizes the closed-loop system
with a positive feedback interconnection.
Putting (61) into the setting of our main results in Theorem IV.3, IV.1, V.1
and V.3, the invariant set is contained in
$M:=\\{(x_{G},x_{C},y_{\Phi})|N\left[\begin{smallmatrix}x_{G}\\\ x_{C}\\\
y_{\Phi}\end{smallmatrix}\right]=0\\}$ where the matrix $N$ can also become a
design parameter for determining $\mathbf{C}$.
## VII Numerical examples
Figure 8: Mass-damper-spring system connected with a hysteretic actuator
As an example, we consider a mass-damper-spring system with a hysteretic
actuator denoted by $\Phi$, as shown in Figure 8, where $m$ is the mass, $b$
is the damping constant, $k$ is the spring constant and $x$ denotes the
displacement of the mass. Let $m=1$, $b=2$ and $k=1$, then the mass-damper-
spring system is given by
$\displaystyle\dot{x}$ $\displaystyle=\left(\begin{array}[]{cc}0&1\\\ -1&-2\\\
\end{array}\right)x+\left(\begin{array}[]{c}0\\\ 1\\\ \end{array}\right)u,$
(66) $\displaystyle y$ $\displaystyle=\left(\begin{array}[]{cc}1&0\\\
\end{array}\right)x+u.$ (68)
### VII-A CCW hysteretic actuator
Let us first consider the case when the hysteretic actuator has CCW I/O
dynamics, such as piezo-actuators [15]. Assume that the actuator is
represented by the Duhem operator (14) where
$f_{1}(\gamma,v)=-\gamma+0.475v+0.3,\ f_{2}(\gamma,v)=\gamma-0.475v+0.3,\
\forall(\gamma,v)\in{\mathbb{R}}^{2}.$ (69)
It can be verified that $f_{an}(v)=0.475v$ and the functions $f_{1}$ and
$f_{2}$ satisfy the hypotheses given in Theorem III.1.
With $A_{c}=\left[\begin{smallmatrix}0&1\\\ -2&-4\end{smallmatrix}\right]$,
$B_{c}=\left[\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right]$,
$C_{c}=\left[\begin{smallmatrix}-1.5&-2\end{smallmatrix}\right]$ and
$D_{c}=1$, conditions (41)-(42) are solvable with
$P=\left[\begin{smallmatrix}1&1&0&-1.5&-2\\\ 1&7.74&5.51&-8.74&-15.86\\\
0&5.51&7.4&-5.51&-14.36\\\ -1.5&-8.74&-5.51&10.24&17.86\\\
-2&-15.86&-14.36&17.86&38.36\end{smallmatrix}\right]$ and $L=[0\ 0\ 1/4\ 0\
0]$. Hence the controller $\mathbf{C}$ can stabilize the closed-loop system
with negative feedback interconnection. In this case, $N=[0\ 1/4\ 0\ 0\ 0]$.
According to Theorem IV.3, the velocity of the mass-damper-spring system
converges to zero and the position of the mass-damper-spring system converges
to a constant. The closed-loop system is simulated in Matlab/Simulink with the
initial condition $x(0)=[-10\ 5]^{T}$ and the results are shown in Figure
9(a).
On the other hand, since we have $f_{an}(v)=0.475v$, then by taking
$A_{c}=\left[\begin{smallmatrix}0&1\\\ -2&-4\end{smallmatrix}\right]$,
$B_{c}=\left[\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right]$,
$C_{c}=\left[\begin{smallmatrix}1&1\end{smallmatrix}\right]$ and $D_{c}=0$, it
can be checked that (36)-(37) holds with $\xi=0.5$ and
$Q=\left[\begin{smallmatrix}6&1&-6&-2\\\ 1&4&-1&-4\\\ -6&-1&7&3\\\
-2&-4&3&7\end{smallmatrix}\right]$. In this case
$N=\left[\begin{smallmatrix}1&0&-2&-3&1\end{smallmatrix}\right]$. Moreover,
$f_{an}$ belongs to the sector $[0,0.5]$. Similar to the previous case, it
follows from Theorem IV.1 that the velocity of the mass-damper-spring system
converges to zero and the position of the mass-damper-spring system converges
to a constant. The simulation results is shown in Figure 9(b).
Figure 9: Simulation results of the numerical example with CCW hysteretic
actuator. (a) The negative feedback interconnection case with the initial
condition $x(0)=[-10\ 5]^{T}$; (b) The positive feedback interconnection case
with the initial condition $x(0)=[-10\ 10]^{T}$.
### VII-B CW hysteretic actuator
For the case of a CW hysteretic actuator, see for example the
magnetorheological (MR) damper used in the structure control [25], the mass-
damper-spring system is given by (68). Assume that the actuator is represented
by the Duhem operator (14) where
$f_{1}(\gamma,v)=0.25(1-\gamma),\ f_{2}(\gamma,v)=0.25(1+\gamma),\
\forall(\gamma,v)\in{\mathbb{R}}^{2}.$ (70)
The anhysteresis function for this Duhem operator is $f_{an}=0$. It can be
shown that $f_{1}\leq 0.25$ and $f_{2}\leq 0.25$ for all $v\in{\mathbb{R}}$.
In addition $f_{1}$ and $f_{2}$ satisfy the hypotheses in Theorem III.3, hence
the Duhem operator with (70) is CW.
With $A_{c}=\left[\begin{smallmatrix}0&1\\\ -2&-4\end{smallmatrix}\right]$,
$B_{c}=\left[\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right]$,
$C_{c}=\left[\begin{smallmatrix}1&1\end{smallmatrix}\right]$ and $D_{c}=0$,
the conditions (52)-(53) are solvable with
$P=\left[\begin{smallmatrix}5&1&-5&-2\\\ 1&3&-1&-3\\\ -5&-1&6&3\\\
-2&-3&3&6\end{smallmatrix}\right]$. Hence the controller $\mathbf{C}$ can
stabilize the closed-loop system with negative feedback interconnection. In
this case, $N=[1\ 0\ -2\ -3\ 1]$. According to Theorem V.1, the velocity of
the mass-damper-spring system converges to zero and the position of the mass-
damper-spring system converges to a constant. The simulation results are shown
in Figure 10(a) with the initial condition $x(0)=[10\ 5]^{T}$.
Since we have $f_{1}\leq 0.25$ and $f_{2}\leq 0.25$ for all
$v\in{\mathbb{R}}$, by taking $A_{c}=\left[\begin{smallmatrix}0&1\\\
-2&-3\end{smallmatrix}\right]$, $B_{c}=\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$,
$C_{c}=\left[\begin{smallmatrix}-3&-1\end{smallmatrix}\right]$ and $D_{c}=2$,
it can be checked that (55)-(56) holds with $\delta=1$, $\eta=0.5$,
$L=\left[\begin{smallmatrix}0&1/4&0&0&0\end{smallmatrix}\right]$ and
$P=\left[\begin{smallmatrix}2&2&0&-3&-1\\\ 2&30.86&15.83&-32.86&-26.9\\\
0&15.83&38.26&-15.83&-51.4\\\ -3&-32.86&-15.83&35.86&27.9\\\
-1&-26.9&-51.4&27.9&74.54\end{smallmatrix}\right]$. It follows from Theorem
V.3 that the velocity of the mass-damper-spring system converges to zero and
the position of the mass-damper-spring system converges to a constant. The
simulation results is shown in Figure 10(b).
Figure 10: Simulation results of the numerical example with CW hysteretic
actuator. (a) The negative feedback interconnection case with the initial
condition $x(0)=[10\ 5]^{T}$; (b) The positive feedback interconnection case
with the initial condition $x(0)=[10\ -5]^{T}$.
## VIII Conclusions
It has been shown in this paper that the stability analysis of a linear system
with hysteresis nonlinearity is accommodated by exploiting the I/O property of
the corresponding hysteresis operator. Furthermore, the stability analysis
enables a straightforward control design methodology for a plant with
hysteresis nonlinearity without having to know precisely the parameters of the
hysteresis operator. It offers a different paradigm in the design of
controller for such systems where we do not need to define an inverse
hysteresis operator which is commonly used in practice. The dissipativity
approach which is used in this paper can be extended directly to nonlinear
plants with hysteresis nonlinearity. One possible class of nonlinear plants
which can be treated with our approach is the CCW systems as studied by Angeli
[1] and by van der Schaft [26].
## References
* [1] D. Angeli, “Systems with Counterclockwise Input-Output Dynamics”, IEEE Transactions on Automatic Control, vol. 51, no. 7, pp. 1130-1143, 2006.
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* [3] G. Bertotti & I. D. Mayergoyz, The Science of Hysteresis: Mathematical Modeling and Applications, Academic Press, San Diego, 2006.
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* [5] Bernard. D. Coleman, Marion. L. Hodgdon, “A Constitutive Relation for Rate-independent Hysteresis in Ferromagnetically Soft Materials”, International Journal of Engineering Science, vol. 24, no. 6, pp. 897-919, 1986.
* [6] P. Dahl, “Solid Friction Damping of Mechanical Vibrations”, AIAA J., vol. 14, no. 2, pp. 1675-1682, 1976.
* [7] D. Damjanovic, Hysteresis in Piezoelectric and Ferroelectric Materials, The Science of Hysteresis, I. Mayergoyz and G. Bertotti (editors), vol. 3, 2005.
* [8] R.B. Gorbet, K.A. Morris, “Generalized Dissipation in Hysteretic Systems”, Proc. IEEE Conf. Dec. Contr., Tampa, 1998.
* [9] B. Jayawardhana, V. Andrieu, “Sufficient Conditions for Dissipativity on Duhem Hysteresis Model”, Proc. IEEE Conf. Dec. Contr., Shanghai, 2009.
* [10] B. Jayawardhana, Ruiyue Ouyang, V. Andrieu, “Dissipativity of General Duhem Hysteresis Models”, Proc. IEEE Conf. Dec. Contr., Orlando, 2011.
* [11] B. Jayawardhana, Ruiyue Ouyang, V. Andrieu, “Stability of Systems with Duhem Hysteresis Operator: Dissipativity Approach”, Automatica, To appear.
* [12] D. C. Jiles, D. L. Atherton, “Theory of Ferromagnetic Hysteresis”, Journal of Magnetism and Magnetic Material, vol. 61, no. 1-2, pp. 48-6, 1986.
* [13] T. Kamada, T. Fujita, T. Hatayama, T. Arikabe, N. Murai, S. Aizawa and K. Tohyama, “Active Vibration Control of Frame Structures with Smart Structures using Piezoelectric Actuators”, Journal of Smart Material and Structures, vol. 6, no. 4, pp. 448-456, 1997.
* [14] H.K. Khalil. Nonlinear Systems, 3rd edition, Prentice-Hall, Upper Saddle River, NJ, 2002.
* [15] C. J. Lin, S. R. Yang, “Precise Positioning of Piezo-actuated Stages using Hysteresis-observer based Control”, Mechatronics, vol. 16, no. 7, pp. 417-426, 2006.
* [16] H. Logemann & E.P. Ryan, “Systems with Hysteresis in the Feedback Loop: Existence, Regularity and Asymptotic Behaviour of Solutions”, ESAIM Control, Optimiz. & Calculus of Variations, vol. 9, pp. 169-196, 2003.
* [17] H. Logemann, E. P. Ryan, “Asymptotic Behaviour of Nonlinear Systems”, American Mathematical Monthly, vol. 111, no. 10, pp. 864-889, 2004.
* [18] J. W. Macki, P. Nistri, P. Zecca, “Mathematical Models for Hysteresis”, SIAM Review, vol. 35, no. 1, pp. 94–123, 1993.
* [19] J. Oh, D. S. Bernstein, “Semilinear Duhem Model for Rate-independent and Rate-dependent Hysteresis”, IEEE Trans. Automat. Contr., vol. 50, no. 5, pp. 631–645, 2005.
* [20] A. K. Padthe, J. Oh and D. S. Bernstein, “Counterclockwise Dynamics of a Rate-independent Semilinear Duhem Model”, Proc. IEEE Conf. Dec. Contr., Seville, 2005.
* [21] A. K. Padthe, B. Drincic, J. Oh, D. D. Rizos, S. D. Fassois and D. S. Bernstein, “Duhem modeling of Friction-Induced Hysteresis”, IEEE Control System Magazine, vol. 28, no. 5, pp. 90-107, 2008.
* [22] T. Pare, A. Hassabi and J. J. How, “A KYP Lemma and Invariance Principle for Systems with Multiple Hysteresis Non-linearities”, Int. J. Contr.,vol. 74, no. 11, pp. 1140-1157, 2001.
* [23] I. R. Petersen and A. Lanzon, “Feedback Control of Negative-imaginary System”, IEEE Control System Magazine, vol. 30, no. 5, pp. 54-72, 2010.
* [24] R. Ouyang, V. Andrieu, Bayu Jayawardhana, “On the Characterization of the Duhem Hysteresis Operator with Clockwise Input-Output Dynamics”, submitted, http://arxiv.org/abs/1201.2035.
* [25] C. Sakai, H. Ohmori, A. Sano “Modeling of MR Damper with Hysteresis for Adaptive Vibration Control”, Proc. IEEE Conf. Dec. Contr., Maui, 2003.
* [26] A.J. van der Schaft, “Positive Feedback Interconnection of Hamiltonian Systems”, Proc. IEEE Conf. Dec. Contr., Orlando, 2011.
* [27] S. Tarbouriech, I. Queinnec, C. Prieur, “Stability Analysis and Stabilization of Systems with Backlash in the Feedback Loop”, IEEE Transactions on Automatic Control, submitted.
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|
arxiv-papers
| 2012-06-14T10:26:09 |
2024-09-04T02:49:31.777261
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruiyue Ouyang and Bayu Jayawardhana",
"submitter": "Bayu Jayawardhana",
"url": "https://arxiv.org/abs/1206.3065"
}
|
1206.3318
|
# On Local Regret
Michael Bowling
Computing Science Department
University of Alberta
Edmonton, Alberta T6G2E8 Canada
bowling@cs.ualberta.ca Martin Zinkevich
Yahoo! Research
Santa Clara, CA 95051 USA
maz@yahoo-inc.com
###### Abstract
Online learning aims to perform nearly as well as the best hypothesis in
hindsight. For some hypothesis classes, though, even finding the best
hypothesis offline is challenging. In such offline cases, local search
techniques are often employed and only local optimality guaranteed. For online
decision-making with such hypothesis classes, we introduce local regret, a
generalization of regret that aims to perform nearly as well as only nearby
hypotheses. We then present a general algorithm to minimize local regret with
arbitrary locality graphs. We also show how the graph structure can be
exploited to drastically speed learning. These algorithms are then
demonstrated on a diverse set of online problems: online disjunct learning,
online Max-SAT, and online decision tree learning.
## 1 Introduction
An online learning task involves repeatedly taking actions and, after an
action is chosen, observing the result of that action. This is in contrast to
offline learning where the decisions are made based on a fixed batch of
training data. As a consequence offline learning typically requires i.i.d.
assumptions about how the results of actions are generated (on the training
data, and all future data). In online learning, no such assumptions are
required. Instead, the metric of performance used is regret: the amount of
additional utility that could have been gained if some alternative sequence of
actions had been chosen. The set of alternative sequences that are considered
defines the notion of regret. Regret is more than just a measure of
performance, though, it also guides algorithms. For specific notions of
regret, no-regret algorithms exist, for which the total regret is growing at
worst sublinearly with time, hence their average regret goes to zero. These
guarantees can be made with no i.i.d., or equivalent assumption, on the
results of the actions.
One traditional drawback of regret concepts is that the number of alternatives
considered must be finite. This is typically achieved by assuming the number
of available actions is finite, and for practical purposes, small. In offline
learning this is not at all the case: offline hypothesis classes are usually
very large, if not infinite. There have been attempts to achieve regret
guarantees for infinite action spaces, but these have all required assumptions
to be made on the action outcomes (e.g., convexity or smoothness). In this
work, we propose new notions of regret, specifically for very large or
infinite action sets, while avoiding any significant assumptions on the
sequence of action outcomes. Instead, the action set is assumed to come
equipped with a notion of locality, and regret is redefined to respect this
notion of locality. This approach allows the online paradigm with its style of
regret guarantees to be applied to previously intractable tasks and hypothesis
classes.
## 2 Background
For $t\in\\{1,2,\ldots\\}$, let $a^{t}\in A$ be the action at time $t$, and
$u^{t}:A\rightarrow\mathbb{R}$ be the utility function over actions at time
$t$.
###### Requirement 1.
For all $t$, $\max_{a,b\in A}|u^{t}(a)-u^{t}(b)|\leq\Delta$.
The basic building block of regret is the additional utility that could have
been gained if some action $b$ was chosen in place of action $a$:
$R^{T}_{a,b}=\sum_{t=1}^{T}1(a^{t}=a)\left(u^{t}(b)-u^{t}(a)\right)$, where
$1(\text{\it condition})$ is equal to $1$ when condition is true and $0$
otherwise. We can use this building block to define the traditional notions of
regret.
$\displaystyle R^{T}_{\text{\rm internal}}=\max_{a,b\in
A}R^{T,+}_{a,b}\quad\quad R^{T}_{\text{\rm swap}}=\sum_{a\in A}\max_{b\in
A}R^{T,+}_{a,b}$ (1) $\displaystyle R^{T}_{\text{\rm external}}=\max_{b\in
A}\left(\sum_{a\in A}R^{T}_{a,b}\right)^{+}$ (2)
where $x^{+}=\max(x,0)$ so that $R^{T,+}_{a,b}=\max(R^{T}_{a,b},0)$. Internal
regret (Hart and Mas-Colell, 2002) is the maximum utility that could be gained
if one action had been chosen in place of some other action. Swap regret
(Greenwald and Jafari, 2003) is the maximum utility gained if each action
could be replaced by another. External regret (Hannan, 1957), which is the
original pioneering concept of regret, is the maximum utility gained by
replacing all actions with one particular action. This is the most relaxed of
the three concepts, and while the others must concern themselves with
$|A|^{2}$ possible regret values (for all pairs of actions) external regret
only need worry about $|A|$ regret values. So although the guarantee is
weaker, it is a simpler concept to learn which can make it considerably more
attractive. These three regret notions have the following relationships.
$\displaystyle R^{T}_{\text{\rm internal}}$ $\displaystyle\leq
R^{T}_{\text{\rm swap}}\leq|A|R^{T}_{\text{\rm internal}}$ $\displaystyle
R^{T}_{\text{\rm external}}$ $\displaystyle\leq R^{T}_{\text{\rm swap}}$ (3)
### 2.1 Infinite Action Spaces
This paper considers situations where $A$ is infinite. To keep the notation
simple, we will use max operations over actions to mean suprema operations and
summations over actions to mean the suprema of the sum over all finite subsets
of actions. Since we will be focused on regret over a finite time period,
there will only ever be a finite set of actually selected actions and, hence
only a finite number of non-zero regrets, $R^{T}_{a,b}$. The summations over
actions will always be thought to be restricted to this finite set.
None of the three traditional regret concepts are well-suited to $A$ being
infinite. Not only does $|A|$ appear in the regret bounds, but one can
demonstrate that it is impossible to have no regret in some infinite cases.
Consider $A=\mathbb{N}$ and let $u^{t}$ be a step function, so $u^{t}(a)=1$ if
$a>y^{t}$ for some $y^{t}$ and $0$ otherwise. Imagine $y^{t}$ is selected so
that $\Pr[a^{t}>y^{t}|u^{1,\ldots,T-1},a^{1,\ldots,T-1}]\leq 0.001$, which is
always possible. Essentially, high utility is always just beyond the largest
action selected. Now, consider $y^{*}=1+\max_{t\leq T}y^{t}$. In expectation
$\frac{1}{T}\sum_{t=1}^{T}u^{t}(a_{t})\leq 0.001$ while
$\frac{1}{T}\sum_{t=1}^{T}u^{t}(y^{*})=1$ (i.e., there is large internal and
external regret for not having played $y^{*}$,) so the average regret cannot
approach zero.
Most attempts to handle infinite action spaces have proceeded by making
assumptions on both $A$ and $u$. For example, if $A$ is a compact, convex
subset of $\mathbb{R}^{n}$ and the utilities are convex with bounded gradient
on $A$, then you can minimize regret even though $A$ is infinite (Zinkevich,
2003). We take an alternative approach where we make use of a notion of
locality on the set $A$, and modify regret concepts to respect this locality.
Different notions of locality then result in different notions of regret.
Although this typically results in a weaker form of regret for finite sets, it
breaks all dependence of regret on the size of $A$ and allows it to even be
applied when $A$ is infinite and $u$ is an arbitrary (although still bounded)
function. Wide range regret methods Lehrer (2003) can also bound regret with
respect to a set of (countably) infinite “alternatives”, but unlike our
results, their asymptotic bound does not apply uniformly across the set, and
uniform finite-time bounds depend upon a finite action space Blum and Mansour
(2007).
## 3 Local Regret Concepts
Let $G=(V,E)$ be a directed graph on the set of actions, i.e., $V=A$. We do
not assume $A$ is finite, but we do assume $G$ has bounded out-degree
$D=\max_{a\in V}|\\{b:(a,b)\in E\\}|$. This graph can be viewed as defining a
notion of locality. The semantics of an edge from $a$ to $b$ is that one
should consider possibly taking action $b$ in place of action $a$. Or rather,
if there is no edge from $a$ to $b$ then one need not have any regret for not
having taken action $b$ when $a$ was taken. By limiting regret only to the
edges in this graph, we get the notion of local regret. Just as with
traditional regret, which we will now refer to as global regret, we can define
different variants of regret.
$\displaystyle R^{T}_{\text{\rm localinternal}}$ $\displaystyle=\max_{(a,b)\in
E}R^{T,+}_{a,b}$ $\displaystyle R^{T}_{\text{\rm localswap}}$
$\displaystyle=\sum_{a\in A}\max_{b:(a,b)\in E}R^{T,+}_{a,b}$ (4)
Local internal and local swap regret just involve limiting regret to edges in
$G$. Local external regret is more subtle and requires a notion of edge
lengths. For all edges $(i,j)\in E$, let $c(i,j)>0$ be the edge’s positive
length. Define $\text{\rm d}(a,b)$ to be the sum of the edge lengths on a
shortest path from vertex $a$ to vertex $b$, and $E^{b}=\\{(i,j)\in
E:d(i,j)=c(i,j)+d(j,b)\\}$ to be the set of edges that are on any shortest
path to vertex $b$.
$R^{T}_{\text{\rm localexternal}}=\max_{b\in A}\left(\sum_{(i,j)\in
E^{b}}R^{T}_{i,j}/D\right)^{+}$ (5)
Global external regret considers changing all actions to some target action,
regardless of locality or distance between the actions. In local external
regret, only adjacent actions are considered, and so actions are only replaced
with actions that take one step toward the target action. The factor of $1/D$
scales the regret of any one action by the out-degree, which is the maximum
number of actions that could be one-step along a shortest path. This keeps
local external regret on the same scale as local swap regret.
It is easy to see that these concepts hold the same relationships between each
other as their global counterparts.
$\displaystyle R^{T}_{\text{\rm localinternal}}$ $\displaystyle\leq
R^{T}_{\text{\rm localswap}}\leq|A|R^{T}_{\text{\rm localinternal}}$ (6)
$\displaystyle R^{T}_{\text{\rm localexternal}}$ $\displaystyle\leq
R^{T}_{\text{\rm localswap}}$ (7)
More interestingly, in complete graphs where there is an edge between every
pair of actions (all with unit lengths) and so everything is local, we can
exactly equate global and local regret.
###### Theorem 1.
If $G$ is a complete graph with unit edge lengths then,
$\displaystyle R^{T}_{\text{\rm localinternal}}$
$\displaystyle=R^{T}_{\text{\rm internal}}$ $\displaystyle R^{T}_{\text{\rm
localswap}}$ $\displaystyle=R^{T}_{\text{\rm swap}}$ and $\displaystyle
R^{T}_{\text{\rm localexternal}}=R^{T}_{\text{\rm external}}/D.$ (8)
###### Proof.
$\displaystyle R^{T}_{\text{\rm localinternal}}$ $\displaystyle=\max_{(a,b)\in
E}R^{T,+}_{a,b}=\max_{a,b\in A}R^{T,+}_{a,b}=R^{T}_{\text{\rm internal}}$ (9)
$\displaystyle R^{T}_{\text{\rm localswap}}$ $\displaystyle=\sum_{a\in
A}\max_{b:(a,b)\in E}R^{T,+}_{a,b}=\sum_{a\in A}\max_{b\in
A}R^{T,+}_{a,b}=R^{T}_{\text{\rm swap}}$ (10) $\displaystyle R^{T}_{\text{\rm
localexternal}}$ $\displaystyle=\max_{b\in A}\sum_{(i,j)\in
E^{b}}R^{T,+}_{i,j}/D$ (11) $\displaystyle=1/D\max_{b\in A}\sum_{a\in
A}R^{T,+}_{a,b}=R^{T}_{\text{\rm external}}/D$ (12)
∎
So our concepts of local regret match up with global regret when the graph is
complete. Of course, we are not really interested in complete graphs, but
rather more intricate locality structures with a large or infinite number of
vertices, but a small out-degree. Before going on to present algorithms for
minimizing local regret, we consider possible graphs for three different
online decision tasks to illustrate where the graphs come from and what form
they might take.
###### Example 1 (Online Max-3SAT).
Consider an online version of Max-3SAT. The task is to choose an assignment
for $n$ boolean variables: $A=\\{0,1\\}^{n}$. After an assignment is chosen a
clause is observed; the utility is 1 if the clause is satisfied by the chosen
assignment, 0 otherwise. Note that $|A|=2^{n}$ which is computationally
intractable for global regret concepts if $n$ is even moderately large. One
possible locality graph for this hypothesis class is the hypercube with an
edge from $a$ to $b$ if and only if $a$ and $b$ differ on the assignment of
exactly one variable (see Figure 1), and all edges have unit lengths. So the
out-degree $D$ for this graph is only $n$. Local regret, then, corresponds to
the regret for not having changed the assignment of just one variable. In
essence, minimizing this concept of regret is the online equivalent of local
search (e.g., WalkSAT (Selman et al., 1993)) on the maximum satisfiability
problem, an offline task where all of the clauses are known up front.
Figure 1: Example graphs. (a) Graph for Max-3SAT and disjuncts ($n=3$). (b)
Part of graph for decision trees ($n=2$), where edges to and from the dashed
boxes represent edges to and from every vertex in the box.
###### Example 2 (Online Disjunct Learning).
Consider a boolean online classification task where input features are boolean
vectors $x\in\\{0,1\\}^{n}$ and the target $y$ is also boolean. Consider
$A=\\{0,1\\}^{n}$, to be the set of all disjuncts such that $a\in A$
corresponds to the disjunct $x_{i_{1}}\vee x_{i_{2}}\vee\ldots\vee x_{i_{k}}$
where $i_{1\leq j\leq k}$ are all of the $k$ indices of $a$ such that
$a_{i_{j}}=1$. In this online task, one must repeatedly choose a disjunct and
then observe an instance which includes a feature vector and the correct
response. There is a utility of 1 if the chosen disjunct over the feature
vector results in the correct response; 0 otherwise. Although a very different
task, the action space $A=\\{0,1\\}^{n}$ is the same as with Online Max-SAT
and we can consider the same locality structure as that proposed for
disjuncts: a hypercube with unit length edges for adding or removing a single
variable to the disjunction (see Figure 1). And as before $|A|=2^{n}$ while
$D=n$.
###### Example 3 (Online Decision Tree Learning).
Imagine the same boolean online classification task for learning disjuncts,
but the hypothesis class is the set of all possible decision trees. The number
of possible decision trees for $n$ boolean variables is more than a staggering
$2^{2^{n}}$, which for any practical purpose is infinite. We can construct a
graph structure that mimics the way decision trees are typically constructed
offline, such as with C4.5 (Quinlan, 1993). In the graph $G$, add an edge from
one decision tree to another if and only if the latter can be constructed by
choosing any node (internal or leaf) of the former and replacing the subtree
rooted at the node with a decision stump or a label. There is one exception:
you cannot replace a non-leaf subtree with a stump splitting on the same
variable as that of the root of the subtree. See Figure 1 for a portion of the
graph. Edges that replace a subtree with a label have length 1, while edges
replacing a subtree with a stump (being a more complex change) have distance
1.1. So, we have local regret for not having further refined a leaf or
collapsing a subtree to a simpler stump or leaf. Notice that the graph edges
in this case are not all symmetric (viz., collapsing edges). In essence, this
is the online equivalent of tree splitting algorithms. While $|A|\geq
2^{2^{n}}$, the out-degree is no more than $(n+1)2^{n+1}$. The maximum size of
the out-degree still appears disconcertingly large, and we will return to this
issue in Section 5 where we show how we can exploit the graph structure to
further simplify learning.
## 4 An Algorithm for Local Swap Regret
We now present an algorithm for minimizing local swap regret, similar to
global swap regret algorithms (Hart and Mas-Colell, 2002; Greenwald and
Jafari, 2003), but with substantial differences. The algorithm essentially
chooses actions according to the stationary distribution of a Markov process
on the graph, with the transition probabilities on the edges being
proportional to the accumulated regrets. However there are two caveats that
are needed for it to handle infinite graphs: it is prevented from playing
beyond a particular distance from a designated root vertex, and there is an
internal bias towards the actual actions chosen.
Formally, let root be some designated vertex. Define $\text{\rm d}_{1}$ to be
the unweighted shortest path distance between two vertices. Define the level
of a vertex as its distance from root: ${\cal L}(v)=\text{\rm d}_{1}(\text{\rm
root},v)$. Note that, ${\cal L}(\text{\rm root})=0$, and $\forall(i,j)\in E$,
${\cal L}(j)\leq{\cal L}(i)+1$. All of the algorithms in this paper take a
parameter $L$, and will never choose actions at a level greater than $L$. In
addition, the algorithms all maintain values $\tilde{R}^{t}_{i,j}$ (which are
biased versions of $R^{t}_{i,j}$) and use these to compute $\pi^{t}_{j}$, the
probability of choosing action $j$ at time $t$. These probabilities are always
computed according to the following requirement, which is a generalization of
(Hart and Mas-Colell, 2002; Greenwald and Jafari, 2003).
###### Requirement 2.
Given a parameter $L$, for all $t\leq T$, and some $\tilde{R}^{t,+}_{i,j}$ let
$\pi^{t+1}$ be such that
1. (a)
$\sum_{j\in V}\pi^{t+1}_{j}=1$, and $\forall j\in V$, $\pi^{t+1}_{j}\geq 0$
2. (b)
$\forall j\in V$ such that ${\cal L}(j)>L$, $\pi^{t+1}_{j}=0$.
3. (c)
$\forall j\in V$ such that $1\leq{\cal L}(j)\leq L$,
$\pi^{t+1}_{j}=\sum_{i:(i,j)\in
E}(\tilde{R}^{t,+}_{i,j}/M)\pi^{t+1}_{i}+(1-\sum_{k:(j,k)\in
E}\tilde{R}^{t,+}_{j,k}/M)\pi^{t+1}_{j}$
4. (d)
$\pi^{t+1}_{\text{\rm root}}=\sum_{i:(i,\text{\rm root})\in
E}(\tilde{R}^{t,+}_{i,\text{\rm root}}/M)\pi^{t+1}_{i}+\sum_{j:{\cal
L}(j)=L+1}\sum_{i:(i,j)\in
E}(\tilde{R}^{t,+}_{i,j}/M)\pi^{t+1}_{i}+(1-\sum_{j:(\text{\rm root},j)\in
E}\tilde{R}^{t,+}_{\text{\rm root},j}/M)\pi^{t+1}_{\text{\rm root}}$
5. (e)
If there exists $j\in V$ such that $\pi^{t+1}_{j}>0$ and $\sum_{k:(j,k)\in
E}\tilde{R}^{t,+}_{j,k}=0$, then for all $j\in V$ where $\pi^{t+1}_{j}>0$,
$\sum_{k:(j,k)\in E}\tilde{R}^{t,+}_{j,k}=0$, and we call such a $\pi^{t+1}$
degenerate.
where $M=\max_{(i,j)\in E}\tilde{R}^{t,+}_{i,j}$. These conditions require
$\pi^{t+1}$ to be the stationary distribution of the transition function whose
probabilities on outgoing edges are proportional to their biased positive
regret, with the root vertex as the starting state, and all outgoing
transitions from vertices in level $L$ going to the root vertex instead.
###### Definition 2.
$(b,L)$-regret matching is the algorithm that initializes
$\tilde{R}^{0}_{i,j}=0$, chooses actions at time $t$ according to a
distribution $\pi^{t}$ that satisfies Requirement 2 and after choosing action
$i$ and observing $u^{t}$ updates
$\tilde{R}^{t}_{i,j}=\tilde{R}^{t-1}_{i,j}+(u^{t}(j)-u^{t}(i)-b)$ for all $j$
where $(i,j)\in E$, and for all other $(k,l)\in E$ where $k\neq i$,
$\tilde{R}^{t}_{k,l}=\tilde{R}^{t-1}_{k,l}$.
There are two distinguishing factors of our algorithm from (Hart and Mas-
Colell, 2002; Greenwald and Jafari, 2003): $\tilde{R}\neq R$, and past a
certain distance from the root, we loop back. $\tilde{R}$ differs from $R$ by
the bias term, $b$. This term can be thought of as a bias toward the action
selected by the algorithm. This is not the same as approaching the negative
orthant with a margin for error. This small amount is only applied to the
action taken, which is very different from adding a small margin of error to
every edge.
###### Theorem 3.
For any directed graph with maximum out-degree $D$ and any designated vertex
root, $(\Delta/(L+1),L)$-regret matching, after $T$ steps, will have expected
local swap regret no worse than,
$\displaystyle\frac{1}{T}E[R^{T}_{\mathrm{localswap}}]\leq\frac{\Delta}{L+1}+\frac{\Delta\sqrt{D|E_{L}|}}{\sqrt{T}}$
(13)
where $E_{L}=\\{(i,j)\in E|{\cal L}(i)\leq L\\}$.
The proof can be found in Appendix A. The overall structure of the proof is
similar to (Blackwell, 1956; Hart and Mas-Colell, 2002; Greenwald and Jafari,
2003) with a few significant changes. As with most algorithms based on
Blackwell, if there is an action you do not regret taking, playing that action
the next round is “safe”. If not, the key quantity in the proof is a flow
$f_{i,j}=\pi^{t+1}_{i}\tilde{R}^{t,+}_{i,j}$ for each edge. On most of the
graph, the incoming flow is equal to the outgoing flow for each node in levels
1 to $L$. Since all the flow out from the nodes on one level is equal to the
flow into the next, the total flow into (and out of) each level is equal.
Thus, the flow out of the last level is only $1/(L+1)$ of the total flow on
all edges since there are $L+1$ levels, including the root.
Traditionally, we wish to show that the incoming flow of an action times the
utility minus the outgoing flow of an action times the utility summed over all
nodes is nonpositive, and then Blackwell’s condition holds. In traditional
proofs, for any given node, the flow in and out are equal, so regardless of
the utility, they cancel. For our problem, the flow out of the last level is
really a flow into the $(L+1)$st level, not the zeroeth level, so the
difference in utilities between the zeroeth level and the $(L+1)$st level
creates a problem. On the other hand, because we subtract $b$ from whatever
action we select, we get to subtract $b$ times the total flow. Since exactly
$1/(L+1)$ fraction of the flow is going into the $(L+1)$st level, these two
discrepancies from the traditional approach exactly cancel. The second term of
Equation (13) is a result of the traditional Blackwell approach. In the final
analysis, we must account for the amount $b$ we subtract from the regret each
round. This means that if we get $\tilde{R}$ to approach the negative orthant,
we only have $bT$ local swap regret left. This is the first term of Equation
(13).
## 5 Exploiting Locality Structure
The local swap regret algorithm in the previous section successfully drops all
dependence on the size of the action set and thus can be applied even for
infinite action sets. However, the appearance of $|E_{L}|$ in the bound in
Theorem 3 is undesirable as $|E_{L}|\in O(D^{L})$, and $L$ is more likely to
be 100 than 2, in order to keep the first term of the bound low. The bound,
therefore, practically provides little beyond an asymptotic guarantee for even
the simplest setting of Example 1. In this section, we will appeal to (i) the
structure in the locality graph, and (ii) local external regret to achieve a
more practical regret bound and algorithm.
### 5.1 Cartesian Product Graphs
We begin by considering the case of $G$ having a very strong structure, where
it can be entirely decomposed into a set of product graphs. In this case, we
can show that by independently minimizing local regret in the product graphs
we can minimize local regret in the full graph.
###### Theorem 4.
Let $G$ be a Cartesian product of graphs, $G=G_{1}\otimes\ldots\otimes G_{k}$
where $G_{l}=(V_{l},E_{l})$. For all $l\in\\{1,\ldots,k\\}$, define
$u^{t}_{l}:V_{l}\rightarrow\mathbb{R}$, such that
$u^{t}_{l}(a_{l})=u^{t}(\left<a^{t}_{1},\ldots,a^{t}_{l-1},a_{l},a^{t}_{l+1},\ldots,a^{t}_{k}\right>)$,
so $u^{t}_{l}$ is a utility function on the $l$th component of the action at
time $t$ assuming the other components remain unchanged. Let $E[l]\subseteq E$
be the set of edges that change only on the $l$th component, so
$\\{E[l]\\}_{l=1,\ldots,k}$ forms a partition of $E$. Let $D_{l}\leq D$ be the
maximum degree of $G_{l}$. Finally, define
$\displaystyle R^{T,l}_{\text{\rm localexternal}}$ $\displaystyle=\max_{b\in
V_{l}}\left(\sum_{(i,j)\in
E^{b}_{l}}\sum_{t=1}^{T}1(a^{t}_{l}=i)(u^{t}_{l}(j)-u^{t}_{l}(i))/D_{l}\right)^{+},$
where $E^{b}_{l}=\\{(i,j)\in E[l]:d(i,b_{l})=c(i,j)+d(j,b_{l})\\}$, i.e., it
contains the edges that moves the $l$th component closer to $b_{l}$. Then,
$R^{T}_{\text{\rm localexternal}}\leq\sum_{l=1}^{k}R^{T,l}_{\text{\rm
localexternal}}$.
###### Proof.
$\displaystyle R^{T}_{\text{\rm localexternal}}$ $\displaystyle=\max_{b\in
V}\left(\sum_{(i,j)\in E^{b}}R^{T}_{i,j}/D\right)^{+}$ (14)
$\displaystyle=\max_{b\in V}\left(\sum_{l=1}^{k}\sum_{(i,j)\in E[l]\cap
E^{b}}R^{T}_{i,j}/D\right)^{+}$ (15)
$\displaystyle\leq\sum_{l=1}^{k}\max_{b\in V}\left(\sum_{(i,j)\in E[l]\cap
E^{b}}R^{T}_{i,j}/D\right)^{+}$ (16) Since $D_{l}\leq D$,
$\displaystyle\leq\sum_{l=1}^{k}\max_{b\in V}\left(\sum_{(i,j)\in E[l]\cap
E^{b}}R^{T}_{i,j}/D_{l}\right)^{+}$ (17)
$\displaystyle=\sum_{l=1}^{k}\max_{b\in V}\left(\sum_{(i,j)\in E[l]\cap
E^{b}}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}(j)-u^{t}(i))/D_{l}\right)^{+}$ (18)
$\displaystyle=\sum_{l=1}^{k}\max_{b\in V}\left(\sum_{(i,j)\in
E^{b}_{l}}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}_{l}(j)-u^{t}_{l}(i))/D_{l}\right)^{+}$
(19) $\displaystyle=\sum_{l=1}^{k}R^{T,l}_{\text{\rm localexternal}}$ (20)
∎
The implication is that we if we apply independent regret minimization to each
factor of our product graph, we can minimize local external regret on the full
graph. For example, consider the hypercube graphs from Example 1 and 2. By
applying $n$ independent external regret algorithms (the component graphs in
this case are 2-vertex complete graphs), the overall local external regret for
the graph is at most $n$ times bigger than the factors’ regrets, so under
regret matching it is bounded by $n\Delta\sqrt{2}/\sqrt{T}$. Hence, we are
able to handle an exponentially large graph (in $n$) with local external
regret only growing linearly (in $n$). If the component graphs are not
complete graphs, then we can simply apply our local swap regret algorithm from
the previous section to the graph factors, which minimizes local external
regret as well.
### 5.2 Color Regret
Cartesian product graphs are a powerful, but not very general structure. We
now substantially generalize the product graph structure, which will allow us
to achieve a similar simplification for very general graphs, such as the graph
on decision trees in Example 3. The key insight of product graphs is that for
any vertex $b$, an edge moves toward $b$ if and only if its corresponding edge
in its component graph moves toward $b_{l}$. In other words, either all of the
edges that correspond to some component edge will be included in the external
regret sum, or none of the eges will. We can group together these edges and
only worry about the regret of the group and not its constituents. We
generalize this fact to graphs which do not have a product structure.
###### Definition 5.
An edge-coloring $\mathbf{C}=\\{C_{i}\\}_{i=1,2,\ldots}$ for an arbitrary
graph $G$ with edge lengths is a partition of $E$: $C_{i}\subseteq E$,
$\bigcup_{i}C_{i}=E$, and $C_{i}\bigcap C_{j}=\emptyset$. We say that
$\mathbf{C}$ is admissble if and only if for all $b\in V$, $C\in\mathbf{C}$,
and $(i,j),(i^{\prime},j^{\prime})\in C$, $\text{\rm d}(i,b)=c(i,j)+\text{\rm
d}(i,b)\Leftrightarrow\text{\rm
d}(i^{\prime},b)=c(i^{\prime},j^{\prime})+\text{\rm d}(j^{\prime},b)$. In
other words, for any arbitrary target, all of the edges with the same color
are on a shortest path, or none of the edges are.
We now consider treating all of the edges of the same color as a single entity
for regret. This gives us the notion of local colored regret.
$\displaystyle R^{T}_{\mathrm{localcolor}}$
$\displaystyle=\sum_{C\in\mathbf{C}}\left(\sum_{(i,j)\in
C}R^{T}_{i,j}\right)^{+}$ (21)
###### Theorem 6.
If $\mathbf{C}$ is admissible then $R^{T}_{\text{\rm localexternal}}\leq
R^{T}_{\text{\rm localcolor}}/D$.
###### Proof.
$\displaystyle R^{T}_{\text{\rm localexternal}}$ $\displaystyle=\max_{b\in
A}\left(\sum_{(i,j)\in E^{b}}R^{T}_{i,j}/D\right)^{+}$ (22)
$\displaystyle=\max_{b\in A}\left(\sum_{C\in\mathbf{C}}\sum_{(i,j)\in C\cap
E^{b}}R^{T}_{i,j}/D\right)^{+}$ (23)
For a particular target $b$ let $\mathbf{C}_{b}=\\{C\in\mathbf{C}:C\subseteq
E^{b}\\}$, i.e., $\mathbf{C}_{b}$ is the set of colors that reduces the
distance to $b$. Then by $\mathbf{C}$’s admissibility,
$\displaystyle R^{T}_{\text{\rm localexternal}}$ $\displaystyle=\max_{b\in
A}\left(\sum_{C\in\mathbf{C}_{b}}\sum_{(i,j)\in C}R^{T}_{i,j}/D\right)^{+}$
(24) $\displaystyle\leq\max_{b\in
A}\sum_{C\in\mathbf{C}_{b}}\left(\sum_{(i,j)\in C}R^{T}_{i,j}/D\right)^{+}$
(25) $\displaystyle\leq\sum_{C\in\mathbf{C}}\left(\sum_{(i,j)\in
C}R^{T}_{i,j}/D\right)^{+}$ (26) $\displaystyle=R^{T}_{\text{\rm
localcolor}}/D$ (27)
∎
So by minimizing local colored regret, we minimize local external regret. The
natural extension of our local swap regret algorithm from the previous section
results in an algorithm that can minimize local colored regret.
###### Definition 7.
$(b,L,\mathbf{C})$-colored-regret-matching is the algorithm that initializes
$\tilde{R}^{0}_{C}=0$, for all $C\in\mathbf{C}$, chooses actions at time $t$
according to a distribution $\pi^{t}$ that satisfies Requirement 2 with
$\tilde{R}^{t}_{i,j}\equiv\tilde{R}^{t}_{c(i,j)}$, and after choosing action
$i$ and observing $u^{t}$ at time $t$ for all $C\in\mathbf{C}$ updates
$\tilde{R}^{t}_{C}=\tilde{R}^{t-1}_{C}+\sum_{j:(i,j)\in
C}(u^{t}(j)-u^{t}(i)-b)$.
###### Theorem 8.
For an arbitrary graph $G$ with maximum degree $D$, arbitrarily chosen vertex
root, and edge coloring $\mathbf{C}$, $(\Delta/(L+1),L,\mathbf{C})$-colored-
regret matching applied after $T$ steps will have expected local colored
regret no worse than,
$\frac{1}{T}E[R^{T}_{\mathrm{localcolor}}]\leq\frac{\Delta
D}{L+1}+\frac{\Delta\sqrt{D|C_{L}|}}{\sqrt{T}}$
where $C_{L}=\\{C\in\mathbf{C}|\exists(i,j)\in C\text{~{}s.t.~{}}{\cal
L}(i)\leq L\\}$.
The proof is in Appendix B. The consequence of this bound depends upon the
number of colors needed for an admissible coloring. Very small admissible
colorings are often possible. The hypercube graph needs only $2n$ colors to
give an admissible coloring, which is exponentially smaller than the total
number of edges, $n2^{n}$. We can also find a reasonably tight coloring for
our decision tree graph example, despite being a complex asymmetric graph.
###### Example 4 (Colored Decision Tree Learning).
Reconsider Example 3 and the graph in Figure 1. Recall that an edge exists
between one decision tree and another if the latter can be constructed from
the former by replacing a subtree at any node (internal or leaf) with a label
(edge length 1) or a stump (edge length 1.1). We will color this edge with the
pair: (i) the sequence of variable assignments that is required to reach the
node being replaced, and (ii) the stump or label that replaces it. This
coloring is admissible. We can see this fact by considering a color: the
sequence of variable assignments and resulting stump or label. If this color
is consistent with the target decision tree (i.e., the sequence exists in the
target decision tree, and the variable of the added stump matches the variable
split on at that point in the target decision tree) then the color must move
you closer to the target tree. A formal proof of its admissibility is very
involved and can be found in Appendix C.
## 6 Experimental Results
The previous section presented algorithms that minimize local swap and local
external regret (by minimizing local colored regret). The regret bounds have
no dependence on the size of the graph beyond the graph’s degree, and so
provide a guarantee even for infinite graphs. We now explore these algorithms’
practicality as well as illustrate the generality of the concepts by applying
them to a diverse set of online problems. The first two tasks we examine,
online Max-3SAT and online decision tree learning, have not previously been
explored in the online setting. The final task, online disjunct learning, has
been explored previously, and will help illustrate some drawbacks of local
regret.
In all three domains we examine two algorithms. The first minimizes local swap
regret by applying $(\Delta/(L+1),L)$-regret matching with $L$ chosen
specifically for the problem. This will be labeled “Local Swap”. The second
focuses on local external regret by using a tight, admissible edge-coloring
and applying $(\Delta/(L+1),L,\mathbf{C})$-colored-regret matching. This will
be labeled simply “Local External”.
### 6.1 Online Max-3SAT
First, we consider Example 1. We randomly constructed problem instances with
$n=20$ boolean variables and 201 clauses each with 3 literals. On each
timestep, the algorithms selected an assignment of the variables, a clause was
chosen at random from the set, and the algorithm received a utility of 1 if
the assignment satisfied the clause, 0 otherwise. This was repeated for 1000
timesteps. The locality graph used was the $n$-dimensional hypercube from
Example 1. The admissible coloring used to minimize local external regret was
the $2n$ coloring that has two colors per variable (one for turning the
variable on, and one for turning the variable off). In both cases we set
$L=\infty$ and $b=0$, since the bounds do not depend on $L$ once it exceeds
20. This also achieved the best performance for both algorithms. The average
results over 200 randomly constructed sets of clauses are shown in Figure 2,
with 95% confidence bars.
|
---|---
(a) | (b)
Figure 2: Results for Online Max-3SAT: (a) regret, (b) fraction of unsatisfied
clauses.
Figure 2 (a) shows the time-averaged colored regret of the two algorithms, to
demonstrate how well the algorithms are actually minimizing regret. Both are
decreasing over time, while external regret is decreasing much more rapidly.
As expected, swap regret may be a stronger concept, but it is more difficult
to minimize. The local external regret algorithm after only one time step can
have regret for not having made a particular variable assignment, while local
swap regret has to observe regret for this assignment from every possible
assignment of the other variables to achieve the same result. This is further
demonstrated by the number of regret values each algorithm is tracking: local
external regret on average had 34 non-zero regret values, while local swap
regret had 4200 non-zero regret values. In summary, external regret provides a
powerful form of generalization. Figure 2 (b) shows the fraction of the
previous 100 clauses that were satisfied. Two baselines are also presented. A
random choice of variable assignments can satisfy $\frac{7}{8}$ of the clauses
in expectation. We also ran WalkSAT (Selman et al., 1993) offline on the set
of 201 clauses, and on average it was able to satisfy all but 4% of the
clauses, which gives an offline lower bound for what is possible. Both
substantially outperformed random, with the external regret algorithm nearing
the performance of the offline WalkSat.
### 6.2 Online Decision Tree Learning
Second, we consider Example 3. We took three datasets from the UCI Machine
Learning Repository (each with categorical inputs and a large number of
instances): nursery, mushroom, and king-rook versus king-pawn (Frank and
Asuncion, 2010). The categorical attributes were transformed into boolean
attributes (which simplified the implementation of the locality graphs) by
having a separate boolean feature for each attribute value.111As a result,
there were $n=28$ features for nursery, $118$ features for mushroom, and $74$
features for king-rook versus king-pawn. We made the problems online
classification tasks by sampling five instances at random (with replacement)
for each timestep, with the utility being the number classified correctly by
the algorithm’s chosen decision tree. This was repeated for 1000 timesteps,
and so the algorithms classified 5,000 instances in total. The locality graph
used was the one described in Example 3. The tight coloring used to minimize
local external regret was the one described in Example 4. $L$ was set to 3 for
local swap regret, and 100 for local external regret, as this achieved the
best performance. Even with the far larger graph, the external regret
algorithm was observing nearly one-eighth of the number of non-zero regret
values observed by the local swap algorithm. The average results over 50
trials are shown in Figure 3(a)-(c) with 95% confidence bars.
|
---|---
(a) | (b)
|
(c) | (d)
Figure 3: Results for online decision tree learning on three UCI datasets: (a)
Nursery, (b) Mushroom, (c) King-Rook/King-Pawn; and (d) a simple sequence of
alternating labels.
The graphs show the average fraction of misclassified instances over the
previous 100 timesteps. Two baselines are also plotted: the best single label
(i.e., the size of the majority class) and the best decision stump. Both
regret algorithms substantially improved on the best label, and local external
regret was selecting trees substantially better than the best stump. As a
further baseline, we ran the batch algorithm C4.5 in an online fashion, by
retraining a decision tree after each timestep using all previously observed
examples. C4.5’s performance was impressive, learning highly accurate trees
after observing only a small fraction of the data. However, C4.5 has no regret
guarantees. As with any offline algorithm used in an online fashion, there is
an implicit assumption that the past and future data instances are i.i.d.. In
our experimental setup, the instances were i.i.d., and as a result C4.5
performed very well. To further illustrate this point, we constructed a simple
online classification task where instances with identical attributes were
provided with alternating labels. The best label (as well as the single best
decision tree) has a 50% accuracy. C4.5 when trained on the previously
observed instances, misclassifies every single instance. This is shown along
with local regret algorithms in Figure 3 (d).
### 6.3 Online Disjunct Learning
Finally, we examine online disjunct learning as described in Example 2. This
task has received considerable attention, notably the celebrated Winnow
algorithm (Littlestone, 1988), which is guaranteed to make a finite number of
mistakes if the instances can be perfectly classified by some disjunction.
Furthermore, the number of mistakes Winnow2 makes, when no disjunction
captures the instances, can be bounded by the number of attribute errors
(i.e., the number of input attributes that must be flipped to make the
disjunction satisfy the instance) made by the best disjunction. In these
experiments we compare our algorithms’ performance to that of Winnow2.
We looked at two learning tasks. In the first, we generated a random
disjunction over $n=20$ boolean variables, where a variable was independently
included in the disjunction with probability $4/n$. Instances were created
with uniform random assignments to all of the variables, with a label being
true if and only if the chosen disjunct is true for the instance’s assignment.
In the second case, we chose instances uniformly at random from a constructed
set of 21 instances: one for each variable with that variable (only) set to
true and the label being true, and one with all of the variables assigned the
value of true and the label being false. We call this task Winnow Killer. For
both tasks, the $n$-dimensional hypercube from Example 1 was used as the
locality graph with the $2n$ coloring as our admissible coloring, and
$L=\infty$ and $b=0$. The average results over 50 trials are shown in Figure
4, with 95% confience bars.
|
---|---
(a) | (b)
Figure 4: Results for online disjunct learning: (a) random disjunct, (b)
Winnow Killer.
The graphs plot error rates over the previous 100 instances. Three baselines
are plotted: randomly assigning a label (guaranteed to get half of the
instances correct on expectation), the best disjunct (which makes no mistakes
for random disjunctions and makes $\frac{1}{21}$ mistakes on the Winnow Killer
task), and Winnow2. Figure 4 (a) shows the results on random disjunctions.
Winnow2 is guaranteed to make a finite number of mistakes and indeed its error
rate drops to zero quickly. The local regret concepts, though, have
difficulties with random disjunctions. The reason can be easily seen for the
case of local external regret. Suppose the first instance is labeled true; the
algorithm now has regret for all of the variables that were true in that
instance (some of these will be in the target disjunction, but many will not).
These variables will now be included in the chosen disjunction for a very long
time, as the only regret that one can have for not removing them is if their
assignment was the sole reason for misclassifying a false instance. In other
words, the problem is that there’s no regret for not removing multiple
variables simultaneously as this is not a local change. Winnow2, though, also
has issues. It performs very poorly in the Winnow Killer task (in fact, if the
instances were ordered it could be made to get every instance wrong), as shown
in Figure 4 (b). Since the mistake bound for Winnow2 is with respect to the
number of attribute errors, a single mistake by the best disjunction can
result in $n$ mistakes by Winnow2. A further issue with Winnow is that while
its peformance is tied to the performance of disjunctions, its own hypothesis
class is not disjunctions but a thresholded linear function, whereas local
regret is playing in the same class of hypotheses that it comparing against.
## 7 Conclusion
We introduced a new family of regret concepts based on restricting regret to
only nearby hypotheses using a locality graph. We then presented algorithms
for minimizing these concepts, even when the number of hypotheses are
infinite. Further we showed that we can exploit structure in the graph to
achieve tighter bounds and better performance. These new regret concepts mimic
local search methods, which are common approaches to offline optimization with
intractably hard hypothesis spaces. As such, our concepts and algorithms
allows us to make online guarantees, with a similar flavor to their offline
counterparts, with these hypothesis spaces.
There is a number of interesting directions for future work as well as open
problems. Admissible colorings can result in radically improved bounds as well
as empirical performance. How can such admissible colorings be constructed for
general graphs? What graph structures lead to exponentially small admissible
colorings compared to the size of the graph? We can easily construct the
minimum admissible coloring for graphs that are recursively constructed as
Cartesian product of graphs and complete graphs. While such graphs can have
exponentially small admissible colorings, they form a very narrow class of
structures. What other structures lead to exponentially small admissible
colorings? Furthermore, edge lengths can have a significant impact on the size
of the minimum admissible coloring. For example, the decision tree graph from
Example 3 was carefully constructed to result in a tight coloring, and, in
fact, unit length edges over the same graph would result in an exponentially
larger admissible coloring. How can edge lengths be defined to allow for small
minimum colorings?
## Acknowledgements
This work was supported by NSERC and Yahoo! Research, where the first author
was a visiting scientist at the time the research was conducted.
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## Appendix A Proof for Local Swap Regret
At its heart, the Hart and Mas-Colell proof for minimizing internal regret
relies on the relationship between Markov chains and flows. The Blackwell
condition is (roughly speaking) that the probability flow into an action
equals the probability flow out of an action. In the variant here, there are
two ways to view this flow. Define $f$ such that for all $(i,j)\in E$,
$f_{i,j}=\pi^{t+1}_{i}\tilde{R}^{t,+}_{i,j}$. Implicitly, $f$ depends on the
time $t$, but we supress this as we always refer to a time $t$. This flow $f$
is similar to the flows in Hart and Mas-Colell as they apply to the Blackwell
condition. However, it lacks the conservation of flow property. Thus, we
consider a second flow $f^{\prime}$ which satisfies the conservation of flow.
To do this, we consider the levels of the graph. To review, root is a distinct
vertex, and, ${\cal L}(v)=d_{1}(\text{\rm root},v)$. If we consider the flow
$f$ as starting from the root, it (roughly) goes from level to level outward
from the root until it reaches level $L$. Then, while $f$ flows to level $L+1$
and reaches a dead end (violating the conservation property), $f^{\prime}$ is
switched, and flows back to the root. In order to make the proof work, we have
to bound the difference between $f$ and $f^{\prime}$. Since this difference is
mostly on the flow from level $L$ to level $L+1$, we need to bound the
fraction of the total flow that is going out of the last level by showing that
this flow is less than the flow going from the root to the first level, and it
is less than the flow from the first level to the second level, et cetera.
First we show that for nodes on most levels, the flow in equals the flow out.
###### Lemma 9.
If Requirement 2 holds, then for all $j\in V$ such that $1\leq{\cal L}(j)\leq
L$,
$\sum_{i:(i,j)\in E}f_{i,j}=\sum_{k:(j,k)\in E}f_{j,k}$
###### Corollary 10.
By summing over the nodes in level $\ell$, for any level $1\leq\ell\leq L$,
$\sum_{(i,j)\in E:{\cal L}(j)=\ell}f_{i,j}=\sum_{(i,j)\in E:{\cal
L}(i)=\ell}f_{i,j}.$
###### Proof.
From Requirement 2(c) we know there exists an $M>0$ such that:
$\displaystyle\pi^{t+1}_{j}$ $\displaystyle=\sum_{i:(i,j)\in
E}(R^{t,+}_{i,j}/M)\pi^{t+1}_{i}+\left(1-\sum_{k:(j,k)\in
E}R^{t,+}_{j,k}/M\right)\pi^{t+1}_{j}$ (28)
$\displaystyle\pi^{t+1}_{j}\left(\sum_{k:(j,k)\in E}R^{t,+}_{j,k}/M\right)$
$\displaystyle=\sum_{i:(i,j)\in E}R^{t,+}_{i,j}\pi^{t+1}_{i}/M$ (29)
$\displaystyle\sum_{k:(j,k)\in E}\pi^{t+1}_{j}R^{t,+}_{j,k}$
$\displaystyle=\sum_{i:(i,j)\in E}R^{t,+}_{i,j}\pi^{t+1}_{i}$ (30)
The lemma follows by the definition of $f_{i,j}$. ∎
If we want the conservation of flow to hold for all nodes, then we need to
define a slightly different flow. We want to say that the flow which is
currently exiting the first $L$ levels (specifically between level $L$ and
level $L+1$) is actually flowing back into the root. So, we want to subtract
the edges $E^{\prime\prime}=\\{(i,j)\in E:{\cal L}(j)\geq L+1\vee{\cal
L}(i)\geq L+1\\}$, and add the edges $E^{\prime}=\\{i\in V:{\cal
L}(i)=L\\}\times\\{\text{\rm root}\\}$. For any edge $e\in E^{\prime}$, define
$f^{\prime}_{i,j}=f_{i,j}+\sum_{k:{\cal L}(k)=L+1,(i,k)\in E}f_{i,k}$., where
$f_{i,j}=0$ if $(i,j)\notin E$. For any edge $(i,j)\in
E\backslash(E^{\prime}\cup E^{\prime\prime})$ where ${\cal L}(i),{\cal
L}(j)\leq L$, $f^{\prime}_{i,j}=f_{i,j}$. Define $\tilde{E}=(E\cup
E^{\prime}\backslash E^{\prime\prime})$.
Thus, we now have a flow over a graph $(V,\tilde{E})$, but we must prove
conservation of flow.
###### Lemma 11.
If Requirement 2 holds, for any $i\in V$,
$\sum_{j:(i,j)\in\tilde{E}}f^{\prime}_{i,j}=\sum_{j:(j,i)\in\tilde{E}}f^{\prime}_{j,i}$
###### Proof.
For ${\cal L}(i)\in\\{1\ldots L-1\\}$, this is a direct result of Requirement
2(c). For when ${\cal L}(i)>L$, there is no flow out or in, making the result
trivial. For ${\cal L}(i)=0$ (when $i=\text{\rm root}$), this is a direct
result of Requirement 2(d). For when ${\cal L}(i)=L$, note that
$\sum_{j:(j,i)\in E}f_{j,i}=\sum_{j:(i,j)\in E}f_{i,j}$, for all $j$ where
$(j,i)\in E$, $f_{j,i}=f^{\prime}_{j,i}$, and for all $(i,j)\in E$ where
${\cal L}(j)\in\\{1\ldots L\\}$, $f_{i,j}=f^{\prime}_{i,j}$ and that the flow
$\sum_{j:(i,j)\in E,{\cal L}(j)\in\\{0,L+1\\}}f_{i,j}=f^{\prime}_{i,\text{\rm
root}}$, so
$\displaystyle\sum_{j:(j,i)\in E}f^{\prime}_{j,i}$
$\displaystyle=\sum_{j:(j,i)\in E}f_{j,i}$ (31)
$\displaystyle=\sum_{j:(i,j)\in E}f_{i,j}$ (32)
$\displaystyle=\sum_{j:(i,j)\in E,{\cal
L}(j)\in\\{0,L+1\\}}f_{i,j}+\sum_{j:(i,j)\in E,{\cal L}(j)\in\\{1\ldots
L\\}}f_{i,j}$ (33) $\displaystyle=f^{\prime}_{i,\text{\rm
root}}+\sum_{j:(i,j)\in E,{\cal L}(j)\in\\{1\ldots L\\}}f^{\prime}_{i,j}$ (34)
$\displaystyle=\sum_{j:(i,j)\in E}f^{\prime}_{i,j}.$ (35)
∎
###### Lemma 12.
If Requirement 2 holds, then:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}\geq(L+1)\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}$ (36) $\displaystyle\sum_{(i,j)\in
E:L(i)=0}f_{i,j}\geq\sum_{(i,j)\in E:{\cal L}(j)=L+1}f_{i,j}$ (37)
###### Proof.
To obtain an intuition, consider the case where all outgoing edges from level
$j$ go to level $j+1$ (modulo the last level). In this case, the flow from
level 0 all goes to level 1, from there goes to level 2, and so forth until it
reaches level $L$ and then returns to level $0$. Thus, the inflows and
outflows of all the levels would be equal. The problem with this is that
outgoing edges from level $j$ can go to other nodes in $j$, or nodes in level
$j-1$, et cetera. At an intuitive level, a backwards flow would not make more
flow through the final level, any more than an eddy would somehow create water
at the mouth of a river, and we must simply formally prove this.
First, we define $g_{i,j}=\sum_{(k,l)\in E:{\cal L}(k)=i,{\cal
L}(l)=j}f^{\prime}_{i,j}$, the total flow between levels. By Lemma 11 for all
$i\in V$, $\sum_{j}f^{\prime}_{i,j}=\sum_{j}f^{\prime}_{j,i}$, so the
aggregate flow satisfies the conservation of flow, namely that for all $i$,
$\sum^{L}_{j=0}g_{i,j}=\sum^{L}_{j=0}g_{j,i}$. Also, if $j>i+1$, then
$g_{i,j}=0$. Define $n_{i}=g_{i,i+1}$, the flow between one level and the
next. Since $f$, $f^{\prime}$, and $g$ are just different groupings of the
total flow throughout the graph, $\sum_{(i,j)\in
E}f_{i,j}=\sum_{(i,j)\in\tilde{E}}f^{\prime}_{i,j}=\sum_{i=0}^{L}\sum_{j=0}^{L}g_{i,j}$.
Since for all $i,j\in V$, $f^{\prime}_{i,j}\geq 0$, then for all $i,j$,
$g_{i,j}\geq 0$. $g_{L,0}+\sum_{i=0}^{L-1}n_{i}\leq\sum_{(i,j)\in E}f_{i,j}$.
Moreover, $n_{0}=g_{0,1}=\sum_{j:(\text{\rm root},j)\in
E}f^{\prime}_{\text{\rm root},j}=\sum_{j:(\text{\rm root},j)\in E}f_{\text{\rm
root},j}$, and $g_{L,0}\geq\sum_{(i,j)\in E:{\cal L}(j)=L+1}f_{i,j}$. So if we
prove that for all $i$, $g_{L,0}\leq n_{i}$, then $g_{L,0}\leq n_{0}$ and that
$g_{L,0}(L+1)\leq g_{L,0}+\sum_{i=0}^{L-1}n_{i}$, we have proven the lemma.
First, we identify this backwards flow. Define $\delta_{i}$ to be the flow
that originates at level $i$ or above and flows back to a lower level.
Formally, define $\delta_{0}=0$, and
$\delta_{i}=\sum_{i^{\prime}<i,j^{\prime}\geq
i}g_{j^{\prime},i^{\prime}}-g_{L,0}$. Note that $\delta_{i}\geq 0$.
Thus, for all $i$ where $0<i<L$:
$\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(\sum_{i^{\prime}<i,j^{\prime}\geq
i}g_{j^{\prime},i^{\prime}}\right)-\left(\sum_{i^{\prime}<i+1,j^{\prime}\geq
i+1}g_{j^{\prime},i^{\prime}}\right)$ (38)
$\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(\sum_{i^{\prime}<i,j^{\prime}=i}g_{j^{\prime},i^{\prime}}\right)+\left(\sum_{i^{\prime}<i,j^{\prime}\geq
i+1}g_{j^{\prime},i^{\prime}}\right)-\left(\sum_{i^{\prime}=i,j^{\prime}\geq
i+1}g_{j^{\prime},i^{\prime}}\right)-\left(\sum_{i^{\prime}<i,j^{\prime}\geq
i+1}g_{j^{\prime},i^{\prime}}\right)$ (39)
$\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(\sum_{i^{\prime}<i,j^{\prime}=i}g_{j^{\prime},i^{\prime}}\right)-\left(\sum_{i^{\prime}=i,j^{\prime}\geq
i+1}g_{j^{\prime},i^{\prime}}\right)$ (40)
$\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(\sum_{i^{\prime}<i}g_{i,i^{\prime}}\right)-\left(\sum_{j^{\prime}\geq
i+1}g_{j^{\prime},i}\right)$ (41) $\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(g_{i,i}+\sum_{i^{\prime}<i}g_{i,i^{\prime}}\right)-\left(g_{i,i}+\sum_{j^{\prime}\geq
i+1}g_{j^{\prime},i}\right)$ (42) $\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(\sum_{i^{\prime}\leq
i}g_{i,i^{\prime}}\right)-\left(\sum_{j^{\prime}\geq
i}g_{j^{\prime},i}\right)$ (43) Since $g_{i,i+1}=n_{i}$, and
$g_{i-1,i}=n_{i-1}$, $\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left((g_{i,i+1}-n_{i})+\sum_{i^{\prime}\leq
i}g_{i,i^{\prime}}\right)-\left((g_{i-1,i}-n_{i-1})+\sum_{j^{\prime}\geq
i}g_{j^{\prime},i}\right)$ (44) $\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(-n_{i}+\sum_{i^{\prime}\leq
i+1}g_{i,i^{\prime}}\right)-\left(-n_{i-1}+\sum_{j^{\prime}\geq
i-1}g_{j^{\prime},i}\right)$ (45) Since $g$ represents the level graph,
$g_{i,i^{\prime}}=0$ if $i^{\prime}>i+1$, or put another way,
$g_{j^{\prime},i}=0$ if $j^{\prime}<i-1$, so
$\displaystyle\delta_{i}-\delta_{i+1}$
$\displaystyle=\left(-n_{i}+\sum_{i^{\prime}}g_{i,i^{\prime}}\right)-\left(-n_{i-1}+\sum_{j^{\prime}}g_{j^{\prime},i}\right)$
(46) $\displaystyle\delta_{i}-\delta_{i+1}$ $\displaystyle=n_{i-1}-n_{i}$ (47)
So, for all $0\leq i<L-1$:
$\displaystyle\delta_{i+1}-\delta_{i+2}$ $\displaystyle=n_{i}-n_{i+1}$ (48)
$\displaystyle n_{i}$ $\displaystyle=\delta_{i+1}-\delta_{i+2}+n_{i+1}$ (49)
For $n_{0}$, note that $\sum_{i}g_{i,0}=g_{0,0}+\delta_{1}+g_{L,0}$, and
$\sum_{i}g_{0,i}=g_{0,0}+n_{0}$, so
$g_{0,0}+\delta_{1}+g_{L,0}=g_{0,0}+n_{0}$, and $g_{L,0}=n_{0}-\delta_{1}$.
This is the base case in a recursive proof that for all $i<L$,
$g_{L,0}=n_{i}-\delta_{i}$. If we wish to prove it holds for $i+1$, then we
assume it holds for $i$, or $g_{L,0}=n_{i}-\delta_{i}$. By Equation (49), for
$i<L-1$:
$\displaystyle g_{L,0}$
$\displaystyle=(\delta_{i+1}-\delta_{i+2}+n_{i+1})-\delta_{i}$ (50)
$\displaystyle=n_{i+1}-\delta_{i+1}$ (51)
Since $\delta_{i}\geq 0$, this implies that for $i<T$, $g_{L,0}\leq n_{i}$,
which completes the proof. ∎
###### Lemma 13.
If Requirements 1 and 2 hold, and $b=\Delta/(L+1)$, then $\sum_{(i,j)\in
E}\tilde{R}^{t,+}_{i,j}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b)\leq 0$.
###### Proof.
First, consider the case where $\pi^{t}$ is degenerate. Then, whenever
$\pi^{t+1}_{i}>0$, we know $R^{t,+}_{i,j}=0$ for all $(i,j)\in E$, and so our
sum of interest is exactly 0. Note that, since
$f_{i,j}=\tilde{R}^{+}_{i,j}\pi^{t+1}_{i}$, what we need to prove is:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i)-b)$
$\displaystyle\leq 0$ (52) $\displaystyle\left(\sum_{(i,j)\in
E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))\right)-b\sum_{(i,j)\in E}f_{i,j}$
$\displaystyle\leq 0.$ (53)
Suppose $\pi^{t}$ is not degenerate. We examine Equation (53)’s two
summations. Notice that only edges $(i,j)$ where $\pi^{t+1}_{i}>0$ have
$f_{i,j}\neq 0$, and by Requirement 2(e) this is only true if ${\cal L}(i)\leq
L$. Also, $f_{i,j}>0$ if and only if $0\leq{\cal L}(i)\leq L$ and $1\leq{\cal
L}(j)\leq L+1$ (because level zero has no incoming edges), so:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))$
$\displaystyle=\sum_{(i,j)\in E}f_{i,j}u^{t+1}(j)-\sum_{(i,j)\in
E}f_{i,j}u^{t+1}(i)$ (54) $\displaystyle=\sum_{\ell=1}^{L+1}\sum_{(i,j)\in
E:{\cal L}(j)=\ell}f_{i,j}u^{t+1}(j)-\sum_{\ell=0}^{L}\sum_{(i,j)\in E:{\cal
L}(i)=\ell}f_{i,j}u^{t+1}(i).$ (55)
Renaming the dummy variables in the second term and then combining:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))$
$\displaystyle=\sum_{\ell=1}^{L+1}\sum_{(i,j)\in E:{\cal
L}(j)=\ell}f_{i,j}u^{t+1}(j)-\sum_{\ell=0}^{L}\sum_{(j,k)\in E:{\cal
L}(j)=\ell}f_{j,k}u^{t+1}(j)$ (56)
$\displaystyle=\sum_{\ell=1}^{L}\left(\sum_{(i,j)\in E:{\cal
L}(j)=\ell}f_{i,j}u^{t+1}(j)-\sum_{(j,k)\in E:{\cal
L}(j)=\ell}f_{j,k}u^{t+1}(j)\right)$ $\displaystyle+\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}u^{t+1}(j)-\sum_{(j,k)\in E:{\cal L}(j)=0}f_{j,k}u^{t+1}(j).$
(57)
First, we show that any term between 1 and $L$ is zero. For any $1\leq\ell\leq
L$, by summing over nodes in level $\ell$:
$\displaystyle\sum_{(i,j)\in E:{\cal
L}(j)=\ell}f_{i,j}u^{t+1}(j)-\sum_{(j,k)\in E:{\cal
L}(j)=\ell}f_{j,k}u^{t+1}(j)$ $\displaystyle=\sum_{j:{\cal
L}(j)=\ell}\left(\sum_{i:(i,j)\in E}f_{i,j}u^{t+1}(j)-\sum_{k:(j,k)\in
E}f_{j,k}u^{t+1}(j)\right)$ (58) $\displaystyle=\sum_{j:{\cal
L}(j)=\ell}u^{t+1}(j)\left(\sum_{i:(i,j)\in E}f_{i,j}-\sum_{k:(j,k)\in
E}f_{j,k}\right).$ (59)
By Lemma 9, $\sum_{i:(i,j)\in E}f_{i,j}=\sum_{k:(j,k)\in E}f_{j,k}$, so these
terms are zero, leaving:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))=$
$\displaystyle\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}u^{t+1}(j)-\sum_{(j,k)\in E:{\cal L}(j)=0}f_{j,k}u^{t+1}(j).$
(60)
If ${\cal L}(j)=0$, then $j=\text{\rm root}$:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))=$
$\displaystyle\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}u^{t+1}(j)-\sum_{(j,k)\in E:{\cal
L}(j)=0}f_{j,k}u^{t+1}(\text{\rm root}).$ (61)
Moreover, for any $j$, $u^{t+1}(j)-u^{t+1}(\text{\rm root})\leq\Delta$, so:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))\leq$
$\displaystyle\sum_{(i,j)\in E:{\cal L}(j)=L+1}f_{i,j}(u^{t+1}(\text{\rm
root})+\Delta)-\sum_{(j,k)\in E:{\cal L}(j)=0}f_{j,k}u^{t+1}(\text{\rm root})$
(62) $\displaystyle\leq$ $\displaystyle\Delta\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}+u^{t+1}(\text{\rm root})\left(\sum_{(i,j)\in E:{\cal
L}(j)=L+1}f_{i,j}-\sum_{(j,k)\in E:{\cal L}(j)=0}f_{j,k}\right).$ (63)
By Lemma 12, Equation (37), the flow into level $L+1$ is less than or equal to
the flow out of level 0, so the last part is nonpositive and:
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i))\leq$
$\displaystyle\Delta\sum_{(i,j)\in E:{\cal L}(j)=L+1}f_{i,j}$ (64)
From Lemm 12, Equation (36), we can show that the second term of Equation (53)
equals:
$\displaystyle b\sum_{(i,j)\in E}f_{i,j}$ $\displaystyle\geq
b(L+1)\sum_{(i,j):{\cal L}(j)=L+1}f_{i,j}$ (65)
Putting Equations (65) and (64) together with the fact that $b=\Delta/(L+1)$,
we get,
$\displaystyle\sum_{(i,j)\in E}f_{i,j}(u^{t+1}(j)-u^{t+1}(i)-b)$
$\displaystyle\leq\Delta\sum_{(i,j):{\cal
L}(j)=L+1}f_{i,j}-b(L+1)\sum_{(i,j):{\cal L}(j)=L+1}f_{i,j}$ (66)
$\displaystyle\leq(\Delta-b(L+1))\sum_{(i,j):{\cal L}(j)=L+1}f_{i,j}=0$ (67)
which is what we were trying to prove. ∎
Lemma 13 is very close to the Blackwell condition, but not identical, so we
sketch a quick variation on a special case of Blackwell’s theorem so we can
apply it to our problem.
###### Fact 14.
$(a+b)^{+}\leq a^{+}+b^{+}$
###### Lemma 15.
$[(a+b)^{+}]^{2}\leq(a^{+}+b)^{2}$
###### Proof.
1. 1.
If $a,b\geq 0$: $(a+b)^{2}\leq(a+b)^{2}$
2. 2.
If $a,b\leq 0$: $[(a+b)^{+}]^{2}=0\leq(a^{+}+b)^{2}$.
3. 3.
If $a\geq 0,b\leq 0$: if $-b\geq a$, then
$[(a+b)^{+}]^{2}=0\leq(a^{+}+b)^{2}$, otherwise
$[(a+b)^{+}]^{2}=(a+b)^{2}=(a^{+}+b)^{2}$.
4. 4.
If $a\leq 0,b\geq 0$: then if $-a\geq b$,
then$[(a+b)^{+}]^{2}=0\leq(a^{+}+b)^{2}$, otherwise,
$[(a+b)^{+}]^{2}=(a+b)^{2}\leq b^{2}=(a^{+}+b)^{2}$.
∎
###### Fact 16.
If $a_{i=1\ldots n}\geq 0$ then
$\sum_{i=1}^{n}a_{i}\leq\sqrt{|n|\sum_{i=1}^{n}a_{i}^{2}}$.
###### Fact 17.
$E\left[X\right]^{2}\leq E\left[X^{2}\right]$
We restate Theorem 3 from Section 4: See 3
###### Proof.
$\displaystyle E[R^{T}_{\mathrm{localswap}}]$ $\displaystyle=E\left[\sum_{i\in
V}\left(\max_{j:(i,j)\in
E}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}(j)-u^{t}(i))\right)^{+}\right]$ (68)
$\displaystyle=E\left[\sum_{i\in V}\left(\max_{j:(i,j)\in
E}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}(j)-u^{t}(i)-b+b)\right)^{+}\right]$ (69)
$\displaystyle=E\left[\sum_{i\in V}\left(\max_{j:(i,j)\in
E}\left(\tilde{R}^{T}_{i,j}+\sum_{t=1}^{T}1(a^{t}=i)b\right)\right)^{+}\right]$
(70) $\displaystyle=E\left[\sum_{i\in
V}\left(\left(\sum_{t=1}^{T}1(a^{t}=i)b\right)+\max_{j:(i,j)\in
E}\tilde{R}^{T}_{i,j}\right)^{+}\right]$ (71) $\displaystyle\leq
E\left[\sum_{i\in
V}\left(\left(\sum_{t=1}^{T}1(a^{t}=i)b\right)+\left(\max_{j:(i,j)\in
E}\tilde{R}^{T}_{i,j}\right)^{+}\right)\right]$ (72)
$\displaystyle=E\left[bT+\sum_{i\in V}\max_{j:(i,j)\in
E}\tilde{R}^{T,+}_{i,j}\right]$ (73) $\displaystyle\leq E\left[bT+\sum_{i\in
V}\sum_{j:(i,j)\in E}\tilde{R}^{T,+}_{i,j}\right]$ (74)
$\displaystyle=bT+\sum_{(i,j)\in E_{L}}E\left[\tilde{R}^{T,+}_{i,j}\right]$
(75) By Facts 16 and 17, $\displaystyle\leq bT+\left(|E_{L}|\sum_{(i,j)\in
E_{L}}E\left[\tilde{R}^{T,+}_{i,j}\right]^{2}\right)^{\frac{1}{2}}$ (76)
$\displaystyle\leq bT+\left(|E_{L}|\sum_{(i,j)\in
E_{L}}E\left[(\tilde{R}^{T,+}_{i,j})^{2}\right]\right)^{\frac{1}{2}}$ (77)
We can bound the inner term as follows, using Lemma 15:
$\displaystyle\sum_{(i,j)\in E_{L}}E\left[(\tilde{R}^{T,+}_{i,j})^{2}\right]$
$\displaystyle\leq\sum_{(i,j)\in
E_{L}}E\left[(\tilde{R}^{T-1,+}_{i,j}+1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b))^{2}\right]$
(78) $\displaystyle=\sum_{(i,j)\in
E_{L}}E\left[\left(\tilde{R}^{T-1,+}_{i,j}\right)^{2}\right]+\sum_{(i,j)\in
E_{L}}E\left[(1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b))^{2}\right]$ (79)
$\displaystyle\quad+\sum_{(i,j)\in
E_{L}}E\left[2\tilde{R}^{T-1,+}_{i,j}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right]$
$\displaystyle=\sum_{(i,j)\in
E_{L}}E\left[\left(\tilde{R}^{T-1,+}_{i,j}\right)^{2}\right]+E\left[\sum_{(i,j)\in
E_{L}}(1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b))^{2}\right]$ (80)
$\displaystyle\quad+2\\!\\!\\!\\!\\!\\!\\!\\!\sum_{a^{1,\ldots,T-1},u^{1,\ldots,T-1}}\left(E\left[\sum_{(i,j)\in
E_{L}}\tilde{R}^{T-1,+}_{i,j}\pi^{T}_{i}(u^{T}(j)-u^{T}(i)-b)\biggr{|}a^{1,\ldots,T-1},u^{1,\ldots,T-1}\right]\times\right.$
$\displaystyle\quad\left.\Pr[a^{1,\ldots,T-1},u^{1,\ldots,T-1}]\right)$ (81)
By Lemma 13, $\sum_{(i,j)\in
E_{L}}\tilde{R}^{T-1,+}_{i,j}\pi^{T}_{i}(u^{T}(j)-u^{T}(i)-b)\leq 0$
regardless of the previous history.
$\displaystyle\sum_{(i,j)\in E_{L}}E\left[(\tilde{R}^{T,+}_{i,j})^{2}\right]$
$\displaystyle\leq\sum_{(i,j)\in
E_{L}}E\left[\left(\tilde{R}^{T-1,+}_{i,j}\right)^{2}\right]+E\left[\sum_{(i,j)\in
E_{L}}(1(a^{T}=i)(\Delta-b))^{2}\right]$ (82) $\displaystyle\leq\sum_{(i,j)\in
E_{L}}E\left[\left(\tilde{R}^{T-1,+}_{i,j}\right)^{2}\right]+D(\Delta-b)^{2}$
(83) $\displaystyle\leq TD(\Delta-b)^{2}\leq
TD\left(\Delta\frac{L}{L+1}\right)^{2}$ (84)
Putting these two pieces together, we get,
$\displaystyle E[R^{T}_{\mathrm{localswap}}]$ $\displaystyle\leq
bT+\left(|E_{L}|\sum_{(i,j)\in
E_{L}}E\left[(R^{T,+}_{i,j})^{2}\right]\right)^{\frac{1}{2}}$ (85)
$\displaystyle\leq bT+\sqrt{|E_{L}|TD\left(\Delta\frac{L}{L+1}\right)^{2}}$
(86) $\displaystyle\leq\frac{\Delta
T}{L+1}+\sqrt{TD|E_{L}|}\Delta\frac{L}{L+1}$ (87)
$\displaystyle\frac{1}{T}E[R^{T}_{\mathrm{localswap}}]$
$\displaystyle\leq\frac{\Delta}{L+1}+\frac{\Delta\sqrt{D|E_{L}|}}{\sqrt{T}}$
(88)
∎
## Appendix B Proof for Color Regret
###### Requirement 3.
Let $C$ be a countable (but possibly infinite) set of colors. The edge
coloring $c:E\rightarrow C$ is such that $c(i,j)=c(i,k)\Leftrightarrow j=k$.
We restate Theorem 8 from Section 5.2: See 8
###### Proof.
First, we show that $\sum_{c\in
C}\tilde{R}^{t,+}_{c}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))\leq 0$.
$\displaystyle\sum_{c\in
C}\tilde{R}^{t,+}_{c}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))$
$\displaystyle=\sum_{c\in C}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\tilde{R}^{t,+}_{c}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))$
(89) $\displaystyle=\sum_{c\in C}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\tilde{R}^{t,+}_{i,j}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))$
(90)
By Lemma 13:
$\displaystyle\sum_{c\in
C}\tilde{R}^{t,+}_{c}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))$
$\displaystyle=\sum_{(i,j)\in
E}\tilde{R}^{t,+}_{i,j}\pi^{t+1}_{i}(u^{t+1}(j)-u^{t+1}(i)-b))\leq 0$ (91)
Now we can bound our quantity of interest.
$\displaystyle E[R^{T}_{\mathrm{localcolor}}]$
$\displaystyle=E\left[\sum_{c\in C}\left(\sum_{\begin{subarray}{c}(i,j)\in
E\\\
c(i,j)=c\end{subarray}}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}(j)-u^{t}(i))\right)^{+}\right]$
(92) $\displaystyle=E\left[\sum_{c\in
C}\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\sum_{t=1}^{T}1(a^{t}=i)(u^{t}(j)-u^{t}(i)-b+b)\right)^{+}\right]$
(93) $\displaystyle=E\left[\sum_{c\in
C}\left(\tilde{R}^{T}_{c}+\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\sum_{t=1}^{T}1(a^{t}=i)b\right)\right)^{+}\right]$
(94) $\displaystyle\leq E\left[\sum_{c\in
C}\left(\tilde{R}^{T,+}_{c}+\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\sum_{t=1}^{T}1(a^{t}=i)b\right)\right)\right]$ (95)
$\displaystyle=E\left[\sum_{c\in C}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\sum_{t=1}^{T}1(a^{t}=i)b+\sum_{c\in
C}R^{T,+}_{c}\right]$ (96) $\displaystyle\leq E\left[bTD+\sum_{c\in
C}\tilde{R}^{T,+}_{c}\right]$ (97) $\displaystyle=bTD+\sum_{c\in
C_{L}}E\left[\tilde{R}^{T,+}_{c}\right]$ (98) $\displaystyle\leq
bTD+\left(|C_{L}|\sum_{c\in
C_{L}}E\left[\tilde{R}^{T,+}_{c}\right]^{2}\right)^{\frac{1}{2}}$ (99)
$\displaystyle\leq bTD+\left(|C_{L}|\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T,+}_{c})^{2}\right]\right)^{\frac{1}{2}}$ (100)
We can bound the inner term as follows,
$\displaystyle\sum_{c\in C_{L}}E\left[(\tilde{R}^{T,+}_{c})^{2}\right]$
$\displaystyle\leq\sum_{c\in C_{L}}E\left[(\tilde{R}^{T}_{c})^{2}\right]$
(101) $\displaystyle=\sum_{c\in
C_{L}}E\left[\left(\tilde{R}^{T-1}_{c}+\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right)^{2}\right]$
(102) $\displaystyle=\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+\sum_{c\in
C_{L}}E\left[\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right)^{2}\right]$
(103) $\displaystyle\quad+\sum_{c\in
C_{L}}E\left[2\tilde{R}^{T-1}_{c}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right]$
$\displaystyle=\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+\sum_{c\in
C_{L}}E\left[\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right)^{2}\right]$
(104) $\displaystyle\quad+2\sum_{c\in
C_{L}}\tilde{R}^{T-1}_{c}\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}\pi^{T}_{i}(u^{T}(j)-u^{T}(i)-b)$
$\displaystyle\leq\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+\sum_{c\in
C_{L}}E\left[\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)(u^{T}(j)-u^{T}(i)-b)\right)^{2}\right]$
(105) $\displaystyle\leq\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+(\Delta-b)^{2}\sum_{c\in
C_{L}}E\left[\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)\right)^{2}\right]$ (106)
Because only one action is taken, and for each color only one edge originating
at an action can have that color, $\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)\in\\{0,1\\}$:
$\displaystyle\sum_{c\in C_{L}}E\left[(\tilde{R}^{T,+}_{c})^{2}\right]$
$\displaystyle\leq\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+(\Delta-b)^{2}\sum_{c\in
C_{L}}E\left[\left(\sum_{\begin{subarray}{c}(i,j)\in E\\\
c(i,j)=c\end{subarray}}1(a^{T}=i)\right)\right]$ (107)
$\displaystyle=\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T-1}_{c})^{2}\right]+D(\Delta-b)^{2}$ (108)
$\displaystyle\leq TD(\Delta-b)^{2}\leq
TD\left(\Delta\frac{L}{L+1}\right)^{2}$ (109)
Putting these two pieces together, we get,
$\displaystyle E[R^{T}_{\mathrm{colorswap}}]$ $\displaystyle\leq
bTD+\left(|C_{L}|\sum_{c\in
C_{L}}E\left[(\tilde{R}^{T,+}_{c})^{2}\right]\right)^{\frac{1}{2}}$ (110)
$\displaystyle\leq bTD+\sqrt{|C_{L}|TD\left(\Delta\frac{L}{L+1}\right)^{2}}$
(111) $\displaystyle\leq\frac{\Delta
DT}{L+1}+\sqrt{TD|E_{L}|}\Delta\frac{L}{L+1}$ (112)
$\displaystyle\frac{1}{T}E[R^{T}_{\mathrm{colorswap}}]$
$\displaystyle\leq\frac{\Delta D}{L+1}+\frac{\Delta\sqrt{D|C_{L}|}}{\sqrt{T}}$
(113)
∎
## Appendix C Decision Tree Graphs
A decision tree is a representation of a hypothesis. Given an instance space
where there are a finite number of binary features, a decision tree can
represent an arbitrary hypothesis. We describe decision trees recursively: the
simplest trees are leaves, which represent constant functions. More complex
trees have two subtrees, and a root node labeled with a variable. A subtree
cannot have a variable that is referred to in the root.
We define $T_{k}(S)$ recursively, where $T_{k}(S)$ will be the set of trees of
depth $k$ or less over the variable set $S$. Define the set
$T_{0}(S)=\\{\mbox{true},\mbox{false}\\}$. Define $T_{k}(S)$ such that:
$\displaystyle T_{k}(S)$ $\displaystyle=T_{k-1}(S)\bigcup_{s\in
S}(\\{s\\}\times T_{k-1}(S\backslash\\{s\\})\times
T_{k-1}(S\backslash\\{s\\}))$ (114)
Define $T^{*}(S)=T_{|S|}(S)$ to be the set of all decision trees over the
variables $S$. Three example decision trees in $T^{*}(\\{x_{1},x_{2}\\})$ are
$(x_{1},\mbox{true},\mbox{false})$, true, and
$(x_{1},(x_{2},\mbox{true},\mbox{false}),\mbox{false})$. Suppose we have an
example $x$, mapping variables to $\\{\mbox{true},\mbox{false}\\}$. For any
tree $t$, we can recursively define $t(x)$:
1. 1.
If $t\in T_{0},$ then $t(x)=t$.
2. 2.
If $t\in T_{k}$ and $x(t_{1})=\mbox{true}$, then $t(x)=t_{2}(x)$.
3. 3.
If $t\in T_{k}$ and $x(t_{1})=\mbox{false}$, then $t(x)=t_{3}(x)$.
Define $P=\\{p\in(S\times\\{\mbox{true},\mbox{false}\\})^{|S|}:\forall i\neq
j,p_{i,1}\neq p_{j,1}\\}$ to be the paths in the trees without repeating
variables. We can talk about whether a path is in a tree. Define $V_{p}(t)$ to
be a function from $T^{*}$ to $S\cup\\{\mbox{true},\mbox{false},\emptyset\\}$,
where $V_{p}(t)=\emptyset$ if the path $p$ is not present in the tree, and
otherwise $V_{p}(t)$ is the value of the node at the end of the path.
Formally,
$\displaystyle V_{\emptyset}(t)$
$\displaystyle=\left\\{\begin{array}[]{@{}l@{}l@{}}{t}&\mbox{ if }{t\in
T_{0}}\\\ {t_{1}}&\mbox{ otherwise}\end{array}\right.$ (117) $\displaystyle
V_{(v,l)\circ p}(t)$
$\displaystyle=\left\\{\begin{array}[]{@{}l@{}l@{}}\emptyset&\mbox{ if }t\in
T_{0}\mbox{ or }t_{1}\neq v\\\ V_{p}(t_{2})&\mbox{ if }t\notin T_{0}\mbox{ and
}t_{1}=v\mbox{ and }l=\mbox{true}\\\ V_{p}(t_{3})&\mbox{ if }t\notin
T_{0}\mbox{ and }t_{1}=v\mbox{ and }l=\mbox{false}\end{array}\right.$ (121)
Given a path $p\in P$, a tree $t^{\prime}\in T^{*}$,define
$R_{p,t^{\prime}}(t)$ to replace the tree at $p$ with $t^{\prime}$ if
$V_{p}(t)\neq\emptyset$. Formally:
$\displaystyle R_{\emptyset,t^{\prime}}(t)$ $\displaystyle=t^{\prime}$ (122)
$\displaystyle R_{(v,l)\circ p,t^{\prime}}(t)$
$\displaystyle=\left\\{\begin{array}[]{@{}l@{}l@{}}t&\mbox{ if }t\in
T_{0}\mbox{ or }t_{1}\neq v\\\ (t_{1},R_{p,t^{\prime}}(t_{2}),t_{3})&\mbox{ if
}t\notin T_{0}\mbox{ and }t_{1}=v\mbox{ and }l=\mbox{true}\\\
(t_{1},t_{2},R_{p}(t_{3}))&\mbox{ if }t\notin T_{0}\mbox{ and }t_{1}=v\mbox{
and }l=\mbox{false}\end{array}\right.$ (126)
Consider the following operations on decision trees:
1. 1.
$ReplaceWithNode(p,v,l_{1},l_{2})=R_{p,(v,l_{1},l_{2})}$ (where it applies):
If there exists a node or leaf at path $p$, replace it with a decision stump
with variable $v$, with label $l_{1}$ on the true branch, and label $l_{2}$ on
the false branch, but only if $V_{p}(t)\neq v$.
2. 2.
$ReplaceWithLeaf(p,l_{1})=R_{p,l_{1}}$: If there exists a node or leaf at path
$p$, replace it with a leaf $l_{1}$.
These operations create the edges between trees: we will determine how to
color them later. Because $ReplaceWithNode$ is a more complex operation, an
edge created by $ReplaceWithNode$ will have length 1.1, whereas
$ReplaceWithLeaf$ will have length 1.0. This weighting is important:
otherwise, consider the following sequence of trees:
$\displaystyle(X,\mbox{true},\mbox{false})$ $\displaystyle(\mbox{false})$
$\displaystyle(X,\mbox{false},\mbox{true})$
If splitting was the same length as changing leaves, this bizarre path would
be a shortest path between $(X,\mbox{true},\mbox{false})$ and
$(X,\mbox{false},\mbox{true})$. In general, when designing this distance
function over trees, a critical concern was whether unnecessary reconstruction
would be on a shortest path. For example, a shortest path from
$(X,(Y,\mbox{true},\mbox{false}),(Z,\mbox{false},\mbox{true}))$ to
$(X,(Y,\mbox{false},\mbox{true}),(Z,\mbox{true},\mbox{false}))$ could pass
through
$\mbox{false},(X,\mbox{true},\mbox{false}),(X,(Y,\mbox{false},\mbox{true}),\mbox{false})$.
But, since replacing something with a decision tree costs slightly more than
changing a leaf, we avoid this.
More generally, if the decision about whether or not an edge is on the
shortest path can be made locally, then this reduces the number of colors
required. Thus, massively reconstructing the root because the leaves are wrong
is not only counterintuitive, it makes the algorithm slower and more complex.
We first hypothesize a shortest path distance function between trees based on
these operations, and then we will prove it satisfies the above operations.
Note that this function is not symmetric, because the shortest path distance
function on a directed graph is not always symmetric.
Given two decision trees $A$ and $B$, a decision node $a$ in $A$ and a
decision node $b$ are in structural agreement if they are on the same path
$p$, and they are labeled with the same variable. A decision node in $B$ that
does not agree with a decision node in $A$ is in structural disagreement with
$A$. Given a leaf in $B$ that has a parent that is in structural agreement
with $A$, if the leaf is not present in $A$, it is in leaf disagreement with
$A$.
Define $d^{*}_{s}(A,B)$ to be the structural disagreement distance between $A$
and $B$, the number of nodes in $B$ that are in structural disagreement with
$A$. Define $d^{*}_{l}(A,B)$ to be the leaf disagreement distance between $A$
and $B$, the number of leaves in $B$ in disagreement with $A$. Define
$d^{*}(A,B)=1.1d^{*}_{s}(A,B)+d^{*}_{l}(A,B)$.
Intuitively, this distance represents the fact that an example shortest path
from $A$ to $B$ can be generated by first fixing all label disagreements
between $A$ and $B$, and then applying $ReplaceWithNode$ to create every node
in $B$ that is in structural disagreement with $A$ (correctly labeling leaves
where appropriate).
###### Fact 18.
If $d:V\times V\rightarrow\mathbf{Z}^{+}$ is the shortest distance function on
a completely connected directed graph $(V,E)$, then for any $i,j\in V$ where
$(i,j)\notin E$, there exists a $k$ such that $(i,k)\in E$ and
$d(i,j)=d(i,k)+d(k,j)$.
###### Theorem 19.
$d^{*}:V\times V\rightarrow\mathbf{Z}^{+}$ corresponds to the shortest
distance function on a completely connected directed graph $(V,E)$ if there
exists a $\Delta>0$ and a $\delta=\Delta/2$ such that the following properties
hold:
1. 1.
For all $a,b\in V$, $d^{*}(a,b)=0$ iff $a=b$.
2. 2.
For all $a,b\in V$, $d^{*}(a,b)>\delta$ iff $a\neq b$.
3. 3.
For all $a,b\in V$, if $a\neq b$ there exists a $c\in V$ such that
$d^{*}(a,c)\leq\Delta$ and $d^{*}(a,b)\geq d^{*}(a,c)+d^{*}(c,b)$.
4. 4.
For all $a,b,c\in V$, if $d^{*}(a,c)\leq\Delta$, then $d^{*}(a,b)\leq
d^{*}(a,c)+d^{*}(c,b)$.
###### Proof.
Observe that the graph $(V,E)$ with edges $E=\\{(i,j)\in
V^{2}:d^{*}(i,j)\leq\Delta\\}$ where the weight of an edge $(i,j)\in E$ is
$d^{*}(i,j)$, is a good candidate for the graph under consideration. We prove
this in two steps. We first prove by induction that $d(i,j)\leq d^{*}(i,j)$.
Then, leveraging this, we prove by induction that $d(i,j)=d^{*}(i,j)$.
First, we prove that if $d^{*}(i,j)\leq\Delta$, then $d(i,j)=d^{*}(i,j)$.
First, observe that if $d^{*}(i,j)=0$, then $i=j$, so $d(i,j)=0$. Secondly, if
$d^{*}(i,j)\in(0,\Delta]$, then there exists an edge $(i,j)\in E$ so
$d(i,j)\leq d^{*}(i,j)$. Since each edge is larger than $\Delta/2$, for any
path of length 2 or greater, the length is larger than $\Delta$, so only a
direct path can be less than or equal to $\Delta$. This establishes that there
is no path between $i$ and $j$ shorter than the direct edge.
For any nonnegative integer $k$, define $P(k)$ to be the property that for any
$i,j\in V$, if the distance $d^{*}(i,j)\leq k\delta$, the shortest distance
between two vertices in this graph $d(i,j)$ is less than or equal to
$d^{*}(i,j)$. This holds for $P(0)$, $P(1)$, and $P(2)$ because of the
paragraph above. Now, suppose that $P(k)$ holds for $k\geq 2$, we need to
establish it holds for $P(k+1)$. Consider some pair $(i,j)\in V$ where
$d^{*}(i,j)\in(k\delta,(k+1)\delta]$, then $i\neq j$, and by condition 3,
there exists a $k$ where $d^{*}(i,k)\leq\Delta$ and $d^{*}(i,j)\geq
d^{*}(i,k)+d^{*}(k,j)$. Since $d^{*}(i,j)\leq(k+1)\delta$ and
$d^{*}(i,j)>\delta$, $d^{*}(k,j)<k\delta$, so $d^{*}(k,j)=d(k,j)$. From the
paragraph above, $d(i,k)=d^{*}(i,k)$, so $d^{*}(i,j)\geq d(i,k)+d(k,j)$, and
by the triangle inequality on $d$, $d^{*}(i,j)\geq d(i,j)$.
Thus, since for all $(i,j)\in V$ there exists a $k$ where $d^{*}(i,j)\leq
k\delta$, for all $(i,j)\in V$, $d(i,j)\leq d^{*}(i,j)$.
Next, we prove that if $d(i,j)\leq\Delta$, then $d(i,j)=d^{*}(i,j)$. First,
observe that if $d(i,j)=0$, then $i=j$, so $d^{*}(i,j)=0$. Secondly, if
$(i,j)\notin E$, then the distance between $i$ and $j$ must be greater than
$\Delta$, because each edge is larger than $\Delta/2$. Therefore, if
$d(i,j)\in(0,\Delta]$ there is a direct edge between $i$ and $j$ with distance
$d^{*}(i,j)$, so $d^{*}(i,j)\leq\Delta$, and so by the second paragraph
$d(i,j)=d^{*}(i,j)$.
Define $Q(k)$ to be the property for any $(i,j)\in V$, if $d(i,j)\leq k\delta$
then $d(i,j)=d^{*}(i,j)$. $Q(0)$, $Q(1)$ and $Q(2)$ hold from the above
paragraph. Now, suppose that $Q(k)$ holds for some $k\geq 2$, we need to
establish the property for $Q(k+1)$. Consider some pair $(i,j)\in V$ where
$d(i,j)\in(k\delta,(k+1)\delta]$, then $i\neq j$, and by condition 18, there
exists a $k$ where there exists an edge from $i$ to $k$ and
$d(i,j)=d(i,k)+d(k,j)$. Since there exists an edge $(i,k)$, then
$d(i,k)\leq\Delta$ and $d(i,k)=d^{*}(i,k)>\delta$. Thus,
$d(k,j)\leq\delta(k+1)-\delta\leq\delta k$. so $d(k,j)=d^{*}(k,j)$. Moreover,
by condition 4, $d^{*}(i,j)\leq d^{*}(i,k)+d^{*}(k,j)=d(i,j)$. Thus, since we
know that $d^{*}(i,j)\geq d(i,j)$, then $d^{*}(i,j)=d(i,j)$.
Therefore, since $d^{*}(i,j)=d(i,j)$, and $d$ is the shortest distance for
graph $(V,E)$, then $d^{*}(i,j)$ is a shortest distance function for a
weighted graph. ∎
###### Lemma 20.
For the decision tree metric $d^{*}$ above, for any two trees $A,B$ where
$A\neq B$, there exists a tree $C$ such that $d^{*}(A,C)\leq 1.1$ and
$d^{*}(A,B)\geq d^{*}(A,C)+d^{*}(C,B)$.
###### Proof.
If $B$ has a leaf at the root, then set $C=B$.
Suppose that, given $A$ and $B$, there is label disagreement. Find the a node
with label disagreement, and correct all the labels in $A$ to form $C$. This
reduces the number of nodes with label disagreement by one, and the decision
node disagreement stays the same.
Suppose that, given $A$ and $B$, there no label disagreement, but there is
structural disagreement. Then select a node $d$ which has decision node
disagreement. Define $C$ to be a tree where we replace node $d$ with the
corresponding node in tree $B$, with leaves that agree with the children of
$d$ if $d$ has children, and arbitrary otherwise. This reduces the structural
disagreement by one. It does not increase the label disagreement, because if
$d$ has children with labels in $B$, it has those same children in $C$.
Finally, if $A$ and $B$ have no label disagreement or structural disagreement,
then they are the same tree and have distance 0. ∎
Before proving a lower bound, we focus on a particular case. Namely, that
changing a correct decision node of a tree to have the wrong variable cannot
decrease the distance.
###### Lemma 21.
Given two trees $A$ and $B$ and a subtree $S$ in $B$, if $n_{S}$ is the number
of nodes in agreement with $B$ in the subtree $S$, and $l_{S}$ is the number
of leaves in disagreement with $A$ in $S$, then $l_{S}\leq n_{S}+1$.
###### Proof.
We prove this by recursion on the size of the subtree $S$ in $B$. If $S$ is of
size 1, then $S$ is a leaf in $B$, then $n_{S}=0$ and $l_{S}\leq 1$, so the
result holds. Suppose we have proven this for all subtrees $S^{\prime}$ of
size less than $S$. If $S$ is rooted at a node in disagreement, then $n_{s}=0$
and $l_{S}=0$, and the result holds (we don’t need induction for this case).
If $S$ is rooted at a node $x$ in agreement, then define $S_{\mbox{true}}$ to
be the subtree of the node down the edge labeled true leaving $x$, and define
$S_{\mbox{false}}$ to be the subtree down the edge labeled false leaving $x$.
$|S_{\mbox{true}}|<|S|$ and $|S_{\mbox{false}}|<|S|$, so by induction
$l_{S_{\mbox{true}}}\leq n_{S_{\mbox{true}}}+1$ and $l_{S_{\mbox{false}}}\leq
n_{S_{\mbox{false}}}+1$. Since $x$ is a node in agreement,
$l_{S}=l_{S_{\mbox{true}}}+l_{S_{\mbox{false}}}$, and therefore:
$\displaystyle l_{S}$ $\displaystyle\leq
n_{S_{\mbox{true}}}+n_{S_{\mbox{false}}}+1+1$ (127)
Again, since $x$ is a node in agreement,
$n_{S_{\mbox{true}}}+n_{S_{\mbox{false}}}+1=n_{S}$, so:
$\displaystyle l_{S}$ $\displaystyle\leq n_{S}+1.$ (129)
∎
We will use this fact in several places in the resulting proofs.
###### Lemma 22.
Given two trees $A$ and $B$ which agree on node $y$, if you change $y$ in $A$
to a node $x$ or leaf to create $C$, then $d^{*}(A,B)<d^{*}(C,B)+1$.
###### Proof.
If $S$ is the subtree rooted at $y$ in $B$, then
$d^{*}_{s}(A,B)+n_{S}=d^{*}_{s}(C,B)$ and
$d^{*}_{l}(A,B)-l_{S}=d^{*}_{l}(C,B)$. By definition,
$d^{*}(A,B)=d^{*}(C,B)+1.1n_{S}-l_{S}$. Since $y$ is in agreement, $n_{S}\geq
1$. By Lemma 21, we know that $l_{S}\leq n_{S}+1$, so
$\displaystyle d^{*}(A,B)$ $\displaystyle=d^{*}(C,B)+1.1n_{S}-(n_{S}+1)$ (130)
$\displaystyle d^{*}(A,B)$ $\displaystyle=d^{*}(C,B)+0.1n_{S}+1$ (131)
Since $n_{s}\geq 1$, $0.1n_{s}\geq 0.1>0$, so:
$\displaystyle d^{*}(A,B)$ $\displaystyle<d^{*}(C,B)+1$ (132)
∎
###### Lemma 23.
For the decision tree metric $d^{*}$ above, for any two trees $A,B$ where
$A\neq B$, then for any $C$ such that $d^{*}(A,C)\leq\Delta$, $d^{*}(A,B)\leq
d^{*}(A,C)+d^{*}(C,B)$.
###### Proof.
First, observe that $C$ has “one” change from $A$, which can be that:
1. 1.
$C$ has a decision node splitting on variable $x$ where $A$ had a decision
node splitting on variable $y$.
2. 2.
$C$ has a decision node splitting on variable $x$ where $A$ had a leaf $l$.
3. 3.
$A$ has a node $x$ that was changed to a leaf.
4. 4.
$C$ has a leaf where $A$ had a node.
In the first case, there is a question of whether or not the decision node $y$
exists in $B$. If so, then the structural disagreement has been reduced by
one. However, the leaf disagreement is unchanged or increased by one, so
$d^{*}(A,B)\leq 1.1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$. If $y$ is not in $B$,
and $x$ is not in $B$, then
$d^{*}(A,B)=d^{*}(C,B)<1.1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$. If $y$ is in
$B$, by Lemma 22, then
$d^{*}(A,B)<d^{*}(C,B)+1<1.1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$.
For the second case, if the new node in $C$ agrees with $B$, then
$d^{*}(A,B)=1.1+d^{*}(C,B)$. If the leaf in $A$ agreed with $B$, then
$d^{*}(A,B)=d^{*}(C,B)-1<1.1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$. If the leaf in
$A$ disagreed with $B$ and the new node in $C$ disagrees with $B$, then
$d^{*}(A,B)=d^{*}(C,B)<1.1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$.
For the third case, if the new leaf in $C$ agrees with $B$, then
$d^{*}(A,B)=1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$. If the node in $A$ agreed
with $B$, then by Lemma 22, $d^{*}(A,C)<d^{*}(C,B)+1=d^{*}(A,C)+d^{*}(C,B)$.
If the node in $A$ disagreed with $B$, and the new leaf in $C$ disagrees with
$B$, then $d^{*}(A,B)=d^{*}(C,B)<1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$.
Finally, for the fourth case, if the new leaf in $C$ agrees with $B$, then
$d^{*}(A,B)=1+d^{*}(C,B)=d^{*}(A,C)+d^{*}(C,B)$. If the leaf in $A$ agreed
with $B$, then by Lemma 22, $d^{*}(A,C)<d^{*}(C,B)+1=d^{*}(A,C)+d^{*}(C,B)$.
If the leaf in $A$ disagreed with $B$, and the new leaf in $C$ disagrees with
$B$, then there was no change, and this is an illegal transition. ∎
###### Theorem 24.
The distance $d^{*}$ as defined above is the distance function for a graph.
###### Proof.
In order to prove this, we use Theorem 19. First $\Delta=1.1$, and
$\delta=0.55$.
Observe that by the definition of $d^{*}$, if two trees are equal, there is no
disagreement, and there is zero distance. Secondly, by the definition of
$d^{*}$, if there is any difference between two trees $A$ and $B$, there will
be disagreement, and $d^{*}(A,B)\geq 1$. Thus, Condition 1 and Condition 2 are
satisfied.
Now, by Lemma 20, Condition 3 is satistfied. By Lemma 23, Condition 4 is
satisfied.
∎
In the graph generated from $d^{*}$, note that a single label disagreement or
a single decision node disagreement results in an edge.
Now, we have to derive colors.
1. 1.
$ReplaceWithNode(p,v,l_{1},l_{2})$: The path, the variable, and the labels
form the color. Note that if the tree already has a decision node with label
$v$ at path $p$, this transition is illegal.
2. 2.
$ReplaceWithLeaf(p,l_{1})$: The path and the leaf form the color.
###### Lemma 25.
$ReplaceWithNode(p,v,l_{1},l_{2})$ is on the shortest path to $B$ if
1. 1.
it can be applied to the current tree
2. 2.
the variable $v$ is at the path $p$ in $B$.
3. 3.
A leaf with the label $\lnot l_{1}$ is not at the path $p\circ(v,\mbox{true})$
in $B$,
4. 4.
A leaf with the label $\lnot l_{2}$ is not at the path
$p\circ(v,\mbox{false})$ in $B$.
If these rules do not apply, it is not on the shortest path.
###### Proof.
Suppose that $A$ is our current tree. Suppose that
$C=R_{p,(v,l_{1},l_{2})}(A)$.
First, we establish that if the conditions are satisfied, the edge is on the
shortest path. Note that if $v$ is at the path $p$ in $B$, and there is a leaf
or another decision node at path $p$ in $A$, then $v$ is in structural
disagreement. Therefore, when we replace that node with $v$, we reduce the
structural disagreement. However, we must be careful not to increase leaf
disagreement. If, for any nodes of $v$ in $B$, they are corrected in $A$, then
leaf disagreement will not increase. Therefore, by reducing the structural
disagreement by 1, we reduce the distance by 1.1, at a cost of 1.1, meaning
the edge is on the shortest path.
Secondly, we can go through the conditions one by one to realize any violated
condition is sufficient. Regarding the first condition: if the operation
cannot be applied to the current tree, then by definition it is not on the
shortest path.
Regarding the second condition: if the variable $v$ is not on path $p$ in $B$,
but $A$ and $B$ are in agreement at the path $p$, then changing the variable
to $v$ will not decrease the distance sufficiently, by Lemma 22, so it is not
on the shortest path. Secondly, if $A$ does not agee with $B$ on path $p$,
then $d^{*}(A,B)=d^{*}(C,B)$, and thus $C$ is not on the shortest path.
Regard the third and fourth conditions. If the variable $v$ is on the path $p$
in $B$, but there is some leaf that is a child of $v$ in $B$ that is set
incorrectly, then the structural distance is decreased, but the leaf
disagreement is increased, so $d^{*}(A,B)=d^{*}(C,B)+0.1$. ∎
###### Lemma 26.
$ReplaceWithLeaf(p,l_{1})$ is on the shortest path to $B$ if it applies to the
current tree, and if the leaf $l_{1}$ is at $p$ in $B$. If these rules do not
apply, it is not on the shortest path.
###### Proof.
Suppose that $A$ is the initial tree, and $C=R_{p,l_{1}}(A)$. If the edge
applies, and there is the wrong label or a decision node at $p$, then the
label is in disagreement in $A$, but not in $C$. There are no other changes,
so $d^{*}(A,B)=d^{*}(C,B)+1=d^{*}(A,C)+d^{*}(C,B)$, and therefore the edge is
on a shortest path.
On the other hand, if there is no leaf at $p$ in $B$, or the leaf has another
label, then this is not the shortest path.
First of all, if the operator does not apply to $A$, it cannot be on the
shortest path.
If the label $V_{p}(B)\neq l_{1}$, but $A$ and $B$ are in agreement at the
path $p$, then by Lemma 22 $d^{*}(A,B)<d^{*}(C,B)+1=d^{*}(A,C)+d^{*}(C,B)$. If
$V_{p}(B)\neq l_{1}$, and $A$ and $B$ are not in agreement at the path $p$,
then $d^{*}(A,B)=d^{*}(C,B)<d^{*}(C,B)+1$. ∎
Thus, we have established our coloring works for decision trees.
|
arxiv-papers
| 2012-06-14T20:07:30 |
2024-09-04T02:49:31.796248
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Bowling, Martin Zinkevich",
"submitter": "Vanessa Burke",
"url": "https://arxiv.org/abs/1206.3318"
}
|
1206.3348
|
# Quantum Compiler Optimizations
Jeff Booth boothjmx@cs.washington.edu
###### Abstract
A quantum computer consists of a set of quantum bits upon which operations
called gates are applied to perform computations. In order to perform quantum
algorithms, physicists would like to design arbitrary gates to apply to
quantum bits. However, the physical limitations of the quantum computing
device restrict the set of gates that physicists are able to apply. Thus, they
must compose a sequence of gates from the permitted gate set, which
approximates the gate they wish to apply - a process called _quantum
compiling_.
Austin Fowler proposes a method [2] that finds optimal gate sequences in
exponential time, but which is tractable for common problems. In this paper, I
present several optimizations to this algorithm. While my optimizations do not
improve its overall exponential behavior, they improve its empirical
performance by one to two orders of magnitude.
## 1 Background
In classical computing, we can generally rely on the correctness of hardware
because of the size of the circuit components. For example, if an atom on a
hard disk drive changed its spin orientation, or lost an electron, the hard
drive’s functionality would not be impaired because it takes many thousands of
atoms to represent and store a single bit of data. However, in quantum
computing, data is stored in quantum bits, which are represented by tiny
particles like trapped ions. These qubits are very easy to perturb,
potentially corrupting calculations based on them. Thus, we use redundancy, in
the form of error-correcting codes, to minimize the impact of individual
errors.
The Steane code is one representation of a quantum bit. It uses seven physical
qubits to represent one Steane code qubit, and can tolerate an arbitrary error
in one of the seven qubits. We can perform any desired operation on a Steane
code qubit by applying a combination of $H$ (Hadamard) and $T$ gate operations
[4]. $T$ gates are generally complicated to implement in quantum computing
hardware, so we seek to use a minimal number. For practical purposes, in
addition to $H$, we can use the Pauli X operator $X$, the Pauli Z operator
$Z$, the single qubit phase gate $S$, and its inverse $S^{\dagger}$
[Fowler2010]. The gates $H$, $X$, $Z$, $S$, and $S^{\dagger}$ generate a group
under multiplication, called the Clifford group. Thus, any sequence of gates
we choose will alternate between a member of the Clifford group and a $T$
gate. A $T^{\dagger}$ gate is also used in this implementation, bringing the
total number of non-identity gates in Fowler’s gate set to 25.
A _single-qubit quantum compiler_ finds sequences of gates which yield
matrices that are “close” to a gate we would like to apply to a quantum bit.
Each gate has a corresponding matrix that represents the operation it would
perform on a quantum bit. How close one gate is to another is given by the
“Fowler” distance:
$dist(U,U_{l})=\sqrt{\frac{2-\left|tr\left(U\cdot
U_{l}^{\dagger}\right)\right|}{2}}$ (1)
The longer the gate sequence is, the more closely it can approximate a desired
target gate that is not in the universal instruction set. However, a longer
gate sequence takes more time to compute on a real quantum computer,
increasing the probability of a computation error. An optimal quantum compiler
will find gate sequences which:
1. 1.
have a minimal Fowler distance from the target gate.
2. 2.
have a minimal length.
## 2 Fowler’s Algorithm
Austin Fowler presents an algorithm that iterates over sequences in order from
smallest to largest [Fowler2010]. For each sequence, it multiplies the
sequence gates’ matrices together to generate a $2\times 2$ unitary matrix
representing the complete operation that sequence would perform. The simple
brute-force iteration runs in time exponential in sequence length, since all
sequences of length $n$ are produced by appending all elements of the
universal instruction set to all sequences of length $n-1$.
To reduce the run time, Fowler’s algorithm intelligently skips redundant
sequences. The algorithm creates a list of unique sequences for all sequences
of length $N$. Then, for each sequence $S$ of length $N+1$, it searches for
sub-sequences of length $N$. If a sub-sequence $Y$ is not in the list, then it
is not unique. That means it performs the same operation as a sequence $V$
which is in the unique sequence list. Since $Y$ and $V$ are the same, then if
you were to replace $Y$ with $V$ in your sequence $S$, you would get a
sequence $W$ that does the same thing $S$ does.
Since the algorithm iterates over all sequences of length $N$, it will
encounter $W$ anyway (or it has already encountered $W$). Therefore, it should
skip sequence $S$. In fact, it should increment the sub-sequence $Y$ until it
is a unique sequence $U$. Fowler’s algorithm contains a tree lookup structure
which, for any given sequence, records the next unique sequence $U$. The
algorithm can determine what sequence to skip to by simply accessing this
tree. It still requires time exponential in sequence length, but interesting
results can now be obtained in mere days using consumer computer hardware.
As I will demonstrate later, Fowler’s tree data structure requires memory that
scales exponentially with the sequence length. Thus, the algorithm consists of
two stages:
1. 1.
During the first stage, it builds the structure until it stores all unique
sequences up to length $W$, where $W=15$ for most of the experiments.
2. 2.
After the structure is built, it enters the second stage, where it generates
sequences – and uses the structure to skip them – but it doesn’t add them to
the structure.
This dramatic change in behavior between stages explains some interesting
features in the following graphs. Also, it means I can only infer behavior on
longer sequences from behavior in the second stage, which explains my focus on
data produced during that stage.
## 3 Experimental Goal
In order to empirically measure the impact of my optimizations, I need a
consistent experimental goal to test on every version of the algorithm. For
this research, I chose to approximate the $\frac{\pi}{6}$ gate
$\exp(i\frac{\pi}{12}\sigma_{z})$ to $10^{-7}$ accuracy. Since this
approximation is currently very time-consuming, I can use it to empirically
evaluate the impact of my enhancements.
## 4 Existing Performance
In order to better understand the performance characteristics of the Fowler
algorithm, I modified its C source code to obtain performance-related
statistics. In this section, I present the data I gathered, along with some
explanations for unusual data features and speculations on how the statistics
should change for a meaningful performance improvement. Most of these
benchmarks ran on an Amazon Elastic Compute Cloud Medium computer, which
contains a 2-2.4 GHz processor and 3.75 GB of memory.
### 4.1 Code Profiling
I ran a profiler (gprof) to determine where the performance bottlenecks are.
Initially, I thought that memory accesses would dominate the program’s
runtime, because of the size of the data structures involved. However, the
program spends 92.49% of its time inside mathematical functions, meaning that
calculation is the dominant operation. In the first stage of the algorithm,
the program spends 98.5% of its time checking for unique matrices. In the
second stage, it spends most of its time multiplying gate matrices together to
calculate the matrix for a given sequence.
### 4.2 Calculation Time vs. Fowler Distance
The figure below indicates how much time will be required to obtain a given
Fowler distance using Fowler’s original source code. For the purposes of this
paper, the Fowler source code is “unoptimized”, as it does not contain my
optimizations and is a baseline for comparison. This graph is perhaps the most
important graph of them all, since we often want gates with a certain specific
precision.
Since there aren’t very many unique distances, there are not enough data
points to establish a clear trend. A power function appears to fit the data
somewhat closely, though. This power function predicts that the unoptimized
version of the program will take about 110 years to approximate the gate to a
distance closer than $10^{-7}$! This massive exponential expansion explains
why Fowler’s original paper had no gates with a precision better than
$10^{-4}$, since it would take at least a day for the $\frac{\pi}{6}$ gate to
compile to even that precision!
### 4.3 Time vs. Sequence Length
This metric is related to the above metric because longer sequences tend to
have better precision. However, the relationship between time and sequence
length is much clearer, as can be witnessed by the much smoother curve. While
this graph may not have as much practical significance, it is much easier to
relate this graph to the underlying implementation of the algorithm.
From sequence length 0 to 2, the line has a steep slope. This feature probably
exists because the processor cache has not warmed up yet. Between 2 and 15,
every sequence generated by the algorithm is checked against a list of unique
sequences, to see if it’s unique. This check only occurs up to a certain
sequence length: 15 in this case. After that, the algorithm speeds up very
rapidly until it reaches about a sequence length of 30. Then, the graph
becomes a clean exponential curve.
To improve performance, I will effectively need to shift this curve down,
producing longer sequences in less time.
### 4.4 Unique Sequences Per Sequence Length
This metric provides insight into the algorithm’s storage requirements. It is
clear that Fowler’s optimizations have not altered the fundamental exponential
nature of the problem. For sequences longer than about 3, the number of unique
sequences grows exponentially with the sequence length. Since I am more
worried about time rather than space, I will not mind if this curve shifts up.
However, I do need to make sure that my optimizations do not consume too much
memory.
## 5 Ways to Improve Performance
To optimize the performance, I need to:
1. 1.
Speed up calculations such as matrix multiplication.
2. 2.
Reduce the number of calculations required for a given gate sequence length.
There are quite a few possible approaches to approaches 1 and 2. Some of these
approaches were taken this quarter, yet others will be left for future work.
### 5.1 “Meet in the Middle” Bidirectional Search
A traditional “uni-directional” search seeks a path from a start state to a
goal state by starting from the start state and exploring all possible paths.
A bidirectional search starts searching from the goal state as well. Thus, the
search paths will “meet in the middle”: each search only has to take
$\frac{N}{2}$ steps to meet the other search. Thus, instead of taking
$O\left(a^{N}\right)$ time, the algorithm only takes
$O\left(a^{\frac{N}{2}}\right)$ time. One will need some data structure to
store the paths, but inserting into this data structure does not require
exponential time. Thus, for a given amount of time, the algorithm could
compute gate sequences that are twice as long. This approach is the most
promising, and it was implemented in software.
### 5.2 Optimized Unique Matrix Lookup
The algorithm checks to see if a matrix is unique by calculating the distance
between it and all other matrices. Since 98.5% of the application’s run time
is spent in this function, optimizing it could yield significant improvements
in performance in the first stage. However, in the second stage, no more
unique matrix checks are performed; therefore, no time will be spent in this
function. Unless the first stage lasts a long time, it may not be worth the
implementation trouble. This optimization was easy to implement since the C++
standard template library provides a red-black binary search tree.
## 6 Bidirectional Search
Searching for the correct gate is like searching through nodes in a tree: for
a given sequence of gates, the computer must choose which gate to add to the
sequence to come closer to the target gate. In the diagrams below, the arrows
represent a choice of gate, and the boxes represent matrices. When an arrow is
drawn from some box A to a box B, box B is the matrix resulting from
multiplying A by some gate matrix.
In the example shown in these figures, the existing code must go through five
levels of searching in order to reach the target gate. At each new level, the
algorithm considers adding all of the available gates to _each_ sequence
generated by the previous level. Thus, each step multiplies the number of
matrices to consider by 25. So, for a sequence of length N, there will be
$25^{N}$ operations. The “meet in the middle” figure reveals that starting the
search from the start and the goal results in the computer exploring fewer
levels. Each side would only have to explore half as many levels since the
searches meet in the middle. Instead of $25^{N}$ operations, the computer can
ideally perform $2\cdot 25^{N/2}$ operations using the MITM (meet in the
middle) algorithm.
### 6.1 The Search Index
The critical component of the MITM algorithm is the structure that allows the
paths to connect. This structure effectively creates the red arrow in the MITM
figure above, matching up left matrices with right matrices. It must be
designed carefully to ensure optimal performance of the algorithm. For a given
left matrix, it should find a minimal number of right matrices which are close
to the left matrix. Thus, the data structure needs a way to parameterize all
of the matrices stored in it, using parameters that are related to the Fowler
distance between two matrices.
The simplest approach is to choose some reference matrix M, and store the
right matrices in a tree map, using their distances from M as keys. Then, to
find right matrices that are “close” to a left matrix L, the algorithm simply
measures the distance from L to M, and performs a range query for all right
matrices that have about the same distance to M. This trick works because the
Fowler distance measure obeys the triangle inequality: if two matrices L and R
are within some distance d of each other, then the difference in their
distances to some other matrix M will not be greater than d. In the figure
below, this fact is true for all matrices inside the circle.
For my implementation, I use the target gate as the reference matrix, and I
choose d to be $10^{-10}$ less than the smallest distance found so far. Since
the left matrix must check its distance from the target gate anyway, we can
re-use the distance calculation without having to cache it. Note that it is
possible for two matrices to be far away from each other while still having
the same distance to M. Thus, the range query may return false positives,
which are shown between the red lines in the figure. The triangle inequality
property simply guarantees that the range query will not leave out potential
candidates.
### 6.2 Building the Structure
For each sequence S the algorithm generates, a corresponding matrix M is
generated. M represents the transformation that S would perform on a quantum
bit. The algorithm usually assumes that S is a prefix of the solution, meaning
that other gates will be added to the end of S to reach the target gate G.
However it’s also possible to consider S as a suffix, in which gates are added
onto the beginning of S. In this case, S would work backwards from G,
attempting to come close to the identity matrix, rather than the other way
around. If the computer knows M, it can work backwards by multiplying the
inverse of M with G to get a matrix M2. Then, prefix sequences can see if S is
their suffix by comparing their matrices to M2. If a prefix matrix is close to
M2, then it would be close to G if it were multiplied by M.
Therefore, the middle structure simply needs to store as many matrices N as
possible, with pointers to their corresponding sequences. It stores a list of
binary search trees by sequence length, so that all short sequences can be
examined before long sequences.
The middle structure only has so much room to store entries, though. Since the
number of unique sequences scales exponentially with the sequence length, the
structure can store entries up to some length L before running out of memory.
Thus, the MITM algorithm does not always cut the number of search levels in
half; instead, it subtracts L from the number of search levels required to
find a solution. This approach replaces the $O(25^{N})$ cost of exploring
sequences of length N with a $O(25^{N-L})$ cost, since a well-optimized middle
structure should not have an exponential lookup time.
### 6.3 Performing the Search
Whenever the algorithm finds a new unique sequence P, it checks the middle
structure to see if one of the suffixes S can connect it to the target gate G.
Since suffixes are searched by ascending length, the first result should be of
optimal length. The search function is given a distance parameter that
indicates the maximum tolerable Fowler distance for the match; all matrices
that are farther away are skipped. If a result is found, the search function
also returns the distance D from P’s matrix to S’s matrix, so that the
distance threshold can be reduced to $D-\epsilon$ (some small value). That
way, future searches will only return more precise matches.
One problem that I noted after obtaining my results is that the real sequence
may not be of optimal length. The Clifford group contains elements that are
composed of multiple real gates, but each Clifford group element is considered
to be one gate in this algorithm. Since every sequence alternates between
Clifford group elements and T gates, the number of real gates in the sequence
of length n returned by the algorithm is about $n/2+3(n/2)$. However, the
resulting sequence will still have an optimal real length: the Clifford group
elements are ordered such that the ones comprised of multiple real gates are
visited later by the algorithm, meaning they are added to the structure at a
later time. Thus, if the structure uses a stable sort, these longer sequences
will be considered later. I am not entirely certain that my structure does so,
however, which would be a good topic for future research.
Another potential problem is that a very good suffix may be skipped because a
“sufficient” suffix was encountered first. For speed, the MITM algorithm
returns the first suffix that is within the desired distance threshold.
Technically, if this event occurs, the improved suffix would be discovered at
the next search level, so this problem should not impact correctness. However,
that means the best result might not be returned as early as possible. One
sufficient correction would be to continue the search; it won’t impact
performance because new sequences are rarely found. This fix could be
implemented in future work.
### 6.4 Results
As the graph below shows, the “meet in the middle” (MITM) optimization
improved performance by an order of magnitude. Instead of taking about one
hour to calculate a gate sequence that is within $10^{-3}$ of the target gate,
it takes about ten minutes. The Unoptimized and MITM Width 15 lines both used
a “width” of 15, meaning that the middle structure and Fowler’s data
structures stored sequences of length 15. The actual improvement appears to
depend on the width of the middle structure: when sequences of length 30 are
stored in it, the time is cut by two orders of magnitude instead of one.
_Note: ”unoptimized” refers to Fowler’s existing algorithm without the MITM
optimization, not to a simple brute-force enumeration._
Fowler’s unoptimized algorithm also improves performance when the width is
increased, because his data structures can cache more data. Thus, it makes
sense that increasing the width to 30 from 15 results in a larger improvement
than just turning on the MITM optimization.
The memory requirements are much clearer as well: the number of unique
sequences increases exponentially with the sequence length. I omitted data for
sequences of length less than five because they adversely affect the
exponential curve fit.
Finally, I noticed that the number of sequences per unit of time was much
larger in the optimized versions than in the unoptimized versions, confirming
my hypothesis. It clearly makes sense to keep expanding the middle structure
if possible: beyond sequences of length 30, the MITM implementation with width
15 slows down relative to the implementation with width 30. However, in the
long run, the MITM optimization does not change the base of the exponential
that governs the algorithm run time: notice that all of the lines are roughly
parallel towards the right side of the graph.
I managed to approximate the $\frac{\pi}{6}$ gate to $6.8\times 10^{-5}$
precision in about 3 hours and 5 minutes. The result is 72 gates long:
$\begin{array}[]{c}HTHT(HS)THTHTHT(HS)THT(HS)T(HS)\\\
T(HS)T(HS)THTHT(HS)THTHT(HXZ)\\\ THTHTHTHTHT(HS)THTHT(HS)THT(HS)\\\
THTHTHTHT(HS)THT(HXS)T^{\dagger}\end{array}$
## 7 Change of Basis
Since the Fowler distance is phase independent, we can adjust gates to remove
their global phase. Thus, it is possible to represent a quantum gate in
$SU\left(2\right)$ by using just four real numbers. In the equation below,
$\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$ are the Pauli basis matrices.
Since they are multiplied by $i$, the basis is called the _modified Pauli
basis_.
$\displaystyle A$ $\displaystyle=$ $\displaystyle a_{0}\cdot
I+a_{1}\cdot\sigma_{x}+a_{2}\cdot\sigma_{y}+a_{3}\cdot\sigma_{z}$ (2)
$\displaystyle=$ $\displaystyle a_{0}\cdot\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)+a_{1}\cdot\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)+a_{2}\cdot\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)+a_{3}\cdot\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right)$ (11) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}a_{0}+a_{3}\cdot i&a_{2}+a_{1}\cdot
i\\\ -a_{2}+a_{1}\cdot i&a_{0}-a_{3}\cdot i\end{array}\right)$ (14)
One advantage of this new basis is that the trace distance between two gates
$A$ and $B$ is just double the dot product of their vectors $a$ and $b$:
$\displaystyle tr\left(A\cdot B^{\dagger}\right)$ $\displaystyle=$
$\displaystyle tr\left(\left(\begin{array}[]{cc}a_{0}+a_{3}\cdot
i&a_{2}+a_{1}\cdot i\\\ -a_{2}+a_{1}\cdot i&a_{0}-a_{3}\cdot
i\end{array}\right)\cdot\left(\begin{array}[]{cc}b_{0}+b_{3}\cdot
i&b_{2}+b_{1}\cdot i\\\ -b_{2}+b_{1}\cdot i&b_{0}-b_{3}\cdot
i\end{array}\right)^{\dagger}\right)$ (19) $\displaystyle=$ $\displaystyle
tr\left(\left(\begin{array}[]{cc}a_{0}+a_{3}\cdot i&a_{2}+a_{1}\cdot i\\\
-a_{2}+a_{1}\cdot i&a_{0}-a_{3}\cdot
i\end{array}\right)\cdot\left(\begin{array}[]{cc}b_{0}-b_{3}\cdot
i&-b_{2}-b_{1}\cdot i\\\ b_{2}-b_{1}\cdot i&b_{0}+b_{3}\cdot
i\end{array}\right)\right)$ (24) $\displaystyle=$ $\displaystyle
tr\left(\left(\begin{array}[]{cc}\begin{array}[]{c}\left(a_{0}+a_{3}i\right)\cdot\left(b_{0}-b_{3}i\right)+\\\
\left(a_{2}+a_{1}i\right)\cdot\left(b_{2}-b_{1}i\right)\end{array}&\ldots\\\
\ldots&\begin{array}[]{c}\left(-a_{2}+a_{1}i\right)\cdot\left(-b_{2}-b_{1}i\right)+\\\
\left(a_{0}-a_{3}i\right)\cdot\left(b_{0}+b_{3}i\right)\end{array}\end{array}\right)\right)$
(31) $\displaystyle=$
$\displaystyle\begin{array}[]{c}\left(a_{0}+a_{3}i\right)\cdot\left(b_{0}-b_{3}i\right)+\left(a_{2}+a_{1}i\right)\cdot\left(b_{2}-b_{1}i\right)+\\\
\left(-a_{2}+a_{1}i\right)\cdot\left(-b_{2}-b_{1}i\right)+\left(a_{0}-a_{3}i\right)\cdot\left(b_{0}+b_{3}i\right)\end{array}$
(34) $\displaystyle=$
$\displaystyle\begin{array}[]{c}\left(a_{0}b_{0}+a_{3}b_{3}-a_{0}b_{3}i+b_{0}a_{3}i\right)+\left(a_{2}b_{2}+a_{1}b_{1}-a_{2}b_{1}i+a_{1}b_{2}i\right)\\\
\left(a_{2}b_{2}+a_{1}b_{1}+a_{2}b_{1}i-a_{1}b_{2}i\right)+\left(a_{0}b_{0}+a_{3}b_{3}+a_{0}b_{3}i-a_{3}b_{0}i\right)\end{array}$
(37) $\displaystyle=$ $\displaystyle
2\left(a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\right)=2\cdot a\cdot b$
(38)
This result is important because multiplication is an expensive operation in
computer calculation, relative to addition. Traditionally, calculating the
trace distance between two $2\times 2$ matrices $A$ and $B$ requires one to
obtain the diagonal elements of the product $AB$, which requires 4 complex
number multiplications. Since every complex number multiplication requires 4
real-number multiplications, 16 real multiplications must be performed in
total. If the matrices are in the modified Pauli basis, on the other hand,
only four real multiplications are required.
The other advantage is that multiplying two gates requires only 16 real
multiplications. A traditional $2\times 2$ matrix multiplication, on the other
hand, requires 8 complex number multiplications, or 32 real multiplications.
The final advantage is storage size: this new basis can be stored in half the
space that a full $2\times 2$ matrix would require.
The advantages of this basis are outlined in this table:
Task | Regular Matrices | Pauli Basis | Improvement
---|---|---|---
Find trace distance | 16 real multiplies | 4 real multiplies | 4x speedup
Multiply matrices | 32 multiplies | 16 multiplies | 2x speedup
Store a matrix | 8 real numbers | 4 real numbers | 1/2 storage
## 8 Future Work
### 8.1 Using multidimensional spatial indices for the bidirectional search
middle structure
The bidirectional search index only uses one parameter to index the right
matrices. For the reference matrices $M$ which I chose, many matrices had
similar Fowler distances to $M$. Thus, while the algorithm was able to avoid
iterating over some right matrices, it still had to iterate over many matrices
that were not close to a given left matrix. In fact, only .0003% of the
matrices returned by the index were actual matches.
The modified Pauli basis offers an excellent way to parameterize the right
matrices in a spatial index:
1. 1.
Its compact representation requires less space than a full matrix would. In
fact, since one can derive one component from any other three components, only
three components are strictly required. Space is not the only advantage;
certain spatial indices, such as k-d trees, perform better with low-
dimensional data. Hardware implementations of the algorithm also benefit from
simpler calculation circuitry.
2. 2.
Since the trace distance is just the dot product of a left matrix vector $a$
with a right matrix vector $b$, all right matrices $b$ that are close to some
left matrix satisfy this equation:
$\displaystyle-D\leq a\cdot b\leq D$ (39)
where $D$ is some constant related to the maximum trace distance between the
two gates. Geometrically, this means all of the close right matrices are
between two parallel hyperplanes. The process of finding points between the
hyperplanes should be straightforward to optimize. Many spatial indices group
points into bounding volumes like boxes or spheres; checking to see if these
volumes are between the parallel hyperplanes is a simple process.
Libraries such as FLANN [3] provide a wide variety of spatial indices to use.
### 8.2 Map-Reduce Parallelism
The Fowler algorithm can be broken down into a cycle for each sequence length.
Each cycle is essentially a map-reduce job. During the map phase, we assign
one gate to each computer, and that computer will consider all sequences of
length $n$ which start with that gate. Once all computers have finished the
cycle, the reduce phase will merge the data structures for unique matrices, as
well as the discovered gate sequences.
There are several advantages to map-reduce parallelism: Unique sequence data
structures can be shared with all the units between cycles. Thus, all units
can benefit from each unit’s work in each subsequent calculation cycle. If you
keep track of the data structure contents after the final stage, you can
restart the algorithm from this final stage. No specialized hardware (such as
a FPGA) is required. Anyone with access to Amazon’s Elastic MapReduce service,
or a Hadoop cluster, can use a map-reduce algorithm.
Map-reduce parallelism will probably divide the algorithm’s run-time for a
given sequence length by the number of computers involved. Thus, if there are
25 computers (for 25 gates), then the algorithm ought to run up to 25 times
faster. However, since all of the computers must merge their data after each
cycle, the faster computers must wait for the slower ones. Due to the
complexity of the map-reduce setup, this method was not implemented this
quarter. However, Amazon provides a map-reduce framework that should be
straightforward to use and scale, should someone decide to adapt the program.
## 9 Related Work
A variation of the MITM algorithm was independently invented by researchers at
the Institute for Quantum Computing at the University of Waterloo [1]. This
group also seeks to find quantum circuits of optimal length implementing a
given quantum gate. There are a few key differences between their research and
the work presented here:
1. 1.
Their work applies the algorithm to multiple-qubit gates, and does not combine
it with Fowler’s algorithm.
2. 2.
They focus on finding _exact_ matches, rather than approximate ones.
Their future work may benefit from the approximate matching technique
discussed in this paper, as well as the brief discussion of using spatial
indices and a change of basis to accelerate matching. My research will benefit
from their more rigorous treatment of the algorithm, as well as its extension
to multiple qubits.
## 10 Summary
I considered a variety of optimizations to Fowler’s quantum compiler
algorithm. Then, I implemented the “meet in the middle” algorithm in software,
as well as a change of basis technique, and I presented the results here.
While the algorithm certainly provides a dramatic performance boost, it also
requires a lot of memory to maintain the middle index structure I introduced.
Future work involves using map-reduce parallelism and better spatial indices
to improve performance.
## 11 Acknowledgements
I performed most of this research independently, but received significant
guidance and assistance from the following individuals and organizations.
Without their involvement, this research project would not have happened!
1. 1.
Paul Pham – the UW graduate student who suggested the research topic for this
project, and who provided essential quantum computing context and advice. I
had weekly meetings with him, and I worked with him on his pulse sequence
board two years ago. He is working on his own quantum compiler based on the
Solovay-Kitaev Theorem.
2. 2.
Austin Fowler – a Research Fellow in Quantum Computer Science at the
University of Melbourne. He wrote the original paper describing the sequence-
skipping optimization, upon which my research is based. He also supplied the C
source code to his algorithm, so that I could test my optimizations.
3. 3.
Aram Harrow – my faculty advisor, who came up with smart suggestions for error
accumulation analysis and calculation optimization. He also indirectly
proposed the MITM algorithm at the beginning of this research project.
## References
* [1] Matthew Amy, Dmitry Maslov, Michele Mosca, and Martin Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits.
* [2] Austin G. Fowler. Constructing arbitrary steane code single logical qubit fault-tolerant gates. Quantum Info. Comput., 11(9-10):867–873, September 2011.
* [3] Marius Muja and David G. Lowe. Fast approximate nearest neighbors with automatic algorithm configuration. In International Conference on Computer Vision Theory and Application VISSAPP’09), pages 331–340. INSTICC Press, 2009.
* [4] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, U.K., 2000.
|
arxiv-papers
| 2012-06-14T23:53:22 |
2024-09-04T02:49:31.808974
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeffrey Booth Jr",
"submitter": "Jeffrey Booth Jr",
"url": "https://arxiv.org/abs/1206.3348"
}
|
1206.3511
|
# Comparison of Bucket Sort and RADIX Sort
Panu Horsmalahti
panu.horsmalahti@tut.fi
###### Abstract
Bucket sort and RADIX sort are two well-known integer sorting algorithms. This
paper measures empirically what is the time usage and memory consumption for
different kinds of input sequences. The algorithms are compared both from a
theoretical standpoint but also on how well they do in six different use cases
using randomized sequences of numbers. The measurements provide data on how
good they are in different real-life situations.
It was found that bucket sort was faster than RADIX sort, but that bucket sort
uses more memory in most cases. The sorting algorithms performed faster with
smaller integers. The RADIX sort was not quicker with already sorted inputs,
but the bucket sort was.
## 1 Introduction
Sorting algorithms have a long history and countless of algorithms have been
devised [7, p. 2], not only because of their real-life applications. Most
programs use sorting and sorting algorithms should therefore be as fast as
possible. One commonplace applicaton is for databases which may need to sort
tens of millions of objects [5, p. 2]. Making a faster sorting algorithm for a
particular use case can save significant amounts of money. It is important to
have good empirical data on sorting algorithms to choose the best sorting
algorithm for each use case. No single sorting algorithm is best for all use
cases [3, p. 1]. The main factors in choosing the algorithm are time usage,
memory consumption and ease of implementation.
This paper compares two well known sorting algorithms, _bucket sort_ and
_RADIX sort_. Bucket sort and RADIX sort are integer sorts, which are used to
sort a sequence of integers instead of the more general comparison sorts.
Related studies by Curtis [3] and by Loeser [6] have focused on comparison
sorting algorithms. Integer sorting algorithms can also be used to sort text
strings for example, since each string can be converted into an integer.
Bucket sort operates by creating a number of buckets. Each integer is placed
into a bucket based on the integer value. A common way is to divide the input
element by the integer range, thus defining an integer range for each of the
bucket. All buckets are then sorted. After that, the buckets are concatenated
into a single list, which is the output of the algorithm. [2, p. 8]
RADIX is an integer sort which sorts all elements using a single digit at a
time. This sorting algorithm was used by very old card-sorting machines [4, p.
150]. The array is first sorted on the basis of the largest digit. After that,
all cards are sorted using the second largest digit and so on. While it is
intuitive to sort using the _MSD_ (most significant digit) first, it is
actually more efficient to sort with respect to the least significant digit.
Although the _LSD_ (least significant digit) RADIX sort is unintuitive, it
works too. This paper uses the LSD version of the RADIX sort.
Theoretical time complexities do not tell much about how fast sorting
algorithms are in real-life applications. The input array does not often have
a uniform distribution, the integer range may be very small or really large
and sometimes the array is already sorted. This paper will empirically measure
the time usage and memory consumption on these inputs and will also compare
them to a uniform distribution of integer values. Each test sorts up to 100
million keys.
This paper answers the problem of how well these sorting algorithms do in
real-life use cases. Input sequences can be biased in numerous ways, therefore
six different cases are selected for the benchmark along with a uniform
distribution of random integers which is used as a control. The algorithms are
benchmarked with increasing size of the input data, and the growth rate of the
time and the absolute time for the largest input will tell which is better for
the selected use cases.
The results show that bucket sort is faster in all test cases, but it also
uses more memory than the RADIX sort in some cases. Bucket sort is slow with
large integer rages. RADIX sort is equally as fast for sorted inputs as it is
for unsorted inputs. The time usage increases linearly with respect to the
size of the input as is predicted by theory.
The paper is organized into four sections. After introduction, the Section 2
briefly describes sorting algorithms theory, and how integer sorting differs
from comparison sorting. Both algorithms are also described in detail. Section
3 describes what input arrays are chosen and how they are generated. The
testing procedure is explained in detail. Finally, the Section 4 describes the
results and how they can help in choosing the correct sorting algorithm.
## 2 Sorting Algorithm Theory
A Sorting algorithm takes as an input a sequence of elements, and outputs a
permutation of the sequence. Additionally, all elements in the output sequence
must be in (e.g. nondecreasing) order, using some come comparison function.
General sorting algorithms, like merge sort, are neutral with respect to the
data type. In essence they work for all kinds of data which can be compared.
### 2.1 Notation
Asymptotic notations are used in asymptotic analysis to describe how an
algorithm behaves with respect to input size. They can describe how much
memory or time consumption increases when the input size is increased. Usually
we are interested in only the asymptotic growth rate, which means that we are
only interested in the largest term and not constants factors.
The _input size_ $n$ depends on the problem to be studied. It can be the
number of items in the input array, the number of bits needed to represent the
input, or it can even be two different numbers if the input would be a graph.
The _running time_ $f(n)$ is the number of operations or steps executed. The
time required to execute each line in pseudocode is constant. In real life,
different computers execute different operations using different amounts of
time, but when $n$ grows large enough, these differences become insignificant.
[4, p. 25]
Definitions 1, 2 and 3 are used to describe the growth rate of an algorithm.
Note that some publications use set theory notation instead of the equality
sign.
###### Definition 1.
We say that $f(n)=O(g(n))$, if there exist constant $c>0$ and $N$ such that
$f(n)\leq cg(n)$ for all $n\geq N$ [1, p. 25]
In practical terms the $O$-notation in Definition 1 describes the worst-case
behaviour of the algorithm. It guarantees that the longest time the algorithm
can use is less than or equal to $g(n)$ as $n\to\infty$. For example, if the
growth rate for the worst case would be $f(n)=2n^{2}$, we can say
$f(n)=O(n^{2})$.
###### Definition 2.
We say that $f(n)=\Omega(g(n))$, if there exist constant $c,N$ such that
$f(n)\geq cg(n)$ for all $n\geq N$ [1, p. 25]
The $\Omega$-notation in Definition 2 tells the best-case behaviour of the
algorithm. It tells us the minimum running time of the algorithm.
###### Definition 3.
We say that $f(n)=\Theta(g(n))$, if $f(n)=O(g(n))$ and $g(n)=O(f(n))$ [1, p.
25]
The $\Theta$-notation in Definition 3 describes both the worst-case and best-
case growth rates for the algorithm.
It has been proven that comparison-based sorting algorithms have a lower bound
of $O(n\log n)$, where $n$ is size of the input [4, p. 146]. However, lower
bound does not apply to integer sorting, which is a special case of sorting
[2, p. 49]. Integer sorts can be faster than comparison based sorting
algorithms because they make assumptions about the input array. Both the RADIX
sort and the bucket sort are integer sorting algorithms.
A sorting algorithm is _stable_ , if the order of equal elements in the input
array remains the same in the output array [4, p. 149]. In the case of RADIX,
stableness depends on the underlying digit sorting algorithm [4, p. 150].
Bucket sort is also stable, if the underlying sorting algorithm is stable.
The time complexities of bucket sort and RADIX sort are well known, but they
vary depending on which variant of the sort is used. Normal bucket sort has
time complexity of $\Theta(n+r)$ where r is the range of numbers [4, p. 155].
RADIX sort has a time complexity of $\Theta(d(n+k))$ [4, p. 151], where $d$ is
the number of digits in the largest integer and each digit can take $k$
values. The average time complexity is the same as the worst case for both
algorithms.
The algorithms in this paper are written in pseudocode notation. This
pseudocode notation is easy to translate into a real programming language.
Assignment is written using $\leftarrow$. Loops begin with for or while and
are closed with end. The to-clause in the for loop is indicated with $\to$.
$length[A]$ refers to the length of the array $A$. Sometimes operations are
described in plain english, e.g. stable sort array A, as any stable sort is
suitable for the algorithm.
### 2.2 Insertion sort
_Insertion sort_ works in a similar way to a human sorting a few playing card
one at a time. Initially, all cards are in the unsorted pile, and they are put
one by one to the left hand in the correct position. The sorted list of cards
grows until all cards are sorted. [4, p. 18]
Sorting is usually done in-place. The algorithm goes through the array $k$
times, and in each iteration places the number A[j] in the correct position of
the already sorted array. It is efficient for small inputs which is why it is
chosen for the bucket sort implementation. It is also simple to implement.
Insertion sort is stable [2, p. 44], as all equal elements are inserted after
the last equal one in the sorted array. This is required for the bucket sort
to be stable. In the worst case the sorted section is completely shifted in
every iteration, resulting in $O(n^{2})$. In the best case the algorithm is
$\Omega(n)$, when input array is already sorted. The pseudocode in Algorithm 1
assumes that indexing starts from zero. [2, p. 44]
Algorithm 1 INSERTION-SORT(A)
for $j\leftarrow 1\to length[A]-1$ do
$key\leftarrow A[j]$
$i\leftarrow j-1$
while $i\geq 0\land A[i]>key$ do
$A[i+1]\leftarrow A[i]$
$i\leftarrow i-1$
end while
$A[i+1]\leftarrow key$
end for
Figure 1: Pseudocode of the insertion sort algorithm [2, p. 44]
### 2.3 Bucket sort
This paper uses the generic form of bucket sort. It is assumed that each
integer is between $0$ and $M$. $B[1\ldots n]$ is an array of buckets (for a
total number of $n$ buckets) which in this implementation are linked lists.
Each input element is inserted into a bucket $B[n\cdot A[i]/M]$. They are then
sorted with the insertion sort, which is decribed in Section 2.2. A pseudo-
code version of bucket sort [4, p. 153] is shown in Algorithm 2.
Algorithm 2 BUCKET-SORT(A)
$n\leftarrow length[A]$
for $i\leftarrow 1\to n$ do
insert $A[i]$ into list $B[n\cdot A[i]/M]$
end for
for $i\leftarrow 0\to n-1$ do
sort list $B[i]$ with insertion sort
end for
concatenate lists $B[0]$,$B[1]$, $\ldots$, $B[n-1]$ together
Figure 2: Pseudocode of the bucket sort algorithm [4, p. 153]
The sorting algorithm assumes that the integers to be sorted tend to have an
uniform distribution, which is the key to the performance of this algorithm.
For example, if $n$ integers are sorted exactly into $n$ buckets, the running
time is $\Theta(n)$. If the integers are not uniformly distributed, the
algorithm may still run in linear time if the sum of the squares of the bucket
sizes is linear in the total number of elements [4, p. 155].
The more uneven the input distribution is, the more the algorithm slows down,
since more elements are put into the same bucket. Bucket sort is stable, if
the underlying sort is also stable, as equal keys are inserted in order to
each bucket.
### 2.4 Counting sort
_Counting sort_ works by determining how many integers are behind each integer
in the input array $A$. Using this information, the input integer can be
directly placed in the output array $B$. Counting sort is stable [4, p. 149],
which is important as it is used in the RADIX sort.
All numbers are assumed to be between $0$ and $k$. In the pseudocode in
Algorith 3, $C[i]$ first holds the number of input integers equal to $i$ after
the second for-loop. Then $C[i]$ is modified to hold the number of integers
less than or equal to $i$, which can be used to place the integers. In the
final loop, integers are directly placed in the correct position to the output
array $B$. $C[A[j]]$ is decremented so that the next $A[j]$ is placed one
position to the left. [4, p. 29]
Algorithm 3 COUNTING-SORT(A, B, k)
for $i\leftarrow 0\to k$ do
$C[i]\leftarrow 0$
end for
for $j\leftarrow 1\to length[A]$ do
$C[A[j]]\leftarrow C[A[j]]+1$
end for
for $i\leftarrow 1\to k$ do
$C[i]\leftarrow C[i]+C[i-1]$
end for
for $j\leftarrow length[A]\to 1$ do
$B[C[A[j]]]\leftarrow A[j]$
$C[A[j]]\leftarrow C[A[j]]-1$
end for
Figure 3: Pseudo code of the counting sort algorithm [4, p. 148]
### 2.5 RADIX sort
RADIX is a sorting algorithm which sorts all elements on the basis of a single
digit at a time. RADIX sort is not limited to sorting integers, because
integers can represent strings. There are two main variations of the RADIX.
The first one starts from the most significant digit (MSD) and the second from
the least significant digit (LSD). RADIX sort is also useful when sorting
records with multiple fields, like year, month and day. RADIX sort could first
sort it on the day, then the month and finally the year. [4, p. 150]
First, the array is sorted using the LSD. For each pass a algorithm sorts it
by a digit. This paper uses the _counting sort_ , described in Section 2.4,
because it is efficient if the integer range is small (e.g. $0\ldots 9$) and
it is also stable. Next step is to sort it using the second LSD and so forth.
The last step sorts it using the MSD. In the end, all the elements are sorted.
RADIX sort is stable, as the chosen underlying digit sorting algorithm is
stable. [4, p. 150]. If the input integers have at most $d$-digits, then the
algorithm will go through the array $d$ times, once for each digit. The
pseudo-code for RADIX sort is shown in Algorithm 4.
Algorithm 4 RADIX-SORT(A, d)
for $i\leftarrow 1\to d$ do
stable sort array $A$ on digit $i$
end for
Figure 4: Pseudo code of RADIX sort algorithm [4, p. 151]
In the pseudocode the array to be sorted is $A$ and the number of digits in
the largest integer is $d$. If we use the counting sort each pass will use
$\Theta(n+k)$ time, where each digit is in the range of $0\ldots k-1$. The
whole sort takes $d$ passes, so the total time usage is $\Theta(d(n+k))$. [4,
p. 151].
## 3 Comparison of the Algorithms
The sorting algorithms are now compared empirically by measuring time usage
and memory consumption. To measure the sorting algorithm following inputs are
used, where $n$ is the number of elements to be sorted. All input cases are
measured using three different sizes: $n=10^{6}$, $n=10^{7}$ and $n=10^{8}$.
All input integers are between $0$ and $M$.
### 3.1 Test cases and implementation
Six different input cases and three different input sizes are empirically
measured to find out how the algorithms perform.
1. 1.
$n$ integers evenly distributed and in random order, $M=10^{6}$
2. 2.
$n$ integers already sorted, $M=10^{6}$
3. 3.
$n$ integers of which 95% already sorted , $M=10^{6}$
4. 4.
$n$ integers with small range of $M=10^{4}$
5. 5.
$n$ integers with large range of $M=10^{8}$
6. 6.
$\frac{n}{3}$ of the integers with the same value $k$, rest of the integers
all with different values, $M=10^{6}$
The above inputs show how the two algorithms work in different real-life use
cases. Time usage and memory consumption is measured. The first input is used
as a control, as sorting algorithms often assume that the data is uniformly
distributed. Sorting algorithms are often analyzed and tested using a uniform
distribution [4, p. 1]. The second input is important to measure, since often
the data is already sorted and the algorithm should still perform well.
The third input array is a common use case, since the inputs are often almost
sorted. The fourth checks how well the algorithm works for a large number of
integers with a small range (e.g. many values might be the same).
The fifth test is for a large range, which is tested since some integer sort
implementations are slow with large integer ranges. The sixth and final case
is to check how well the algorithm copes with a large number of the same
value. Bucket sort is expected to suffer from this use case a lot.
The algorithms were implemented using the C++ programming language. The same
C++ program also created the inputs. Random integers are created using rand()
function. The seed number was not randomized, so that each test could be run
multiple times with the same input.
The implementation uses the vector$<$int$>$ container. Time usage is measured
using clock() function. When sorting the same input array using the same
algorithm multiple times, the time usage varied less than 5%, so time
measurement is assumed to be sufficiently accurate.
Memory consumption is measured using the task manager. A standard laptop with
a Intel i5 processor was used for the benchmark using the GNU/Linux operating
system. The results are shown in Table 1 and Table 2.
### 3.2 Results
Table 1: Measured time consumptions [s] | RADIX sort | Bucket sort
---|---|---
Input no. | $n=10^{6}$ | $n=10^{7}$ | $n=10^{8}$ | $n=10^{6}$ | $n=10^{7}$ | $n=10^{8}$
1 | 1.78 | 17.66 | 176.84 | 0.74 | 5.84 | 45.92
2 | 1.88 | 18.93 | 174.85 | 0.61 | 4.14 | 26.83
3 | 1.86 | 17.69 | 178.68 | 0.61 | 4.12 | 28.27
4 | 1.05 | 11.10 | 105.31 | 0.31 | 3.00 | 31.54
5 | 2.45 | 24.56 | 246.28 | 0.78 | 8.54 | 97.43
6 | 1.77 | 17.66 | 176.65 | 0.61 | 5.18 | 41.06
Table 2: Measured memory usage [MB] | RADIX sort | Bucket sort
---|---|---
Input no. | $n=10^{6}$ | $n=10^{7}$ | $n=10^{8}$ | $n=10^{6}$ | $n=10^{7}$ | $n=10^{8}$
1 | 5. | 1 | 39. | 4 | 382. | 7 | 24. | 4 | 117. | 4 | 382. | 8
2 | 20. | 6 | 79. | 2 | 382. | 9 | 47. | 2 | 117. | 4 | 382. | 8
3 | 19. | 5 | 76. | 7 | 382. | 9 | 24. | 4 | 109. | 4 | 382. | 8
4 | 5. | 1 | 39. | 4 | 382. | 7 | 31. | 9 | 39. | 5 | 382. | 8
5 | 5. | 1 | 39. | 4 | 382. | 7 | 24. | 4 | 232. | 8 | 2344. | 0
6 | 5. | 1 | 39. | 4 | 382. | 7 | 19. | 9 | 96. | 7 | 382. | 8
Table 1 summarises the time consumption and Table 2 summarises the memory
consumption. Figure 5 shows a graph for input cases 1, 2 and 3 and the Figure
6 for inputs 4, 5 and 6. The diagrams are in logarithmic scale. The time usage
increases linearly according to the results as is predicted by theory.
On the whole, bucket sort is faster in all cases, but uses significantly more
memory, except when $n=10^{8}$. In the fifth input case the bucket sort uses
an order of magnitude more memory than RADIX. The reason why bucket sort is
faster in all cases may be implementation specific. A further study with a
different implementation might clarify whether or not the reason is
implementation specific.
The results for inputs no. 2 and 3 indicate that RADIX sort performs as well
for unsorted and sorted inputs. Bucket sort on the other hand is clearly
faster for sorted arrays. The reason might be that the underlying insertion
sort is fast for sorted input arrays. The results for the fourth input
sequence indicates that RADIX sort performs well with smaller integers,
because then $d$ is smaller. Bucket sort is also faster than the default case.
Figure 5: Time consumption for inputs 1, 2 and 3 Figure 6: Time consumption
for inputs 4, 5 and 6
Both algorithms are quite slow in the fifth input array with the large range
of $M=10^{8}$, and RADIX is slow because of the higher number of digits $d$.
This is the slowest input array for both of the algorithms.
RADIX performs just as fast for the sixth input sequence as it does for the
control sequence with uniform distribution. The same is true for bucket sort,
even though bucket sort has to use the insertion sort for third of the
numbers.
## 4 Summary
Two sorting algorithms were compared both empirically and theoretically. Six
different use cases were identified and measurements were made using three
different input sizes, ranging three orders of magnitude. The first input had
a uniform distribution of random numbers. The second input was sorted, which
tested how well the algorithms perform with fully sorted input sequences.
The third input had 95% of numbers sorted, which tested for nearly sorted
input arrays. The fourth and fifth inputs had a small range and a large range,
respectively, to test how the algorithm react. The last case tested an input
with a large amount of numbers with a same value.
RADIX sort used the least significant digit version and the counting sort.
Bucket sort used the insertion sort as the underlying sorting algorithm.
The algorithms and the creation of input arrays were implemented using the C++
programming language. Time consumption and memory usage were empirically
measured. The sorting took up to a hundred seconds with the largest input.
It was found out that bucket sort is faster in all cases. The performance of
the RADIX sort is slow only when the range of the integers is rather large.
The bucket sort was found also to be slow with large integer ranges. The
bucket sort was found to be quite fast with small integer ranges, which is
also true for the RADIX sort. RADIX sort is as quick for unsorted inputs as it
is for sorted inputs. The memory usage of the RADIX sort is slightly better
than the bucket sort when sorting a small number of integers. Bucket sort uses
large amounts of memory when sorting numbers with a large range.
More research should be made in the future to compare these sorting algorithms
with comparison based ones like the merge sort. Studies should also be made
using parallel versions of bucket sort and RADIX sort.
## References
* [1] Eric Bach and Jeffrey Shallit. “Algorithmic Number Theory: Efficient Algorithms”, volume 1 of Foundations of Computing. August 1996.
* [2] C. Canaan, M. S. Garai, and M. Daya:. “Popular sorting algorithms”. World Applied Programming, 1(1):42–50, April 2011.
* [3] Curtis R. Cook and Do Jin Kim. “Best sorting algorithm for nearly sorted lists”. Commun. ACM, 23(11):620–624, November 1980.
* [4] T. H. Cormen, C. E. Leiserson, R.L Rivest, and C. Stein. “Introduction to Algorithms”. MIT Press, 2nd edition edition, August 2001.
* [5] G. Graefe. “Implementing sorting in database systems”. ACM Comput. Surv., 38, September 2006.
* [6] Rudolf Loeser. “Some performance tests of “quicksort” and descendants”. Commun. ACM, 17(3):143–152, March 1974.
* [7] N. Satish, M. Harris, and M. Garland. “Designing efficient sorting algorithms for manycore gpus”. In Parallel & Distributed Processing, pages 1–10, May 2009.
|
arxiv-papers
| 2012-06-15T16:39:51 |
2024-09-04T02:49:31.820270
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Panu Horsmalahti",
"submitter": "Panu Horsmalahti",
"url": "https://arxiv.org/abs/1206.3511"
}
|
1206.3582
|
# Decentralized Learning for Multi-player Multi-armed Bandits ††thanks:
Dileep Kalathil, Naumaan Nayyar and Rahul Jain ((manisser,nnayyar,rahul.jain)
@usc.edu) are with the Department of Electrical Engineering, University of
Southern California, Los Angeles, CA, USA. This research is supported by AFOSR
grant FA9550-10-1-0307 and NSF CAREER award CNS-0954116.
††thanks: A preliminary version of this paper is under submission to IEEE CDC
2012. This version contains proofs of all theorems as well as new results on
Markovian MABs.
Dileep Kalathil, Naumaan Nayyar and Rahul Jain
###### Abstract
We consider the problem of distributed online learning with multiple players
in multi-armed bandits (MAB) models. Each player can pick among multiple arms.
When a player picks an arm, it gets a reward. We consider both i.i.d. reward
model and Markovian reward model. In the i.i.d. model each arm is modelled as
an i.i.d. process with an unknown distribution with an unknown mean. In the
Markovian model, each arm is modelled as a finite, irreducible, aperiodic and
reversible Markov chain with an unknown probability transition matrix and
stationary distribution. The arms give different rewards to different players.
If two players pick the same arm, there is a “collision”, and neither of them
get any reward. There is no dedicated control channel for coordination or
communication among the players. Any other communication between the users is
costly and will add to the regret. We propose an online index-based
distributed learning policy called ${\tt dUCB_{4}}$ algorithm that trades off
exploration v. exploitation in the right way, and achieves expected regret
that grows at most as near-$O(\log^{2}T)$. The motivation comes from
opportunistic spectrum access by multiple secondary users in cognitive radio
networks wherein they must pick among various wireless channels that look
different to different users. This is the first distributed learning algorithm
for multi-player MABs to the best of our knowledge.
###### Index Terms:
Distributed adaptive control, multi-armed bandit, online learning, multi-agent
systems.
## I Introduction
In [1], Lai and Robbins introduced the classical non-Bayesian multi-armed
bandit model. Such models capture the essence of the learning problem that
players face in an unknown environment, where the players must not only
explore to learn but also exploit in choosing the best arm. Specifically,
suppose a player can choose between $N$ arms. Upon choosing an arm $i$, it
gets a reward from a distribution with density $f(x,\theta_{i})$. Time is
slotted, and players do not know the distributions (nor any statistics about
them). The problem is to find a learning policy that minimizes the expected
regret over some time horizon $T$. It was shown by Lai and Robbins [1] that
there exists an index-type policy that achieves expected regret that grows
asymptotically as $\log T$, and this is order-optimal, i.e., there exists no
causal policy that can do better. This was generalized by Anantharam, et al
[2] to the case of multiple plays, i.e., when the player can pick multiple
arms at the same time. In [3], Agrawal proposed a sample mean based index
policy which achieves $\log T$ regret asymptotically. Assuming that the
rewards are coming from a distribution of bounded support, Auer, et al [4]
proposed a much simpler sample mean based index policy, called ${\tt
UCB_{1}}$, which achieves $\log T$ uniformly over time, not only
asymptotically. Also, unlike the policy in [3], the index doesn’t depend on
the specific family of distributions that the rewards come from.
In [5], Anantharam, et al proposed a policy to the case where the arms are
modelled as Markovian, not i.i.d. The rewards are assumed to come from a
finite, irreducible and aperiodic Markov chain represented by a single
parameter probability transition matrix. The state of each arm evolves
according to an underlying transition probability matrix when the arm is
played and remains frozen when passive. Such problems are called _rested
Markovian bandit problems_ (where _rested_ refers to no state evolution until
the arm is played). In [6], Tenkin and Liu extended the ${\tt UCB_{1}}$ policy
to the case of rested Markovian bandit problems. If some non-trivial bounds on
the underlying Markov chains are known a priori, they showed that the policy
achieves $\log T$ regret uniformly over time. Also, if no information about
the underlying Markov chains is available, the policy can easily be modified
to get a _near_ -$O(\log T)$ regret asymptotically. The models in which the
state of an arm continues to evolve even when it is not played are called
_restless Markovian bandit problems_. Restless models are considerably more
difficult than the rested models and have been shown to be P-SPACE hard [7].
This is because the optimal policy no longer will be to “play the arm with the
highest mean reward”. [8] employs a weaker notion of regret (weak regret)
which compares the reward of a policy to that of a policy which always plays
the the arm with the highest mean reward. They propose a policy which achieves
$\log T$ (weak) regret uniformly over time if certain bounds on the underlying
Markov model are known a priori and achieves a near-$O(\log T)$ (weak) regret
asymptotically when no such knowledge is available. [9] proposes another
simpler policy which achieves the same bounds for weak regret. [10] proposes a
policy based on deterministic sequence of exploration and exploitation and
achieves the same bounds for weak regret. In [11], the authors consider the
notion of strong regret and propose a policy which achieves near-$\log T$
(strong) regret for some special cases of the restless model.
Recently, there is an increasing interest in multi-armed bandit models, partly
because of opportunistic spectrum access problems. Consider a user who must
choose between $N$ wireless channels. Yet, it knows nothing about the channel
statistics, i.e., has no idea of how good or bad the channels are, and what
rate it may expect to get from each channel. The rates could be learnt by
exploring various channels. Thus, these have been formulated as multi-armed
bandit problems, and index-type policies have been proposed for choosing
spectrum channels. In many scenarios, there are multiple users accessing the
channels at the same time. Each of these users must be matched to a different
channel. These have been formulated as a combinatorial multi-armed bandit
problem [12] [13], and it was shown that an “index-matching” algorithm that at
each instant determines a matching by solving a sum-index maximization problem
achieves $O(\log T)$ regret uniformly over time, and this is indeed order-
optimal.
In other settings, the users cannot coordinate, and the problem must be solved
in a decentralized manner. Thus, settings where all channels (arms) are
identical for all users with i.i.d. rewards have been considered, and index-
type policies that can achieve coordination have been proposed that get
$O(\log T)$ regret uniformly over time [14, 15, 16, 10]. A similar result for
Markovian reward model with weak regret has been shown by [10], assuming some
non-trivial bounds on the underlying Markov chains are known a priori. The
regret scales only polynomially in the number of users and channels.
Surprisingly, the lack of coordination between the players asymptotically
imposes no additional cost or regret.
In this paper, we consider the decentralized multi-armed bandit problem with
distinct arms for each players. We consider both the i.i.d. reward model and
the _rested_ Markovian reward model. All players together must discover the
best arms to play as a team. However, since they are all trying to learn at
the same time, they may collide when two or more pick the same arm. We propose
an index-type policy ${\tt dUCB_{4}}$ based on a variation of the ${\tt
UCB_{1}}$ index. At its’ heart is a distributed bipartite matching algorithm
such as Bertsekas’ auction algorithm [17]. This algorithm operates in rounds,
and in each round prices for various arms are determined based on bid-values.
This imposes communication (and computation) cost on the algorithm that must
be accounted for. Nevertheless, we show that when certain non-trivial bounds
on the model parameters are known a priori, the ${\tt dUCB_{4}}$ algorithm
that we introduce achieves (at most) near-$O(\log^{2}T)$ growth non-
asymptotically in expected regret. If no such information about the model
parameters are available, ${\tt dUCB_{4}}$ algorithm still achieves (at most)
near-$O(\log^{2}T)$ regret asymptotically. A lower bound, however, is not
known at this point, and a work in progress.
The paper is organized as follows. In Section II, we present the model and
problem formulation. In section III and IV we present some variations on
single player MAB with i.i.d. rewards and Markovian rewards respectively. In
section V, we introduce the decentralized MAB problem with i.i.d. rewards. We
then extend the results to the decentralized cases with Markovian rewards in
section VI. In section VII we present the distributed bipartite matching
algorithm which is used in our main algorithm for decentralized MAB. In
section VIII, we present some simulation results to numerically evaluate the
performance of our algorithm.
## II Model and Problem Formulation
### II-A Arms with i.i.d. rewards
We consider an $N$-armed bandit with $M$ players. In a wireless cognitive
radio setting [18], each arm could correspond to a channel, and each player to
a user who wants to use a channel. Time is slotted, and at each instant each
player picks an arm. There is no dedicated control channel for coordination
among the players. So, potentially more than one players can pick the same arm
at the same instant. We will regard that as a collision. Player $i$ playing
arm $k$ at time $t$ yields i.i.d. reward $S_{ik}(t)$ with univariate density
function $f(s,\theta_{ik})$, where $\theta_{ik}$ is a parameter in the set
$\Theta_{ik}$. We will assume that the rewards are bounded, and without loss
of generality lie in $[0,1]$. Let $\mu_{i,k}$ denote the mean of $S_{ik}(t)$
w.r.t. the pdf $f(s,\theta_{ik})$. We assume that the parameter vector
$\theta=(\theta_{ij},1\leq i\leq M,1\leq j\leq N)$ is unknown to the players,
i.e., the players have no information about the mean, the distributions or any
other statistics about the rewards from various arms other than what they
observe while playing. We also assume that each player can only observe the
rewards that they get. When there is a collision, we will assume that all
players that choose the arm on which there is a collision get zero reward.
This could be relaxed where the players share the reward in some manner though
the results do not change appreciably.
Let $X_{ij}(t)$ be the reward that player $i$ gets from arm $j$ at time $t$.
Thus, if player $i$ plays arm $k$ at time $t$ (and there is no collision),
$X_{ik}(t)=S_{ik}(t)$, and $X_{ij}(t)=0,j\neq k$. Denote the action of player
$i$ at time $t$ by $a_{i}(t)\in\mathcal{A}:=\\{1,\ldots,N\\}$. Then, the
history seen by player $i$ at time $t$ is
$\mathcal{H}_{i}(t)=\\{(a_{i}(1),X_{i,a_{i}(1)}(1)),\cdots,(a_{i}(t),X_{i,a_{i}(t)}(t))\\}$
with $\mathcal{H}_{i}(0)=\emptyset$. A policy
$\alpha_{i}=(\alpha_{i}(t))_{t=1}^{\infty}$ for player $i$ is a sequence of
maps $\alpha_{i}(t):\mathcal{H}_{i}(t)\to\mathcal{A}$ that specifies the arm
to be played at time $t$ given the history seen by the player. Let
$\mathcal{P}(N)$ be the set of vectors such that
$\displaystyle\mathcal{P}(N):=\\{\mathbf{a}=(a_{1},\ldots,a_{M}):a_{i}\in\mathcal{A},a_{i}\neq
a_{j},\text{for}~{}i\neq j\\}.$
The players have a team objective: namely over a time horizon $T$, they want
to maximize the expected sum of rewards
$\mathbb{E}[\sum_{t=1}^{T}\sum_{i=1}^{M}X_{i,a_{i}(t)}(t)]$ over some time
horizon $T$. If the parameters $\mu_{i,j}$ are known, this could easily be
achieved by picking a bipartite matching
$\mathbf{k}^{**}\in\arg\max_{\mathbf{k}\in\mathcal{P}(N)}\sum_{i=1}^{M}\mu_{i,k_{i}},$
(1)
i.e., the optimal bipartite matching with expected reward from each match.
Note that this may not be unique. Since the expected rewards, $\mu_{i,j}$, are
unknown, the players must pick learning policies that minimize the expected
regret, defined for policies $\alpha=(\alpha_{i},1\leq i\leq M)$ as
$\mathcal{R}_{\alpha}(T)=T\sum_{i}\mu_{i,k_{i}^{**}}-\mathbb{E}_{\alpha}\left[\sum_{t=1}^{T}\sum_{i=1}^{M}X_{i,\alpha_{i}(t)}(t)\right].$
(2)
Our goal is to find a decentralized algorithm that players can use such that
together they minimize the expected regret.
### II-B Arms with Markovian rewards
Here we follow the model formulation introduced in the previous subsection,
with the exception that the rewards are now considered Markovian. The reward
that player $i$ gets from arm $j$ (when there is no collision) $X_{ij}$, is
modelled as an irreducible, aperiodic, reversible Markov chain on a finite
state space $\mathcal{X}^{i,j}$ and represented by a transition probability
matrix
$P^{i,j}:=\left(p^{i,j}_{x,x^{{}^{\prime}}}:x,x^{{}^{\prime}}\in\mathcal{X}^{i,j}\right)$.
We assume that rewards are bounded and strictly positive, and without loss of
generality lie in $(0,1]$. Let
$\mathbf{\pi}^{i,j}:=\left(\pi^{i,j}_{x},x\in\mathcal{X}^{i,j}\right)$ be the
stationary distribution of the Markov chain $P^{i,j}$. The mean reward from
arm $j$ for player $i$ is defined as
$\mu_{i,j}:=\sum_{x\in\mathcal{X}^{i,j}}x\pi^{i,j}_{x}$. Note that the Markov
chain represented by $P^{i,j}$ makes a state transition only when player $i$
plays arm $j$. Otherwise it remains _rested_.
We note that although we use the ‘big $O$’ notation to emphasis the regret
order, unless otherwise noted results are non-asymptotic.
## III Some variations on single player multi-armed bandit with i.i.d.
rewards
We first present some variations on the single player non-Bayesian multi-armed
bandit model. These will prove useful later for the multi-player problem
though they should also be of independent interest.
### III-A ${\tt UCB_{1}}$ with index recomputation every $L$ slots
Consider the classical single player non-Bayesian $N$-armed bandit problem. At
each time $t$, the player picks a particular arm, say $j$, and gets a random
reward $X_{j}(t)$. The rewards $X_{j}(t),1\leq t\leq T$ are independent and
identically distributed according to some unknown probability measure with an
unknown expectation $\mu_{j}$. Without loss of generality, assume that
$\mu_{1}>\mu_{i}>\mu_{N},$ for $i=2,\cdots N-1$. Let $n_{j}(t)$ denote the
number of times arm $j$ has been played by time $t$. Denote
$\Delta_{j}:=\mu_{1}-\mu_{j}$, $\Delta_{min}:=\min_{j,j\neq 1}\Delta_{j}$ and
$\Delta_{max}:=\max_{j}\Delta_{j}$. The regret for any policy $\alpha$ is
$\mathcal{R}_{\alpha}(T):=\mu_{1}T-\sum_{j=1}^{N}\mu_{j}\mathbb{E}_{\alpha}[n_{j}(T)].$
(3)
${\tt UCB_{1}}$ index [4] is defined as
$g_{j}(t):=\overline{X}_{j}(t)+\sqrt{\frac{2\log(t)}{n_{j}(t)}},$ (4)
where $\overline{X}_{j}(t)$ is the average reward obtained by playing arm $j$
by time $t$. It is defined as
$\overline{X}_{j}(t)=\sum_{m=1}^{t}r_{j}(m)/n_{j}(t)$, where $r_{j}(m)$ is the
reward obtained from arm $j$ at time $m$. If the arm $j$ is played at time $t$
then $r_{j}(m)=X_{j}(m)$ and otherwise $r_{j}(t)=0$. Now, an index-based
policy called ${\tt UCB_{1}}$ [4] is to pick the arm that has the highest
index at each instant. It can be shown that this algorithm achieves regret
that grows logarithmically in $T$ non-asymptotically.
An easy variation of the above algorithm which will be useful in our analysis
of subsequent algorithms is the following. Suppose the index is re-computed
only once every $L$ slots. In that case, it is easy to establish the
following.
###### Theorem 1.
Under the ${\tt UCB_{1}}$ algorithm with recomputation of the index once every
$L$ slots, the expected regret by time $T$ is given by
$\mathcal{R}_{\tt UCB_{1}}(T)\leq\sum_{j>1}^{N}\frac{8L\log
T}{\Delta_{j}}+L\left(1+\frac{\pi^{2}}{3}\right)\sum_{j>1}^{N}\Delta_{j}.$ (5)
The proof follows [4] and taking into account the fact that every time a
suboptimal arm is selected, it is played for the next $L$ time slots. We omit
it due to space consideration.
### III-B ${\tt UCB_{4}}$ Algorithm when index computation is costly
Often, learning algorithms pay a penalty or cost for computation. This is
particularly the case when the algorithms must solve combinatorial
optimization problems that are NP-hard. Such costs also arise in decentralized
settings wherein algorithms pay a communication cost for coordination between
the decentralized players. This is indeed the case, as we shall see later when
we present an algorithm to solve the decentralized multi-armed bandit problem.
Here, however, we will just consider an “abstract” communication or
computation cost. The problem we formulate below can be solved with better
regret bounds than what we present. At this time though we are unable to
design algorithms with better regret bounds, that also help in
decentralization.
Consider a computation cost every time the index is recomputed. Let the cost
be $C$ units. Let $m(t)$ denote the number of times the index is computed by
time $t$. Then, under policy $\alpha$ the expected regret is now given by
$\tilde{\mathcal{R}}_{\alpha}(T):=\mu_{1}T-\sum_{j=1}^{N}\mu_{j}\mathbb{E}_{\alpha}[n_{j}(T)]+C\mathbb{E}_{\alpha}[m(T)].$
(6)
It is easy to argue that the ${\tt UCB_{1}}$ algorithm will give a regret
$\Omega(T)$ for this problem. We present an alternative algorithm called ${\tt
UCB_{4}}$ algorithm, that gives sub-linear regret. Define the ${\tt UCB_{4}}$
index
$g_{j}(t):=\overline{X}_{j}(t)+\sqrt{\frac{3\log(t)}{n_{j}(t)}}.$ (7)
We define an arm $j^{*}(t)$ to be the best arm if $j^{*}(t)\in\arg\max_{1\leq
i\leq N}g_{i}(t).$
Algorithm 1 : $\tt UCB_{4}$
1: Initialization: Select each arm $j$ once for $t\leq N$. Update the $\tt
UCB_{4}$ indices. Set $\eta=1$.
2: while ($t\leq T$) do
3: if ($\eta=2^{p}$ for some $p=0,1,2,\cdots$) then
4: Update the index vector $g(t)$;
5: Compute the best arm $j^{*}(t)$;
6: if $(j^{*}(t)\neq j^{*}(t-1))$ then
7: Reset $\eta=1$;
8: end if
9: else
10: $j^{*}(t)=j^{*}(t-1)$;
11: end if
12: Play arm $j^{*}(t)$;
13: Increment counter $\eta=\eta+1$; $t=t+1$;
14: end while
We will use the following concentration inequality.
Fact 1: Chernoff-Hoeffding inequality [19]
Let $X_{1},\ldots,X_{t}$ be random variables with a common range such that
$\mathbb{E}[X_{t}|X_{1},\ldots,X_{t-1}]=\mu$. Let $S_{t}=\sum_{i=1}^{t}X_{i}$.
Then for all $a\geq 0$,
$\displaystyle\mathbb{P}\left(S_{t}\geq t\mu+a\right)\leq
e^{-2a^{2}/t},~{}~{}\text{and}~{}~{}\mathbb{P}\left(S_{t}\leq
t\mu-a\right)\leq e^{-2a^{2}/t}.$ (8)
###### Theorem 2.
The expected regret for the single player multi-armed bandit problem with per
computation cost $C$ using the ${\tt UCB_{4}}$ algorithm is given by
$\displaystyle\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle(\Delta_{max}+C(1+\log T))\cdot\left(\sum_{j>1}^{N}\frac{12\log
T}{\Delta_{j}^{2}}+2N\right).$
Thus, $\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)=O(\log^{2}T).$
###### Proof.
We prove this in two steps. First, we compute the expected number of times a
suboptimal arm is played and then the expected number of times we recompute
the index.
Consider any suboptimal arm $j>1$. Denote $c_{t,s}=\sqrt{3\log t/s}$ and the
indicator function of the event $A$ by $I\\{A\\}$. let $\tau_{j,m}$ be the
time at which the player makes the $m$th transition to the arm $j$ from
another arm and $\tau_{j,m}^{{}^{\prime}}$ be the time at which the player
makes the $m$th transition from the arm $j$ to another arm. Let
$\tilde{\tau}_{j,m}^{{}^{\prime}}=\min\\{\tau_{j,m}^{{}^{\prime}},T\\}$. Then,
$n_{j}(T)\leq
1+\sum_{m=1}^{T}(\tilde{\tau}_{j,m}^{{}^{\prime}}-\tau_{j,m})I\\{\text{Arm}~{}j~{}\text{is
picked at time}~{}\tau_{j,m},\tau_{j,m}\leq T\\}$
$\displaystyle\leq$ $\displaystyle
1+\sum_{m=1}^{T}(\tilde{\tau}_{j,m}^{{}^{\prime}}-\tau_{j,m})I\\{g_{j}(\tau_{j,m}-1)\geq
g_{1}(\tau_{j,m}-1),\tau_{j,m}\leq T\\}$ $\displaystyle\leq$ $\displaystyle
l+\sum_{m=1}^{T}(\tilde{\tau}_{j,m}^{{}^{\prime}}-\tau_{j,m})I\\{g_{j}(\tau_{j,m}-1)\geq
g_{1}(\tau_{j,m}-1),\tau_{j,m}\leq T,n_{j}(\tau_{j,m}-1)\geq l\\}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
l+\sum_{m=1}^{T}\sum_{p=0}^{\infty}2^{p}I\\{g_{j}(\tau_{j,m}+2^{p}-2)\geq
g_{1}(\tau_{j,m}+2^{p}-2),\tau_{j,m}+2^{p}\leq T,n_{j}(\tau_{j,m}-1)\geq l\\}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
l+\sum_{m=2}^{T}\sum_{p=0}^{\infty}2^{p}I\\{g_{j}(m+2^{p}-2)\geq
g_{1}(m+2^{p}-2),m+2^{p}\leq T,n_{j}(m-1)\geq l\\}$ $\displaystyle\leq$
$\displaystyle l+\sum_{m=1}^{T}\sum_{p\geq 0,m+2^{p}\leq
T}2^{p}I\\{\overline{X}_{j}(m+2^{p}-1)+c_{m+2^{p}-1,n_{j}(m+2^{p}-1)}\geq$
$\displaystyle\hskip
113.81102pt\overline{X}_{1}(m+2^{p}-1)+c_{m+2^{p}-1,n_{1}(m+2^{p}-1)},n_{j}(m-1)\geq
l\\}$ $\displaystyle\leq$ $\displaystyle l+\sum_{m=1}^{T}\sum_{p\geq
0,m+2^{p}\leq T}2^{p}I\\{\max_{l\leq
s_{j}<m+2^{p}}\overline{X}_{j}(m+2^{p}-1)+c_{m+2^{p}-1,s_{j}}\geq$
$\displaystyle\hskip 113.81102pt\min_{1\leq
s_{1}<m+2^{p}}\overline{X}_{1}(m+2^{p}-1)+c_{m+2^{p}-1,s_{1}}\\}$ (10)
$\displaystyle\leq$ $\displaystyle l+\sum_{m=1}^{\infty}\sum_{p\geq
0,m+2^{p}\leq
T}2^{p}\sum_{s_{1}=1}^{m+2^{p}}\sum_{s_{j}=l}^{m+2^{p}}I\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}.$
In Algorithm 1 ($\tt UCB_{4}$), if an arm is for the $p$th time consecutively
(without switching to any other arms in between), it is be played for the next
$2^{p}$ slots. Inequality (a) uses this fact. In the inequality (b), we
replace $\tau_{j,m}$ by $m$ which is clearly an upper bound. Now, observe that
the event
$\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}$
implies at least one of the following events,
$\displaystyle A:=\big{\\{}\overline{X}_{1}(m+2^{p})$ $\displaystyle\leq$
$\displaystyle\mu_{1}-c_{m+2^{p},s_{1}}\big{\\}},\hskip
14.22636ptB:=\big{\\{}\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}}\big{\\}},$
(11)
$\displaystyle\text{or}~{}C:=\big{\\{}\mu_{1}<\mu_{j}+2c_{m+2^{p},s_{j}}\big{\\}}.$
Now, using the Chernoff-Hoeffding bound, we get
$\displaystyle\mathbb{P}\left(\overline{X}_{1}(m+2^{p})\leq\mu_{1}-c_{m+2^{p},s_{1}}\right)\leq(m+2^{p})^{-6},~{}\mathbb{P}\left(\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}}\right)\leq(m+2^{p})^{-6}.$
For $l=\left\lceil\frac{12\log T}{\Delta^{2}_{j}}\right\rceil$, the last event
in (11) is false. In fact, $\mu_{1}-\mu_{j}-2c_{m+2^{p},s_{j}}$
$\displaystyle=\mu_{1}-\mu_{j}-2\sqrt{3\log(m+2^{p})/s_{j}}$
$\displaystyle\geq\mu_{1}-\mu_{j}-\Delta_{j}=0,~{}\text{for}~{}s_{j}\geq\left\lceil
12\log T/\Delta_{j}^{2}\right\rceil.$ $\displaystyle\text{So, we
get,}~{}\mathbb{E}[n_{j}(T)]\leq\left\lceil 12\log
T/\Delta_{j}^{2}\right\rceil$ $\displaystyle+$
$\displaystyle\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1}=1}^{m+2^{p}}\sum_{s_{j}=1}^{m+2^{p}}2(m+2^{p})^{-6}$
$\displaystyle\leq\left\lceil 12\log T/\Delta_{j}^{2}\right\rceil$
$\displaystyle+$ $\displaystyle
2\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}(m+2^{p})^{-4}\leq\frac{12\log
T}{\Delta^{2}_{j}}+2.$ (12)
Next, we upper-bound the expectation of $m(T)$, the number of index
computations performed by time $T$. We can write $m(T)=m_{1}(T)+m_{2}(T)$,
where $m_{1}(T)$ is the number of index updates that result in an optimal
allocation, and $m_{2}(T)$ is the number of index updates that result in a
suboptimal allocation. Clearly, the number of updates resulting in a
suboptimal allocation is less than the number of times a suboptimal arm is
played. Thus,
$\mathbb{E}[m_{2}(T)]\leq\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)].$ (13)
To bound $\mathbb{E}[m_{1}(T)]$, let $\tau_{l}$ be the time at which the
player makes the $l$th transition to an optimal arm from a suboptimal arm and
$\tau_{l}^{\prime}$ be the time at which the player makes the $l$th transition
from an optimal arm to a suboptimal arm. Then,
$m_{1}(T)\leq\sum_{l=1}^{n_{sub}(T)}\log|\tau_{l}-\tau_{l}^{\prime}|$, where
$n_{sub}(T)$ is the total number of such transitions by time $T$. Clearly,
$n_{sub}(T)$ is upper-bounded by the total number of times the player picks a
sub-optimal arm. Also, $\log|\tau_{l}-\tau_{l}^{\prime}|\leq\log T$. So,
$\mathbb{E}[m_{1}(T)]\leq\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot\log T.$ (14)
Thus, from bounds (13) and (14), we get
$\mathbb{E}[m(T)]\leq\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot(1+\log T).$ (15)
Now, using equation (6), the expected regret is
$\displaystyle\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)$ $\displaystyle=$
$\displaystyle\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot\Delta_{j}+C\mathbb{E}[m(T)]\leq\Delta_{max}\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]+C\mathbb{E}[m(T)]$
$\displaystyle\leq$ $\displaystyle\left(\Delta_{max}+C(1+\log
T)\right)\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)].$
by using (15). Now, by bound (12), we get the desired bound on the expected
regret. ∎
Remarks. 1\. It is easy to show that the lower bound for the single player MAB
problem with computation costs is $\Omega(\log T)$. This can be achieved by
the ${\tt UCB_{2}}$ algorithm [4]. To see this, note that the number of times
the player selects a suboptimal arm when using ${\tt UCB_{2}}$ is $O(\log T)$.
Since $\mathbb{E}[n_{j}(T)]=O(\log T)$, we get
$\mathbb{E}[\sum_{j>1}^{N}n_{j}(T)]=O(\log T),$ and also
$\mathbb{E}[m_{2}(T)]=O(\log T).$ Now, since the epochs are not getting reset
after every switch and are exponentially spaced, the number of updates that
result in the optimal allocation, $m_{1}(T)\leq\log T.$ These together yield
$\tilde{\mathcal{R}}_{\tt
UCB_{2}}(T)\leq\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot\Delta_{j}+C\mathbb{E}[m(T)]=O(\log
T).$
2\. Variations of the ${\tt UCB_{2}}$ algorithm that use a deterministic
schedule can also be used [20]. But it is unknown at this time if these can be
used in solving the decentralized MAB problem that we introduce in the next
section. This is the main reason for introducing the ${\tt UCB_{4}}$
algorithm.
### III-C Algorithms with finite precision indices
Often, there might be a cost to compute the indices to a particular precision.
In that case, indices may be known upto some $\epsilon$ precision, and it may
not possible to tell which of two indices is greater if they are within
$\epsilon$ of each other. The question then is how is the performance of
various index-based policies such as ${\tt UCB_{1},UCB_{4}}$, etc. affected if
there are limits on index resolution, and only an arm with an
$\epsilon$-highest index can be picked. We first show that if $\Delta_{min}$
is known, we can fix a precision $0<\epsilon<\Delta_{min}$, so that ${\tt
UCB_{4}}$ algorithm will achieve order log-squared regret growth with $T$. If
$\Delta_{min}$ is not known, we can pick a positive monotone sequence
$\\{\epsilon_{t}\\}$ such that $\epsilon_{t}\to 0$, as $t\to\infty$. Denote
the cost of computation for $\epsilon$-precision be $C(\epsilon)$. We assume
that $C(\epsilon)\rightarrow\infty$ monotonically as $\epsilon\rightarrow 0$.
###### Theorem 3.
(i) If $\Delta_{min}$ is known, choose an $0<\epsilon<\Delta_{min}$. Then, the
expected regret of the ${\tt UCB_{4}}$ algorithm with $\epsilon$-precise
computations is given by
$\displaystyle\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle\left(\Delta_{max}+C(\epsilon)(1+\log
T)\right)\cdot\left(\sum_{j>1}^{N}\frac{12\log
T}{(\Delta_{j}-\epsilon)^{2}}+2N\right).$
Thus, $\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)=O(\log^{2}T).$
(ii) If $\Delta_{min}$ is unknown, denote $\epsilon_{min}=\Delta_{min}/2$ and
choose a positive monotone sequence $\\{\epsilon_{t}\\}$ such that
$\epsilon_{t}\to 0$ as $t\to\infty$. Then, there exists a $t_{0}>0$ such that
for all $T>t_{0}$,
$\displaystyle\tilde{\mathcal{R}}_{\tt UCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle\left(\Delta_{max}+C(\epsilon_{min})\right)t_{0}+(\Delta_{max}+C(\epsilon_{T})(1+\log
T))\cdot\left(\sum_{j>1}^{N}\frac{12\log
T}{(\Delta_{j}-\epsilon_{min})^{2}}+2N\right)$
where $t_{0}$ is the smallest $t$ such that $\epsilon_{t_{0}}<\epsilon_{min}$.
Thus by choosing an arbitrarily slowly increasing sequence
$\\{\epsilon_{t}\\}$, we can make the regret arbitrarily close to
$O(\log^{2}T)$ asymptotically.
###### Proof.
(i) The proof is only a slight modification of the proof given in Theorem 2.
Due to the $\epsilon$ precision, the player will pick a suboptimal arm if the
event
$\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}+\epsilon\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}$
occurs. Thus equation (III-B) becomes, $n_{j}(T)$
$\leq l+\sum_{m=1}^{\infty}\sum_{p\geq 0,m+2^{p}\leq
T}2^{p}\sum_{s_{1}=1}^{m+2^{p}}\sum_{s_{j}=l}^{m+2^{p}}I\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}+\epsilon\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}.$
Now, the event
$\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}+\epsilon\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}$
implies that at least one of the following events must occur:
$\displaystyle
A:=\big{\\{}\overline{X}_{1}(m+2^{p})\leq\mu_{1}-c_{m+2^{p},s_{1}}\big{\\}},$
$\displaystyle
B:=\big{\\{}\overline{X}_{j}(m+2^{p})\geq\mu_{j}+\epsilon+c_{m+2^{p},s_{j}}\big{\\}},$
$\displaystyle
C:=\big{\\{}\mu_{1}<\mu_{j}+\epsilon+2c_{m+2^{p},s_{j}}\big{\\}},$
$\displaystyle\text{or}~{}D:=\big{\\{}\mu_{1}<\mu_{j}+\epsilon\big{\\}}.$ (16)
Since
$\\{\overline{X}_{j}(m+2^{p})\geq\mu_{j}+\epsilon+c_{m+2^{p},s_{j}}\\}\subseteq\\{\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}}\\}$,
we have
$\displaystyle\mathbb{P}(\\{\overline{X}_{j}(m+2^{p})$ $\displaystyle\geq$
$\displaystyle\mu_{j}+\epsilon+c_{m+2^{p},s_{j}}\\})\leq\mathbb{P}(\\{\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}}\\}).$
Also, for $l=\left\lceil 12\log T/(\Delta_{j}-\epsilon)^{2}\right\rceil$, the
event $C$ cannot happen. In fact,
$\mu_{1}-\mu_{j}-\epsilon-2c_{t+2^{p},s_{j}}=\mu_{1}-\mu_{j}-\epsilon-2\sqrt{\frac{3\log(t+2^{p})}{s_{j}}}\geq\mu_{1}-\mu_{j}-\epsilon-(\Delta_{j}-\epsilon)=0,$
for $s_{j}\geq\left\lceil 12\log T/(\Delta_{j}-\epsilon)^{2}\right\rceil$. If
$\epsilon<\Delta_{min}$, the last event (D) in equation (III-C) is also not
true. Thus, for $0<\epsilon<\Delta_{min}$, we get
$\mathbb{E}[n_{j}(T)]\leq\frac{12\log(n)}{(\Delta_{j}-\epsilon)^{2}}+2.$ (17)
The rest of the proof is the same as in Theorem 2. Now, if $\Delta_{min}$ is
known, we can choose $0<\epsilon<\Delta_{min}$ and by Theorem 2 and bound
(17), we get the desired result.
(ii) If $\Delta_{min}$ is unknown, we can choose a positive monotone sequence
$\\{\epsilon_{t}\\}$ such that $\epsilon_{t}\to 0$ as $t\to\infty$. Thus,
there exists a $t_{0}$ such that for $t>t_{0}$, $\epsilon_{t}<\epsilon_{min}$.
We may get a linear regret upto time $t_{0}$ but after that the analysis
follows that in the proof of Theorem 2, and regret grows only sub-linearly.
Since $C(\cdot)$ is monotone, $C(\epsilon_{T})>C(\epsilon_{t})$ for all $t<T$.
The last part can now be trivially established using the obtained bound on the
expected regret. ∎
## IV Single Player Multi-armed Bandit with Markovian Rewards
Now, we consider the scenario where the rewards obtained from an arm are not
i.i.d. but come from a Markov chain. Reward from each arm is modelled as an
irreducible, aperiodic, reversible Markov chain on a finite state space
$\mathcal{X}^{i}$ and represented by a transition probability matrix
$P^{i}:=\left(p^{i}_{x,x^{{}^{\prime}}}:x,x^{{}^{\prime}}\in\mathcal{X}^{i}\right)$.
Assume that the reward space $\mathcal{X}^{i}\subseteq(0,1]$. Let
$X_{i}(1),X_{i}(2),\ldots$ denote the successive rewards from arm $i$. All
arms are mutually independent. Let
$\mathbf{\pi^{i}}:=\left(\pi^{i}_{x},x\in\mathcal{X}^{i}\right)$ be the
stationary distribution of the Markov chain $P^{i}$. Since the Markov chains
are ergodic under these assumptions, the mean reward from arm $i$ is given by
$\mu_{i}:=\sum_{x\in\mathcal{X}^{i}}x\pi^{i}_{x}$. Without loss of generality,
assume that $\mu_{1}>\mu_{i}>\mu_{N},$ for $i=2,\cdots N-1$. As before,
$n_{j}(t)$ denotes the number of times arm $j$ has been played by time $t$.
Denote $\Delta_{j}:=\mu_{1}-\mu_{j}$, $\Delta_{min}:=\min_{j,j\neq
1}\Delta_{j}$ and $\Delta_{max}:=\max_{j}\Delta_{j}$. Denote
$\pi_{min}:=\min_{1\leq i\leq N,x\in\mathcal{X}^{i}}\pi^{i}_{x}$,
$x_{max}:=\max_{1\leq i\leq N,x\in\mathcal{X}^{i}}x$ and $x_{min}:=\min_{1\leq
i\leq N,x\in\mathcal{X}^{i}}x$. Let
${\hat{\pi}^{i}}_{x}:=\max\\{\pi^{i}_{x},1-\pi^{i}_{x}\\}$ and
$\hat{\pi}_{max}:=\max_{1\leq i\leq
N,x\in\mathcal{X}^{i}}{\hat{\pi}^{i}}_{x}$. Let $|\mathcal{X}^{i}|$ denote the
cardinality of the state space $\mathcal{X}^{i}$,
$|\mathcal{X}|_{max}:=\max_{1\leq i\leq N}|\mathcal{X}^{i}|$. Let $\rho^{i}$
be the eigenvalue gap, $1-\lambda_{2}$, where $\lambda_{2}$ is the second
largest eigenvalue of the matrix ${P^{i}}^{2}$. Denote
$\rho_{max}:=\max_{1\leq i\leq N}\rho^{i}$ and $\rho_{min}:=\min_{1\leq i\leq
N}\rho^{i}$, where $\rho^{i}$ is the eigenvalue gap of the $i$th arm.
The total reward obtained by the time $T$ is then given by
$S_{T}=\sum_{j=1}^{N}\sum_{s=1}^{n_{j}(T)}X_{j}(s)$. The regret for any policy
$\alpha$ is defined as
$\tilde{\mathcal{R}}_{M,\alpha}(T):=\mu_{1}T-\mathbb{E}_{\alpha}\sum_{j=1}^{N}\sum_{s=1}^{n_{j}(T)}X_{j}(s)+C\mathbb{E}_{\alpha}[m(T)]$
(18)
where $C$ is the cost per computation and $m(T)$ is the number of times the
index is computed by time $T$, as described in section III. Define the index
$g_{j}(t):=\overline{X}_{j}(t)+\sqrt{\frac{\kappa\log(t)}{n_{j}(t)}},$ (19)
where $\overline{X}_{j}(t)$ is the average reward obtained by playing arm $j$
by time $t$, as defined in the previous section. $\kappa$ can be any constant
satisfying $\kappa>168|\mathcal{X}|_{max}^{2}/\rho_{min}$.
We introduce one more notation here. If $\mathcal{F}$ and $\mathcal{G}$ are
two $\sigma$-algebras, then $\mathcal{F}\vee\mathcal{G}$ denotes the smallest
$\sigma$-algebra containing $\mathcal{F}$ and $\mathcal{G}$. Similarly, if
$\\{\mathcal{F}_{t},t=1,2,\ldots\\}$ is a collection of $\sigma$-algebras,
then $\vee_{t\geq 1}F_{t}$ denotes the smallest $\sigma-$algebra containing
$\mathcal{F}_{1},\mathcal{F}_{2},\ldots$
The following can be derived easily from Lemma 4 [5], reproduced in the
appendix.
###### Lemma 1.
If the reward of each arm is given by a Markov chain satisfying the hypothesis
of Lemma 4, then under any policy $\alpha$ we have
$\tilde{\mathcal{R}}_{M,\alpha}(T)\leq\sum_{j=2}^{N}\Delta_{j}\mathbb{E}_{\alpha}[n_{j}(T)]+K_{\mathcal{X},P}+C\mathbb{E}_{\alpha}[m(T)]$
(20)
where
$K_{\mathcal{X},P}=\sum_{j=1}^{N}\sum_{x\in\mathcal{X}^{j}}x/\pi^{j}_{min}$
and $\pi^{j}_{min}=\min_{x\in\mathcal{X}^{j}}\pi^{j}_{x}$
###### Proof.
Let $X_{j}(1),X_{j}(2),\ldots$ denote the successive rewards from arm $j$. Let
$\mathcal{F}^{j}_{t}$ denotes the $\sigma$-algebra generated by
$\left(X_{j}(1),\ldots,X_{j}(t)\right)$. Let $\mathcal{F}^{j}=\vee_{t\geq
1}\mathcal{F}^{j}_{t}$ and $\mathcal{G}^{j}=\vee_{i\neq j}F^{i}$. Since arms
are independent, $\mathcal{G}^{j}$ is independent of $\mathcal{F}^{j}$.
Clearly, $n_{j}(T)$ is a stopping time with respect to
$\mathcal{G}^{j}\vee\mathcal{F}^{j}_{T}$. The total reward is
$S_{T}=\sum_{j=1}^{N}\sum_{s=1}^{n_{j}(T)}X_{j}(s)=\sum_{j=1}^{N}\sum_{x\in\mathcal{X}^{j}}xN(x,n_{j}(T))$
where $N(x,n_{j}(T)):=\sum_{t=1}^{n_{j}(T)}I\\{X_{j}(t)=x\\}$. Taking the
expectation and using the Lemma 4, we have
$\left\lvert\mathbb{E}[S_{T}]-\sum_{j=1}^{N}\sum_{x\in\mathcal{X}^{j}}x\pi^{j}_{x}\mathbb{E}[n_{j}(T)]\right\rvert\leq\sum_{j=1}^{N}\sum_{x\in\mathcal{X}^{j}}x/\pi^{j}_{min}$,
which implies
$\left\lvert\mathbb{E}[S_{T}]-\sum_{j=1}^{N}\mu_{j}\mathbb{E}[n_{j}(T)]\right\rvert\leq
K_{\mathcal{X},P},$ where
$K_{\mathcal{X},P}=\sum_{j=1}^{N}\sum_{x\in\mathcal{X}^{j}}x/\pi^{j}_{min}$.
Since regret
$\tilde{\mathcal{R}}_{M,\alpha}(T)=\mu_{1}T-\mathbb{E}_{\alpha}\sum_{j=1}^{N}\sum_{t=1}^{n_{j}(T)}X_{j}(t)+C\mathbb{E}_{\alpha}[m(T)]$
(c.f. equation (18)), we get
$|\tilde{\mathcal{R}}_{M,\alpha}(T)-\left(\mu_{1}T-\sum_{j=1}^{N}\mu_{j}\mathbb{E}[n_{j}(T)]+C\mathbb{E}_{\alpha}[m(T)]\right)|\leq
K_{\mathcal{X},P}.$
∎
We will use a concentration inequality for Markov chains (Lemma 5, from [21]),
reproduced in the appendix.
###### Theorem 4.
(i) If $|\mathcal{X}|_{max}$ and $\rho_{min}$ are known, choose
$\kappa>168|\mathcal{X}|_{max}^{2}/\rho_{min}$. Then, the expected regret
using the ${\tt UCB_{4}}$ algorithm with the index defined as in (19) for the
single player multi-armed bandit problem with Markovian rewards and per
computation cost $C$ is given by
$\displaystyle\tilde{\mathcal{R}}_{M,\tt UCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle(\Delta_{max}+C(1+\log
T))\cdot\left(\sum_{j>1}^{N}\frac{4\kappa\log
T}{\Delta_{j}^{2}}+N(2D+1)\right)+K_{\mathcal{X},P}$
where $D=\frac{|\mathcal{X}|_{max}}{\pi_{min}}$. Thus,
$\tilde{\mathcal{R}}_{M,\tt UCB_{4}}(T)=O(\log^{2}T).$
(ii) If $|\mathcal{X}|_{max}$ and $\rho_{min}$ are not known, choose a
positive monotone sequence $\\{\kappa_{t}\\}$ such that
$\kappa_{t}\rightarrow\infty$ as $t\rightarrow\infty$ and $\kappa_{t}\leq t$.
Then, $\tilde{\mathcal{R}}_{M,\tt UCB_{4}}(T)=O(\kappa_{T}\log^{2}T)$. Thus,
by choosing an arbitrarily slowly increasing sequence $\\{\kappa_{t}\\}$ we
can make the regret arbitrarily close to $\log^{2}T$.
###### Proof.
(i) Consider any suboptimal arm $j>1$. Denote $c_{t,s}=\sqrt{\kappa\log t/s}$.
As in the proof of Theorem 2, we start by bounding $n_{j}(T)$. The initial
steps are the same as in the proof of Theorem 2. So, we skip those steps and
start from the inequality (III-B) there.
$\displaystyle n_{j}(T)$ $\displaystyle\leq$ $\displaystyle
l+\sum_{m=1}^{\infty}\sum_{p\geq 0,m+2^{p}\leq
T}2^{p}\sum_{s_{1}=1}^{m+2^{p}}\sum_{s_{j}=l}^{m+2^{p}}I\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}.$
The event
$\\{\overline{X}_{j}(m+2^{p})+c_{m+2^{p},s_{j}}\geq\overline{X}_{1}(m+2^{p})+c_{m+2^{p},s_{1}}\\}$
is true only if at least one of the events shown in display (11) are true. We
note that, for any initial distribution $\lambda^{j}$ for arm $j$,
$N_{\lambda^{j}}=\left\lVert\left(\frac{\lambda^{j}_{x}}{\pi^{j}_{x}},x\in\mathcal{X}^{j}\right)\right\rVert_{2}\leq\sum_{x\in\mathcal{X}^{j}}\left\lVert\left(\frac{\lambda^{j}_{x}}{\pi^{j}_{x}}\right)\right\rVert_{2}\leq\frac{1}{\pi_{min}}.$
(21)
Also, $x_{max}\leq 1$. Let $n^{j}_{x}(s_{j})$ be the number of times the state
$x$ is observed when arm $j$ is pulled $s_{j}$ times. Then, the probability of
the first event in (11),
$\mathbb{P}(\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}})$
$\displaystyle=\mathbb{P}\left(\sum_{x\in\mathcal{X}^{j}}xn^{j}_{x}(s_{j})\geq
s_{j}\sum_{x\in\mathcal{X}^{j}}x\pi^{j}_{x}+s_{j}c_{m+2^{p},s_{j}}\right)=\mathbb{P}\left(\sum_{x\in\mathcal{X}^{j}}(n^{j}_{x}(s_{j})-s_{j}\pi^{j}_{x})\geq
s_{j}c_{m+2^{p},s_{j}}/x\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\sum_{x\in\mathcal{X}^{j}}\mathbb{P}\left(n^{j}_{x}(s_{j})-s_{j}\pi^{j}_{x}\geq\frac{s_{j}c_{m+2^{p}}}{x|\mathcal{X}^{j}|}\right)=\sum_{x\in\mathcal{X}^{j}}\mathbb{P}\left(\frac{\sum_{t=1}^{s_{j}}I\\{X_{j}(t)=x\\}-s_{j}\pi^{j}_{x}}{s_{j}{\hat{\pi}^{j}}_{x}}\geq\frac{c_{m+2^{p},s_{j}}}{x|\mathcal{X}^{j}|\hat{\pi}^{j}_{x}}\right)$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{x\in\mathcal{X}^{j}}N_{\lambda^{j}}(m+2^{p})^{-\kappa\rho^{i}/28x^{2}|\mathcal{X}^{j}|^{2}(\hat{\pi}^{j}_{x})^{2}}~{}~{}~{}\stackrel{{\scriptstyle(c)}}{{\leq}}\frac{|\mathcal{X}|_{max}}{\pi_{min}}(m+2^{p})^{-\kappa\rho_{min}/28|\mathcal{X}|_{max}^{2}}.$
The inequality (a) follows after some simple algebra, which we skip due to
space limitations. The inequality (b) follows by defining the function
$f(X_{j}(t))=(I\\{X_{j}(t)=x\\}-\pi^{j}_{x})/{\hat{\pi}^{j}}_{x}$ and using
the Lemma 5. For inequality (c) we used the facts that $N_{\lambda^{j}}\leq
1/\pi_{min}$, $x_{max}\leq 1$ and ${\hat{\pi}_{max}}\leq 1$. Thus,
$\mathbb{P}(\overline{X}_{j}(m+2^{p})\geq\mu_{j}+c_{m+2^{p},s_{j}})\leq
D(m+2^{p})^{-\kappa\rho_{min}/28|\mathcal{X}|_{max}|^{2}}$ (22)
where $D=\frac{|\mathcal{X}|_{max}}{\pi_{min}}$. Similarly we can get,
$\mathbb{P}(\overline{X}_{1}(m+2^{p})\leq\mu_{1}-c_{m+2^{p},s_{1}})\leq
D(m+2^{p})^{-\kappa\rho_{min}/28|\mathcal{X}|_{max}|^{2}}$ (23)
For $l=\left\lceil 4\kappa\log T/\Delta^{2}_{j}\right\rceil$, the last event
in (11) is false. In fact, $\mu_{1}-\mu_{j}-2c_{m+2^{p},s_{j}}$
$\displaystyle=\mu_{1}-\mu_{j}-2\sqrt{\kappa\log(m+2^{p})/s_{j}}\geq\mu_{1}-\mu_{j}-\Delta_{j}=0,~{}\text{for}~{}s_{j}\geq\left\lceil
4\kappa\log T/\Delta^{2}_{j}\right\rceil.~{}\text{Thus},$
$\displaystyle\mathbb{E}[n_{j}(T)]$ $\displaystyle\leq$
$\displaystyle\left\lceil\frac{4\kappa\log
T}{\Delta_{j}^{2}}\right\rceil+\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1}=1}^{m+2^{p}}\sum_{s_{j}=1}^{m+2^{p}}2D(m+2^{p})^{-\frac{\kappa\rho_{min}}{28|\mathcal{X}|_{max}|^{2}}}$
$\displaystyle=\left\lceil\frac{4\kappa\log
T}{\Delta_{j}^{2}}\right\rceil+2D\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}(m+2^{p})^{-\frac{\kappa\rho_{min}-56|\mathcal{X}|_{max}^{2}}{28|\mathcal{X}|_{max}^{2}}}.$
(24)
When $\kappa>168|\mathcal{X}|_{max}^{2}/\rho_{min}$, the above summation
converges to a value less that $1$ and we get
$\mathbb{E}[n_{j}(T)]\leq\frac{4\kappa\log T}{\Delta^{2}_{j}}+(2D+1).$ (25)
Now, from the proof of Theorem 2 (equation (15)),
$\mathbb{E}[m(T)]\leq\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot(1+\log T).$ (26)
Now, using inequality (20), the expected regret $\tilde{\mathcal{R}}_{M,\tt
UCB_{4}}(T)=$
$\displaystyle=$
$\displaystyle\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]\cdot\Delta_{j}+C\mathbb{E}[m(T)]+K_{\mathcal{X},P}\leq\Delta_{max}\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]+C\mathbb{E}[m(T)]+K_{\mathcal{X},P}$
$\displaystyle\leq$ $\displaystyle\left(\Delta_{max}+C(1+\log
T)\right)\sum_{j>1}^{N}\mathbb{E}[n_{j}(T)]+K_{\mathcal{X},P}.$
by using (26). Now, by bound (25), we get the desired bound on the expected
regret.
(ii) Replacing $\kappa$ with $\kappa_{t}$, equation (IV) becomes
$\displaystyle\mathbb{E}[n_{j}(T)]$ $\displaystyle\leq$
$\displaystyle\left\lceil\frac{4\kappa_{T}\log
T}{\Delta_{j}^{2}}\right\rceil+2D\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}(m+2^{p})^{-\frac{\kappa_{m+2^{p}}\rho_{min}-56|\mathcal{X}|_{max}^{2}}{28|\mathcal{X}|_{max}^{2}}}$
Since, $\kappa_{t}\rightarrow\infty$ as $t\rightarrow\infty$, the exponent
${-\frac{\kappa_{m+2^{p}}\rho_{min}-56|\mathcal{X}|_{max}^{2}}{28|\mathcal{X}|_{max}^{2}}}$
becomes smaller that $-4$ for sufficiently large $m$ and $p$, and the above
summation converges, yielding the desired result. ∎
We note that we have used the results in [22] in the above proof. We note that
the results for Markovian reward just presented extend easily even with finite
precision indices. As before, suppose the cost of computation for
$\epsilon$-precision is $C(\epsilon)$. We assume that
$C(\epsilon)\rightarrow\infty$ monotonically as $\epsilon\rightarrow 0$. We
formally state the following result, which we will use in section VI.
###### Theorem 5.
(i) If $\Delta_{min}$, $|\mathcal{X}|_{max}$ and $\rho_{min}$ are known,
choose an $0<\epsilon<\Delta_{min}$, and a
$\kappa>168|\mathcal{X}|_{max}^{2}/\rho_{min}$. Then, the expected regret
using the ${\tt UCB_{4}}$ algorithm with the index defined as in (19) for the
single player multi-armed bandit problem with Markovian rewards with
$\epsilon$-precise computations is given by
$\displaystyle\tilde{\mathcal{R}}_{M,\tt UCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle\left(\Delta_{max}+C(\epsilon)(1+\log
T)\right)\cdot\left(\sum_{j>1}^{N}\frac{4\kappa\log
T}{(\Delta_{j}-\epsilon)^{2}}+N(2D+1)\right).$
where $D=\frac{|\mathcal{X}|_{max}}{\pi_{min}}$. Thus,
$\tilde{\mathcal{R}}_{M,\tt UCB_{4}}(T)=O(\log^{2}T).$
(ii) If $\Delta_{min}$, $|\mathcal{X}|_{max}$ and $\rho_{min}$ are unknown,
choose a positive monotone sequences $\\{\epsilon_{t}\\}$ such that and
$\\{\kappa_{t}\\}$ such that $\kappa_{t}\leq t$, $\epsilon_{t}\to 0$ and
$\kappa_{t}\rightarrow\infty$ as $t\to\infty$. Then,
$\tilde{\mathcal{R}}_{M,\tt
UCB_{4}}(T)=O(C(\epsilon_{T})\kappa_{T}\log^{2}T)$. We can choose
$\\{\epsilon_{t}\\}$ and $\\{\kappa_{t}\\}$ as two arbitrarily slowly
increasing sequences and thus the regret can be made arbitrarily close to
$\log^{2}(T)$.
The proof follows by a combination of the proof of the theorems 3 and 4, and
is omitted.
## V The Decentralized MAB problem with i.i.d. rewards
We now consider the decentralized multi-armed bandit problem with i.i.d.
rewards wherein multiple players play at the same time. Players have no
information about means or distribution of rewards from various arms. There
are no dedicated control channels for coordination or communication between
the players. If two or more players pick the same arm, we assume that neither
gets any reward. Tshis is an online learning problem of distributed bipartite
matching.
Distributed algorithms for bipartite matching algorithms are known [23, 24]
which determine an $\epsilon$-optimal matching with a ‘minimum’ amount of
information exchange and computation. However, every run of this distributed
bipartite matching algorithm incurs a cost due to computation, and
communication necessary to exchange some information for decentralization. Let
$C$ be the cost per run, and $m(t)$ denote the number of times the distributed
bipartite matching algorithm is run by time t. Then, under policy $\alpha$ the
expected regret is
$\mathcal{R}_{\alpha}(T)=T\sum_{i=1}^{M}\mu_{i,k_{i}^{**}}-\mathbb{E}_{\alpha}\left[\sum_{t=1}^{T}\sum_{i=1}^{M}X_{i,\alpha_{i}(t)}(t)\right]+C\mathbb{E}[m(T)].$
(27)
where $\mathbf{k}^{**}$ is the optimal matching as defined in equation (1) in
section II-A.
Temporal Structure. We divide time into frames. Each frame is one of two
kinds: a decision frame, and an exploitation frame. In the decision frame, the
index is recomputed, and the distributed bipartite matching algorithm run
again to determine the new matching. The length of such a frame can be seen as
cost of the algorithm. We further divide the decision frame into two phases, a
negotiation phase and an interrupt phase (see Figure 1). The information
exchange needed to compute an $\epsilon$-optimal matching is done in the
negotiation phase. In the interrupt phase, a player signals to other players
if his allocation has changed. In the exploitation frame, the current matching
is exploited without updating the indices. Later, we will allow the frame
lengths to increase with time.
We now present the ${\tt dUCB_{4}}$ algorithm, a decentralized version of
${\tt UCB_{4}}$. For each player $i$ and each arm $j$, we define a ${\tt
dUCB_{4}}$ index at the end of frame $t$ as
$g_{i,j}(t):=\overline{X}_{i,j}(t)+\sqrt{\frac{(M+2)\log
n_{i}(t)}{n_{i,j}(t)}},$ (28)
where $n_{i}(t)$ is the number of successful plays (without collisions) of
player $i$ by frame $t$, $n_{i,j}(t)$ is the number of times player $i$ picks
arm $j$ successfully by frame $t$. $\overline{X}_{i,j}(t)$ is the sample mean
of rewards from arm $j$ for player $i$ from $n_{i,j}(t)$ samples. Let $g(t)$
denote the vector $(g_{i,j}(t),1\leq i\leq M,1\leq j\leq N)$. Note that $g$ is
computed only in the decision frames using the information available upto that
time. Each player now uses the ${\tt dUCB_{4}}$ algorithm. We will refer to an
$\epsilon$-optimal distributed bipartite matching algorithm as ${\tt
dBM_{\epsilon}}(g(t))$ that yields a solution
$\mathbf{k}^{*}(t):=(k_{1}^{*}(t),\ldots,k_{M}^{*}(t))\in\mathcal{P}(N)$ such
that
$\sum_{i=1}^{M}g_{i,k^{*}_{i}(t)}(t)\geq\sum_{i=1}^{M}g_{i,k_{i}}(t))-\epsilon,~{}\forall\mathbf{k}\in\mathcal{P}(N),\mathbf{k}\neq\mathbf{k}^{*}$.
Let $\mathbf{k}^{**}\in\mathcal{P}(N)$ be such that
$\mathbf{k}^{**}\in\arg\max_{\mathbf{k}\in\mathcal{P}(N)}\sum_{i=1}^{M}\mu_{i,\mathbf{k}_{i}},$
i.e., an optimal bipartite matching with expected rewards from each matching.
Denote $\mu^{**}:=\sum_{i=1}^{M}\mu_{i,\mathbf{k}_{i}^{**}}$, and define
$\Delta_{\mathbf{k}}:=\mu^{**}-\sum_{i=1}^{M}\mu_{i,\mathbf{k}_{i}},~{}\mathbf{k}\in\mathcal{P}(N)$.
Let
$\Delta_{min}=\min_{\mathbf{k}\in\mathcal{P}(N),\mathbf{k}\neq\mathbf{k}^{**}}\Delta_{\mathbf{k}}$
and $\Delta_{max}=\max_{\mathbf{k}\in\mathcal{P}(N)}\Delta_{\mathbf{k}}$. We
assume that $\Delta_{min}>0$.
Algorithm 2 $\tt dUCB_{4}$ for User $i$
1: Initialization: Play a set of matchings so that each player plays each arm
at least once. Set counter $\eta=1$.
2: while ($t\leq T$) do
3: if ($\eta=2^{p}~{}\text{for some}~{}p=0,1,2,\cdots$) then
4: //Decision frame:
5: Update $g(t)$;
6: Participate in the ${\tt dBM_{\epsilon}}(g(t))$ algorithm to obtain a match
$k_{i}^{*}(t)$;
7: if $(k_{i}^{*}(t)\neq k_{i}^{*}(t-1))$ then
8: Use interrupt phase to signal an INTERRUPT to all other players about
changed allocation;
9: Reset $\eta=1$;
10: end if
11: if (Received an INTERRUPT) then
12: Reset $\eta=1$;
13: end if
14: else
15: // Exploitation frame:
16: $k_{i}^{*}(t)=k_{i}^{*}(t-1)$;
17: end if
18: Play arm $k_{i}^{*}(t)$;
19: Increment counter $\eta=\eta+1$, $t=t+1$;
20: end while
In the $\tt dUCB_{4}$ algorithm, at the end of every decision frame, the ${\tt
dBM_{\epsilon}}(g(t))$ will give a legitimate matching with no two players
colliding on any arm. Thus, the regret accrues either if the matching
$\mathbf{k}(t)$ is not the optimal matching $\mathbf{k}^{**}$, or if a
decision frame is employed by the players to recompute the matching. Every
time a frame is a decision frame, it adds a cost $C$ to the regret. The cost
$C$ depends on two parameters: (a) the precision of the bipartite matching
algorithm $\epsilon_{1}>0$, and (b) the precision of the index representation
$\epsilon_{2}>0$. A bipartite matching algorithm has an
$\epsilon_{1}$-precision if it gives an $\epsilon_{1}$-optimal matching. This
would happen, for example, when such an algorithm is run only for a finite
number of rounds. The index has an $\epsilon_{2}$-precision if any two indices
are not distinguishable if they are closer than $\epsilon_{2}$. This can
happen for example when indices must be communicated to other players in a
finite number of bits.
Thus, the cost $C$ is a function of $\epsilon_{1}$ and $\epsilon_{2}$, and can
be denoted as $C(\epsilon_{1},\epsilon_{2})$, with
$C(\epsilon_{1},\epsilon_{2})\rightarrow\infty$ as $\epsilon_{1}$ or
$\epsilon_{2}\rightarrow 0$. Since, $\epsilon_{1}$ and $\epsilon_{2}$ are the
parameters that are fixed a priori, we consider
$\epsilon=\min(\epsilon_{1},\epsilon_{2})$ to specify both precisions. We
denote the cost as $C(\epsilon)$.
We first show that if $\Delta_{min}$ is known, we can choose an
$\epsilon<\Delta_{min}/(M+1)$, so that ${\tt dUCB_{4}}$ algorithm will achieve
order log-squared regret growth with $T$. If $\Delta_{min}$ is not known, we
can pick a positive monotone sequence $\\{\epsilon_{t}\\}$ such that
$\epsilon_{t}\to 0$, as $t\to\infty$. In a decentralized bipartite matching
algorithm, the precision $\epsilon$ will depend on the amount of information
exchanged in the decision frames. It, thus, is some monotonically decreasing
function $\epsilon=f(L)$ of their length $L$ such that $\epsilon\to 0$ as
$L\to\infty$. Thus, we must pick a positive monotone sequence $\\{L_{t}\\}$
such that $L_{t}\to\infty$. Clearly, $C(f(L_{t}))\to\infty$ as $t\to\infty$.
This can happen arbitrarily slowly.
###### Theorem 6.
(i) Let $\epsilon>0$ be the precision of the bipartite matching algorithm and
the precision of the index representation. If $\Delta_{min}$ is known, choose
$\epsilon>0$ such that $\epsilon<\Delta_{min}/(M+1)$. Let $L$ be the length of
a frame. Then, the expected regret of the ${\tt dUCB_{4}}$ algorithm is
$\displaystyle\tilde{\mathcal{R}}_{\tt dUCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle(L\Delta_{max}+C(f(L))(1+\log
T))\cdot\left(\frac{4M^{3}(M+2)N\log
T}{(\Delta_{min}-((M+1)\epsilon)^{2}}+NM(2M+1)\right).$
Thus, $\tilde{\mathcal{R}}_{\tt dUCB_{4}}(T)=O(\log^{2}T).$
(ii) When $\Delta_{min}$ is unknown, denote
$\epsilon_{min}=\Delta_{min}/(2(M+1))$ and let $L_{t}\rightarrow\infty$ as
$t\rightarrow\infty$. Then, there exists a $t_{0}>0$ such that for all
$T>t_{0}$,
$\displaystyle\tilde{\mathcal{R}}_{\tt dUCB_{4}}(T)$
$\displaystyle\leq(L_{t_{0}}\Delta_{max}+C(f(L_{t_{0}}))t_{0}+(L_{T}\Delta_{max}+C(f(L_{T}))(1+\log
T))\cdot$ $\displaystyle\hskip 56.9055pt\left(\frac{4M^{3}(M+2)N\log
T}{(\Delta_{min}-\epsilon_{min})^{2}}+NM(2M+1)\right),$
where $t_{0}$ is the smallest $t$ such that $f(L_{t_{0}})<\epsilon_{min}$.
Thus by choosing an arbitrarily slowly increasing sequence $\\{L_{t}\\}$ we
can make the regret arbitrarily close to $\log^{2}T$.
###### Proof.
(i) First, we obtain a bound for $L=1$. Then, appeal to a result like Theorem
1 to obtain the result for general $L$. The implicit dependence between
$\epsilon$ and $L$ through the function $f(\cdot)$ does not affect this part
of the analysis. Details are omitted due to space limitations.
We first upper bound the number of sub-optimal plays. We define
$\tilde{n}_{i,j}(t),1\leq i\leq M,1\leq j\leq N$ as follows: Whenever the
${\tt dBM_{\epsilon}}(g(t))$ algorithm gives a non-optimal matching
$\mathbf{k}(t)$, $\tilde{n}_{i,j}(t)$ is increased by one for some
$(i,j)\in\arg\min_{1\leq i\leq M,1\leq j\leq N}n_{i,j}(t)$. Let $\tilde{n}(T)$
denote the total number of suboptimal plays. Then, clearly,
$\tilde{n}(T)=\sum_{i=1}^{M}\sum_{j=1}^{N}\tilde{n}_{i,j}(T)$. So, in order to
get a bound on $\tilde{n}(T)$ we first get a bound on $\tilde{n}_{i,j}(T)$.
Let $\tilde{I}_{i,j}(t)$ be the indicator function which is equal to $1$ if
$\tilde{n}_{i,j}(t)$ is incremented by one, at time $t$. When
$\tilde{I}_{i,j}(t)=1$, there will be a corresponding matching
$\mathbf{k}(t)\neq\mathbf{k}^{**}$ such that $k_{i}(t)=j$. In the following,
we denote it as $\mathbf{k}$, omitting the time index. A non-optimal matching
$\mathbf{k}$ is selected if the event
$\bigg{\\{}\sum_{i=1}^{M}g_{i,k^{**}_{i}}(m+2^{p}-1)\leq(M+1)\epsilon+\sum_{i=1}^{M}g_{i,k_{i}}(m+2^{p}-1)\bigg{\\}}$
happens. If each index has an error of at most $\epsilon$, the sum of $M$
terms may introduce an error of atmost $M\epsilon$. In addition, the
distributed bipartite matching algorithm ${\tt dBM_{\epsilon}}$ itself yields
only an $\epsilon$-optimal matching. This accounts for the term
$(M+1)\epsilon$ above. Since the initial steps are similar to that in Theorem
2, we skip those steps. Thus, similar to the equation (III-B), we get
$\tilde{n}_{i,j}(T)\leq$
$\displaystyle l$ $\displaystyle+$
$\displaystyle\sum_{m=1}^{T}\sum_{p=0}^{\infty}2^{p}I\bigg{\\{}\sum_{i=1}^{M}g_{i,k^{**}_{i}}(m+2^{p}-1)\leq(M+1)\epsilon+\sum_{i=1}^{M}g_{i,k_{i}}(m+2^{p}-1),\tilde{n}_{i,j}(m-1)\geq
l\bigg{\\}}$ $\displaystyle\leq$ $\displaystyle
l+\sum_{m=1}^{T}\sum_{p=0}^{\infty}2^{p}I\bigg{\\{}\sum_{i=1}^{M}\bigg{(}\overline{X}_{i,k^{**}_{i}}(m+2^{p}-1)+c_{m+2^{p}-1,n_{i,k^{**}_{i}}(m+2^{p}-1)}\bigg{)}$
$\displaystyle\hskip
56.9055pt\leq(M+1)\epsilon+\sum_{i=1}^{M}\overline{X}_{i,k_{i}}(m+2^{p}-1)+c_{m+2^{p}-1,n_{i,k_{i}}(m+2^{p}-1)},\tilde{n}_{i,j}(m-1)\geq
l\bigg{\\}}$ (29) $\displaystyle\leq$ $\displaystyle
l+\sum_{m=1}^{T}\sum_{p=0}^{\infty}2^{p}I\bigg{\\{}\min_{1\leq
s_{1,k^{**}_{1}},\ldots,s_{M,k^{**}_{M}}<m+2^{p}}\sum_{i=1}^{M}\left(\overline{X}_{i,k^{**}_{i}}(m+2^{p}-1)+c_{m+2^{p}-1,s_{i,k^{**}_{i}}}\right)$
$\displaystyle\hskip 56.9055pt\leq(M+1)\epsilon+\max_{l\leq
s_{1,k_{1}}^{{}^{\prime}},\ldots,s_{M,k_{M}}^{{}^{\prime}}<m+2^{p}}\sum_{i=1}^{M}\left(\overline{X}_{i,k_{i}}(m+2^{p}-1)+c_{m+2^{p}-1,s_{i,k_{i}}^{{}^{\prime}}}\right)\bigg{\\}}$
$\displaystyle\leq
l+\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1,k^{**}_{1}}=1}^{m+2^{p}}\ldots\sum_{s_{M,k^{**}_{M}}=1}^{m+2^{p}}\sum_{s^{{}^{\prime}}_{1,k_{1}}=1}^{m+2^{p}}\ldots\sum_{s^{{}^{\prime}}_{M,k_{M}}=1}^{m+2^{p}}I\bigg{\\{}\sum_{i=1}^{M}\left(\overline{X}_{i,k^{**}_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k^{**}_{i}}}\right)$
$\displaystyle\hskip
113.81102pt\leq(M+1)\epsilon+\sum_{i=1}^{M}\left(\overline{X}_{i,k_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}\right)\bigg{\\}}.$
Now, it is easy to observe that the event
$\displaystyle\bigg{\\{}\sum_{i=1}^{M}\left(\overline{X}_{i,k^{**}_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k^{**}_{i}}}\right)\leq(M+1)\epsilon+\sum_{i=1}^{M}\left(\overline{X}_{i,k_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}\right)\bigg{\\}}$
implies at least one of the following events:
$\displaystyle A_{i}$ $\displaystyle:=$
$\displaystyle\bigg{\\{}\overline{X}_{i,k^{**}_{i}}(m+2^{p})\leq\mu_{i,k^{**}_{i}}-c_{m+2^{p},s_{i,k^{**}_{i}}}\bigg{\\}},$
$\displaystyle B_{i}$ $\displaystyle:=$
$\displaystyle\bigg{\\{}\overline{X}_{i,k_{i}}(m+2^{p})\geq\mu_{i,k_{i}}+c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}\bigg{\\}},1\leq
i\leq M,$ $\displaystyle C$ $\displaystyle:=$
$\displaystyle\bigg{\\{}\sum_{i=1}^{M}\mu_{i,k^{**}_{i}}<(M+1)\epsilon+\sum_{i=1}^{M}\mu_{i,k_{i}}+2\sum_{i=1}^{M}c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}\bigg{\\}}$
(30) $\displaystyle D$ $\displaystyle:=$
$\displaystyle\bigg{\\{}(M+1)\epsilon>\sum_{i=1}^{M}\mu_{i,k^{**}_{i}}-\sum_{i=1}^{M}\mu_{i,k_{i}}\bigg{\\}}.$
Using the Chernoff-Hoeffding inequality, we get
$\mathbb{P}(A_{i})\leq(m+2^{p})^{-2(M+2)},~{}~{}\mathbb{P}(B_{i})\leq(m+2^{p})^{-2(M+2)},~{}1\leq
i\leq M.$ For $l\geq\left\lceil\frac{4M^{2}(M+2)\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}\right\rceil$, we get
$\sum_{i=1}^{M}\mu_{i,k^{**}_{i}}-\sum_{i=1}^{M}\mu_{i,k_{i}}-(M+1)\epsilon-2\sum_{i=1}^{M}c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}$
$\displaystyle\geq$
$\displaystyle\sum_{i=1}^{M}\mu_{i,k^{**}_{i}}-\sum_{i=1}^{M}\mu_{i,k_{i}}-(M+1)\epsilon-2M\sqrt{\frac{(M+2)\log(m+2^{p})}{l}}$
(31) $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{M}\mu_{i,k^{**}_{i}}-\sum_{i=1}^{M}\mu_{i,k_{i}}-(M+1)\epsilon-(\Delta_{min}-(M+1)\epsilon)\geq
0$
The event $D$ is false by assumption. So, we get,
$\mathbb{E}[\tilde{n}_{i,j}(T)]$
$\displaystyle\leq\left\lceil\frac{4M^{2}(M+2)\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}\right\rceil+\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1,k^{**}_{1}}=1}^{m+2^{p}}\ldots\sum_{s_{M,k^{**}_{M}}=1}^{m+2^{p}}\sum_{s^{{}^{\prime}}_{1,k_{1}}=1}^{m+2^{p}}\ldots\sum_{s^{{}^{\prime}}_{M,k_{M}}=1}^{m+2^{p}}2M(m+2^{p})^{-2(M+2)}$
$\displaystyle\leq\left\lceil\frac{4M^{2}(M+2)\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}\right\rceil+2M\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}(m+2^{p})^{-4}$
$\displaystyle\leq\frac{4M^{2}(M+2)\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}+(2M+1).$ (32)
Now, putting it all together, we get
$\displaystyle\mathbb{E}[\tilde{n}(T)]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{N}\mathbb{E}[\tilde{n}_{i,j}(T)]\leq\frac{4M^{3}(M+2)N\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}+(2M+1)MN.$
Now, by the proof of Theorem 2 (c.f. equation(15),
$\mathbb{E}[m(T)]\leq\mathbb{E}[\tilde{n}(T)](1+\log T).$ We can now bound the
regret, $\tilde{\mathcal{R}}_{\tt dUCB_{4}}(T)=\sum_{k\in\mathcal{P}(N),k\neq
k^{**}}\Delta_{k}\sum_{i=1}^{M}\mathbb{E}[\tilde{n}_{i,k_{i}}(T)]+C\mathbb{E}[m(T)]$
$\displaystyle\leq$ $\displaystyle\Delta_{max}\sum_{k\in\mathcal{P}(N),k\neq
k^{**}}\sum_{i=1}^{M}\mathbb{E}[\tilde{n}_{i,k_{i}}(T)]+C\mathbb{E}[m(T)]$
$\displaystyle=$
$\displaystyle\Delta_{max}\mathbb{E}[\tilde{n}(T)]+C\mathbb{E}[m(t)].$
For a general $L$, by Theorem 1 we get
$\displaystyle\tilde{\mathcal{R}}_{\tt dUCB_{4}}(T)\leq
L\Delta_{max}\mathbb{E}[\tilde{n}(T)]+C(f(L))\mathbb{E}[m(T)]\leq(L\Delta_{max}+C(f(L))(1+\log
T))\mathbb{E}[\tilde{n}(T)].$
Now, using the bound (V), we get the desired upper bound on the expected
regret.
(ii) Since $\epsilon_{t}=f(L_{t})$ is a monotonically decreasing function of
$L_{t}$ such that $\epsilon_{t}\to 0$ as $L_{t}\to\infty$, there exists a
$t_{0}$ such that for $t>t_{0}$, $\epsilon_{t}<\epsilon_{min}$. We may get a
linear regret upto time $t_{0}$ but after that by the analysis of Theorem 2,
regret grows only sub-linearly. Since $C(\cdot)$ is monotonically increasing,
$C(f(L_{T}))\geq C(f(L_{t})),\forall t\leq T$, we get the desired result. The
last part is illustrative and can be trivially established using the obtained
bound on the regret in (ii). ∎
Remarks. 1\. We note that in the initial steps, our proof followed the proof
of the main result in [12].
2\. The ${\tt UCB_{2}}$ algorithm described in [4] performs computations only
at exponentially spaced time epochs. So, it is natural to imagine that a
decentralized algorithm based on it could be developed, and get a better
regret bound. Unfortunately, the single player ${\tt UCB_{2}}$ algorithm has
an obvious weakness: regret is linear in the number of arms. Thus, the
decentralized/combinatorial extension of ${\tt UCB_{2}}$ would yield regret
growing exponentially in the number of players and arms. We use a similar
index but a different scheme, allowing us to achieve poly-log regret growth
and a linear memory requirement for each player.
## VI The Decentralized MAB problem with Markovian rewards
Now, we consider the decentralized MAB problem with $M$ players and $N$ arms
where the rewards obtained each time when an arm is pulled are not i.i.d. but
come from a Markov chain. The reward that player $i$ gets from arm $j$ (when
there is no collision) $X_{ij}$, is modelled as an irreducible, aperiodic,
reversible Markov chain on a finite state space $\mathcal{X}^{i,j}$ and
represented by a transition probability matrix
$P^{i,j}:=\left(p^{i,j}_{x,x^{{}^{\prime}}}:x,x^{{}^{\prime}}\in\mathcal{X}^{i,j}\right)$.
Assume that $\mathcal{X}^{i,j}\in(0,1]$. Let $X_{i,j}(1),X_{i,j}(2),\ldots$
denote the successive rewards from arm $j$ for player $i$. All arms are
mutually independent for all players. Let
$\mathbf{\pi}^{i,j}:=\left(\pi^{i,j}_{x},x\in\mathcal{X}^{i,j}\right)$ be the
stationary distribution of the Markov chain $P^{i,j}$. The mean reward from
arm $j$ for player $i$ is defined as
$\mu_{i,j}:=\sum_{x\in\mathcal{X}^{i,j}}x\pi^{i,j}_{x}$. Note that the Markov
chain represented by $P^{i,j}$ makes a state transition only when player $i$
plays arm $j$. Otherwise, it remains _rested_. As described in the previous
section, $n_{i}(t)$ is the number of successful plays (without collisions) of
player $i$ by frame $t$, $n_{i,j}(t)$ is the number of times player $i$ picks
arm $j$ successfully by frame $t$ and $\overline{X}_{i,j}(t)$ is the sample
mean of rewards from arm $j$ for player $i$ from $n_{i,j}(t)$ samples. Denote
$\Delta_{min}:=\min_{\mathbf{k}\in\mathcal{P}(N),\mathbf{k}\neq\mathbf{k}^{**}}\Delta_{\mathbf{k}}$
and $\Delta_{max}:=\max_{\mathbf{k}\in\mathcal{P}(N)}\Delta_{\mathbf{k}}$.
Denote $\pi_{min}:=\min_{1\leq i\leq M,1\leq j\leq
N,x\in\mathcal{X}^{i,j}}\pi^{i,j}_{x}$, $x_{max}:=\max_{1\leq i\leq M,1\leq
j\leq N,x\in\mathcal{X}^{i,j}}x$ and $x_{min}:=\min_{1\leq i\leq M,1\leq j\leq
N,x\in\mathcal{X}^{i,j}}x$. Let
${\hat{\pi}^{i,j}}_{x}:=\max\\{\pi^{i,j}_{x},1-\pi^{i,j}_{x}\\}$ and
$\hat{\pi}_{max}:=\max_{1\leq i\leq M,1\leq j\leq
N,x\in\mathcal{X}^{i,j}}{\hat{\pi}^{i,j}}_{x}$. Let $|\mathcal{X}^{i,j}|$
denote the cardinality of the state space $\mathcal{X}^{i,j}$,
$|\mathcal{X}|_{max}:=\max_{1\leq i\leq M,1\leq j\leq N}|\mathcal{X}^{i,j}|$.
Let $\rho^{i,j}$ be the eigenvalue gap, $1-\lambda_{2}$, where $\lambda_{2}$
is the second largest eigenvalue of the matrix ${P^{i,j}}^{2}$. Denote
$\rho_{max}:=\max_{1\leq i\leq M,1\leq j\leq N}\rho^{i,j}$ and
$\rho_{min}:=\min_{1\leq i\leq M,1\leq j\leq N}\rho^{i,j}$.
The total reward obtained by time $T$ is
$S_{T}=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{s=1}^{n_{i,j}(T)}X_{i,j}(s)$ and the
regret is
$\tilde{\mathcal{R}}_{M,\alpha}(T):=T\sum_{i=1}^{M}\mu_{i,k_{i}^{**}}-\mathbb{E}_{\alpha}\left[\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{s=1}^{n_{i,j}(T)}X_{i,j}(s)\right]+C\mathbb{E}[m(T)].$
(33)
Define the index
$g_{i,j}(t):=\overline{X}_{i,j}(t)+\sqrt{\frac{\kappa\log
n_{i}(t)}{n_{i,j}(t)}}$ (34)
where $\kappa$ be any constant such that
$\kappa>(112+56M)|\mathcal{X}|_{max}^{2}/\rho_{min}$.
We need the following lemma to prove the regret bound.
###### Lemma 2.
If the reward of each player-arm pair $(i,j)$ is given by a Markov chain,
satisfying the properties of Lemma 4, then under any policy $\alpha$
$\tilde{\mathcal{R}}_{M,\tt\alpha}(T)\leq\sum_{k\in\mathcal{P}(N),k\neq
k^{**}}\Delta_{k}\mathbb{E}[n^{k}(T)]+C\mathbb{E}[m(T)]+\tilde{K}_{\mathcal{X},P}$
(35)
where $n^{k}(T)$ is the number of times that the matching $k$ occurred by the
time $T$ and $\tilde{K}_{\mathcal{X},P}$ is defined as
$\tilde{K}_{\mathcal{X},P}=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{x\in\mathcal{X}^{i,j}}x/\pi^{j}_{min}$
###### Proof.
Let $(X_{i,j}(1),X_{i,j}(2),\ldots)$ denote the successive rewards for player
$i$ from arm $j$. Let $\mathcal{F}^{i,j}_{t}$ denote the $\sigma$-algebra
generated by $(X_{i,j}(1),\ldots,X_{i,j}(t))$, $\mathcal{F}^{i,j}=\vee_{t\geq
1}\mathcal{F}^{i,j}_{t}$ and $\mathcal{G}^{i,j}=\vee_{(k,l)\neq(i,j)}F^{k,l}$.
Since arms are independent, $\mathcal{G}^{i,j}$ is independent of
$\mathcal{F}^{i,j}$. Clearly, $n_{i,j}(T)$ is a stopping time with respect to
$\mathcal{F}^{i,j}\vee\mathcal{G}^{i,j}_{T}$. The total reward is
$S_{T}=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{t=1}^{n_{i,j}(T)}X_{i,j}(t)=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{x\in\mathcal{X}^{i,j}}xN(x,n_{i,j}(T))$
where $N(x,n_{i,j}(T)):=\sum_{t=1}^{n_{i,j}(T)}I\\{X_{i,j}(t)=x\\}$. Taking
expectations and using the Lemma 4,
$\left\lvert\mathbb{E}[S_{T}]-\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{x\in\mathcal{X}^{i,j}}x\pi^{i,j}_{x}\mathbb{E}[n_{i,j}(T)]\right\rvert\leq\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{x\in\mathcal{X}^{i,j}}x/\pi^{i,j}_{min}$
which implies,
$\left\lvert\mathbb{E}[S_{T}]-\sum_{j=1}^{N}\sum_{i=1}^{M}\mu_{i,j}\mathbb{E}[n_{i,j}(T)]\right\rvert\leq\tilde{K}_{\mathcal{X},P}$
where
$\tilde{K}_{\mathcal{X},P}=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{x\in\mathcal{X}^{i,j}}x/\pi^{i,j}_{min}$.
Now,
$\displaystyle\sum_{j=1}^{N}\sum_{i=1}^{M}\mu_{i,j}\mathbb{E}[n_{i,j}(T)]$
$\displaystyle=\sum_{j=1}^{N}\sum_{i=1}^{M}\sum_{k\in\mathcal{P}(N),(i,j)\in
k}\mu_{i,k_{i}}\mathbb{E}[n_{i,k_{i}}(T)]=\sum_{k\in\mathcal{P}(N)}\sum_{i=1}^{M}\mu_{i,k_{i}}\mathbb{E}[n_{i,k_{i}}(T)]$
$\displaystyle=\sum_{k\in\mathcal{P}(N)}\mu^{k}\mathbb{E}[n^{k}(T)]$
where $\mu^{k}=\sum_{i=1}^{M}\mu_{i,k_{i}}$. Since regret is defined as in the
equation (33),
$\left\lvert\tilde{\mathcal{R}}_{M,\alpha}(T)-\left(T\mu^{**}-\sum_{k\in\mathcal{P}(N),(i,j)\in
k}\mu_{i,k_{i}}\mathbb{E}[n_{i,k_{i}}(T)]+C\mathbb{E}_{\alpha}[m(T)]\right)\right\rvert\leq\tilde{K}_{\mathcal{X},P}.$
(36)
∎
The main result of this section is the following.
###### Theorem 7.
(i) Let $\epsilon>0$ be the precision of the bipartite matching algorithm and
the precision of the index representation. If $\Delta_{min}$,
$|\mathcal{X}|_{max}$ and $\rho_{min}$ are known, choose $\epsilon>0$ such
that $\epsilon<\Delta_{min}/(M+1)$ and
$\kappa>(112+56M)|\mathcal{X}|_{max}^{2}/\rho_{min}$. Let $L$ be the length of
a frame. Then, the expected regret of the ${\tt dUCB_{4}}$ algorithm with
index (34) for the decentralized MAB problem with Markovian rewards and per
computation cost $C$ is given by
$\tilde{\mathcal{R}}_{M,\tt dUCB_{4}}(T)$
$\displaystyle\leq$ $\displaystyle(L\Delta_{max}+C(f(L))(1+\log
T))\cdot\left(\frac{4M^{3}\kappa N\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}+(2MD+1)MN\right)+\tilde{K}_{\mathcal{X},P}.$
Thus, $\tilde{\mathcal{R}}_{M,\tt dUCB_{4}}(T)=O(\log^{2}T).$
(ii) If $\Delta_{min}$, $|\mathcal{X}|_{max}$ and $\rho_{min}$ are unknown,
denote $\epsilon_{min}=\Delta_{min}/(2(M+1))$ and let $L_{t}\rightarrow\infty$
as $t\rightarrow\infty$. Also, choose a positive monotone sequence
$\\{\kappa_{t}\\}$ such that $\kappa_{t}\rightarrow\infty$ as
$t\rightarrow\infty$ and $\kappa_{t}\leq t$. Then, $\tilde{\mathcal{R}}_{M,\tt
dUCB_{4}}(T)=O(C(f(L_{T}))\kappa_{T}\log^{2}T)$. Thus by choosing an
arbitrarily-slowly increasing sequences, we can make the regret arbitrarily
close to $\log^{2}T$.
###### Proof.
(i) We skip the initial steps as they are same as in the proof of Theorem 6.
We start by bounding $\tilde{n}_{i,j}(T)$ as defined in the proof of Theorem
6. Then, from equation (V), we get $\tilde{n}_{i,j}(T)$
$\displaystyle\leq
l+\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1,k^{**}_{1}}=1}^{m+2^{p}}\ldots\sum_{s_{M,k^{**}_{M}}=1}^{m+2^{p}}\sum_{s^{{}^{\prime}}_{1,k_{1}}=1}^{m+2^{p}}\ldots\sum_{s^{{}^{\prime}}_{M,k_{M}}=1}^{m+2^{p}}I\\{\sum_{i=1}^{M}\left(\overline{X}_{i,k^{**}_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k^{**}_{i}}}\right)$
$\displaystyle\hskip
113.81102pt\leq(M+1)\epsilon+\sum_{i=1}^{M}\left(\overline{X}_{i,k_{i}}(m+2^{p})+c_{m+2^{p},s_{i,k_{i}}^{{}^{\prime}}}\right)\\}$
(37)
Now, the event in the parenthesis $\\{\cdot\\}$ above implies at least one of
the events ($A_{i},B_{i},C,D$) given in the display (V). From the proof of
Theorem 4 (equations (22, 23), $\mathbb{P}(A_{i})\leq
D(m+2^{p})^{-\frac{\kappa\rho_{min}}{28|\mathcal{X}|_{max}|^{2}}},\hskip
28.45274pt\mathbb{P}(B_{i})\leq
D(m+2^{p})^{-\frac{\kappa\rho_{min}}{28|\mathcal{X}|_{max}|^{2}}},1\leq i\leq
M.$ Similar to the steps in display (31), we can show that the event $C$ is
false. Also, the event $D$ is false by assumption. So, similar to the proof of
the Theorem 6 (c.f. display (V) we get,
$\displaystyle\mathbb{E}[\tilde{n}_{i,j}(T)]$ $\displaystyle\leq$
$\displaystyle\left\lceil\frac{4M^{2}\kappa\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}\right\rceil+\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}\sum_{s_{1,k^{**}_{1}}=1}^{m+2^{p}}\ldots\sum_{s_{M,k^{**}_{M}}=1}^{m+2^{p}}$
$\displaystyle\hskip
85.35826pt\sum_{s^{{}^{\prime}}_{1,k_{1}}=1}^{m+2^{p}}\ldots\sum_{s^{{}^{\prime}}_{M,k_{M}}=1}^{m+2^{p}}2MD(m+2^{p})^{-\frac{\kappa\rho_{min}}{28|\mathcal{X}|_{max}|^{2}}}$
$\displaystyle\leq$ $\displaystyle\left\lceil\frac{4M^{2}\kappa\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}\right\rceil+2MD\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}2^{p}(m+2^{p})^{-\frac{\kappa\rho_{min}-56M|\mathcal{X}|_{max}^{2}}{28|\mathcal{X}|_{max}^{2}}}$
$\displaystyle\leq$ $\displaystyle\frac{4M^{2}\kappa\log
T}{(\Delta_{min}-(M+1)\epsilon)^{2}}+(2MD+1).$
when $\kappa>(112+56M)|\mathcal{X}|_{max}^{2}/\rho_{min}$. Now, putting it all
together, we get
$\displaystyle\mathbb{E}[\tilde{n}(T)]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{N}\mathbb{E}[\tilde{n}_{i,j}(T)]\leq\frac{4M^{3}\kappa
N\log T}{(\Delta_{min}-(M+1)\epsilon)^{2}}+(2MD+1)MN.$
Now, by proof of the Theorem 2 (equation (15)),
$\mathbb{E}[m(T)]\leq\mathbb{E}[\tilde{n}(T)](1+\log T).$ We can now bound the
regret,
$\displaystyle\tilde{\mathcal{R}}_{M,\tt dUCB_{4}}(T)$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{P}(N),k\neq
k^{**}}\Delta_{k}\sum_{i=1}^{M}\mathbb{E}[\tilde{n}_{i,k_{i}}(T)]+C\mathbb{E}[m(T)]+\tilde{K}_{\mathcal{X},P}$
$\displaystyle\leq$ $\displaystyle\Delta_{max}\sum_{k\in\mathcal{P}(N),k\neq
k^{**}}\sum_{i=1}^{M}\mathbb{E}[\tilde{n}_{i,k_{i}}(T)]+C\mathbb{E}[m(T)]+\tilde{K}_{\mathcal{X},P}$
$\displaystyle=$
$\displaystyle\Delta_{max}\mathbb{E}[\tilde{n}(T)]+C\mathbb{E}[m(T)]+\tilde{K}_{\mathcal{X},P}.$
For a general $L$, by Theorem 1
$\displaystyle\tilde{\mathcal{R}}_{M,\tt dUCB_{4}}(T)$ $\displaystyle\leq$
$\displaystyle
L\Delta_{max}\mathbb{E}[\tilde{n}(T)]+C(f(L))\mathbb{E}[m(T)]+\tilde{K}_{\mathcal{X},P}.$
$\displaystyle\leq$ $\displaystyle(L\Delta_{max}+C(f(L))(1+\log
T))\mathbb{E}[\tilde{n}(T)]+\tilde{K}_{\mathcal{X},P}.$
Now, using the bound (VI), we get the desired upper bound on the expected
regret.
(ii) This can now easily be obtained using the above and following Theorem 6.
∎
## VII Distributed Bipartite Matching: Algorithm and Implementation
In the previous section, we referred to an unspecified distributed algorithm
for bipartite matching ${\tt dBM}$, that is used by the ${\tt dUCB_{4}}$
algorithm. We now present one such algorithm, namely, Bertsekas’ auction
algorithm [17], and its distributed implementation. We note that the presented
algorithm is not the only one that can be used. The ${\tt dUCB_{4}}$ algorithm
will work with a distributed implementation of any bipartite matching
algorithm, e.g. algorithms given in [24].
Consider a bipartite graph with $M$ players on one side, and $N$ arms on the
other, and $M\leq N$. Each player $i$ has a value $\mu_{i,j}$ for each arm
$j$. Each player knows only his own values. Let us denote by $k^{**}$, a
matching that maximizes the matching surplus $\sum_{i,j}\mu_{i,j}x_{i,j}$,
where the variable $x_{i,j}$ is 1 if $i$ is matched with $j$, and 0 otherwise.
Note that $\sum_{i}x_{i,j}\leq 1,\forall j$, and $\sum_{j}x_{i,j}\leq
1,\forall i$. Our goal is to find an $\epsilon$-optimal matching. We call any
matching $k^{*}$ to be $\epsilon$-optimal if
$\sum_{i}\mu_{i,k^{**}(i)}-\sum_{i}\mu_{i,k^{*}(i)}\leq\epsilon$.
Algorithm 3 : ${\tt dBM_{\epsilon}}$ ( Bertsekas Auction Algorithm)
1: All players $i$ initialize prices $p_{j}=0,\forall~{}\text{channels}~{}j$;
2: while (prices change) do
3: Player $i$ communicates his preferred arm $j_{i}^{*}$ and bid
$b_{i}=\max_{j}(\mu_{ij}-p_{j})-\text{2max}_{j}(\mu_{ij}-p_{j})+\frac{\epsilon}{M}$
to all other players.
4: Each player determines on his own if he is the winner $i_{j}^{*}$ on arm
$j$;
5: All players set prices $p_{j}=\mu_{i_{j}^{*},j}$;
6: end while
Here, $\text{2max}_{j}$ is the second highest maximum over all $j$. The best
arm for a player $i$ is arm $j_{i}^{*}=\arg\max_{j}(\mu_{i,j}-p_{j})$. The
winner $i_{j}^{*}$ on an arm $j$ is the one with the highest bid.
The following lemma in [17] establishes that Bertsekas’ auction algorithm will
find the $\epsilon$-optimal matching in a finite number of steps.
###### Lemma 3.
[17] Given $\epsilon>0$, Algorithm 3 with rewards $\mu_{i,j}$, for player $i$
playing the $j$th arm, converges to a matching $k^{*}$ such that
$\sum_{i}\mu_{i,k^{**}(i)}-\sum_{i}\mu_{i,k^{*}(i)}\leq\epsilon$ where
$k^{**}$ is an optimal matching. Furthermore, this convergence occurs in less
than $(M^{2}\max_{i,j}\\{\mu_{i,j}\\})/\epsilon$ iterations.
The temporal structure of the ${\tt dUCB_{4}}$ algorithm is such that time is
divided into frames of length $L$. Each frame is either a decision frame, or
an exploitation frame. In the exploitation frame, each player plays the arm it
was allocated in the last decision frame. The distributed bipartite matching
algorithm (e.g. based on Algorithm 3), is run in the decision frame. The
decision frame has an interrupt phase of length $M$ and negotiation phase of
length $L-M$. We now describe an implementation structure for these phases in
the decision frame.
Figure 1: Structure of the decision frame
Interrupt Phase: The interrupt phase can be implemented very easily. It has
length $M$ time slots. On a pre-determined channel, each player by turn
transmits a ‘1’ if the arm with which it is now matched has changed, ‘0’
otherwise. If any user transmits a ‘1’, everyone knows that the matching has
changed, and they reset their counter $\eta=1$.
Negotiation Phase: The information needed to be exchanged to compute an
$\epsilon$-optimal matching is done in the negotiation phase. We first provide
a packetized implementation of the negotiation phase. The negotiation phase
consists of $J$ subframes of length $M$ each (See figure 1). In each subframe,
the users transmit a packet by turn. The packet contains bid information:
(channel number, bid value). Since all users transmit by turn, all the users
know the bid values by the end of the subframe, and can compute the new
allocation, and the prices independently. The length of the subframe $J$
determines the precision $\epsilon$ of the distributed bipartite matching
algorithm. Note that in the packetized implementation, $\epsilon_{1}=0$, i.e.,
bid values can be computed exactly, and for a given $\epsilon_{2}$, we can
determine $J$, the number of rounds the ${\tt dBM}$ algorithm 3 runs for, and
returns an $\epsilon_{2}$-optimal matching.
If a packetized implementation is not possible, we can give a physical
implementation. Our only assumption here is going to be that each user can
observe a channel, and determine if there was a successful transmission on it,
a collision, or no transmission, in a given time slot. The whole negotiation
phase is again divided into $J$ sub-frames. In each sub-frame, each user
transmits by turn. It simply transmits $\lceil{\log M}\rceil$ bits to indicate
a channel number, and then $\lceil{\log 1/\epsilon_{1}}\rceil$ bits to
indicate its bid value to precision $\epsilon_{1}$. The number of such sub-
frames $J$ is again chosen so that the ${\tt dBM}$ algorithm (based on
Algorithm 3) returns an $\epsilon_{2}$-optimal matching.
## VIII Simulations
Figure 2: (i) Cumulative regret : $2$ users, $2$ channels; i.i.d. channels;
Mean reward matrix = $[0.8,0.6;0.6,0.35]$. (ii) Cumulative regret : $2$ users,
$2$ channels; Markovian channels.
We illustrate the empirical performance of the ${\tt dUCB_{4}}$ algorithm when
the successive rewards from a channel are i.i.d. and when they are Markovian.
Consider two users and two channels. In the i.i.d. case, each channel has
rewards that are generated by a Bernoulli distribution taking values $0$ and
$1$. The first user has mean rewards of $0.8$ and $0.6$ for channels $1$ and
$2$ respectively. The second user has mean rewards of $0.6$ and $0.35$. The
algorithm’s performance, averaged over 50 runs, is shown in Figure 2 (i). It
shows cumulative regret with time. The red bold curve is the theoretical upper
bound we derived, while the blue curve is the observed regret. The algorithm
seems to perform much better than even the poly-log regret upper bound we
derived.
In the Markovian case, rewards are generated by a Markov chain having states
$0$ and $1$. The mean reward on a channel is given by its stationary
distribution, i.e., the probability the Markov chain is in state $1$,
$\pi_{1}$. The properties of the Markov chains are given in Table I. The
performance of the ${\tt dUCB_{4}}$ algorithm on this model, averaged over 50
runs, is shown in Figure 2 (ii). Once again, the algorithm seems to perform
much better than even the poly-log regret upper bound we derived.
TABLE I: Markov Chain Parameters : Transition probability and Stationary distribution User | Channel | $p_{01}$,$p_{10}$ | $\pi$
---|---|---|---
1 | 1 | $0.3$,$0.5$ | $0.3/0.8$
1 | 2 | $0.2$,$0.6$ | $0.2/0.8$
2 | 1 | $0.6$,$0.3$ | $0.6/0.9$
2 | 2 | $0.7$,$0.2$ | $0.7/0.9$
## IX Conclusions
We have proposed a ${\tt dUCB_{4}}$ algorithm for decentralized learning in
multi-armed bandit problems that achieves a regret of near-$O(\log^{2}(T))$.
Finding a lower bound is usually quite difficult, and currently a work in
progress.
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Let $\left(X_{t},t=1,2,\ldots\right)$ be an irreducible, aperiodic and
reversible Markov chain on a finite state space $\mathcal{X}$ with transition
probability matrix $P$, a stationary distribution $\pi$ and an initial
distribution $\lambda$. Let $\mathcal{F}_{t}$ be the $\sigma$-algebra
generated by $\left(X_{1},X_{2},\ldots,X_{t}\right)$. Denote
$N_{\lambda}=\left\lVert\left(\frac{\lambda_{x}}{\pi_{x}},x\in\mathcal{X}\right)\right\rVert_{2}$.
###### Lemma 4.
[5] Let $\mathcal{G}$ be a $\sigma$-algebra independent of
$\mathcal{F}=\vee_{t\geq 1}F_{t}$. Let $\tau$ be a stopping time of
$\mathcal{F}_{t}\vee\mathcal{G}$. Let
$N(x,\tau):=\sum_{t=1}^{\tau}I\\{X_{t}=x\\}$. Then,
$|\mathbb{E}[N(x,\tau)]-\pi_{x}\mathbb{E}[\tau]|\leq K,$ where $K\leq
1/\pi_{min}$ and $\pi_{min}=\min_{x\in\mathcal{X}}\pi_{x}$. $K$ depends on
$P$.
###### Lemma 5.
[21] Denote
$N_{\lambda}=\left\lVert\left(\frac{\lambda_{x}}{\pi_{x}},x\in\mathcal{X}\right)\right\rVert_{2}$.
Let $\rho$ be the eigenvalue gap, $1-\lambda_{2}$, where $\lambda_{2}$ is the
second largest eigenvalue of the matrix $P^{2}$. Let
$f:\mathcal{X}\rightarrow\mathbf{R}$ be such that
$\sum_{x\in\mathcal{X}}\pi_{x}f(x)=0$, $\left\lVert f\right\rVert_{\infty}\leq
1,{\left\lVert f\right\rVert}^{2}_{2}\leq 1$. Then, for any $\gamma>0$,
$\mathbb{P}\left(\sum_{a=1}^{t}f(X_{a})/t\geq\gamma\right)\leq
N_{\lambda}e^{-t\rho\gamma^{2}/28}.$
|
arxiv-papers
| 2012-06-14T07:07:58 |
2024-09-04T02:49:31.832424
|
{
"license": "Public Domain",
"authors": "Dileep Kalathil, Naumaan Nayyar and Rahul Jain",
"submitter": "Dileel Kalathil",
"url": "https://arxiv.org/abs/1206.3582"
}
|
1206.3595
|
# Observational consequences of the Partially Screened Gap
Andrzej Szary, George Melikidze, and Janusz Gil Kepler Institute of
Astronomy, University of Zielona Góra, Lubuska 2, 65-265, Zielona Góra, Poland
###### Abstract
Observations of the thermal X-ray emission from old radio pulsars implicate
that the size of hot spots is much smaller then the size of the polar cap that
follows from the purely dipolar geometry of pulsar magnetic field. Plausible
explanation of this phenomena is an assumption that the magnetic field at the
stellar surface differs essentially from the purely dipolar field. Using the
conservation of the magnetic flux through the area bounded by open magnetic
field lines we can estimate the surface magnetic field as of the order of
$10^{14}$G. Based on observations that the hot spot temperature is about a few
million Kelvins the Partially Screened Gap (PSG) model was proposed which
assumes that the temperature of the actual polar cap equals to the so called
critical temperature. We discuss correlation between the temperature and
corresponding area of the thermal X-ray emission for a number of pulsars.
We have found that depending on the conditions in a polar cap region the gap
breakdown can be caused either by the Curvature Radiation (CR) or by the
Inverse Compton Scattering (ICS). When the gap is dominated by ICS the density
of secondary plasma with Lorentz factors $10^{2}-10^{3}$ is at least an order
of magnitude higher then in a CR scenario. We believe that two different gap
breakdown scenarios can explain the mode-changing phenomenon and in particular
the pulse nulling. Measurements of the characteristic spacing between sub-
pulses ($P_{2}$) and the period at which a pattern of pulses crosses the pulse
window ($P_{3}$) allowed us to determine more strict conditions for avalanche
pair production in the PSG.
## 1 Introduction
The Standard model of radio pulsars assumes that there exists the Inner
Acceleration Region (IAR) above the polar cap where the electric field has a
component along the opened magnetic field lines. In this region particles
(electrons and positrons) are accelerated in both directions: outward and
toward the stellar surface (Ruderman & Sutherland (1975)). Consequently,
outflowing particles are responsible for generation of the magnetospheric
emission (radio and high-frequency) while the backflowing particles heat the
surface and provide required energy for the thermal emission. The Vacuum Gap
model assumes that ions cannot be extracted from stellar surface due to huge
surface magnetic field of a pulsar. On the other hand it predicts the surface
temperature of few million Kelvins (heating by backflowing particles). As
shown by Medin & Lai (2007) for such high temperatures the ions extraction
from surface cannot be ignored. In fact for surface temperature few million
Kelvins the gap can form only if surface magnetic field is much stronger than
the dipolar component ($B_{s}=10^{14}G$).
The analysis of X-ray radiation is an excellent method to get insight into the
most intriguing region of the neutron star (NS). X-ray emission seems to be a
quite common feature of radio pulsars. In general X-ray radiation from an
isolated NS can consist of two distinguishable components: the thermal
emission and the nonthermal emission. The thermal emission can originate
either from the entire surface of cooling NS or the spots around the magnetic
poles on stellar surface (polar caps and adjacent areas). The nonthermal
component is usually attributed to radiation produced by Synchrotron Radiation
and/or Inverse Compton Scattering from charged relativistic particles
accelerated in the pulsar magnetosphere. For most observations it is very
difficult to distinguish contribution of different components (thermal and
nonthermal). To get an information about polar cap of radio pulsars we
analysed X-ray radiation from old pulsars as their surface is already cooled
down and their magnetospheric radiation (nonthermal component) is also
significantly weaker.
The blackbody fit allows us to obtain directly the temperature ($T_{s}$) of
the hot spot. Using the distance ($D$) to the pulsar and the luminosity of
thermal emission ($L_{bol}$) we can estimate the area ($A_{pc}$) of the hot
spot. In most cases ($A_{pc}$) differs from the conventional polar cap area
$A_{dp}\approx 6.2\times 10^{4}P^{-1}\,{\rm m^{2}}$, where $P$ is the pulsar
period. We use parameter $b=A_{dp}/A_{pc}$ to describe the difference between
$A_{dp}$ and $A_{pc}$. Pulsars for which it is possible to determine polar cap
size (old NSs) show that the actual polar cap size is much smaller ($b\gg 1$)
than the size of conventional polar cap (see Tab. 1).
The surface magnetic field can be estimated by the magnetic flux conservation
law as $b=A_{dp}/A_{pc}=B_{s}/B_{d}$, where $B_{d}=2.02\times
10^{12}\left(P\dot{P}_{-15}\right)^{0.5}$, and
$\dot{P}_{-15}=\dot{P}/10^{-15}$ is the period derivative. The X-ray
observations suggest that surface magnetic field strength at polar cap should
be of the order of $10^{14}$ G. On the other hand we know from radio
observations that magnetic field at altitudes where radio emission is
generated should be dipolar. To meet both these requirements Partially
Screened Gap model assumes the existence of crust-anchored local magnetic
anomalies which affect magnetic field only on short distances. According to
our model the actual surface temperature equals to the critical value
($T_{s}\sim T_{crit}$) which leads to the formation of Partially Screened Gap.
## 2 Partially Screened Gap
The PSG model assumes existence of heavy (Fe56) ions with density near but
still below corotational charge density ($\rho_{{\rm GJ}}$), thus the actual
charge density causes partial screening of the potential drop just above the
polar cap. The degree of shielding can be described by shielding factor
$\eta=1-\rho_{i}/\rho_{{\rm GJ}}$, where $\rho_{i}$ is the charge density of
heavy ions in the gap. The thermal ejection of ions from surface causes
partial screening of the acceleration potential drop $\Delta V=\eta\Delta
V_{max}$, where $\Delta V_{max}$ is the potential drop in vacuum gap. Using
calculations of Medin & Lai (2007) we can express the dependence of the
critical temperature on pulsar parameters as $T_{{crit}}=1.1\times
10^{6}\left(B_{14}^{1.1}+0.3\right)$, where $B_{14}=B_{s}/10^{14}$,
$B_{s}=bB_{d}$ is surface magnetic field (applicable only if hot spot is
observed i.e. $b>1$).
The actual potential drop $\Delta V$ should be thermostatically regulated and
there should be established a quasi-equilibrium state, in which heating due to
electron/positron bombardment is balanced by cooling due to thermal radiation
(see Gil et al. (2003) for more details). The necessary condition for
formation of this quasi-equilibrium state is
$\sigma T_{s}^{4}=\eta e\Delta Vcn_{{\rm GJ}},$ (1)
where $\sigma$ is the Stefan-Boltzmann constant, $e$ \- the electron charge,
$n_{{\rm GJ}}=\rho_{{\rm GJ}}/e=6.93\times 10^{12}B_{14}P^{-1}$ is the
corotational number density.
Using the Gauss’s law and Faraday’s law of induction we can find the formula
for potential drop in a gap region
$\Delta V/h^{2}+\Delta V/h_{\perp}^{2}=4\pi\eta
B_{s}\cos\left(\alpha\right)/cP$ (2)
where $h$ is gap height, $h_{\perp}$ is spark width and $\alpha$ is the
inclination angle between rotation and magnetic axis. We have found that the
main parameter that determines the process responsible for gamma-ray photon
emission in gap region is spark width ($h_{\perp}$). For narrower sparks
(higher shielding factor) acceleration potential drop is lower, which results
in smaller Lorentz factors of primary particles ($\gamma\sim 10^{3}-10^{4}$).
In this regime the gap will be dominated by ICS. Wider sparks (smaller
shielding factor) corresponds to higher acceleration ($\gamma\sim
10^{5}-10^{6}$) and results in gap dominated by CR. In this case the particles
will be accelerated to higher energies before they would upscatter x-ray
photons emitted from the hot polar cap. As the determination of spark width is
not possible by only using X-ray data we decided to use radio observations to
put more strict constrains on PSG model.
## 3 The drift model
The existence of IAR in general causes rotation of plasma relative to the NS.
The power spectrum of radio emission must have a feature due to this plasma
rotation. This feature is indeed observed and it is called drifting sub-pulse
phenomenon. Using assumption that the spark width and distance between sparks
are of the same order, we can define the drifting velocity as
$v_{dr}=2h_{\perp}/\left(PP_{3}\right)$ (3)
where $P_{3}$ is the period at which a pattern of pulses crosses the pulse
window (in units of pulsar period). In our model drift is caused by lack of
charge in IAR, then knowing that ${\bf v_{\perp}}=c{\bf\Delta E}\times{\bf
B}/B^{2}$ we can use calculation of circulation of electric field to find the
dependence of drift velocity on shielding factor
$v_{dr}=4\pi\eta h_{\perp}\cos\alpha/P$ (4)
Finally we can find dependence of shielding factor on observed drift
parameters
$\eta=1/2\pi P_{3}\cos\alpha$ (5)
Knowing that heating luminosity $L_{heat}=\eta n_{{\rm GJ}}\left(\Delta
Ve\right)c\pi R_{pc}^{2}$ we can use Eqs. 2 and 5 to find the dependence of
heating efficiency ($\xi=L_{heat}/L_{sd}$) on sub-pulse drift parameters
$\xi\approx 0.15\left(P_{2}^{\circ}/\left(P_{3}W_{\beta 0}\right)\right)^{2},$
(6)
where $W_{\beta 0}$ is the pulse width in degrees calculated with an
assumption that impact angle is zero ($\beta=0$). Thus, radio data allow not
only to determine shielding factor (and hence width of the sparks, see Eq. 2)
but also observations of sub-pulse drift allow to predict polar cap x-ray
luminosity. Tab. 1 presents observed and derived parameters of PSG for pulsars
with available radio and x-ray data. Please note that we consider only pulsars
with visible hot spot component (old NS). Despite the fact that sample is very
small we still managed to determine that for observed pulsars ICS is
responsible for gamma-photon generation in IAR.
Table 1.: Observed and derived parameters of PSG for pulsars with available radio observations of sub-pulse drift ($P_{2}^{\circ}$, $P_{3}$) and X-ray observations of actual polar cap (hot spot). $T_{s}$, $R_{pc}$ and $B_{s}$ was chosen to fit $1\sigma$ uncertainty. Please note that for calculations $\tilde{P}_{2}^{\circ}$ was used as the predicted value of sub-pulse separation (the observed value is greater than pulse width and can not be interpreted as the actual sub-pulse separation). Name | $P_{3}$ | $\eta$ | $\tilde{P}_{2}^{\circ}$ | $\log\xi$ | $\log\xi_{bol}$ | $T_{s}$ | $B_{s}$ | $R_{pc}$ | $h_{\perp}$
---|---|---|---|---|---|---|---|---|---
| $\left(P\right)$ | | $\left({\rm deg}\right)$ | $\left({\rm radio}\right)$ | $\left({\rm x-ray}\right)$ | $\left(10^{6}{\rm K}\right)$ | $\left(10^{14}{\rm G}\right)$ | $\left({\rm m}\right)$ | $\left({\rm m}\right)$
B0628–28 | $7.0$ | $0.07$ | $7.6$ | $-4.0$ | $-3.6$ | $2.5$ | $2.0$ | $23$ | $3.9$
B0834+06 | $2.2$ | $0.15$ | $1.1$ | $-3.6$ | $-3.3$ | $3.0$ | $2.4$ | $20$ | $1.8$
B0943+10 | $1.8$ | $0.09$ | $8.9$ | $-3.2$ | $-3.3$ | $3.2$ | $2.5$ | $17$ | $2.0$
B0950+08 | $6.5$ | $0.09$ | $2.8$ | $-5.1$ | $-4.5$ | $2.6$ | $2.1$ | $14$ | $0.7$
B1133+16 | $3.0$ | $0.09$ | $2.7$ | $-3.3$ | $-3.1$ | $3.4$ | $2.7$ | $17$ | $2.9$
B1929+10 | $9.8$ | $0.02$ | $5.2$ | $-5.1$ | $-4.2$ | $4.2$ | $2.0$ | $22$ | $1.6$
ICS in strong magnetic fields is very efficient process i.e. particle loses
most of its energy during scattering. This is the cause of very high
multiplicity, $M$, (number of secondary particles produced by one primary
particle). The number density of secondary plasma in PSG model can be
described as $n_{sec}=\eta n_{{\rm GJ}}M$. ICS dominated gap produces two
populations of secondary plasma. The first population (higher Lorentz factors)
is produced when primary particles lose most of their energy in ICS process.
The second population corresponds to particles produced by gamma-ray photons
above the gap (lower Lorentz factors).
## 4 Conclusions
To follow both theoretical predictions and observational data PSG model was
proposed. Recent studies on the model showed that cascade scenario in a gap
(CR or ICS) strongly depends on spark width. X-ray observations in combination
with sub-pulse drift analysis allowed to determine that for observed pulsars
ICS is responsible for gamma-ray photon generation in a gap. The exact density
of secondary plasma can be calculated only by performing full cascade
simulation with inclusion of heating by backstreaming particles. Nevertheless
we can still find dependence of multiplicity factor on number of photons
upscatterd by one primary particle. We were able to find two populations of
secondary plasma with different energy distribution. It turns out that ICS
dominated gap creates conditions suitable for generation of radio emission at
altitudes several tens of stellar radii.
## References
* Gil et al. (2003) Gil, J., Melikidze, G. I., & Geppert, U. 2003, A&A, 407, 315. arXiv:astro-ph/0305463
* Medin & Lai (2007) Medin, Z., & Lai, D. 2007, MNRAS, 382, 1833
* Ruderman & Sutherland (1975) Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51
|
arxiv-papers
| 2012-06-15T21:02:01 |
2024-09-04T02:49:31.844361
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrzej Szary, George Melikidze, Janusz Gil",
"submitter": "Andrzej Szary M.Sc.",
"url": "https://arxiv.org/abs/1206.3595"
}
|
1206.3611
|
2012 Vol. 12 No. 9, 1185–1190
11institutetext: Department of Astronomy, Peking University, Beijing 100871,
China; wuxb@pku.edu.cn
22institutetext: Yunnan Astronomical Observatory, National Astronomical
Observatories, Chinese Academy of Sciences, Kunmin 650011, China
33institutetext: Key Laboratory for Research in Galaxies and Cosmology, The
University of Science and Technology of China, Chinese Academy of Sciences,
Hefei, Anhui, 230026, China
44institutetext: Polar Research Institute of China, Jinqiao Rd. 451, Shanghai,
200136, China
Received 2012 June 12; accepted 2012 July 27
# Discovery of six high-redshift quasars with the Lijiang 2.4m telescope and
the Multiple Mirror Telescope
Xue-Bing Wu 11 Wenwen Zuo 11 Qian Yang 11 Weimin Yi 22 Chenwei Yang 3344
Wenjuan Liu 3344 Peng Jiang 3344 Xinwen Shu 3344 Hongyan Zhou 3344
###### Abstract
Quasars with redshifts greater than 4 are rare, and can be used to probe the
structure and evolution of the early universe. Here we report the discovery of
six new quasars with $i$-band magnitudes brighter than 19.5 and redshifts
between 2.4 and 4.6 from the YFOSC spectroscopy of the Lijiang 2.4m telescope
in February, 2012. These quasars are in the list of $z>3.6$ quasar candidates
selected by using our proposed $J-K/i-Y$ criterion and the photometric
redshift estimations from the SDSS optical and UKIDSS near-IR photometric
data. Nine candidates were observed by YFOSC, and five among six new quasars
were identified as $z>3.6$ quasars. One of the other three objects was
identified as a star and the other two were unidentified due to the lower
signal-to-noise ratio of their spectra. This is the first time that $z>4$
quasars have been discovered using a telescope in China. Thanks to the Chinese
Telescope Access Program (TAP), the redshift of 4.6 for one of these quasars
was confirmed by the Multiple Mirror Telescope (MMT) Red Channel spectroscopy.
The continuum and emission line properties of these six quasars, as well as
their central black hole masses and Eddington ratios, were obtained.
###### keywords:
quasars: general — quasars: emission lines — galaxies: active — galaxies:
high-redshift
## 1 Introduction
The number of known quasars has increased steadily in the past four decades
since their discovery in 1963 (Schmidt 1963). In particular, a large number of
quasars have been discovered in two large spectroscopic surveys, namely, the
Two-degree Field (2dF) survey (Boyle et al. 2000) and the Sloan Digital Sky
Survey (SDSS) (York et al. 2000). 2dF mainly selected low redshift ($z<2.2$)
quasar candidates with UV-excess (Smith et al. 2005) and has discovered more
than 20,000 quasars (Croom et al. 2004). SDSS adopted a multi-band optical
color selection method for quasars mainly by excluding the point sources in
the stellar locus of the color-color diagrams (Richards et al. 2002) and has
identified more than 120,000 quasars (Schneider et al. 2010). 90% of SDSS
quasars have low redshifts ($z<2.2$), though some dedicated methods were also
proposed for finding high redshift quasars ($z>3.5$) (Fan et al. 2001a,b;
Richards et al. 2002).
High-redshift quasars are rare, and those with redshifts greater than 4
represent only 1% in the total quasar population. In the SDSS DR7 quasar
catalog (Schneider et al. 2010), only 1248 (392) among 105783 quasars have
redshifts greater than 4 (4.5). Since these $z\sim 4$ quasars exist when the
universe is at age of 1.57 Gyr, they can be used to probe the structure and
evolution of the early universe (Smith et al. 1994; Constantin et al. 2002).
In particular, the absorption line spectra of these quasars can give valuable
information on the nature of intergalactic medium at high redshift. However,
discovering $z\sim 4$ quasars is a big challenge because they are fainter than
the low redshift quasars due to their larger distances. Moreover, the
Ly$\alpha$ emission lines for $z\sim 4$ quasars move to the red end of optical
spectra, making them hard to be distinguishable from stars due to similar
optical colors. Recently, Wu & Jia (2010) proposed using the $Y-K/g-z$
criterion to select $z<4$ quasars and using the $J-K/i-Y$ criterion to select
$z<5$ quasars with the SDSS optical and UKIDSS (UKIRT Infrared Deep Sky
Survey)111The UKIDSS project is defined in Lawrence et al. (2007). UKIDSS uses
the UKIRT Wide Field Camera (WFCAM; Casali et al. 2007) and a photometric
system described in Hewett et al. (2006). The pipeline processing and science
archive are described in Hambly et al. (2008). near-IR data based on a K-band
excess technique (Warren et al. 2000; Hewett et al. 2006; Chiu et al. 2007;
Maddox et al. 2008). With these two criteria, we expect to obtain more
complete quasar samples than previous ones. Recent optical spectroscopic
observations made by the GuoShouJing Telescope (LAMOST) and MMT have
demonstrated the success of finding the missing quasars with redshifts between
2.2 and 3 using the Y-K/g-z criterion (Wu et al. 2010a,b; Wu et al. 2011). We
also hope to discover some $z\sim 4$ quasars with the J-K/i-Y criterion, which
is expect to be applicable for selecting the candidates of quasars with
redshifts up to 5 (Wu & Jia 2010).
In this letter, we report our discovery of six new high redshift quasars from
the spectroscopic observations with the Lijiang 2.4m telescope and MMT in
February, 2012. The successful identifications of these high redshft quasars
further demonstrate the effectiveness of using our newly proposed criteria for
discovering the missing quasars including high-redshift ones.
Figure 1: The YFOSC spectra of six new quasars. From the left to right, the
red dashed lines mark the wavelengths of Ly$\alpha$, SiIV and CIV emission
lines at the estimated redshift for five $z>3.6$ quasars, while for SDSS
J113816.85+045023.6 they mark the wavelengths of CIV and CIII].
## 2 Target Selection and Spectroscopic Observations
Richards et al. (2009) presented a catalog of about 1million quasar candidates
selected from the SDSS DR6 photometric data using Bayesian methods.
Photometric redshifts for these candidates were also provided based on the
SDSS $urgiz$ magnitudes. From this catalog we selected all unidentified
candidates with the photometric redshift greater than 3.6, the photometric
redshift probability larger than 0.6 and the $i$-band magnitude brighter than
19.5. Then we cross-matched them with the UKIDSS Large Area Survey (LAS) DR7
catalog using a positional offset of 3 arcsec to find the closest
counterparts. From this sample with both SDSS $ugriz$ data and UKIDSS $YJHK$
data, we adopted our $J-K/i-Y$ criterion (Wu & Jia 2010), namely,
$J-K>0.45(i-Y)+0.48$ (where $YJK$ are the Vega magnitudes and $i$ is the AB
magnitude) , to make further selection of $z\sim 4$ quasar candidates. We also
used our own program to estimate the photometric redshifts of these candidates
with SDSS and UKIDSS 9-band photometric data (Wu & Jia 2010; Wu, Zhang & Zhou
2004), and excluded the sources whose photometric redshifts estimated from the
5-band SDSS photometric data in Richards et al. (2009) are inconsistent with
ours. After these procedures we obtained a final list of about 20 high-
redshift ($z>3.6$) quasar candidates.
The spectroscopic observations were carried out on February 26-28, 2012, with
the Yunnan Faint Object Spectrograph and Camera (YFOSC) instrument of the
Lijiang 2.4m telescope in Yunnan Astronomical Observatory. Due to the cloudy
weather, nine candidates were observed with YFOSC using a low resolution grism
with the central wavelength around $6500\AA$ , the spectral resolving power of
870, and a long slit with of 2.5′′ width. The typical seeing is around 2′′. In
Table 1 we summarize the details of the observations for these 9 candidates.
Six of them were idenfied as quasars, one as a G-type star and two as
unidentified due to the lower signal-to-noise ratios of their spectra.
Table 1: Parameters of 9 objects observed by YFOSC
Name | Date | Exposure | $u$ | $g$ | $r$ | $i$ | $z$ | $Y$ | $J$ | $H$ | $K$ | Result
---|---|---|---|---|---|---|---|---|---|---|---|---
(SDSS J) | | ($s$) | | | | | | | | | |
075733.86+190403.1 | 2012-02-26 | 2700 | 21.37 | 20.40 | 19.45 | 19.00 | 18.53 | 17.88 | 17.02 | 16.63 | 15.68 | low S/N
085203.84+020437.7 | 2012-02-27 | 6000 | 21.67 | 20.99 | 19.69 | 19.09 | 18.66 | 17.82 | 17.47 | 16.66 | 15.72 | low S/N
092740.04-023347.5 | 2012-02-28 | 4200 | 21.01 | 20.50 | 19.55 | 19.08 | 18.86 | 18.42 | 18.01 | 17.11 | 16.42 | G star
093345.70-020439.5 | 2012-02-27 | 3600 | 23.77 | 20.72 | 19.55 | 19.47 | 19.41 | 19.21 | 18.66 | 18.46 | 17.88 | quasar
095023.74+004419.7 | 2012-02-27 | 6600 | 23.97 | 20.97 | 19.75 | 19.48 | 19.36 | 18.83 | 18.40 | 17.76 | 17.25 | quasar
113816.85+045023.6 | 2012-02-27 | 6000 | 21.26 | 20.70 | 19.67 | 19.27 | 19.06 | 18.30 | 17.96 | 16.90 | 16.46 | quasar
120312.63-001118.8 | 2012-02-28 | 5400 | 25.38 | 22.34 | 20.29 | 19.14 | 18.95 | 18.32 | 18.00 | 17.19 | 16.64 | quasar
125052.11+074919.6 | 2012-02-26 | 5400 | 23.56 | 20.08 | 18.75 | 18.63 | 18.43 | 18.14 | 17.35 | 16.81 | 16.15 | quasar
145115.89+015843.3 | 2012-02-27 | 4800 | 23.72 | 20.42 | 19.30 | 19.23 | 19.09 | 18.61 | 18.02 | 17.56 | 16.92 | quasar
0.86The $ugriz$ magnitudes are given in AB system and the $YJHK$ magnitudes
are given in Vega system.
The spectra of six new quasars, after the flat-field correction and both
wavelength and flux calibrations, are shown in Fig.1. The strongest emission
lines for five $z>3.6$ quasar are Ly$\alpha$ lines, while for SDSS
J113816.85+045023.6 the strongest line around $6400\AA$ is CIII]. The
redshifts for these quasars are the average values given mostly by the
Ly$\alpha$ and CIV lines for five $z>3.6$ quasar and by the III] and CIV lines
for SDSS J113816.85+045023.6.
Thanks to the Chinese Telescope Access Program222http://info.bao.ac.cn/tap/,
SDSS J120312.63-001118.8 was also observed with the Red Channel Spectrograph
on the MMT 6.5m telescope333Observation reported here was obtained at the MMT
Observatory, a joint facility of the Univeristy of Arizona and the Smithsonian
Institution. at Mt. Hopkins, Arizona, USA on Feb. 29, 2012, with a wavelength
coverage of 5100-8600$\AA$ and a spectral resolution of 1.6Å. It was observed
twice, with the exposures of 10 minutes and 15 minutes respectively. The
spectrum was processed using the IRAF Echellette package and is shown in Fig.
2. The average redshift estimated from Ly$\alpha$ and SiIV (1400Å) emission
lines is 4.601$\pm$0.008, consistent with the result (z=4.592$\pm$0.048)
obtained from the YFOSC observation.
Figure 2: The MMT Red Channel spectrum of SDSS J120312.63-001118.8. The
average redshift estimated from Ly$\alpha$ and SiIV (1400Å) emission lines is
4.601. The red line refer to the smoothed spectrum with a binsize of 8Å.
## 3 Spectral analyses and properties of six high-redshift quasars
The redshift corrected rest-frame YFOSC spectra of six quasars are first
corrected for the Galactic extinction using the extinction map of Schlegel et
al. (1998). They are then fitted with an IDL code MPFIT (Markwardt , 2009). We
fit the spectra with the pseudo-continuum model consisting of the featureless
nonstellar continuum and FeII emissions. The continuum is assumed to be a
power-law, so two free parameters (the amplitude and the slope) are required.
The UV FeII template (Vestergaard & Wilkes 2001; Tsuzuki et al. 2006) is
convolved with a velocity dispersion and shifted with a velocity, assuming the
line width of FeII lines are same with emission lines in the corresponding
wavelength range. During the fitting, the amplitude and slope of the power-law
continuum and the amplitude, velocity shift and broadening width of the FeII
emission, are set to be free parameters. The initial value of the power-law
continuum is obtained by fitting a simple power law to the data in the chosen
windows, which are free from emission-line contamination. The initial value of
broadening width of the FeII emissions is set to be the line width of the
strong emission line CIV. Then we use the pseudo-continuum model to fit a set
of continuum windows with strong FeII emissions but no other emission lines,
as mentioned in Hu et al. (2008), slightly adjusted interactively for each
individual spectrum in order to avoid broad absorption features or extended
wings of emission lines.
After constructing the pseudo-continuum, the CIV line should be fitted with
two Gaussians, one for the narrow component and another for the broad
component. However, except SDSS J095023.74+004419.7, the spectra of other five
quasars have low signal-to-noise ratio, so we used only one Gaussian to fit
the CIV emission line. Absorption features are evident in the spectra of four
quasars, and one more negative Gaussian was added in the fitting. We measure
the Full Width at Half Magnitude of CIV line (FWHMCIV), luminosity at 1350
Å($L_{1350}$) from the spectra (except for SDSS J113816.65+004419.7, where
1350$\AA$ is not within the wavelength coverage, we estimate the luminosity at
1500$\AA$ instead). The black hole mass is estimated based on FWHMCIV and
$L_{1350}$ with Eq. (7) in Vestergaard & Peterson (2006)(see also Kong et al.
2006). Using a scaling between $L_{1350}$ and bolometric luminosity $L_{\rm
bol}$, $L_{\rm bol}=4.62L_{1350}$, we estimated the bolometric luminosity for
the six quasars. Based on the estimated black hole mass and bolometric
luminosity, we also estimated their Eddington ratios ($R_{\rm EDD}$). The
results are summarized in Table 2. Although the uncertainties of these values
are probably quite large due to the low spectral quality and the unusual
properties of CIV, we noticed that the overall properties of these six quasars
are consistent with those of typical SDSS quasars at high redshift (Shen et
al. 2011).
Table 2: Spectral parameters and black hole masses of six new quasars
Name | Redshift | FWHM(CIV) | $\log(L_{1350})$ | $\log(M_{BH})$ | $\log(L_{\rm bol})$ | $\log R_{\rm EDD}$
---|---|---|---|---|---|---
(SDSS J) | | ($km~{}s^{-1}$) | ($erg~{}s^{-1}$) | ($M_{\odot}$) | ($erg~{}s^{-1}$) |
093345.70-020439.5 | 3.701$\pm$0.011 | 6749 | 46.457 | 9.621 | 47.122 | -0.588
095023.74+004419.7 | 3.967$\pm$0.035 | 4500 | 46.300 | 9.185 | 46.964 | -0.310
113816.85+045023.6 | 2.368$\pm$0.011 | 5144 | 46.000 | 9.118 | 46.664 | -0.543
120312.63-001118.8 | 4.592$\pm$0.048 | 4865 | 46.431 | 9.323 | 47.096 | -0.316
125052.11+074919.6 | 3.748$\pm$0.030 | 5424 | 46.805 | 9.615 | 47.469 | -0.235
145115.89+015843.3 | 3.940$\pm$0.006 | 4507 | 45.682 | 8.859 | 46.346 | -0.602
## 4 Discussion
A complete quasar sample is crucial for studying the large scale structure of
the universe. The current available quasar samples are mostly biased towards
low redshifts ($z<2.2$) and more efforts are needed to find quasars at high
redshift. Wu & Jia (2010) proposed to obtain a large complete quasar sample
with redshifts up to five by combining the $J-K/i-Y$ criterion with the
$Y-K/g-z$ criterion to select quasar candidates. Some recent optical
spectroscopic observations have demonstrated the success of finding the
missing quasars with redshifts between 2.2 and 3 using the $Y-K/g-z$ criterion
(Wu et al. 2010a,b; Wu et al. 2011). Our discovery of six high redshift
quasars (five with $z>3.6$) from the spectroscopic observations with the
Lijiang 2.4m telescope and MMT further demonstrates the effectiveness of using
the $J-K/i-Y$ criterion for discovering quasars with redshifts up to five.
Moreover, the identification of five quasars with $z>3.6$ from nine candidates
with photometric redshift larger than 3.6 also confirms the robustness of the
photometric redshifts estimated by the SDSS and UKIDSS photometric data. We
noticed that two among our five $z>3.6$ new quasars do not meet the SDSS $gri$
or $riz$ selection critrion for $z>3.6$ quasars (Fan et al. 2001a,b; Richards
et al. 2002), which suggests that about 40% of such quasars may be missed in
the SDSS spectroscopic survey. This obviously needs to be confirmed by future
observations of a large sample of $z>3.6$ quasars.
Our identification of a $z=4.6$ quasar demonstrates that $z>4$ quasars can be
identified with the 2-meter size telescopes in China. We hope more high
redshift quasars will be discovered by the future LAMOST quasar survey (Wu
2011), which is aiming at discovering 0.3-0.4 million quasars from 1 million
quasar candidates with $i<20.5$, by taking the advantages of 4000 fibers and 5
degree field of view of LAMOST (Su et al. 1998; Zhao et al. 2012). The new
quasar selection criteria, such as those based on SDSS, UKIDSS and the Wide-
field Infrared Survey Explorer (WISE; Wright et al. 2010) data (Wu et al.
2012), will be applied for selecting quasar candidates in the LAMOST quasar
survey. This will hopefully provide the largest quasar sample in the next five
years for further studies of AGN physics, large scale structure and cosmology.
###### Acknowledgements.
We thank the referee for a constructive report and Jianguo Wang, Cheng Hu and
Zhaoyu Chen for great helps on the spectral analysis. This work was supported
by the National Natural Science Foundation of China (11033001). We acknowledge
the use of Lijiang 2.4m telescope and the MMT 6.5m telescope, as well as the
archive data from SDSS and UKIDSS. This research uses data obtained through
the Telescope Access Program (TAP), which is funded by the National
Astronomical Observatories, Chinese Academy of Sciences, and the Special Fund
for Astronomy from the Ministry of Finance. Funding for the SDSS and SDSS-II
has been provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Science Foundation, the U.S. Department of Energy,
the National Aeronautics and Space Administration, the Japanese
Monbukagakusho, the Max Planck Society, and the Higher Education Funding
Council for England. The SDSS Web Site is http://www.sdss.org/.
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|
arxiv-papers
| 2012-06-15T23:50:17 |
2024-09-04T02:49:31.852559
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-Bing Wu, Wen-Wen Zuo, Qian Yang, Wei-Min Yi, Chen-Wei Yang,\n Wen-Juan Liu, Peng Jiang, Xin-Wen Shu, Hong-Yan Zhou",
"submitter": "Xue-Bing Wu",
"url": "https://arxiv.org/abs/1206.3611"
}
|
1206.3793
|
# A distributed classification/estimation algorithm for sensor
networks††thanks: A preliminary version of some of the results has appeared in
the proceedings of the 50st IEEE Conference on Decision and Control and
European Control Conference, Orlando, Florida, 12-15 December 2011.
Fabio Fagnani) DISMA (Dipartimento di Scienze Matematiche), Politecnico di
Torino, Corso Duca degli Abruzzi, 24, I-10129 TO, (e-mail:
fabio.fagnani@polito.it Sophie M. Fosson DET (Dipartimento di Elettronica e
Telecomunicazioni), Politecnico di Torino, Corso Duca degli Abruzzi, 24,
I-10129 TO (e-mail: sophie.fosson@polito.it) Chiara Ravazzi DET (Dipartimento
di Elettronica e Telecomunicazioni), Politecnico di Torino, Corso Duca degli
Abruzzi, 24, I-10129 TO (e-mail: chiara.ravazzi@polito.it)
###### Abstract
In this paper, we address the problem of simultaneous classification and
estimation of hidden parameters in a sensor network with communications
constraints. In particular, we consider a network of noisy sensors which
measure a common scalar unknown parameter. We assume that a fraction of the
nodes represent faulty sensors, whose measurements are poorly reliable. The
goal for each node is to simultaneously identify its class (faulty or non-
faulty) and estimate the common parameter.
We propose a novel cooperative iterative algorithm which copes with the
communication constraints imposed by the network and shows remarkable
performance. Our main result is a rigorous proof of the convergence of the
algorithm and a characterization of the limit behavior. We also show that, in
the limit when the number of sensors goes to infinity, the common unknown
parameter is estimated with arbitrary small error, while the classification
error converges to that of the optimal centralized maximum likelihood
estimator. We also show numerical results that validate the theoretical
analysis and support their possible generalization. We compare our strategy
with the Expectation-Maximization algorithm and we discuss trade-offs in terms
of robustness, speed of convergence and implementation simplicity.
###### keywords:
Classification, Consensus, Gaussian mixture models, Maximum-likelihood
estimation, Sensor networks, Switching systems.
## 1 Introduction
Sensor networks are one of the most important technologies introduced in our
century. Promoted by the advances in wireless communications and by the
pervasive diffusion of smart sensors, wireless sensor networks are largely
used nowadays for a variety of purposes, e.g., environmental and habitat
surveillance, health and security monitoring, localization, targeting, event
detection.
A sensor network basically consists in the deployment of a large numbers of
small devices, called sensors, that have the ability to perform measurements
and simple computations, to store few amounts of data, and to communicate with
other devices. In this paper, we focus on _ad hoc_ networks, in which
communication is local: each sensor is connected only with a restricted number
of other sensors. This kind of cooperation allows to perform elaborate
operations in a self-organized way, with no centralized supervision or data
fusion center, with a substantial energy and economic saving on processors and
communication links. This allows to construct large sensor networks at
contained cost.
A problem that can be addressed through ad hoc sensor networking is the
distributed estimation: given an unknown physical parameter (e.g., the
temperature in a room, the position of an object), one aims at estimating it
using the sensing capabilities of a network. Each sensor performs a (not
exact) measurement and shares it with the sensors with which it can establish
a communication; in turn, it receives information and consequently updates its
own estimate. If the network is connected, by iterating the sharing procedure,
the information propagates and a consensus can be reached. Neither centralized
coordinator nor data fusion center is present. The mathematical model of this
problem must envisage the presence of noise in measurements, which are
naturally corrupted by inaccuracies, and possible constraints on the network
in terms of communication, energy or bandwidth limitations, and of necessity
of quantization or data compression.
_Distributed estimation_ in ad hoc sensor networks has been widely studied in
the literature. For the problem of estimating an unknown common parameter,
typical approach is to consider distributed versions of classical maximum
likelihood (ML) or maximum-a-posteriori (MAP) estimators. Decentralization can
be obtained, for instance, through consensus type protocols (see [1], [2],
[3]) adapted to the communication graph of the network, or by belief
propagation methods [4] and [5].
A second important issue is _sensors’ classification_ , which we define as
follows [6]. Let us imagine that sensors can be divided into different classes
according to peculiar properties, e.g., measurements’ or processing
capabilities, and that no sensor knows to which class it belongs: by
classification, we then intend the labeling procedure that each sensor
undertakes to determine its affiliation. This task is addressed to a variety
of clustering purposes, for example, to rebalance the computation load in a
network where sensors can be distinguished according to their processing
power. On most occasions, sensors’ classification is faced through some
distributed estimation, the underlying idea being the following: each sensor
performs its measurement of a parameter, then iteratively modifies it on the
basis of information it receives; during this iterative procedure the sensor
learns something about itself which makes it able to estimate its own
configuration.
In this paper, we consider the following model: each sensor $i$ performs a
measurement $y_{i}=\theta^{\star}+\omega_{i}^{\star}\eta_{i}$, where
$\theta^{\star}\in\mathbb{R}$ is the unknown global parameter,
$\omega_{i}^{\star}>0$ is the unknown status of the sensor, and $\eta_{i}$ is
a Gaussian random noise. The more $\omega_{i}^{\star}$ is large, the more the
sensor $i$ is malfunctioning, that is, the quality if its measurement is low.
The $\omega_{i}^{\star}$ parameter is supposed to belong to a discrete set, in
particular in this paper we consider the binary case.
The goal of each unit $i$ is to estimate the parameter $\theta^{\star}$ and
the specific configuration $\omega^{\star}_{i}$. The presence of the common
unknown parameter $\theta^{\star}$ imposes a coupling between the different
nodes and makes the problem interesting.
An additive version of the aforementioned model has been studied in [7], where
measurement is given by $y_{i}=\theta^{\star}+\omega_{i}^{\star}+\eta_{i}$.
Another related problem is the so-called calibration problem [8, 9]: sensor
$i$ performs a noisy linear measurement $y_{i}=A_{i}\theta+\eta_{i}$ where the
unknown $\theta$ and $A_{i}$ are a vector and a matrix, respectively, while
$\eta_{i}$ is a noise; the goal consists in the estimation of $\theta$ and of
$A_{i}$, the latter being known as calibration problem.
All these are particular cases of the problem of the estimation of Gaussian
mixtures’ parameters [10, 11]. This perspective has been studied for sensor
networks in [12], [13], [14], and [15] where distributed versions of the
Expectation-Maximization (EM) algorithm have been proposed. A network is given
where each node independently performs the E-step through local observations.
In particular, in [14] a consensus filter is used to propagate the local
information. The tricky point of such techniques is the choice of the number
of averaging iterations between two consecutive M-steps, which must be
sufficient to reach consensus.
The aim of this paper is the development of a distributed, iterative procedure
which copes with the communication constraints imposed by the network and
computes an estimation ($\widehat{\theta},\widehat{\omega}$) approximating the
maximum likelihood optimal solution of the proposed problem. The core of our
methodology is an Input Driven Consensus Algorithm (IA for short), introduced
in [16], which takes care of the estimation of the parameter $\theta^{*}$. IA
is coupled with a classification step where nodes update the estimation of
their own type $\omega^{*}_{i}$ by a simple threshold estimator based on the
current estimation of $\theta^{*}$. The fact of using a consensus protocol
working on inputs instead, as more common, on initial conditions, is a key
strategic fact: it serves the purpose of using the innovation coming from the
units who are modifying the estimation of their status, as time passes by. Our
main theoretical contribution is a complete analysis of the algorithm in terms
of convergence and of behaviour with respect to the size of the network. With
respect to other approaches like distributed EM for which convergence results
are missing, this makes an important difference. We also present a number of
numerical simulations showing the remarkable performance of the algorithm
which, in many situations, outperform classical choices like EM.
The outline of the paper is the following. In Section 2 we shortly present
some graph nomenclature needed in the paper. Section 3 is devoted to a formal
description of the problem and to a discussion of the classical centralized
maximum likelihood solution. In Section 4, we present the details and the
analysis of our IA. Our main results are Theorems 1 and 2: Theorem 1 ensures
that, under suitable assumptions on the graph, the algorithm converges to a
local maximum of the log-likelihood function; Theorem 2 is a concentration
result establishing that when the number of nodes $N\to+\infty$, the estimate
$\widehat{\theta}$ converges to the true value $\theta^{*}$ (a sort of
asymptotic consistency). Finally, we also study the behavior of the discrete
estimate $\widehat{\omega}$ by analyzing the performance index the relative
classification error over the network when $N\to+\infty$ (see Corollary 4).
Section 5 contains a set of numerical simulations carried on different graph
architectures: complete, circulant, grids, and random geometric graphs.
Comparisons are proposed with respect to the optimal centralized ML solution
and also with respect to the EM solution. Finally, a long Appendix contains
all the proofs.
## 2 General notation and graph theoretical preliminaries
Throughout this paper, we use the following notational convention. We denote
vectors with small letters, and matrices with capital letters. Given a matrix
$M$, $M^{T}$ denotes its transpose. Given a vector $v$, $||v||$ denotes its
Euclidean norm. $\mathbf{1}_{A}$ is the indicator function of set $A$. Given a
finite set $\mathcal{V}$, $R^{\mathcal{V}}$ denotes the space of real vectors
with components labelled by elements of $\mathcal{V}$. Given two vectors
$x,z\in\mathbb{R}^{\mathcal{V}}$,
$\mathrm{d_{H}}(x,z)=|\\{i\in\mathcal{V}:x_{i}\neq z_{i}\\}|$. We use the
convention that a summation over an empty set of indices is equal to zero,
while a product over an empty set gives one.
A symmetric graph is a pair $\mathcal{G}=(\mathcal{V,E})$ where $\mathcal{V}$
is a set, called the set of vertices, and
$\mathcal{E}\subseteq\mathcal{V\times V}$ is the set of edges with the
property that $(i,i)\not\in\mathcal{E}$ for all $i\in\mathcal{V}$ and
$(i,j)\in\mathcal{E}$ implies $(j,i)\in\mathcal{E}$. $\mathcal{G}$ is strongly
connected if, for all $i,j\in\mathcal{V}$, there exist vertices $i_{1},\dots
i_{s}$ such that $(i,i_{1}),(i_{1},i_{2}),\dots,(i_{s},j)\in\mathcal{E}$. To
any symmetric matrix $P\in\mathbb{R}^{\mathcal{V}\times\mathcal{V}}$ with non-
negative elements, we can associate a graph
$\mathcal{G}_{P}=(\mathcal{V},\mathcal{E}_{P})$ by putting
$(i,j)\in\mathcal{E}_{P}$ if and only if $P_{ij}>0$. $P$ is said to be adapted
to a graph $\mathcal{G}$ if $\mathcal{G}_{P}\subseteq\mathcal{G}$. A matrix
with non-negative elements $P$ is said to be stochastic if
$\sum_{j\in\mathcal{V}}P_{ij}=1$ for every $i\in\mathcal{V}$. Equivalently,
denoting by ${\mathbbm{1}}$ the vector of all $1$ in
$\mathbb{R}^{\mathcal{V}}$, $P$ is stochastic if
$P{\mathbbm{1}}={\mathbbm{1}}$. $P$ is said to be primitive if there exists
$n_{0}\in\mathbb{N}$ such that $P^{n_{0}}_{ij}>0$ for every
$i,j\in\mathcal{V}$. A sufficient condition ensuring primitivity is that
$\mathcal{G}_{P}$ is strongly connected and $P_{ii}>0$ for some
$i\in\mathcal{V}$.
## 3 Bayesian modeling for estimation and classification
### 3.1 The model
In our model, we consider a network, represented by a symmetric graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$. $\mathcal{G}$ represents the system
communication architecture. We denote the number of nodes by
$N=|\mathcal{V}|$. We assume that each node $i\in\mathcal{V}$ measures the
observable
$y_{i}=\theta^{\star}+\omega^{\star}_{i}\eta_{i}$ (1)
where $\theta^{\star}\in\mathbb{R}$ is an unknown parameter, $\eta_{i}$’s
Gaussian noises $\mathsf{N}(0,1)$, $\omega^{\star}_{i}$’s Bernoulli random
variables taking values in $\\{\alpha,\beta\\}$ (with
${\mathbb{P}}(\omega^{\star}_{i}=\beta)=p$). We assume all the random
variables $\eta_{i}$’s and $\omega^{\star}_{i}$’s to be mutually independent.
Notice that each $y_{i}\in\mathbb{R}$ is a Gaussian mixture distributed
according to the probability density function
$\displaystyle
f(y_{i})=(1-p)f(y_{i}|\theta^{\star},\alpha)+pf(y_{i}|\theta^{\star},\beta)$
(2) $\displaystyle
f(y_{i}|\theta^{\star},x)=\frac{1}{x\sqrt{2\pi}}\mathrm{e}^{-\frac{(y_{i}-\theta^{\star})^{2}}{2x^{2}}}\quad
x\in\\{\alpha,\beta\\}.$ (3)
The binary model of $\omega^{\star}$ is motivated by different scenarios: as
an example, if $0<\alpha<<\beta$, the nodes of type $\beta$ may represent a
subset of faulty sensors, whose measurements are poorly reliable; the aim may
be the detection of faulty sensors in order to switch them off or neglect
their measurements, or for other clustering purposes. It is also realistic to
assume that some a-priori information about the quantity of faulty sensors is
extracted, e.g., from experimental data on the network, and it is conceivable
to represent such information as an a-priori distribution. This is why we
assume a Bernoulli distribution on each $\omega^{\star}_{i}$; on the other
hand, we suppose that no a-priori information is available on the unknown
parameter $\theta^{\star}$. However, the addition of an a priori probability
distribution on $\theta^{*}$ does not significantly alter our analysis and our
results.
### 3.2 The maximum likelihood solution
The goal is to estimate the parameter $\theta^{\star}$ and the specific
configuration $\omega^{\star}_{i}$ of each unit. Disregarding the network
constraints, a natural solution to our problem would be to consider a joint ML
in $\theta^{\star}$ and MAP in the $\omega^{\star}_{i}$’s (see [17, 18]). Let
$f(y,\omega|\theta)$ be the joint distribution of $y$ and $\omega$ (density in
$y$ and probability in $\omega$) given the parameter $\theta$, and consider
the rescaled log-likelihood function
$\displaystyle\begin{split}L_{N}(\theta,\omega)&:=\frac{1}{N}\log
f(y,\omega|\theta).\end{split}$ (4)
The hybrid ML/MAP solution, which for simplicity for now on we will refer to
as the ML solution, prescribes to choose $\theta$ and $\omega$ which maximize
$L_{N}(\theta,\omega)$
$(\widehat{\theta}^{\mathrm{ML}},\widehat{\omega}^{\mathrm{ML}}):=\underset{\theta\in\mathbb{R},\leavevmode\nobreak\
\omega\in\\{\alpha,\beta\\}^{\mathcal{V}}}{\mathrm{argmax\,}}L_{N}(\theta,\omega).$
(5)
Standard calculations lead us to
$L_{N}(\theta,\omega)=-\frac{1}{N}\sum_{j\in\mathcal{V}}\left(\frac{(y_{j}-\theta)^{2}}{2\beta^{2}}+{\mathbf{1}}_{\\{\omega_{j}=\alpha\\}}\left(\frac{(y_{j}-\theta)^{2}}{2}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)+\log\frac{1-p}{p}\frac{\beta}{\alpha}\right)\right)+c$
(6)
where $c$ is a constant. It can be noted that partial maximizations of
$L_{N}(\theta,\omega)$ with respect to just one of the two variables have
simple representation. Let
$\widehat{\theta}(\omega):=\underset{\theta}{\mathrm{argmax\,}}{L}_{N}(\theta,\omega)\qquad\widehat{\omega}(\theta):=\underset{\omega}{\mathrm{argmax\,}}L_{N}(\theta,\omega).$
(7)
Then
$\widehat{\theta}(\omega)=\frac{\sum_{j}y_{j}/\omega_{j}^{2}}{\sum_{j}1/\omega_{j}^{2}}\qquad\widehat{\omega}(\theta)_{i}=\begin{cases}\alpha&\text{if
}|y_{i}-{{\theta}}|<\delta\\\ \beta&\text{otherwise}\end{cases}$ (8)
where
$\delta=\sqrt{2\frac{\ln\left(\frac{1-p}{p}\frac{\beta}{\alpha}\right)}{\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}}}.$
The ML solution can then be obtained, for instance, by considering
$\widehat{\theta}^{\mathrm{ML}}=\underset{\theta}{\mathrm{argmax\,}}L(\theta,\widehat{\omega}(\theta))\,,\quad\widehat{\omega}^{\mathrm{ML}}=\widehat{\omega}(\widehat{\theta}^{\mathrm{ML}}).$
(9)
It should be noted how the computation of the
$(\widehat{\omega}^{\mathrm{ML}})_{i}$’s becomes totally decentralized once
$\widehat{\theta}^{\mathrm{ML}}$ has been computed. For the computation of
$\widehat{\theta}^{\mathrm{ML}}$ instead one needs to gather information from
all units to compute $L_{N}(\theta,\widehat{\omega}(\theta))$ and it is not at
all evident how this can be done in a decentralized way. Moreover, further
difficulties are caused by the fact that
$L_{N}(\theta,\widehat{\omega}(\theta))$ may contain many local maxima, as
shown in Figure 1.
It should be noted that $L_{N}(\theta,\widehat{\omega}(\theta))$ is
differentiable except at a finite number of points, and between two successive
non-differentiable points the function is concave. Therefore, the local maxima
of the function coincide with its critical points. On the other hand, the
derivative, where it exists, is given by
$\begin{split}\frac{d}{d\theta}L_{N}(\theta,\widehat{\omega}(\theta))&=\left(\frac{1}{\beta^{2}}-\frac{1}{\alpha^{2}}\right)\frac{1}{N}\sum_{i\in\mathcal{V}}(\theta-
y_{i}){\mathbf{1}}_{\\{|y_{i}-\theta|<\delta\\}}-\frac{1}{\beta^{2}}\left(\theta-\frac{1}{N}\sum_{i\in\mathcal{V}}y_{i}\right).\end{split}$
(10)
Stationary points can therefore be represented by the relation
$\theta=\frac{\frac{1}{\beta^{2}}\sum_{i}y_{i}+\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)\sum_{i}y_{i}{\mathbf{1}}_{\\{|y_{i}-\theta|<\delta\\}}}{N\frac{1}{\beta^{2}}+\sum_{i}{\mathbf{1}}_{\\{|y_{i}-\theta|<\delta\\}}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)}.$
(11)
A moment of thought shows us that (11) is equivalent to the relation
$\theta=\widehat{\theta}(\widehat{\omega}(\theta))$.
This representation will play a key role in the sequel of this paper.
Figure 1: $\alpha=0.3,\beta=10,p=0.25$: Plot of function
$L_{N}(\theta,\widehat{\omega}(\theta))$ as a function of $\theta$ and size
$N\in\\{50,100,400,500,1000,5000\\}$.
### 3.3 Iterative centralized algorithms
The computational complexity of the optimization problem (5) is practically
unfeasible in most situations. However, relations (8) suggest a simple way to
construct an iterative approximation of the ML solution (which we will denote
IML). The formal pattern is the following: fixed
$\widehat{\omega}^{(0)}=\alpha\mathbbm{1}$, for $t=0,1,\dots$, we consider the
dynamical system
$\displaystyle\widehat{\theta}^{(t+1)}=\frac{\sum_{j=1}^{N}y_{j}\left[\widehat{\omega}_{j}^{(t)}\right]^{-2}}{\sum_{j=1}^{N}\left[\widehat{\omega}_{j}^{(t)}\right]^{-2}}$
$\displaystyle\widehat{\omega}(\theta)^{(t+1)}_{i}=\begin{cases}\alpha&\text{if
}|y_{i}-{{\theta}}|<\delta\\\
\beta&\text{otherwise}\end{cases}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for any }i=1,\dots,N.$
The algorithm stops whenever
$|\widehat{\theta}^{(t+1)}-\widehat{\theta}^{(t)}|<\varepsilon$, for some
fixed tolerance $\varepsilon>0$.
A more refined iterative solution is given by the so-called Expectation-
Maximization (EM) algorithm [19]. The main idea is to introduce a hidden (say,
unknown and unobserved) random variable in the likelihood; then, at each step,
one computes the mean of the likelihood function with respect to the hidden
variable and finds its maximum. Such a method seeks to find the maximum
likelihood solution, which in many cases cannot be formulated in a closed
form. EM is widely and successfully used in many frameworks and in principle
it could also be applied to our problem. In our context, making the variable
$\omega$ to play the part of the hidden variable, equations for EM become (see
the tutorial [20] for their derivation)
Given $\widehat{\theta}^{(0)}\in\mathbb{R}$, for $t=0,1,\dots$,
1. 1.
E-step: for all node $i\in\mathcal{V}$,
$q_{i}^{(t)}=\mathbb{P}\left(\widehat{\omega}_{i}^{(t)}=\alpha|y,\widehat{\theta}^{(t)}\right)=\frac{(1-p)f\left(y|\widehat{\omega}_{i}^{(t)}=\alpha,\widehat{\theta}^{(t)}\right)}{(1-p)f\left(y|\widehat{\omega}_{i}^{(t)}=\alpha,\widehat{\theta}^{(t)}\right)+pf\left(y|\widehat{\omega}_{i}^{(t)}=\beta,\widehat{\theta}^{(t)}\right)}.$
2. 2.
M-step:
$\widehat{\theta}^{(t+1)}=\frac{\sum_{j\in\mathcal{V}}y_{j}\left(q_{j}^{(t)}\alpha^{-2}+(1-q_{j}^{(t)})\beta^{-2}\right)}{\sum_{j\in\mathcal{V}}q_{j}^{(t)}\alpha^{-2}+(1-q_{j}^{(t)})\beta^{-2}}.$
The algorithm stops whenever
$|\widehat{\theta}^{(t+1)}-\widehat{\theta}^{(t)}|<\varepsilon$, for some
fixed tolerance $\varepsilon>0$. It is worth to notice that $q_{i}^{(t)}$
computed in the E-step actually is the expectation of the binary random
variable $\mathbf{1}_{\\{\widehat{\omega}_{i}^{(t)}=\alpha\\}}$. On the other
hand $\widehat{\theta}^{(t+1)}$ computed in the M-step is the maximum of such
expectation.
An important feature of EM is that it is possible to prove the convergence of
the sequence $\\{\widehat{\theta}^{(t)}\\}_{\in\mathbb{N}}$ to a local maximum
of the expected value of the log-likelihood with respect to the unknown data
$\omega$, a result which is instead not directly available for IML. Both
algorithms however share the drawback of requiring centralization. Distributed
versions of the EM have been proposed (see, e.g., [12], [14]) but convergence
is not guaranteed for them. In Section 5 we will compare both these algorithms
against the distributed IA we are going to present in the next section. While
it is true that EM always outperforms IML, algorithm IA outperforms both of
them for small size algorithms, while shows comparable performance to EM for
large networks.
## 4 Input driven consensus algorithm
### 4.1 Description of the algorithm
In this section we propose a distributed iterative algorithm approximating the
centralized ML estimator. The algorithm is suggested by the expressions in (8)
and consists of the iteration of two steps: an averaging step where all units
aim at computing $\widehat{\theta}$ through a sort of Input Driven Consensus
Algorithm (IA) followed by an update of the classification estimation
performed autonomously by all units.
Formally, IA is parametrized by a symmetric stochastic matrix $P$, adapted to
the communication graph $\mathcal{G}$ ($P_{ij}>0$ if and only if,
$(i,j)\in\mathcal{E}$), and by a real sequence $\gamma^{(t)}\rightarrow 0$.
Every node $i$ has three messages stored in its memory at time $t$, denoted
with $\mu_{i}^{(t)},\nu_{i}^{(t)}$, and $\widehat{\omega}^{(t)}_{i}$. Given
the initial conditions $\mu_{i}^{(0)}=0,\nu_{i}^{(0)}=0$ and the initial
estimate $\widehat{\omega}_{i}^{(0)}=\alpha$, the dynamics consists of the
following steps.
1. 1.
Average step:
$\displaystyle\mu_{i}^{(t+1)}$
$\displaystyle=(1-\gamma^{(t)}){\sum_{j}P_{ij}\mu_{j}^{(t)}}+\gamma^{(t)}{{y_{i}}{\left(\widehat{\omega}_{i}^{(t)}\right)^{-2}}}$
(12a) $\displaystyle\nu_{i}^{(t+1)}$
$\displaystyle=(1-\gamma^{(t)}){\sum_{j}P_{ij}\nu_{j}^{(t)}}+\gamma^{(t)}{{\left(\widehat{\omega}_{i}^{(t)}\right)^{-2}}}$
(12b) $\displaystyle\widehat{\theta}^{(t+1)}_{i}$
$\displaystyle={\mu_{i}^{(t+1)}}/{\nu_{i}^{(t+1)}}.$ (12c)
2. 2.
Classification step:
$\widehat{\omega}_{i}^{(t+1)}=\widehat{\omega}_{i}(\widehat{\theta}^{(t+1)})=\left\\{\begin{array}[]{cl}\alpha&\text{if
}|y_{i}-\widehat{\theta}_{i}^{(t+1)}|<\delta\\\
\beta&\text{otherwise.}\end{array}\right.$ (13)
It should be noted that the algorithm provides a distributed protocol: each
node only needs to be aware of its neighbours and no further information about
the network topology is required.
### 4.2 Convergence
The following theorem ensures the convergence of IA. The proof is rather
technical and therefore deferred to Appendix A.
###### Theorem 1.
Let
1. (a)
$\gamma^{(t)}\rightarrow 0$, $\gamma^{(t)}\geq 1/t$, and
$\gamma^{(t)}=\gamma^{(t+1)}+o(\gamma^{(t+1)})$ for $t\to+\infty$;
2. (b)
$P\in\mathbb{R}_{+}^{\mathcal{V}\times\mathcal{V}}$ be a stochastic,
symmetric, and primitive matrix with positive eigenvalues.
Then, there exist $\widehat{\omega}^{{IA}}\in\\{\alpha,\beta\\}^{\mathcal{V}}$
and $\widehat{\theta}^{{IA}}\in{\mathbb{R}}$ such that
1. 1.
$\lim_{t\rightarrow+\infty}\widehat{\omega}^{(t)}\stackrel{{\scriptstyle\text{a.s.}}}{{=}}\widehat{\omega}^{IA}\,,\qquad\lim_{t\rightarrow+\infty}\widehat{\theta}^{(t)}_{i}\stackrel{{\scriptstyle\text{a.s.}}}{{=}}\widehat{\theta}^{IA}$
for all $i\in\mathcal{V}$;
2. 2.
they satisfy the relations
$\widehat{\theta}^{IA}=\widehat{\theta}(\widehat{\omega}^{IA})\,,\
\widehat{\omega}^{IA}=\widehat{\omega}(\widehat{\theta}^{IA}).$
A number of remarks are in order.
* •
The assumption on the eigenvalues of $P$ is essentially a technical one: in
simulations it does not seem to have a crucial role, but we need it in our
proof of convergence. On the other hand, given any symmetric stochastic
primitive $P$, we cam consider a ’lazy’ version of it $P_{\tau}=(1-\tau)I+\tau
P$ and notice that for $\tau\in(0,1)$ sufficiently small, indeed $P_{\tau}$
will satisfy the assumption on the eigenvalues.
* •
The requirement $\gamma^{(t)}\geq 1/t$ is not new in decentralized algorithms
(see for instance the Robbins-Monro algorithm, introduced in [21]) and serves
the need of maintaining ’active’ the system input for sufficiently long time.
Less classical is the assumption $\gamma^{(t)}\sim\gamma^{(t+1)}$ which is
essentially a request of regularity in the decay of $\gamma^{(t)}$ to $0$.
Possible choices of $\gamma^{(t)}$ satisfying the above conditions are
$\gamma^{(t)}=t^{-\zeta}$ for $\zeta\in(0,1)$, or $\gamma^{(t)}=t^{-1}(\ln
t)^{\alpha}$ for any $\alpha>0$.
* •
The proof (see Appendix A) will also give an estimation on the speed of
convergence: indeed it will be shown that
$||\widehat{\theta}^{(t)}-\widehat{\theta}^{IA}||=O(\gamma^{(t)})$ for
$t\rightarrow\infty$.
* •
Relations in item 2. implies that $\widehat{\theta}^{IA}$ is a local maximum
of the function $L_{N}(\theta,\widehat{\omega}(\theta))$ (see (11)).
### 4.3 Limit behavior
In this section we present results on the behavior of our algorithm for
$N\to+\infty$. All quantities derived so far are indeed function of network
size $N$. In order to emphasize the role of $N$, we will add an index $N$ when
dealing with quantities like $\theta^{\star}$ (e.g.
$\widehat{\theta}^{\text{ML}}_{N}$). Instead we will not add anything to
expressions where there are vectors $\omega$ involved since their dimension is
itself $N$.
Figure 1 shows a sort of concentration of the local maxima of
$L_{N}(\theta,\widehat{\omega}(\theta))$ to a global maximum for large $N$.
Considering that IA converges to a local maximum, this observation would lead
to the conclusion that, for large $N$, the IA resembles the optimal ML
solution. This section provides some results which make rigorous these
considerations.
Notice first that, applying the uniform law of large numbers [22] to the
expression (6), we obtain that, for any compact $K\subseteq\mathbb{R}$, almost
surely
$\lim\limits_{N\to+\infty}\max_{\theta\in
K}\left|L_{N}(\theta,\widehat{\omega}(\theta))-\int_{\mathbb{R}}\mathcal{J}(s,\theta)f(s)\mathrm{d}s\right|=0$
(14)
where
$\mathcal{J}(s,\theta)=-\left(\frac{(s-\theta)^{2}}{2\beta^{2}}+{\mathbf{1}}_{\\{\omega_{j}=\alpha\\}}\left(\frac{(s-\theta)^{2}}{2}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)+\log\frac{1-p}{p}\frac{\beta}{\alpha}\right)\right)+c$
(15)
where $c$ is the same constant as in (6). The limit function
$\int_{\mathbb{R}}\mathcal{J}(s,\theta)f(s)\mathrm{d}s$ turns out to be
differentiable for every value of $\theta$ and to have a unique stationary
point for $\theta=\theta^{*}$ which turns out to be the global minimum.
Unfortunately, this fact by itself does not guarantee that global and local
minima will indeed converge to $\theta^{*}$. In our derivations the properties
of the function $\int_{\mathbb{R}}\mathcal{J}(s,\theta)f(s)\mathrm{d}s$ will
not play any direct role and therefore they will not be proven here. The main
technical result which will be proven in Appendix B is the following:
###### Theorem 2.
Denote by $\mathcal{S}_{N}$ the set of local maxima of
$L(\theta,\widehat{\omega}(\theta))$. Then,
$\lim\limits_{N\to+\infty}\max\limits_{\xi\in{\mathcal{S}}_{N}}|\xi-\theta^{*}|=0$
(16)
almost surely and in mean square sense.
This has an immediate consequence,
###### Corollary 3.
$\lim\limits_{N\to+\infty}\widehat{\theta}^{IA}_{N}=\lim\limits_{N\to+\infty}\widehat{\theta}^{\mathrm{ML}}_{N}=\theta^{\star}$
(17)
almost surely and in mean square sense.
Regarding the classification error, we have instead the following result:
###### Proposition 4.
$\begin{split}\lim_{N\rightarrow+\infty}\
\frac{1}{N}\mathbb{E}d_{H}(\widehat{\omega}^{IA},\omega^{\star})&=\lim_{N\rightarrow+\infty}\
\frac{1}{N}\mathbb{E}d_{H}(\widehat{\omega}^{{\rm ML}},\omega^{\star})\\\
&=q(p,\alpha,\beta)\\\ \end{split}$ (18)
where
$q(p,\alpha,\beta)=(1-p)\mathrm{erfc}\left(\frac{\delta}{\alpha\sqrt{2}}\right)+p\left[1-\mathrm{erfc}\left(\frac{\delta}{\beta\sqrt{2}}\right)\right]$
and
$\mathrm{erfc}(x):=\frac{2}{\sqrt{\pi}}\int_{x}^{+\infty}\mathrm{e}^{-t^{2}}\mathrm{d}t$
is the complementary error function.
These results ensure that the IA performs, in the limit of large number of
units $N$, as the centralized optimal ML estimator. Moreover, they also show,
consistency in the estimation of the parameter $\theta^{\star}$. As expected,
for $N\to+\infty$ the classification error does not go to $0$ since the
increase of measurements is exactly matched by the same increase of variables
to be estimated. Consistency however is obtained when $p$ goes to zero since
we have that $\lim_{p\rightarrow 0}q(p,\alpha,\beta)=0.$ Moreover, notice that
the dependence of function $q$ on the parameters $\alpha$ and $\beta$ is
exclusively through their ratio $\beta/\alpha$. In particular, we have
$\lim_{\beta/\alpha\rightarrow+\infty}q(p,\alpha,\beta)=0\qquad\lim_{\beta/\alpha\rightarrow
1}q(p,\alpha,\beta)=1.$
## 5 Simulations
In this section, we propose some numerical simulations. We test our algorithm
for different graph architectures and dimensions, and we compare it with the
IML and EM algorithms. Our goal is to give evidence of the theoretical
results’ validity and also to evaluate cases that are not included in our
analysis: the good numerical outcomes we obtain suggest that convergence
should hold in broader frameworks. The numerical setting for our simulations
is now presented.
Model: the sensors perform measurements according to the model (1) with
$\theta^{\star}=0$, $\alpha=0.3$, $\beta=10$; the prior probability
$\mathbb{P}(\omega^{\star}_{i}=\beta)$ is equal to $p=0.25$.
Communication architectures: given a strongly connected symmetric graph
$\mathcal{G}=({\mathcal{V}},\mathcal{E})$, we use the so-called Metropolis
random walk construction for $P$ (see [23]) which amounts to the following: if
$i\neq j$
$P_{ij}=\left\\{\begin{array}[]{ll}0&{\rm if}\,(i,j)\not\in\mathcal{E}\\\
\left(\max\\{\mathrm{deg}(i)+1,\mathrm{deg}(j)+1\\}\right)^{-1}&{\rm
if}\,(i,j)\in\mathcal{E}\end{array}\right.$
where $\mathrm{deg}(i)$ denotes the degree (the number of neighbors) of unit
$i$ in the graph $\mathcal{G}$. $P$ constructed in this way is automatically
irreducible and aperiodic.
We consider the following topologies:
1. 1.
Complete graph: $P_{ij}=\frac{1}{N}$ for every $i,j=1,\dots,N$; it actually
corresponds to the centralized case.
2. 2.
Ring: $N$ agents are disposed on a circle, and each agent communicates with
its first neighbor on each side (left and right). The corresponding circulant
symmetric matrix $P$ is given by $P_{ij}=\frac{1}{3}$ for every
$i=2,\dots,N-1$ and $j\in\\{i-1,i,i+1\\}$; $P_{11}=P_{12}=P_{1N}=\frac{1}{3}$;
$P_{N1}=P_{NN-1}=P_{NN}=\frac{1}{3}$; $P_{ij}=0$ elesewhere.
3. 3.
Torus-grid graph: sensors are deployed on a two dimensional grid and are each
connected with their four neighbors; the last node of each row of the grid is
connected with the first node of the same row, and analogously on columns, so
that a torus is obtained. The so-obtained graph is regular.
4. 4.
Random Geometric Graph with radius $r=0.3$: sensors are deployed in the square
$[0,1]\times[0,1]$, their positions being randomly generated with a uniform
distribution; links are switched on between two sensors whenever the distance
is less than $r$. We only envisage connected realizations.
From Theorem 1, $P$ is required to possess positive eigenvalues: our intuition
is that these hypotheses, that are useful to prove the convergence of the IA,
are not really necessary. We test this conjecture on the ring graph, whose
eigenvalues are known [24] to be
$\lambda_{m}=\frac{1}{3}\left(1+2\cos\left(\frac{2\pi
m}{N}\right)\right),\quad m=0,\ldots,N-1$ and which are not necessarily
positive.
Algorithms: We implement and compare the following algorithms: IA with
$\gamma^{(t)}=1/t^{\zeta}$ for different choices of
$\zeta\in\\{0.5,0.7,0.9\\}$, and the two centralized iterative algorithms IML,
and EM described in Section 3.3.
(a) Complete graph.
(b) Ring.
(c) Grid.
(d) Random geometric graph.
Figure 2: Relative classification error
Outcomes: we show the performance of the aforementioned algorithms in terms of
classification error and of mean square error on the global parameter, in
function of the number of sensors $N$. All the outcomes are obtained averaging
over $400$ Monte Carlo runs.
We observe that the classification error (Figure 2) converges for $N\to\infty$
for all the considered algorithms. On the other hand, when $N$ is small, IA
performs better than IML and EM, no matter which graph topology has been
chosen: this suggests that decentralization is then not a drawback for IA.
Moreover, for smaller $\gamma^{(t)}$ (i.e., slowing down the procedure), we
obtain better IA performance in terms of classification. Nevertheless, this is
not universally true: in other simulations, in fact, we have noticed that if
$\gamma^{(t)}$ is too small, the performance are worse. This is not
surprising, since $\gamma^{(t)}$ determines the weights assigned to the
consensus and input driven parts, whose contributions must be somehow balanced
in order to obtain the best solution. An important point that we will study in
future is the optimization of $\gamma^{(t)}$, whose choice may in turn depend
on the graph topology and on the weights assigned in the matrix $P$.
Analogous considerations can be done for the mean square error on $\theta$:
when $N$ increases, the mean square error decays to zero.
We remark that convergence is numerically shown also for the ring topology,
which is not envisaged by our theoretical analysis. Hence, our guess is that
convergence should be proved even under weaker hypotheses on matrix $P$.
For the interested reader, a graphical user interface of our algorithm is
available and downloadable on
http://calvino.polito.it/$\sim$fosson/software.html.
## 6 Concluding remarks
In this paper, we have presented a fully distributed algorithm for the
simultaneous estimation and classification in a sensor network, given from
noisy measurements. The algorithm only requires the local cooperation among
units in the network. Numerical simulations show remarkable performance. The
main contribution includes the convergence of the algorithm to a local maximum
of the centralized ML estimator. The performance of the algorithm has been
also studied when the network size is large, proving that the solution of the
proposed algorithm concentrates around the classical ML solution.
Different variants are possible, for example the generalization to multiple
classes with unknown prior probabilities should be inferred. The choice of
sequence $\\{\gamma^{(t)}\\}_{t\in\mathbb{N}}$ is critical, since it
influences both convergence time and final accuracy; the determination of a
protocol for the adaptive search of sequence
$\\{\gamma^{(t)}\\}_{t\in\mathbb{N}}$ is left for a future work.
## 7 Acknowledgment
The authors wish to thank Sandro Zampieri for bringing the problem to our
attention and Luca Schenato for useful discussions. F. Fagnani and C. Ravazzi
further acknowledge the financial support provided by MIUR under the PRIN
project no. 20087W5P2K.
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## Appendix A Proof of Theorem 1
Consider the discrete-time dynamical system defined by the update equations
(12) and (13): the proof of its convergence is obtained through intermediate
steps.
1. 1.
First, we show that, for sufficiently large $t$, vectors
$\mu^{(t)},\nu^{(t)}$, and $\widehat{\theta}^{(t)}$ are close to consensus
vectors and we prove their convergence, assuming $\widehat{\omega}^{(t)}$ has
already stabilized.
2. 2.
Second, we prove the stabilization of $\widehat{\omega}^{(t)}$ in finite time,
by modelling the system in (12) and (13) as a switching dynamical system.
3. 3.
Finally, combining these facts together we conclude the proof.
### A.1 Towards consensus
We start with some notation: let
$\Omega:=I-N^{-1}\mathbbm{1}\mathbbm{1}^{\mathsf{T}}$; given
$x\in\mathbb{R}^{\mathcal{V}}$, let
$\overline{x}:=N^{-1}\mathbbm{1}^{\mathsf{T}}x$ so that
$x=\overline{x}\mathbbm{1}+\Omega x$.
Given a bounded sequence $u^{(t)}\in\mathbb{R}^{N}$, consider the dynamics
$\begin{split}x^{(t+1)}=\left(1-\gamma^{(t)}\right)Px^{(t)}+\gamma^{(t)}u^{(t)}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ t\in\mathbb{N}\end{split}$ (19)
where $x^{(0)}$ is any fixed vector, and where, we recall the standing
assumptions,
1. (a)
$\gamma^{(t)}\in(0,1)$, $\gamma^{(t)}\geq 1/t$, $\gamma^{(t)}\searrow 0$ and
$\gamma^{(t)}=\gamma^{(t+1)}+o(\gamma^{(t+1)})$ for $t\to+\infty$;
2. (b)
$P\in\mathbb{R}_{+}^{\mathcal{V}\times\mathcal{V}}$ is a stochastic,
symmetric, primitive matrix with positive eigenvalues.
A useful fact consequence of the assumptions on $\gamma^{(t)}$, is the
following:
$\prod\limits_{s=t_{0}}^{t-1}(1-\gamma^{(s)})\leq
e^{-\sum\limits_{s=t_{0}}^{t-1}1/s}\leq\frac{t_{0}}{t}\leq t_{0}\gamma^{(t)}$
(20)
for any choice of $t\geq t_{0}>0$.
On the other hand, as a consequence of the assumptions of $P$ (see [25]) we
have that $P^{t}\to N^{-1}\mathbbm{1}\mathbbm{1}^{T}$, or equivalently that
$P^{t}\Omega\to 0$ for $t\to+\infty$. More precisely, we can order the
eigenvalues of $P$ as $1=\mu_{1}>\mu_{2}\geq\cdots\geq\mu_{N}\geq 0$, and we
have that $||P^{t}\Omega||\leq\mu_{2}^{t}$.
###### Lemma 5.
It holds
$\Omega x^{(t)}=O(\gamma^{(t)})\,,\quad{\rm for}\;t\to+\infty.$
###### Proof.
From (19) and the fact that $\Omega P=P\Omega$ we get, for any fixed $t_{0}$
and $t\geq t_{0}$,
$\Omega x^{(t+1)}=\prod_{s=t_{0}}^{t}\left(1-\gamma^{(s)}\right)P^{t}\Omega
x^{(t_{0})}+\sum_{s=t_{0}}^{t}\prod_{k=s+1}^{t}\left(1-\gamma^{(k)}\right)\gamma^{(s)}P^{t-s}\Omega
u^{(s)}.$ (21)
This yields
$\displaystyle||\Omega x^{(t+1)}||_{2}$
$\displaystyle\leq\prod_{s=t_{0}}^{t}\left(1-\gamma^{(s)}\right)||\Omega
x^{(t_{0})}||_{2}+\sum_{s=t_{0}}^{t}\prod_{k=s+1}^{t}\left(1-\gamma^{(k)}\right)\gamma^{(s)}|\mu_{2}|^{t-s}||u^{(s)}||_{2}$
$\displaystyle\leq\prod_{s=t_{0}}^{t}\left(1-\gamma^{(s)}\right)||\Omega
x^{(t_{0})}||_{2}+K\sum_{s=t_{0}}^{t}\prod_{k=s+1}^{t}\left(1-\gamma^{(k)}\right)\gamma^{(s)}|\mu_{2}|^{t-s}$
(22)
with $K:=\max_{s}||u^{(s)}||_{2}$.
Fix now $0<\varepsilon<1-|\mu_{2}|$ and let $t_{0}\in\mathbb{N}$ be such that
$\frac{\gamma^{(t+1)}}{\gamma^{(t)}}\in(1-\varepsilon,1)$ for all $t\geq
t_{0}$. Hence, for $t\geq s\geq t_{0}$, we have that
$\gamma^{(s)}<\frac{\gamma^{(t)}}{(1-\varepsilon)^{t-s}}$. Consider now the
estimation (A.1) with this choice of $t_{0}$. We get
$\displaystyle||\Omega x^{(t+1)}||_{2}$
$\displaystyle\leq\prod_{s=t_{0}}^{t}\left(1-\gamma^{(s)}\right)||\Omega
x^{(t_{0})}||_{2}+K\gamma^{(t)}\sum_{s=t_{0}}^{t}\left(\frac{|\mu_{2}|}{1-\varepsilon}\right)^{t-s}$
$\displaystyle\leq\prod_{s=t_{0}}^{t}\left(1-\gamma^{(s)}\right)||\Omega
x^{(t_{0})}||_{2}+\frac{K\gamma^{(t)}}{1-\frac{|\mu_{2}|}{1-\varepsilon}}.$
Using now (20) the proof is completed. ∎
###### Proposition 6.
If $\exists\ t_{0}\in\mathbb{N}$ s.t. $u^{(t)}=u$ $\forall t\geq t_{0}$ then
$\lim_{t\rightarrow+\infty}x^{(t)}=\overline{u}\mathbbm{1}.$
###### Proof.
Write $x^{(t)}=\overline{x}^{(t)}\mathbbm{1}+\Omega x^{(t)}$ and notice that
from Lemma 5 it is sufficient to prove that
$\lim_{t\rightarrow+\infty}\overline{x}^{(t)}\mathbbm{1}=\overline{u}\mathbbm{1}.$
From (19) and the fact that ${\mathbbm{1}}^{T}P={\mathbbm{1}}^{T}$, we obtain
$\overline{x}^{(t)}-\overline{u}=\prod_{s=t_{0}}^{t-1}(1-\gamma^{(s)})(\overline{x}^{(s)}-\overline{u})$
which goes to zero from the non-summability of $\gamma^{(t)}$. ∎
We now apply these results to the analysis of $\widehat{\theta}^{(t)}$. We
start with a representation result.
###### Lemma 7.
It holds, for $t\to+\infty$,
$\widehat{\theta}^{(t)}=\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}\mathbbm{1}+\frac{1}{\bar{\nu}^{(t)}}\Omega\left(\mu^{(t)}-\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}\nu^{(t)}\right)+o\left(\gamma^{(t)}\right).$
(23)
###### Proof.
For any $i\in\mathcal{V}$,
$\displaystyle\frac{\mu_{i}^{(t)}}{\nu_{i}^{(t)}}-\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}$
$\displaystyle=\frac{\mu_{i}^{(t)}}{\nu_{i}^{(t)}}-\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}+\frac{\mu_{i}^{(t)}}{\bar{\nu}^{(t)}}-\frac{\mu_{i}^{(t)}}{\bar{\nu}^{(t)}}$
$\displaystyle=\frac{\mu_{i}^{(t)}-\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}+\mu_{i}^{(t)}\left(\frac{1}{\nu_{i}^{(t)}}-\frac{1}{\bar{\nu}^{(t)}}\right)$
$\displaystyle=\frac{1}{\bar{\nu}^{(t)}}\left(\Omega\mu^{(t)}\right)_{i}-\frac{\mu_{i}^{(t)}}{\nu_{i}^{(t)}\bar{\nu}^{(t)}}\left(\Omega\nu^{(t)}\right)_{i}.$
It follows from Lemma 5 that
$\mu^{(t)}=\bar{\mu}^{(t)}\mathbbm{1}+O(\gamma^{(t)})$ and
$\nu^{(t)}=\bar{\nu}^{(t)}\mathbbm{1}+O(\gamma^{(t)})$ for $t\to+\infty$. This
and the fact that $\bar{\nu}^{(t)}$ is bounded away from $0$ (indeed
$\bar{\nu}^{(t)}\geq\alpha^{-2}$ for all $t>0$), yields
$\displaystyle\frac{\mu_{i}^{(t)}}{\nu_{i}^{(t)}\bar{\nu}^{(t)}}\left(\Omega\nu^{(t)}\right)_{i}$
$\displaystyle=\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}\left[\frac{\left(\Omega\nu^{(t)}\right)_{i}}{\bar{\nu}^{(t)}}\right]\left(1+O\left(\gamma^{(t)}\right)\right)$
from which thesis follows. ∎
We can now present our first convergence result.
###### Corollary 8.
It holds, for $t\to+\infty$,
$\bar{\widehat{\theta}}^{(t)}=\frac{\bar{\mu}^{(t)}}{\bar{\nu}^{(t)}}+o(\gamma^{(t)})\,,\quad\Omega\widehat{\theta}^{(t)}=O(\gamma^{(t)}).$
###### Proof.
Both relations are obtained from (23). The first one is immediate. The second
one follows from Lemma 5 and the fact that $\bar{\nu}^{(t)}$ stays bounded
away from $0$. ∎
Corollary 8 says that the estimate $\widehat{\theta}^{(t)}$ is close to a
consensus for sufficiently large $t$. Something more precise can be stated if
we know that if $\widehat{\omega}^{(t)}$ stabilizes at finite time as
explained in the next result.
###### Corollary 9.
If $\exists\ t_{0}\in\mathbb{N}$ s.t.
$\widehat{\omega}^{(t)}=\widehat{\omega}^{IA}$ $\forall t\geq t_{0}$ then
$\lim_{t\rightarrow+\infty}\widehat{\theta}^{(t)}=\widehat{\theta}(\widehat{\omega}^{IA})=\frac{\sum_{i\in\mathcal{V}}{y_{i}}{\left[\widehat{\omega}^{IA}_{i}\right]^{-2}}}{\sum_{i\in\mathcal{V}}{\left[\widehat{\omega}^{IA}_{i}\right]^{-2}}}\mathbbm{1}.$
###### Proof.
Proposition 6 guarantees that $\mu^{(t)}$ and $\nu^{(t)}$ converge to
$\frac{1}{N}\sum_{i\in\mathcal{V}}y_{i}[\widehat{\omega}^{IA}_{i}]^{-2}\mathbbm{1}$
and
$\frac{1}{N}\sum_{i\in\mathcal{V}}[\widehat{\omega}^{IA}_{i}]^{-2}\mathbbm{1}$,
respectively. This yields the thesis. ∎
### A.2 Stabilization of $\widehat{\omega}^{(t)}$
We are going to prove that vector $\widehat{\omega}^{(t)}$ almost surely
stabilizes in finite time: this, by virtue of previous considerations will
complete our proof. To prove this fact will take lots of effort and will be
achieved through several intermediate steps.
We start observing that, since $\widehat{\omega}^{(t)}$ can only assume values
in a finite set, equations in (12) and (13) can be conveniently modeled by a
switching system as shown below.
For reasons which will be clear below, in this subsection we will replace the
configuration space $\\{\alpha,\beta\\}^{\mathcal{V}}$ with the augmented
state space $\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}$. If
$\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}$, define
$\displaystyle\Theta_{\omega}=\\{x\in\mathbb{R}^{\mathcal{V}}:$
$\displaystyle|x_{i}-y_{i}|<\delta,\text{if }\omega_{i}=\alpha,x_{i}\geq
y_{i}+\delta,\text{if }\omega_{i}=\beta+,x_{i}\leq y_{i}-\delta,\text{if
}\omega_{i}=\beta-\\}.$
We clearly have
$\mathbb{R}^{\mathcal{V}}=\bigcup_{\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}}\Theta_{\omega}$.
On each $\Theta_{\omega}$ the dynamical system is linear. Indeed, define the
maps
$f_{\omega}:\mathbb{R}\times\mathbb{R}^{\mathcal{V}}\rightarrow\mathbb{R}^{\mathcal{V}}$
and
$g_{\omega}:\mathbb{R}\times\mathbb{R}^{\mathcal{V}}\rightarrow\mathbb{R}^{\mathcal{V}}$
by
$\displaystyle[f_{\omega}(t,x)]_{i}$
$\displaystyle=(1-\gamma^{(t)})[Px^{(t)}]_{i}+\gamma^{(t)}\frac{y_{i}}{\omega_{i}^{2}}$
$\displaystyle[g_{\omega}(t,x)]_{i}$
$\displaystyle=(1-\gamma^{(t)})[Px^{(t)}]_{i}+\gamma^{(t)}\frac{1}{\omega_{i}^{2}}$
where, conventionally, $\omega_{i}^{2}=\beta^{2}$ if
$\omega_{i}=\beta+,\beta-$. Then, if
$\widehat{\theta}^{(t)}\in\Theta_{\omega}$, (12a), (12b), and (12c) can be
written as
$\mu^{(t+1)}=f_{{{\omega}}}(t,\mu^{(t)})\qquad\nu^{(t+1)}=g_{{{\omega}}}(t,\nu^{(t)})$
$\widehat{\theta}^{(t+1)}_{i}={\mu_{i}^{(t+1)}}/{\nu_{i}^{(t+1)}}.$
Notice that this is a closed-loop switching system, since the switching policy
is determined by $\widehat{\theta}^{(t)}$. It is clear that the stabilization
of $\widehat{\omega}^{(t)}$ is equivalent to the fact that there exist an
${\omega}\in\\{\alpha,\beta+,\beta-\\}^{N}$ and a time $\widetilde{t}$ such
that $\widehat{\theta}^{(t)}\in\Theta_{{\omega}}$ for all
$t\geq\widetilde{t}$.
From Corollary 9 candidate limit points for $\widehat{\theta}^{(t)}$ are
${\widehat{\theta}}(\omega)\mathbbm{1}=\frac{\sum_{i\in\mathcal{V}}y_{i}\omega_{i}^{-2}}{\sum_{i\in\mathcal{V}}\omega_{i}^{-2}}\mathbbm{1}\qquad\omega\in\\{\alpha,\beta+,\beta-\\}^{N}.$
Also, from Proposition 8, the dynamics can be conveniently analysed by
studying it in a neighborhood of the line
$\Lambda=\\{\lambda\mathbbm{1}|\lambda\in\mathbb{R}\\}$.
We now make an assumption which holds almost everywhere with respect to the
choice of $y_{i}$’s and, consequently, does not entail any loss of generality
in our proof.
ASSUMPTION:
* •
$y_{i}-y_{j}\not\in\\{0,\pm\delta,\pm 2\delta\\}$ for all $i\neq j$;
* •
${\widehat{\theta}}(\omega)-y_{i}\not\in\\{\pm\delta\\}$ for all
$\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}$ and for all $i$.
This assumption has a number of consequences which will be used later on:
1. (C1)
${\widehat{\theta}}(\omega){\mathbbm{1}},y_{i}{\mathbbm{1}}\in\bigcup_{\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}}{\rm
int}(\Theta_{\omega})$ for all
$\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}$ and for all
$i\in\mathcal{V}$;
2. (C2)
$\Lambda\cap\bar{\Theta}_{\omega}\cap\bar{\Theta}_{\omega^{\prime}}\cap\bar{\Theta}_{\omega^{\prime\prime}}=\emptyset$
for any triple of distinguished
$\omega,\omega^{\prime},\omega^{\prime\prime}$. In other terms, $\Lambda$
always crosses boundaries among regions $\Theta_{\omega}$ at internal point of
faces.
We now introduce some further notation, which will be useful in the rest of
the paper.
$\displaystyle\Theta^{\epsilon}:=\\{x\in\mathbb{R}^{\mathcal{V}}:||\Omega
x||_{2}<\epsilon\\},\quad\Theta^{\epsilon}_{\omega}:=\Theta^{\epsilon}\cap\Theta_{\omega}$
$\displaystyle\Gamma:=\\{\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}:\
\Theta_{\omega}\cap\Lambda\neq\emptyset\\}.$
For any $\omega\in\Gamma$ consider
$\Pi_{\omega}=\\{\pi=\bar{\Theta}_{\omega}\cap\bar{\Theta}_{\omega^{\prime}}:\
\mathrm{d_{H}}(\omega,\omega^{\prime})=1,\ \pi\cap\Lambda=\emptyset\\}$
and define
$\sigma_{\omega}:=\min_{\pi\in\Pi_{\omega}}\mathrm{d}(\Theta_{\omega}\cap\Lambda,\pi)>0$111
$\mathrm{d}(\Theta_{\omega}\cap\Lambda,\pi)$ denotes the distance between the
two sets $\Theta_{\omega}\cap\Lambda$ and the set $\pi$.
In the sequel, we will use the natural ordering on $\Lambda$: given the sets
$X,Y\subseteq\Lambda$, $X<Y$ means that $x<y$ for all $x\in X$ and $y\in Y$.
###### Definition 10.
Given two elements $\omega,\ \omega^{\prime}\in\Gamma$, we say that
$\omega^{\prime}$ is the future-follower of $\omega$ (or also that $\omega$ is
the past-follower of $\omega^{\prime}$) if the following happens:
1. (A)
There exists $i_{0}$ such that $\omega_{i}=\omega^{\prime}_{i}\text{ for all
}\,i\neq i_{0}\text{ and }\omega_{i_{0}}\neq\omega^{\prime}_{i_{0}}$;
2. (B)
$\Theta_{\omega}\cap\Lambda<\Theta_{\omega^{\prime}}\cap\Lambda$.
Notice that, in order for $\omega$ and $\omega^{\prime}$ to satisfy definition
above, it must necessarily happen that either $\omega_{i_{0}}=\alpha$ and
$\omega^{\prime}_{i_{0}}=\beta+$, or $\omega_{i_{0}}=\beta-$ and
$\omega^{\prime}_{i_{0}}=\alpha$. Given $\omega\in\Gamma$, its future-follower
(if it exists) will be denoted by $\omega^{+}$. It is clear that (because of
property (C2) described above) that we can order elements in $\Gamma$ as
$\omega^{1},\omega^{2},\dots,\omega^{M}$ in such a way that
$\omega^{r+1}=(\omega^{r})^{+}$ for every $r=1,\dots,M-1$.
Given $\omega\in\Gamma$, consider the following subsets of $\mathbb{R}^{N}$
(see Fig. 3):
$\displaystyle\mathcal{M}_{\omega}^{\epsilon}$
$\displaystyle:=\left\\{x\in\Theta_{\omega}^{\epsilon}:\overline{x}\mathbbm{1}+\Omega
z\in\Theta_{\omega}^{\epsilon},\leavevmode\nobreak\ \forall
z:||z||_{2}<\epsilon\right\\}$
$\displaystyle\mathcal{L}_{\omega,\omega^{+}}^{\epsilon}$
$\displaystyle:=\left\\{x\in\Theta^{\epsilon}:\mathcal{M}_{\omega}^{\epsilon}\cap\Lambda<\bar{x}<\mathcal{M}_{\omega^{+}}^{\epsilon}\cap\Lambda\right\\}.$
(with the implicit assumption that
$\mathcal{L}_{\omega,\omega^{+}}^{\epsilon}=\emptyset$ if $\omega^{+}$ does
not exist.) We clearly have
$\Theta^{\epsilon}=\bigcup_{\omega\in\Gamma}\mathcal{M}_{\omega}^{\epsilon}\cup\mathcal{L}_{\omega,\omega^{+}}^{\epsilon}$.
Figure 3: Given the couple $(\omega,\omega^{\prime})$ the sets
$\mathcal{L}_{\omega,\omega^{\prime}}^{\epsilon}$ and
$\mathcal{M}_{\omega}^{\epsilon}$ are visualized.
Notice that, because of property (C1), we can always choose
$\epsilon_{0}\in(0,\min_{\omega\in\Gamma}\sigma_{\omega})$ such that
$\displaystyle{\widehat{\theta}}(\omega)\mathbbm{1},y_{i}\mathbbm{1}\in\bigcup_{\omega^{\prime}\in\Gamma}\mathcal{M}_{\omega^{\prime}}^{\epsilon_{0}}\qquad\forall\omega\in\Gamma,\forall
i\in\mathcal{V}.$
This implies that there exists $\tilde{c}>0$ such that
$\mathrm{d}\left(\bigcup_{\omega^{\prime}\in\Gamma}\partial_{\Lambda}\left(\mathcal{M}_{\omega^{\prime}}^{\epsilon}\cap\Lambda\right),\\{{\widehat{\theta}}(\omega),y_{i}\\}\right)\geq\tilde{c},\quad\forall\epsilon\leq\epsilon_{0}$
(24)
where $\partial_{\Lambda}(\cdot)$ denotes the boundary of a set in the
relative topology of $\Lambda$.
Fix now $\epsilon\leq\epsilon_{0}$ and choose $t_{\epsilon}$ such that
$\widehat{\theta}^{(t)}\in\Theta^{\epsilon}$ for all $t\geq t_{\epsilon}$ (it
exists by Corollary 8). From now on we consider times $t\geq t_{\epsilon}.$
Our aim is to prove through intermediate steps the following facts
* F1)
if ${\widehat{\theta}}(\omega)\in\mathcal{M}^{\epsilon}_{\omega}$ then
$\mathcal{M}^{\epsilon}_{\omega}$ is an _asymptotically invariant_ set for
$\widehat{\theta}^{(t)}$, namely, when $t$ is sufficiently large, if
$\widehat{\theta}^{(t)}\in\mathcal{M}^{\epsilon}_{\omega}$ then
$\widehat{\theta}^{(t+1)}\in\mathcal{M}^{\epsilon}_{\omega}$;
* F2)
if ${\widehat{\theta}}(\omega)\notin\mathcal{M}^{\epsilon}_{\omega}$ then
$\widehat{\theta}^{(t)}\notin\mathcal{M}^{\epsilon}_{\omega}$ for $t$
sufficiently large;
* F3)
$\widehat{\theta}^{(t)}\notin\bigcup_{\omega\in\\{\alpha,\beta+,\beta-\\}^{\mathcal{V}}}\mathcal{L}^{\epsilon}_{\omega,\omega^{+}}$
for $t$ sufficiently large.
#### F1) Asymptotic invariance of $\mathcal{M}_{\omega}^{\epsilon}$ when
${\widehat{\theta}}(\omega)\mathbbm{1}\in\mathcal{M}^{\epsilon}_{\omega}$
###### Lemma 11.
If $\widehat{\theta}^{(t)}\in\Theta_{\omega}$ then there exists
$c^{(t)}\in[\alpha^{2}/\beta^{2},\beta^{2}/\alpha^{2}]$ and
$r^{(t)}=o(\gamma^{(t)})$ for $t\to+\infty$ such that
$\overline{\widehat{\theta}}^{(t+1)}=\overline{\widehat{\theta}}^{(t)}+c^{(t)}\gamma^{(t)}\left({\widehat{\theta}}(\omega)-\overline{\widehat{\theta}}^{(t)}\right)+r^{(t)}$
(25)
###### Proof.
If $\widehat{\theta}^{(t)}\in\Theta_{\omega}$ then
$\displaystyle\frac{\overline{\mu}^{(t+1)}}{\overline{\nu}^{(t+1)}}-\frac{\overline{\mu}^{(t)}}{\overline{\nu}^{(t)}}$
$\displaystyle=\frac{(1-\gamma^{(t)})\overline{\mu}^{(t)}+\gamma^{(t)}N^{-1}\sum_{i=1}^{N}y_{i}\omega_{i}^{-2}}{(1-\gamma^{(t)})\overline{\mu}^{(t)}+\gamma^{(t)}N^{-1}\sum_{i=1}^{N}\omega_{i}^{-2}}-\frac{\overline{\mu}^{(t)}}{\overline{\nu}^{(t)}}$
$\displaystyle=\frac{\overline{\nu}^{(t)}\gamma^{(t)}N^{-1}\sum_{i=1}^{N}y_{i}\omega_{i}^{-2}-\overline{\mu}^{(t)}\gamma^{(t)}N^{-1}\sum_{i=1}^{N}\omega_{i}^{-2}}{\overline{\nu}^{(t+1)}\overline{\nu}^{(t)}}$
$\displaystyle=\gamma^{(t)}\frac{N^{-1}\sum_{i=1}^{N}\omega_{i}^{-2}}{\overline{\nu}^{(t+1)}}{\left({\widehat{\theta}}(\omega)-\frac{\overline{\mu}^{(t)}}{\overline{\nu}^{(t)}}\right)}$
Choosing
$c^{(t)}=\frac{N^{-1}\sum_{i=1}^{N}\omega_{i}^{-2}}{\overline{\nu}^{(t+1)}}\in[\alpha^{2}/\beta^{2},\beta^{2}/\alpha^{2}]$
and using Corollary 8 thesis easily follows. ∎
###### Proposition 12 (Proof of F1)).
There exists $t^{\prime}\geq t_{\epsilon}$ such that, if
${\widehat{\theta}}(\omega)\mathbbm{1}\in\Theta_{\omega}$, then
$\widehat{\theta}^{(t)}\in\mathcal{M}^{\epsilon}_{\omega}\;\Rightarrow\;\widehat{\theta}^{(t+1)}\in\mathcal{M}^{\epsilon}_{\omega}\quad\forall
t\geq t^{\prime}\,.$
###### Proof.
Consider the relation (25). If
$\widehat{\theta}^{(t)}\in\mathcal{M}^{\epsilon}_{\omega}$ and if $t$ is large
enough so that $c^{(t)}\gamma^{(t)}<1$ , we have, by convexity, that
$z:=\overline{\widehat{\theta}}^{(t)}+c^{(t)}\gamma^{(t)}\left({\widehat{\theta}}(\omega)-\overline{\widehat{\theta}}^{(t)}\right)\in\mathcal{M}^{\epsilon}_{\omega}.$
Moreover, because of (24) and the fact that $c^{(t)}$ is bounded away from
$0$, there exists $c^{\prime}>0$ such that
$\mathrm{d}(z,\partial(\mathcal{M}^{\epsilon}_{\omega}\cap\Lambda))\geq
c^{\prime}\gamma^{(t)}$. Proof is then completed by selecting $t^{\prime}\geq
t_{\epsilon}$ such that $c^{(t)}\gamma^{(t)}<1$ and
$|r(t)|<c^{\prime}\gamma^{(t)}/2$ for all $t\geq t^{\prime}$. ∎
### F2) Transitivity of $\mathcal{M}^{\epsilon}_{\omega}$ when
${\widehat{\theta}}(\omega)\mathbbm{1}\notin\mathcal{M}_{\omega}^{\epsilon}$
Our next goal is to prove that if
${\widehat{\theta}}(\omega)\mathbbm{1}\notin\mathcal{M}^{\epsilon}_{\omega}$,
then, at a certain time $t$, $\widehat{\theta}^{(t)}$ will definitively be
outside $\mathcal{M}^{\epsilon}_{\omega}$. A technical lemma based on
convexity arguments is required.
###### Lemma 13.
Let $\omega\in\Gamma$ be such that there exists its future-follower
$\omega^{+}$. Then,
$\begin{array}[]{lcl}{\widehat{\theta}}(\omega)\mathbbm{1}>\Theta_{\omega}\cap\Lambda&\Rightarrow&{\widehat{\theta}}(\omega^{+})\mathbbm{1}>\Theta_{\omega}\cap\Lambda\\\
{\widehat{\theta}}(\omega^{+})\mathbbm{1}<\Theta_{\omega^{+}}\cap\Lambda&\Rightarrow&{\widehat{\theta}}(\omega)\mathbbm{1}<\Theta_{\omega^{+}}\cap\Lambda.\end{array}$
###### Proof.
Suppose $\omega_{i}=\omega^{+}_{i},\forall i\neq i_{0}$ and
$\omega_{i_{0}}=\beta-$, $\omega^{+}_{i_{0}}=\alpha$ (the other case can be
treated in an analogous way). Pick $x^{\prime}\in\Theta_{\omega}\cap\Lambda$
and $x^{\prime\prime}\in\Theta_{\omega^{+}}\cap\Lambda$. From
$|x^{\prime\prime}-y_{i_{0}}|<\delta$, and $|x^{\prime}-y_{i_{0}}|>\delta$ it
immediately follows that $x^{\prime\prime}>y_{i_{0}}-\delta,\
x^{\prime}<y_{i_{0}}-\delta$ and, in particular, the fact
$y_{i_{0}}\mathbbm{1}>\Theta_{\omega}\cap\Lambda\,.$ (26)
Notice now that
$\displaystyle{\widehat{\theta}}(\omega^{+})$
$\displaystyle=\frac{y_{i_{0}}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}+\frac{\sum\limits_{i\in\mathcal{V}\setminus{i_{0}}}\frac{y_{i}}{{\omega^{+}_{i}}^{2}}+\frac{y_{i_{0}}}{\beta^{2}}}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}$
$\displaystyle=\frac{y_{i_{0}}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}+\frac{{\widehat{\theta}}(\omega)\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega_{i}}^{2}}}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}$
$\displaystyle=\frac{y_{i_{0}}\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}+\frac{{\widehat{\theta}}(\omega)\left[\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}-\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}\right)\right]}{\sum\limits_{i\in\mathcal{V}}\frac{1}{{\omega^{+}_{i}}^{2}}}.$
In Figures 4 and 5 a picture of the various points is depicted when
${\widehat{\theta}}(\omega)>\Theta_{\omega}\cap\Lambda$.
$\Lambda$$\Theta_{\omega}$$\Theta_{\omega^{+}}$$y_{i_{0}}\mathbbm{1}$${\widehat{\theta}}(\omega^{+})\mathbbm{1}$${\widehat{\theta}}(\omega)\mathbbm{1}$
(a) $y_{i_{0}}<y_{\omega^{+}}<y_{\omega}$.
$\Lambda$$\Theta_{\omega}$$\Theta_{\omega^{+}}$${\widehat{\theta}}(\omega)\mathbbm{1}$${\widehat{\theta}}(\omega^{+})\mathbbm{1}$$\bar{y}_{i_{0}}\mathbbm{1}$
(b) $y_{i_{0}}<{\widehat{\theta}}(\omega^{+})<{\widehat{\theta}}(\omega)$.
Figure 4:
$\omega_{i_{0}}=\beta,{\widehat{\theta}}(\omega)\mathbbm{1}>\Theta_{\omega}\cap\Lambda$
$\Lambda$$\Theta_{\omega}$$\Theta_{\omega^{+}}$${\widehat{\theta}}(\omega)\mathbbm{1}$${\widehat{\theta}}(\omega^{+})\mathbbm{1}$$\bar{y}_{i_{0}}\mathbbm{1}$
Figure 5: ${\widehat{\theta}}(\omega)\mathbbm{1}>\Theta_{\omega}\cap\Lambda$
A convexity argument and the use of (26) now allow to conclude. ∎
###### Proposition 14 (Proof of F2)).
If ${\widehat{\theta}}(\omega)\mathbbm{1}\notin\Theta_{\omega}$, then there
exists $t^{\prime\prime}$ such that
$\widehat{\theta}^{(t)}\notin\Theta_{\omega}^{\epsilon}$ $\forall
t>t^{\prime\prime}$.
###### Proof.
Suppose ${\widehat{\theta}}(\omega)\mathbbm{1}>\Theta_{\omega}\cap\Lambda$
(the case when is $<$ can be treated analogously). Lemma 13 implies that
${\widehat{\theta}}(\omega^{+})\mathbbm{1}>\Theta_{\omega}\cap\Lambda$. Let
$\tilde{c}$ be the constant given in (24) and put
$A:=\\{x\in\Theta^{\epsilon}_{\omega}\cup\Theta^{\epsilon}_{\omega^{+}}\;|\;\overline{x}\leq\alpha:=\min\\{{\widehat{\theta}}(\omega),{\widehat{\theta}}(\omega^{+})\\}-\tilde{c}/2\\}.$
Consider the relation (25) and choose $t_{1}$ in such a way that
$\bar{\widehat{\theta}}^{(t+1)}-\bar{\widehat{\theta}}^{(t)}\leq
c_{2}(\max\\{y_{i}\\}-\min\\{y_{i}\\})\gamma^{(t)}+r(t)<\tilde{c}/2$ (27)
and $|r(t)|<\alpha^{2}\tilde{c}\gamma^{(t)}/4\beta^{2}$ for all $t\geq t_{1}$.
It also follows from (25) that, if for some $t\geq t_{1}$
$\widehat{\theta}^{(t)}\in A$, then,
$\overline{\widehat{\theta}}^{(t+1)}\geq\overline{\widehat{\theta}}^{(t)}+\alpha^{2}{\tilde{c}}\
\gamma^{(t)}/{4\beta^{2}}.$ (28)
Owing to the non-summability of $\gamma^{(t)}$ it follows that if
$\widehat{\theta}^{(t)}$ enters in $\Theta^{\epsilon}_{\omega}$ for some
$t\geq t_{1}$, then, in finite time it will enter into
$A\setminus\Theta^{\epsilon}_{\omega}$ and then it will finally exit $A$. In
particular there must exist $t_{2}\geq t_{1}$ such that
$\overline{\widehat{\theta}}^{(t_{2})}>\alpha$. We now prove that
$\overline{\widehat{\theta}}^{(t_{2})}{\mathbbm{1}}>\Theta_{\omega}$ for every
$t\geq t_{2}$. If not there must exist a first time index $t_{3}>t_{2}$ such
that $\overline{\widehat{\theta}}^{(t_{3})}<\alpha-\tilde{c}$. Because of
(27), it must be that
$\overline{\widehat{\theta}}^{(t_{3}-1)}<\alpha-\tilde{c}/2$ but this
contradicts the fact that on $A$, $\overline{\widehat{\theta}}^{(t)}$ is
increasing (28). ∎
### F3) Transitivity of
$\bigcup_{\omega,\omega^{+}\in\\{\alpha,\beta+\beta-\\}^{N}}\mathcal{L}_{\omega,\omega^{+}}$
We start with the following technical result concerning the general system
(19).
###### Lemma 15.
Let $x^{(t)}$ be the sequence defined in (19) and suppose that there exists a
strictly increasing sequence of switching times
$\\{\tau_{k}\\}_{k=0}^{+\infty}$ such that
$u_{i}^{(s+1)}=u_{i}^{(s)}\quad\forall i\neq i_{0}\quad\text{and}\quad\forall
s\in[\tau_{0},+\infty[$ $u_{i_{0}}^{(s)}=\begin{cases}v^{\prime}&\forall s\in
I^{\prime}:=\bigcup_{k=0}^{+\infty}[\tau_{2k},\tau_{2k+1})\\\
v^{\prime\prime}&\forall s\in
I^{\prime\prime}:=\bigcup_{k=0}^{+\infty}[\tau_{2k+1},\tau_{2k+2}).\end{cases}$
Then, for every $\delta>0$, there exists $\bar{t}_{\delta}$ and two sequences
$a_{\delta}^{(t)}\geq 0$ and $b_{\delta}^{(t)}\leq\delta\gamma^{(t)}$, such
that
$\displaystyle\left(\Omega\left(x^{(t+1)}-x^{(t)}\right)\right)_{i_{0}}\\!\\!\\!=A_{\delta}^{(t)}\gamma^{(t)}\left(v^{\prime}-v^{\prime\prime}\right)+b_{\delta}^{(t)}$
for $t\in I^{\prime}$ with $t\geq\bar{t}_{\delta}$.
###### Proof.
Let $\phi_{j}\in\mathbb{R}^{\mathcal{V}}$ be an orthonormal basis of
eigenvectors for $P$ relative to the eigenvalues
$1=\lambda_{1}>\lambda_{2}\geq\cdots\geq\lambda_{N}\geq 0$. Also assume we
have chosen $\phi_{1}=N^{-1/2}\mathbbm{1}$.
We put
$F^{(t)}:=\frac{\prod_{k=0}^{t}\left(1-\gamma^{(k)}\right)}{\gamma^{(t)}}$
and we notice that
$\frac{F^{(s+1)}}{F^{(s)}}=(1-\gamma^{(s+1)})\frac{\gamma^{(s)}}{\gamma^{(s+1)}}\to
1\,,\;{\rm for}\;s\to+\infty.$
Fix $\epsilon$ in such a way that $\lambda_{2}(1+\epsilon)<1$ and choose
$s_{0}$ such that
$\frac{F^{(s+1)}}{F^{(s)}}\leq 1+\epsilon\,,\;\forall s\geq s_{0}.$
Let $t_{0}\geq s_{0}$ to be fixed later. From (21) we can write
$\displaystyle\Omega(x^{(t+1)}-x^{(t)})=$
$\displaystyle=\prod_{s=t_{0}}^{t-1}\left(1-\gamma^{(s)}\right)\left[\left(1-\gamma^{(t)}\right)P-I\right]P^{t-t_{0}}\Omega
x^{(t_{0})}$
$\displaystyle+\sum_{s=t_{0}}^{t}\prod_{k=s+1}^{t}\left(1-\gamma^{(k)}\right)\gamma^{(s)}P^{t-s}\Omega
u^{(s)}-\sum_{s=t_{0}}^{t-1}\prod_{k=s+1}^{t-1}\left(1-\gamma^{(k)}\right)\gamma^{(s)}P^{t-s-1}\Omega
u^{(s)}v$
$\displaystyle=\prod_{s=t_{0}}^{t-1}\left(1-\gamma^{(s)}\right)\left[\left(1-\gamma^{(t)}\right)P-I\right]P^{t-t_{0}}\Omega
x^{(t_{0})}$
$\displaystyle+\gamma^{(t)}\sum_{s=t_{0}-1}^{t-1}P^{t-s-1}\frac{F^{(t)}}{F^{(s+1)}}\Omega
u^{(s+1)}-\gamma^{(t-1)}\sum_{s=t_{0}}^{t-1}P^{t-s-1}\frac{F^{(t-1)}}{F^{(s)}}\Omega
u^{(s)}$
$\displaystyle=\prod_{s=t_{0}}^{t-1}\left(1-\gamma^{(s)}\right)\left[\left(1-\gamma^{(t)}\right)P-I\right]P^{t}\Omega
x^{(t_{0})}$ (29)
$\displaystyle+(\gamma^{(t)}-\gamma^{(t-1)})\sum_{s=t_{0}}^{t-1}P^{t-s-1}\frac{F^{(t-1)}}{F^{(s)}}\Omega
u^{(s)}+\gamma^{(t)}P^{t-t_{0}}\frac{F^{(t-1)}}{F^{(t_{0})}}\Omega
u^{(t_{0})}$ (30)
$\displaystyle+\gamma^{(t)}\sum_{s=t_{0}}^{t-1}P^{t-s-1}\left(\frac{F^{(t)}}{F^{(s+1)}}-\frac{F^{(t-1)}}{F^{(s)}}\right)\Omega
u^{(s+1)}$ (31)
$\displaystyle+\gamma^{(t)}\sum_{s=t_{0}}^{t-1}P^{t-s-1}\frac{F^{(t-1)}}{F^{(s)}}\Omega\left(u^{(s+1)}-u^{(s)}\right).$
(32)
It follows from the assumptions on $P$, the assumptions on $\gamma^{(t)}$ and
relation (20) that the terms (29) and (30) are both $o(\gamma^{(t)})$ for
$t\to+\infty$. We now estimate (31):
$\displaystyle\left|\left|\sum_{s=t_{0}}^{t-1}P^{t-s-1}\left(\frac{F^{(t)}}{F^{(s+1)}}-\frac{F^{(t-1)}}{F^{(s)}}\right)\Omega
u^{(s+1)}\right|\right|_{2}=$
$\displaystyle\left|\left|\sum_{s=t_{0}}^{t-1}\frac{F^{(t-1)}}{F^{(s)}}\left(\frac{F^{(t)}}{F^{(t-1)}}\frac{F^{(s)}}{F^{(s+1)}}-1\right)P^{t-s-1}\Omega
u^{(s+1)}\right|\right|_{2}\leq$
$\displaystyle\sum_{s=t_{0}}^{t-1}[\lambda_{2}(1+\epsilon)]^{t-s-1}\left|\left(\frac{F^{(s)}}{F^{(s+1)}}-1\right)\right|K\leq$
$\displaystyle\frac{K}{1-\lambda_{2}(1+\epsilon)}\beta_{t_{0}}$ (33)
where
$K=\max||u^{(s)}||_{2}\,,\quad\beta_{t_{0}}:=\sup\limits_{t\geq s\geq
t_{0}}\left|\left(\frac{F^{(t)}}{F^{(t-1)}}\frac{F^{(s)}}{F^{(s+1)}}-1\right)\right|.$
We now concentrate on the component $i_{0}$ of the term (32). Using the
spectral decomposition of $P$ and the assumptions on $u^{(t)}$, we can write,
$\displaystyle\left[\sum_{s=t_{0}}^{t-1}P^{t-s-1}\frac{F^{(t-1)}}{F^{(s)}}\Omega\left(u^{(s+1)}-u^{(s)}\right)\right]_{i_{0}}=$
(34) $\displaystyle\sum\limits_{j\geq
2}(\phi_{j})_{i_{0}}^{2}\sum\limits_{h:t_{0}\leq\tau_{h}\leq
t-1}\lambda_{j}^{t-\tau_{h}}\frac{F(t-1)}{F(\tau_{h}-1)}(-1)^{h}(v^{\prime}-v^{\prime\prime}).$
(35)
If $t\in I^{\prime}$, the above expression can be rewritten as
$\sum\limits_{j\geq
2}(\phi_{j})_{i_{0}}^{2}\sum\limits_{k:t_{0}\leq\tau_{2k}\leq
t-1}\left[\lambda_{j}^{t-\tau_{2k}}\frac{F(t-1)}{F(\tau_{2k}-1)}-\lambda_{j}^{t-\tau_{2k-1}}\frac{F(t-1)}{F(\tau_{2k-1}-1)}\right](v^{\prime}-v^{\prime\prime}).$
Notice that
$\lambda_{j}^{t-\tau_{2k}}\frac{F(t-1)}{F(\tau_{2k}-1)}-\lambda_{j}^{t-\tau_{2k-1}}\frac{F(t-1)}{F(\tau_{2k-1}-1)}=\lambda_{j}^{t-\tau_{2k}}\frac{F(t-1)}{F(\tau_{2k}-1)}\left(1-\lambda_{j}^{\tau_{2k}-\tau_{2k-1}}\frac{F(\tau_{2k}-1)}{F(\tau_{2k-1}-1)}\right)>0$
(we have used the fact that $0\leq\lambda_{j}(1+\epsilon)<1$ for all $j\geq
2$). To complete the proof now proceed as follows. For a fixed $\delta>0$,
choose $t_{0}\geq s_{0}$ in such a way that (33) is below $\delta/2$. Then,
fix $\bar{t}_{\delta}\geq t_{0}$ in such a way that the summation of (29) and
(30) is below $\delta\gamma^{(t)}/2$ for $t\geq\bar{t}_{\delta}$. It is now
sufficient to define
$a_{\delta}^{(t)}:=\sum\limits_{j}(\phi_{j})_{i_{0}}^{2}\sum\limits_{k:t_{0}\leq\tau_{2k}\leq
t-1}\left[\lambda_{j}^{t-\tau_{2k}-1}\frac{F(t-1)}{F(\tau_{2k}-1)}-\lambda_{j}^{t-\tau_{2k-1}}\frac{F(t-1)}{F(\tau_{2k-1}-1)}\right]$
and $b_{\delta}^{(t)}$ equal to the sum of the terms (29), (30), and (31), ∎
###### Proposition 16 (Proof of F3)).
There exists $t^{\prime\prime\prime}\in\mathbb{N}$ such that
$\widehat{\theta}^{(t)}\not\in\bigcup_{\omega,\omega^{\prime}\in\\{\alpha,\beta\\}^{N}}\mathcal{L}_{\omega,\omega^{+}}$
(36)
for all $t>t^{\prime\prime\prime}$.
###### Proof.
In view of the results in Propositions 12 and 14, and the fact that
$\widehat{\theta}^{(t+1)}-\widehat{\theta}^{(t)}$ goes to $0$ for
$t\to+\infty$, if (36) negation of (36) yields that there exists
$\omega\in\Gamma$ such that
$\widehat{\theta}^{(t)}\in\mathcal{L}_{\omega,\omega^{+}}$ for $t$ large
enough. Now, if
$\widehat{\theta}^{(t)}\in\mathcal{L}_{\omega,\omega^{+}}\cap\Theta_{\omega}$
(or if
$\widehat{\theta}^{(t)}\in\mathcal{L}_{\omega,\omega^{+}}\cap\Theta_{\omega^{+}}$)
for $t$ sufficiently large, a straightforward application of (25) would imply
that $\widehat{\theta}^{(t)}$ would necessarily exit
$\mathcal{L}_{\omega,\omega^{+}}$ in finite time. Therefore, it must hold that
$\widehat{\theta}^{(t)}$ keeps switching, for large $t$, between
$\mathcal{L}_{\omega,\omega^{+}}\cap\Theta_{\omega}$ and
$\mathcal{L}_{\omega,\omega^{+}}\cap\Theta_{\omega^{+}}$.
From Lemma 7 and Corollary 8 we can write
$\displaystyle\widehat{\theta}^{(t+1)}-\widehat{\theta}^{(t)}=$
$\displaystyle\left(\frac{{\bar{\widehat{\theta}}}^{(t+1)}}{{\bar{\nu}}^{(t+1)}}-\frac{{\bar{\widehat{\theta}}}^{(t)}}{{\bar{\nu}}^{(t)}}\right){\mathbbm{1}}+\frac{1}{{\bar{\nu}}^{(t)}}\left[\Omega\left(\mu^{(t+1)}-\mu^{(t)}\right)-\frac{{\bar{\mu}}^{(t)}}{{\bar{\nu}}^{(t)}}\Omega\left(\nu^{(t+1)}-\nu^{(t)}\right)\right]+o(\gamma^{(t)})$
Define now
$I^{\prime}:=\\{t\,|\,\widehat{\theta}^{(t)}\in\Theta_{\omega}\\}\,,\quad
I^{\prime\prime}:=\\{t\,|\,\widehat{\theta}^{(t)}\in\Theta_{\omega^{+}}\\}$
and put $v^{\prime}=1/\omega_{i_{0}}^{2}$ and
$v^{\prime\prime}=1/\omega_{i_{0}}^{+2}$. From Lemma 11, and applying Lemma 15
to $\mu^{(t)}$ and $\nu^{(t)}$, we get that for $t\in I^{\prime}$ sufficiently
large, it holds
$\widehat{\theta}^{(t+1)}_{i_{0}}-\widehat{\theta}^{(t)}_{i_{0}}=c^{(t)}\gamma^{(t)}(\bar{y}_{\omega}-\overline{\widehat{\theta}}^{(t)})+\frac{1}{\bar{\nu}^{(t)}}\gamma^{(t)}a^{(t)}_{\delta}\left(v^{\prime}-v^{\prime\prime}\right)(y_{i_{0}}-\overline{\widehat{\theta}}^{(t)})+a^{(t)}_{\delta}+r^{(t)}.$
(37)
If ${\widehat{\theta}}(\omega)>\Theta_{\omega}\cap\Lambda$, then also, by
Lemma 13, $\bar{y}_{\omega^{+}}>\Theta_{\omega}\cap\Lambda$. This, using (25),
would imply that $\widehat{\theta}^{(t)}$ would necessarily exit
$\mathcal{L}_{\omega,\omega^{+}}$ in finite time. Therefore, we must have
${\widehat{\theta}}(\omega)<\Theta_{\omega}\cap\Lambda$. Hence,
$\bar{y}_{\omega}-\overline{\widehat{\theta}}^{(t)}<0$. Moreover, it is easy
to check that in any case
$\left(v^{\prime}-v^{\prime\prime}\right)(y_{i_{0}}-\overline{\widehat{\theta}}^{(t)})<0$.
Recall now the definition of the constant $\tilde{c}$ in (24) and notice that,
since $\widehat{\theta}^{(t)}\in\mathcal{L}_{\omega,\omega^{+}}$,
$c^{(t)}\gamma^{(t)}(\bar{y}_{\omega}-\overline{\widehat{\theta}}^{(t)})\leq-\alpha^{2}\tilde{c}/4\beta^{2}\gamma^{(t)}.$
Choose now $\delta$ such that $\delta<\alpha^{2}\tilde{c}/16\beta^{2}$ and
$\bar{t}\geq\bar{t}_{\delta}$ such that $r(t)<\delta\gamma^{(t)}$. It then
follows from (37) that for $t\in I^{\prime}$ and $t\geq\bar{t}$, it holds
$\widehat{\theta}^{(t+1)}_{i_{0}}-\widehat{\theta}^{(t)}_{i_{0}}\leq-\alpha^{2}\tilde{c}/8\beta^{2}\gamma^{(t)}<0.$
This says that as long as $\widehat{\theta}^{(t)}\in\Theta_{\omega}$, its
$i_{0}$-th component decreases. But this entails that $\widehat{\theta}^{(t)}$
can never leave $\Theta_{\omega}$, which contradicts the infinite switching
assumption and thus implies the thesis. ∎
### A.3 Proof of Theorem 1
Propositions 12, 14, and 16 imply that there exists
$\widehat{\omega}^{IA}\in\\{\alpha,\beta\\}^{\mathcal{V}}$ such that
$\widehat{\theta}^{(t)}\in\Theta_{\widehat{\omega}^{IA}}$ for $t$ sufficiently
large. This immediately implies that
$\widehat{\omega}^{(t)}=\widehat{\omega}^{IA}$ for $t$ sufficiently large.
Corollary 9 implies that
$\widehat{\theta}^{IA}=\lim_{t\rightarrow+\infty}\widehat{\theta}^{(t)}=\hat{\theta}(\widehat{\omega}^{IA})$
Finally, since
$\hat{\theta}(\widehat{\omega}^{IA})\in\Theta_{\widehat{\omega}^{IA}}$, we
also have that $\widehat{\omega}^{IA}=\hat{\omega}(\widehat{\theta}^{IA})$.
## Appendix B Proof of concentration results
### B.1 Preliminaries
For a more efficient parametrization of the stationary points, we introduce
the notation:
$\omega\in\\{\alpha,\beta\\}^{\mathcal{V}}\quad\Theta_{\omega}:=\\{x\in\mathbb{R}\,|\,|x-y_{i}|<\delta\,\Leftrightarrow\,\omega_{i}=\alpha\\}$
(38)
It is then straightforward to check from (11) that the set of local maxima
${\mathcal{S}}_{N}$ can be represented as
${\mathcal{S}}_{N}:=\\{\theta=\widehat{\theta}(\omega)\,|\,\omega\in\\{\alpha,\beta\\}^{\mathcal{V}},\;\widehat{\theta}(\omega)\in\Theta_{\omega}\\}.$
(39)
Since,
$\Theta_{\omega}\neq\emptyset\;\Leftrightarrow\;\omega=\widehat{\omega}(x)\,\hbox{\rm
for some}\;x\in\mathbb{R}$ (40)
for analysing the set ${\mathcal{S}}_{N}$ we can restrict to consider $\omega$
of type $\omega=\widehat{\omega}(x)$. Consider the sequence of random
functions $\gamma_{N}(x):=\widehat{\theta}(\widehat{\omega}(x))$.
From (8), applying the strong law of large numbers, we immediately get that
$\lim_{N\rightarrow+\infty}\gamma_{N}(x)\stackrel{{\scriptstyle\mathrm{a.s.}}}{{=}}\gamma_{\infty}(x):=\frac{\mathbb{E}(y_{1}\widehat{\omega}(x)_{1}^{-2})}{\mathbb{E}(\widehat{\omega}(x)_{1}^{-2})}.$
(41)
Something stronger can indeed be said by a standard use of Chernoff bound
[26]:
###### Lemma 17.
For every $\epsilon>0$, there exists $q<1$ such that, for any
$x\in\mathbb{R}$,
$\mathbb{P}\left(\left|\gamma_{N}(x)-\gamma_{\infty}(x)\right|>\epsilon\right)\leq
2q^{N}.$
###### Proof.
Let $a_{i}=y_{i}\omega_{N}(x)_{i}^{-2}$ and $b_{i}=\omega_{N}(x)_{i}^{-2}$
with $i\in\\{1,\ldots,N\\}$ and let $a$ and $b$ denote the corresponding
expected values.
By Chernoff’s bound and by Hoeffding’s inequality we have, respectively, that
$\displaystyle\mathbb{P}\left(\left|\frac{1}{N}\sum_{i=1}^{N}a_{i}-a\right|\geq\epsilon_{1}\right)\leq
q_{1}^{N}\qquad\mathbb{P}\left(\left|\frac{1}{N}\sum_{i=1}^{N}b_{i}-b\right|\geq\epsilon_{2}\right)\leq
2q_{2}^{N}$
with
$q_{1}=e^{-\frac{\alpha^{2}\epsilon_{1}^{2}}{4}}\qquad
q_{2}=e^{-2\epsilon_{2}^{2}\left(\alpha^{-2}-\beta^{-2}\right)^{-2}}.$ (42)
Fix $\epsilon_{1}<\frac{\epsilon}{2b\beta^{4}}$ and
$\epsilon_{2}<\frac{\epsilon}{2|a|\beta^{4}}$, then
$\displaystyle\mathbb{P}\left(\left|\bar{y}_{\omega_{N}(x)}-y_{\infty}(x)\right|>\epsilon\right)$
$\displaystyle\leq\mathbb{P}\left(\frac{\left|\frac{1}{N}\sum_{i=1}^{N}a_{i}-a\right|b+|a|\left|b-\frac{1}{N}\sum_{i=1}^{N}b_{i}\right|}{b\frac{1}{N}\sum_{i=1}^{N}b_{i}}>\epsilon\right)$
$\displaystyle\leq
q_{1}^{N}+q_{2}^{N}+\mathbf{1}_{\left\\{\beta^{4}({\epsilon_{1}b+|a|\epsilon_{2}})>\epsilon\right\\}}$
$\displaystyle=q_{1}^{N}+q_{2}^{N}$
where the last step follows by the way $\epsilon_{1}$ and $\epsilon_{2}$ have
been chosen.
There is still a point to be understood: in our derivation $q_{1}$ and $q_{2}$
depend on the choice of $x$ through $a$ and $b$. However, it is immediate to
check that $a$ and $b$ are both bounded in $x$. This allows to conclude. ∎
From (41) is immediate to see that $\gamma_{\infty}$ is a bounded function of
class $C^{1}$ and it has an important property which will be useful later on.
###### Lemma 18.
There exists a constant $C>0$ such that
$\displaystyle x-\gamma_{\infty}(x)\geq C(x-\theta^{*})\;\quad{\rm
if}\,x\in(\theta^{\star},+\infty)$ $\displaystyle\gamma_{\infty}(x)-x\geq
C(\theta^{*}-x)\;\quad{\rm if}\,x\in(-\infty,\theta^{\star})$
$\displaystyle\gamma_{\infty}(\theta^{*})=\theta^{*}$
###### Proof.
If $x\in(\theta^{\star},+\infty)$ and $f$ is the density of each $y_{i}$ (a
mixture of two Gaussians) then
$\displaystyle x-y_{\infty}(x)$
$\displaystyle=\frac{\frac{1}{\alpha^{2}}\int_{x-\delta}^{x+\delta}{(x-t)f(t)}\mathrm{d}t+\frac{1}{\beta^{2}}\int_{\mathbb{R}\setminus(x-\delta,x+\delta)}{(x-t)f(t)}\mathrm{d}t}{\frac{1}{\alpha^{2}}\int_{x-\delta}^{x+\delta}{f(t)}\mathrm{d}t+\frac{1}{\beta^{2}}\int_{\mathbb{R}\setminus(x-\delta,x+\delta)}{f(t)}\mathrm{d}t}$
$\displaystyle\geq\frac{\frac{1}{\beta^{2}}\int_{\mathbb{R}}{(x-t)f(t)}\mathrm{d}t}{\frac{1}{\alpha^{2}}\int_{x-\delta}^{x+\delta}{f(t)}\mathrm{d}t+\frac{1}{\beta^{2}}\int_{\mathbb{R}\setminus(x-\delta,x+\delta)}{f(t)}\mathrm{d}t}$
where the last inequality follows from the fact that
$\int_{x-\delta}^{x+\delta}{(x-t)f(t)}\mathrm{d}t\geq 0$. We conclude that
$\displaystyle x-y_{\infty}(x)$
$\displaystyle\geq\frac{\frac{1}{\beta^{2}}(x-\theta^{\star})}{\frac{1}{\alpha^{2}}\int_{x-\delta}^{x+\delta}{f(t)}\mathrm{d}t+\frac{1}{\beta^{2}}\int_{\mathbb{R}\setminus(x-\delta,x+\delta)}{f(t)}\mathrm{d}t}>0.$
Second statement if $x\in(-\infty,\theta^{\star})$ can be verified in a
completely analogous way. The third statement then simply follows by
continuity. ∎
We now come to a key result.
###### Lemma 19.
For any fixed $\epsilon>0$, there exist $\tilde{q}\in(0,1)$ and $\chi>0$ such
that
$\mathbb{P}\left(\gamma_{N}(x)\in\Theta_{\widehat{\omega}(x)}\right)\leq\chi\tilde{q}^{N}$
(43)
for all $x$ such that $|x-\theta^{\star}|>\epsilon$.
###### Proof.
We assume $x>\theta^{\star}+\epsilon$ (the other case
$x<\theta^{\star}-\epsilon$ being completely equivalent). Fix
$\epsilon^{\prime}\in(0,C\epsilon)$ where $C$ was defined in Lemma 18 and
estimate as follows
$\begin{split}\mathbb{P}\left(\gamma_{N}(x)\in\Theta_{\widehat{\omega}(x)}\right)&\leq\mathbb{P}\left(\gamma_{N}(x)\in\Theta_{\widehat{\omega}(x)}\,,\;|\gamma_{N}(x)-\gamma_{\infty}(x)|\leq\epsilon^{\prime}\right)\\\
&+\mathbb{P}\left(|\gamma_{N}(x)-\gamma_{\infty}(x)|>\epsilon^{\prime}\right).\end{split}$
(44)
Using Lemma 18 we get
$\left\\{|\gamma_{N}(x)-\gamma_{\infty}(x)|\leq\epsilon^{\prime}\right\\}\subseteq\left\\{\gamma_{N}(x)\leq
x-(C\epsilon-\epsilon^{\prime})\right\\}.$
Thus
$\begin{split}&\left\\{\gamma_{N}(x)\in\Theta_{\widehat{\omega}(x)},|\gamma_{N}(x)-\gamma_{\infty}(x)|\leq\epsilon^{\prime}\right\\}\\\
&\qquad\subseteq\\{\nexists
i\,:\,y_{i}\in(\gamma_{N}(x)-\delta,\gamma_{N}(x)-\delta+\min\\{C\epsilon-\epsilon^{\prime},\delta\\})\\}\end{split}$
and, consequently, the first term in (44) can be estimated as
$\begin{split}\mathbb{P}&\left(\gamma_{N}(x)\in\Theta_{\widehat{\omega}_{N}(x)}\,,\;|\gamma_{N}(x)-\gamma_{\infty}(x)|\leq\epsilon^{\prime}\right)\leq\left(1-\int_{\gamma_{N}(x)-\delta}^{\gamma_{N}(x)-\delta+\min\\{C\epsilon-\epsilon^{\prime},\delta\\}}f(y)dy\right)^{N}\end{split}$
(45)
where $f(y)$ is the density of each $y_{i}$. Considering now that $f(y)$ is
bounded away from $0$ on any bounded interval, that
$|\gamma_{N}(x)-\gamma_{\infty}(x)|\leq\epsilon^{\prime}$ and that
$\gamma_{\infty}(x)$ is a bounded function, we deduce that the right hand side
of (45) can be uniformly bounded as $\tilde{q}^{N}$ for some
$\tilde{q}\in(0,1)$. Substituting in (44), and using Lemma 17 we finally
obtain the thesis. ∎
### B.2 Proof of Theorem 2
Define
$\mathcal{A}_{N}(\epsilon):=\left\\{\exists\omega\in\\{\alpha,\beta\\}^{\mathcal{V}}:\widehat{\theta}(\omega)\in\Theta_{\omega},|\widehat{\theta}(\omega)-\theta^{\star}|>\epsilon\right\\}$
for any $\epsilon>0$ and
$\displaystyle\mathcal{B}_{1}$ $\displaystyle:=\left\\{\exists
i\in\mathcal{V}:|y_{i}-\theta^{\star}|>N\right\\}$
$\displaystyle\mathcal{B}_{2}$
$\displaystyle:=\left\\{\exists(i,j)\in\mathcal{V}\times{\mathcal{V}}:|y_{i}-y_{j}|<N^{-4}\right\\}$
$\displaystyle\mathcal{B}_{3}$
$\displaystyle:=\\{\exists(i,j)\in\mathcal{V}\times{\mathcal{V}}:|y_{i}-y_{j}|\in\left(2\delta,2\delta+{N^{-4}}\right\\}$
and estimate
$\mathbb{P}\left(\mathcal{A}_{N}(\epsilon)\right)\leq\mathbb{P}\left(\mathcal{A}_{N}(\epsilon),\mathcal{B}_{1}^{c}\cap\mathcal{B}_{2}^{c}\cap\mathcal{B}_{3}^{c}\right)+\mathbb{P}(\mathcal{B}_{1})+\mathbb{P}(\mathcal{B}_{2})+\mathbb{P}(\mathcal{B}_{3})$.
Standard considerations allow to upper bound the probability of each event
$\mathcal{B}_{i}$ by a common term $K/N^{2}$. We now focus on the estimation
of the first term. The crucial point is that, the condition
$\mathcal{B}_{1}^{c}\cap\mathcal{B}_{2}^{c}\cap\mathcal{B}_{3}^{c}$ allow us
to reinforce condition (40) in the sense that all $\omega$ for which
$\Theta_{\omega}\neq\emptyset$ can be obtained as $\omega=\widehat{\omega}(x)$
as $x$ varies in a set whose cardinality is polynomial in $N$. Specifically,
define
$Z=\\{\zeta_{j}=\theta^{\star}-N-\delta+{j}{N^{-4}}:j\in\mathbb{N},j\leq
j_{\rm max}\\}$
where $j_{\rm max}:=\lceil N^{4}(2N+2\delta)\rceil$ and notice that, assuming
that the $y_{i}$’s satisfy $\mathcal{B}_{2}^{c}\cap\mathcal{B}_{3}^{c}$, we
have that $\widehat{\omega}(\zeta_{j})$ and $\widehat{\omega}(\zeta_{j+1})$
differ in at most one component and that
$\widehat{\omega}(x)\in\\{\widehat{\omega}(\zeta_{j}),\widehat{\omega}(\zeta_{j+1})\\}$
for every $x\in[\zeta_{j},\zeta_{j+1}]$. Moreover, because of
$\mathcal{B}_{1}^{c}$ we have that
$\widehat{\omega}(x)_{i}=\widehat{\omega}(\zeta_{0})_{i}=\beta$ for all
$x\leq\theta^{\star}_{N}-\delta$ and for all $i$. Similarly,
$\widehat{\omega}(x)_{i}=\widehat{\omega}(\zeta_{j_{\rm max}})_{i}=\beta$ for
all $x\geq\theta^{\star}+N+\delta$ and for sll $i$. In other terms, under the
assumption that the $y_{i}$’s satisfy
$\mathcal{B}_{1}^{c}\cap\mathcal{B}_{2}^{c}\cap\mathcal{B}_{3}^{c}$, it holds
$\\{\omega\in\\{\alpha,\beta\\}^{\mathcal{V}}\;|\;\Theta_{\omega}\neq\emptyset\\}=\\{\widehat{\omega}(x)\;|\;x\in
Z\\}$. Hence,
$\displaystyle\mathbb{P}$
$\displaystyle\left(\mathcal{A}_{N}(\epsilon),\mathcal{B}_{1}^{c}\cap\mathcal{B}_{2}^{c}\cap\mathcal{B}_{3}^{c}\right)\leq$
$\displaystyle\leq\mathbb{P}\left(\bigcup_{\zeta\in
Z}\left\\{\gamma_{N}(\zeta)\in\Theta_{\widehat{\omega}(\zeta)},|\gamma_{N}(\zeta)-\theta^{\star}|>\epsilon\right\\}\right)$
$\displaystyle\leq\mathbb{P}\left(\bigcup_{\zeta\in
Z}\left\\{\gamma_{N}(\zeta)\in\Theta_{\widehat{\omega}(\zeta)},|\gamma_{\infty}(\zeta)-\theta^{\star}|>\epsilon/2\right\\}\right)+$
$\displaystyle+\mathbb{P}\left(|\gamma_{N}(\zeta)-\gamma_{\infty}(\zeta)|\leq\epsilon/2\right).$
Notice that, because of the continuity of $\gamma_{\infty}$, there exists
$\tilde{\epsilon}>0$ such that
$|\gamma_{\infty}(\zeta)-\theta^{\star}|>\epsilon/2\;\Rightarrow\;|\zeta-\theta|>\tilde{\epsilon}$.
We can then use Lemma 19,
$\displaystyle\mathbb{P}\left(\bigcup_{\zeta\in
Z}\left\\{\gamma_{N}(\zeta)\in\Theta_{\widehat{\omega}(\zeta)},|\gamma_{\infty}(\zeta)-\theta^{\star}|>\epsilon/2\right\\}\right)$
$\displaystyle\qquad\leq|Z|\mathbb{P}\left(\gamma_{N}(\zeta)\in\Theta_{\widehat{\omega}(\zeta)},|\gamma_{N}(\zeta)-\theta^{\star}|>\epsilon\right)\leq
cN^{5}\tilde{q}^{N}$
where $c$ and $\tilde{q}$ are those coming from Lemma 19 relatively to
$\tilde{\epsilon}$. Putting together all the estimations we have obtained and
using Lemma 17, we finally obtain that there exists $\chi>0$ such that
$\mathbb{P}\left(\mathcal{A}_{N}(\epsilon)\right)\leq\chi/N^{2}$. Using Borel-
Cantelli Lemma and standard arguments, it follows now that the relation (16)
hold in an almost surely sense.
It remains to be shown convergence in mean square sense. For this we need to
go back to the form (10) of the derivative of
$L(\theta,\widehat{\omega}(\theta))$. The key observation is that the second
additive term in the right hand side of (10) can be bounded uniformly in
modulus by some constant $C$. If we denote
$\bar{\gamma}_{N}=N^{-1}\sum_{i}y_{i}$, this implies that the function is
increasing for $\theta>\bar{\gamma}_{N}+\beta^{2}C$ and decreasing for
$\theta<\bar{\gamma}_{N}-\beta^{2}C$. Hence, necessarily,
$|\xi-\bar{\gamma}_{N}|\leq\beta^{2}C\;\ \forall\xi\in{\mathcal{S}}_{N}.$ (46)
On the other hand, by the law of large numbers, $\bar{\gamma}_{N}$ almost
surely converges to $\theta^{\star}$ and this implies, by the previous part of
the theorem that $\max\limits_{\xi\in{\mathcal{S}}_{N}}|\xi-\bar{\gamma}_{N}|$
converges to $0$. This, together with (46), yields
$\mathbb{E}\max\limits_{\xi\in{\mathcal{S}}_{N}}|\xi-\bar{\gamma}_{N}|^{2}\to
0$ for $N\to+\infty$. Since by the ergodic theorem also
$\mathbb{E}|\bar{\gamma}_{N}-\theta^{\star}|^{2}\to 0$ for $N\to+\infty$, the
proof is complete.
### B.3 Proof of Proposition 4
We prove it for $\widehat{\omega}^{\mathrm{IA}}$, the other verification being
completely equivalent). If $\sigma\in\\{\alpha,\beta\\}$, we define
$\begin{split}f(\theta,\sigma)&=\mathbb{P}(\widehat{\omega}(\theta)_{i}\neq\sigma\;|\;\omega^{\star}_{i}=\sigma)\\\
&=\begin{cases}\frac{1}{\sqrt{2\pi\sigma^{2}}}\int_{{\theta}-\delta}^{{\theta}+\delta}\mathrm{e}^{-\frac{(s-\theta^{\star})^{2}}{2\sigma^{2}}}\mathrm{d}s\qquad\
\ \text{if }\sigma=\beta\\\
1-\frac{1}{\sqrt{2\pi\sigma^{2}}}\int_{{\theta}-\delta}^{{\theta}+\delta}\mathrm{e}^{-\frac{(s-\theta^{\star})^{2}}{2\sigma^{2}}}\mathrm{d}s\quad\text{if
}\sigma=\alpha\end{cases}\end{split}$
(notice that $f$ does not depend on $i$). We can compute
$\begin{split}\frac{1}{N}\mathbb{E}d_{H}(\widehat{\omega}^{\mathrm{IA}},\omega^{\star})&=\frac{1}{N}\sum\limits_{i}\mathbb{P}(\widehat{\omega}^{\mathrm{IA}}_{i}\neq\omega^{\star}_{i})\\\
&=p\mathbb{E}f(\widehat{\theta}^{\mathrm{IA}},\alpha)+(1-p)\mathbb{E}f(\widehat{\theta}^{\mathrm{IA}},\beta).\end{split}$
Since $f(\theta,\sigma)$ is a $C^{1}$ function of $\theta$, we immediately
obtain that
$|\mathbb{E}f(\widehat{\theta}^{\mathrm{IA}},\sigma)-\mathbb{E}f(\theta^{\star},\sigma)|\leq
C\mathbb{E}|\widehat{\theta}^{\mathrm{IA}}-\theta^{\star}|$
and, by Corollary 3, this last expression converges to $0$, for $N\to+\infty$.
Hence,
$\frac{1}{N}\mathbb{E}d_{H}(\widehat{\omega}^{\mathrm{IA}},\omega^{\star})=p\mathbb{E}f(\theta^{\star},\alpha)+(1-p)\mathbb{E}f(\theta^{\star},\beta).$
Straightforward computation now proves the thesis.
|
arxiv-papers
| 2012-06-17T21:19:45 |
2024-09-04T02:49:31.868278
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabio Fagnani, Sophie M. Fosson, and Chiara Ravazzi",
"submitter": "Sophie Fosson",
"url": "https://arxiv.org/abs/1206.3793"
}
|
1206.3864
|
# Topological superfluid of spinless Fermi gases in p-band honeycomb optical
lattices with on-site rotation
Beibing Huang
Department of Experiment Teaching, Yancheng Institute of Technology, Yancheng,
224051, China
Xiaosen Yang and ShaoLong Wan
Institute for Theoretical Physics and Department of Modern Physics
University of Science and Technology of China, Hefei, 230026, China
Abstract
In this paper, we put forward to another route realizing topological
superfluid (TS). In contrast to conventional method, spin-orbit coupling and
external magnetic field are not requisite. Introducing an experimentally
feasible technique called on-site rotation (OSR) into p-band honeycomb optical
lattices for spinless Fermi gases and considering CDW and pairing on the same
footing, we investigate the effects of OSR on superfluidity. The results
suggest that when OSR is beyond a critical value, where CDW vanishes, the
system transits from a normal superfluid (NS) with zero TKNN number to TS
labeled by a non-zero TKNN number. In addition, phase transitions between
different TS are also possible.
PACS number(s): 67.85.Lm, 03.65.Vf, 74.20.-z
## 1 Introduction
Topological superfluid (or superconductor) (TS) has a full pairing gap in the
bulk and is labeled by a non-zero integer topological invariant [1, 2]. From
the famous bulk-boundary correspondence such a topological integer ensures the
existence of gapless excitations on the boundary of the system, in other words
Majorana fermions (MF) [3] in vortex core of pairing order parameter. Roughly,
MFs are neither fermions nor bosons but non-Abelian anyons [4] and play an
important role for the realization of fault-tolerant topological quantum
computation (TQC) [5]. The application prospect of MFs makes TS become one of
the hottest frontiers.
In the condensed matter physics some practical two-dimensional systems have
been theoretically proposed to realize TS [6, 7, 8, 9, 10, 11, 12, 13]. In
terms of these systems the entrance into TS requires subtle adjustment of
Hamiltonian and it is very difficult in condensed matter physics, although MFs
have been detected in InSb nanowires contacted with one normal (Au) and one
superconducting electrode (NbTiN) [14]. In the light of the disadvantage for
condensed matter, TS has been also suggested in cold Fermi gases owing to
their many controllable advantages and operabilities. Following the successful
observation p-wave Feshbach resonance (FR), Gurarie et al. [15] show that
degenerate Fermi gases near a p-wave FR naturally give a concrete realization
of TS. Zhang et al. [16] propose to create TS directly from an s-wave
interaction making use of an artificially generated spin-orbit coupling (SOC).
In fact, SOC have been realized in a neutral atomic Bose-Einstein condensate
(BEC) by dressing two atomic spin states with a pair of lasers and the same
technique is also feasible for cold Fermi gases [17, 18]. Realizing that in a
dual transformation SOC is formally equivalent to a p-wave superfluid gap,
Sato et al. [19] suggest to artificially generate the vortices of SOC by using
lasers carrying orbital angular momentum. In terms of the latter two ways, SOC
and a large magnetic field are crucial in order to enter into TS.
In this paper we suggest to create TS from spinless Fermi gases in p-band
honeycomb optical lattices with so-called on-site rotation (OSR), that rotates
every lattice site around its own center but keeps the whole lattice intact
and has been realized for triangular optical lattices [20]. As a matter of
fact p-band Fermi gases in honeycomb optical lattices in absence of OSR have
shown many interesting characteristics, such as ferromagnetism [21] and Wigner
crystallization [22, 23] associating with flat bands, f-wave superfluidity
with conventional pairing interaction [24]. The motivation for this paper
comes from the Wu’s work on quantum anomalous Hall effect in the same system
[25]. Under single particle picture, Wu found that an arbitrary non-zero OSR
not only breaks time-reversal symmetry and changes the topological properties
of the system, but also drives a topological phase transition when OSR is
beyond a critical value. Here we add on-site attraction interaction between
p-band Fermi atoms into Hamiltonian and ask a question whether OSR can drive a
phase transition into TS. The results are positive and OSR brings phase
transitions not only from normal superfluid (NS) to TS, but also among
different TS. From another perspective our work also can be considered as an
extension to [24], where f-wave superfluidity without OSR is discussed. Thus
we also investigate the effects of OSR on f-wave superfluidity.
Experimentally the route to realize TS suggested here is also feasible. On the
one hand by placing two electro-optic modulators at two of three laser beams
which coherently superpose to form a honeycomb lattice, OSR is available as
illustrated in [26]. On the other hand due to Pauli exclusion principle the
occupation of p-band is very convenient as long as the lowest s-band is
fulfilled. In addition, on-site attraction interaction can be enhanced by
using atoms with large magnetic moments, such as 167Er with $m=7\mu_{B}$ on
which laser cooling has been performed [27]. In contrast to [16, 19], where a
pair of extra lasers and a large magnetic field are needed to produce an
effective SOC and split two SOC bands respectively, our system is much
simpler.
The organization of this paper is as follows. In section 2, we give the model
and at the mean-field level investigate the ground state of the system by
numerically minimizing the thermodynamic potential. In section 3 by
calculating TKNN number $I_{TKNN}$ [28] of occupied bands addressing the
topological properties of the model, the topological phase diagram is
obtained. In addition we also investigate the properties of edge states to
prove our results. A brief conclusion is given in section 4.
## 2 Model and Mean-Field Ground State
The honeycomb optical lattice was realized experimentally by using three laser
beams with co-planar propagating wavevectors quite some time age [29]. It is
well known that a honeycomb lattice is not a Bravais lattice and there are two
inequivalent sites in a unit cell, denoted by A and B respectively. Fulfilling
the lowest s-band and defining three unit vectors
$\vec{e}_{1}=\frac{\sqrt{3}}{2}\vec{e}_{x}+\frac{1}{2}\vec{e}_{y}$,
$\vec{e}_{2}=-\frac{\sqrt{3}}{2}\vec{e}_{x}+\frac{1}{2}\vec{e}_{y}$ and
$\vec{e}_{3}=-\vec{e}_{y}$, the Hamiltonian of p-band honeycomb optical
lattices with OSR is
$\displaystyle H=t_{\|}\sum_{\vec{r}\in
A,i}\left[p_{\vec{r},i}^{{\dagger}}p_{\vec{r}+\vec{e}_{i},i}+H.C.\right]-U\sum_{\vec{r}\in
A\oplus
B}p_{\vec{r}x}^{{\dagger}}p_{\vec{r}y}^{{\dagger}}p_{\vec{r}y}p_{\vec{r}x}-\Omega\sum_{\vec{r}\in
A\oplus B}\hat{l}_{\vec{r},z}-\mu\sum_{\vec{r}\in A\oplus
B}\hat{n}_{\vec{r}},$ (1)
where
$p_{\vec{r},i}=(p_{\vec{r},x}\vec{e}_{x}+p_{\vec{r},y}\vec{e}_{y})\cdot\vec{e}_{i}$
and $p_{\vec{r},x}$ ($p_{\vec{r},y}$) is the annihilation operator for $p_{x}$
($p_{y}$) band at the lattice site $\vec{r}$.
$\hat{n}_{\vec{r}}=p_{\vec{r},x}^{{\dagger}}p_{\vec{r},x}+p_{\vec{r},y}^{{\dagger}}p_{\vec{r},y}$
and
$\hat{l}_{\vec{r},z}=-i(p_{\vec{r},x}^{{\dagger}}p_{\vec{r},y}-p_{\vec{r},y}^{{\dagger}}p_{\vec{r},x})$
represent particle number and orbital angular moment operators. $t_{\|}$ is
the nearest-neighbor hopping matrix element of atoms in $\sigma$ bonds and
positive due to the odd parity of the p-orbital. $U$ ($>0$), $\Omega$ ($>0$)
and $\mu$ are the on-site interaction strength, on-site rotation angular
velocity and chemical potential, respectively. Note that we have neglected the
nearest-neighbor atom hopping of $\pi$ bonds and supposed the nearest neighbor
distance in the lattice to be unit.
When $U=0$, introducing operator
$\phi(k)=[p_{Ax}(k),p_{Ay}(k),p_{Bx}(k),p_{By}(k)]^{T}$ and making a unitary
transformation $\phi_{n}(k)=U_{nm}(k)\Psi_{m}(k)$, Hamiltonian can be
diagonalized exactly. Meanwhile four energy bands can be obtained. Wu found
two of four bands always are topological for any nonzero OSR and the others
can be topological only if OSR is beyond a critical value [25]. On the basis
of this findings, Wu proposed an orbital analogue of the quantum anomalous
Hall effect, arising from orbital angular momentum polarization due to OSR.
With $\Omega=0$, Lee et al. discussed f-wave superfluidity and charge density
wave (CDW) in this system at the mean-field level [24]. Their results show
that away from the half filling the system is f-wave superfluidity, while
around the half filling superfluidity and CDW coexist and the system is a
supersolid. Although superfluidity exists all the time, it is not topological
as stated below.
Following the same spirit in [24] we decouple interaction term into CDW
channel
$\displaystyle H_{int}^{CDW}=\sum_{\tau=x,y}\left[\sum_{\vec{r}\in
A}(-\frac{n}{2}U-\frac{\Delta_{CDW}}{2})p_{\vec{r},\tau}^{{\dagger}}p_{\vec{r},\tau}+\sum_{\vec{r}\in
B}(-\frac{n}{2}U+\frac{\Delta_{CDW}}{2})p_{\vec{r},\tau}^{{\dagger}}p_{\vec{r},\tau}\right]$
(2)
and pairing channel
$\displaystyle H_{int}^{pairing}$ $\displaystyle=$
$\displaystyle-\sum_{k}\left[\Delta_{A}p_{Ax}^{{\dagger}}(k)p_{Ay}^{{\dagger}}(-k)+\Delta_{B}p_{Bx}^{{\dagger}}(k)p_{By}^{{\dagger}}(-k)+H.C.\right]$
(3) $\displaystyle=$
$\displaystyle-\sum_{k^{\prime}}\left[\Delta_{nm}(k^{\prime})\Psi_{n}^{{\dagger}}(k^{\prime})\Psi_{m}^{{\dagger}}(-k^{\prime})+H.C.\right]$
where $n=<\hat{n}_{\vec{r}_{A}}+\hat{n}_{\vec{r}_{B}}>/2$ is filling factor of
every site, $\Delta_{CDW}=U<\hat{n}_{\vec{r}_{A}}-\hat{n}_{\vec{r}_{B}}>/2$,
$\Delta_{A}=U\sum_{k}<p_{Ay}(-k)p_{Ax}(k)>$,
$\Delta_{B}=U\sum_{k}<p_{By}(-k)p_{Bx}(k)>$ are order parameters for CDW and
superfluidity. In (3) we also express the pairing channel using quasiparticle
$\Psi(k)$. In this representation
$\displaystyle\Delta_{nm}(k)$ $\displaystyle=$
$\displaystyle\Delta_{A}\left[U_{1n}^{\ast}(k)U_{2m}^{\ast}(-k)-U_{2n}^{\ast}(k)U_{1m}^{\ast}(-k)\right]$
(4) $\displaystyle+$
$\displaystyle\Delta_{B}\left[U_{3n}^{\ast}(k)U_{4m}^{\ast}(-k)-U_{4n}^{\ast}(k)U_{3m}^{\ast}(-k)\right].$
After the mean-field approximation, the Hamiltonian (1) becomes a BdG
Hamiltonian
$H=[\phi^{{\dagger}}(k),\phi(-k)]H_{k}[\phi(k),\phi^{{\dagger}}(-k)]^{T}$ and
the properties of system are completely decided by the $8\times 8$ matrix
$H_{k}$. Diagonalizing $H_{k}$, we attain spectrum $\epsilon_{i}(k)$ and
correspondingly eigenvectors $\varphi_{i}(k)$ ($i=1,2,\cdot\cdot\cdot,8$). Due
to particle-hole symmetry inherent in this BdG Hamiltonian, the spectrum are
symmetric about zero energy and we assume
$\epsilon_{1}(k)=-\epsilon_{8}(k)>0$, $\epsilon_{2}(k)=-\epsilon_{7}(k)>0$,
$\epsilon_{3}(k)=-\epsilon_{6}(k)>0$, $\epsilon_{4}(k)=-\epsilon_{5}(k)>0$.
Then the thermodynamical potential at zero temperature is
$\displaystyle
F=\frac{1}{2}\sum_{k}\left[-4\mu-\epsilon_{1}(k)-\epsilon_{2}(k)-\epsilon_{3}(k)-\epsilon_{4}(k)\right]+\frac{N}{U}|\Delta_{A}|^{2}+\frac{N}{U}|\Delta_{B}|^{2}+\frac{N}{2U}\Delta_{CDW}^{2},$
(5)
where $N$ is the number of the unit cell. Below we numerically minimize
thermodynamic potential $F$ about $\Delta_{A}$, $\Delta_{B}$ and
$\Delta_{CDW}$ for fixed interaction strength $U$. Without loss of generality
we choose $\Delta_{A}$ to be real, $\Delta_{B}=|\Delta_{B}|e^{i\theta}$ and
$U/t_{\|}=3.0$.
Fig.1 shows the solutions of the Hamiltonian (1) at the mean-field level for
changing chemical potential $\mu$ and OSR $\Omega$. Due to the particle-hole
symmetry we only concentrate on negative chemical potential. Fig.1(a)
describes the variation of $\Delta_{CDW}$. For $\Omega=0$ CDW is robust, but
when $\Omega$ is beyond a critical value $\Omega_{c}$ it vanishes suddenly.
This is due to the fact that the appearance of OSR changes the band structures
of single particle and breaks the nesting condition for CDW. Numerically we
find $\Omega_{c}/t_{\|}\approx 0.4\sim 0.6$ and is monotonically increasing as
the function of chemical potential. Fig.1(b) shows the effect of OSR on
particle density and further exemplifies that the variations of band
structures driven by OSR cause nonmonotonic behavior of particle density. In
contrast, superfluid order parameters $\Delta_{A}$, $\Delta_{B}$ are more
interesting and shown in (c) and (d). On the one hand for $\Omega>\Omega_{c}$,
$\Delta_{A}=|\Delta_{B}|$ and with the increase of OSR superfluid order
smoothly decreases until disappearance. This suppression mechanism of
superfluidity consists in time-reversal symmetry broken caused by OSR. While
on the other hand for $\Omega<\Omega_{c}$ $\Delta_{A}$ is still decreasing but
$\Delta_{B}$ is increasing with $\Omega$. In fact the increase of $\Delta_{B}$
originates from the redistribution of particle density between sites A and B,
in other words the decrease of $\Delta_{CDW}$ as seen in (a). Thus at the
mean-field level our calculation suggests that (1) OSR weakens stabilities of
CDW and superfluidity and (2) for $\Omega<\Omega_{c}$, superfluidity and CDW
coexist and the system is a supersolid.
The optimization of $\theta$ leads to $\theta=\pi$ for all parameters we
choose. Below we discuss the effects of OSR on pairing symmetry for
$\Omega>\Omega_{c}$. From [24] without OSR and away from the half filling the
intraband pairings in (4) have f-wave symmetry with three nodal lines of
$k_{x}=0$, $k_{y}=\pm k_{x}/\sqrt{3}$ and $\pi/3$ rotation symmetry
[Fig.2(d)]. On introducing OSR, in terms of the pairing magnitude, nodal lines
degenerate into some disconnected regions where intraband gap disappears, and
$\pi/3$ rotation symmetry retains. However real and imaginary parts of pairing
break $\pi/3$ into $\pi$ rotation symmetry. In Fig.2(a) (b) and (c) as an
example we show the magnitude, real and imaginary parts of $\Delta_{11}$.
## 3 Topological Phase Diagram and Majorana Fermion Modes
In this section we discuss topological properties of Hamiltonian (1). In terms
of our system, it explicitly breaks the time-reversal symmetry due to OSR.
Thus TKNN number $I_{TKNN}$ plays a central role in deciding topological
nature of the system [28]. TKNN number is defined, by eigenvectors
$\varphi_{i}(k)$ ($i=5,6,7,8$) corresponding to negative energy spectrm of the
matrix $H_{k}$, into $I_{TKNN}=\frac{1}{2\pi i}\int d^{2}k\,Tr\,dA$, where $A$
is a matrix one-form $A_{ij}=A_{ij}^{\nu}(k)dk_{\nu}$ with
$A_{ij}^{\nu}(k)=\varphi_{i}^{{\dagger}}(k)\nabla_{k_{\nu}}\varphi_{j}(k)$. By
numerically calculating TKNN number [30], we show the topological phase
diagram of the system in Fig.3. For parameter region we choose, there are four
different subregions labeled by $I_{TKNN}=1,0,-1,2$ respectively. Moreover by
comparison with Fig.1(a) it is easily found that the boundary between
$I_{TKNN}=0$ and other TKNN numbers in the direction of $\Omega$ coincides
with that of CDW disappearance. This finding is very important and ensures
that topological order of our system is not topological CDW [31]. According to
the criteria for TS [31] $I_{TKNN}=2$ corresponds to Abelian TS while
$I_{TKNN}=1,-1$ are non-Abelian TS. Thus Fig.3 tells us that OSR drives
topological phase transition not only from NS to TS, but also between
different TS. Here we mention a fact the energy gap of the bulk spectrum
closes when topological phase transitions between topologically distinct
phases occur.
From the bulk-edge correspondence, a non-trivial bulk topological number
implies the existence of gapless edge states localized on open edges of the
system. Cold Fermi gases with sharp edges may be realized along the lines
proposed in [32]. In order to understand the relation between $I_{TKNN}$ and
the number of edge states, we study the Hamiltonian (1) with the open boundary
condition along the zigzag edge of the honeycomb lattice. The resulting
excitation spectrum are depicted in Fig.4 for representative parameter
choices. Very explicitly the number of gapless states for every edge is one-
to-one correspondence with the TKNN number. For $I_{TKNN}=\pm 1$
($I_{TKNN}=2$) there are one (two) pair(s) of gapless states, while for
$I_{TKNN}=0$, gapless state does not exist. Due to particle-hole symmetry, in
terms of gapless states, they are Majorana fermion modes. It should also be
remembered that the core of a vortex is topologically equivalent to an edge
which has been closed on itself. The edge modes we describe are therefore
equivalent to the Majorana fermions known to exist in the core of vortices of
p-wave superfluids [33].
## 4 Conclusions
In conclusion at the mean-field level we have investigated the effects of OSR
on CDW and superfluidity for p-band spinless Fermi gases in honeycomb optical
lattices. We found that OSR weakens the stabilities of CDW and superfluidity
simultaneously, although superfluidity can survives a larger OSR. This
conclusion leads to another important result that once CDW drops out the
system enters into topological superfluidity. By numerically calculating the
TKNN number we obtained topological phase diagram of the system. In addition
edge states, i.e. bulk-boundary correspondence are also investigated.
## Acknowledgement
The work was supported by National Natural Science Foundation of China under
Grant No. 10675108 and Foundation of Yancheng Institute of Technology under
Grant No. XKR2010007.
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Figure 1: The mean-field solution of the Hamiltonian (1). Parameter
$U/t_{\|}=3.0$.
Figure 2: The symmetry of intraband pairing $\Delta_{11}$. In (a) the
magnitude, (b) real part and (c) imaginary part of $\Delta_{11}$ for
$\Omega/t_{\|}=1.0$ are shown. For comparison (d) plots $\Delta_{11}$ for
$\Omega/t_{\|}=0$. Figure 3: Topological phase diagram of the Hamiltonian (1)
at the mean-field level. The light grey, dark grey, black and blue colors
correspond to $I_{TKNN}=1,0,-1,2$ respectively. Parameter $U/t_{\|}=3.0$.
Figure 4: The gapless edge states with the open boundary condition along the
zigzag edge of the honeycomb lattice. In (a) $I_{TKNN}=0$, $\mu/t_{\|}=-0.75$,
$\Omega/t_{\|}=0.3$, $\Delta_{A}/t_{\|}=0.969$, $\Delta_{B}/t_{\|}=0.063$,
$\Delta_{CDW}/t_{\|}=1.462$, (b) $I_{TKNN}=2$, $\mu/t_{\|}=-0.5$,
$\Omega/t_{\|}=0.8$, $\Delta_{A}/t_{\|}=\Delta_{B}/t_{\|}=0.288$,
$\Delta_{CDW}/t_{\|}=0$, (c) $I_{TKNN}=-1$, $\mu/t_{\|}=-0.85$,
$\Omega/t_{\|}=0.8$, $\Delta_{A}/t_{\|}=\Delta_{B}/t_{\|}=0.523$,
$\Delta_{CDW}/t_{\|}=0$, (d) $I_{TKNN}=1$, $\mu/t_{\|}=-0.65$,
$\Omega/t_{\|}=1.1$, $\Delta_{A}/t_{\|}=\Delta_{B}/t_{\|}=0.307$,
$\Delta_{CDW}/t_{\|}=0$.
|
arxiv-papers
| 2012-06-18T09:27:05 |
2024-09-04T02:49:31.882724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Beibing Huang, xiaoseng yang, shaolong wan",
"submitter": "Beibing Huang",
"url": "https://arxiv.org/abs/1206.3864"
}
|
1206.3879
|
# Gromov-Witten theory and cycle-valued modular forms
Todor Milanov & Yongbin Ruan & Yefeng Shen Kavli IPMU (WPI)
The University of Tokyo
Kashiwa
Chiba 277-8583
Japan todor.milanov@ipmu.jp Department of Mathematics
University of Michigan
Ann Arbor
MI 48105
USA ruan@umich.edu Department of Mathematics
University of Michigan
Ann Arbor
MI 48105
USA yfschen@umich.edu
###### Contents
1. 1 Introduction
1. 1.1 Acknowledgements
2. 2 Cohomological field theory and quantization
1. 2.1 Cohomological field theories
2. 2.2 Examples of CohFTs
3. 2.3 Givental’s formalism
4. 2.4 Cycle-valued Quantization
1. 2.4.1 Coordinate Change
2. 2.4.2 Feynman type sum
3. 2.4.3 Classification of semi-simple CohFT
4. 2.4.4 Higher-genus reconstruction
3. 3 Global Frobenius manifolds for simple elliptic singularities
1. 3.1 Saito theory
2. 3.2 Global Frobenius manifold structures for simple elliptic singularities.
1. 3.2.1 Primitive forms and global moduli of Frobenius manifolds
3. 3.3 The action of the monodromy group on flat coordinates
4. 4 Global B-model CohFT and anti-holomorphic completion
1. 4.1 Global B-model CohFT
1. 4.1.1 Givental’s semisimple quantization operator
2. 4.1.2 Global B-model CohFT
2. 4.2 Monodromy group action on $\Lambda_{g,n}^{W}(t)$
3. 4.3 Anti-holomorphic completion and modular transformation.
1. 4.3.1 Anti-holomorphic completion of $\Lambda_{g,n}^{W}(t)$
2. 4.3.2 Cycle-valued quasi-modular forms from $\Lambda_{g,n}^{W}(t)$
5. 5 A-model CohFT and cycle valued modular forms
1. 5.1 A-model
2. 5.2 Analyticity and generic semisimplicity
3. 5.3 Convergence of $\Lambda^{\mathcal{X}}_{g,n}(\mathbf{t})$
4. 5.4 Extension property
5. 5.5 Quasi-modularity
## 1\. Introduction
A remarkable phenomenon in Gromov-Witten theory is the appearance of (quasi)
modular forms. Classically, modular forms arise as a counting function of
points, i.e., zero dimensional objects. A Gromov-Witten generating function
can be thought as a counting function for the virtual number of holomorphic
curves, i.e., one dimensional objects. Therefore, it is natural to speculate
if modular forms appear here too. One can attempt to compute them explicitly.
If one is lucky enough, the answers can be organized as modular forms. Indeed,
this strategy has been carried out for elliptic curves [OP] and the so called
reduced Gromov-Witten theory of K3-surfaces [MPT]. However, we should
emphasize that both steps of the strategy are highly nontrivial. In fact, the
above modularity results are some of the most sophisticated works in Gromov-
Witten theory. Generally speaking, it is very difficult to compute Gromov-
Witten invariants. Even if you can compute, it is not clear how to organize
them into modular forms. Unlike the case of counting points, it is impractical
to try to compute a large number of coefficients and then guess the general
pattern.
In the middle of the 90’s, by studying the physical B-model of Gromov-Witten
theory, BCOV boldly conjectured that Gromov-Witten generating function of any
Calabi-Yau manifolds are in fact quasi-modular forms. A key idea in [BCOV] is
that the B-model Gromov-Witten function should be modular but non-holomorphic.
Furthermore, its anti-holomorphic dependence is governed by the famous
holomorphic anomaly equations. During the last decade, Klemm and his
collaborators have put forth a series of papers to solve the holomorphic
anomaly equations [ABK, HKQ]. One upshot is a stunning predication of Gromov-
Witten invariants of quintic 3-fold up to genus 51. Indeed, this is a great
achievement since mathematicians can only compute Gromov-Witten invariants for
genus zero and one. Motivated by the physical intuition, there were two
independent works recently in mathematics to establish the modularity of
Gromov-Witten theory rigorously for local $\mathbb{P}^{2}$ [CI2] and elliptic
orbifolds $\mathbb{P}^{1}$ [KS, MR]. Let’s briefly describe the authors’ work
on the elliptic orbifolds $\mathbb{P}^{1}$. The current article can be thought
as a sequel.
Let $X$ be a projective manifold and $\overline{\mathcal{M}}_{g,n}(X,\beta)$
be the moduli space of genus-$g$, degree-$\beta$ stable maps with $n$
markings, where $\beta$ is a nef class in $H^{2}(X,\mathbb{Z})$, i.e.,
$\beta\in{\rm NE}(X)$. Let ${\rm ev}_{i}$ be the evaluation map at the $i$-th
marked point $p_{i}$ and $\psi_{i}\in H^{*}(\overline{\mathcal{M}}_{g,n})$ be
the first Chern class of the cotangent line bundle at $p_{i}$. Choose elements
$\gamma_{i}$ in $H^{*}(X,\mathbb{Q})$ with $\gamma_{0}=1\in
H^{0}(X,\mathbb{Q})$.
$\pi:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}$ be
the stabilization of the forgetful morphism. The numerical GW invariants with
ancestors are defined by
(1.1)
$\langle\tau_{\iota_{1}}(\gamma_{1}),\cdots,\tau_{\iota_{n}}(\gamma_{n})\rangle^{X}_{g,n,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\rm
vir}}\prod_{i=1}^{n}{\rm
ev}^{*}_{i}(\gamma_{i})\cup\pi^{*}\psi^{\iota_{i}}_{i}.$
The above invariant is zero unless
$\sum_{i=1}^{n}({\rm
deg}_{\mathbb{C}}(\gamma_{i})+\iota_{i})=c_{1}(TX)\cdot\beta+(3-\mathop{\rm
dim}\nolimits_{\mathbb{C}}X)(g-1)+n.$
The advantage of Calabi-Yau manifolds, such as the elliptic curve $E$, is that
$c_{1}(TX)=0$ and hence the dimension constraint is independent of $\beta$.
For the elliptic curve $E$, the degree $\beta=d\cdot\mathcal{D}$, where $d$ is
a non-negative integer and $\mathcal{D}$ is a nef generator of
$H^{2}(E,\mathbb{Z})$. Then, it is natural to define
(1.2)
$\langle\tau_{\iota_{1}}(\gamma_{1}),\cdots,\tau_{\iota_{n}}(\gamma_{n})\rangle^{E}_{g,n}(q)=\sum_{d\geq
0}\langle\tau_{\iota_{1}}(\gamma_{1}),\cdots,\tau_{\iota_{n}}(\gamma_{n})\rangle^{E}_{g,n,d}\,q^{d},$
where $q$ is the Novikov variable that we use to keep track of the degree
$\beta$. In our case, the function (1.2) can be seen as an ancestor Gromov-
Witten function along $t\cdot\mathcal{D}\in H^{2}(X,\mathbb{Z})$ by setting
$q=e^{t}$ (see Section 5). The authors proved the modularity for the elliptic
orbifolds $\mathbb{P}^{1}$ with weights of non-trivial orbifold points are
$(3,3,3),(2,4,4),(2,3,6)$. These orbifolds are the quotients of some elliptic
curve $E$ by
$\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/6\mathbb{Z}$
respectively.
To state the theorem, let $\mathcal{X}$ be one of the three elliptic orbifolds
$\mathbb{P}^{1}$. Again, $c_{1}(T\mathcal{X})=0$ in these cases. We can choose
elements $\gamma_{i}$ of $H^{*}_{CR}(\mathcal{X})$ and define
(1.3)
$\langle\tau_{\iota_{1}}(\gamma_{i_{1}}),\cdots,\tau_{\iota_{n}}(\gamma_{i_{n}})\rangle^{\mathcal{X}}_{g,n}(q)$
similarly. The main result of [KS, MR] is the following modularity theorem.
###### Theorem 1.1.
[MR] Suppose that $\mathcal{X}$ is one of the three elliptic orbifolds
$\mathbb{P}^{1}$ from above. For any multi-indices $\iota_{j},i_{j}$, the GW
invariant (1.3) converges to a quasi-modular form of an appropriate weight for
a finite index subgroup $\Gamma$ of $SL_{2}(\mathbb{Z})$ under the change of
variables $q=e^{2\pi i\tau/3}$, $e^{2\pi i\tau/4},$ $e^{2\pi i\tau/6},$
respectively (see [MR] for the subgroup $\Gamma$ and the weights of the quasi-
modular forms).
The same theorem for elliptic curves were proved ten years ago by Okounkov-
Pandharipande [OP].
Recall that one can construct Gromov-Witten cycles (cohomological field
theories) by a partial integration, i.e., pushforward via the forgetfull
morphism
(1.4)
$\Lambda_{g,n,\beta}^{X}(\gamma_{1},\cdots,\gamma_{n})=\pi_{*}\big{(}\prod_{i=1}^{n}{\rm
ev}^{*}_{i}(\gamma_{i})\big{)}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q}).$
The degree of the cycle is computed from the dimension axiom,
$\mathop{\rm
deg}\nolimits_{\mathbb{C}}\Lambda^{X}_{g,n,\beta}(\gamma_{1},\cdots,\gamma_{n})=(g-1)\mathop{\rm
dim}\nolimits_{C}(X)+\sum_{i=1}^{n}\mathop{\rm
deg}\nolimits_{\mathbb{C}}(\gamma_{i})-c_{1}(TX)\cdot\beta.$
The numerical Gromov-Witten invariants are obtained by
$\langle\tau_{\iota_{1}}(\gamma_{1}),\cdots,\tau_{\iota_{c}}(\gamma_{n})\rangle^{X}_{g,n,\beta}=\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g,n,\beta}^{X}(\gamma_{1},\cdots,\gamma_{n})\cup\prod_{i=1}^{n}\psi^{\iota_{i}}_{i}.$
Motivated by the corresponding work in number theory [Z], we want to consider
the generating function of Gromov-Witten cycles
(1.5) $\Lambda_{g,n}^{X}(\gamma_{1},\cdots,\gamma_{n})(q)=\sum_{\beta\in{\rm
NE}(X)}\Lambda^{X}_{g,n,\beta}(\gamma_{1},\cdots,\gamma_{n})\,q^{\beta}.$
We view the RHS of (1.5) as a function on $q$ taking value in
$H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$. To emphasise this
perspective, we sometimes refer to it as cycle-valued generating function. The
main theorem of this paper is
###### Theorem 1.2.
Suppose that $\mathcal{X}$ is one of the three elliptic orbifolds
$\mathbb{P}^{1}$ with three non-trivial orbifold points; then
$\Lambda_{g,n}^{\mathcal{X}}(\gamma_{1},\cdots,\gamma_{n})(q)$ converges to a
cycle-valued quasi-modular form of an appropriate weight for a finite index
subgroup $\Gamma$ of $SL_{2}(\mathbb{Z})$ under the change of variables
$q=e^{2\pi i\tau/3}$, $e^{2\pi i\tau/4},$ $e^{2\pi i\tau/6},$ respectively.
We should mention that the above cycle-valued modularity theorem is not yet
known for elliptic curve.
We obtain the modularity of numerical Gromov-Witten invariants by integrating
the $\Lambda_{g,n}^{\mathcal{X}}(\gamma_{1},\cdots,\gamma_{n})$ with psi-
classes over the fundamental cycle $[\overline{\mathcal{M}}_{g,n}]$. On the
other hand, we can also use other interesting classes of
$\overline{\mathcal{M}}_{g,n}$ such as $\kappa_{i}$’s or Hodge class
$\lambda_{i}$’s.
Suppose that $P$ is a polynomial of $\psi_{i},\kappa_{i},\lambda_{i}$. We
define a generalized numerical Gromov-Witten invariants
$\langle\gamma_{1},\cdots,\gamma_{n};P\rangle^{X}_{g,n,\beta}=\int_{\overline{\mathcal{M}}_{g,n}}P\cup\Lambda_{g,n,\beta}^{X}(\gamma_{1},\cdots,\gamma_{n})$
and its generating function
$\langle\gamma_{1},\cdots,\gamma_{n};P\rangle^{X}_{g,n}(q)=\sum_{\beta\in{\rm
NE}(X)}\langle\gamma_{1},\cdots,\gamma_{n};P\rangle^{X}_{g,n,\beta}\,q^{\beta}.$
Here, we set it to be zero if the dimension constraint are not satisfied.
###### Corollary 1.3.
Suppose that $\mathcal{X}$ is one of the above three elliptic orbifolds
$\mathbb{P}^{1}$. Then, the above generalized numerical Gromov-Witten
generating functions are quasi-modular forms for the same modular group and
weights given by the main theorem.
Recall that the proof of the numerical version consists of two steps. The
first step is to construct a higher genus B-model theory (modulo an extension
problem) and prove its modularity. Then, the second step is to prove mirror
theorems to match it with a Gromov-Witten theory which will solve the
extension property as well as inducing the modularity for a Gromov-Witten
theory. In this paper, we follow the same outline, i.e., our strategy can be
carried out on the cycle level. Our main new ingredient is Teleman’s
reconstruction theorem [T].
The paper is organized as follows. In Section 2, we will review the action of
upper-triangular symplectic operators on a cohomological field theory, which
will be the main tool of the paper. In Section 3, we review the construction
of global Frobenius manifold structures from [MR]. Using it, we can define
Givental B-model cohomological field theory as indicated by Telemann [T]. In
Section 4, we calculate the action of the monodromy group on the Givental’s
B-model cohomological field theory and prove the (quasi-)modularity. Finally,
in Section 5, we prove the mirror theorems on the cycle level. Here, the
original $g$-reduction argument does not apply. We replace it by Teleman’s
reconstruction theorem [T].
### 1.1. Acknowledgements
The work of the first author is supported by Grant-In-Aid and by the World
Premier International Research Center Initiative (WPI Initiative), MEXT,
Japan. The second author is partially supported by a NSF grant. The second and
third authors would like to thank Hiroshi Iritani for interesting discussions
on the convergence of Gromov-Witten theory. The third author would like to
thank Emily Clader, Nathan Priddis and Mark Shoemaker for helpful discussions
on Givental’s theory. Finally, three of us would like to thank IPMU for
hospitality where the part of this work is carried out. We thank Arthur
Greenspoon for editorial assistance.
## 2\. Cohomological field theory and quantization
The quantization formalism in Gromov-Witten theory was introduced by Givental
in [G1] and then revisited by Teleman at the cohomological field theory level
in [T]. The latter will be used in this article. For the readers’ convenience,
we give a brief introduction here.
$\pi_{g,n,k}:\overline{\mathcal{M}}_{g,n+k}\rightarrow\overline{\mathcal{M}}_{g,n}$
be the stabilization of the morphism that forgets the last $k$ marked points.
For simplicity, we will omit the subscripts if they are indicated in the
context.
### 2.1. Cohomological field theories
Let $H$ be a vector space of dimension $N$ with a unit 1 and a non-degenerate
paring $\eta$. Without loss of generality, we always fix a basis of $H$, say
$\mathscr{S}:=\\{\partial_{i},i=0,\cdots,N-1\\}$, and we set
$\partial_{0}={\bf 1}$. Let $\\{\partial^{j}\\}$ be the dual basis in the dual
space $H^{\vee}$, (i.e., $\eta(\partial_{i},\partial^{j})=\delta_{i}^{j}$). A
_Cohomological field theory_ (or CohFT) is a set of multi-linear maps
$\Lambda=\\{\Lambda_{g,n}\\}$, with
$\Lambda_{g,n}:H^{\otimes n}\longrightarrow
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C}),$
or equivalently,
$\Lambda_{g,n}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes(H^{\vee})^{\otimes n},$
defined for each stable genus $g$ curve with $n$ marked points,i.e.,
$2g-2+n>0$. Furthermore, $\Lambda$ satisfies a set of axioms (CohFT axioms)
described below:
1. (i)
($S_{n}$-invariance) For any $\sigma\in S_{n}$, and
$\gamma_{1},\cdots,\gamma_{n}\in H$; then
$\Lambda_{g,n}(\gamma_{\sigma(1)},\cdots,\gamma_{\sigma(n)})=\Lambda_{g,n}(\gamma_{1},\cdots,\gamma_{n}).$
2. (ii)
(Gluing tree) Let
$\rho_{tree}:\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\to\overline{\mathcal{M}}_{g,n}$
where $g=g_{1}+g_{2},n=n_{1}+n_{2}$, be the morphism induced from gluing the
last marked point of the first curve and the first marked point of the second
curve; then
$\displaystyle\rho_{tree}^{*}$
$\displaystyle\big{(}\Lambda_{g,n}(\gamma_{1},\cdots,\gamma_{n})\big{)}$
$\displaystyle=\sum_{\alpha,\beta\in\SS}\Lambda_{g_{1},n_{1}+1}(\gamma_{1},\cdots,\gamma_{n_{1}},\alpha)\eta^{\alpha,\beta}\Lambda_{g_{2},n_{2}+1}(\beta,\gamma_{n_{1}+1},\cdots,\gamma_{n}).$
Here $\big{(}\eta^{\alpha,\beta}\big{)}_{N\times N}$ is the inverse matrix of
$\big{(}\eta(\alpha,\beta)\big{)}_{N\times N}$.
3. (iii)
(Gluing loop) Let
$\rho_{loop}:\overline{\mathcal{M}}_{g-1,n+2}\to\overline{\mathcal{M}}_{g,n},$
be the morphism induced from gluing the last two marked points; then
$\rho_{loop}^{*}\big{(}\Lambda_{g,n}(\gamma_{1},\cdots,\gamma_{n})\big{)}=\sum_{\alpha,\beta\in\SS}\Lambda_{g-1,n+2}(\gamma_{1},\cdots,\gamma_{n},\alpha,\beta)\eta^{\alpha,\beta}.$
4. (iv)
(Pairing)
$\int_{\overline{\mathcal{M}}_{0,3}}\Lambda_{0,3}({\bf
1},\gamma_{1},\gamma_{2})=\eta(\gamma_{1},\gamma_{2}).$
If in addition the following axiom holds
* (v)
(Flat identity) Let
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ be the
forgetful morphism; then
$\Lambda_{g,n+1}(\gamma_{1},\cdots,\gamma_{n},{\bf
1})=\pi^{*}\Lambda_{g,n}(\gamma_{1},\cdots,\gamma_{n}).$
then we say that $\Lambda$ is a CohFT with a flat identity.
Note that $\Lambda_{0,3}$ will induce a Frobenius multiplication $\bullet$ on
$(H,\eta)$, defined by
(2.1)
$\eta(\alpha\bullet\beta,\gamma)=\int_{\overline{\mathcal{M}}_{0,3}}\Lambda_{0,3}(\alpha,\beta,\gamma);$
We refer to $(H,\eta,\bullet)$ as the Frobenius algebra underlying $\Lambda$,
or simply as the state space of $\Lambda$. The CohFT is called semisimple if
the underlying Frobenius algebra is semisimple.
### 2.2. Examples of CohFTs
Let $\mathbb{C}^{N}$ be the complex vector space equipped with the standard
bi-linear pairing: $(e_{i},e_{j})=\delta_{i,j}$. Let
$\Delta=(\Delta_{1},\cdots,\Delta_{N})$ be a sequence of non-zero complex
numbers. The following definition
(2.2)
$I^{N,\Delta}_{g,n}(e_{i_{1}},\dots,e_{i_{n}}):=\begin{cases}\Delta_{i}^{g-1+\frac{n}{2}}&\mbox{if
}i=i_{1}=i_{2}=\cdots=i_{n},\\\ 0&\mbox{otherwise},\end{cases}$
induces a CohFT on $\mathbb{C}^{N}$ which we call a rank $N$ trivial CohFT.
The Frobenius algebra underlying $I^{N,\Delta}$ will be denoted by
$(\mathbb{C}^{N},\Delta)$. Note that the Frobenius multiplication is given by
$e_{i}\bullet e_{j}=\delta_{ij}\,\sqrt{\Delta_{i}}\,e_{i}.$
Another famous example comes from Gromov-Witten theory (cf. [KM, CheR]). Let
$X$ be a projective variety (or orbifold), let $H$ be its cohomology
$H^{*}(X)$ (or Chen-Ruan cohomology $H^{*}_{\rm CR}(X)$), $\eta$ be the
Poincaré pairing. Then $\Lambda_{g,n}^{X}(q)$ defined in (1.5) gives a CohFT
for $q=0$. The above axioms make sense for cohomology classes
$\Lambda^{X}_{g,n}(q)$ that have coefficients in some ring of formal power
series. In such a case we say that we have a formal cohomological field
theory. A priori, the CohFT in (1.5) is only formal.
### 2.3. Givental’s formalism
Following Givental, we introduce the vector space $\mathcal{H}=H((z))$ of
formal Laurent series in $z^{-1}$. Furthermore, $\mathcal{H}$ is equipped with
the following symplectic structure $\Omega$:
$\displaystyle\Omega(f(z),g(z))={\rm res}_{z=0}(f(-z),g(z))dz,\quad
f(z),g(z)\in\mathcal{H},$
where for brevity we put $(a,b)=\eta(a,b)$ for $a,b\in H$. Note that
$\mathcal{H}$ has a polarization
$\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$
with $\mathcal{H}_{+}=H[z]$ and $\mathcal{H}_{-}=z^{-1}H[[z^{-1}]]$, which
allows us to identify $\mathcal{H}\cong T^{*}\mathcal{H}_{+}$. We fix a
Darboux coordinate system $q_{k}^{i},p_{l,j}$ for $\mathcal{H}$ via
$f(z)=\sum_{k=0}^{\infty}\sum_{i=0}^{N-1}q^{i}_{k}\,\partial_{i}z^{k}+\sum_{l=0}^{\infty}\sum_{j=0}^{N-1}p_{l,j}\,\partial^{j}(-z)^{-l-1}\in\mathcal{H},$
For convenience, we put
(2.3) $\mathbf{q}_{k}:=(q_{k}^{1},\cdots,q_{k}^{N})\hskip
28.45274pt\text{and}\hskip
28.45274pt\mathbf{q}:=(\mathbf{q}_{0},\mathbf{q}_{1},\cdots).$
In this paper, we focus on the subgroup $\mathcal{L}^{(2)}{\rm GL}(H)$ of the
loop group $\mathcal{L}{\rm GL}(H)$ consisting of symplectomorphisms
$T:\mathcal{H}\to\mathcal{H}$. Note that such symplectomorphisms are defined
by the following equation:
${}^{*}T(-z)T(z)=\rm{Id},$
where ${}^{*}T$ is the adjoint operator with respect to the bi-linear pairing
$\eta$, i.e.,
$(^{*}Tf,g)=(f,Tg).$
We will allow symplectomorphism $E$ of the following form:
$E:={\rm Id}+E_{1}z+E_{2}z^{2}+\cdots\in{\rm End}(H)[[z]].$
They form a group which we denote by $\mathcal{L}^{(2)}_{+}{\rm GL}(H)$ and we
refer to its elements as _upper-triangular_ transformations.
Next, we want to define the quantization $\widehat{E}$. Note that $A=\log E$
is a well-defined infinitesimal symplectomorphism, i.e., ${}^{*}A=-A$. For any
infinitesimal symplectomorphism $A$, we can associate a quadratic Hamiltonian
$h_{A}$ on $\mathcal{H}$,
(2.4) $h_{A}(f)=\frac{1}{2}\Omega(Af,f).$
The quadratic Hamiltonians are quantized by the rules:
(2.5) $(p_{k,i}p_{l,j})^{^}=\hbar\frac{\partial^{2}}{\partial
q_{k}^{i}\partial
q_{l}^{j}},\quad(p_{k,i}q_{l}^{j})^{^}=(q_{l}^{j}p_{k,i})^{^}=q_{l}^{j}\frac{\partial}{\partial
q_{k}^{i}},\quad(q_{k}^{i}q_{l}^{j})^{^}=\frac{q_{k}^{i}q_{l}^{j}}{\hbar}.$
The quantization of $E$ is defined by
$\widehat{E}=e^{\widehat{A}}:=e^{\widehat{h_{A}}}.$
For an upper-triangular symplectomorphism $E$, there is an explicit formula
for the quantization $\widehat{E}$. Put
$\mathbf{q}(z)=\sum_{k=0}^{\infty}\sum_{i=0}^{N-1}q^{i}_{k}\,\partial_{i}z^{k}\in
H[[z]].$
Denote the dilaton shift by $\widetilde{\mathbf{q}}(z)=\mathbf{q}(z)+{\bf
1}z$, i.e., $\widetilde{q}^{i}_{k}=q^{i}_{k}+\delta^{1}_{k}\delta^{i}_{0}$.
Recall that the ancestor GW potential of $X$ is
(2.6)
$\mathcal{A}^{X}(\hbar,\mathbf{q}(z)):=\exp\Big{(}\sum_{g,n}\sum_{\beta\in{\rm
NE}(X)}\sum_{\iota_{i},k_{i}=0}^{\infty}\frac{\hbar^{g-1}\langle\tau_{\iota_{1}}\partial_{k_{1}},\cdots,\tau_{\iota_{n}}\partial_{k_{n}}\rangle_{g,n,\beta}^{X}\,q^{\beta}}{n!}\prod_{i=1}^{n}\widetilde{q}_{k_{i}}^{\iota_{i}}\Big{)}.$
$\mathcal{A}^{X}(\hbar,\mathbf{q}(z))$ belongs to a Fock space
$\mathbb{C}[[\mathbf{q}_{0},\widetilde{\mathbf{q}}_{1},\mathbf{q}_{2},\cdots]]$.
The action of the quantization operator $\widehat{E}$, whenever it makes
sense, is given by the following formula:
(2.7)
$\widehat{E}\big{(}\mathcal{A}^{X}(\hbar,\mathbf{q}(z))\big{)}=\left.\big{(}e^{W_{E}}\mathcal{A}^{X}(\hbar,\mathbf{q}(z))\big{)}\right|_{\mathbf{q}\mapsto
E^{-1}\mathbf{q}},$
where $E^{-1}\mathbf{q}$ is the change of $\mathbf{q}$-coordinate
$(E^{-1}\mathbf{q})_{k}^{i}=\sum_{l=0}^{k}\sum_{j=0}^{N-1}(E^{-1})_{l}^{ji}q_{k-l}^{j}.$
And $W_{E}$ is the quadratic differential operator
(2.8)
$W_{E}:=\frac{\hbar}{2}\sum_{k,l=0}^{\infty}\sum_{i,j=0}^{N-1}\left(\partial^{i},V_{kl}(\partial^{j})\right)\frac{\partial^{2}}{\partial
q_{k}^{i}\partial q_{l}^{j}},$
whose coefficients $V_{kl}\in{\rm End}(H)$ are given by
(2.9) $\sum_{k,l\geq 0}V_{kl}(-z)^{k}(-w)^{l}=\frac{E^{*}(z)E(w)-{\rm
Id}}{z+w}.$
###### Remark 2.1.
Givental also considered the quantization of a general symplectomorphism of
the form $e^{A}$. For example, $A$ could be lower triangular in the sense
containing the negative power of $z$. The lower triangular one can not be lift
to cycle level. Hence, it will not be considered here.
### 2.4. Cycle-valued Quantization
Teleman [T] was able to lift the quantization of an upper triangular
symplectic transformation to the level of cohomological field theory. Let us
describe his construction. According to formula (2.7), the action of
$\widehat{E}$ is a composition of two operations: exponential of the Laplace
type operator (2.8) followed by the coordinate change $\mathbf{q}\mapsto
E^{-1}\mathbf{q}$.
#### 2.4.1. Coordinate Change
Let $\Lambda_{g,n}$ be any multi-linear function on $H^{\otimes n}$ with
values in the cohomology ring of $\overline{\mathcal{M}}_{g,n}$. We can extend
$\Lambda_{g,n}$ from $H^{\otimes n}$ to $\mathcal{H}^{\otimes n}_{+}$ uniquely
so that multiplication by $z$ is compatible with the multiplication by psi-
classes, i.e.,,
(2.10) $\Lambda_{g,n}(\sum_{i\geq 0}\gamma_{1}z^{i},\cdots)=\sum_{i\geq
0}\Lambda_{g,n}(\gamma_{1},\cdots)\psi^{i}_{1}.$
Given an isomorphis of $\mathbb{C}[z]$-modules
$\Phi(z):H_{1}[[z]]\to H_{2}[[z]],$
we define
$(\Phi(z)\circ\Lambda)_{g,n}(\gamma_{1},\cdots,\gamma_{n})=\Lambda_{g,n}(\Phi(z)^{-1}(\gamma_{1}),\cdots,\Phi(z)^{-1}(\gamma_{n}))\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C}).$
Note that even if $\Lambda$ is a CohFT, $\Phi(z)\circ\Lambda$ might fail to be
a CohFT.
#### 2.4.2. Feynman type sum
The action of the exponential of the Laplacian (2.8) can be described in terms
of sum over graphs. Let us explain this in some more details. For a given
graph $\Gamma$ let us denote by $V(\Gamma)$ the set of vertices, $E(\Gamma)$
the set of edges, and by $T(V)$ the set of tails. For a fixed vertex $v\in
V(\Gamma)$ we denote by $E_{v}(\Gamma)$ and $T_{v}(\Gamma)$ respectively the
set of edges and tails incident with $v$. The graph is decorated in the
following way: each vertex $v$ is assigned a non-negative number $g_{v}$
called genus of $v$; there is a bijection $t\mapsto m(t)$ between the set of
tails and the set of integers $\\{1,2,\dots,{\rm Card}(T(\Gamma))\\}$, and
finally every flag $(v,e)$ (i.e., a pair consists a vertex and an incident
edge) is decorated with a vector $z^{k}\partial^{i}$ ($k\geq 0$).
Furthermore, for a given edge $e$ we define a propagator $V_{e}$ as follows.
Let $v^{\prime}$, $v^{\prime\prime}$ be the two vertexes incident with $e$ and
let $z^{k^{\prime}}\partial^{i^{\prime}}$ and
$z^{k^{\prime\prime}}\partial^{i^{\prime\prime}}$ be the labels respectively
of the flags $(v^{\prime},e)$ and $(v^{\prime\prime},e)$; then we define
$\displaystyle
V_{e}=\left(\partial^{i^{\prime}},V_{k^{\prime},k^{\prime\prime}}\partial^{i^{\prime\prime}}\right).$
Note that since
${}^{*}V_{k^{\prime},k^{\prime\prime}}=V_{k^{\prime\prime},k^{\prime}}$ the
definition of $V_{e}$ is independent of the orientation of the edge $e$. For
every vertex $v$ we define the differential operator
$\displaystyle D_{\mathbf{q}}^{v}=\prod_{e\in
E_{v}(\Gamma)}\,{\partial}/{\partial q_{k(e)}^{i(e)}},$
where $z^{k(e)}\partial^{(i(e)}$ is the label of the flag $(v,e)$.
Given any formal function
$\mathcal{A}(\hbar;\mathbf{q})=\exp\Big{(}\sum\hbar^{g-1}\mathcal{F}^{(g)}(\mathbf{q})\Big{)}$
we have
(2.11) $e^{W_{E}}\
\mathcal{A}(\hbar;\mathbf{q})=\exp\Big{(}\sum_{\Gamma}\frac{1}{|{\rm
Aut}(\Gamma)|}\,\prod_{e\in E(\Gamma)}\,V_{e}\prod_{v\in
V(\Gamma)}\,D_{\mathbf{q}}^{v}\mathcal{F}^{(g_{v})}(\mathbf{q})\ \Big{)},$
where the sum is over all connected decorated graphs $\Gamma$ and $|{\rm
Aut}(\Gamma)|$ is the number of automorphisms of $\Gamma$ compatible with the
decoration.
Motivated by formula (2.11) we define
(2.12)
$(e^{W_{E}}\circ\Lambda)_{g,n}(\gamma_{1}\otimes\cdots\otimes\gamma_{n})$
by the following formula:
$\displaystyle\sum_{\Gamma}\frac{1}{|{\rm Aut}(\Gamma)|}\,\prod_{e\in
E(\Gamma)}\,V_{e}\prod_{v\in
V(\Gamma)}\,\Lambda_{g_{v},r_{v}+n_{v}}\Big{(}\otimes_{e\in
E_{v}(\Gamma)}\partial_{i(e)}\psi^{k(e)}\otimes_{t\in
T_{v}(\Gamma)}\gamma_{m(t)}\Big{)},$
where $r_{v}={\rm Card}(E_{v}(\Gamma))$, $n_{v}={\rm Card}(T_{v}(\Gamma))$,
and the sum is over all connected, decorated, genus-g graphs $\Gamma$ with $n$
tails. Note that this definition is compatible with (2.11) in a sense that the
potential of the multi-linear maps (2.12) coincides with (2.11).
For an upper-triangular symplectic transformation $E$, we define
(2.13) $\widehat{E}\circ\Lambda:=E\circ(e^{W_{E}}\circ\Lambda).$
Using induction on the number of nodes, it is not hard to check that
$\widehat{E}\circ\Lambda$ is a CohFT (see [T]).
#### 2.4.3. Classification of semi-simple CohFT
Let $(H,\eta,\bullet)$ be a semi-simple Frobenius algebra. We pick an
orthonormal basis $\\{e_{i}\\}$ of $H$, which allows us to identify
$(H,\eta,\bullet)$ with the Frobenius algebra of a trivial CohFT, i.e.,, the
state space of $I^{N,\Delta}$ for a particular $\Delta$ (see (2.2)). In this
section we would like to recall the classification of all CohFTs whose state
space is $(H,\eta,\bullet)$.
The space of such CohFTs admits the action (2.13) of the group
$\mathcal{L}^{(2)}_{+}{\rm GL}(H)$. Note that this action does not change the
Frobenius multiplication on $H$. On the other hand, the Abelian group
$z^{2}H[z]$ (with group operation addition) acts on the space of CohFTs via
translations. Namely, given $a(z)\in z^{2}H[z]$, we define
$\displaystyle(T_{a(z)}\circ\Lambda)_{g,n}(\gamma_{1},\cdots,\gamma_{n})$
by the fomula
(2.14) $\sum_{k\geq
0}\frac{(-1)^{k}}{k!}\pi_{*}\big{(}\Lambda_{g,n+k}(\gamma_{1},\cdots,\gamma_{n},a(z),\cdots,a(z))\big{)},$
where for each $k$, the map $\pi$ in the $k$-th summand is the map forgetting
the last $k$ marked points. This action also preserves the Frobenius
multiplication. Moreover, the following formula holds:
(2.15) $T_{a(z)}\circ\widehat{E}\circ T^{-1}_{a(z)}=\widehat{E}\circ
T_{a(z)-E^{-1}a(z)},$
i.e.,, we have an action of the group
$z^{2}H[z]\rtimes\mathcal{L}^{(2)}_{+}{\rm GL}(H)$ on the set of CohFTs with
state space $(H,\eta,\bullet)$. According to Teleman (see [T], Theorem 2)
###### Theorem 2.2.
([T]) The orbit of the group $z^{2}H[z]\rtimes\mathcal{L}^{(2)}_{+}{\rm
GL}(H)$ containing $I^{N,\Delta}$ consists of all CohFTs whose underlying
Frobenius algebra is $(H,\eta,\bullet)$.
Let $a(z)\in zH[z]$ be arbitrary. Although the translation $T_{a(z)}$ is
singular and will be not well defined after replacing multiplication by $z$ in
terms of multiplication by psi-classes, the RHS of formula (2.15) always makes
sense since $a(z)-E^{-1}a(z)\in z^{2}H[z]$. Therefore, we can define the
following subgroup of $z^{2}H[z]\rtimes\mathcal{L}^{(2)}_{+}{\rm GL}(H)$:
$\displaystyle\mathcal{L}^{(2)}_{a(z)}{\rm
GL}(H)=T_{a(z)}\circ\mathcal{L}^{(2)}_{+}{\rm GL}(H)\circ T^{-1}_{a(z)}.$
The following fact follows easily from Theorem 2.2
###### Corollary 2.3.
Let $a(z)={\bf 1}\,z$; then the orbit of the subgroup
$\mathcal{L}^{(2)}_{a(z)}{\rm GL}(H)$ containing $I^{N,\Delta}$ consists of
all CohFTs with a flat identity whose underlying Frobenius algebra is
$(H,\eta,\bullet)$.
#### 2.4.4. Higher-genus reconstruction
For $\mathbf{t}=\sum_{i=0}^{N-1}t_{i}\partial_{i}\in H,$ we define a
translation operator $T_{\mathbf{t}}$ acting on a CohFT $\Lambda$ by
(2.16)
$(T_{\mathbf{t}}\circ\Lambda)_{g,n}(\gamma_{1},\cdots,\gamma_{n}):=\sum_{k\geq
0}\frac{(-1)^{k}}{k!}\pi_{*}\big{(}\Lambda_{g,n+k}(\gamma_{1},\cdots,\gamma_{n},\mathbf{t},\cdots,\mathbf{t})\big{)}.$
For brevity we put ${}_{\mathbf{t}}\Lambda:=T_{\mathbf{t}}\circ\Lambda$.
According to Teleman (see [T], Proposition 7.1), ${}_{\mathbf{t}}\Lambda$ is a
formal CohFT, i.e.,,
$({}_{\mathbf{t}}\Lambda)_{g,n}\in\big{(}H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes\mathbb{C}[[\mathbf{t}]]\big{)}\otimes(H^{\vee})^{\otimes
n},\quad\mathbb{C}[[\mathbf{t}]]:=\mathbb{C}[[t_{0},\cdots,t_{N-1}]].$
It induces a ring structure on $H$ with the multiplication
$\bigstar_{\mathbf{t}}$ defined by
(2.17)
$\gamma_{1}\bigstar_{\mathbf{t}}\gamma_{2}=\sum_{\alpha,\beta\in\mathscr{S}}\int_{\overline{\mathcal{M}}_{0,3}}({}_{\mathbf{t}}\Lambda)_{0,3}(\gamma_{1},\gamma_{2},\alpha)\,\eta^{\alpha,\beta}\,\beta,\quad\gamma_{1},\gamma_{2}\in
H.$
Let us assume that the vector space $H$ is graded and that
$\\{\partial_{i}\\}$ is a homogeneous basis with ${\rm
deg}(\partial_{i})=1-d_{i}$. We further assume that we are given an Euler
vector field of the form
$\displaystyle\mathcal{E}=\sum_{i=0}^{N-1}\,d_{i}\,t_{i}\,\frac{\partial}{\partial
t_{i}}+\sum_{j:d_{j}=0}\ r_{j}\,\partial_{j},$
where $r_{j}$ are some constants, so that the ring of formal power series
$\mathbb{C}[[\mathbf{t}]]$ is a graded ring: an element $f(\mathbf{t})$ is
homogeneous of degree $d_{f}$ iff $E(f(\mathbf{t}))=d_{f}\,f(\mathbf{t})$. The
CohFT is called homogeneous of conformal weight $d$ if $H$ is graded, there
exists an Euler vector field, and the maps:
$\displaystyle{}_{\mathbf{t}}\Lambda_{g,n}:H^{\otimes n}\rightarrow
H^{*}(\overline{\mathcal{M}}_{g,n};\mathbb{C})\otimes\mathbb{C}[[\mathbf{t}]]$
are homogeneous of weight $d(g-1)+n$. Here the source of
${}_{\mathbf{t}}\Lambda_{g,n}$ inherits the grading from $H$, the first tensor
factor in the target is graded by halfing the degree of a cohomology class,
and $\mathbb{C}[[\mathbf{t}]]$ is graded by $E$.
Let us assume further that the CohFT $\Lambda$ is _generically semisimple_ ,
i.e.,, the ring structure $\bigstar_{\mathbf{t}}$ is semisimple for generic
$\mathbf{t}$. We denote by $u_{i}(\mathbf{t})$ the corresponding canonical
coordinates, so that the map
$\displaystyle\Psi(\mathbf{t}):\mathbb{C}^{N}\rightarrow H,\quad
e_{i}\mapsto\sqrt{\Delta_{i}(\mathbf{t})}\partial/\partial u_{i}(\mathbf{t})$
identifies the Frobenius algebras $(H,\eta,\bigstar_{\mathbf{t}})$ and
$(\mathbb{C}^{N},\Delta(\mathbf{t}))$, where
$\displaystyle\Delta(\mathbf{t}):=(\Delta_{1}(\mathbf{t}),\dots,\Delta_{N}(\mathbf{t})).$
Let $U(\mathbf{t})$ be the diagonal matrix with entries
$u_{i}(\mathbf{t}),0\leq i\leq N-1$. Following Givental we define an upper-
triangular symplectic transformation $R(\mathbf{t})$, such that the formal
asymptotical series $\Psi(\mathbf{t})R(\mathbf{t})e^{U(\mathbf{t})}$ is a
solution to the differential equations (3.6) and (3.7). In fact, these
equations determine $R(\mathbf{t})$ uniquely in terms of $\Psi(\mathbf{t})$
and $U(\mathbf{t})$ (see [G2]). According to Teleman (see [T], Theorem 1) we
have the following higher-genus reconstruction result.
###### Theorem 2.4.
([T]) If $\Lambda$ is a homogeneous CohFT with flat identity and
$\bigstar_{\mathbf{t}}$ is (formal) semi-simple; then
$\displaystyle{}_{\mathbf{t}}\Lambda=\widehat{\Psi}(\mathbf{t})\circ(T_{z}\circ\widehat{R}(\mathbf{t})\circ
T^{-1}_{z})\circ I^{N,\Delta(\mathbf{t})},$
where $T_{z}:=T_{{\bf 1}\,z}$.
Let us finish this section by drawing an important corrolary from the above
theorem. The ancestor potential of a (singular) CohFT $\Lambda$ is a function
of $a(z)=\sum_{i\geq 0}a_{i}z^{i}\in H[[z]]$, defined by
(2.18)
$\mathcal{A}\big{(}\Lambda,\hbar,a(z)\big{)}=\exp\Big{(}\sum_{g,n}\frac{\hbar^{g-1}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g,n}(a(z),\cdots,a(z))\Big{)}.$
Note that the translation $T_{z}^{-1}$ induces the so called dilaton shift,
i.e, $\mathbf{q}(z)\mapsto\widetilde{\mathbf{q}}(z)=:\mathbf{q}(z)+{\bf 1}z$,
(2.19)
$\mathcal{A}\big{(}T_{z}^{-1}\circ\Lambda,\hbar,\mathbf{q}(z)\big{)}=\mathcal{A}\big{(}\Lambda,\hbar,\widetilde{\mathbf{q}}(z)\big{)}.$
Furthermore, following Givental, the formal ancestor potential
$\displaystyle\mathcal{A}^{{formal}}(\mathbf{t})(\hbar,\mathbf{q}(z))$
of (the germ of) the Frobenius structure $(H,\eta,\bigstar_{\mathbf{t}})$ is
defined by
(2.20) $\widehat{\Psi}(\mathbf{t})\ \widehat{R}(\mathbf{t})\
e^{(U(\mathbf{t})/z)^{^}}\ \prod_{i=1}^{\mu}\mathcal{A}^{\rm
pt}\big{(}\hbar\Delta_{i}(\mathbf{t}),{}^{i}\widetilde{\mathbf{q}}(z)\sqrt{\Delta_{i}(\mathbf{t})}\big{)},$
where $\widehat{\Psi}$ means change of the variables
$\mathbf{q}(z)\mapsto\Psi^{-1}(\mathbf{t})\,\mathbf{q}(z)$ and
$\mathcal{A}^{\rm pt}$ is the total ancestor potential of the CohFT
$I^{N=1,\Delta=1}$.
###### Corollary 2.5.
Under the same assumption as in Theorem 2.4 the following formula holds:
$\mathcal{A}({}_{\mathbf{t}}\Lambda,\hbar,\widetilde{\mathbf{q}}(z))=\mathcal{A}^{{formal}}(\mathbf{t})(\hbar,\mathbf{q}(z)).$
Finally, let us point out that if $\Lambda^{X}$ is the CohFT induced from the
Gromov-Witten theory of $X$; then
(2.21)
$\mathcal{A}^{X}(\hbar,\mathbf{q}(z)):=\mathcal{A}(\Lambda^{X},\hbar,\widetilde{\mathbf{q}}(z)),$
coincides with the so called total ancestor potential of $X$. In particular,
$\mathcal{A}^{\rm pt}$ in formula (2.20) is the total ancestor potential of a
point.
## 3\. Global Frobenius manifolds for simple elliptic singularities
In this section, we review the construction of global Frobenius manifolds for
simple elliptic singularities. We follow [MR].
### 3.1. Saito theory
K. Saito’s theory of primitive forms [S1] yields a certain flat structure on
the space of miniversal deformations of a singularity, which is known to be a
Frobenius structure cf. [He, ST]. We refer to it as the Saito’s Frobenius
manifold structure. Let us recall the general set up.
Recall (see [ArGV]) the action of the group of germs of holomorphic changes of
the coordinates $(\mathbb{C}^{N},0)\to(\mathbb{C}^{N},0)$ on the space of all
germs at $0$ of holomorphic functions. Let ${\bf
x}=(x_{1},\cdots,x_{N})\in\mathbb{C}^{N}$. Given a holomorphic germ $f({\bf
x})$ with an isolated critical point at ${\bf x}=0$, we say that the family of
functions $F(\mathbf{s},{\bf x})$ is a miniversal deformation of $f$ if it is
transversal to the orbit of $f$. One way to construct a miniversal deformation
is to choose a $\mathbb{C}$-linear basis $\\{\phi_{i}({\bf
x})\\}_{i=0}^{\mu-1}$ in the Jacobi algebra
$\mathcal{O}_{\mathbb{C}^{N},0}/\langle\partial_{x_{1}}f,\cdots,\partial_{x_{N}}f\rangle$.
Here $\mu$ is the rank of the Jacobi algebra as a vector space, also known as
the Milnor number or the multiplicity of the critical point. Then the
following family provides a miniversal deformation:
(3.1) $F(\mathbf{s},{\bf x})=f({\bf x})+\sum_{i=0}^{\mu-1}s_{i}\phi_{i}({\bf
x}),\quad\mathbf{s}=(s_{0},s_{1},\dots,s_{\mu-1})\in\mathcal{S},$
where $\mathcal{S}\subset\mathbb{C}^{\mu}$ is a small ball around
$0\in\mathbb{C}^{\mu}$. The domain of the function $F(s,\,)$ is chosen
uniformly for $s\in\mathcal{S}$ to be a certain open naighbourhood of
$0\in\mathbb{C}^{N}$, such that its boundary satisfies certain transversality
conditions (see [ArGV]). Slightly abusing the notation we write
$\mathbb{C}^{N}$, but we really mean an appropriate chosen open neighborhood
of $0.$ This should not cause any confusion. Moreover, we will apply Saito’s
theory only to singularities for which this special domain does coincide with
$\mathbb{C}^{N}$.
Put $X=\mathcal{S}\times\mathbb{C}^{N}$ and let $C\subset X$ be the critical
set of $F$, that is the support of the sheaf
$\displaystyle\mathcal{O}_{C}:=\mathcal{O}_{X}/\langle\partial_{x_{0}}F,\cdots,\partial_{x_{N-1}}F\rangle.$
We have the following maps:
$\displaystyle\begin{CD}\mathcal{S}\times\mathbb{C}^{N}&&\\\
@V{\varphi}V{}V\searrow&\\\
\mathcal{S}\times\mathbb{C}&@>{}>{p}>&\mathcal{S}\end{CD}\qquad\begin{tabular}[]{rl}&$\varphi(\mathbf{s},{\bf
x})=(\mathbf{s},F(\mathbf{s},{\bf x}))$,\\\ &\\\
&$p(\mathbf{s},\lambda)=\mathbf{s},$\end{tabular}$
The map $\partial/\partial s_{i}\mapsto\partial F/\partial s_{i}$ induces an
isomorphism between the sheaf $\mathcal{T}_{\mathcal{S}}$ of holomorphic
vector fields on $\mathcal{S}$ and $q_{*}\mathcal{O}_{C}$, where
$q=p\circ\varphi.$ In particular, for any $\mathbf{s}\in\mathcal{S},$ the
tangent space $T_{\mathbf{s}}\mathcal{S}$ is equipped with an associative
commutative multiplication $\bullet_{\mathbf{s}}$ depending holomorphically on
$\bf{s}\in\mathcal{S}$. In addition, if we have a volume form
$\omega=g(\mathbf{s},{\bf x})d^{N}{\bf x},$ where $d^{N}{\bf
x}=dx_{1}\wedge\cdots\wedge dx_{N}$ is the standard volume form; then
$q_{*}\mathcal{O}_{C}$ (hence $\mathcal{T}_{\mathcal{S}}$ as well) is equipped
with the residue pairing:
(3.3) $\langle\psi_{1},\psi_{2}\rangle=\frac{1}{(2\pi i)^{N}}\
\int_{\Gamma_{\epsilon}}\frac{\psi_{1}({\bf s,y})\psi_{2}({\bf
s,y})}{F_{y_{1}}\cdots F_{y_{N}}}\,\omega,$
where ${\bf y}=(y_{1},\cdots,y_{N})$ are unimodular coordinates for the volume
form, i.e., $\omega=d^{N}{\bf y}$, and $\Gamma_{\epsilon}$ is a real
$N$-dimensional cycle supported on $|F_{x_{i}}|=\epsilon$ for $1\leq i\leq N.$
Given a holomorphic function $f$ on $\mathbb{C}^{N}$ and a real number $m$ we
define
$\displaystyle{\rm Re}^{m}_{f}(\mathbb{C}^{N}):=\Big{\\{}x\in\mathbb{C}^{N}\
:\ {\rm Re}(f(x))\leq m\Big{\\}}.$
Let
(3.4) $J_{\mathcal{A}}(\mathbf{s},z)=(-2\pi
z)^{-N/2}\,zd_{\mathcal{S}}\,\int_{\mathcal{A}}e^{F({\bf s,x})/z}\omega,$
where $d_{\mathcal{S}}$ is the de Rham differential on $\mathcal{S}$ and
$\mathcal{A}$ is a semi-infinite cycle from
(3.5) $\lim_{\longleftarrow}H_{N}(\mathbb{C}^{N},{\rm
Re}_{F(\mathbf{s},\cdot)/z}^{-m}(\mathbb{C}^{N});\mathbb{C})\cong\mathbb{C}^{\mu}.$
By definition, the oscillatory integrals $J_{\mathcal{A}}$ are sections of the
cotangent sheaf $\mathcal{T}_{\mathcal{S}}^{*}$. According to Saito’s theory
of primitive forms [S1], there exists a volume form $\omega$ such that the
residue pairing is flat and the oscillatory integrals satisfy a system of
differential equations, which in flat-homogeneous coordinates ${\bf
t}=(t_{0},\dots,t_{\mu-1})$ have the form
(3.6) $z\partial_{i}J_{\mathcal{A}}({\bf t},z)=\partial_{i}\bullet_{\bf
t}J_{\mathcal{A}}({\bf t},z),$
where $\partial_{i}:=\partial/\partial t_{i}\ (0\leq i\leq\mu-1)$ and the
multiplication is defined by identifying vectors and covectors via the residue
pairing. Due to homogeneity the integrals satisfy a differential equation with
respect to the parameter $z\in\mathbb{C}^{*}$:
(3.7) $(z\partial_{z}+\mathcal{E})J_{\mathcal{A}}({\bf
t},z)=\Theta\,J_{\mathcal{A}}({\bf t},z),$
where
$\displaystyle\mathcal{E}=\sum_{i=0}^{\mu-1}d_{i}t_{i}\partial_{i},\quad(d_{i}:={\rm
deg}\,t_{i}={\rm deg}\,s_{i}),$
is the Euler vector field and $\Theta$ is the so-called Hodge grading operator
. The latter is defined by
$\displaystyle\Theta:\mathcal{T}^{*}_{S}\rightarrow\mathcal{T}^{*}_{S},\quad\Theta(dt_{i})=\Big{(}1-\frac{D}{2}-d_{i}\Big{)}dt_{i},$
where $D$ is the so called conformal dimension of the Frobenius manifold,
uniquely determined by the symmetry of the degree spectrum: the numbers
$d_{i}$ are symmetric with respect to the point $1-D/2$. The compatibility of
the system (3.6)–(3.7) implies that the residue pairing, the multiplication,
and the Euler vector field give rise to a conformal Frobenius structure of
conformal dimension $D$. We refer to [D, M] for the definition and more
details on Frobenius structures.
###### Theorem 3.1 ([He, ST]).
Let $f$ be an isolated singularity, a primitive form $\omega$ induces a germ
of Frobenius manifold structures
$(T_{\mathbf{s}}\mathcal{S},\langle,\rangle,\bullet_{\mathbf{s}},\mathcal{E},\partial_{0})$
with an Euler vector $\mathcal{E}$ and a flat identity $\partial_{0}$ for any
$\mathbf{s}\in\mathcal{S}$. It is homogeneous and generically semisimple.
### 3.2. Global Frobenius manifold structures for simple elliptic
singularities.
Simple elliptic singularities are classified by K.Saito (cf. [S2]) into three
different types, $\widetilde{E}_{6},\widetilde{E}_{7},\widetilde{E}_{8}$. In
this paper, we consider three families of simple elliptic singularities by
choosing a particular normal form in each family, see (3.8) below. The
differential equations for the primitive forms will be the same under our
choice, see (3.12). Besides, all the possible elliptic orbifolds
$\mathbb{P}^{1}$ with three singular points can be seen as mirrors of our
families at infinity of the complex plane, referred to as large complex
structure limit point. The method also works for other normal forms although
the mirrors of those elliptic orbifolds may appear in singular points on the
complex plane other than the large complex structure limit points. For the
framework of global mirror symmetry, see [ChiR]. Such global mirror symmetry
phenomena for simple elliptic singularities are studied in [MS]. However, our
choices here are enough for describing the quasi-modularity properties of
CohFTs for those elliptic orbifolds. Only modular subgroups will be different
for different normal forms. Let $W$ be one of the following three polynomials
(3.8)
$\widetilde{E}_{6}:=x_{1}^{3}+x_{2}^{3}+x_{3}^{3},\quad\widetilde{E}_{7}:=x_{1}^{2}x_{3}+x_{1}x_{2}^{3}+x_{3}^{2},\quad\widetilde{E}_{8}:=x_{1}^{3}x_{3}+x_{2}^{3}+x_{3}^{2}.$
Let us analyze the case $W=\widetilde{E}_{6}$. The other cases are similar. We
define a 1-dimensional family by
$W_{\sigma}=W+\sigma x_{1}x_{2}x_{3}.$
Note that $W_{\sigma}$ has an isolated singularity of the same rank $\mu$ iff
$\sigma\in\Sigma$,
$\Sigma=\Big{\\{}\sigma\in\mathbb{C}\Big{|}\sigma^{3}+27\neq 0\Big{\\}}.$
Now we can replace $f$ in section 3.1 by $W_{\sigma}$. Its miniversal
deformation is
(3.9)
$F:=W_{\sigma}(\textbf{s},\textbf{x})=W_{\sigma}+\sum_{i=0}^{\mu-1}s_{i}\phi_{i}.$
Here $\\{\phi_{i}\\}_{i=0}^{\mu-1}$ is a basis of homogeneous polynomials of
the Milnor ring $\mathscr{Q}_{W_{\sigma}}$. We always set
$\phi_{\mu-1}=x_{1}x_{2}x_{3},\phi_{0}=1$ and identify the index $\mu-1$ by
$-1$. Thus $\phi_{-1}=x_{1}x_{2}x_{3}$ and $s_{-1}=s_{\mu-1}$.
#### 3.2.1. Primitive forms and global moduli of Frobenius manifolds
Recall that for a generic $(\mathbf{s},\lambda)$, the fiber
$X_{\mathbf{s},\lambda}=\big{\\{}\mathbf{x}\in\mathbb{C}^{N}\big{|}\varphi(\mathbf{s},\mathbf{x})=\lambda\big{\\}}$
is homotopic to $\mu$ copies of $N-1$ dimensional sphere. The non-generic
$(\mathbf{s},\lambda)$ are the ones for which $X_{\mathbf{s},\lambda}$ has a
singularity. They form an analytic hypersurface called discirminant. The
complement of the latter is a base for the middle cohomology bundle formed by
the middle cohomology groups $H^{N-1}(X_{\mathbf{s},\lambda};\mathbb{C})$. In
addition the integral structure in cohomology induces a flat Gauss-Manin
connection.
Let us denote by $E_{\sigma}$ be the curve defined by $W_{\sigma}$,
$E_{\sigma}:=\big{\\{}[x_{1},x_{2},x_{3}]\in\mathbb{C}P^{2}\big{|}W_{\sigma}(x_{1},x_{2},x_{3})=0\big{\\}}.$
One may compactify the family $X\to\mathcal{S}\times\mathbb{C}$ to
$\overline{X}\to\mathcal{S}\times\mathbb{C}$ so that
$E_{\sigma}=\overline{X}-X$ is the boundary. $E_{\sigma}$ is also known as the
elliptic curve at infinity, cf.[L]. According to K. Saito (see [S1]), the
primitive forms for simple elliptic singularity $W_{\sigma}$ are homogenous of
degree 0 and can be expressed as
$\omega=\frac{d^{3}{\bf x}}{\pi_{A}(\sigma)}.$
They are parametrized by the periods of $E_{\sigma}$,
(3.10) $\pi_{A}(\sigma):=2\pi i\int_{A_{\sigma}}{\rm
Res}_{E_{\sigma}}[d^{3}\mathbf{x}/dF]\ ,$
where we fix a reference point $\sigma_{0}\in\Sigma$, $A\in
H_{1}(E_{\sigma_{0}},\mathbb{C})$ is some fixed non-zero 1-cycle and
$A_{\sigma}$ is a flat family of cycles uniquely determined by $A$ for all
$\sigma$ in a small neighborhood of $\sigma_{0}$. $d^{3}\mathbf{x}/dF$ is a
holomorphic 2-form on $X_{\sigma,\lambda}$ and ${\rm Res}$ is the residue
along $E_{\sigma}$. The boundary of any tubular neighborhood of $E_{\sigma}$
in $\overline{X}_{\mathbf{s},\lambda}$ is a circle bundle over $E_{\sigma}$
that induces via pullback an injective tube map $L:H_{1}(E_{\sigma})\to
H_{2}(X_{\mathbf{s},\lambda})$. Let $\alpha=L(A);$ then we have
(3.11) $\pi_{A}(\sigma)=\int_{\alpha}\frac{d^{3}\mathbf{x}}{dF}.$
We refer to $\alpha$ as a tube or toroidal cycle. The space of all toroidal
cycles coincides with the kernel of the intersection pairing on
$H_{2}(X_{\mathbf{s},\lambda};\mathbb{C})$.
The space of all periods $\pi_{A}(\sigma)$ coincides with the space of
solutions of the following differential equation (see the Appendix in [MR]),
(3.12) $\frac{d^{2}}{d\sigma^{2}}+\frac{3\sigma^{2}}{\sigma^{3}+27}\
\frac{d}{d\sigma}+\frac{\sigma}{\sigma^{3}+27}=0.$
Take $\lambda=-\sigma^{3}/27$, equation (3.12) is just a Gauss hypergeometric
equation,
(3.13)
$\lambda(1-\lambda)\frac{d^{2}}{d\lambda^{2}}+(\frac{2}{3}-\frac{5\lambda}{3})\frac{d}{d\lambda}-\frac{1}{9}=0$
Now let us describe the global Frobenius manifold structure for those normal
forms. We fix a symplectic basis $\\{A^{\prime},B^{\prime}\\}$ of
$H_{1}(E_{\sigma_{0}};\mathbb{Z})$ once and for all. Then the primitive form
is a multi-valued function on $\Sigma$. Thus it is more natural to replace
$\Sigma$ by its universal cover. The latter is naturally identified with the
upper half-plane $\mathbb{H}.$ The points in the universal cover
$\widetilde{\Sigma}$ of $\Sigma$ are pairs consisting of a point
$\sigma\in\Sigma$ and a homotopy class of paths $l(s)\in\Sigma$ with
$l(0)=\sigma_{0},l(1)=\sigma.$ The map
$(\sigma,l(s))\mapsto\tau=\frac{\pi_{B^{\prime}}(\sigma)}{\pi_{A^{\prime}}(\sigma)},$
where the periods $\pi_{B^{\prime}}$ and $\pi_{A^{\prime}}$ are analytically
continued along the path $l(s)$, defines an analytic isomorphism between the
universal cover of $\widetilde{\Sigma}$ and the upper half-plane $\mathbb{H}$.
Let $\mathcal{M}=\mathbb{H}\times\mathbb{C}^{\mu-1}$. A global Frobenius
structure exists on $\mathcal{M}$ for any non-zero cycle
(3.14) $A=dA^{\prime}+cB^{\prime}\in
H_{1}(E_{\sigma_{0}};\mathbb{C}),\quad-d/c\notin\mathbb{H}.$
Now let us describe the choice of a coordinate on $\mathbb{H}$, which we use
through out the paper. Let $M$ be the classical monodromy operator on the
middle homology bundle. By definition, $M$ is the linear operator induced by
the parallel transport with respect to the Gauss-Manin connection along a loop
in $\mathbb{C}^{*}\equiv\\{\sigma_{0}\\}\times(\mathbb{C}\backslash\\{0\\})$
based at $\lambda=1$. The operator $M$ is diagonalizable and one can find an
eigenbasis $\\{\alpha_{i}\\}_{i=-1}^{\mu-2},\alpha_{i}\in
H^{*}(X_{\sigma,1},\mathbb{C})$, s.t., the eigenvalue of $\alpha_{i}$ is
$e^{2\pi id_{i}}$. Here
$(\sigma,1):=(\sigma,0,\cdots,0,1)\in\mathcal{S}\times\mathbb{C}.$
In particular, the invariant subspace of $M$ is spanned by $\alpha_{-1}$ and
$\alpha_{0}$. Put $\alpha_{0}=-(-2\pi)^{3/2}L(A)$ and
$\alpha_{-1}=-(-2\pi)^{3/2}L(B)$, where the cycle $B=bA^{\prime}+aB^{\prime}$
is chosen to be any cycle linearly independent from $A.$ Then it was proved in
[MR] that the function
(3.15) $t:=\frac{\pi_{B}(\sigma)}{\pi_{A}(\sigma)}=\frac{a\tau+b}{c\tau+d}$
is a flat coordinate. Slightly abusing the notation we simply write
$t\in\mathbb{H}$ instead of saying that $t$ is given by formula (3.15) for
some $\tau\in\mathbb{H}$. The entire flat coordinate system can be described
in a similar way (see Section 2.2.2 in [MR]). Hence, $\mathcal{M}$ is a moduli
space of global Frobenius manifold structures. For convenience, we denote the
flat coordinates by $\mathbf{t}=(t,\mathbf{t}_{\geq 0})\in\mathcal{M}$, with
$\mathbf{t}_{\geq 0}=(t_{0},\cdots,t_{\mu-2})\in\mathbb{C}^{\mu-1}.$
### 3.3. The action of the monodromy group on flat coordinates
The monodromy group $\Gamma$ acts on $\mathcal{M}$ by covering
transformations. In this subsection, we recall its action on flat coordinates.
Let $\nu$ be a _monodromy transformation_ in the vanishing homology along a
given loop $C$ in $\Sigma$ based at $\sigma_{0}$. According to [MR], the $\nu$
action on $\\{\alpha_{i}\\}_{i=-1}^{\mu-2}$ has a matrix form with respect to
the vector of basis $(\alpha_{-1},\cdots,\alpha_{\mu-2})^{T}$,
$g\oplus{\rm Diag}(e^{2\pi id_{1}k},\dots,e^{2\pi id_{\mu-2}k})\in{\rm
SL}(2;\mathbb{C})\times\mathbb{Z}^{\mu-2},$
where
$g(\alpha_{-1})=n_{11}\alpha_{-1}+n_{12}\alpha_{0},\quad\mbox{and}\quad
g(\alpha_{0})=n_{21}\alpha_{-1}+n_{22}\alpha_{0},$
and the matrix $(n_{ij})\in{\rm SL}(2;\mathbb{C})$.
From now on we fix a flat coordinate system $t_{a}=t_{a}(\mathbf{s})$ $(-1\leq
a\leq\mu-2)$, multi-valued on $\mathcal{S}$ and holomorphic on the cover
$\mathcal{M}$, and denote by $H$ the space of flat vector fields on
$\mathcal{M}$. We further assume that the flat coordinates are chosen in such
a way that the residue pairing assumes the form:
$\displaystyle(\partial_{i},\partial_{j})=\delta_{i,j^{\prime}},\quad-1\leq
i,j\leq\mu-2,$
where $\partial_{a}:=\partial/\partial t_{a}$ and ′ is the involution defined
by
$-1\mapsto 0,\quad 0\mapsto-1,\quad i\mapsto\mu-1-i,\quad 1\leq i\leq\mu-2.$
According to [MR] the flat coordinates can be expressed via certain period
integrals as rational functions on the vanishing homology. It follows that the
monodromy group $\Gamma$ acts on the flat coordinates as well and that this
action coincides with the analytic continuation along $C$. According to [MR]
if the flat coordinate system is such that the residue pairing has the above
form; then the monodromy transformation (or equivalently the analytic
continuation) of the flat coordinates has the following form. Put
(3.16) $j_{\nu}(t):=j(g,t):=n_{21}t+n_{22};$
then
(3.17) $\nu(\mathbf{t})_{-1}=g(t):=\frac{n_{11}t+n_{12}}{n_{21}t+n_{22}}$
and
(3.18)
$\nu(\textbf{t})_{0}=t_{0}+\frac{n_{12}}{2j_{\nu}(t)}\sum_{i=1}^{\mu-2}t_{i}t_{i^{\prime}},\
\nu(\textbf{t})_{i}=\frac{e^{2\pi id_{i}k}}{j_{\nu}(t)}\ t_{i},\ 1\leq
i\leq\mu-2.$
## 4\. Global B-model CohFT and anti-holomorphic completion
The core of our paper is global B-model CohFTs for simple elliptic
singularities, which we will construct in this section. The basic idea is that
the global higher genus B-model theory of [MR] can be enhanced to a global
B-model CohFT using the construction of Teleman (see section two). The
modularity will follow essentially from the monodromy calculations in [MR].
### 4.1. Global B-model CohFT
#### 4.1.1. Givental’s semisimple quantization operator
Suppose that $W$ is one of the three families of simple elliptic singularities
under consideration. Recall the global Frobenius manifold structures on
$\mathcal{M}.$ First we recall the definiton of Givental’s quantization
operator and then we use it to define a CohFT $\Lambda^{W}(\mathbf{t})$ over
the semisimple loci $\mathcal{M}_{ss}.$
Let $\mathcal{K}\subset\mathcal{M}$ be the set of points $\mathbf{t}$ such
that $u_{i}(\mathbf{s}(\mathbf{t}))=u_{j}(\mathbf{s}(\mathbf{t}))$ for some
$i\neq j$. We call this set the caustic and put $\mathcal{M}_{ss}$ for its
complement. Note that the points $\mathbf{t}\in\mathcal{M}_{ss}$ are
semisimple, i.e., the critical values $u_{i}(\mathbf{s}(\mathbf{t}))$ ($1\leq
i\leq\mu$) form a coordinate system locally near $\mathbf{t}$. Let
$\mathbf{t}\in\mathcal{M}_{ss}$; then we have an isomorphism
$\Psi(\mathbf{t}):\mathbb{C}^{\mu}\to T_{\mathbf{t}}\mathcal{M},\quad
e_{i}\mapsto\sqrt{\Delta_{i}(\mathbf{s}(\mathbf{t}))}\,\frac{\partial}{\partial
u_{i}(\mathbf{s}(\mathbf{t}))},$
where $\Delta_{i}(\mathbf{s}(\mathbf{t}))$ is defined by
$\Big{(}\frac{\partial}{\partial
u_{i}(\mathbf{s}(\mathbf{t}))},\frac{\partial}{\partial
u_{j}(\mathbf{s}(\mathbf{t}))}\Big{)}=\frac{\delta_{ij}}{\Delta_{i}(\mathbf{s}(\mathbf{t}))},$
and we identify $T_{\mathbf{t}}\mathcal{M}$ with $H$ via the flat metric,
i.e.,
$\displaystyle\frac{\partial}{\partial
u_{i}}=\sum_{j=0}^{\mu-1}\,\frac{\partial t_{j}}{\partial
u_{i}}\,\partial_{j},\quad 1\leq i\leq\mu.$
$\Psi_{\mathbf{t}}$ diagonalizes the Frobenius multiplication and the residue
pairing:
$e_{i}\bullet e_{j}=\delta_{i,j}\sqrt{\Delta_{i}(\mathbf{s}(\mathbf{t}))}\
e_{i},\quad(e_{i},e_{j})=\delta_{ij}.$
The system of differential equations (3.6) and (3.7) admits a unique formal
solution of the type
$\displaystyle\Psi(\mathbf{t})R(\mathbf{t})\,e^{U(\mathbf{t})/z},\quad
R(\mathbf{t})={\rm Id}+\sum_{k=1}^{\infty}R_{k}(\mathbf{t})z^{k}\in{\rm
End}(\mathbb{C}^{\mu})[[z]].$
where $U(\mathbf{t})$ is a diagonal matrix with entries
$u_{1}(\mathbf{s}(\mathbf{t})),\dots,u_{\mu}(\mathbf{s}(\mathbf{t}))$ on the
diagonal, cf.[D, G1].
#### 4.1.2. Global B-model CohFT
Givental used $R(\mathbf{t})$ to define a higher genus generating function
over $\mathcal{M}_{ss}$. We would like to enhance his definition to CohFT. The
main difficulty is to extend our definition to non-semisimple points in
$\mathcal{K}.$
For any semisimple point $\mathbf{t}\in\mathcal{M}_{ss}$, we define a CohFT
with a flat identity and a state space $H$ (see Sect. 2)
(4.1) $\Lambda^{W}(\mathbf{t}):=\Psi(\mathbf{t})\circ
T_{z}\circ\widehat{R}(\mathbf{t})\circ T_{z}^{-1}\circ
I^{\mu,\Delta(\mathbf{t})}.$
We are interested in the loci of points
$\mathbf{t}=(t,0)\in\mathbb{H}\times\mathbb{C}^{\mu-1}$, which are never
semisimple. To continue our B-model discussion, we need to prove that
$\Lambda^{W}(\mathbf{t})$ extends holomorphically for all
$\mathbf{t}\in\mathcal{M}.$ To begin with, let us fix $g$, $n$, and
$\gamma_{i}\in H$; for convenience, we denote by
$\Lambda_{g,n}^{W}(\mathbf{t}):=(\Lambda^{W}(\mathbf{t}))_{g,n}.$
$\Lambda^{W}_{g,n}(\mathbf{t})(\gamma_{1},\dots,\gamma_{n})$ is a linear
combination of cohomology classes on $\overline{\mathcal{M}}_{g,n}$ whose
coefficients are functions on $\mathcal{M}$.
###### Lemma 4.1.
The coefficients of
$\Lambda^{W}_{g,n}(\mathbf{t})(\gamma_{1},\dots,\gamma_{n})$ are meromorphic
functions on $\mathcal{M}$ with at most finite order poles along the caustic
$\mathcal{K}$.
###### Proof.
By definition, the CohFT (4.1) depends only on the choice of a canonical
coordinate system
$u(\mathbf{t}):=(u_{1}(\mathbf{s}(\mathbf{t}),\dots,u_{\mu}(\mathbf{s}(\mathbf{t}))$.
The latter is uniquely determined up to permutation. Note that (4.1) is
permutation-invariant, i.e., it does not matter how we order the canonical
coordinates. On the other hand, up to a permutation $u(\mathbf{t})$ is
invariant under the analytical continuation along a closed loop in
$\mathcal{M}_{ss}.$ It follows that $\Lambda^{W}_{g,n}(\mathbf{t})$ is a
single valued function on $\mathcal{M}_{ss}.$
We need only to prove that the poles along $\mathcal{K}$ have finite order.
Note that according to the definition of the class (2.12) only finitely many
graphs $\Gamma$ contribute. The reason for this is that in order to have a
non-zero contribution, we must have
$\sum_{e\in E_{v}(\Gamma)}\,k(e)\ \leq\ 3g_{v}-3+r_{v}+n_{v}.$
Summing up these inequalities, we get
$\sum_{v}\sum_{e\in E_{v}(\Gamma)}\,k(e)\,\leq 3(g-1)-3{\rm
Card}(E(\Gamma))+\sum_{v}r_{v}+n,$
However
$\sum_{v}r_{v}=2{\rm Card}(E(\Gamma)),$
which implies that the number of edges of $\Gamma$ is bounded by $3g-3+n$.
This proves that there are finitely many possibilities for $\Gamma$. Moreover,
there are only finitely many possibilities for $k(e)$, i.e., our class is a
rational function on the entries of only finitely many $R_{k}$. Since each
$R_{k}$ has only a finite order pole along the caustic the Lemma follows. ∎
We will prove below that $\Lambda_{g,n}^{W}(\mathbf{t})$ is convergent near
the point
$(\sqrt{-1}\,\infty,0)\in\overline{\mathbb{H}}\times\mathbb{C}^{\mu-1}$ and
that it extends holomorphically through the caustic (see Theorem 5.3 and
Proposition 5.5). Thus $\Lambda^{W}(\mathbf{t})$ is a CohFT for all
$\mathbf{t}\in\mathcal{M}$. In particular,
(4.2)
$\Lambda_{g,n}^{W}(t)=\lim_{\mathbf{t}\in\mathcal{M}_{ss}\rightarrow(t,0)}\Lambda_{g,n}^{W}(\mathbf{t})$
for all $t\in\mathbb{H}=\mathbb{H}\times\\{0\\}\subset\mathcal{M}$.
### 4.2. Monodromy group action on $\Lambda_{g,n}^{W}(t)$
Using the residue pairing we identify $T^{*}\mathcal{M}$ and $T\mathcal{M}$,
i.e., $dt_{i}=\partial_{i^{\prime}}$. We also identify ${\rm End}(H)$ with the
space of $\mu\times\mu$ matrices via $A\mapsto(A_{ij})$, where the entries
$A_{ij}$ are defined in the standard way, i.e.,
$\displaystyle A(dt_{j})=\sum_{i=-1}^{\mu-2}A_{ij}dt_{i}\,.$
Recall the notation from Section 3.3: a loop $C$ in $\Sigma$, inducing via the
Gauss-Manin connection a monodromy transformation $\nu$ on vanishing homology
and a transformation of the flat coordinates via analytic continuation
$\mathbf{t}\mapsto\nu(\mathbf{t})$. The latter induces a monodromy
transformation of the stationary phase asymptotics, which was computed in
[MR], Lemma 4.1. In case $W=\widetilde{E}_{6},$ let
(4.3) $M_{\nu}(\mathbf{t})=\begin{bmatrix}j_{\nu}(t)^{-1}&*&*&*\\\
0&j_{\nu}(t)&0&0\\\ 0&*&e^{4\pi i\,k/3}\,I_{3}&0\\\ 0&*&0&e^{2\pi
i\,k/3}\,I_{3}\end{bmatrix}\in{\rm End}(H)[[z]].$
where
$M_{-1,j}=-e^{2\pi id_{j}k}\,n_{12}\,j^{-1}_{\nu}(t)\,t_{j},\quad 1\leq j\leq
6$
and
$M_{-1,0}=-n_{12}z-\frac{n_{12}^{2}}{2j_{\nu}(t)}\sum_{i=1}^{6}t_{i}t_{i^{\prime}},\quad
M_{i,0}=n_{12}t_{i^{\prime}},\quad 1\leq i\leq 6.$
###### Lemma 4.2 ([MR] ).
The analytic continuation along $C$ transforms
$\displaystyle\Psi(\mathbf{t})R(\mathbf{t})e^{U(\mathbf{t})/z}\quad\mbox{ into
}\quad{\vphantom{M}}^{{\rm
T}}{M}_{\nu}(\mathbf{t})\Psi_{\mathbf{t}}R(\mathbf{t})e^{U(\mathbf{t})/z}\,P,$
where $P$ is a permutation matrix and ${\vphantom{}}^{{\rm T}}{}$ means
transposition. ∎
The CohFT constructed by the analytical continuation along $C$ of
$\Lambda^{W}(\mathbf{t})$ will be denoted by
$\Lambda_{g,n}^{W}\big{(}\nu(\mathbf{t})\big{)}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes(H^{\vee})^{\otimes n}.$
Restricting to $\mathbf{t}_{\geq 0}=0$, we have
${\vphantom{M}}^{{\rm T}}{M}_{\nu}(t):=\lim_{\mathbf{t}_{\geq 0}\to
0}{\vphantom{M}}^{{\rm T}}{M}_{\nu}(\mathbf{t})=j^{-1}_{\nu}(t)\ J_{\nu}(t).$
With
(4.4) $J_{\nu}(t):=\begin{bmatrix}1&0\\\ 0&j^{2}_{\nu}(t)\end{bmatrix}\oplus
j_{\nu}(t)\,e^{4\pi i\,k/3}\,I_{3}\oplus j_{\nu}(t)\,e^{2\pi
i\,k/3}\,I_{3}\in{\rm End}(H)[[z]].$
Now let
(4.5) $X_{\nu,t}(z)=\begin{bmatrix}1&-n_{12}z/j_{\nu}(t)\\\
0&1\end{bmatrix}\bigoplus I_{6}\in{\rm End}(H)[[z]].$
###### Theorem 4.3.
The analytic continuation transforms the Coh FT as follows:
(4.6)
$\Lambda^{W}\big{(}\nu(t)\big{)}=J_{\nu}^{-1}(t)\circ\widehat{X}_{\nu,t}(z)\circ\Lambda^{W}(t).$
###### Proof.
The calculation in [MR] also works on cycle-valued level. ∎
Now we give a lemma which is very useful later on.
###### Lemma 4.4.
Let $E(z)\in\mathcal{L}^{(2)}_{+}{\rm GL}(H)$; then it intertwines with
$J^{-1}_{\nu}(t)$ by
$J^{-1}_{\nu}(t)\circ\widehat{E}(z)=\widehat{E}(j^{2}_{\nu}(t)z)\circ
J^{-1}_{\nu}(t).$
###### Proof.
From (4.4) and the definition of $J^{-1}_{\nu}(t)\circ$, we know that the
pairing $\eta$ is scaled by $j^{2}_{\nu}(t)$ when applying
$J^{-1}_{\nu}(t)\circ$. Thus the quadratic differential action
$\widehat{E}(z)$ becomes $\widehat{E}(j^{2}_{\nu}(t)z)$. ∎
### 4.3. Anti-holomorphic completion and modular transformation.
Let $\mathscr{R}$ or $\mathcal{R}$ be a cohomology ring of any fixed Deligne-
Mumford moduli space of stable curves of genus $g$ with $n$ marked points,
i.e., $\mathscr{R}=H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})$ for some
$2g-2+n>0$.
###### Definition 4.5.
We say that a $\mathscr{R}$-valued function $f:\mathbb{H}\to\mathscr{R}$ is a
$\mathscr{R}$-valued quasi-modular form of weight $m$ with respect to some
finite-index subgroup $\Gamma\subset{\rm SL}_{2}(\mathbb{Z})$ if there are
$\mathscr{R}$-valued functions $f_{i}$, $1\leq i\leq K$, holomorphic on
$\mathbb{H},$ such that
1. (i)
The functions $f_{0}:=f$ and $f_{i}$ are holomorphic near cusp $\tau=i\infty$.
2. (ii)
The following $\mathscr{R}$-valued function
$f(\tau,\bar{\tau})=f_{0}(\tau)+f_{1}(\tau)(\tau-\overline{\tau})^{-1}+\dots+f_{K}(\tau)(\tau-\overline{\tau})^{-K}.$
is modular, i.e., there exists some $m\in\mathbb{N}$ such that for any
$g\in\Gamma$,
$\displaystyle
f(g\tau,g\overline{\tau})=j(g,\tau)^{m}f(\tau,\overline{\tau}).$
$f(\tau,\overline{\tau})$ is called the anti-holomorphic completion of
$f(\tau)$.
#### 4.3.1. Anti-holomorphic completion of $\Lambda_{g,n}^{W}(t)$
Let $W$ be the homogeneous polynomial as in (3.8). Denote by
(4.7) $X_{t,\bar{t}}(z)=\begin{bmatrix}1&-z(t-\bar{t})^{-1}\\\ &\\\
0&1\end{bmatrix}\oplus I_{6}\in{\rm End}(H)[[z]],$
where $\bar{t}$ is the anti-holomorphic coordinate on $\mathbb{H}$ defined by
(cf. formula (3.15))
$\displaystyle\bar{t}:=\frac{a\overline{\tau}+b}{c\overline{\tau}+d}\ .$
We define the _anti-holomorphic completion_ of Coh FT $\Lambda^{W}(t)$ by:
(4.8) $\Lambda^{W}(t,\bar{t}):=\widehat{X}_{t,\bar{t}}(z)\circ\Lambda^{W}(t).$
###### Theorem 4.6.
Under the assumption of extension property, the analytic continuation of the
anti-holomorphic completion $\Lambda_{g,n}^{W}(t,\bar{t})$ along $\nu$ is
$J_{\nu}^{-1}(t)\circ\Lambda_{g,n}^{W}\big{(}t,\bar{t}\big{)}.$
###### Proof.
We define an operator $\widehat{X}_{\nu,t,\bar{t}}(z)$, s.t., the following
diagram is commutative:
$\begin{CD}&&\Lambda^{W}(t)&@>{\widehat{X}_{t,\bar{t}}(z)}>{}>&\Lambda^{W}(t,\bar{t})\\\
&&@V{}V{J_{\nu}(t)\circ\widehat{X}_{\nu,t}(z)}V&&@V{}V{\widehat{X}_{\nu,t,\bar{t}}(z)}V\\\
&&\Lambda^{W}\big{(}\nu(t)\big{)}&@>{\widehat{X}_{\nu(t),\nu(\bar{t})}(z)}>{}>&\Lambda^{W}\big{(}\nu(t),\nu(\bar{t})\big{)}\\\
\end{CD}$
We need to prove that
$\widehat{X}_{\nu,t,\bar{t}}(z)=J^{-1}_{\nu}(t).$
Let us consider the analytic continuation for $X_{t,\bar{t}}(z)$. Analytic
continuation acts on $(t-\bar{t})^{-1}$ by
$\frac{1}{\nu(t)-\nu(\bar{t})}=-\Big{(}\frac{n_{12}}{j_{\nu}(t)}+\frac{1}{t-\bar{t}}\Big{)}\,j^{2}_{\nu}(t).$
By definition (4.7), this implies
(4.9)
$X_{\nu(t),\nu(\bar{t})}(z)=X_{t,\bar{t}}(j^{2}_{\nu}(t)z)\,X^{-1}_{\nu,t}(j^{2}_{\nu}(t)z).$
Recalling Lemma 4.4, we get,
(4.10)
$J^{-1}_{\nu}(t)\circ\widehat{X}_{\nu,t}(z)\circ\widehat{X}_{t,\bar{t}}^{-1}(z)=\widehat{X}_{\nu,t}(j^{2}_{\nu}(t)z)\circ\widehat{X}_{t,\bar{t}}^{-1}(j^{2}_{\nu}(t)z)\circ
J^{-1}_{\nu}(t).$
Thus the result follows from (4.9) and (4.10). ∎
#### 4.3.2. Cycle-valued quasi-modular forms from $\Lambda_{g,n}^{W}(t)$
We consider a pair
$(\vec{\gamma}_{I},\iota_{I})=\big{(}(\gamma_{1},\cdots,\gamma_{n}),(\iota_{1},\cdots,\iota_{n})\big{)}\in
H^{\otimes n}\times\mathbb{Z}_{\geq 0}^{n}$
where each
$\gamma_{i}\in\mathscr{S}=\\{\partial_{-1}=\partial_{\mu-1},\partial_{0},\cdots,\partial_{\mu-2}\\}$.
$I$ is a multi-index
$I=(i_{-1},i_{0},\cdots,i_{\mu-2})\in\mathbb{Z}_{\geq 0}^{\mu},\quad
i_{-1}+\cdots+i_{\mu-2}=n.$
$i_{j}$ is the number of $i\in\\{1,\cdots,n\\}$ such that
$\gamma_{i}=\partial_{j}$. Under the assumption of extension property, we
define a cycle-valued function $f^{W}_{I,\iota_{I}}$ on $\mathbb{H}$,
(4.11) $f^{W}_{I,\iota_{I}}(t)=\Lambda_{g,n}^{W}(t)(\vec{\gamma}_{I})\ \in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C}).$
and its anti-holomorphic completion
$f^{W}_{I,\iota_{I}}(t,\bar{t}):=\Lambda_{g,n}^{W}(t,\bar{t})(\vec{\gamma}_{I}).$
For $\iota_{I}=(0,\cdots,0)$, we simply denote them by $f^{W}_{I}(t)$ and
$f^{W}_{I}(t,\bar{t})$. Let
(4.12) $m(I):=2i_{-1}+\sum_{j=1}^{\mu-2}i_{j}.$
###### Theorem 4.7.
Let $W$ be a simple elliptic singularity. Then $f^{W}_{I,\iota_{I}}(t)$
satisfies the transformation law of cycle-valued quasi-modular forms of weight
$m(I)$.
###### Proof.
First we consider $\iota_{I}=(0,\cdots,0)$. It is easy to see
$f^{W}_{I}(t,\bar{t})$ is an anti-holomorphic completion for $f^{W}_{I}(t)$
and for monodromy $\nu$ described as before, we have
$\displaystyle f^{W}_{I}(\nu(t),\nu(\bar{t}))$ $\displaystyle=$
$\displaystyle\big{(}\widehat{X}_{\nu,t,\bar{t}}(z)\circ\Lambda^{W}(t,\bar{t})\big{)}_{g,n}(\vec{\gamma}_{I})$
$\displaystyle=$ $\displaystyle
j_{\nu}^{m(I)}(t)\,\Lambda^{W}_{g,n}(t,\bar{t})(\vec{\gamma}_{I})$
$\displaystyle=$ $\displaystyle j_{\nu}^{m(I)}(t)\,f^{W}_{I}(t,\bar{t}).$
Now the statement follows from monodromy acts trivially on psi-classes. ∎
###### Remark 4.8.
For $f^{W}_{I}(t)$ to be a cycle-valued modular form, it needs to be
holomophic at $\tau=\sqrt{-1}\,\infty$ (cf. formula (3.15)). This will be
achieved by the mirror theorem in section 5. Hence, by combining A-model with
B-model, we produce cycle-valued quasi-modular forms.
## 5\. A-model CohFT and cycle valued modular forms
In the last section we constructed an anti-holomorphic modification of the
B-model CohFT, such that it has the correct transformation property under
analytic continuation. However, we still have to prove the following two
properties: (1) the CohFT extends holomorphically through the caustic; (2) the
quasi-modular forms are holomorphic at the cusp $\tau=\sqrt{-1}\,\infty$. We
address both issues using an A-model (Gromov-Witten) CohFT and mirror
symmetry. As a byproduct we obtain a geometric interpretation of the B-model
CohFT as the Gromov-Witten CohFT of an elliptic orbifold $\mathbb{P}^{1}$ and
we obtain a proof of our main result Theorem 1.2.
The hard part of the argument is already completed in [KS, MR]. Our goal is to
recall the appropriate results and to show in what order they have to be used.
The idea is as follows. We first establish analyticity and generic
semisimplicity of the genus zero Gromov-Witten theory. This is done by using
an estimate for the GW invariants and genus zero mirror symmetry. Then, we
make use of a result of Coates-Iritani in order to prove the convergence of
the Gromov-Witten ancestor CohFT of all genera. The last major step is a
higher genus mirror symmetry that allows us to match the Gromov-Witten
ancestor CohFT with the B-model CohFT near the large complex limit. This
implies the extension property at $\tau=\sqrt{-1}\,\infty$. Finally, we use
Lemma 3.2 from [MR] to conclude the extension property over entire B-model
moduli space $\mathcal{M}$.
### 5.1. A-model
Let us recall a general mirror symmetry construction, called Berglund-Hübsch-
Krawitz mirror symmetry. For a quasi-homogeneous polynomial $W$ with a
suitable symmetry group $G$, a pair of mirror $(W^{T},G^{T})$ is constructed,
[BH, K]. In our case, we choose a cubic polynomial $W^{T}$ with the maximal
admissible group $G^{T}=G_{W^{T}}$, and consider this pair in A-model side.
Its mirror will be the pair $(W,G=\\{{\rm Id}\\})$. So the B-model will be
Saito-Givental’s theory on the miniversal deformation of the family
$W_{\sigma}$.
For $W=\widetilde{E}_{i},i=6,7,8$ (see (3.8)) the mirror $W^{T}$ is given
respectively by the following cubic polynomials:
(5.1) $W^{T}=x_{1}^{3}+x_{2}^{3}+x_{3}^{3},\quad
x_{1}^{2}x_{2}+x_{2}^{3}+x_{1}x_{3}^{2},\quad
x_{1}^{3}+x_{2}^{3}+x_{1}x_{3}^{2}.$
The weights are $q_{i}=1/3$, for all $i=1,2,3.$ Consider a hypersurface in the
projective space,
$X_{W^{T}}=\\{(x_{1},x_{2},x_{3})|W^{T}(x_{1},x_{2},x_{3})=0\\}\hookrightarrow\mathbb{P}^{2}.$
Its maximal admissible group is
$G_{W^{T}}:=\big{\\{}(\lambda_{1},\lambda_{2},\lambda_{3})\in\mathbb{C}^{3}\big{|}\,W(\lambda_{1}\,x_{1},\lambda_{2}\,x_{2},\lambda_{3}\,x_{3})=W^{T}(x_{1},x_{2},x_{3})\big{\\}}.$
It contains a subgroup $\langle J\rangle$, generated by the exponential
grading element
$J:=(\exp(2\pi i\cdot q_{1}),\exp(2\pi i\cdot q_{2}),\exp(2\pi i\cdot
q_{3}))\in G_{W^{T}}.$
$\langle J\rangle$ acts trivially on $X_{W^{T}}$. We denote by
$\widetilde{G}=G/\langle J\rangle.$
The quotient space
$\mathcal{X}_{W^{T}}:=X_{W^{T}}/\widetilde{G_{W^{T}}}$
is an elliptic orbifold with $\mathbb{P}^{1}$ as its underlying space. The
A-model is the orbifold Gromov-Witten theory of
$\mathcal{X}:=\mathcal{X}_{W^{T}}$.
### 5.2. Analyticity and generic semisimplicity
Let $H$ be the Chen-Ruan cohomology of $\mathcal{X}$ with unit ${\bf 1}$ and
Poincaré pairing $\eta$. Let the divisor $\mathcal{D}$ be a nef generator in
$H^{2}(\mathcal{X},\mathbb{Z})\subset H^{2}_{\rm CR}(\mathcal{X},\mathbb{Z})$
and let $\mathbf{t}=(t,t_{0},t_{1}\dots,t_{\mu-2})$ be a linear coordinate
system on $H$, such that $t$ is the coordinate along $\mathcal{D}$. Recall the
Gromov-Witten CohFT ${}_{\mathbf{t}}\Lambda^{\mathcal{X}}$, which a priori is
only formal. Due to the so called divisor axiom we can identify $q=e^{t}$,
i.e.,
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{g,n}(\gamma_{1},\cdots,\gamma_{n})\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes\mathbb{C}[[e^{t},t_{0},\cdots,t_{\mu-2}]].$
For every $\alpha,\beta,\gamma\in H$, the big quantum product
$\alpha\star_{\mathbf{t}}\beta$ is defined by the relation
(5.2)
$\langle\alpha\star_{\mathbf{t}}\beta,\gamma\rangle={}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{0,3}(\alpha,\beta,\gamma).$
The product is only formal in $\mathbf{t}$. We would like to prove that
$\star_{\mathbf{t}}$ is convergent in the open polydisk
$D_{\epsilon}\subset\mathbb{C}^{\mu}$ with center the origin and radius
$\epsilon$, i.e, $(q=e^{t},\mathbf{t}_{\geq 0})\in D_{\epsilon}$. More
precisely, our goal is to prove the following theorem.
###### Theorem 5.1.
The following statements hold:
* (1)
There exists an $\epsilon>0$ such that
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{0,3}(\alpha,\beta,\gamma)$ is
convergent for all $(q=e^{t},\mathbf{t}_{\geq 0})\in D_{\epsilon}$ and
$\alpha,\beta,\gamma\in H$.
* (2)
The quantum product $\star_{\mathbf{t}}$ is generically semisimple.
Part (1) follows from Theorem 1.2 in [KS]. For the reader’s convenience, we
sketch the proof here as well. First let us denote by
(5.3) $I_{0,n,d}^{GW}:=\max_{-1\leq
i_{j}\leq\mu-2}\big{|}\langle\partial_{i_{1}},\cdots,\partial_{i_{n}}\rangle_{0,n,d}^{\mathcal{X}}\big{|}.$
By direct computation, $I_{0,3,0}^{GW}\leq 1$.
###### Lemma 5.2 (Lemma 4.16 in [KS]).
For $n+d\geq 4,$ we have:
$I_{0,n,d}^{GW}\leq\left\\{\begin{array}[]{ll}d^{n-5}C^{n+d-4},&d\geq 1.\\\
C^{n-4},&d=0.\end{array}\right.$
Here $C$ is some positive constant depending only on $n$.
Since $H^{*}(\overline{\mathcal{M}}_{0,3},\mathbb{C})\cong\mathbb{C}$, it is
enough to prove the convergence of the corresponding ancestor Gromov-Witten
invariants. The divisor axiom implies
$\int_{\overline{\mathcal{M}}_{0,3}}{}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{0,3}(\alpha,\beta,\gamma)=\sum_{d\geq
0}q^{d}\,\sum_{k=0}^{\infty}\sum_{k_{0}+\cdots+k_{\mu-2}=k}\frac{\langle\alpha,\beta,\gamma,\cdots\rangle^{\mathcal{X}}_{0,3+k,d}}{k_{0}!\cdots
k_{\mu-2}!}\prod_{0\leq i\leq\mu-2}t_{i}^{k_{i}}$
where the dots stand for the insertion $\partial_{0},\cdots,\partial_{\mu-2}$
with multiplicities respectively $k_{0},\dots,k_{\mu-2}$. For dimensional
reasons the Gromov-Witten invariants in the above formula vanish except for
finitely many $k$. In another words,
$\int_{\overline{\mathcal{M}}_{0,3}}{}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{0,3}(\alpha,\beta,\gamma)\in\mathbb{C}[t_{0},\cdots,t_{\mu-2}]\otimes\mathbb{C}[[q]].$
Thus the convergence of
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{0,3}(\alpha,\beta,\gamma)$ in
$(q,\mathbf{t}_{\geq 0})\in D_{\epsilon}$ follows from the convergence of the
following series near $q=0$,
$\sum_{d\geq 0}q^{d}\,d^{n-5}C^{n+d-4}.$
Part (2) is not so easy to prove directly in the settings of Gromov-Witten
theory. We use the genus-0 part of the mirror symmetry theorem of [KS]. We
recall the genus-$0$ ancestor Gromov-Witten potential constructed from
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}$
$\mathcal{F}_{0}^{GW}(\mathcal{X})(\mathbf{t}):=\sum_{n}\sum_{\iota_{j},i_{j},d}\frac{1}{n!}\langle\tau_{\iota_{1}}(\partial_{i_{1}}),\cdots,\tau_{\iota_{n}}(\partial_{i_{n}})\rangle^{\mathcal{X}}_{0,n,d}(\mathbf{t})\prod_{j=1}^{n}\widetilde{q}_{i_{j}}^{\iota_{j}}.$
We can expand it as a formal series (due to the divisor axiom) as follows:
$\mathcal{F}_{0}^{GW}(\mathcal{X})(\mathbf{t})\in\mathbb{C}[[q=e^{t},t_{0},\cdots,t_{\mu-2}]]\otimes\mathbb{C}[[\mathbf{q}_{0},\widetilde{\mathbf{q}}_{1},\mathbf{q}_{2},\cdots]].$
The space $\mathcal{M}=\mathbb{H}\times\mathbb{C}^{\mu-1}$ can be equipped
with flat coordinates $\mathbf{t}^{B}$ corresponding to a choice of a primitve
form (for $W$). In fact, $\mathcal{M}$ has a generically semi-simple Frobenius
structure, which allows us to define the Saito–Givental ancestor potentials
$\mathcal{F}_{g}^{SG}(W)(\mathbf{t}^{B})$ for all genera $g$ (see 2.20 ).
The genus-0 mirror symmetry can be stated as follows: the primitive form can
be chosen in such a way that there exists an analytic embedding
$D_{\epsilon}\hookrightarrow\mathcal{M}$, called a mirror map, s.t.,
* (1)
The linear coordinates $\mathbf{t}$ on $D_{\epsilon}$ correspond to flat
coordinates $\mathbf{t}^{B}$ on $\mathcal{M}.$
* (2)
We have
$\mathcal{F}_{0}^{GW}(\mathcal{X}_{W^{T}})(\mathbf{t})=\mathcal{F}_{0}^{SG}(W)(\mathbf{t}^{B}).$
We denote the image of $D_{\epsilon}$ by $D_{\epsilon}^{B}$. Let us recall
(see [MR]) that under the mirror map the modulus $\tau$ (cf. Sect. 3.2.1) is a
flat coordinate on $\mathcal{M}$ and we have
(5.4) $t=\frac{2\pi\sqrt{-1}}{N}\,\tau,$
where $N=3,4$, and $6$ respectively for
$W=\widetilde{E}_{6},\widetilde{E}_{7},$ and $\widetilde{E}_{8}.$ It follows
that the large volume limit point $e^{t}=0$, i.e., $t=-\infty$ corresponds to
the large complex structure limit point $\tau=\sqrt{-1}\,\infty.$
The proof can be splitted into two parts: choice of a primitive form, s.t.,
(1) holds and prove that the ancestor potentials on both sides are uniquely
determined from a finite set of correlators, which agree under the mirror map.
The first step was done in [MR] and the second one in [KS].
### 5.3. Convergence of $\Lambda^{\mathcal{X}}_{g,n}(\mathbf{t})$
We identify via the mirror map the flat coordinates $\mathbf{t}^{B}$ on
$\mathcal{M}$ and the linear coordinates $\mathbf{t}$ on $D_{\epsilon}$.
Recall the CohFT $\Lambda_{g,n}^{W}(\mathbf{t}^{B})$ defined by formula (4.1)
for all semisimple points $\mathbf{t}^{B}$.
###### Theorem 5.3.
The CohFT $\Lambda_{g,n}^{W}(\mathbf{t}^{B})$ extends holomorphically for all
$\mathbf{t}^{B}\in D^{B}_{\epsilon}$, the ancestor Gromov–Witten CohFT
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}$ is convergent for all $\mathbf{t}\in
D_{\epsilon}$, and we have
$\displaystyle{}_{\mathbf{t}}\Lambda_{g,n}^{\mathcal{X}}=\Lambda_{g,n}^{W}(\mathbf{t}^{B}),\quad\forall
t\in D_{\epsilon}.$
###### Proof.
The Frobenius structure of the quantum cohomology is generically semi-simple
(cf. Theorem 5.1, (2)). In particular, if we think of the CohFT
${}_{\mathbf{t}}\Lambda^{\mathcal{X}}$ as a CohFT over the field
$\displaystyle\overline{{\rm
Frac}\,\mathbb{C}[[e^{t},t_{0},\cdots,t_{\mu-2}]]},$
where overline means algebraic closure and Frac stands for the field of
fractions; then ${}_{\mathbf{t}}\Lambda^{\mathcal{X}}$ is a semi-simple CohFT
with a flat identity. Teleman’s reconstruction Theorem 2.4 applies and we get
that
(5.5)
${}_{\mathbf{t}}\Lambda_{g,n}^{\mathcal{X}}=\Lambda_{g,n}^{W}(\mathbf{t}^{B}),$
where the equality should be interpreted as equality in the space
$\displaystyle
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes\overline{{\rm
Frac}\,\mathbb{C}[[e^{t},t_{0},\cdots,t_{\mu-2}]]}.$
On the other hand, according to Lemma (4.1),
$\Lambda_{g,n}^{W}(\mathbf{t}^{B})$ is meromorphic for $\mathbf{t}\in
D^{B}_{\epsilon}$, thus
(5.6)
$\Lambda_{g,n}^{W}(\mathbf{t}^{B})={}_{\mathbf{t}}\Lambda^{\mathcal{X}}_{g,n}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes\overline{{\rm
Frac}\,\mathbb{C}\\{e^{t},t_{0},\cdots,t_{\mu-2}\\}},$
where $\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ is the ring of convergent power
series at $x_{1}=\cdots=x_{n}=0$ (the overline means algebraic closure). On
the other hand, by definition
(5.7) ${}_{\mathbf{t}}\Lambda_{g,n}^{\mathcal{X}}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})\otimes\mathbb{C}[[e^{t},t_{0},\cdots,t_{\mu-2}]].$
Now we apply the following lemma of Coates–Iritani,
###### Lemma 5.4 ([CI1], Lemma 6.6).
The intersection
$\displaystyle\overline{{\rm
Frac}\,\mathbb{C}\\{x_{1},\cdots,x_{n}\\}}\cap\mathbb{C}[[x_{1},\cdots,x_{n}]]\subset\overline{{\rm
Frac}\,\mathbb{C}[[x_{1},\cdots,x_{n}]]}$
coincides with $\mathbb{C}\\{x_{1},\cdots,x_{n}\\}.$
This completes the proof. ∎
### 5.4. Extension property
In this subsection, we use Lemma 3.2 from [MR] to derive the extension
property.
###### Proposition 5.5.
The coefficients of
$\Lambda_{g,n}^{W}(\mathbf{t}^{B})(\gamma_{1},\dots,\gamma_{n})$ extend
holomorphically through $\mathcal{K}$, i.e., they are holomorphic functions on
$\mathcal{M}$.
###### Proof.
Let us define an action of $\mathbb{C}^{*}$ on
$\mathcal{M}=\mathbb{H}\times\mathbb{C}^{\mu-1}$ according to the weights of
the coordinates $\mathbf{t}^{B}$. Since $\Lambda^{W}(\mathbf{t}^{B})$ is a
homogeneous CohFT, the domain $\widetilde{\mathcal{K}}$ of all
$\mathbf{t}^{B}$ where the theory does not extend analytically is
$\mathbb{C}^{*}$-invariant. Since $\widetilde{\mathcal{K}}$ is the set of
points $\mathbf{t}^{B}\in\mathcal{M}$, such that $\Lambda^{W}(\mathbf{t}^{B})$
has a pole, $\widetilde{\mathcal{K}}$ must be an analytic subset. Let us
assume that $\widetilde{\mathcal{K}}$ is non-empty. The Hartogues extension
theorem implies that the codimension of $\widetilde{\mathcal{K}}$ is at most 1
and hence precisely one. On the other hand, according to Theorem 5.3, the
polydisk $D_{\epsilon}$ is disjoint from $\widetilde{\mathcal{K}}$. In
particular, $\mathbb{H}\times\\{0\\}$ is not contained in
$\widetilde{\mathcal{K}}$ and hence the two subvarieties interesect
transversely. This combined with the $\mathbb{C}^{*}$ invariance of
$\widetilde{\mathcal{K}}$ implies that the connected components of
$\widetilde{\mathcal{K}}$ have the form
$\\{\tau_{0}\\}\times\mathbb{C}^{\mu-1}$. This is a contradiction, because
$\widetilde{\mathcal{K}}\subset\mathcal{K}$, while
$\\{\tau_{0}\\}\times\mathbb{C}^{\mu-1}\not\subset\mathcal{K}.$ ∎
### 5.5. Quasi-modularity
Finally, let us complete the proof of our main theorem. According to Theorem
5.3 the Gromov–Witten CohFT $\Lambda_{g,n}^{\mathcal{X}}(q)$ is convergent and
it coincides with $\Lambda_{g,n}^{W}(\tau)$, under the mirror map (5.4). The
latter transforms as a quasi-modular form according to Theorem 4.7, it is
analytic for all $\tau\in\mathbb{H}$ due to Proposition 5.5, and finally it
extends holomorphically over the cusp $\tau=i\,\infty$ because
$\Lambda_{g,n}^{\mathcal{X}}(q)$ extends holomorphically over $q=0$. This
completes the proof of Theorem 1.2.
## References
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|
arxiv-papers
| 2012-06-18T10:31:24 |
2024-09-04T02:49:31.890257
|
{
"license": "Public Domain",
"authors": "Todor Milanov, Yongbin Ruan, Yefeng Shen",
"submitter": "Yefeng Shen",
"url": "https://arxiv.org/abs/1206.3879"
}
|
1206.3977
|
# Upper bounds of depth of monomial ideals
Dorin Popescu Dorin Popescu, ”Simion Stoilow” Institute of Mathematics of
Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
dorin.popescu@imar.ro
###### Abstract.
Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field,
generated by square free monomials. We show that some inequalities among the
numbers of square free monomials of $I\setminus J$ of different degrees give
upper bounds of $\operatorname{depth}_{S}I/J$.
Key words : Square free monomial ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55.
The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.
## Introduction
Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$ variables over
a field $K$, $d\leq t$ be two positive integers and $I\supsetneq J$, be two
square free monomial ideals of $S$ such that $I$ is generated in degrees $\geq
d$, respectively $J$ in degrees $\geq d+1$. By [2, Theorem 3.1] and [4, Lemma
1.1] $\operatorname{depth}_{S}I/J\geq d$. Let $\rho_{t}(I\setminus J)$ be the
number of all square free monomials of degree $t$ of $I\setminus J$.
###### Theorem 0.1.
([4, Theorem 2.2]) If $\rho_{d}(I)>\rho_{d+1}(I\setminus J)$ then
$\operatorname{depth}_{S}I/J=d$, independently of the characteristic of $K$.
The aim of this paper is to extend this theorem. Our Theorem 1.3 says that
$\operatorname{depth}_{S}I/J=t$ if $\operatorname{depth}_{S}I/J\geq t$ and
$\rho_{t+1}(I\setminus J)<\sum_{i=0}^{t-d}(-1)^{t-d+i}\rho_{d+i}(I\setminus
J).$
If $t=d$ then this result is stated in Theorem 0.1 (a previous result is given
in [3]). If $t=d+1$ then the above result says that
$\operatorname{depth}_{S}I/J\leq d+1$ if
$\rho_{d+1}(I\setminus J)>\rho_{d+2}(I\setminus J)+\rho_{d}(I\setminus J).$
A particular case with $I$ principal is given, with a different proof, in our
Proposition 1.1. Theorem 0.1 is a small step in an attempt to show Stanley’s
Conjecture for some classes of factors of square free monomial ideals (see our
Remark 1.6 for some details) and we hope that our Theorem 1.3 will be useful
in the same frame.
## 1\. Upper bounds of depth
The aim of this section is to show the extension of Theorem 0.1 stated in the
introduction. We start with a particular case.
###### Proposition 1.1.
Suppose that $I$ is generated by a square free monomial $f$ of degree $d$, and
$s=\rho_{d+1}(I\setminus J)>\rho_{d+2}(I\setminus J)+1$. Then
$\operatorname{depth}_{S}I/J=d+1$.
###### Proof.
First suppose that $q=\rho_{d+2}(I\setminus J)>0$. Let $g\in I\setminus J$ be
a square free monomial of degree $d+2$. Renumbering the variables $x$ we may
suppose that $I$ is generated by $f=x_{1}\cdots x_{d}$ and
$g=fx_{d+1}x_{d+2}$. Since $g\not\in J$ we see that $b_{1}=fx_{d+1}$,
$b_{2}=fx_{d+2}$ are not in $J$. Again renumbering $x$ we may suppose that
$b_{i}=fx_{d+i}$, $i\in[s]$ are all the square free monomials of degree $d+1$
from $I\setminus J$. Set $T=(b_{3},\ldots,b_{s})$ (by hypothesis $s\geq 3$).
In the exact sequence
$0\rightarrow T/T\cap J\rightarrow I/J\rightarrow I/(T+J)\rightarrow 0$
we see that the left end has depth $d+1$ by Theorem 0.1 since $T\cap J$ is
generated in degree $\geq d+2$ and
$\rho_{d+1}(T)=s-2>q-1=\rho_{d+2}(T\setminus T\cap J)$. On the other hand,
$(T+J):f=(x_{d+3},\ldots,x_{n})$ because $b_{1},b_{2},g\not\in T+J$. It
follows that $\operatorname{depth}_{S}I/(T+J)=d+2$ and so the Depth Lemma says
that $\operatorname{depth}_{S}I/J=d+1$.
Now suppose that $q=0$. As above we may assume that $b_{i}=fx_{d+i}$,
$i\in[s]$ are the square free monomials of degree $d+1$ of $I\setminus J$.
Then $J:f=(x_{d+s+1},\dots,x_{n})+L$, where $L$ is the Veronese ideal
generated by all square free monomials of degree $2$ in
$x_{d+1},\ldots,x_{d+s}$. It follows that $I/J\cong K[x_{1},\ldots,x_{d+s}]/L$
which has depth $d+1$.
Next we present some details on the Koszul homology (see [1]) which we need
for the proof of our main result. Let $\partial_{i}:K_{i}(x;I/J)\rightarrow
K_{i-1}(x;I/J)$, $K_{i}(x;I/J)\cong S^{{n\choose i}}$, $i\in[n]$ be the Koszul
derivation given by
$\partial_{i}(e_{j_{1}}\wedge\ldots\wedge
e_{j_{i}})=\sum_{k=1}^{i}(-1)^{k+1}x_{j_{k}}e_{j_{1}}\wedge\ldots\wedge
e_{j_{k-1}}\wedge e_{j_{k+1}}\wedge\ldots\wedge e_{j_{i}}.$
Fix $0\leq i<n-d$. Let $f_{1},\ldots,f_{r}$, $r=\rho_{d+i}(I\setminus J)$ be
all square free monomials of degree $d+i$ from $I\setminus J$ and
$b_{1},\cdots,b_{s}$, $s=\rho_{d+i+1}(I\setminus J)$ be all square free
monomials of degree $d+i+1$ from $I\setminus J$. Let
$\operatorname{supp}f_{i}=\\{j\in[n]:x_{j}|f_{i}\\}$,
$e_{\sigma_{i}}=\wedge_{j\in([n]\setminus\operatorname{supp}f_{i})}\ e_{j}$
and $\operatorname{supp}b_{k}=\\{j\in[n]:x_{j}|b_{k}\\}$,
$e_{\tau_{k}}=\wedge_{j\in([n]\setminus\operatorname{supp}b_{k})}\ e_{j}$. We
consider the element $z=\sum_{q=1}^{r}y_{q}f_{q}e_{\sigma_{q}}$ of
$K_{n-d-i}(x;I/J)$, where $y_{q}\in K$. Then
$\partial_{n-d-i}(z)=\sum_{k=1}^{s}(\sum_{q\in[r]}\varepsilon_{kq}y_{q})b_{k}e_{\tau_{k}},$
where $\varepsilon_{kq}\in\\{1,-1\\}$ if $f_{q}|b_{k}$, otherwise
$\varepsilon_{kq}=0$. Thus $\partial_{n-d-i}(z)=0$ if and only if
$\sum_{q\in[r]}\varepsilon_{kq}y_{q}=0$ for all $k\in[s]$, that is
$y=(y_{1},\ldots,y_{r})$ is in the kernel of the linear map
$h_{n-d-i}:K^{r}\rightarrow K^{s}$ given by the matrix $\varepsilon_{kq}$.
Now we will see when $z\in\operatorname{Im}\partial_{n-d-i+1}$. Since the
Koszul derivation is a graded map we note that
$z\in\operatorname{Im}\partial_{n-d-i+1}$ if and only if
$z=\partial_{n-d-i+1}(w)$ for a $w=\sum_{p=1}^{c}u_{p}g_{p}e_{\nu_{p}}$, where
$c=\rho_{d+i-1}(I\setminus J)$, $u_{p}\in K$, $g_{1},\ldots,g_{c}$ are all
square free monomials of degree $d+i-1$ from $I\setminus J$ and
$e_{\nu_{p}}=\wedge_{j\in([n]\setminus\operatorname{supp}g_{p})}\ e_{j}$. It
follows
$z=\sum_{q=1}^{r}(\sum_{p\in[c]}\gamma_{qp}u_{p})f_{q}e_{\sigma_{q}},$
where $\gamma_{qp}\in\\{1,-1\\}$ if $g_{p}|f_{q}$, otherwise $\gamma_{qp}=0$.
Thus $z\in\operatorname{Im}\partial_{n-d-i+1}$ if and only if $y$ belongs to
the image of the linear map $h_{n-d-i+1}:K^{c}\rightarrow K^{r}$ given by the
matrix $\gamma_{qp}$. When $i=0$ we have $h_{n-d-i+1}=0$.
Note that $\operatorname{Im}h_{n-d-i+1}\subset\operatorname{Ker}h_{n-d-i}$ and
the inclusion is strict if and only if there exists $y\in K^{r}$ such that $z$
induces a nonzero element in $H_{n-d-i}(x;I/J)$. This implies
$\operatorname{depth}_{S}I/J\leq d+i$ by [1, Theorem 1.6.17]. If
$\operatorname{depth}I/J>d+i$ then
$\operatorname{Im}\partial_{n-d-i+1}=\operatorname{Ker}\partial_{n-d-i}$ and
it follows $\operatorname{Im}h_{n-d-i+1}=\operatorname{Ker}h_{n-d-i}$.
###### Lemma 1.2.
Let $0\leq i<n-d$, then the following statements hold independently of the
characteristic of $K$.
1. (1)
the complex $K^{c}\xrightarrow{h_{n-d-i+1}}K^{r}\xrightarrow{h_{n-d-i}}K^{s}$
is exact if $\operatorname{depth}I/J>d+i$,
2. (2)
if $\operatorname{depth}_{S}I/J>d+i$ then
$r=\operatorname{rank}h_{n-d-i+1}+\operatorname{rank}h_{n-d-i}$,
3. (3)
if $r>\operatorname{rank}h_{n-d-i+1}+\operatorname{rank}h_{n-d-i}$ then
$\operatorname{depth}_{S}I/J\leq d+i.$
###### Proof.
The first statement follows from above and the second one is only a
consequence. If
$r>\operatorname{rank}h_{n-d-i+1}+\operatorname{rank}h_{n-d-i}$ then
$\operatorname{Im}h_{n-d-i+1}\subsetneq\operatorname{Ker}h_{n-d-i}$ and the
last statement follows also from above.
###### Theorem 1.3.
Let $d\leq t\leq n$ be two integers and set
$\alpha_{j}=\sum_{i=0}^{j-d}(-1)^{j-d+i}\rho_{d+i}(I\setminus J),$
for $d\leq j\leq t$. Suppose that $\operatorname{depth}_{S}I/J\geq t$ and
$\rho_{t+1}(I\setminus J)<\alpha_{t}$. Then $\operatorname{depth}_{S}I/J=t$
independently of the characteristic of $K$.
###### Proof.
We have $\alpha_{j}=\rho_{j}(I\setminus J)-\alpha_{j-1}$ for $d<j\leq t$. By
Lemma 1.2 (2) we get $h_{n-d}$ injective and $\rho_{d+i}(I\setminus
J)=\operatorname{rank}h_{n-d-i+1}+\operatorname{rank}h_{n-d-i}$ for $0<i<t-d.$
It follows that $\rho_{d}(I\setminus
J)=\operatorname{rank}h_{n-d}=\alpha_{d}$, $\rho_{d+1}(I\setminus
J)=\operatorname{rank}h_{n-d}+\operatorname{rank}h_{n-d-1}=\rho_{d}(I\setminus
J)+\operatorname{rank}h_{n-d-1}$, and so
$\operatorname{rank}h_{n-d-1}=\alpha_{d+1}$. By recurrence we get
$\operatorname{rank}h_{n-t+1}=\alpha_{t-1}$. Clearly,
$\operatorname{rank}h_{n-t}\leq\rho_{t+1}(I\setminus J)$. By hypothesis,
$\rho_{t+1}(I\setminus J)<\alpha_{t}=\rho_{t}(I\setminus J)-\alpha_{t-1}$. It
follows that $\operatorname{rank}h_{n-t}<\rho_{t}(I\setminus
J)-\alpha_{t-1}=\rho_{t}(I\setminus J)-\operatorname{rank}h_{n-t+1}$ which
gives $\operatorname{depth}_{S}I/J=t$ by Lemma 1.2 (3).
Next example shows that the above theorem is tight.
###### Example 1.4.
Let $n=4$, $I=(x_{1},x_{3})$, $J=(x_{1}x_{4})$. Note that
$x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{3}x_{4}$ are all square free monomials of
degree $2$ from $I\setminus J$ and $x_{1}x_{2}x_{3},x_{2}x_{3}x_{4}$ are all
square free monomials of degree $3$ from $I\setminus J$. Thus
$\rho_{2}(I\setminus J)=4=\rho_{1}(I)+\rho_{3}(I\setminus J)$, but
$\operatorname{depth}_{S}I/J=3$. On the other hand, taking
$J^{\prime}=J+(x_{2}x_{3}x_{4})$ we see that
$\operatorname{depth}_{S}I/J^{\prime}=2$ which is given also by Theorem 1.3
since $\rho_{3}(I\setminus J^{\prime})=1$ and we have $\rho_{2}(I\setminus
J^{\prime})=4>3=\rho_{1}(I)+\rho_{3}(I\setminus J^{\prime})$.
###### Corollary 1.5.
Suppose that $\operatorname{depth}_{S}I/J\geq d+2$. Then
$\rho_{d}(I)\leq\rho_{d+1}(I\setminus J)\leq\rho_{d}(I)+\rho_{d+2}(I\setminus
J)$. Moreover, if $\rho_{d+2}(I\setminus J)=0$ then
$\rho_{d}(I)=\rho_{d+1}(I\setminus J)$.
###### Remark 1.6.
Consider the poset $P_{I\setminus J}$ of all square free monomials of
$I\setminus J$ (a finite set) with the order given by the divisibility. Let
${\mathcal{P}}$ be a partition of $P_{I\setminus J}$ in intervals
$[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$, let us say $P_{I\setminus
J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I\setminus J}$.
This is the Stanley depth of $I/J$, in the idea of [2] (see also [5]).
Stanley’s Conjecture says that
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. In the above
corollary $\rho_{d+2}(I\setminus J)=0$ implies
$\operatorname{sdepth}_{S}I/J\leq d+1$ and so $\operatorname{depth}_{S}I/J\leq
d+1$ if Stanley’s Conjecture holds. This shows the weakness of the above
corollary, which accepts the possibility to have
$\operatorname{depth}_{S}I/J\geq d+2$ when $\rho_{d}(I)=\rho_{d+1}(I\setminus
J)$.
## References
* [1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition. Cambridge University Press (1998).
* [2] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [3] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19(3), (2011), 163-166, arXiv:AC/1107.2621.
* [4] D. Popescu, Depth of factors of square free monomial ideals, arXiv:AC/1110.1963.
* [5] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
|
arxiv-papers
| 2012-06-18T16:12:23 |
2024-09-04T02:49:31.916684
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1206.3977"
}
|
1206.4036
|
# Characterization of a 6Li-loaded liquid organic scintillator for fast
neutron spectrometry and thermal neutron detection
C.D. Bass E.J. Beise H. Breuer C.R. Heimbach T. Langford J.S. Nico
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Department of Physics, University of Maryland, College Park, MD 20742, USA
Institute for Research in Electronics and Applied Physics, University of
Maryland, College Park, MD 20742, USA
###### Abstract
The characterization of a liquid scintillator incorporating an aqueous
solution of enriched lithium chloride to produce a scintillator with 0.40% 6Li
is presented, including the performance of the scintillator in terms of its
optical properties and neutron response. The scintillator was incorporated
into a fast neutron spectrometer, and the light output spectra from 2.5 MeV,
14.1 MeV, and 252Cf neutrons were measured using capture-gated coincidence
techniques. The spectrometer was operated without coincidence to perform
thermal neutron measurements. Possible improvements in spectrometer
performance are discussed.
###### keywords:
Fast neutron spectrometry , Thermal neutron detection , Organic scintillator ,
Lithium-6 , Capture-gated coincidence
###### PACS:
29.30.Hs Neutron spectrometry , 29.40.Mc Scintillation detectors
††journal: Applied Radiation and Isotopes
## 1 Introduction
Precise knowledge of the fast neutron flux and spectrum is essential for
several experimental endeavors requiring a low-background underground
environment (Formaggio and Martoff, 2004). These include searches for WIMP
dark matter (Akerib _et al._ , 2003, Angloher _et al._ , 2005, Gaitskell,
2004), neutrinoless double beta decay (Aalseth _et al._ , 2005, Elliott and
Vogel, 2002, Schönert _et al._ , 2006), and solar neutrinos (Abdurashitov _et
al._ , 2009, Aharmim _et al._ , 2007, Cleveland _et al._ , 1998, Fukada _et
al._ , 2001, Hampel _et al._ , 1999, McKinsey and Coakley, 2005). The
technological challenges associated with fast neutron spectrometry in
underground labs are similar to those presented to fast neutron flux
measurements for detecting and identifying fissile materials with low-level
neutron activity. Both applications require a detector with a low energy
threshold, high sensitivity, and good energy resolution.
Liquid organic scintillators are often used for fast neutron spectrometry
because of their fast response times, good pulse-height response, and modest
cost. However, organic scintillators have a high gamma-ray sensitivity with
comparable detection probabilities for neutrons and gamma-rays. For certain
types of organic scintillators, pulse-shape discrimination techniques can be
used to distinguish between particle types (Söderström _et al._ , 2008). Even
so, complicated unfolding procedures (Klein and Neumann, 2002) are needed for
obtaining energy information from pulse-height spectra because only a fraction
of the high-energy particles are brought to rest in the scintillator.
One technique for performing fast neutron spectrometry that overcomes spectral
unfolding procedures involves a capture-gated coincidence between an incident
fast neutron that is completely thermalized within an organic scintillator and
its subsequent capture on a nucleus with a large neutron capture cross
section, which is loaded within the scintillator volume (Czirr _et al._ ,
2002, Drake _et al._ , 1986, Kim _et al._ , 2010). Neutron thermalization is
fast (on the order of a few nanoseconds), but the capture time is typically
tens to hundreds of microseconds and depends on the diffusion time needed for
a thermalized neutron to propagate through the scintillator and capture on a
nucleus. A capture signal preceded by a thermalization signal within a
characteristic time can be used to select those fast neutrons that have
deposited all of their kinetic energy into the scintillator, and the initial
thermalization signal provides energy information about the incident neutron.
Both 10B and 6Li have large cross sections for thermal neutron capture. The
10B$(n,\alpha)^{7}$Li reaction has a _Q_ -value energy of 2.79 MeV and
produces either a 1.78 MeV alpha (ground state, 6% branching ratio) or a 1.47
MeV alpha and a 477 keV gamma-ray (first excited state, 94% branching ratio).
The 6Li$(n,\alpha)^{3}$H reaction has a _Q_ -value energy of 4.78 MeV and
produces a 2.05 MeV alpha and a 2.73 MeV triton. Because the fluorescent
efficiency in organic scintillators generally decreases for heavier particles
(Birks, 1951), a 6Li loading should produce a higher light output111It is
convenient to express light output in terms of the scintillator’s response to
electrons, as this can be taken to be linear at least above about 100 keV
(Flynn _et al._ , 1964). for neutron capture than a 10B loading. For example,
the calculated light output for the neutron capture products from 6Li and 10B
in BC501A222Certain commercial equipment, instruments, or materials are
identified in this paper in order to specify the experimental procedure
adequately. Such identification is not intended to imply recommendation or
endorsement by the National Institute of Standards and Technology, nor is it
intended to imply that the materials or equipment identified are necessarily
the best available for the purpose. (a liquid organic scintillator based on
xylene, which is commonly used in neutron spectroscopy and produced by Saint-
Gobain) is shown in Table 1. The light output of the triton from
${}^{6}\textrm{Li}(n,\alpha)^{3}\textrm{H}$ is around an order of magnitude
larger than that of the alpha from
${}^{10}\textrm{B}(n,\alpha)^{7}\textrm{Li}$.
While organic scintillators loaded with boron or gadolinium are available, to
our knowledge there are currently no commercially-available sources of
lithium-loaded organic scintillators. There are other methods for adding 6Li
to a scintillator (e.g. immersion of 6Li-glass plates in BC-501A (Hayashi _et
al._ , 2006)), but those techniques differ significantly in that the 6Li is
not distributed uniformly throughout the volume, as is the case with most
other commercially-available scintillators that are loaded with boron or
gadolinium. This is particularly important if one wants to scale up a
detector, where optical clarity becomes important. In this paper, we discuss
the development, production, and optical characterization of a 6Li-loaded
liquid organic scintillator along with measurements of its response to fast
and thermal neutrons.
| neutron | | Light
---|---|---|---
| capture | Energy | Output
| product | (MeV) | (keVee)
10B$(n,\alpha)^{7}$Li | $\alpha$ | 1.78 | 56
10B$(n,\alpha)^{7}$Li∗ | $\alpha$ | 1.47 | 37
| $\gamma$ | 0.477 | 311
6Li$(n,\alpha)^{3}$H | $\alpha$ | 2.05 | 74
| _t_ | 2.73 | 409
Table 1: Calculated light output for products of thermal neutron capture on
10B and 6Li in the organic liquid scintillator BC501A. The light output of the
charged particles was calculated using the Bethe-Bloch formula and measured
data (Verbinski _et al._ , 1968, Nakao _et al._ , 1995).
## 2 Lithium-loaded liquid scintillator chemistry
Loading an organic scintillator with 6Li can be accomplished by either
dissolving a lithium compound directly into the scintillator solvent or by
incorporating a medium containing a lithium compound into the scintillator
bulk, where the medium does not dissolve into the solvent. Aqueous solutions
of lithium compounds are technically straightforward to produce although they
will not dissolve into the aromatic solvents of organic scintillators.
However, there exists a class of organic liquid scintillators that was
developed specifically for accepting aqueous solutions by the inclusion of
surfactants (Wiel and Hegge, 1991). These so-called “scintillator cocktails”
are predominantly used for biological and health physics applications and they
possess good light output characteristics, low toxicity, a high flashpoint,
and are economically priced.
Zinsser Analytic developed a high-efficiency scintillator cocktail under the
commercial name Quickszint 164 that could incorporate aqueous solutions up to
40% water by volume. It is a mixture of an aromatic solvent (di-isopropyl
naphthalene), organic fluors (PPO and bis-MSB), a non-ionic surfactant
(ethoxylated nonylphenol), and mineral oil. Quickszint 164 has a high
flashpoint ($>$100∘ C), is nontoxic, and is biodegradable. It has a density of
0.92 g cm-1, viscosity of 27.6 cP at 20∘ C, index of refraction of 1.57, and a
hydrogen-to-carbon ratio of 1.48 (Zinsser, 2008).
It should be noted that Quickzint 164 was specifically developed by Zinsser
Analytic as an experimental scintillator cocktail under the project name
XLS164H and is a non-stock item. Zinsser Analytic currently produces
Quickszint Flow 302+, which is a commercially-available liquid scintillator
cocktail that is formulated using the same aromatic solvent, organic fluors,
and non-ionic surfactant as Quickszint 164 with similar proportional
composition. We have not evaluated Quickszint Flow 302+ as a direct
replacement for Quickszint 164.
| | maximum
---|---|---
| solubility | Molarity
| (mass % of solute ) | (M)
Li2CO3 | 1.3 | 0.2
LiCl | 45.8 | 13.9
LiBr | 65.4 | 17.6
Table 2: Solubility of lithium compounds in water at 25∘ C. Data taken from
CRC Handbook (Lide, 2010).
Lithium chloride was chosen to incorporate into Quickszint 164 because of its
solubility and maximum molar concentration in water (Fisher _et al._ , 2011)
(see Table 2). For purposes of initially characterizing the optical and
physical properties of a loaded scintillator, unenriched lithium compounds
were used (isotope abundance for naturally-occurring lithium: 7.7% 6Li, 92.3 %
7Li). Lithium carbonate was reacted with hydrochloric acid to produce aqueous
lithium chloride. Excess water was removed from the solution by a combination
of heating and low vacuum, and the resulting slurry was diluted with deionized
water to produce three batches of aqueous lithium chloride in different molar
concentrations: 2.26 M, 4.72 M, and 9.48 M. Each of these solutions was added
to Quickszint 164 in varying volume fractions to create samples of lithium-
loaded scintillator over a range of loadings. The prepared samples were mixed
on a roller-mixer for 10 minutes and then allowed to settle for 24 hours to
allow trapped air bubbles to escape.
When aqueous solutions are added to scintillator cocktails, they form water-
in-oil microemulsions that are uniform mixtures of oil, water, and surfactant,
which occur with minimal mixing and are highly stable (Fanun, 2009).
Microemulsions are dynamical systems in which droplets undergo collision,
fusion, and core material exchange. However, the exchange process is
controlled by activation energy and not diffusion, so an equilibrium droplet
size and shape is maintained. Typical microemulsions have structural
dimensions between 2 nm and 50 nm and are transparent in the near-UV and
visible light range.
The amount of lithium chloride solution that Quickszint 164 was able to accept
was initially determined by visual inspection of the prepared samples.
Complete emulsification resulted in a water-clear, colorless liquid with a
slight bluish fluorescent tinge. Observation of cloudiness, films, or
suspended slurry was evidence of phase separation at the macroscopic level or
incomplete emulsification. Based on these criteria, Quickszint 164 had a
definite lower and upper limit of the volume fraction of solution that could
be emulsified. Prepared samples were monitored for long-term stability, and
all samples that were completely emulsified remained stable after one year.
Figure 1: Transmittance spectra of Quickszint 164 (black line) and the 0.40%
6Li-loaded scintillator (red line). Vertical lines at 400 nm and 550 nm denote
the relevant spectral band for fluorescence. Normalization was accomplished by
taking a null baseline transmittance measurement of the empty sample cuvette.
Normalized transmittance values over 100% are possible due to differences in
refractive indices between the (quartz) sample cuvette, scintillator, and air.
The ratio of the transmittance integrals over the spectral band was 88%, which
indicated acceptable optical quality for the 6Li-loaded scintillator.
Lithium bromide was also investigated as a possible choice for an aqueous
lithium compound because it has a larger maximum solubility in water and can
form a higher-concentration solution than lithium chloride, which would allow
a larger lithium-loading within Quickszint 164. A 12.3 M lithium bromide
solution was prepared and added to Quickszint 164 in varying concentrations
using the same procedures as the lithium chloride solutions. However, when
incorporated into Quickszint 164 over a range of lithium mass fractions
equivalent to the upper range of the water-clear emulsions formed with lithium
chloride, the lithium bromide solution formed a viscous, milky liquid that
failed to emulsify. In terms of emulsification, aqueous lithium chloride
appears to be superior to aqueous lithium bromide for loading Quickszint 164,
despite the higher solubility and larger maximum molar concentration of
lithium bromide.
## 3 Optical properties of the lithium-loaded scintillator
The optical quality of each sample was evaluated by measuring its light
transmittance across the UV-Vis spectrum. The light transmittance _T_ for a
sample is defined as
$T(\lambda)=\frac{I(\lambda)}{I_{o}(\lambda)},$ (1)
where _I_ and $I_{o}$ are (respectively) the intensities of light transmitted
through and incident upon a sample for a given wavelength $\lambda$.
Transmittance measurements were performed with a spectrophotometer using a
fused quartz, 10-mm pathlength cuvette.
Quickszint 164 uses bis-MSB as a waveshifter, which has an absorbance spectrum
that ranges from near-UV to almost 400 nm and a fluorescence spectrum that
ranges between 370 nm and 550 nm with prominent peaks at 400 nm and 424 nm.
The relevant spectral band for optical quality was therefore selected from 400
nm to 550 nm.
Figure 2: Plot of the optical performance of the Li-loaded scintillator as a
function of the volume fraction of aqueous lithium chloride for various molar
concentrations. The minimal loading for acceptable transmittance is 0.4% -
0.6% aqueous lithium chloride by volume and is independent of molar
concentration, while the maximal loading depends proportionally on the molar
concentration of aqueous lithium chloride. Error bars indicate the accuracy of
the UV-Vis transmittance measurements and physical measurements during
chemistry procedures.
Transmittance measurements were taken for each sample, and a typical spectrum
is shown in Figure 1. The transmittance integral in the spectral band for each
sample was compared to that of Quickszint 164, and the ratio of the integrals
$\frac{\int{T_{Li}(\lambda)\,d\lambda}}{\int{T_{pure}(\lambda)\,d\lambda}},$
(2)
provided a metric of the optical degradation of Quickszint 164 due to the
inclusion of aqueous lithium chloride.
Light transmittance studies indicate a range of loadings in Quickszint 164
that display minimal optical degradation. As shown in Figure 2, the lower
loading limit depends primarily on the volume fraction of aqueous lithium
chloride present in Quickszint 164. In contrast, the upper loading limit
depends proportionally on the molar concentration of the aqueous lithium
chloride – highly-concentrated solutions extend the range of allowed loadings.
The choice of loading within the acceptable transmittance range is constrained
by the neutron response of the loaded scintillator. Monte Carlo simulations
using MCNP5 (Brown _et al._ , 2003) of a small volume detector employing 6Li-
loaded scintillator indicate that detector efficiency increases with the mass
fraction of 6Li while neutron diffusion time decreases. Therefore, the optimal
6Li-loaded scintillator should incorporate enriched aqueous lithium chloride
with a high molar concentration using the largest possible volume fraction in
a scintillator cocktail, which possesses acceptable optical transmittance.
## 4 Production of 6Li-loaded liquid scintillator
A 9.40 M lithium chloride solution was prepared using lithium carbonate with a
95.5% enrichment of 6Li. It was mixed into Quickszint 164 to yield a lithium-
loading of 0.40% 6Li by mass and a calculated hydrogen-to-carbon ratio of
1.57. UV-Vis measurements indicated a transmittance integral ratio of 88%.
Because the addition of aqueous lithium chloride into Quickszint 164 reduced
its light output, fluorescence measurements were performed to ascertain the
level of decrease in scintillation light. Fluorescence spectra were measured
with a multifrequency phase fluorometer using single wavelength excitations at
300 nm and at 350 nm, which matched the fluorescence spectrum of the primary
fluor, PPO. As seen in Figure 3, the addition of aqueous lithium chloride
decreased the light output of Quickszint 164 by a factor of two.
The microemulsion droplet size of the 6Li-loaded scintillator was measured
using a dynamic light scattering instrument. The measured hydrodynamic droplet
mean radius was 4.5 nm, which included the extent of the surfactant molecules
from the droplet into the scintillator bulk. The calculated mean length of the
hydrophobic tail of the ethoxylated nonylphenol surfactant is approximately 1
nm, so the calculated diameter for a droplet of lithium chloride solution in
Quickszint 164 is approximately 8 nm.
A test neutron detector was assembled by filling a 5 cm diameter by 6 cm
cylindrical borosilicate glass cell (approximately 100 ml) with the prepared
6Li-loaded scintillator. The volume and geometry of the glass cell was chosen
1) to couple with available 5 cm PMTs with characteristics deemed compatible
with the expected performance of the 6Li-loaded scintillator, and 2) to
provide roughly the same dimensions in terms of diameter and length so that
the detector performance would be largely independent of neutron-field
direction and easier to model in simulation. The cell was externally coated
with Bicron 622A reflective paint and coupled to a 5-cm Burle 8850
photomultiplier tube (PMT) using optical grease. PMT signals were recorded as
waveforms by a GaGe 8-channel 125 MHz digital oscilloscope card with 14-bit
resolution. The data acquisition electronics (see Figure 4) allowed events to
be recorded either in singles mode, where the digitizer triggered on signals
above a threshold, or in capture-gated coincidence mode, where the digitizer
would be triggered on any event (a “start event”) above a threshold that was
followed within 40 $\mu$s by a second event (a “stop event”) above the
threshold. All of the digitized waveforms were recorded to disk for later
analysis.
## 5 Fast neutron response
The PMT and glass cell containing the 6Li-loaded scintillator were inserted
into a 0.64 cm thick lead cylinder to reduce the gamma-ray background. This
assembly was surrounded by 6 mm of borated-silicone to reduce the thermal
neutron background. Borated-silicone is a neutron shielding material based on
a silicone elastomer that has boron carbide powder homogeneously mixed
throughout its matrix. The boron content attenuates thermal neutron flux due
to its high thermal neutron absorption cross section.
Fast neutron irradiation of the detector was performed in the NIST Californium
Neutron Irradiation Facility (Grundl _et al._ , 1977), and neutrons were
produced either by P-325 Neutron Generators manufactured by Thermo Electron
Corporation (2.5 MeV neutrons from DD-fusion or 14.1 MeV neutrons from DT-
fusion) or by spontaneous fission of 252Cf sources. During irradiation, the
detector was positioned approximately 2 m from a neutron source, and the
resulting fast neutron field was a combination of an isotropic distribution
from the source and neutron return from the environment, including albedo from
boundaries of the irradiation room. Data for each irradiation was acquired in
capture-gated coincidence mode, which allowed the detector to function as a
fast neutron spectrometer.
Figure 3: Fluorescence spectra for pure scintillator cocktail and 6Li-loaded
scintillator using excitation energies of 300 nm and 350 nm. The addition of
aqueous lithium chloride decreases the fluorescence intensity of Quickszint
164 by about a factor of two. Figure 4: Block diagram of the data acquisition
electronics. In capture-gated coincidence mode, the digitizer card is
triggered by the time-to-amplitude converter (TAC) whenever a pair of PMT
signals above threshold (set by a discriminator) are produced within 40 $\mu$s
of each other. The TAC is used as a coincidence analyzer with a longer
resolving time than is possible with typical coincidence analyzers. A 450 ns
delay ensures that the coincidence trigger is due to a pair of pulses and not
from the long tail of a single high-energy PMT signal. In singles-mode, the
digitizer card is triggered by any PMT signal above threshold (set by the
card). All digitized waveforms are recorded to disk for offline analysis.
Figure 5: Background subtraction scheme used in offline analysis. The top
graph shows a typical distribution of time delays between the start and stop
signals for capture-gated coincidence measurement of fast neutrons. This
distribution is for an irradiation by 2.5 MeV neutrons and has a decay
constant of 4.0 ${\mu}$s, which corresponds to the mean diffusion time in the
scintillator for a thermalized neutron to capture on a 6Li nucleus. The
horizontal line indicates the level of uncorrelated background events due to
random coincidences, and the vertical line corresponds to a time $t=4.6\tau$,
which is used to separate data from the background. The middle and bottom
graph shows the recoil and capture spectrum (respectively) for 2.5 MeV
neutrons before and after the background subtraction cut in analysis.
Data runs of 50,000 events for each of the neutron sources were taken. Off-
line analysis of the runs determined the relative time delays between start
and stop signals for each digitized waveform as well as the pulse heights for
the start and stop signals. Histograms of the time delays for each run were
created and fit with an exponential curve. The decay constant $\tau$ for the
fit curve corresponds to the mean diffusion time for a thermalized neutron to
propagate through the scintillator and capture on a 6Li nucleus. For this
detector geometry and scintillator loading, the measured decay constant was
4.0 $\mu$s. Because coincidences with delay times greater than several mean
diffusion times can be rejected as probable background events, a cut was
applied to those coincidences that had delay times larger than
$t=\tau\ln{100}\approx 4.6\tau$, which should retain 99% of non-background
events (Figure 5 shows a typical distribution). Start and stop pulse height
histograms of those cut events were used to perform a weighted background
subtraction on the start and stop pulse height histograms for the remaining
events. This background subtraction scheme is used to improve both the signal-
to-noise on the capture signal and energy resolution of the spectrometer.
Because the data were acquired in event mode, additional cuts on stop events
that did not correspond to a neutron capture on 6Li could be made in analysis.
The detector was calibrated by removing the lead and borated-silicone and
acquiring data in singles mode of irradiations by 60Co, 137Cs, 22Na, and 133Ba
gamma-ray sources. Pulse height information from each waveform was used to
build histograms for each source, and the Compton edge of the gamma-rays for
each source provided light output calibrations for the detector across a range
of energies. The calibrations were then used to convert the start and stop
event histograms into light output spectra for the proton recoil and neutron
capture events (respectively).
The measured light output spectra for 2.5 MeV neutrons, 14.1 MeV neutrons, and
neutrons from 252Cf decay are shown in Figure 6, which shows the proton recoil
and neutron capture spectra for each neutron source after performing
background subtraction in analysis. Monte Carlo calculations of the light
output spectra in Figure 6 were performed using MCNP5 for the response
functions and MCNP5’s PTRAC option for identifying energy deposition with
particle type. The energy-to-light response functions from Verbinski _et al._
(1968) for various charged particles were used to convert deposited energy to
light on an event-by-event basis.
Neutron capture on 6Li verifies that all of the neutron interactions occurred
within the scintillator and would indicate that the response should be a peak
in the recoil spectra for each monoenergetic source and an approximately
Maxwellian recoil spectrum for the 252Cf source. Monte Carlo calculations of
the recoil spectra included the free-field for each neutron source and the
geometry and composition of the irradiation facility, and show a low-energy
tail that primarily arises due to the neutron return in addition to the peaks
for the monoenergetic sources and the Maxwellian distribution for the 252Cf
sources. This low-energy tail comes close to merging with the peak resulting
from the 2.5 MeV neutrons but is well-separated from the 14.1 MeV peak; the
low-energy tail merges completely with the Maxwellian distribution of the
recoil spectrum for the 252Cf sources. The calculations were smoothed using a
Gaussian convolution and then overlaid on the measured spectra; the
calculations show good agreement to the measured recoil spectra.
It should be noted that the measured data are of the light production in the
scintillator and not of energy deposition. The non-linear light yield in an
organic scintillator (Birks, 1951) for hydrogen (protons) cause an apparent
loss of energy, and the light yield for carbon recoils is significantly less
than for proton recoils, which causes an additional loss of response that
further contributes to the low-energy tail in the recoil spectra. At higher
energies, neutron inelastic reactions generate 4.4 MeV gamma-rays that are
lost to the scintillator, which contribute to a gap between the peak and the
low-energy tail in the 14.1 MeV neutron recoil spectrum.
2.5 MeV Neutrons
14.1 MeV Neutrons
Neutrons from 252Cf Decay
(a) (b) (c) (d) (e) (f)
Figure 6: Background-subtracted data from the 6Li-loaded scintillator test
detector irradiated with 2.5 MeV neutrons (top graphs), 14.1 MeV neutrons
(middle graphs), and neutrons from 252Cf decay (bottom graphs). The left
graphs shows the proton recoil light output spectra (black lines) that
correspond to the energies of the incident neutrons and the Gaussian-smoothed
Monte Carlo calculations (red lines) of the light output spectra (note: the
low-energy portion of recoil spectra for the 14.1 MeV data and Monte Carlo
were truncated because they reduce the clarity for seeing the peak around 6
MeVee, and because the exponential shape of data and Monte Carlo are
unremarkable). The right graphs shows the neutron capture light output
spectra, which corresponds to the total energy of the 6Li(n,$\alpha$)3H
reaction products. Data was collected at rates ranging between 0.4 events/s
and 20 events/s. For clarity, the bin widths for the recoil spectra for 14.1
MeV neutrons and neutrons from 252Cf decay were increased by a factor of 10
and 4 (respectively) relative to their corresponding neutron capture light
output spectra.
## 6 Thermal neutron studies
Calculations done in MCNP5 indicate that the mean free path of thermal
neutrons in the 6Li-loaded scintillator is 3.5 mm, so the test neutron
detector could function as a high-efficiency thermal neutron detector as well
as a fast neutron spectrometer. Calculations indicate that approximately 25%
of thermal neutrons incident on the 6Li-loaded liquid scintillator will
backscatter out from the incident surface of the scintillator. The neutrons
that do penetrate into the scintillator volume will undergo neutron capture
with nearly 100% efficiency.
Ambient thermal neutron flux measurements were performed at two locations in
the vicinity of the 20 MW research reactor at the NIST Center for Neutron
Research (NCNR): outside the concrete shield of the research reactor in the
confinement building, and at the end station of the NG-6 cold neutron beamline
(Nico _et al._ , 2005) in the experimental hall. The beam line extends
approximately 70 m from the reactor, and the measurement was taken off-axis of
the beam and with the beam shutter closed. The data acquisition of the test
neutron detector was operated without requiring a capture-gated coincidence.
When running in this mode, the detector is also sensitive to gamma-rays so
three different shielding configurations were investigated: 1) unshielded; 2)
enclosed within a 10-cm thick lead house to suppress the gamma-ray background;
and 3) surrounded by 9 mm of borated-silicone to reduce the thermal neutron
flux, and enclosed within a 10-cm thick lead house to reduce the gamma-ray
background.
For comparison, an additional measurement at each location and shielding
configuration was taken by replacing the 6Li-loaded scintillator filled glass
cell in the test neutron detector with a 5 cm diameter by 6 cm cylindrical
block of Saint-Gobain BC-454 natural boron-loaded plastic scintillator.
Calculations indicate the mean free path of thermal neutrons in BC-454 is 2.6
mm and the backscattering of thermal neutrons out of the incident surface is
6%. Gamma-ray energy calibrations were performed on each detector using a 60Co
gamma-ray source, and the calibrations were used to convert the pulse height
histograms from the thermal neutron flux measurements into light output
spectra. The spectra for the lead and borated-silicone shielding configuration
were used as a background subtraction for the lead-only shielding spectra, and
the resulting spectra were integrated to yield an ambient thermal neutron flux
measurement at each location. The measured and background-subtracted spectra
in the reactor confinement building are shown in Figure 7.
Because the test neutron detector uses a borosilicate glass cell to contain
the 6Li-loaded liquid scintillator, some fraction of thermal neutrons capture
on the naturally-occurring 10B nuclei in the glass, thus reducing the fluence
of thermal neutrons incident on the 6Li-loaded scintillator. To quantify the
lost fraction, neutron transmission measurements were performed on an empty
cell using the Alpha-Gamma neutron detector (Gilliam, Greene, and Lamaze,
1989) and the NG-6M neutron beam line at the NCNR (Nico _et al._ , 2005). The
NG-6M beam line produces a $\lambda=0.496$ nm monochromatic neutron beam, and
the Alpha-Gamma neutron detector can count the number of neutrons for a sample
in-beam and out-beam. Because the measured neutron fluence can be reduced by
both absorption and scattering in the sample, gamma-ray and beta-ray rates
were measured with hand-held monitors near the glass cell. No excessive gamma-
ray or beta-ray production was observed, so neutron losses through the glass
cell were predominantly due to absorption.
The measured transmission of 0.496 nm neutrons passing through the glass cell
was 7.3%. However, the 6Li-loaded scintillator is contained within the
interior of the cell so neutrons incident on the scintillator nominally pass
through only a single wall thickness of glass, thus the corrected transmission
value for the borosilicate glass in the test neutron detector is 27.0%. In
addition, the neutron capture cross section for 10B is proportional to $1/v$,
so the energy-correction for thermal neutrons ($\lambda=0.18$ nm) yields a
transmission of 44.9% for the glass cell in the test neutron detector.
After correcting for transmission losses through the borosilicate glass (6Li-
loaded scintillator only) and back-scattering (approximately 25% for the 6Li-
loaded scintillator and 6% for BC-454), both detectors measured an ambient
thermal neutron flux of approximately 2.0 $\textrm{cm}^{-2}\textrm{s}^{-1}$ in
the confinement building of the research reactor. A measurement using a 4-atm
3He neutron detector-proportional counter measured the ambient thermal neutron
flux at this location as 1.9 $\textrm{cm}^{-2}\textrm{s}^{-1}$. At the end
station of the NG-6 cold neutron beamline, both detectors measured an ambient
thermal neutron flux of approximately 0.11 $\textrm{cm}^{-2}\textrm{s}^{-1}$.
6Li-loaded Scintillator
Boron-loaded Plastic Scintillator
(a) (b) (c) (d)
Figure 7: Ambient thermal neutron flux measured outside the concrete shielding
of the research reactor at the NCNR. The test neutron detectors employed 6Li-
loaded scintillator (top graphs) and BC-454 boron-loaded plastic scintillator
(bottom graphs). The left graphs shows the light output spectra for the
different shielding configurations designed to suppress gamma-ray backgrounds
and isolate thermal neutrons. The right graphs show the ambient thermal
neutron capture signal spectra after subtracting the borated-silicone and lead
shielding spectra from the lead-only shielding spectra.
## 7 Discussion
Because of the microemulsion nature of the loaded scintillator, the 6Li nuclei
are not dispersed uniformly throughout the scintillator bulk but are localized
in suspended droplets. This means that charged particle products from neutron
capture on 6Li – an alpha and triton – must first escape from a droplet and
then pass through other droplets as they propagate through the scintillator.
The particles deposit energy during their transit through the droplets but do
not produce scintillation light. This loss of light output corresponds to a
downward shift in the measured spectra of the neutron capture on 6Li. The same
process affects the measured proton recoil light output spectra and the gamma-
ray calibration spectra.
An estimate of the effect can be calculated based on the volume fraction of
aqueous lithium chloride in the 6Li-loaded scintillator and the ranges of the
charged particles in the lithium chloride solution and Quickszint 164. The
ranges for each of the charged particles in 9.40 M lithium chloride solution
and Quickszint 164 as calculated using the _Stopping and Range of Ions in
Matter_ 2008 software package (Ziegler, 2003) are large compared to the
droplet size and are roughly equivalent (see Table 3). An estimate of the loss
of light output when compared to unloaded scintillator cocktail is given by
the volume fraction of lithium chloride solution, which for the 6Li-loaded
scintillator is 7.5%. However, at these energies this reduction is
approximately the same for the energy spectra from neutron capture, proton
recoil, and gamma-ray sources, so energy calibrations are not significantly
affected.
| 9.40 M LiCl(aq) | Quickszint
---|---|---
2.05 MeV alpha | 10.8 $\mu$m | 10.4$\mu$m
2.73 MeV triton | 66.6 $\mu$m | 62.5 $\mu$m
Table 3: Ranges of the charged particle products of neutron capture on 6Li
through the components of the 0.40% 6Li-loaded scintillator as calculated
using SRIM. The calculation assumes a Bragg correction of -6.0% for the
lithium chloride solution and +5.0% for the liquid scintillator.
While the 100 ml test detector demonstrated the ability of 6Li-loaded
scintillator to unambiguously detect capture neutrons and provide incident
neutron energy information from the proton recoil spectra, the estimated
efficiency was of order $10^{-3}$. For a fast neutron spectrometer to be of
practical value, it would require good efficiency and energy resolution with
low sensitivity to uncorrelated background events. Detector efficiency could
be improved by increasing the volume of scintillator because the mean
diffusion time for a thermalized neutron to propagate out of the scintillator
would increase, which would increase the probability of neutron capture. In
addition, efficiency could be improved by increasing the concentration of 6Li
within the scintillator, which would decrease the mean thermal neutron capture
time.
However, increasing the volume of the detector would require optimizing the
optical performance of the 6Li-loaded scintillator to obtain good energy
resolution. The volume fraction of lithium chloride solution that was added to
Quickszint 164 was chosen to maximize the mass fraction of 6Li within the
scintillator, maintain microemulsion stability, and still retain good optical
properties. For a spectrometer that incorporates large volumes of
scintillator, the choice of volume fraction of lithium chloride should also
consider the attenuation length of the scintillator. In addition, the loss of
energy resolution due to the non-linear light yield of organic scintillators
could be addressed by employing a multichannel, optically-segmented detector
(Abdurashitov _et al._ , 2002), which would allow better resolution of the
energy deposition from multiple elastic scattering during neutron
thermalization.
This type of fast-neutron detector should have good efficiency and have energy
resolution capable of detecting low-rate signals from fissile materials.
Additional rejection of false events could be accomplished through pulse-shape
discrimination techniques (Flaska and Pozzi, 2007, Fisher _et al._ , 2011,
Klein and Neumann, 2002, Wolski _et al._ , 1995) and using energy information
in analysis. It should be noted that acquisition time needed to produce the
fast neutron energy spectra detailed in this paper was of order 10 hours and
was a consequence of the small volume of the 6Li-loaded scintillator. Any
practical device for detecting low-rate signals from fissile materials would
require a spectrometer that incorporates several liters of 6Li-loaded
scintillator. MCNP5 studies indicate that a fast neutron spectrometer
containing about 10 liters of 6Li-loaded scintillator in a cubic geometry
could detect neutrons from unmoderated 252Cf at a rate of 0.3
$\textrm{ng}^{-1}\textrm{s}^{-1}$ at a distance of 2 m in the energy range of
1 MeV to 20 MeV.
As a thermal neutron detector, the 6Li-loaded scintillator performed
comparable to commercially-available boron-loaded scintillator, and both
scintillators yielded ambient thermal neutron flux measurements that agree at
the 10% level. However, the neutron capture signal for the 6Li-loaded
scintillator was well-separated from the low-energy gamma-ray tail in the
unshielded detector configuration, while the neutron capture signal (the
primary alpha peak) for the BC-454 boron-loaded plastic was well within the
low-energy gamma-ray tail in its unshielded detector configuration (see Figure
7). This separation of the neutron capture by the 6Li-loaded scintillator
could allow an analysis of the gamma-ray spectra concurrently with the thermal
neutron flux measurement, which might not be possible with boron-loaded
scintillators (and is not possible with a 3He neutron detector-proportional
counter).
## 8 Conclusion
We have developed a 6Li-loaded liquid scintillator suitable for use in fast
neutron spectrometry and thermal neutron detection. We chose the 6Li-loading
based on optical transmittance within a wavelength band appropriate for PMT
operation and detector efficiency. Irradiations of a test spectrometer by 2.5
MeV neutrons, 14.1 MeV neutrons, and neutrons from 252Cf decay demonstrated
that the scintillator is capable of cleanly identifying neutron capture events
and generating a proton recoil light output spectrum that is related to the
incident neutron energy. In addition, the 6Li-loaded scintillator was used to
measure ambient thermal neutron flux and performed comparably to a boron-
loaded plastic scintillator.
A 6Li-loaded liquid scintillator has some advantages over scintillators loaded
with other neutron capture isotopes and may be the preferred agent for some
applications. The products from the 6Li$(n,\alpha)^{3}$H reaction are charged
particles and not gamma-rays, so their energy deposits are completely
contained within the scintillating medium. This is particularly advantageous
if the detection volume is not large. The _Q_ -value of the
6Li$(n,\alpha)^{3}$H reaction is larger than that for 10B$(n,\alpha)^{7}$Li,
so the energy deposit peak is much better separated from the noise threshold.
The production of aqueous lithium chloride uses straightforward chemical
procedures and does not require elaborate facilities. In addition, the cost of
loading the liquid scintillator with 6Li is significantly less than
commercially available boron-loaded scintillators. We do note that acquiring
enriched 6Li may prove difficult for some institutions.
The present investigations show promise for 6Li-loaded scintillator, but
further research into its properties is warranted. As a thermal neutron
detector, one should quantify the effect of gamma-ray contamination in the
capture peak; pulse shape discrimination methods could optimize the neutron-
to-gamma signal in that region. The optical and physical quality of the
loaded-scintillator has been observed over the course of approximately one-
year, but for most applications, it is reasonable to think that researchers
would want stability over periods of several years. In addition, these
investigations have used only relatively small samples; for a large-scale
detector, one must know the attenuation length of the loaded scintillator and
its stability over time.
## 9 Acknowledgements
The authors acknowledge the support of the National Institute of Standards and
Technology, U.S. Department of Commerce, in providing the neutron research and
chemistry facilities used in this work. This work is supported in part by NSF
PHY-0809696 and NSF PHY-0757690. Tom Langford acknowledges support under the
National Institute of Standards and Technology American Recovery and
Reinvestment Act Measurement Science and Engineering Fellowship Program Award
70NANB10H026 through the University of Maryland.
We acknowledge Dr. Andrew Yue at the NIST Center of Neutron Research for his
help with cold neutron transmission studies. We acknowledge Dr. Paul DeRose at
the NIST Advanced Chemical Sciences Laboratory for his help with fluoroscopic
measurements. We thank Dr. Vladimir Gavrin and Dr. Johnrid Abdurashitov of the
Institute for Nuclear Research - Russian Academy of Sciences for useful
discussions.
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|
arxiv-papers
| 2012-06-18T19:49:48 |
2024-09-04T02:49:31.924155
|
{
"license": "Public Domain",
"authors": "C. D. Bass, E. J. Beise, H. Breuer, C. R. Heimbach, T. Langford and J.\n S. Nico",
"submitter": "Christopher Bass",
"url": "https://arxiv.org/abs/1206.4036"
}
|
1206.4087
|
# Building non-coding RNA families
Lars Barquist Wellcome Trust Sanger Institute, Hinxton, UK Sarah W. Burge
Wellcome Trust Sanger Institute, Hinxton, UK Paul P. Gardner University of
Canterbury, Christchurch, NZ
###### Abstract
Homology detection is critical to genomics. Identifying homologous sequence
allows us to transfer information gathered in one organism to another quickly
and with a high degree of confidence. Non-coding RNA (ncRNA) presents a
challenge for homology detection, as the primary sequence is often poorly
conserved and de novo structure prediction remains difficult. This chapter
introduces methods developed by the Rfam database for identifying “families”
of homologous ncRNAs from single “seed” sequences using manually curated
alignments to build powerful statistical models known as covariance models
(CMs). We provide a brief overview of the state of alignment and secondary
structure prediction algorithms. This is followed by a step-by-step iterative
protocol for identifying homologs, then constructing an alignment and
corresponding CM. We also work through an example, building an alignment and
CM for the bacterial small RNA MicA, discovering a previously unreported
family of divergent MicA homologs in Xenorhabdus in the process. This chapter
will provide readers with the background necessary to begin defining their own
ncRNA families suitable for use in comparative, functional, and evolutionary
studies of structured RNA elements.
## 1 Introduction
Alignment is a central problem in bioinformatics. A wide range of critical
applications in genomics rely on our ability to produce “good” alignments.
Single-sequence homology search as implemented in tools such as BLAST[1] is an
(often heuristic) application of alignment. The sensitivity and specificity of
homology search can be improved by the use of evolutionary information in the
form of accurate substitution and insertion-deletion (indel) rates derived
from multiple sequence alignments (MSAs), captured in the statistical models
used by HMMER[2] and Infernal[3] for protein and RNA alignments respectively.
These models can be interpreted as defining “families” of homologous
sequences, as in the Pfam and Rfam databases[4, 5]. By using these models to
classify sequences, we can infer functional and structural properties of
uncharacterized sequences.
Unfortunately, producing the high-quality “seed” alignments of RNA these
methods require remains difficult. While proteins can be aligned accurately
using only primary sequence information with pairwise sequence identities as
low as 20% for an average-length sequence[6, 7], it appears that the “twilight
zone” where blatantly erroneous alignments occur between RNA sequences may
begin at above 60% identity[8]. The inclusion of secondary structure
information can improve alignment accuracy[9], but predicting secondary
structure is not trivial[10]. An instructive example of the difficulties this
can lead to is the case of the 6S gene, a bacterial RNA which modulates
$\sigma^{70}$ activity during the shift from exponential to stationary growth.
The _Escherichia coli_ 6S sequence was determined in 1971[11] and its function
determined in 2000[12]. However, the extent of this gene’s phylogenic
distribution was not realized until 2005 when Barrick and colleagues carefully
constructed an alignment from a number of deeply diverged putative 6S
sequences, and through successive secondary-structure aware homology searches
demonstrated its presence across large swaths of the bacterial phylogeny[13].
Even now, new homologs are discovered on a regular basis[14, 15], and 6S
appears to be an ancient and important component of the bacterial regulatory
machinery.
It is our hope to make these techniques accessible to sequence analysis
novices. This chapter aims to introduce the techniques necessary to construct
a high-quality RNA alignment from a single seed sequence, and then use the
information contained in this alignment to identify additional more distant
homologs, expanding the alignment in an iterative fashion. These methods,
while time-consuming, can be far more sensitive than a BLAST search[16]. We
will briefly review the state of the art in RNA sequence alignment and
structure prediction. We then present a brief protocol which starts with a
single sequence, and then uses a collection of web and command-line based
tools for alignment, structure prediction, and search to construct an Infernal
covariance model (CM), a probabilistic model which captures many important
features of structured RNA sequence variation[3]. These models may then be
used in the iterative expansion of alignments or for homology search and
genome annotation. CMs are also are used by the Rfam database in defining RNA
sequence families, and are the subject of a dedicated RNA families track at
the journal _RNA Biology_[17]. Finally, we present as an instructive example
the construction of an RNA family for the enterobacterial small RNA MicA,
discovering a convincing divergent clade of homologs in the process.
### 1.1 RNA Alignment and Secondary Structure Prediction
RNA sequence alignment remains a challenge despite at least 25 years of work
on the problem. As discussed above, alignments based on primary sequence
become highly untrustworthy below 60% pair-wise sequence identity, likely due
to the lower information content of individual nucleic acids as compared to
amino acids in protein alignments. This can be intuitively understood by
recalling the fact that 3 nucleic acids are required to encode an individual
amino acid; so, an amino acid carries 3 times as much information as a nucleic
acid (a bit less, actually, due to the redundancy of the genetic code). In
addition, the larger alphabet size of protein sequences allows for the easy
deployment of more complex substitution models, and a glut of protein sequence
data allows for highly effective parameterization of these models.
The incorporation of secondary structure, i.e. base-pairing, information has
been proposed as a means to make up for these difficulties in RNA alignment
methods. The first proposal for such a method is now known as the Sankoff
algorithm[18]. The Sankoff algorithm uses dynamic programming, an optimization
technique long central to to sequence analysis111A full explanation of dynamic
programming is beyond the scope of this book chapter, but for a brief
introduction see two excellent primers by Sean Eddy covering applications to
alignment[19] and secondary structure prediction[20]; for those seeking a
deeper understanding Durbin et al.[21] provides coverage of dynamic
programming as well as covariance models.. Dynamic programming had previously
been applied to the problems of sequence alignment[22] and RNA folding[23].
Sankoff proposed a union of these two methods. Unfortunately, the resulting
algorithm has a time requirements of $\mathcal{O}(L^{3N})$ and space
requirements of $\mathcal{O}(L^{2N})$ where $L$ is the sequence length and $N$
is the number of sequences aligned. This is impractical, even for small
numbers of short sequences. A number of faster algorithms have been developed
to approximate Sankoff alignment. Recent examples include CentroidAlign[24],
mLocARNA[25], and FoldalignM[26]. These methods can push the RNA alignment
twilight zone as low as 40 percent identity[8].
However, for the purpose of family-building, we are often starting with a
single sequence of unknown secondary structure, and have to gather additional
homologs using a fast alignment tool, such as BLAST. Such methods are not able
to reliably detect homologs below 60 percent sequence identity. In this range
of pair-wise sequence identities, the slight increases in accuracy of Sankoff-
type algorithms over non-structural alignment is only rarely worth the
additional computational costs involved222For recent benchmarks of alignment
tools on ncRNA sequences see Hamada et al.[24] and the supplementary
information of Bradley et al.[27]; Hamada includes comparisons of aligner
runtimes, while Bradley examines relative performance over a range of pair-
wise sequence percent identities.. Alignments generated with standard
alignment tools can then be used as a basis for predictions of secondary
structure using tools like Pfold[28], RNAalifold[29], or CentroidFold[30].
Regardless, all modern alignment tools, Sankoff-type or standard, suffer from
a number of known problems. Most alignment tools use _progressive alignment_.
This means that the aligner decomposes the alignment problem in to a series of
pair-wise alignment problems along a guide tree. This greatly reduces the
computational complexity of the alignment problem, but means that any error in
an early pair-wise alignment step is propagated through the entire alignment.
A number of solutions have been proposed to this problem, such as explicitly
modeling insertion-deletion histories[31] or using modified or alternative
optimization methods such as consistency-guided progressive alignment[32] or
sequence annealing[33]. A second issue is that it is not clear which function
of the alignment aligners should be optimizing, and many appear to over-
predict homology[34, 27, 35]. Finally, many parameters commonly used in
alignment, such as gap opening and closing probabilities and substitution
matrices, appear to vary across organisms, sequences, and even positions
within an alignment. All of this leads to considerable uncertainty in
alignment[36], which is not easily captured by most current alignment methods.
The additional parameters introduced by RNA secondary structure prediction
only compounds these these problems.
A final problem with alignment is the issue of determining whether two
sequences are similar due to _homology_ or _analogy_. Homology describes a
similarity in features based on common descent; for instance, all bird wings
are homologous wings. Analogy, on the other hand, describes a similarity in
features based on common function without common descent; bat and bird wings
perform the same function, and appear superficially similar. However, their
evolutionary histories are quite different. In sequence analysis, we often
assume that aligned residues within an alignment share common ancestors, but
this assumption can be confounded by analogous sequence. These analogs often
take the form of _motifs_ , short sequences which perform specific functions
within the RNA molecule and can arise easily through convergent evolution. An
example of such a motif is the bacterial rho-independent terminator[37], a
short hairpin responsible for halting transcription in many species. While
such motifs can be a boon in discovering novel ncRNA genes[38] or aligning
homologs which contain them, they can also be a source of false-positives when
attempting to build an alignment of homologous sequences.
Rfam has developed a pipeline designed to address many of these problems[39].
Starting from a single sequence, we iteratively expand an alignment using
Infernal covariance models. During each iteration, we use a variety of
automatic alignment and secondary structure prediction tools together with
manual curation and editing in an effort to avoid many of the issues raised
above. While the Rfam pipeline is designed to run on a high-end computational
cluster, we have adapted the process here to make it accessible to anyone with
a commodity PC and an internet connection.
## 2 Materials
### 2.1 Single Sequence Search
We rely on NCBI BLAST[1] to quickly identify close homologs of RNA sequences
in this protocol. NCBI and EMBL-EBI both maintain servers[40, 41] with
slightly different interfaces, though there are no substantive differences in
the implementations. We use the NCBI server here. EBI also maintains servers
for a number of BLAST and FASTA derivatives, which may be helpful. Both sites
also allow users to BLAST against databases of expressed sequences including
GEO at NCBI, and high throughput cDNA and transcriptome shotgun assembly
databases at EMBL-EBI. Such searches can be helpful for gathering comparative
expression data for your ncRNA.
A nucleotide version of the HMMER3 package[2] for sequence search is currently
in development which promises both increased sensitivity and specificity over
BLAST at little additional computational cost. We hope that a web server
similar to the one currently available for protein sequences[42] will be
forthcoming. If it is possible that homologous sequences are spliced (e.g.
introns in the U3 snoRNA[43]), then a splice-site aware search method may be
useful, such as BLAT[44] or GenomeWise[45], but there are not publicly
available webservers for them that we are aware of.
Resource | Reference | URL
---|---|---
NCBI-BLAST | [40] | http://blast.ncbi.nlm.nih.gov/Blast.cgi
EMBL-EBI NCBI-BLAST | [41] | http://www.ebi.ac.uk/Tools/sss/ncbiblast/
EMBL-EBI Sequence Search | [41] | http://www.ebi.ac.uk/Tools/sss/
HMMER3333Currently amino acid only | [42] | http://hmmer.janelia.org/search
### 2.2 Alignment and Secondary Structure Prediction Tools
We find it best to run a variety of alignment and secondary structure
prediction tools simultaneously. Each has its own peculiarities, and our hope
is that by looking for shared homology and secondary structure predictions we
can mitigate some of the problems discussed in the introduction. In this
protocol, we use the WAR webserver[46] which allows the user to run 14
different methods simultaneously. These include Sankoff-type methods:
FoldalignM[26], LocARNA[25], MXSCARNA[47], Murlet[48], and StrAL[49] \+
PETcofold[50]; Align-then-fold methods, which use a traditional alignment tool
(ClustalW[51, 52] or MAFFT[53, 54]) followed by structure prediction
(RNAalifold[29, 55] or Pfold[28]); Fold-then-align methods, which predict
structures in all the input sequences and attempt to align these structures
(RNAcast[56] \+ RNAforester[57]); Sampling methods which attempt to
iteratively refine alignment and structure: MASTR[58] and RNASampler[59]; and
other methods which do not fit in to the above traditional categories:
CMfinder[60] and LaRA[61]. Finally, WAR also computes a consensus alignment
using the alignments produced by all user-selected methods as input to the
T-Coffee consistency-based aligner[32].
However, WAR is by no means exhaustive, and the applications may not be the
most recent versions available. A number of groups maintain their own servers
for RNA sequence analysis. Notable servers include the Vienna RNA
WebServers[62], the Freiburg RNA Tools[63], the CBRC Functional RNA
Project[64], and the Center for Non-Coding RNA in Technology and Health (RTH)
Resources page. In addition, EMBL-EBI maintains a number of webservers for
popular multiple sequence alignment alignment tools. Ultimately, as you become
more comfortable with RNA sequence analysis you may want to begin installing
and running new tools on a local *NIX machine; however, this is beyond the
scope of the current chapter.
Resource | Reference | URL
---|---|---
WAR | [46] | http://genome.ku.dk/resources/war/
Vienna RNA | [62] | http://rna.tbi.univie.ac.at/
Freiburg RNA Tools | [63] | http://rna.informatik.uni-freiburg.de
CBRC Functional RNA Project | [64] | http://software.ncRNA.org
RTH Resources | NA | http://rth.dk/pages/resources.php
EMBL-EBI Alignment | NA | http://www.ebi.ac.uk/Tools/msa/
### 2.3 Genome Browsers
Genome browsers are essential for checking the context of putative homologs.
The ENA[41] provides a no-frills sequence browser perfect for quickly checking
annotations. For deeper annotations, the UCSC genome broswer[65] and
Ensembl[66] both contain a wide range of information for the organisms they
cover. For bacterial and archaeal genomes, the Lowe lab maintains a modified
version of the UCSC genome browser[67] which provides a number of tracks of
particular interest to those working with ncRNA. The CBRC Functional RNA
Project maintains a UCSC genome browser mirror[64] for a number of eukaryotic
organisms with a larger number of ncRNA-related tracks.
Resource | Reference | URL
---|---|---
European Nucleotide Archive | [41] | http://www.ebi.ac.uk/ena/
UCSC Genome Browser | [65] | http://genome.ucsc.edu/
Ensembl | [66] | http://www.ensembl.org
UCSC Microbial Genome Browser | [67] | http://microbes.ucsc.edu/
CBRC UCSC Genome Browser for Functional RNA | [64] | http://www.ncrna.org/glocal/cgi-bin/hgGateway
### 2.4 Alignment Editors
It is possible to edit alignments in any text editor; however we highly
recommend using a secondary structure-aware editor such as Emacs with the
RALEE major mode[68]. RALEE allows you to color bases according to base
identity, secondary structure, or base conservation. It also allows the easy
manipulation of sequences which are involved in structural interactions but
are not close in sequence space through the use of split screens. A number of
other specialized RNA editors are available: BoulderALE[69] and S2S[70] both
allow the end user to visualize and manipulate tertiary structure in addition
to secondary structure, and may be particularly useful if crystallographic
information is available for your RNA. Other alternatives for editing RNA
secondary structure are SARSE[71] and MultiSeq[72]. Recent versions of
JalView[73] have begun to support RNA secondary structure as well, though this
functionality isn’t completely mature at the time of writing (late 2011.)
Resource | Reference | URL
---|---|---
RALEE | [68] | http://personalpages.manchester.ac.uk/staff/sam.griffiths-jones/software/ralee/
BoulderALE | [69] | http://www.microbio.me/boulderale
S2S | [70] | http://bioinformatics.org/S2S/
SARSE | [71] | http://sarse.ku.dk/
MultiSeq | [72] | http://www.ks.uiuc.edu/Research/vmd/plugins/multiseq/
JalView | [73] | http://www.jalview.org
### 2.5 Infernal
The centerpiece of our protocol is the Infernal package for constructing
covariance models(CMs) from RNA multiple alignments[3]. We will use this to
construct models of our RNA family. CMs model the conservation of positions in
an alignment similar to a hidden Markov model(HMM), while also capturing
_covariation_ in structured regions[74, 75, 21]. Covariation is the process
whereby a mutation of a single base in a hairpin structure will lead to
selection in subsequent generations for compensatory mutations of its
structural partner in order to preserve canonical base-pairing, ie: Watson-
Crick plus G-U pairs, and a functional structure. This combination of
structural-evolutionary information has been shown to provide the most
sensitive and specific homology search for RNA of any tools currently
available[9, 76]. Unfortunately, this sensitivity and specificity come at a
high computational cost, and Infernal searches can be time-consuming with
genome-scale searches often taking hours on desktop computers. The development
of heuristics to reduce this computational cost is an area of active research
for the Infernal team, and has already been mitigated to some extent by the
use of HMM filters and query-dependent banding of alignment matrices[77]. We
refer the reader to Eric Nawrocki’s excellent primer on annotating functional
RNAs in genomic sequence for a friendly introduction to the mechanics of the
Infernal package[78].
Resource | Reference | URL
---|---|---
Infernal | [3, 78] | http://infernal.janelia.org/
## 3 Methods
We assume for the sake of this protocol that you are starting with a single
sequence of interest. If you already have a set of putative homologs, you may
wish to further diversify your collection of sequences using the methods
described in section 3.1, or you may skip directly to section 3.2, or 3.4 if a
secondary structure is known. No matter how many sequences you are starting
with, it is always a good idea to run the tools available on the Rfam website
(rfam.sanger.ac.uk) on them. This will verify that there isn’t already a CM
available that covers your sequences. There are a number of other specialist
databases that may also be worth searching if you have reason to believe your
RNA sequence is a member of a well-defined class of RNAs, i.e. microRNAs,
snoRNAs, rRNAs, tRNAs, etc. We have previously reviewed these databases in
another book chapter[79]. A generic RNA sequence database aiming to capture
all known RNA sequences, RNAcentral[80] is currently in development and will
provide a resource for easily identifying similar sequences with some evidence
of transcription.
### 3.1 Gathering an initial set of homologous sequence
Now that you’ve confirmed that your sequence is novel, we will use NCBI-BLAST
to identify additional homologous sequences. Once you’ve navigated to the
nucleotide BLAST server there are a number of important options to set.
#### 3.1.1 Setting NCBI-BLAST Parameters
First, it is important to choose a search set appropriate to your sequence. At
this initial phase, we want to limit our exposure to sequences which are very
distant from ours to avoid the number of obviously spurious alignments we will
need to examine, increasing the power of our search. So, if your initial
sequence is of human origin, you may want to limit your search to Mammalia,
Tetrapoda, or Vertebrata depending on sequence conservation. Similarly, if you
are working with an Escherichia coli sequence, you may want to limit your
initial searches to Enterobacteriaceae or the Gammaproteobacteria. NCBI-BLAST
searches are relatively fast, so try several search sets to get a feel for how
conserved your sequence is.
The second set of options to set is the “Program Selection” and the “Algorithm
Parameters”. We recommend blastn as it allows for smaller word sizes. The word
size describes the minimum length of an initial perfect match needed to
trigger an alignment between our query sequence and a target. Smaller word
sizes provide greater sensitivity, and seem to perform better for non-coding
RNAs. We recommend a word size of 7, the smallest the NCBI-BLAST server
allows.
Finally, you should set “Max Target Sequences” parameter to at least 1000.
NCBI-BLAST returns hits in a ranked list from best match to worst by E-value
(or the number of matches with the same quality expected to be found in a
search over a database of this size), and will only display as many as “Max
Target Sequences” is set to. We are primarily interested in matches on the
edge of what NCBI-BLAST is capable of detecting reliably, and these will
naturally fall towards the end of this list.
#### 3.1.2 Selecting Sequences
Our goal at this stage is to pick a representative set of homologous sequences
to “seed” our alignment with. As discussed in the introduction, single
sequence alignment for nucleotides is generally only reliable to approximately
60 percent pair-wise sequence identity. At the same time, picking a large
number of sequences with high percent identity can lead to _overfitting_ of
the secondary structure; that is, if our sequences are too similar we can end
up predicting alignments and secondary structures which capture accidental
features of a narrow clade, rather than the biologically relevant structure
and sequence variation.
There are 3 major criteria we pick additional sequences based on, in rough
order of importance: percent sequence identity, taxonomy, and sequence
coverage. Handily, the NCBI-BLAST output displays measures of all of these.
Our first selection criterion, percent identity, should fall between 65% and
95%; much lower and the sequence will be difficult to align, higher and it
will be too similar to have any meaningful variation.
The second selection criterion, taxonomy, will depend somewhat on the
organisms your sequence is associated with, but we generally want to limit the
inclusion to a single (orthologous) instance per species. The exception to
this rule is for diverged paralogous sequences within the species; if paralogs
exist, you will need to decide how broadly you wish to define your family.
Additionally, it may be useful to further limit the maximum percent identity
to, say, 90% within a genus to further limit the number of highly similar
sequences in your initial alignment.
Finally, assuming that you are sure of your sequence boundaries, we want to
select sequences that cover the entire starting sequence. If you see many
matches covering only a short section of your sequence, this may be due to the
matching of a short convergent motif. This most commonly happens with the
relatively long, highly-constrained bacterial rho-independent terminators, but
may occur with other motifs. Alternatively, if you do not have well-defined
sequence boundaries, you will need to determine these from the conservation
you see in your BLAST hits – look for taxonomically diverse hits covering the
same segment of your query sequence. In some cases, such as the long non-
coding RNAs, conserved domains may be much shorter than the complete
transcribed sequence, but stay aware of the potential motif issue. A taxonomic
distribution of sequences that makes biological sense given your knowledge of
the molecule’s function and that can be explained by direct inheritance of the
sequence will be your best guide.
#### 3.1.3 Examining Your Initial Homolog Set
Once you have assembled a set of sequences fitting the criteria described
above, it is worth taking a closer look at them. Remember that these sequences
will form the core of your alignment and CM, and errors at this stage can
dramatically bias your results. A good first test is to examine the taxonomy
of your sequences, and make sure it makes sense. Can you identify a clear
pattern of inheritance that might explain the taxonomic distribution you see
at this stage? Another good check is to examine your sequences in the ENA
browser, or a domain-specific browser if one exists for your organisms. For
many independently transcribed RNAs, genomic context is more conserved than
sequence, and ncRNA genes will often fall in homologous intergenic or intronic
regions even at large evolutionary distances. If you are particularly
ambitious, and the tools are available for your organisms of interest, you may
wish to try to identify promoter sequence upstream of your candidate or
terminator sequence downstream. If your sequence is a putative cis-regulatory
element, such as a riboswitch, thermosensor, or attenuator, you may want to
check that it occurs upstream of genes with similar functions or in similar
pathways. Finally, it is always worth searching your putative homologs through
the Rfam website even if your initial sequence had no matches – Rfam’s models
are not perfect, and may miss distant homologs of known families.
### 3.2 Aligning and predicting secondary structure
We will use the WAR servers to construct an initial alignment. Because of the
criteria we’ve set for sequence similarity in our gathering step, all of the
sequences in our initial homolog set should have at least 60% pairwise
sequence identity with at least one other sequence in the set. Under these
conditions sequence-only alignment methods using primary sequence information
only can preform adequately, as discussed previously. These methods combined
with alignment folding tools which identify for conserved structural signals
and covariation can produce reasonable predicted secondary structures[10].
However it is still often useful to observe the behavior of as many alignment
tools as possible. Using WAR, for a fairly fast alignment we recommend running
CMfinder[60], StrAL+PETfold[49, 50], ClustalW[51, 52] and MAFFT[53, 54] with
RNAalifold[29, 55] and Pfold[28]. WAR will also produce a consensus alignment
using T-Coffee[32], which will attempt to find an alignment consistent with
all of the individual alignments produced by other methods.
Figure 1: T-coffee consensus alignment for close MicA homologs produced by
WAR, colored for alignment consistency between methods. Due to the high
percent identity in these sequence, the alignments are highly consistent,
though even here the areas of lower consistency are informative for manual
refinement - see section 4.
Once WAR returns your alignment results, there are a number of things you
should take a note of that will assist you in picking an alignment and further
in manual refinement. First, the consensus alignment page will display a
graphical representation of the consistency of the alignments which will allow
you to quickly tell which areas of the alignment may require attention during
manual refinement, or areas that may harbor structure not captured by the
majority consensus. The consensus can be recomputed based on differing subsets
alignment methods, if you believe one method (or set of methods) may be unduly
influencing the consensus. Once you’ve carefully looked over the consensus
alignment, examine each alignment produced by WAR in turn: What structures are
shared? Where do the alignments differ from each other? Can you identify any
sequence or structural motifs which may help to guide your alignment? At this
level of sequence identity, you should hope to see fairly consistent
alignments in functional regions of the sequence, interspersed with more
difficult to align regions, presumably under less severe selective pressure.
Often the consensus alignment is a good choice to move forward with. However,
there are cases where certain classes of tools will obviously mis-align
regions of the sequence and bias the consensus. Keep in mind what you’ve seen
in the alternative alignments as well; this information may be useful in
manual refinement. You will want to save the stockholm file for the alignment
you’ve chosen to your local computer at this point.
Later in the family-building process when you have identified more distant
homologs, the average pair-wise identity of the sequences in your data set may
have dropped below 60%. At this point, you may want to begin including some of
the Sankoff-type alignment methods available in WAR. Using these methods can
dramatically increase the runtime for your sequence alignment jobs, though,
particularly for sequences over a couple of hundred of bases long. We will
discuss alternatives to re-aligning sequences during the iterative expansion
of the alignment in section 3.5.
### 3.3 Manually refining alignments
Our goal in manual refinement is to attempt to correct errors made by
automatic alignment tools. We generally use RALEE[68], an RNA editing mode for
Emacs, for editing alignments. However, any editor you are comfortable with in
which you can easily visualize sequence and structural conservation will work;
a number of alternative editors are listed in the Materials section.
A good place to start editing is around the edges of predicted hairpin
structures. Are there base-pairs which appear to be misaligned? Can you add
base-pairs to the structure? Are there predicted base-pairs which don’t appear
to be well conserved that should be trimmed? Can individual bases be moved in
the alignment to create more convincing support for the predicted structure?
Once you are satisfied with your manual refinement of predicted secondary
structure elements, next you should turn your attention to areas identified as
uncertain in the WAR/T-Coffee consensus alignment. Were there alternative
structures predicted in these regions? Do you see support for these structures
in the sequences? If these regions are unstructured, can you identify any
conserved sequence motifs in the region? If you will be regularly working with
a particular class of ncRNA, it can be useful to familiarize yourself with
predicted binding motifs of associated RNA-binding proteins as these are
likely to be conserved but can have many variable positions.
At this stage, it is also possible to include information from experimental
data. Crystal structure information from a single sequence in the SEED
alignment can be used to validate and improve a predicted secondary structure.
Tertiary structure-aware editors such as BoulderAle[69] can help in applying
this information to the alignment. Other experimental evidence, such as
chemical footprinting can also provide valuable information. Knowing whether
even a single base is involved in a pairing interaction can drastically reduce
the space of possible structures the sequence can fold in to, simplifying the
problem of predicting secondary structure. Both the RNAfold and RNAalifold web
servers available through the Vienna RNA website[62] are capable of taking
advantage of this information in the form of folding constraints. We hope that
these sorts of datasets will become widely available in consistent formats in
the near future[81].
### 3.4 Building a covariance model
For those comfortable with the *NIX command line, building an Infernal CM is
fairly straight-forward. We refer the reader to the User’s Guide available
from the Infernal website (http://infernal.janelia.org) for installation
instructions and a detailed tutorial. The basic syntax to build and calibrate
a family is:
> cmbuild my.cm my.sto
> cmcalibrate my.cm
The first command constructs the CM (my.cm) from the alignment you’ve
carefully curated (my.sto). The second command calibrates the various filters
Infernal uses to accelerate its search using simulated sequences generated
from the CM. Note that calibration can take a long time – hours for longer
models. You can get a quick estimate of the time calibration will take using
the command:
> cmcalibrate --forecast 1 my.cm
Congratulations! You should now have a working CM for your RNA family. This is
a fully capable model, and can be used as is for homology search and genome
annotation. However, as it stands, your CM will only capture the sequence
diversity which was able to be detected by our initial BLAST search. In order
to fully take advantage of the power of CMs, it is necessary to expand the
diversity of the sequence it is trained on through iterative expansion of our
initial set of sequence homologs.
### 3.5 Strategies for expanding model coverage
#### 3.5.1 Plan A: Iterative search of sequence databases
The method Rfam uses to identify more divergent homologs to seed sequences is
to pre-filter CM-based searches with sequence-based homology search tools.
This allows us to cover a large sequence space with a (comparatively) modest
investment of computational time. Any of the single sequence search tools
mentioned in section 2.1 would make an effective pre-filter.
The easiest way to preform filtering yourself is to use the NCBI BLAST
webserver to search each sequence in your seed alignment following the methods
outlined for collecting your initial set of homologs in section 3.1. You may
wish to relax the criteria slightly, then use the CM to preform a more
sensitive search on this set of filtered sequences. This will enable you to
detect more distantly related sequences, though you should always examine
sequence context and the phylogenetic relationship between sequences as a
sanity check before including them in your seed. These methods can be
automated with basic scripting and bioinformatics modules such as BioPerl[82]
or Biopython[83], though this is beyond the scope of this chapter.
Once you have identified a new set of homologs, you can align them to your
previous CM using Inferal’s cmalign:
> cmalign my.cm newsequences.fasta > newsequences.sto
This alignment can then be merged with your original alignment:
> cmalign --merge my.cm my.sto newsequences.sto > combined_alignment.sto
This alignment can then be used to build a new CM, which will capture the
additional sequence variation you have discovered in your BLAST searches.
The disadvantage of this method is that each search only uses the information
available in a single sequence, meaning that valuable information about
variation is lost and as a result the power of the search suffers. Fast
profile-based methods such as HMMER3[2] will hopefully remedy this problem in
the near future, but these methods are not mature for DNA and RNA sequence at
the present. Older versions of HMMER can be used to search DNA sequence with
increased power, but they require more computational resources than BLAST
(though far less than Infernal) and need to be used at the command-line.
#### 3.5.2 Plan B: Directed search of chosen sequences
Another approach is to run the unfiltered CM over selected genomes or genomic
regions. While the greater sensitivity and specificity of this method can help
identify more distant homologs than is possible with BLAST, it has the
disadvantage that it requires a much larger investment of computational
resources to provide an equivalent phylogenetic coverage. This method can be
particularly powerful in bacterial and archaeal genomes, where small genome
size allows us to search a phylogenetically-representative sample of genomes
in less than a day on a desktop computer. In the case of larger eukaryotic
genomes, it may be necessary to search a few genomes to determine if homologs
of your RNA are likely to exist in certain lineages, then extract homologous
intergenic regions to continue searching. Our rationale here is much the same
as in limiting the database for our initial BLAST search: by only looking in
genomes where we have some prior belief that they may contain homologous
sequence we reduce the noise in our low-scoring hits, meaning that we have to
manually examine less hits to establish a score threshold for likely homologs.
Once you have examined candidates following the principles outlined earlier,
it is easy to incorporate your new sequences using the easel package included
with Infernal. First, search the genome generating a tabfile:
> cmsearch --tabfile searchfile.tab my.cm genome.fasta
Then use easel to index the genome and extract the hits:
> esl-sfetch --index genome.fasta
> esl-sfetch --tabfile genome.fasta searchfile.tab > hits.fasta
These sequences can then be aligned and merged as with BLAST hits.
Alternatively, if you discover a divergent lineage, it may be easiest to
construct a separate alignment for these sequences, then use shared structural
and sequence motifs to manually combine the two alignments. Sankoff-type
alignment method may also be useful for aligning divergent clades.
#### 3.5.3 Plan C: When A and B fail…
In some cases, it will be very difficult to identify homologs of a candidate
RNA across its full phylogenetic range. This can be because of high sequence
variability, as in the Vault RNAs[84]. Alternatively, some longer RNAs, such
as the RNA component of the telomerase ribonuceloprotein, consist of well-
conserved segments interspersed with long variable regions which can’t be
easily discovered by standard search with naive covariance models.
A number of computational techniques exist for approaching these difficult
cases, reviewed by Mosig and colleagues[85]. These methods include
fragrep2[86], which allows the user to search fragmented conserved regions,
fragrep3, which allows the user to incorporate custom structural motifs with
fragmented search, and GotohScan[87], which implements a semi-global alignment
algorithm that will align a query sequence to a (potentially) extended genomic
region.
## 4 An example: MicA
We will now illustrate some of the concepts we’ve discussed using the example
of MicA, an Hfq-dependent bacterial trans-acting antisense small RNA (sRNA).
Many bacterial sRNAs are similar in function to eukaryotic microRNAs, pairing
to target mRNA transcripts through a short antisense-binding region, generally
targeting the transcript for degradation[88]. MicA is known to target a wide-
range of outer membrane protein mRNAs using a $5^{\prime}$ binding-region in
both E. coli[89] and S. enterica[90] in response to membrane stress. The
current covariance model for MicA (accession RF00078) in Rfam (release 10.1)
is largely restricted to E. coli, S. enterica, and Y. pestis. Here, as an
example, we will attempt to improve on this model using the methods we’ve
described in this chapter. In the process, we discover previously unreported
homologs in the nematode symbionts of the Gammaproteobacterial genus
Xenorhabdus.
For our starting point, we are using the MicA sequence from Gisela Storz’s
spreadsheet of known E. coli sRNAs[91]:
MicA: GAAAGACGCGCATTTGTTATCATCATCCCTGAATTCAGAGATGAAATTTTGGCCACTCACGAGTGGCCTTTTT
It is a useful exercise to compare the single sequence predicted secondary
structures for this sequence and the E. coli sequence from the current Rfam
SEED alignment(see Figure 2). This illustrates that even for nearly identical
sequences, single sequence structure prediction methods can give divergent
results. Other important features to notice are that the $3^{\prime}$ hairpin
shared by the predicted structures appears to be a rho-independent terminator,
and this could be confirmed with a motif hunting tool[37] and used during
manual curation.
Figure 2: Alternative structures predicted by the RNAfold webserver for single
MicA sequences. A) E. coli APEC sequence from the current Rfam seed alignment.
B) E. coli sequence from Storz’s sRNA spreadsheet. C) A likely homolog from
Erwinia pyrifoliae. Notice the differences in the secondary structure of the
first two examples, despite only differing by two extra nucleotides at the
gene boundaries. The Erwinia prediction only shares a single stem with the E.
coli predictions, despite relatively high sequence similarity.
We now begin by following the guidance in section 3.1 to collect an initial
set of putative homologs. To obtain an initial set of sequences, we BLAST the
E. coli MicA sequence over the nucleotide collection database limited to the
enterobacteria (taxonomy id: 543) using the blastn algorithm. The BLAST search
returns a number of highly similar E. coli sequences, as well as related
sequences from the closely related S. enterica. As we move down to less
similar sequences (as judged by their E-values) we identify progressively more
evolutionarily distant organisms.
Figure 3: Truncated results from a NCBI-BLAST search of the E. coli MicA
sequence, showing the low E-value results. We are primarily interested in
column 2 for genus and species information, column 5 for sequence coverage
information, and column 7 for percent identity informations.
From these sequences, we want to select a group of sequences with a reasonably
diverse taxonomic range and as much sequence diversity as possible, while
being reasonably confident that they are true homologs. In this case we will
choose based on maximzing genus diversity, a percent id between 75% and 90%,
and 100% sequence coverage as we’re fairly confident in the MicA gene
boundaries. For our initial alignment, we have chosen sequences from
Salmonella typhimurium (EMBL-Bank accession: FQ312003), Klebsiella pneumoniae
(CP002910), Enterobacter cloaca (CP002272), Yersinia pestis (AM286415),
Pantoea sp. At-9b (CP002433), and Erwinia pyrifoliae (FP236842). From a quick
examination with the ENA browser, it appears that all of these sequences fall
in a intergenic region between a luxS protein homolog and a gshA protein
homolog, further increasing our confidence that these are true homologs. From
our results, we can also see a few promising hits that don’t quite meet our
criteria, such as Dickeya, Xenorhabdus, Photorhabdus and Wigglesworthia. We
will keep these in mind later to expand our coverage.
Now that we have a starting set of sequences, we can assemble them in to a
fasta file:
>U00096.2
GAAAGACGCGCATTTGTTATCATCATCCCTGAATTCAGAGATGAAATTTTGGCCACTCACGAGTGGCCTTTTT
>FQ312003
GAAAGACGCGCATTTGTTATCATCATCCCTGTTTTCAGCGATGAAATTTTGGCCACTCCGTGAGTGGCCTTTTT
>CP002272
GAAAGACGCGCATTTGTTATCATCATCCCTGACTTCAGAGATGAAATGTTTGGCCACAGTGATGTGGCCTTTTT
>CP002910
GAAAGACGCGCATTTATTATCATCATCATCCCTGAATCAGAGATGAAAGTTTGGCCACAGTGATGTGGCCTTTTT
>AM286415
GAAAGACGCGCATTTGTTATCATCATCCCTGTTATCAGAGATGTTAATTTGGCCACAGCAATGTGGCCTTTT
>CP002433
GAAAGACGCGCATTTGTTATCATCATCCCTGACAACAGAGATGTTAATTCGGCCACAGTGATGTGGCCTTTT
>FP236842
GAAAGACGCGTATTTGTTATCATCATCTCATCCCTGACAACAGAGATGTTAATTTAGGCCACAGTGACGTGGCCTTTTT
We can use this to run WAR, and look at the secondary structures predicted by
each method. One secondary structure appears to dominates the predictions.
However, it s important to check the other predicted secondary structures - do
any of them pick up convincing substructures that may have been missed by
other methods?
Figure 4: Alternative structures predicted by the WAR server based on
different alignment methods. A) T-Coffee consensus alignment, B) CMfinder, and
C) StrAL+PETfold. While these structures and alignments share some features,
the differences in predicted structure illustrate the hazard of relying on a
single method, even for a short, well-conserved sequence.
In this case, the consensus alignment (see Figure 1) seems to agree well with
the majority of alignment and structure prediction methods, and is consistent
with previous experimental probing[92]. We can improve the alignment manually.
The first basepair in the first stem in CP002433 can be rescued by shifting a
few nucleotides, and by pulling apart the alignment between the first and
second stem we reveal what appears to be a well-conserved AAUUU sequence motif
that was previously hidden (Figure 5). The RNA chaperone Hfq is known to bind
to A/U rich sequences, so this motif may have some functional significance.
The strong conservation of the $5^{\prime}$ antisense-binding domain provides
more confidence that these are in fact homologous RNAs.
Figure 5: MicA alignment before(top) and after(bottom) manual alignment in
RALEE, colored for secondary structure and sequence conservation.
Now we will follow Plan B to add sequences to our alignment using the genomes
for the low-scoring BLAST hits we had previously made a note of while
collecting our initial set of sequences, though you could also choose these
sequences based on your knowledge of your organisms phylogeny or the suspected
function of your RNA. The genomes we’ve chosen here are Dickeya zeae
(CP001655), Sodalis Glossinidius (AP008232), Xenorhabdus nematophila
(FN667742) and Wiggglesworthia glosinidia (BA000021). Searching these genomes
allows us to identify strong hits in D. zeae and S. glossinidius with E-values
of $10^{-12}$ and $10^{-10}$ which we can merge in to our alignment using the
methods in section 3.5.1. You should then manually refine the resulting merged
alignment with an eye towards maintaining conserved sequence motifs and
structure. Already at this distance, there have been some apparent small decay
in secondary structure, as well as an expansion of the sequence contained in
the loop region of the second stem in D. zeae (Figure 6).
Figure 6: MicA alignment including merged sequences from D. zeae and S.
glossinidius.
We observe a number of hits in X. nematophila with E-values in the range of
$10^{-2}$. By checking each of these individually in the ENA browser, we can
identify one that falls in the same genomic context as our previous MicA
homologs (Figure 7). By using this sequence as the starting point for a BLAST
search, we are able to identify a number of other divergent Xenorhabdus
homologs. As these are quite diverged from the E. coli sequence, we first
construct an alignment for them using WAR (Figure 8), then attempt to merge
our alignments manually (Figure 9) using shared structural features as our
guide. Interestingly, the target-binding region of MicA appears to have
suffered a poly-A insertion down this lineage, suggesting that there may be
changes in the regulon it targets. Using this model to search all of the
bacterial genomes in EMBL-Bank (approximately 6GB of sequence, taking 30 hours
on a 2.26 GHz Intel Core 2 Duo processor) shows that our CM now has high-
scoring hits exclusively in Enterobacteriales, while covering a broader range
than our BLAST searches. This search also reveals a number of possible sources
of additional diversity: Photorhabdus asymbiotica and Edwardsiella ictaluri
both have strong hits below the average score for other Enterobacterial
genomes – incorporating them may further increase the sensitivity of our
model, and is left as an exercise to the reader.
Figure 7: Context of a marginal X. nematophila hit viewed in the ENA genome
browser. Figure 8: An alignment of Xenorhabdus homologs. Figure 9: Divergent
Xenorhabdus homologs manually merged with the MicA alignment. Notice the
variation in both secondary structure and sequence conservation added by these
sequences.
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|
arxiv-papers
| 2012-06-18T22:02:35 |
2024-09-04T02:49:31.939852
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Lars Barquist, Sarah W. Burge and Paul P. Gardner",
"submitter": "Paul Gardner",
"url": "https://arxiv.org/abs/1206.4087"
}
|
1206.4110
|
# ConeRANK: Ranking as Learning Generalized Inequalities
Truyen T. Tran$\dagger$ and Duc-Son Pham$\ddagger$
$\dagger$ Center for Pattern Recognition and Data Analytics (PRaDA),
Deakin University, Geelong, VIC, Australia
$\ddagger$ Department of Computing, Curtin University, Western Australia
Email: truyen@vietlabs.com dspham@ieee.org
###### Abstract
We propose a new data mining approach in ranking documents based on the
concept of cone-based generalized inequalities between vectors. A partial
ordering between two vectors is made with respect to a proper cone and thus
learning the preferences is formulated as learning proper cones. A pairwise
learning-to-rank algorithm (ConeRank) is proposed to learn a non-negative
subspace, formulated as a polyhedral cone, over document-pair differences. The
algorithm is regularized by controlling the ‘volume’ of the cone. The
experimental studies on the latest and largest ranking dataset LETOR 4.0 shows
that ConeRank is competitive against other recent ranking approaches.
## 1 Introduction
_Learning to rank_ in information retrieval (IR) is an emerging subject [7,
11, 9, 4, 5] with great promise to improve the retrieval results by applying
machine learning techniques to learn the document relevance with respect to a
query. Typically, the user submits a query and the system returns a list of
related documents. We would like to learn a ranking function that outputs the
position of each returned document in the decreasing order of relevance.
Generally, the problem can be studied in the supervised learning setting, in
that for each query-document pair, there is an extracted feature vector and a
position label in the ranking. The feature can be either _query-specific_
(e.g. the number of matched keywords in the document title) or _query-
independent_ (e.g. the PageRank score of the document, number of in-links and
out-links, document length, or the URL domain). In training data, we have a
groundtruth ranking per query, which can be in the form of a relevance score
assigned to each document, or an ordered list in decreasing level of
relevance.
The learning-to-rank problem has been approached from different angles, either
treating the ranking problem as ordinal regression [10, 6], in which an
ordinal label is assigned to a document, as pairwise preference classification
[11, 9, 4] or as a listwise permutation problem [14, 5].
We focus on the pairwise approach, in that ordered pairs of document per query
will be treated as training instances, and in testing, predicted pairwise
orders within a query will be combined to make a final ranking. The advantage
of this approach is that many existing powerful binary classifiers that can be
adapted with minimal changes - SVM [11], boosting [9], or logistic regression
[4] are some choices.
We introduce an entirely new perspective based on the concept of cone-based
_generalized inequality_. More specifically, the inequality between two
multidimensional vectors is defined with respect to a cone. Recall that a cone
is a geometrical object in that if two vectors belong to the cone, then any
non-negative linear combination of the two vectors also belongs to the cone.
Translated into the framework of our problem, this means that given a cone
$\mathcal{K}$, when document $l$ is ranked higher than document $m$, the
feature vector $\mathbf{x}_{l}$ is ‘greater’ than the feature vector
$\mathbf{x}_{m}$ with respect to $\mathcal{K}$ if
$\mathbf{x}_{l}-\mathbf{x}_{m}\in\mathcal{K}$. Thus, given a cone, we can find
the correct order of preference for any given document pair. However, since
the cone $\mathcal{K}$ is not known in advance, it needs to be estimated from
the data. Thus, in our paper, we consider polyhedral cones constructed from
basis vectors and propose a method for learning the cones via the estimation
of this set of basis vectors.
This paper makes the following contributions:
* •
A novel formulation of the learning to rank problem, termed as ConeRank, from
the angle of cone learning and generalized inequalities;
* •
A study on the generalization bounds of the proposed method;
* •
Efficient online cone learning algorithms, scalable with large datasets; and,
* •
An evaluation of the algorithms on the latest LETOR 4.0 benchmark dataset
111Available at: http://research.microsoft.com/en-
us/um/beijing/projects/letor/letor4dataset.aspx.
Figure 1: Illustration of ConeRank. Here the pairwise differences are
distributed in 3-dimensional space, most of which however lie only on a
surface and can be captured most effectively by a ‘minimum’ cone plotted in
green. Red stars denotes noisy samples.
## 2 Previous Work
Learning-to-rank is an active topic in machine learning, although ranking and
permutations have been studied widely in statistics. One of the earliest paper
in machine learning is perhaps [7]. The seminal paper [11] stimulates much
subsequent research. Machine learning methods extended to ranking can be
divided into:
_Pointwise approaches_ , that include methods such as ordinal regression [10,
6]. Each query-document pair is assigned a ordinal label, e.g. from the set
$\\{0,1,2,...,L\\}$. This simplifies the problem as we do not need to worry
about the exponential number of permutations. The complexity is therfore
linear in the number of query-document pairs. The drawback is that the
ordering relation between documents is not explicitly modelled.
Pairwise approaches, that span preference to binary classification [11, 9, 4]
methods, where the goal is to learn a classifier that can separate two
documents (per query). This casts the ranking problem into a standard
classification framework, wherein many algorithms are readily available. The
complexity is quadratic in number of documents per query and linear in number
of queries.
_Listwise approaches_ , modelling the distribution of permutations [5]. The
ultimate goal is to model a full distribution of all permutations, and the
prediction phase outputs the most probable permutation. In the statistics
community, this problem has been long addressed [14], from a different angle.
The main difficulty is that the number of permutations is exponential and thus
approximate inference is often used.
However, in IR, often the evaluation criteria is different from those employed
in learning. So there is a trend to optimize the (approximate or bound) IR
metrics [8].
## 3 Proposed Method
### 3.1 Problem Settings
We consider a training set of $P$ queries $q_{1},q_{2},\ldots,q_{P}$ randomly
sampled from a query space $\mathcal{Q}$ according to some distribution
${P}_{\mathcal{Q}}$. Associated with each query $q$ is a set of documents
represented as pre-processed feature vectors
$\\{\mathbf{x}^{q}_{1},\mathbf{x}^{q}_{2}\ldots\\},\mathbf{x}^{q}_{l}\in\mathbb{R}^{N}$
with relevance scores $r^{q}_{1},r^{q}_{2},\ldots$ from which ranking over
documents can be based. We note that the values of the feature vectors may be
query-specific and thus the same document can have different feature vectors
according to different queries. Document $\mathbf{x}^{q}_{l}$ is said to be
more preferred than document $\mathbf{x}^{q}_{m}$ for a given query $q$ if
$r^{q}_{l}>r^{q}_{m}$ and vice versa. In the pairwise approach, pursued in
this paper, equivalently we learn a ranking function $f$ that takes input as a
pair of different documents $\mathbf{x}^{q}_{l},\mathbf{x}^{q}_{m}$ for a
given query $q$ and returns a value $y\in\\{+1,-1\\}$ where $+1$ corresponds
to the case where $\mathbf{x}^{q}_{l}$ is ranked above $x^{q}_{m}$ and vice
versa. For notational simplicity, we may drop the superscript q where there is
no confusion.
### 3.2 Ranking as Learning Generalized Inequalities
In this work, we consider the ranking problem from the viewpoint of
generalized inequalities. In convex optimization theory [3, p.34], a
generalized inequality $\succ_{\mathcal{K}}$ denotes a partial ordering
induced by a proper cone $\mathcal{K}$, which is convex, closed, solid, and
pointed:
$\mathbf{x}_{l}\succ_{\mathcal{K}}\mathbf{x}_{m}\Longleftrightarrow\mathbf{x}_{l}-\mathbf{x}_{m}\in\mathcal{K}.$
Generalized inequalities satisfy many properties such as preservation under
addition, transitivity, preservation under non-negative scaling, reflexivity,
anti-symmetry, and preservation under limit.
We propose to learn a generalized inequality or, equivalently, a proper cone
$\mathcal{K}$ that best describes the training data (see Fig. 1 for an
illustration). Our important assumption is that this proper cone, which
induces the generalized inequality, is not query-specific and thus prediction
can be used for unseen queries and document pairs coming from the same
distributions.
From a fundamental property of convex cones, if $\mathbf{z}\in\mathcal{K}$
then $w\mathbf{z}\in\mathcal{K}$ for all $w>0$, and any non-negative
combination of the cone elements also belongs to the cone, i.e. if
$\mathbf{u}_{k}\in\mathcal{K}$ then
$\sum_{k}w_{k}\mathbf{u}_{k}\in\mathcal{K},\forall w_{k}>0$.
In this work, we restrict our attention to polyhedral cones for the learning
of generalized inequalities. A polyhedral cone is a polyhedron and a cone. A
polyhedral cone can be defined as sum of rays or intersection of halfspaces.
We construct the polyhedral cone $\mathcal{K}$ from ‘basis’ vectors
$\mathbf{U}=[\mathbf{u}_{1},\mathbf{u}_{2},\ldots,\mathbf{u}_{K}]$. They are
the extreme vectors lying on the intersection of hyperplanes that define the
halfspaces. Thus, the cone $\mathcal{K}$ is a conic hull of the basis vectors
and is completely specified if the basis vectors are known. A polyhedral cone
with $K$ basis vectors is said to have an order $K$ if one basis vector cannot
be expressed as a conic combination of the others. It can be verified that
under these regular conditions, a polyhedral cone is a proper cone and thus
can induce a generalized inequality. We thus propose to learn the basis
vectors $\mathbf{u}_{k},k=1,\ldots,K$ for the characterization of
$\mathcal{K}$.
A projection of $\mathbf{z}$ onto the cone $\mathcal{K}$, denoted by
$\mathsf{P}_{\mathcal{K}}(\mathbf{z})$, is generally defined as some
$\mathbf{z}^{\prime}\in\mathcal{K}$ such that a certain criterion on the
distance between $\mathbf{z}$ and $\mathbf{z}^{\prime}$ is met. As
$\mathbf{z}^{\prime}\in\mathcal{K}$, it follows that it admits a conic
representation
$\mathbf{z}^{\prime}=\sum_{k=1}^{K}w_{k}\mathbf{u}_{k}=\mathbf{U}\mathbf{w},\
w_{k}\geq 0$. By restricting the order $K\leq N$, it can be shown that when
$\mathbf{U}$ is full-rank then the conic representation is unique.
Define an ordered document-pair ($l,m$) difference as
$\mathbf{z}=\mathbf{x}_{l}-\mathbf{x}_{m}$ where, without loss of generality,
we assume that $r_{l}\geq r_{m}$. The linear representation of
$\mathbf{z}^{\prime}\in\mathcal{K}$ can be found from
$\displaystyle\min_{\mathbf{w}}$
$\displaystyle\|\mathbf{z}-\mathbf{U}\mathbf{w}\|^{2}_{2},\ \
\mathbf{w}\geq{\mathbf{0}}$ (1)
where the inequality constraint is element-wise. It can be seen that
$\mathsf{P}_{\mathcal{K}}(\mathbf{z})=\mathbf{z},\forall\mathbf{z}\in\mathcal{K}$.
Otherwise, if $\mathbf{z}\not\in\mathcal{K}$ then it can be easily proved by
contradiction that the solution $\mathbf{w}$ is such that
$\mathbf{U}\mathbf{w}$ lies on a facet of $\mathcal{K}$. Let $\mathcal{K}^{-}$
be the cone with the basis $-\mathbf{U}$ then it can be easily shown that if
$\mathbf{z}\in\mathcal{K}^{-}$ then
$\mathsf{P}_{\mathcal{K}}(\mathbf{z})={\mathbf{0}}$.
Returning to the ranking problem, we need to find a $K$-degree polyhedral cone
$\mathcal{K}$ that captures most of the training data. Define the $\ell_{2}$
distance from $\mathbf{z}$ to $\mathcal{K}$ as
$d_{\mathcal{K}}(\mathbf{z})=\|\mathbf{z}-\mathsf{P}_{\mathcal{K}}(\mathbf{z})\|_{2}$
then we define the document-pair-level loss as
$\displaystyle l(\mathcal{K};\mathbf{z},y)=d_{\mathcal{K}}(\mathbf{z})^{2}.$
(2)
Suppose that for a query $q$, a set of document pair differences
$S_{q}=\\{\mathbf{z}^{q}_{1},\ldots,\mathbf{z}^{q}_{n_{q}}\\}$ with relevance
differences $\phi^{q}_{1},\ldots,\phi^{q}_{n_{q}},\phi^{q}_{j}>0$ can be
obtained. Following [13], we define the empirical query-level loss as
$\displaystyle\hat{L}(\mathcal{K};q,S_{q})=\frac{1}{n_{q}}\sum_{j=1}^{n_{q}}l(\mathcal{K};\mathbf{z}^{q},y^{q}).$
(3)
For a full training set of $P$ queries and
$S=\\{S_{q_{1}},\ldots,S_{q_{P}}\\}$ samples, we define the query-level
empirical risk as
$\displaystyle\hat{R}(\mathcal{K};S)=\frac{1}{P}\sum_{i=1}^{P}\hat{L}(\mathcal{K};q_{i},S_{q_{i}}).$
(4)
Thus, the polyhedral cone $\mathcal{K}$ can be found from minimizing this
query-level empirical risk. Note that even though other performance measures
such as mean average precision (MAP) or normalized discounted cumulative gain
(NDCG) is the ultimate assessment, it is observed that good empirical risk
often leads to good MAP/NDCG and simplifies the learning. We next discuss some
additional constraints for the algorithm to achieve good generalization
ability.
### 3.3 Modification
Normalization. Using the proposed approach, the direction of the vector
$\mathbf{z}$ is more important than its magnitude. However, at the same time,
if the magnitude of $\mathbf{z}$ is small it is desirable to suppress its
contribution to the objective function. We thus propose the normalization of
input document-pair differences as follows
$\mathbf{z}\leftarrow\rho\mathbf{z}/(\alpha+\|\mathbf{z}\|_{2}),\ \
\alpha,\rho>0.$ (5)
The constant $\rho$ is simply the scaling factor whilst $\alpha$ is to
suppress the noise when $\|\mathbf{z}\|_{2}$ is too small. With this
normalization, we note that
$\displaystyle\|\mathbf{z}\|_{2}\leq\rho.$ (6)
Relevance weighting. In the current setting, we consider all ordered document-
pairs equally important. This is however a disadvantage because the cost of
the mismatch between the two vectors which are close in rank is less than the
cost between those distant in rank. To address this issue, we propose an
extension of (2)
$l(\mathcal{K};\mathbf{z},y)=\phi d_{\mathcal{K}}(\mathbf{z})^{2}.$ (7)
where $\phi>0$ is the corresponding ordered relevance difference.
Conic regularization. From statistical learning theory [15, ch.4], it is known
that in order to obtain good generalization bounds, it is important to
restrict the hypothesis space from which the learned function is to be found.
Otherwise, the direct solution from an unconstrained empirical risk
minimization problem is likely to overfit and introduces large variance
(uncertainty). In many cases, this translates to controlling the complexity of
the learning function. In the case of support vector machines (SVMs), this has
the intuitive interpretation of maximizing the margin, which is the inverse of
the norm of the learning function in the Hilbert space.
In our problem, we seek a cone which captures most of the training examples,
i.e. the cone that encloses the conic hull of most training samples. In the
SVM case, there are many possible hyperplanes that separates the samples
without a controlled margin. Similarly, there is also a large number of
polyhedral cones that can capture the training samples without further
constraints. In fact, minimizing the empirical risk will tend to select the
cone with larger solid angle so that the training examples will have small
loss (see Fig. 2). In our case, the complexity is translated roughly to the
size (volume) of the cone. The bigger cone will likely overfit (enclose) the
noisy training samples and thus reduces generalization. Thus, we propose the
following constraint to indirectly regularize the size of the cone
$\displaystyle 0\leq\lambda_{l}\leq\|\mathbf{w}\|_{1}\leq\lambda_{u},\ \
\mathbf{w}\geq{\mathbf{0}}$ (8)
where $\mathbf{w}$ is the coefficients defined as in (1) and for simplicity we
set $\lambda_{l}=1$. To see how this effectively controls $\mathcal{K}$,
consider a 2D toy example in Fig. 2. If $\lambda_{u}=1$, the solution is the
cone $\mathcal{K}_{1}$. In this case, the loss of the positive training
examples (within the cone) is the distance from them to the simplex define
over the basis vectors $\mathbf{u}_{1},\mathbf{u}_{2}$ (i.e.
$\\{\mathbf{z}:\mathbf{z}=\lambda\mathbf{u}_{1}+(1-\lambda)\mathbf{u}_{2},0\leq\lambda\leq
1\\}$) and the loss of the negative training example is the distance to the
cone. With the same training examples, if we let $\lambda_{u}>1$ then there
exists a cone solution $\mathcal{K}_{2}$ such that all the losses are
effectively zero. In particular, for each training example, there exists a
corresponding $\|\mathbf{w}\|_{1}=\lambda$ such that the corresponding simplex
$\\{\mathbf{z}:\mathbf{z}=w_{1}\mathbf{u}_{1}+w_{2}\mathbf{u}_{2},w_{1}+w_{2}=\lambda\\}$,
passes all positive training examples.
Finally, we note that as the product $\mathbf{U}\mathbf{w}^{q_{i}}_{j}$
appears in the objective function and that both $\mathbf{U}$ and
$\mathbf{w}^{q_{i}}_{j}$ are variables then there is a scaling ambiguity in
the formulation. We suggest to address this scale ambiguity by considering the
norm constraint $\|\mathbf{u}_{k}\|_{2}=c>0$ on the basis vectors.
Figure 2: Illustration of different cone solutions. For simplicity, we plot
for the case $c=1$ and $\|\mathbf{z}\|_{2}\approx 1$.
In summary, the proposed formulation can be explicitly written as
$\displaystyle\min_{\mathbf{U}}\left\\{\frac{1}{P}\sum_{i=1}^{P}\frac{1}{n_{q_{i}}}\left(\sum_{j=1}^{n_{q_{i}}}\min_{\mathbf{w}^{q_{i}}_{j}}\phi^{q_{i}}_{j}\|\mathbf{z}^{q_{i}}_{j}-\mathbf{U}\mathbf{w}^{q_{i}}_{j}\|_{2}^{2}\right)\right\\}$
(9) $\displaystyle\mbox{s.t.}\
\|\mathbf{u}_{k}\|_{2}=c,\mathbf{w}^{q_{i}}_{j}\geq{\mathbf{0}},0<\lambda_{l}\leq\|\mathbf{w}^{q_{i}}_{j}\|_{1}\leq\lambda_{u}.$
### 3.4 Generalization bound
We restrict our study on generalization bound from an algorithmic stability
viewpoint, which is initially introduced in [2] and based on the concentration
property of random variables. In the ranking context, generalization bounds
for point-wise ranking / ordinal regression have been obtained [1, 8].
Recently, [13] show that the generalization bound result in [2] still holds in
the ranking context. More specifically, we would like to study the variation
of the expected query-level risk, defined as
$\displaystyle
R(\mathcal{K})=\int_{\mathcal{Q}\times\mathcal{Y}}L(\mathcal{K};q){P}_{\mathcal{Q}}(dq).$
(10)
where $L(\mathcal{K};q)$ denotes the expected query-level loss defined as
$\displaystyle
L(\mathcal{K};q)=\int_{\mathcal{Z}}l(\mathcal{K};\mathbf{z}^{q},y^{q}){P}_{\mathcal{Z}}(d\mathbf{z}^{q})$
(11)
and ${P}_{\mathcal{Z}}$ denotes the probability distribution of the (ordered)
document differences.
Following [2] and [13] we define the uniform leave-one-query-out document-
pair-level stability as
$\displaystyle\beta=\sup_{q\in\mathcal{Q},i\in[1,\ldots,P]}|l(\mathcal{K}_{S};\mathbf{z}^{q},y^{q})-l(\mathcal{K}_{S^{-i}};\mathbf{z}^{q},y^{q})|$
(12)
where $\mathcal{K}_{S}$ and $\mathcal{K}_{S^{-i}}$ are respectively the
polyhedral cones learned from the full training set and that without the $i$th
query. As stated in [13], it can be easily shown the following query-level
stability bounds by integration or average sum of the term on the left hand
side in the above definition
$\displaystyle|L(\mathcal{K}_{\mathcal{S}};q)-L(\mathcal{K}_{\mathcal{S}^{-i}};q)|\leq\beta,\forall
i$ (13)
$\displaystyle|\hat{L}(\mathcal{K}_{\mathcal{S}};q)-\hat{L}(\mathcal{K}_{\mathcal{S}^{-i}};q)|\leq\beta,\forall
i.$ (14)
Using the above query-level stability results and by considering $S_{q_{i}}$
as query-level samples, one can directly apply the result in [2] (see also
[13]) to obtain the following generalization bound
###### Theorem 1
For the proposed ConeRank algorithm with uniform leave-one-query-out document-
pair-level stability $\beta$, with probability of at least $1-\varepsilon$ it
holds
$\displaystyle
R(\mathcal{K}_{S})\leq\hat{R}(\mathcal{K}_{S})+2\beta+(4P\beta+\gamma)\sqrt{\frac{\ln(1/\varepsilon)}{2P}},$
(15)
where $\gamma=\sup_{q\in\mathcal{Q}}l(\mathcal{K}_{S};\mathbf{z}^{q},y^{q})$
and $\varepsilon\in[0,1]$.
As can be seen, the bound on the expected query-level risk depends on the
stability. It is of practical interest to study the stability $\beta$ for the
proposed algorithm. The following result shows that the change in the cone due
to leaving one query out can provide an effective upper bound on the uniform
stability $\beta$. For notational simplicity, we only consider the non-
weighted version of the loss, as the weighted version is simply a scale of the
bound by the maximum weight.
###### Theorem 2
Denote as $\mathbf{U}$ and $\mathbf{U}^{-i}$ the ‘basis’ vectors of the
polyhedral cones $\mathcal{K}_{S}$ and $\mathcal{K}_{S^{-i}}$ respectively.
For a ConeRank algorithm with non-weighted loss, we have
$\displaystyle\beta\leq 2s_{\rm
max}\lambda_{u}(\rho+\sqrt{K}c\lambda_{u})+s_{\rm max}^{2}\lambda_{u}^{2},$
(16)
where $s_{\rm max}=\max_{i}\|\mathbf{U}-\mathbf{U}^{-i}\|$, $\|\bullet\|$
denotes the spectral norm, and $\rho$ is the normalizing factor of
$\mathbf{z}$ (c.f. (6)).
Proof. Following the proposed algorithm, we equivalently study the bound of
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sup_{q\in\mathcal{Q}\atop\|\mathbf{z}^{q}\|_{2}\leq\rho}\left|\min_{\mathbf{w}\in\mathcal{C}}\|\mathbf{z}^{q}-\mathbf{U}\mathbf{w}\|_{2}^{2}-\min_{\mathbf{w}\in\mathcal{C}}\|\mathbf{z}^{q}-\mathbf{U}^{-i}\mathbf{w}\|_{2}^{2}\right|$
where the constraint set
$\mathcal{C}=\\{\mathbf{w}:\mathbf{w}\geq{\mathbf{0}},\lambda_{l}\leq\|\mathbf{w}\|_{1}\leq\lambda_{u}\\}$.
Without loss of generality, we can assume that
$\min_{\mathbf{w}\in\mathcal{C}}\|\mathbf{z}^{q}-\mathbf{U}\mathbf{w}\|_{2}^{2}>\min_{\mathbf{w}\in\mathcal{C}}\|\mathbf{z}^{q}-\mathbf{U}^{-i}\mathbf{w}\|_{2}^{2}$
and the minima are attained at $\mathbf{w}$ and $\mathbf{w}^{-i}$
respectively. Due to the definition, it follows that
$\displaystyle\beta$ $\displaystyle\leq$
$\displaystyle\sup_{q\in\mathcal{Q}\atop\|\mathbf{z}^{q}\|_{2}\leq\rho}\left(\|\mathbf{z}^{q}-\mathbf{U}\mathbf{w}^{-i}\|_{2}^{2}-\|\mathbf{z}^{q}-\mathbf{U}^{-i}\mathbf{w}^{-i}\|_{2}^{2}\right).$
Expanding the term on the left, and using matrix norm inequalities, one
obtains
$\displaystyle\beta$ $\displaystyle\leq$
$\displaystyle\sup_{q\in\mathcal{Q}}\left(2\|\mathbf{U}\|\|\bm{\Delta}\|+\|\bm{\Delta}\|^{2})\|\mathbf{w}^{-i}\|_{2}^{2}\right.$
(17)
$\displaystyle\left.+2\|\mathbf{z}^{q}\|_{2}\|\bm{\Delta}\|\|\mathbf{w}^{-i}\|_{2}\right)$
where $\bm{\Delta}=\mathbf{U}-\mathbf{U}^{-i}$. The proof follows by the
following facts
* •
$\|\mathbf{U}\|\leq\sqrt{K}c$ due to each $\|\mathbf{u}_{k}\|_{2}\leq c$ and
that $\|\mathbf{U}\|\leq\|\mathbf{U}\|_{F}$ where $\|\bullet\|_{F}$ denotes
the Frobenius norm.
* •
$\|\mathbf{w}\|_{2}^{2}\leq{\|\mathbf{w}\|_{1}^{2}}$ for
$\mathbf{w}\geq{\mathbf{0}}$
* •
$\|\mathbf{z}^{q}\|_{2}\leq\rho$ due to the normalization
and that $\|\bm{\Delta}\|\leq s_{\rm max}$ by definition.
It is more interesting to study the bound on $s_{\rm max}$. We conjecture that
this will depend on the sample size as well as the nature of the proposed
conic regularization. However, this is still an open question and such an
analysis is beyond the scope of the current work.
We note importantly that as the stability bound can be made small by lowering
$\lambda_{u}$. Doing so definitely improves stability at the cost of making
the empirical risk large and hence the bias becomes significantly undesirable.
In practice, it is important to select proper values of the parameters to
provide optimal bias-variance trade-off. Next, we turn the discussion on
practical implementation of the ideas, taking into account the large-scale
nature of the problem.
## 4 Implementation
In the original formulation (9), the scaling ambiguity is resolved by placing
a norm constraint on $\mathbf{u}_{k}$. However, a direct implementation seems
difficult. In what follows, we propose an alternative implementation by
resolving the ambiguity on $\mathbf{w}$ instead. We fix $\|\mathbf{w}\|_{1}=1$
and consider the norm inequality constraint on $\mathbf{u}_{k}$ as
$\|\mathbf{u}_{k}\|_{2}\leq c$ (i.e. convex relaxation on equality constraint)
where $c$ is a constant of $\mathcal{O}(\|\mathbf{z}^{q}\|_{2})$. This leads
to an approximate formulation
$\displaystyle\min_{\mathbf{U},\mathbf{w}^{q_{i}}_{j}}\left\\{\frac{1}{P}\sum_{i=1}^{P}\frac{1}{n_{q_{i}}}\left(\sum_{j=1}^{n_{q_{i}}}\phi^{q_{i}}_{j}\|\mathbf{z}^{q_{i}}_{j}-\mathbf{U}\mathbf{w}^{q_{i}}_{j}\|_{2}^{2}\right)\right\\}$
(18) $\displaystyle\mbox{s.t.}\ \|\mathbf{u}_{k}\|_{2}\leq
c,\mathbf{w}^{q_{i}}_{j}\geq{\mathbf{0}},\|\mathbf{w}^{q_{i}}_{j}\|_{1}=1.$
The advantage of this approximation is that the optimization problem is now
convex with respect to each $\mathbf{u}_{k}$ and still convex with respect to
each ${\mathbf{w}^{q_{i}}_{j}}$. This suggests an alternating and iterative
algorithm, where we only vary a subset of variables and fix the rest. The
objective function should then always decrease. As the problem is not strictly
convex, there is no guarantee of a global solution. Nevertheless, a locally
optimal solution can be obtained. The additional advantage of the formulation
is that gradient-based methods can be used for each sub-problem and this is
very important in large-scale problems.
Algorithm 1 Stochastic Gradient Descent
Input: queries $q_{i}$ and pair differences $\mathbf{z}^{q_{i}}_{j}$.
Randomly initialize $\mathbf{u}_{k},\ \forall k\leq K$; set $\mu>0$
repeat
1\. The _folding-in_ step (fixed $\mathbf{U}$):
Randomly initialize
$\mathbf{w}^{q_{i}}_{j}:\mathbf{w}^{q_{i}}_{j}\geq{\mathbf{0}};\|\mathbf{w}^{q_{i}}_{j}\|_{1}=1$;
repeat
1a. Compute
$\mathbf{w}^{q_{i}}_{j}\leftarrow\mathbf{w}^{q_{i}}_{j}-\mu{\partial\hat{R}(\mathbf{w}^{q_{i}}_{j}})/{\partial\mathbf{w}^{q_{i}}_{j}}$
1b. Set
$\mathbf{w}^{q_{i}}_{j}\leftarrow\max\\{\mathbf{w}^{q_{i}}_{j},{\mathbf{0}}\\}$
(element-wise)
1c. Normalize
$\mathbf{w}^{q_{i}}_{j}\leftarrow\mathbf{w}^{q_{i}}_{j}/\|\mathbf{w}^{q_{i}}_{j}\|_{1}$
until converged
2\. The _basis-update_ step (fixed $\mathbf{w}$):
for $k=1$ to $K$ do
2a. Update
$\mathbf{u}_{k}\leftarrow\mathbf{u}_{k}-\mu{\partial\hat{R}(\mathbf{u}_{k}})/{\partial\mathbf{u}_{k}}$
2b. Normalize $\mathbf{u}_{k}$ to norm $c$ if violated.
end for
until converged
### 4.1 Stochastic Gradient
Since the number of pairs may be large for typically real datasets, we do not
want to store every $\mathbf{w}^{q}_{j}$. Instead, for each iteration, we
perform a _folding-in_ operation, in that we fix the basis $\mathbf{U}$, and
estimate the coefficients $\mathbf{w}^{q}_{j}$. Since this is a convex
problem, it is possible to apply the stochastic gradient (SG) method as shown
in Algorithm 1. Note that we express the empirical risk as the function of
only variable of interest when other variables are fixed for notational
simplicity. In practice, we also need to check if the cone is proper and we
find this is always satisfied.
### 4.2 Exponentiated Gradient
Exponentiated Gradient (EG) [12] is an algorithm for estimating distribution-
like parameters. Thus, Step 1a can be replaced by
$\displaystyle\mathbf{w}^{q_{i}}_{j}\leftarrow\mathbf{w}^{q_{i}}_{j}\exp\left\\{-\mu{\partial\hat{R}(\mathbf{w}^{q_{i}}_{j})}/{\partial\mathbf{w}^{q_{i}}_{j}}\right\\}(\mbox{element-
wise}).$
For faster numerical computation (by avoiding the exponential), as shown in
[12], this step can be approximated by
$\displaystyle(\mathbf{w}^{q_{i}}_{j})_{k}\leftarrow(\mathbf{w}^{q_{i}}_{j})_{k}\left(1-\mu\left({\partial\hat{R}(\hat{\mathbf{z}}^{q_{i}}_{j})}/{\partial\hat{\mathbf{z}}^{q_{i}}_{j}}\right)^{\top}\
(\mathbf{u}_{k}-\hat{\mathbf{z}}^{q_{i}}_{j})\right)$
where the empirical risk $\hat{R}$ is parameterized in terms of
$\hat{\mathbf{z}}^{q_{i}}_{j}=\mathbf{U}\mathbf{w}^{q_{i}}_{j}$. When the
learning rate $\mu$ is sufficiently small, this update readily ensures the
normalization of $\mathbf{w}^{q_{i}}_{j}$. The main difference between SG and
EG is that, update in SG is _additive_ , while it is _multiplicative_ in EG.
Algorithm 2 Query-level Prediction
Input: New query $q$ with pair differences
$\\{\mathbf{z}^{q}_{j}\\}_{j=1}^{n_{q}}$
Maintain a scoring array $A$ of all pre-computed feature vectors, initialize
$A_{l}=0$ for all $l$.
Set $\phi^{q}_{j}=1,\forall j\leq n_{q}$.
for $j=1$ to $n_{q}$ do
Perform _folding-in_ to estimate the coefficients without the non-negativity
constraints.
Check if the sum of the coefficients is positive, then $A_{l}\leftarrow
A_{l}+1$ ; otherwise $A_{m}\leftarrow A_{m}+1$
end for
Output the ranking based on the scoring array $A$.
### 4.3 Prediction
Assume that the basis
$\mathbf{U}=(\mathbf{u}_{1},\mathbf{u}_{2},...,\mathbf{u}_{K})$ has been
learned during training. In testing, for each query, we are also given a set
of feature vectors, and we need to compute a ranking function that outputs the
appropriate positions of the vectors in the list.
Unlike the training data where the order of the pair $(l,m)$ is given, now
this order information is missing. This breaks down the conic assumption, in
that the difference of the two vectors is the non-negative combination of the
basis vectors. Since the either preference orders can potentially be
incorrect, we relax the constraint of the non-negative coefficients. The idea
is that, if the order is correct, then the coefficients are mostly positive.
On the other hand, if the order is incorrect, we should expect that the
coefficients are mostly negative. The query-level prediction is proposed as
shown in Algorithm 2. As this query-level prediction is performed over a
query, it can address the shortcoming of logical discrepancy of document-level
prediction in the pairwise approach.
## 5 Discussion
RankSVM [11] defines the following loss function over ordered pair differences
$\displaystyle L(\mathbf{u})$ $\displaystyle=$
$\displaystyle\frac{1}{P}\sum_{j}\max(0,1-\mathbf{u}^{\top}\mathbf{z}_{j})+\frac{C}{2}\|\mathbf{u}\|_{2}^{2}$
where $\mathbf{u}\in\mathbb{R}^{N}$ is the parameter vector, $C>0$ is the
penalty constant and $P$ is the number of data pairs.
Being a pairwise approach, RankNet instead uses
$\displaystyle L(\mathbf{u})$ $\displaystyle=$
$\displaystyle\frac{1}{P}\sum_{j}\log(1+\exp\\{-\mathbf{u}^{\top}\mathbf{z}_{j}\\})+\frac{C}{2}\|\mathbf{u}\|_{2}^{2}.$
This is essentially the 1-class SVM applied over the ordered pair differences.
The quadratic regularization term tends to push the separating hyperplane away
from the origin, i.e. maximizing the 1-class margin.
It can be seen that the RankSVM solution is the special case when the cone
approaches a halfspace. In the original RankSVM algorithm, there is no
intention to learn a non-negative subspace where ordinal information is to be
found like in the case of ConeRank. This could potentially give ConeRank more
analytical power to trace the origin of preferences.
## 6 Experiments
### 6.1 Data and Settings
We run the proposed algorithm on the latest and largest benchmark data LETOR
4.0. This has two data sets for supervised learning, namely MQ2007 (1700
queries) and MQ2008 (800 queries). Each returned document is assigned a
integer-valued relevance score of $\\{0,1,2\\}$ where $0$ means that the
document is irrelevant with respect to the query. For each query-document
pair, a vector of $46$ features is pre-extracted, and available in the
datasets. Example features include the term-frequency and the inverse document
frequency in the body text, the title or the anchor text, as well as link-
specific like the PageRank and the number of in-links. The data is split into
a training set, a validation set and a test set. We normalize these features
so that they are roughly distributed as Gaussian with zero means and unit
standard deviations. During the folding-in step, the parameters
$\mathbf{w}^{q}_{j}$ corresponding to pair $j$th of query $q$ are randomly
initialized from the non-negative uniform distribution and then normalized so
that $\|\mathbf{w}^{q}_{j}\|_{1}=1$. The basis vectors $\mathbf{u}_{k}$ are
randomly initialized to satisfy the relaxed norm constraint. The learning rate
is $\mu=0.001$ for the SG and $\mu=0.005$ for the EG. For normalization, we
select $\alpha=1$ and $\rho=\sqrt{N}$ where $N$ is the number of features, and
we set $c=2\rho$.
Figure 3: Performance versus basis number
### 6.2 Results
The two widely-used evaluation metrics employed are the Mean Average Precision
(MAP) and the Normalized Discounted Cumulative Gain (NDCG). We use the
evaluation scripts distributed with LETOR 4.0.
In the first experiment, we investigate the performance of the proposed method
with respect to the number of basis vectors $K$. The result of this experiment
on the MQ2007 dataset is shown in Fig. 3. We note an interesting observation
that the performance is highest at about $K=10$ out of 46 dimensions of the
original feature space. This seems to suggest that the idea of capturing an
informative subspace using the cone makes sense on this dataset. Furthermore,
the study on the eigenvalue distribution of the non-centralized ordered
pairwise differences on on the MQ2007 dataset, as shown in Fig. 4, also
reveals that this is about the dimension that can capture most of the data
energy.
Figure 4: Eigenvalue distribution on the MQ2007 dataset.
We then compare the proposed and recent base-line methods222from
http://research.microsoft.com/en-
us/um/beijing/projects/letor/letor4baseline.aspx in the literature and the
results on the MQ2007 and MQ2008 datasets are shown in Table 1. The proposed
ConeRank is studied with $K=10$ due to the previous experiment. We note that
all methods tend to perform better on MQ2007 than MQ2008, which can be
explained by the fact that the MQ2007 dataset is much larger than the other,
and hence provides better training.
On the MQ2007 dataset, ConeRank compares favourably with other methods. For
example, ConeRank-SG achieves the highest MAP score, whilst its NDCG score
differs only less than 2% when compared with the best (RankSVM-struct). On the
MQ2008 dataset, ConeRank still maintains within the 3% margin of the best
methods on both MAP and NDCG metrics.
Table 1: Results on LETOR 4.0.
| MQ2007 | MQ2008
---|---|---
Algorithms | MAP | NDCG | MAP | NDCG
AdaRank-MAP | 0.482 | 0.518 | 0.463 | 0.480
AdaRank-NDCG | 0.486 | 0.517 | 0.464 | 0.477
ListNet | 0.488 | 0.524 | 0.450 | 0.469
RankBoost | 0.489 | 0.527 | 0.467 | 0.480
RankSVM-struct | 0.489 | 0.528 | 0.450 | 0.458
ConeRank-EG | 0.488 | 0.514 | 0.444 | 0.456
ConeRank-SG | 0.492 | 0.517 | 0.454 | 0.464
## 7 Conclusion
We have presented a new view on the learning to rank problem from a
generalized inequalities perspective. We formulate the problem as learning a
polyhedral cone that uncovers the non-negative subspace where ordinal
information is found. A practical implementation of the method is suggested
which is then observed to achieve comparable performance to state-of-the-art
methods on the LETOR 4.0 benchmark data.
There are some directions that require further research, including a more
rigorous study on the bound of the spectral norm of the leave-one-query-out
basis vector difference matrix, a better optimization scheme that solves the
original formulation without relaxation, and a study on the informative
dimensionality of the ranking problem.
## References
* [1] S. Agarwal, _Lecture notes in artificial intelligence_. Springer-Verlag, 2008, ch. Generalization bounds for some ordinal regression algorithms, pp. 7–21.
* [2] O. Bousquet and A. Elisseff, “Stability and generalization,” _Journal of Machine Learning Research_ , pp. 499–526, 2002.
* [3] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
* [4] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender, “Learning to rank using gradient descent,” in _Proc. ICML_ , 2005.
* [5] Z. Cao, T. Qin, T. Liu, M. Tsai, and H. Li, “Learning to rank: from pairwise approach to listwise approach,” in _Proc. ICML_ , 2007.
* [6] W. Chu and Z. Ghahramani, “Gaussian processes for ordinal regression,” _Journal of Machine Learning Research_ , vol. 6, no. 1, p. 1019, 2006.
* [7] W. Cohen, R. Schapire, and Y. Singer, “Learning to order things,” _J Artif Intell Res_ , vol. 10, pp. 243–270, 1999.
* [8] D. Cossock and T. Zhang, “Statistical analysis of Bayes optimal subset ranking,” _IEEE Transactions on Information Theory_ , vol. 54, pp. 5140–5154, 2008.
* [9] Y. Freund, R. Iyer, R. Schapire, and Y. Singer, “An efficient boosting algorithm for combining preferences,” _Journal of Machine Learning Research_ , vol. 4, no. 6, pp. 933–969, 2004.
* [10] R. Herbrich, T. Graepel, and K. Obermayer, “Large margin rank boundaries for ordinal regression,” in _Proc. KDD_ , 2000.
* [11] T. Joachims, “Optimizing search engines using clickthrough data,” in _Proc. KDD_ , 2002.
* [12] J. Kivinen and M. Warmuth, “Exponentiated gradient versus gradient descent for linear predictors,” _Information and Computation_ , 1997.
* [13] Y. Lan, T.-Y. Liu, T. Quin, Z. Ma, and H. Li, “Query-level stability and generalization in learning to rank,” in _Proc. ICML_ , 2008.
* [14] R. Plackett, “The analysis of permutations,” _Applied Statistics_ , pp. 193–202, 1975.
* [15] B. Schölkopf and Smola, _Learning with kernels_. MIT Press, 2002.
|
arxiv-papers
| 2012-06-19T02:24:55 |
2024-09-04T02:49:31.951542
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Truyen T. Tran and Duc Son Pham",
"submitter": "Duc Son Pham",
"url": "https://arxiv.org/abs/1206.4110"
}
|
1206.4291
|
# On The Sub-Mixed Fractional Brownian Motion
Charles El-Nouty and Mounir Zili
###### Abstract
Let $\\{S_{t}^{H},\,t\geq 0\\}$ be a linear combination of a Brownian motion
and of an independent sub-fractional Brownian motion with Hurst index $0<H<1$.
Its main properties are studied and it is shown that $S^{H}$ can be considered
as an intermediate process between a sub-fractional Brownian motion and a
mixed fractional Brownian motion. Finally, we determine the values of $H$ for
which $\;S^{H}$ is not a semi-martingale.
## 1 Introduction
Let $\\{B_{t}^{H},t\in{\mathbb{R}}\\}$ be a fractional Brownian motion (fBm)
with Hurst index $0<~{}H<~{}1$, i.e. a centered Gaussian process with
stationary increments satisfying $B_{0}^{H}=0$, with probability 1, and
${\mathbb{E}}{(B_{t}^{H})}^{2}=\;{\mid t\mid}^{2H},t\in{\mathbb{R}}$. We
obviously have for any real numbers $t$ and $s$
(1.1) $cov\Bigl{(}B_{t}^{H},B_{s}^{H}\Bigr{)}=\frac{1}{2}\;\Bigl{(}\mid
t\mid^{2H}+\mid s\mid^{2H}-\mid t-s\mid^{2H}\Bigr{)}.$
Consider $\\{B_{t},t\in{\mathbb{R}}\\}$ an independent Brownian motion (Bm)
and $(a,b)$ two real numbers such that $(a,b)\neq(0,0)$.
The mixed-fractional Brownian motion (mfBm) is an extension of a Bm and a fBm.
It was introduced in [3] in order to solve some problems in mathematical
finance, such as modelling some arbitrage-free and complete markets. The mfBm
$\displaystyle M^{H}=\\{M_{t}^{H}(a,b);t\geq 0\\}=\\{M_{t}^{H};t\geq 0\\}$ of
parameters $a,b$ and $H$ is defined as follows:
$\forall t\in{\mathbb{R}}_{+},\hskip
14.22636ptM_{t}^{H}=M_{t}^{H}(a,b)=a\;B_{t}+b\;B_{t}^{H}.$
We refer also to [5] and [12] for further information on this process. Let us
recall some of its main properties.
###### Lemma 1
The mfBm $\;(M_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ satisfies the following
properties:
* •
$M^{H}$ is a centered Gaussian process.
* •
$\forall s\in{{\mathbb{R}}}_{+},\forall t\in{{\mathbb{R}}}_{+},$
$Cov\Big{(}M_{t}^{H}(a,b),M_{s}^{H}(a,b)\Big{)})=a^{2}(t\wedge
s)+\frac{b^{2}}{2}\Big{(}t^{2H}+s^{2H}-\mid t-s\mid^{2H}\Big{)},$
where $\displaystyle t\wedge s=\frac{1}{2}\Big{(}t+s-\mid t-s\mid\Big{)}.$
* •
The increments of the mfBm are stationary.
In [2], the authors suggested a second extension of a Bm, called the sub-
fractional Brownian motion (sfBm), that preserves most of the properties of
the fBm, but not the stationarity of the increments. It is the stochastic
process $\displaystyle\xi^{H}=\\{\xi_{t}^{H};t\geq 0\\}$, defined by:
(1.2) $\forall t\in{\mathbb{R}}_{+},\hskip
14.22636pt\xi_{t}^{H}=\frac{B_{t}^{H}+B_{-t}^{H}}{\sqrt{2}},$
This process arises from occupation time fluctuations of branching particle
systems with Poisson initial condition (see [2]). Let us state some results on
the sfBm.
###### Lemma 2
The sfBm $\;(\xi_{t}^{H})_{t\in{\mathbb{R}}_{+}}$ satisfies the following
properties:
* •
$\xi^{H}$ is a centered Gaussian process.
* •
$\displaystyle\forall s\in{{\mathbb{R}}}_{+},\forall t\in{{\mathbb{R}}}_{+},$
$Cov\Big{(}\xi_{t}^{H},\xi_{s}^{H}\Big{)})=s^{2H}+t^{2H}-\frac{1}{2}\Big{(}(s+t)^{2H}+\mid
t-s\mid^{2H}\Big{)}.$
* •
The increments of the smfBm are not stationary.
We can easily remark that, when $H=1/2,\;\xi^{1/2}$ is a Bm.
We refer to [2, 6, 11] for further information on this process.
In the spirit of [2] and [12], we introduce a new process, that we will call
the sub-mixed fractional Brownian motion (smfBm). More precisely, the smfBm of
parameters $a,b$ and $H$, is a process $\;S^{H}=\\{S_{t}^{H}(a,b);t\geq
0\\}=\\{S_{t}^{H};t\geq 0\\}$, defined by:
(1.3) $\forall t\in{\mathbb{R}}_{+},\hskip
14.22636ptS_{t}^{H}=S_{t}^{H}(a,b)=\frac{a\;(B_{t}+B_{-t})+b\;(B_{t}^{H}+B_{-t}^{H})}{\sqrt{2}}=a\;\xi_{t}+b\
\xi_{t}^{H},$
where $\xi$ is a Bm, obviously independent of $\xi^{H}$.
When $a=0$ and $b=1,\;S^{H}=\xi^{H}$ is a sfBm. When $a=1$ and
$b=0,\;S^{H}=\xi$ is a Bm.
So the smfBm is clearly an extension of the sfBm and the Bm. This is the
flavor of this process. We will show first that it has the same properties as
the sfBm. Then, we will prove that it has also some of the main properties of
the mfBm, but that its increments are not stationary; they are more weakly
correlated on non-overlapping intervals. Hence $S^{H}$ may be considered as
being intermediate between the sfBm and the mfBm. This is why we call it the
smfBm.
The aim of this paper is to study on one hand some key properties of the smfBm
and on the other hand its martingale properties. The motivation of the authors
is to measure the consequences of the lack of increments stationarity.
In section 2, the main properties of the smfBm are studied, namely:
* •
the mixed-self-similarity property (see [12]),
* •
the non Markovian property,
* •
the increments non stationarity property,
* •
the correlation coefficient and the influence of the parameters $a$ and $b$ on
it,
* •
the comparison between the mfBm and the smfBm covariance properties.
Finally it is shown in section 3 that the smfBm is a semi-martingale if and
only if
$b=0\;\;\;\mbox{or}\;\;\ H\,\in\,\\{1/2\\}\,\cup\,]3/4,1[.$
## 2 Main properties
### 2.1 Basic properties
The following lemmas describe the basic properties of the smfBm.
###### Lemma 3
The smfBm $\displaystyle(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ satisfies the
following properties:
* •
$S^{H}$ is a centered Gaussian process.
* •
$\forall s\in{\mathbb{R}}_{+},\;\forall t\in{\mathbb{R}}_{+}$,
(2.1)
$\begin{array}[]{rcl}&&Cov\Big{(}S_{t}^{H}(a,b),S_{s}^{H}(a,b)\Big{)}=a^{2}\;\left(s\wedge
t\right)\\\ \vskip
8.53581pt\cr&+&b^{2}\;\left(t^{2H}+s^{2H}-\frac{1}{2}\left(\left(s+t\right)^{2H}+\mid
t-s\mid^{2H}\right)\right).\end{array}$
* •
(2.2) $\forall t\in{\mathbb{R}}_{+},\hskip
8.53581pt{\mathbb{E}}\Big{(}\big{(}S_{t}^{H}(a,b)\big{)}^{2}\Big{)}=a^{2}t+b^{2}\;\left((2-2^{2H-1})\quad
t^{2H}\right).$
###### Proof.
It is a direct consequence of lemma 2. ∎
NOTATION. Let $(X_{t})_{t\in{\mathbb{R}}_{+}}$ and
$(Y_{t})_{t\in{\mathbb{R}}_{+}}$ be two processes defined on the same
probability space $(\Omega,F,{\mathbb{P}})$. The notation
$\\{X_{t}\\}\overset{\Delta}{=}\\{Y_{t}\\}$ will mean that
$(X_{t})_{t\in{\mathbb{R}}_{+}}$ and $(Y_{t})_{t\in{\mathbb{R}}_{+}}$ have the
same law.
Let us check the mixed-self-similarity property of the smfBm, which was
introduced in [12] in the mfBm case.
###### Lemma 4
For any $h>0$,
$\displaystyle\\{S_{ht}^{H}(a,b)\\}\stackrel{{\scriptstyle\Delta}}{{=}}\Big{\\{}S_{t}^{H}\Big{(}ah^{1/2},bh^{H}\Big{)}\Big{\\}}.$
###### Proof.
For fixed $h>0$ , the processes $\\{S_{ht}^{H}(a,b)\\}$, and
$\Big{\\{}S_{t}^{H}\Big{(}ah^{1/2},bh^{H}\Big{)}\Big{\\}}$ are centered
Gaussian. Therefore, one has only to prove that they have the same covariance
function. We have for any $s$ and $t$ in ${\mathbb{R}}_{+}$:
$\begin{array}[]{rcl}Cov\Big{(}S_{ht}^{H}(a,b),S_{hs}^{H}(a,b)\Big{)}&=&(h^{1/2}a)^{2}\;\left(s\wedge
t\right)\\\ \vskip
8.53581pt\cr&+&(h^{H}b)^{2}\;\left(t^{2H}+s^{2H}-\frac{1}{2}\left(\left(s+t\right)^{2H}+\mid
t-s\mid^{2H}\right)\right)\\\ \vskip
8.53581pt\cr&=&Cov\Bigg{(}S_{t}^{H}(ah^{1/2},bh^{H}),S_{s}^{H}(ah^{1/2},bh^{H})\Bigg{)}.\end{array}$
This ends the proof of the lemma.
∎
###### Lemma 5
For any $H\in\Big{]}0,1\Big{[}\setminus\Big{\\{}\frac{1}{2}\Big{\\}}$,
$a\in{\mathbb{R}}$ and $b\in{\mathbb{R}}^{\ast}$,
$(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ is not a Markovian process.
###### Proof.
By lemma 3, $S^{H}$ is a centered Gaussian process such that
$\mathbb{E}\left(S_{t}^{H}\right)^{2}>0$ for all $t>0$. Then, if $S^{H}$ were
a Markovian process, according to [9], for all $0<s<t<u$ we would have:
(2.3)
$Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}=Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}.$
We get by lemma 3,
$\displaystyle Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}s+b^{2}s^{2H}+b^{2}\left(t^{2H}-\frac{1}{2}\left(t+s\right)^{2H}-\frac{1}{2}\left(t-s\right)^{2H}\right),$
$\displaystyle Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle a^{2}t+b^{2}\left(2-2^{2H-1}\right)\quad t^{2H},$
$\displaystyle Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}t+b^{2}t^{2H}+b^{2}\left(u^{2H}-\frac{1}{2}\left(u+t\right)^{2H}-\frac{1}{2}\left(u-t\right)^{2H}\right),$
$\displaystyle Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}s+b^{2}s^{2H}+b^{2}\left(u^{2H}-\frac{1}{2}\left(u+s\right)^{2H}-\frac{1}{2}\left(u-s\right)^{2H}\right).$
Let $s$ be fixed and set $u=e^{t}$. When $t\rightarrow+\infty,$ Taylor
expansions yield
$t^{2H}-\frac{1}{2}\left(t+s\right)^{2H}-\frac{1}{2}\left(t-s\right)^{2H}=-H\left(2H-1\right)\frac{s^{2}}{t^{2-2H}}+o\left(\frac{s^{2}}{t^{2-2H}}\right),$
and
$u^{2H}-\frac{1}{2}\left(u+t\right)^{2H}-\frac{1}{2}\left(u-t\right)^{2H}=-H\left(2H-1\right)\frac{t^{2}}{e^{(2-2H)t}}+o\left(\frac{t^{2}}{e^{(2-2H)t}}\right).$
Therefore, for $(h,x)\in\\{(s,t),(t,u),(s,u)\\}$,
$\underset{x\rightarrow\infty}{\lim}\left(x^{2H}-\frac{1}{2}\left(x+h\right)^{2H}-\frac{1}{2}\left(x-h\right)^{2H}\right)=0.$
To verify (2.3), a necessary condition is that, when $b\neq 0$,
$\underset{t\rightarrow\infty}{\lim}\left(Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}-Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}\right)=0,$
that is
$\left(a^{2}s+b^{2}s^{2H}\right)\underset{t\rightarrow\infty}{\lim}\left(\left(a^{2}t+b^{2}\left(2-2^{2H-1}\right)t^{2H}\right)-\left(a^{2}t+b^{2}t^{2H}\right)\right)=0.$
The last equality is satisfied when
$2-2^{2H-1}=1\Leftrightarrow H=\frac{1}{2}.$
The proof of lemma 5 is complete. ∎
###### Proposition 6
Second moment of increments:
We have for all $(s,t)\in{\mathbb{R}}_{+}^{2},$ $s\leq t$,
* •
(2.4)
$\begin{array}[]{rcl}&&E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}=a^{2}(t-s)\\\
\vskip
5.69054pt\cr&+&b^{2}\Bigg{(}-2^{2H-1}(t^{2H}+s^{2H})+(t+s)^{2H}+(t-s)^{2H}\Bigg{)}.\end{array}$
* •
(2.5) $a^{2}(t-s)+b^{2}\gamma(t-s)^{2H}\leq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}\leq
a^{2}(t-s)+b^{2}\nu(t-s)^{2H},$
where
$\gamma=\left\\{\begin{array}[]{rcl}\displaystyle 2-2^{2H-1}&if&\displaystyle
H>\frac{1}{2},\\\ \vskip 5.69054pt\cr\displaystyle 1&if&\displaystyle
H\leq\frac{1}{2},\\\ &&\end{array}\right.$
and
$\nu=\left\\{\begin{array}[]{rcl}\displaystyle 1&if&\displaystyle
H\geq\frac{1}{2},\\\ \vskip 5.69054pt\cr\displaystyle
2-2^{2H-1}&if&\displaystyle H<\frac{1}{2}.\\\ &&\end{array}\right.$
###### Proof.
Equality (2.4) is a direct consequence of equalities (2.1) and (2.2). So let
us check the inequalities (2.5). Setting
(2.6) $A(s,t)=\Bigg{(}\frac{t+s}{2}\Bigg{)}^{2H}-\frac{t^{2H}+s^{2H}}{2},$
we can write
(2.7)
$E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}-a^{2}(t-s)=b^{2}\Bigg{(}(t-s)^{2H}+2^{2H}A(s,t)\Bigg{)}.$
We get by convexity that, if $H\leq\frac{1}{2}$, then $A(s,t)\geq 0$ and
consequently
(2.8) $a^{2}(t-s)+b^{2}(t-s)^{2H}\leq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2},$
and if $\displaystyle H\geq\frac{1}{2}$, then $A(s,t)\leq 0$ and consequently
(2.9) $a^{2}(t-s)+b^{2}(t-s)^{2H}\geq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}.$
To complete the proof of proposition 6, we need a technical lemma.
###### Lemma 7
Consider, for any $s>0$, the function $f$ defined as follows
$f(x)=-2^{2H-1}((x+s)^{2H}+s^{2H})+(x+2s)^{2H}-(1-2^{2H-1})\ x^{2H},\quad
x\geq 0.$
If $H<\frac{1}{2}$, $f$ is a negative decreasing function, whereas, if
$H>\frac{1}{2}$, $f$ is a positive increasing one.
###### Proof.
(of lemma 7 ) It is clear that $\ f(0)=0$. We get for $x>0$
$f^{\prime}\left(x\right)=H\ x^{2H-1}g(x),$
where
$g(x)=-2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-1}+2\Big{(}\frac{2s}{x}+1\Big{)}^{2H-1}-(2-2^{2H}).$
We have
$g^{\prime}(x)=\frac{(2H-1)s}{x^{2}}\Bigg{(}2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-2}-4\Big{(}\frac{2s}{x}+1\Big{)}^{2H-2}\Bigg{)}.$
Let us consider the two following cases:
Case $1$: $H<\frac{1}{2}$. Since $2H-1<0$, $2-2^{2H}>0$ and consequently
$\lim_{x\rightarrow
0^{+}}g(x)=-(2-2^{2H})<0\;\;\;\mbox{and}\;\;\;\lim_{x\rightarrow+\infty}g(x)=0.$
Set
$\ell(x)=\frac{s+x}{2s+x}=\frac{\frac{s}{x}+1}{\frac{2s}{x}+1}.$
Since $\ell$ increases from $\frac{1}{2}$ to $1$, $\ell^{2H-2}$ decreases from
$2^{2-2H}$ to $1$. Then $\;\ell(x)^{2H-2}\leq~{}2^{2-2H}$, which is equivalent
to
$2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-2}-4\Big{(}\frac{2s}{x}+1\Big{)}^{2H-2}\leq
0,$
and consequently $g^{\prime}(x)\geq 0$. Since $g$ increases from $-(2-2^{2H})$
to $0$, $g(x)\leq 0$ and therefore $f^{\prime}(x)\leq 0$. Hence $f$ decreases
and $f(x)\leq 0$.
Case $2$: $H>\frac{1}{2}$. Following the same lines as in case $1$, we get
$g^{\prime}(x)\leq 0$. Since the function $g$ decreases from $-(2-2^{2H})$ to
$0$, $f$ increases and $f(x)\geq 0$. This completes the proof of lemma 7.
∎
Combining $(\ref{eq17})$ and $(\ref{eq18})$ with (2.7) and lemma 7, we
complete the proof of proposition 6.
∎
###### Remark 8
As a consequence of proposition 6, we insist on the fact that the smfBm does
not have stationary increments, but this property is replaced by inequalities
(2.5).
### 2.2 Study of the correlation coefficient of the smfBm increments
NOTATION. Let $\displaystyle X$ and $\displaystyle Y$ be two random variables
defined on the same probability space $(\Omega,F,{\mathbb{P}})$ such that
$V(X)\times V(Y)\neq 0$. We denote the correlation coefficient $\rho(X,Y)$ by:
$\rho(X,Y)=\frac{Cov(X,Y)}{\sqrt{V(X)}\sqrt{V(Y)}}.$
###### Lemma 9
We have for
$a\in{\mathbb{R}},b\in{\mathbb{R}}^{*},s\in{\mathbb{R}}_{+},t\in{\mathbb{R}}_{+}$
and $h\in{\mathbb{R}}_{+}$ such that $\displaystyle 0<h\leq t-s$,
(2.10)
$\rho\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=\frac{\gamma(s,t,h)}{\sqrt{\Big{(}2\frac{a^{2}}{b^{2}}h+\alpha(s,h)\Big{)}\Big{(}2\frac{a^{2}}{b^{2}}h+\alpha(t,h)\Big{)}}},$
where
$\begin{array}[]{rcl}\displaystyle\gamma(s,t,h)&=&\displaystyle\Bigg{(}(t-s+h)^{2H}-2(t-s)^{2H}+(t-s-h)^{2H}\\\
\vskip
8.53581pt\cr&-&\displaystyle(t+s)^{2H}+2(t+s+h)^{2H}-(t+s+2h)^{2H}\Bigg{)},\end{array}$
and
$\;\alpha(s,h)=-2^{2H}\Bigl{(}(s+h)^{2H}+s^{2H}\Bigr{)}+2\,(2s+h)^{2H}+2h^{2H}.$
###### Proof.
We have by equality (2.4)
(2.11)
$\begin{array}[]{rcl}{\mathbb{E}}\Bigl{(}S_{t+h}^{H}-S_{t}^{H}\Bigr{)}^{2}&=&a^{2}\,h+b^{2}\;\Biggl{(}-2^{2H-1}\;\Bigl{(}(t+h)^{2H}+t^{2H}\Bigr{)}\\\
&+&(2t+h)^{2H}+h^{2H}\Biggr{)}\\\
&=&a^{2}\,h+\frac{b^{2}}{2}\;\alpha(t,h).\end{array}$
Recall that a Bm has independent increments and that the processes $\xi^{H}$
and $\xi$ are independent. Then, we have
$Cov\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=b^{2}\;Cov\Big{(}\xi_{t+h}^{H}-\xi_{t}^{H},\xi_{s+h}^{H}-\xi_{s}^{H}\Big{)}$,
and we get by using lemma 2
(2.12)
$Cov\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=\frac{b^{2}}{2}\;\gamma(s,t,h).$
Combining (2.11) with (2.12), we complete the proof of lemma 9.
∎
###### Corollary 10
Let $a\in{\mathbb{R}}$ and $b\in{\mathbb{R}}^{*}$. Then, the increments of
$\displaystyle(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ are positively
correlated for $\displaystyle\frac{1}{2}<H<1$, uncorrelated for $\displaystyle
H=\frac{1}{2}$, and negatively correlated for $\displaystyle 0<H<\frac{1}{2}$.
###### Proof.
Let us write the function $\gamma$ given in (2.10) as
$\displaystyle\gamma(s,t,h)=f(t-s)-f(t+s+h),$
where $f:x\longmapsto(x+h)^{2H}-2x^{2H}+(x-h)^{2H}$. We have for every $x>0$
$f^{\prime}(x)=2H\Big{(}(x+h)^{2H-1}-2x^{2H-1}+(x-h)^{2H-1}\Big{)}.$
The study of the convexity of the function $\displaystyle x\longmapsto
x^{2H-1}$ enables us to determine the sign of $f^{\prime}$ and therefore the
monotony of $f$. This ends the proof of corollary 10.
∎
As a direct consequence of lemma 9, we get the following corollary.
###### Corollary 11
Assume that $b\neq 0$. Then,
$\mid\rho\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}\mid$ is a
decreasing function of $\frac{a^{2}}{b^{2}}$.
Thus, to model some phenomena, we can choose the parameters $H,a$ and $b$ in
such a manner that $\\{S_{t}^{H}(a,b),\,t\geq 0\\}$ yields a good model,
taking the sign and the level of correlation of the phenomenon of interest
into account. For example, let us assume that the parameters $H$ and $a$ are
known with $H>1/2$, and $b\neq 0$ is not known. Combining corollary 10 with
corollary 11, we obtain that the correlation of the increments of $S_{H}$
increases with $\mid b\mid$.
### 2.3 Some comparisons between mfBm and smfBm
Set for any $\displaystyle s,t>0$
$R_{H}(s,t)=Cov\Big{(}M_{t}^{H}(a,b),M_{s}^{H}(a,b)\Big{)}\hskip
5.69054pt\mathrm{and}\hskip
5.69054ptC_{H}(s,t)=Cov\Big{(}S_{t}^{H}(a,b),S_{s}^{H}(a,b)\Big{)}.$
Let us compare $R_{H}$ and $C_{H}$.
###### Lemma 12
* •
$\displaystyle C_{H}(s,t)\geq 0.$
* •
If $\displaystyle H>\frac{1}{2},$ $\displaystyle C_{H}(s,t)<R_{H}(s,t)$.
* •
If $\displaystyle H=\frac{1}{2},$ $\displaystyle C_{1/2}(s,t)=R_{1/2}(s,t)$.
* •
If $\displaystyle H<\frac{1}{2},$ $\displaystyle C_{H}(s,t)>R_{H}(s,t)$.
###### Proof.
Let us show the first assertion. We have by equality (2.4)
$\frac{1}{2}\Bigg{(}-2^{2H}(t^{2H}+s^{2H})+2\,(t+s)^{2H}+2\mid
t-s\mid^{2H}\Bigg{)}=E\Big{(}S_{t}^{H}(0,1)-S_{s}^{H}(0,1)\Big{)}^{2}\geq 0.$
Thus, we get for every $0<s^{{}^{\prime}}<t^{{}^{\prime}}$
$2\,(t^{{}^{\prime}}+s^{{}^{\prime}})^{2H}+2\,(t^{{}^{\prime}}-s^{{}^{\prime}})^{2H}\geq
2^{2H}(t^{{}^{\prime}2H}+s^{{}^{\prime}2H}).$
By applying this inequality with $t^{{}^{\prime}}=t+s$ and
$s^{{}^{\prime}}=t-s$, we obtain
$2\,(t^{2H}+s^{2H})\geq(t+s)^{2H}+(t-s)^{2H}.$
This implies by equality (2.1) that $\displaystyle C_{H}(s,t)\geq 0.$
For the next three assertions, we observe that, by using the expressions of
$C_{H}$ and $R_{H}$,
$C_{H}(s,t)-R_{H}(s,t)=\frac{b^{2}}{2}\;\Big{(}t^{2H}+s^{2H}-(s+t)^{2H}\Big{)}.$
When $H=\frac{1}{2},C_{1/2}=R_{1/2}$. When $H\neq\frac{1}{2}$, set
$u=\frac{s}{t},\;0\leq u\leq 1$. We get
$C_{H}(s,t)-R_{H}(s,t)=\frac{b^{2}}{2}\;t^{2H}\;g(u),$
where $g(u)=1+u^{2H}-(1+u)^{2H}$.
The study of the function $g$ completes the proof of the lemma.
∎
Let us turn to the expressions of the covariances of the mfBm and the smfBm
increments on non-overlapping intervals. To this aim, denote for $0\leq
u<v\leq s<t,$
$R_{u,v,s,t}=Cov\Big{(}M_{v}^{H}(a,b)-M_{u}^{H}(a,b),M_{t}^{H}(a,b)-M_{s}^{H}(a,b)\Big{)}$
and
$C_{u,v,s,t}=Cov\Big{(}S_{v}^{H}(a,b)-S_{u}^{H}(a,b),S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}.$
We deduce easily from lemma 1 and lemma 3 the following result.
###### Lemma 13
We have
(2.13)
$R_{u,v,s,t}=\frac{b^{2}}{2}\;\Big{(}(t-u)^{2H}+(s-v)^{2H}-(t-v)^{2H}-(s-u)^{2H}\Big{)},$
(2.14) $\begin{array}[]{rcl}\displaystyle
C_{u,v,s,t}&=&\displaystyle\frac{b^{2}}{2}\;\Big{(}(t+u)^{2H}+(t-u)^{2H}+(s+v)^{2H}+(s-v)^{2H}\\\
\vskip
5.69054pt\cr&-&\displaystyle(t+v)^{2H}-(t-v)^{2H}-(s+u)^{2H}-(s-u)^{2H}\Big{)}.\end{array}$
Let us show that the covariances of the mfBm and the smfBm increments on non-
overlapping intervals have the same sign but, those of the smfBm are smaller
in absolute value than those of the mfBm.
###### Corollary 14
We have for $0\leq u<v\leq s<t,$ that $R_{u,v,s,t}$ and $C_{u,v,s,t}$ are
strictly positive (respectively strictly negative) for $H>1/2$ (respectively
$H<1/2$). Moreover, $C_{u,v,s,t}<R_{u,v,s,t}$ (respectively $>$).
###### Proof.
First, we have $0\leq u<v\leq s<t$
$R_{u,v,s,t}=\frac{b^{2}}{2}\;\Bigl{(}g_{1}(v)-g_{1}(u)\Bigr{)},$
where $g_{1}(x)=(s-x)^{2H}-(t-x)^{2H},\;u\leq x\leq v$.
We have
$g_{1}^{{}^{\prime}}(x)=2H\;\Bigl{(}-(s-x)^{2H-1}+(t-x)^{2H-1}\Bigr{)}.$
When $H<1/2,g_{1}^{{}^{\prime}}\leq 0$. Then $g_{1}$ decreases and therefore
$R_{u,v,s,t}\leq 0$. When $H>1/2,g_{1}^{{}^{\prime}}\geq 0$. Then $g_{1}$
increases and therefore $R_{u,v,s,t}\geq 0$.
Next we have for $0\leq\ u<v\leq s<t$
$C_{u,v,s,t}=\frac{b^{2}}{2}\Bigl{(}g_{2}(t)-g_{2}(s)\Bigr{)}$
where $g_{2}(x)=-(x+v)^{2H}-(x-v)^{2H}+(x+u)^{2H}+(x-u)^{2H},\;s\leq x\leq t$.
We have
$g_{2}^{{}^{\prime}}(x)=2H\;\Bigl{(}g_{3}(u)-g_{3}(v)\Bigr{)},$
where $g_{3}(y)=(x+y)^{2H-1}+(x-y)^{2H-1},\;u\leq y\leq v$.
We have
$g_{3}{{}^{\prime}}(y)=(2H-1)\;\Bigl{(}(x+y)^{2H-2}-(x-y)^{2H-2}\Bigr{)}.$
When $H<1/2,g_{3}^{{}^{\prime}}>0$. Since $g_{3}$ increases,
$g_{2}^{{}^{\prime}}<0$ and therefore $g_{2}$ decreases. Thus $C_{u,v,s,t}\leq
0$. When $H>1/2,g_{3}^{{}^{\prime}}<0$. Since $g_{3}$ decreases,
$g_{2}^{{}^{\prime}}>0$ and therefore $g_{2}$ increases. Thus $C_{u,v,s,t}\geq
0$.
Finally let us denote by $D_{(}u,v,s,t)$ the quantity defined as follows
(2.15) $\begin{array}[]{crl}D_{u,v,s,t}&=&C_{u,v,s,t}-R_{u,v,s,t}\\\
&=&\frac{b^{2}}{2}\;\Bigl{(}(t+u)^{2H}-(t+v)^{2H}+(s+v)^{2H}-(s+u)^{2H}\Bigr{)}\\\
&=&\frac{b^{2}}{2}\;\Bigl{(}g_{4}(t)-g_{4}(s)\Bigr{)},\end{array}$
where $g_{4}(x)=(x+u)^{2H}-(x+v)^{2H},\;s\leq x\leq t.$
Let us remark that, when $H>1/2,\;g_{4}$ decreases, and when $H<1/2,\;g_{4}$
increases. This ends the proof of the lemma.
∎
###### Corollary 15
We have
* •
$\displaystyle\lim_{s,t\rightarrow+\infty}R_{u,v,s,t}=0$ if and only if
$0<H\leq\frac{1}{2}$.
* •
For every $\displaystyle 0<H<1$,
$\displaystyle\lim_{s,t\rightarrow+\infty}C_{u,v,s,t}=0$.
###### Proof.
Combining (2.13) with Taylor expansions, we have as $s,t\rightarrow+\infty$,
$R_{u,v,s,t}=b^{2}\;H\;(v-u)\;\Bigl{(}\frac{1}{t^{1-2H}}-\frac{1}{s^{1-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{t^{1-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{s^{1-2H}}\Bigr{)},$
which proves the first assertion of the corollary.
Let us turn to $C_{u,v,s,t}$. Combining (2.14) with Taylor expansions, we have
as $s,t\rightarrow~{}+\infty$,
$C_{u,v,s,t}=b^{2}\;H\;(2H-1)\;(v^{2}-u^{2})\;\Bigl{(}\frac{1}{s^{2-2H}}-\frac{1}{t^{2-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{s^{2-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{t^{2-2H}}\Bigr{)},$
which completes the proof of corollary 15.
∎
In the next lemma, we will show that the increments of the smfBm on intervals
$[u,u+r]$ and $[u+r,u+2r]$ are more weakly correlated than those of the mfBm.
###### Lemma 16
Assume $H\neq 1/2$. We have for $u\geq 0$ and $r>0$,
(2.16)
$\Big{|}\rho\Bigl{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Bigr{)}\Big{|}\leq\Big{|}\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}\Big{|}.$
###### Proof.
Combining the definition of $R_{u,v,s,t}$ with (2.13), we get
(2.17)
$\begin{array}[]{crl}\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}&=&\frac{R_{u,u+r,u+r,u+2r}}{\sqrt{V(M_{u+r}^{H}-M_{u}^{H})\;V(M_{u+2r}^{H}-M_{u+r}^{H})}}\\\
&&\\\
&=&\frac{b^{2}(2^{2H-1}-1)r^{2H}}{\sqrt{V(M_{u+r}^{H}-M_{u}^{H})\;V(M_{u+2r}^{H}-M_{u+r}^{H}\Bigr{)}}}.\end{array}$
Moreover, we get by lemma 1
(2.18)
$V(M_{u+r}^{H}-M_{u}^{H})=V(M_{u+2r}^{H}-M_{u+r}^{H})=V(M_{r}^{H})=a^{2}\;r+b^{2}\;r^{2H}.$
Then, combining (2.17) with (2.18), we have
(2.19)
$\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}=\frac{b^{2}(2^{2H-1}-1)r^{2H}}{a^{2}\;r+b^{2}\;r^{2H}}.$
Let us turn to
$\rho\Bigl{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Bigr{)}$. We have
(2.20)
$\rho\Big{(}S_{u+2r}^{H}-S_{u+r}^{H},S_{u+r}^{H}-S_{u}^{H}\Big{)}=\frac{C_{u,u+r,u+r,u+2r}}{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}.$
Let us consider the two following cases.
Case 1. $H<\frac{1}{2}$
By using (2.19) and (2.20), we can rewrite inequality (2.16) as follows:
(2.21)
$\Big{|}\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\Big{|}\leq\frac{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}{a^{2}\;r+b^{2}\;r^{2H}}.$
Note that by corollary 14 and equality (2.15)
$\Big{|}\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\Big{|}=\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}=1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}.$
Then, (2.21) can be rewritten as follows
(2.22) $0\leq
1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\leq\frac{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}{a^{2}\;r+b^{2}\;r^{2H}}.$
The second part of proposition 6 implies that
$a^{2}r+b^{2}r^{2H}\leq\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})\;V(S_{u+r}^{H}-S_{u}^{H})}.$
Then, to prove (2.22), it suffices to show that
$1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\leq 1.$
By corollary 14, $R_{u,u+r,u+r,u+2r}<0$ and $D_{u,u+r,u+r,u+2r}>0$. The proof
of case 1 is complete.
Case 2. $H>\frac{1}{2}$
Combining (2.19) with (2.20), we get
$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}=\frac{C_{u,u+r,u+r,u+2r}}{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}\;\;\frac{a^{2}\;r+b^{2}\;r^{2H}}{b^{2}(2^{2H-1}-1)r^{2H}}.$
Recall that we have precise expressions of $V(S_{u+r}^{H}-S_{u}^{H})$ and of
$V(S_{u+2r}^{H}-S_{u+r}^{H})$ by the first part of proposition 6 and of
$C_{u,v,s,t}$ by equality (2.14).
Set $x=\frac{2u}{r}$ and denote by $A,B$ and $C$ the functions defined as
follows :
$A(x)=2(x+2)^{2H}+(2^{2H}-2)-(x+3)^{2H}-(x+1)^{2H},$
$B(x)=2-x^{2H}-(x+2)^{2H}+2(x+1)^{2H}$
and $C(x)=2-(x+2)^{2H}-(x+4)^{2H}+2(x+3)^{2H}.$
Easy computations yield
$C_{u,u+r,u+r,u+2r}=\frac{b^{2}}{2}\;r^{2H}\;A(x)$
$V(S_{u+r}^{H}-S_{u}^{H})=\frac{1}{2}\;\Bigl{(}2a^{2}r+b^{2}r^{2H}B(x)\Bigr{)}$
$V(S_{u+2r}^{H}-S_{u+r}^{H})=\frac{1}{2}\;\Bigl{(}2a^{2}r+b^{2}r^{2H}C(x)\Bigr{)}$
Then, we have
$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}=\frac{A(x)\;(a^{2}\;r+b^{2}\;r^{2H})}{(2^{2H-1}-1)\;\sqrt{(2a^{2}r+b^{2}r^{2H}B(x))(2a^{2}r+b^{2}r^{2H}C(x))}}$
$=\frac{A(x)}{(2^{2H-1}-1)\;\sqrt{B(x)\,C(x)}}\;\frac{a^{2}\;r+b^{2}\;r^{2H}}{\sqrt{(2a^{2}r/B(x)+b^{2}r^{2H})(2a^{2}r/C(x)+b^{2}r^{2H})}}.$
Since it has been proved in [[, see]p. 412]TB, that
$\frac{A(x)}{(2^{2H-1}-1)\;\sqrt{B(x)\,C(x)}}\leq 1,$
we get
$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}\leq\frac{a^{2}\;r+b^{2}\;r^{2H}}{\sqrt{(2a^{2}r/B(x)+b^{2}r^{2H})(2a^{2}r/C(x)+b^{2}r^{2H})}}.$
Therefore it suffices to show
$a^{2}\;r+b^{2}\;r^{2H}\leq
2a^{2}r/B(x)+b^{2}r^{2H}\;\;\;\mbox{and}\;\;\;a^{2}\;r+b^{2}\;r^{2H}\leq
2a^{2}r/C(x)+b^{2}r^{2H},$
that is
$0<B(x)\leq 2\;\;\;\mbox{and}\;\;\;0<C(x)\leq 2.$
Let us show the first double inequality. Since by lemma 2
$b^{2}\;r^{2H}\;B(x)=2\;V\Bigl{(}b(\xi^{H}(u+r)-\xi^{H}(u))\Bigr{)},$
$B(x)>0$. Moreover, since the function $x\rightarrow x^{2H}$ is convex for
$H>1/2,\;B(x)\leq 2$. Similarly, we can establish $0<C(x)\leq 2$.
The proof of the lemma is complete.
∎
In [12], it was proved that the increments of the mfBm $(M_{t}^{H}(a,b))$ are
short-range dependent if, and only if $H<\frac{1}{2}$. To end this subsection,
let us show that for every $\displaystyle H\in]0,1[$, the increments of
$\displaystyle(S^{H}_{t}(a,b))_{t\in{{\mathbb{R}}_{+}}}$ are short-range
dependent. For convenience, let us introduce the following notation
$C(p,n)=C_{p,p+1,p+n,p+n+1},$
where $p$ and $n$ are integers with $n\geq 1$.
We get by (2.14)
$C(p,n)=\frac{b^{2}}{2}\Bigg{(}(n+1)^{2H}-2n^{2H}+(n-1)^{2H}-(2p+n+2)^{2H}+2(2p+n+1)^{2H}-(2p+n)^{2H}\Bigg{)}.$
A third-order Taylor expansion enables us to state the following lemma.
###### Lemma 17
For any $0<H<1$ and $p\in{\mathbb{N}}$, we have when $n\rightarrow+\infty$
$C(p,n)=\Big{(}2(1-H)H(2H-1)(2p+1)b^{2}\Big{)}\;n^{2H-3}+o(n^{2H-3}),$
and consequently
$\sum_{n\geq 1}\mid C(p,n)\mid<+\infty.$
## 3 Semi-martingale properties
In the sequel, we assume $b\neq 0$. For any process $X$, set
$\Delta_{j}^{n}X(t)=X(jt/n)-X((j-1)t/n),\;j\in\\{1,..n\\}.$
Denote by $A_{n}$ the quantity defined as follows :
$A_{n}={\mathbb{E}}\Biggl{(}\sum_{j=1}^{n}\;\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}\Biggr{)}=\sum_{j=1}^{n}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}.$
###### Lemma 18
* •
If $H<\frac{1}{2}$, then
$\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}=+\infty$.
* •
If $H=\frac{1}{2}$, then $\;A_{n}=(a^{2}+b^{2})\;t$.
* •
If $H>\frac{1}{2}$, then
$\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}=a^{2}\;t$.
###### Proof.
Since the processes $B$ and $B_{H}$ are independent, we have
${\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}=\frac{a^{2}}{2}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}B(t)+\Delta_{j}^{n}B(-t)\Biggr{)}^{2}+\frac{b^{2}}{2}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}B_{H}(t)+\Delta_{j}^{n}B_{H}(-t)\Biggr{)}^{2}.$
Using equality (1.1), direct computations imply
${\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}=a^{2}\;\frac{t}{n}+\;b^{2}\;\frac{t^{2H}}{n^{2H}}+2^{2H}\;b^{2}\;\frac{t^{2H}}{n^{2H}}\;\Biggl{(}\Biggl{(}\frac{2j-1}{2}\Biggr{)}^{2H}-\frac{j^{2H}+(j-1)^{2H}}{2}\Biggr{)},$
and hence
$A_{n}=a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;\frac{t^{2H}}{n^{2H}}\;\sum_{j=1}^{n}\;\Biggl{(}\Biggl{(}\frac{2j-1}{2}\Biggr{)}^{2H}-\frac{j^{2H}+(j-1)^{2H}}{2}\Biggr{)}.$
Let us consider the function $f$ defined as follows :
$f(x)=\Biggl{(}\frac{2x-1}{2}\Biggr{)}^{2H}-\frac{x^{2H}+(x-1)^{2H}}{2},\;\;\;x\geq
0.$
We deduce from convexity properties that, when $H<1/2,\;f(x)>0$, when
$H=1/2,\;f(x)=0$ and when $H>1/2,\;f(x)<0$. We have also
$f^{{}^{\prime}}(x)=2\;H\;\Biggl{(}\Biggl{(}\frac{2x-1}{2}\Biggr{)}^{2H-1}-\frac{x^{2H-1}+(x-1)^{2H-1}}{2}\Biggr{)},\;\;\;x\geq
0.$
To determine $\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}$, we have to
consider the following three cases.
Case 1. $H<1/2$
Since $f^{{}^{\prime}}\leq 0$, for every $j\in\\{1,..,n\\}$,
$f(j)\geq
f(n)=\Biggl{(}\frac{2n-1}{2}\Biggr{)}^{2H}-\frac{n^{2H}+(n-1)^{2H}}{2}>0.$
When $n$ is large enough, we get
$a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;t^{2H}\;\Biggl{(}-\frac{H(2H-1)}{4n}+o\Bigl{(}\frac{1}{n}\Bigr{)}\Biggr{)}\leq
A_{n},$
and therefore, since $b\neq 0$,
$\lim_{n\rightarrow+\infty}\;A_{n}=+\infty.$
Case 2. $H=1/2$
We obviously have
$A_{n}=(a^{2}\;+\;b^{2})\;t.$
Case 3. $H>1/2$
Since $f^{{}^{\prime}}\geq 0,\;f$ increases from
$f(1)=\frac{1}{2^{2H}}-\frac{1}{2}$ to
$f(n)=\Biggl{(}\frac{2n-1}{2}\Biggr{)}^{2H}-\frac{n^{2H}+(n-1)^{2H}}{2}<0.$
When $n$ is large enough, we get
$\displaystyle
a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;t^{2H}\;n^{1-2H}\;\Biggl{(}\frac{1}{2^{2H}}-\frac{1}{2}\Biggr{)}$
$\leq A_{n}\leq
a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H+}\;b^{2}\;t^{2H}\;\Biggl{(}-\frac{H(2H-1)}{4n}+o\Bigl{(}\frac{1}{n}\Bigr{)}\Biggr{)}$
and therefore
$\lim_{n\rightarrow+\infty}\;A_{n}=a^{2}\;t.$
This completes the proof of the lemma.
∎
Let us now recall the Bichteler-Dellacherie theorem [[, see]section
VIII.4]DELL.
###### Theorem 19
Assume that a filtration $\displaystyle\mathcal{F}=({\cal F}_{t})_{0\leq t\leq
T},0<T<+\infty,$ satisfies the usual assumptions, i.e. it is right-continuous,
${\cal F}_{T}$ is complete and ${\cal F}_{0}$ contains all null sets of ${\cal
F}_{T}$. An a.s. right-continuous, $\displaystyle{\cal F}$-adapted stochastic
process $\\{X_{t},0\leq t\leq T\\}$ is a $\displaystyle{\cal F}$-semi-
martingale if and only if
$I_{X}\Bigl{(}\beta(\mathcal{F})\Bigr{)}$
is bounded in $L^{0}$, where
$\beta(\mathcal{F})=\Biggl{\\{}\sum_{j=0}^{n-1}\;f_{j}\;\mathbf{1}_{]t_{j},t_{j+1}]},\;\;n\in\mathbb{N},\;0\leq
t_{0}\leq..\leq t_{n}\leq T,\;$ $\forall\,j,\,f_{j}\hskip
2.84526pt\mathrm{is}\hskip 2.84526pt\mathcal{F}_{t_{j}}\hskip
2.84526pt\mathrm{measurable}\hskip 2.84526pt\mathrm{and}\hskip 2.84526pt\mid
f_{j}\mid\leq 1,\mathrm{with}\hskip 2.84526pt\mathrm{probability}\hskip
2.84526pt1\Biggr{\\}},$
and
$I_{X}(\theta)=\sum_{j=0}^{n-1}\;f_{j}\;\Bigl{(}X_{t_{j+1}}-X_{t_{j}}\Bigr{)},\;\;\theta\in\beta(\mathcal{F}).$
Following the same lines as those of [3], we introduce two definitions.
###### Definition 20
A stochastic process $\\{X_{t},0\leq t\leq T\\}$ is a weak semi-martingale
with respect to a filtration $\displaystyle\mathcal{F}=({\cal F}_{t})_{0\leq
t\leq T}$ if $X$ is $\displaystyle{\cal F}$-adapted and $\displaystyle
I_{X}\Bigl{(}\beta(\mathcal{F})\Bigr{)}$ is bounded in $L^{0}$.
We insist on the fact that if a process $X$ is not a weak semi-martingale with
respect to its own filtration, then it is not a weak semi-martingale with
respect to any other filtration.
###### Definition 21
Let $\\{X_{t},0\leq t\leq T\\}$ be a stochastic process. We call $X$ a weak
semi-martingale if it is a weak semi-martingale with respect to its own
filtration $\displaystyle\mathcal{F}^{X}=({\cal F}_{t}^{X})_{0\leq t\leq T}$.
We call $X$ a semi-martingale if it is a semi-martingale with respect to the
smallest filtration that contains $\displaystyle\mathcal{F}^{X}$ and satisfies
the usual assumptions.
Let us determine now the values of $H$ for which the smfBm is not a semi-
martingale.
###### Corollary 22
If $\;0<H<\frac{1}{2}$, then the smfBm $S^{H}(a,b)$ is not a weak semi-
martingale.
###### Proof.
A direct consequence of lemma 18 is that since $0<H<\frac{1}{2}$ and $b\neq
0$, the quadratic variation of the smfBm is infinity. To complete the proof of
the corollary, it suffices to apply proposition 2.2 of [3, pp. 918-919].
∎
The study of the case $H>3/4$ is based on a result of [1, p. 348]. We insist
on the fact that this method is different from the one which was used in [3].
###### Proposition 23
For every $T>0,\;H\in]\frac{3}{4},1[$, and $\;a\neq 0$, the smfBm
$S^{H}(a,b)=\\{S_{t}^{H}(a,b),t\in[0,T]\\}$
is a semi-martingale equivalent in law to $a\times B_{t}$, where
$\\{B_{t},t\in[0,T]\\}$ is a Bm.
###### Proof.
The smfBm $S^{H}$ can be rewritten as follows
$\forall\;t\in{\mathbb{R}}^{+}\;\;\;S_{t}^{H}(a,b)=a\Big{(}\xi_{t}+\frac{b}{a}\xi_{t}^{H}\Big{)}$
where $\xi$ and $\xi^{H}$ have been introduced by equation (1.3). Recall that
the processes $\xi$ and $\xi^{H}$ are independent.
The covariance function of the Gaussian process
$\displaystyle\frac{b}{a}\;\xi^{H}$
$R(s,t)=\frac{b^{2}}{a^{2}}\Bigg{(}t^{2H}+s^{2H}-\frac{1}{2}\Big{(}(s+t)^{2H}+\mid
t-s\mid^{2H}\Big{)}\Bigg{)},$
is twice continuously differentiable on
$\displaystyle[0,T]^{2}\setminus\\{(s,t);t=s\\}$.
According to [1, p. 348], it suffices to verify
$\displaystyle\frac{\partial^{2}R}{\partial s\partial t}\in L^{2}([0,T]^{2})$,
in order to show that the process
$\\{\xi_{t}+\frac{b}{a}\xi_{t}^{H},t\in[0,T]\\}$
is a semi-martingale equivalent in law to a Bm.
We have for any $\displaystyle(s,t)\in[0,T]^{2}\setminus\\{(s,t);t=s\\}$
$\frac{\partial^{2}R(s,t)}{\partial s\partial
t}=\frac{b^{2}}{a^{2}}\;H(2H-1)\;\Big{(}\mid
t-s\mid^{2H-2}-(s+t)^{2H-2}\Big{)}.$
It is easy to check that if $\displaystyle H>\frac{3}{4},$ then
$\displaystyle\frac{\partial^{2}R}{\partial s\partial t}\in L^{2}([0,T]^{2})$.
This completes the proof of the proposition.
∎
To study the case $H\in]1/2,3/4]$, we follow the same lines as those of [3].
But many technical results have to be proved. Let us first recall the
definition of a quasi-martingale.
###### Definition 24
A stochastic process $\\{X_{t},0\leq t\leq T\\}$ is a quasi-martingale if
$\displaystyle X_{t}\in L^{1}$ for all $t\in[0,T],$ and
$\sup_{\tau}\sum_{j=0}^{n-1}\left\|{\mathbb{E}}\Big{(}X_{t_{j+1}}-X_{t_{j}}|{\cal
F}_{t_{j}}^{X}\Big{)}\right\|_{1}<+\;\infty,$
where $\tau$ is the set of all finite partitions
$0=t_{0}<t_{1}<...<t_{n}=T\hskip 2.84526ptof\hskip 2.84526pt[0,T].$
In the following key lemma, we will specify the relation between quasi-
martingale and weak semi-martingale in the case of our process $S^{H}$.
###### Lemma 25
If $S^{H}$ is not a quasi-martingale, then it is not a weak semi-martingale.
###### Proof.
Let us assume that $S^{H}$ is a weak semi-martingale. Then, by theorem 1 of
[10], we have
$I_{S^{H}}\Bigl{(}\beta(\mathcal{F}^{S^{H}})\Bigr{)},$
which was defined in theorem 19, is bounded in $L^{2}$, and therefore in
$L^{1}$.
But, for any partition $0=t_{0}<t_{1}<..<t_{n}=T$,
$\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\mathbf{1}_{]t_{j},t_{j+1}]}\in\beta(\mathcal{F}^{S^{H}}),$
and
$\begin{array}[]{rcl}&&\displaystyle\Bigg{|}\Bigg{|}I_{S^{H}}\Biggl{(}\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\mathbf{1}_{]t_{j},t_{j+1}]}\Biggr{)}\Bigg{|}\Bigg{|}_{1}\\\
\vskip
8.53581pt\cr&=&\displaystyle\Bigg{|}\Bigg{|}\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\Bigg{|}_{1}\\\
\vskip
8.53581pt\cr&\geq&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}{\mathbb{E}}\Biggl{(}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}{\mathbb{E}}\Biggl{(}\;\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;\Bigg{|}\Bigg{|}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Bigg{|}\Bigg{|}_{1}.\end{array}$
Then, $S^{H}$ is a quasi-martingale. The proof of the lemma is complete.
∎
The following lemmas deal with the two last cases $1/2<H<3/4$ and $H=3/4$.
###### Proposition 26
If $\displaystyle H\in\Big{]}\frac{1}{2},\frac{3}{4}\Big{[}$, then the smfBm
$S^{H}(a,b)$ is not a quasi-martingale.
###### Proof.
For $n\in{\mathbb{N}}$ and $j\in\\{1,2,...,n\\}$, let us denote
$\Delta_{j}^{n}S^{H}=S^{H}_{\frac{Tj}{n}}-S^{H}_{\frac{T(j-1)}{n}}.$
Since conditional expectation is a contraction with respect to the $L^{1}-$
norm, we have for all $n\in{\mathbb{N}}$ and all $j=1,...,n-1$,
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{1}\leq\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\mathcal{F}_{\frac{Tj}{n}}^{S^{H}}\Big{)}\right\|_{1}.$
Moreover, since
$\displaystyle{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}$
is a centered Gaussian random variable,
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{1}=\sqrt{\frac{2}{\pi}}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{2}.$
Consequently,
(3.1)
$\begin{array}[]{rcl}\displaystyle\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\mathcal{F}_{\frac{Tj}{n}}^{S^{H}}\Big{)}\right\|_{1}&\geq&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|\frac{Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}{Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}\Delta_{j}^{n}S^{H}\right\|_{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\frac{Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}{\sqrt{Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}}:=\displaystyle\sqrt{\frac{2}{\pi}}\;I_{n}.\end{array}$
We have by lemma 13,
$\begin{array}[]{rcl}\displaystyle
Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}&=&\displaystyle
C_{\frac{T(j-1)}{n},\frac{Tj}{n},\frac{Tj}{n},\frac{T(j+1)}{n}}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{b^{2}\;T^{2H}}{2\;n^{2H}}\;\Big{(}2^{2H}(2j^{2H}+1)-2-(2j+1)^{2H}-(2j-1)^{2H}\Big{)}.\end{array}$
Combining proposition 6 with the fact that $2H>1$, we get
$\begin{array}[]{rcl}&&\displaystyle
Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}\\\ \vskip
5.69054pt\cr&=&\displaystyle
a^{2}\frac{T}{n}+\frac{b^{2}\;T^{2H}}{n^{2H}}\Bigg{(}-2^{2H-1}(j^{2H}+(j-1)^{2H})+(2j-1)^{2H}+1\Bigg{)}\\\
\vskip
5.69054pt\cr&\leq&\displaystyle\displaystyle\frac{1}{n}\Bigg{(}a^{2}\;T+b^{2}\;T^{2H}\Big{(}-2^{2H-1}(j^{2H}+(j-1)^{2H})+(2j-1)^{2H}+1\Big{)}\Bigg{)}.\\\
&&\end{array}$
Then,
$I_{n}\geq\frac{b^{2}\;T^{2H}}{2\;n^{2H-\frac{1}{2}}}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}=\frac{b^{2}\;T^{2H}}{2\;n^{2H-\frac{3}{2}}}\times\Biggl{(}\frac{1}{n}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}\Biggr{)},$
where we have for any $n\in{\mathbb{N}}^{*}$,
$u_{n}=2^{2H}(2n^{2H}+1)-2-(2n+1)^{2H}-(2n-1)^{2H}$
and
$v_{n}=\sqrt{a^{2}\;T+b^{2}\;T^{2H}\Big{(}-2^{2H-1}(n^{2H}+(n-1)^{2H})+(2n-1)^{2H}+1\Big{)}}.$
Since
$\lim_{n\rightarrow+\infty}\frac{u_{n}}{v_{n}}=\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}},$
we have by Césaro theorem that
$\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}=\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}}.$
Hence, since $\displaystyle\frac{1}{2}<H<\frac{3}{4}$ and
$\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}}>0$, we have
$\displaystyle\lim_{n\rightarrow\infty}I_{n}=+\infty$. Then, we get by using
(3.1) that
$\sup_{\tau}\sum_{j=0}^{n-1}\left\|{\mathbb{E}}\Big{(}S^{H}_{t_{j+1}}-S^{H}_{t_{j}}|\mathcal{F}_{t_{j}}^{S^{H}}\Big{)}\right\|_{1}=+\;\infty.$
This completes the proof of the lemma.
∎
###### Proposition 27
The smfBm $\displaystyle S^{\frac{3}{4}}(a,b)$ is not a quasi-martingale.
###### Proof.
Since conditional expectation is a contraction with respect to the $L^{1}-$
norm, we have for all $n\in{\mathbb{N}}$ and all $j=1,...,n-1$,
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{1}\leq\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\mathcal{F}_{\frac{Tj}{n}}^{S^{\frac{3}{4}}}\Big{)}\right\|_{1}.$
Moreover, since
$\displaystyle{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}$
is a centered Gaussian random variable,
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{1}=\sqrt{\frac{2}{\pi}}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}.$
Consequently,
$\begin{array}[]{rcl}\displaystyle\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\mathcal{F}_{\frac{j}{n}}^{S^{\frac{3}{4}}}\Big{)}\right\|_{1}&\geq&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2},\end{array}$
and the lemma is proved if we show that
(3.2)
$\lim_{n\rightarrow\infty}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}=+\;\infty.$
For $n\in{\mathbb{N}}$ and $j=1,...,n-1$,
$\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}$
is a Gaussian vector. Therefore,
(3.3)
${\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}=\sum_{k=1}^{j}\;b_{k}\;\Delta_{k}^{n}\;S^{\frac{3}{4}},$
where the vector $\displaystyle b=\left(\begin{array}[]{c}b_{1}\\\ \vdots\\\
b_{j}\end{array}\right)$ solves the system of linear equations
(3.4) $m=Ab,$
in which $m$ is a $j-$vector whose $k-th$ component $m_{k}$ is
$Cov\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{k}^{n}S^{\frac{3}{4}}\Big{)}$
and $A$ is the covariance matrix of the Gaussian vector
$\Big{(}\Delta_{1}^{n}S^{\frac{3}{4}},...,\Delta_{j}^{n}S^{\frac{3}{4}}\Big{)}.$
Note that $A$ is symmetric and, since the random variables
$\Delta_{1}^{n}S^{\frac{3}{4}},...,\Delta_{j}^{n}S^{\frac{3}{4}}$
are lineary independent, $A$ is also positive definite. It follows from (3.3)
and (3.4) that
(3.5)
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}^{2}=b^{T}Ab=m^{T}A^{-1}m\geq\left\|m\right\|_{2}^{2}\lambda^{-1},$
where $\lambda$ is the largest eigenvalue of the matrix $A$. Set $I$ the
identity matrix and $C=(C_{i,k})_{1\leq i,k\leq j}$ the covariance matrix of
the increments of the sfBm with index $3/4$. We have
$A=\frac{a^{2}T}{n}\;I+b^{2}\;C,$
and consequently
(3.6) $\lambda=\frac{a^{2}T}{n}+b^{2}\;\mu,$
where $\mu$ is the largest eigenvalue of the matrix $C$. We deduce also from
lemma 13
$\begin{array}[]{rcl}\displaystyle
C_{ik}&=&\displaystyle\frac{T^{3/2}}{2\;n^{3/2}}\;\Biggl{(}\Bigl{(}\mid
k-i\mid+1\Bigr{)}^{3/2}-2\mid k-i\mid^{3/2}+\mid\mid k-i\mid-1\mid^{3/2}\\\
\vskip
8.53581pt\cr&&\displaystyle+2\;(k+i-1)^{3/2}-(k+i)^{3/2}-(k+i-2)^{3/2}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{2\;n^{3/2}}\;(E_{ik}+F_{ik}),\end{array}$
where
$E_{ik}=2\Biggl{(}\frac{\Bigl{(}\mid k-i\mid+1\Bigr{)}^{3/2}+\mid\mid
k-i\mid-1\mid^{3/2}}{2}-\mid k-i\mid^{3/2}\Biggr{)}$
and
$F_{ik}=2\Biggl{(}(k+i-1)^{3/2}-\frac{(k+i)^{3/2}+(k+i-2)^{3/2}}{2}\Biggr{)}.$
Note that the convexity of the function $\displaystyle x\rightarrow
x^{3/2},\;x\geq 0$, implies that $\displaystyle E_{ik}\geq 0$ and
$\displaystyle F_{ik}\leq 0$. Moreover, since $H=3/4>1/2$, corollary 14 yields
$\;C_{ik}\geq 0$.
So, using the Gershgorin circle theorem [7] we obtain
$\mu\leq\max_{k=1,..,j}\;\sum_{k=1}^{j}\;\mid
C_{ik}\mid\leq\frac{T^{3/2}}{2\;n^{3/2}}\max_{k=1,..,j}\;\sum_{k=1}^{j}E_{ik},$
and consequently
$\begin{array}[]{rcl}\displaystyle\mu&\leq&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k=1}^{j}\;\Biggl{(}2\Biggl{(}\frac{\Bigl{(}\mid
k-1\mid+1\Bigr{)}^{3/2}+\mid\mid k-1\mid-1\mid^{3/2}}{2}-\mid
k-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k=1}^{j}\;\Biggl{(}\Bigl{(}\mid
k-1\mid+1\Bigr{)}^{3/2}+\mid\mid k-1\mid-1\mid^{3/2}-2\mid
k-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k^{{}^{\prime}}=0}^{j-1}\;\Biggl{(}(k^{{}^{\prime}}+1)^{3/2}-2\;k^{{}^{\prime}3/2}+\mid
k^{{}^{\prime}}-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\Bigl{(}1+j^{3/2}-(j-1)^{3/2}\Bigr{)}\\\
\vskip 8.53581pt\cr&\leq&\displaystyle
T^{3/2}\Bigg{(}\frac{1}{n^{3/2}}+\frac{1}{n^{3/2}}\;\max_{j-1\leq x\leq
j}\;\frac{d(x^{3/2})}{dx}\Bigg{)}\\\ \vskip
8.53581pt\cr&\leq&\displaystyle\frac{5}{2\;n}\;T^{3/2}.\end{array}$
Hence combining equality (3.6) with the above result, we obtain
(3.7) $\lambda^{-1}\geq\alpha\,n,$
where $\displaystyle\alpha=\frac{2}{T(2a^{2}+5b^{2}T^{1/2})}.$
Next, let us determine a suitable lower bound of $\|m\|_{2}^{2}$. From the
lemma 13 we have
(3.8)
$\begin{array}[]{rcl}\displaystyle\|m\|_{2}^{2}&=&\displaystyle\sum_{k=1}^{j}\;\Bigl{(}Cov\Bigl{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{k}^{n}S^{\frac{3}{4}}\Bigr{)}\Bigr{)}^{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3}b^{4}}{4n^{3}}\sum_{k=1}^{j}\;\Bigl{(}f_{1}(k)-f_{2}(k)\Bigr{)}^{2},\end{array}$
where
$f_{1}(k)=(j-k+2)^{3/2}-2(j-k+1)^{3/2}+(j-k)^{3/2}$
and
$f_{2}(k)=(j+k+1)^{3/2}-2(j+k)^{3/2}+(j+k-1)^{3/2}.$
The functions $f_{1}$ and $f_{2}$ satisfy three properties, which we shall use
at the end of the proof. We will state them in the following technical lemma.
###### Lemma 28
For any $k\in\\{1,..,j\\}$
* •
$f_{1}(k)\geq 0$ and $f_{2}(k)\geq 0$,
* •
$f_{1}(k)-f_{2}(k)>0$,
* •
(3.9)
$f_{1}(k)-f_{2}(k)\geq\frac{3}{4}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}\geq
0.$
###### Proof.
(of lemma 28) The first assertion of the lemma is due to the fact that the
function $\displaystyle x\longmapsto x^{3/2}$ is convex on the interval
$\displaystyle[0,+\infty[$.
Now, let us prove the second assertion of the lemma. Consider the function $g$
defined by
$g(x)=(x+1)^{3/2}-2\;x^{3/2}+(x-1)^{3/2},\;\;x\geq 1.$
Since the function $\displaystyle x\longmapsto x^{1/2}$ is concave on
$\displaystyle[1,+\infty[$, $g$ decreases on this interval and consequently
$f_{1}(k)=g(j-k+1)>g(j+k)=f_{2}(k).$
Finally, let us prove inequality (3.9). For every $a\geq 1$, let us consider
the function $g_{a}$ defined by
$g_{a}(x)=(a+x)^{3/2}-2\;a^{3/2}+(a-x)^{3/2},\;\;0\leq x\leq 1\leq a.$
We have
$\displaystyle\;g_{a}(0)=0,g_{a}^{{}^{\prime}}(x)=\frac{3}{2}\;((a+x)^{1/2}-(a-x)^{1/2})$
and therefore $g_{a}^{{}^{\prime}}(0)=0$.
On the otherhand, by Taylor-Lagrange theorem, we get that there exists
$c\in]0,1[$ such that
$g_{a}(1)=g_{a}(0)+g_{a}^{{}^{\prime}}(0)+\frac{1}{2}g_{a}^{"}(c)=\frac{1}{2}g_{a}^{"}(c),$
where
$g_{a}^{"}(x)=\frac{3}{4}\;\Bigl{(}(a+x)^{-1/2}+(a-x)^{-1/2}\Bigr{)}.$
Next, it is easy to check that the function $g_{a}^{"}$ increases, and
consequently
$g_{a}^{"}(0)\leq g_{a}^{"}(c)\leq g_{a}^{"}(1).$
So, we have
$\frac{3}{4\;a^{1/2}}\leq g_{a}(1)\leq\frac{3}{4\;(a-1)^{1/2}},$
and therefore
$f_{1}(k)=g_{j-k+1}(1)\geq\frac{3}{4}\,(j-k+1)^{-1/2}\;\;\;\mbox{and}\;\;\;f_{2}(k)=g_{j+k}(1)\leq\frac{3}{4}\,(j+k-1)^{-1/2},$
which ends the proof of the lemma.
∎
Let us turn back to the proof of proposition 27. Combining (3.8) with (3.9),
we get
(3.10)
$\|m\|_{2}^{2}\geq\frac{9\,b^{4}\,T^{3}}{64\,n^{3}}\sum_{k=2}^{j}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}^{2}.$
For every integer $j\geq 1$, let us consider the function
$f_{j}(x)=(j-x+1)^{-1/2}-(j+x-1)^{-1/2},\;1\leq x\leq j.$
Since $f_{j}$ increases, we have
(3.11)
$\sum_{k=2}^{j}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}^{2}\geq\int_{1}^{j}\;f_{j}(x)^{2}\;dx.$
But
(3.12)
$\begin{array}[]{rcl}\displaystyle\int_{1}^{j}\;f_{j}(x)^{2}\;dx&=&\displaystyle\int_{1}^{j}\;\Biggl{(}\frac{1}{j-x+1}+\frac{1}{j+x-1}-2\frac{1}{\sqrt{j^{2}-(x-1)^{2}}}\Biggr{)}\;dx\\\
\vskip
8.53581pt\cr&=&\displaystyle\ln(2j-1)-2\;\int_{1}^{j}\;\frac{1}{\sqrt{j^{2}-(x-1)^{2}}}\;dx\\\
\vskip
8.53581pt\cr&=&\displaystyle\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi.\end{array}$
Hence, combining (3.7) with (3.10), (3.11) and (3.12), we get
(3.13)
$\|m\|_{2}^{2}\;\lambda^{-1}\geq\frac{\beta}{n^{2}}\;\Bigl{(}\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi\Bigr{)},$
where $\displaystyle\beta=\frac{\alpha}{64}(9\;T^{3}\;b^{4}).$
Combining (3.13) with (3.5), we have,
$\sum_{j=1}^{n-1}\;\|E\Bigl{(}\Delta_{j+1}^{n}S_{3/4}\mid\Delta_{j}^{n}S_{3/4},..,\Delta_{1}^{n}S_{3/4}\Bigr{)}\|_{2}$
$\geq\frac{\sqrt{\beta}}{n}\;\sum_{j=1}^{n-1}\;\sqrt{\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi}.$
Since
$\displaystyle\lim_{n\rightarrow\infty}\sqrt{\ln(2n-1)+2\arccos\Bigl{(}\frac{n-1}{n}\Bigr{)}-\pi}=+\;\infty$,
we have by Césaro theorem
$\lim_{n\rightarrow\infty}\frac{\sqrt{\beta}}{n}\;\sum_{j=1}^{n-1}\;\sqrt{\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi}=+\;\infty,$
which completes the proof of proposition 27.
∎
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* [11] C. Tudor, Some properties of the sub-fractional Brownian motion, Stochastics 79 (2007), pp. 431–448.
* [12] M. Zili, On the mixed fractional Brownian motion, J. Appl. Math. Stoch. Anal. (2006), Article ID 32435, 9 pp.
Charles EL-NOUTY
LAGA, université Paris XIII, 99 avenue J-B Clément, 93430 Villetaneuse, FRANCE
Email: elnouty@math.univ-paris13.fr
Mounir ZILI
Preparatory Institute to the Military Academies, Research unit UR04DN04,
Avenue Maréchal Tito, 4029 Sousse, TUNISIA
Email: zilimounir@yahoo.fr
|
arxiv-papers
| 2012-06-19T18:52:04 |
2024-09-04T02:49:31.968432
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Charles El-Nouty and Mounir Zili",
"submitter": "Charles El-Nouty",
"url": "https://arxiv.org/abs/1206.4291"
}
|
1206.4383
|
# Lepton Masses and Flavor Violation in Randall Sundrum Model
Abhishek M Iyer abhishek@cts.iisc.ernet.in Sudhir K Vempati
vempati@cts.iisc.ernet.in Centre for High Energy Physics, Indian Institute of
Science, Bangalore 560012
###### Abstract
Lepton masses and mixing angles via localization of 5D fields in the bulk are
revisited in the context of Randall-Sundrum models. The Higgs is assumed to be
localized on the IR brane. Three cases for neutrino masses are considered: (a)
The higher dimensional LH.LH operator (b) Dirac masses (c) Type I see-saw with
bulk Majorana mass terms. Neutrino masses and mixing as well as charged lepton
masses are fit in the first two cases using $\chi^{2}$ minimisation for the
bulk mass parameters, while varying the $\mathcal{O}(1)$ Yukawa couplings
between $0.1$ and $4$. Lepton flavour violation is studied for all the three
cases. It is shown that large negative bulk mass parameters are required for
the right handed fields to fit the data in the LH LH case. This case is
characterized by a very large Kaluza-Klein (KK) spectrum and relatively weak
flavour violating constraints at leading order. The zero modes for the charged
singlets are composite in this case and their corresponding effective 4-D
Yukawa couplings to the KK modes could be large. For the Dirac case, good fits
can be obtained for the bulk mass parameters, $c_{i}$, lying between $0$ and
$1$. However, most of the ‘best fit regions’ are ruled out from flavour
violating constraints. In the bulk Majorana terms case, we have solved the
profile equations numerically. We give example points for inverted hierarchy
and normal hierarchy of neutrino masses. Lepton flavor violating rates are
large for these points. We then discuss various minimal flavor violation (MFV)
schemes for Dirac and bulk Majorana cases. In the Dirac case with MFV
hypothesis, it is possible to simultaneously fit leptonic masses and mixing
angles and alleviate lepton flavor violating constraints for Kaluza-Klein
modes with masses of around 3 TeV. Similar examples are also provided in the
Majorana case.
###### pacs:
73.21.Hb, 73.21.La, 73.50.Bk
## I Introduction
One of the most interesting solutions of the hierarchy problem is the Randall-
Sundrum model RS which proposes a warped extra space dimension compactified
on an $S_{1}/Z_{2}$ orbifold. Two branes representing the UV and the IR scales
are located at the two end points of the orbifold. In the simplest models, the
Standard Model matter and gauge fields are localized on the IR brane along
with the Higgs field. Massive Planck scale modes are exponentially suppressed
at the IR brane, due to the warped bulk geometry, caused by the presence of a
large negative cosmological constant111 The RS metric is given by
$ds^{2}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2},$ where
$\sigma(y)=k|y|$. For recent reviews on RS models, please see gherghetta ..
Variations of this set up have been considered in several different
contexts222The phenomenology of RS models has been extensively studied. A
recent review on collider phenomenology concentrating on LHC can be found in
shrihari ..
For example, introducing gauge fields in the bulk facilitates unification of
couplings Agashe:2002pr . But this leads to large corrections to the
electroweak precision observables and places a lower bound on the mass of the
lightest gauge Kaluza-Klein (KK) mode to be around 25 TeV. This is because the
coupling of brane localized fermions to the gauge KK states is enhanced by a
factor $\sim~{}8.5$ compared to the SM coupling Davou ; Hisano ; Huber:2000fh
. A similar study in terms of oblique parameters was reported in Csaki:2002gy
; Burdman:2002gr . Boundary kinetic terms for the gauge fields can lower the
bound Davoudiasl:2002ua ; Carena:2002me , but this might spoil the
unification. Alternatively, allowing the fermions to propagate in the bulk
eases the constraint of 25 TeV on the lightest KK mode, to about 10 TeV
Huber:2001gw . Having a bulk Higgs further eases the bound Huber:2000fh . On
the other hand, scenarios with extended particle content and a bulk custodial
symmetry with a brane localized Higgs boson were found to lower the bounds on
the KK gauge boson mass to $\sim\text{3 TeV}$ Agashe:2003zs . In Hewett:2002fe
the authors explored a mixed scenario where part of fermions, the third
generation quarks are localized on the IR brane. It was shown that such a
scenario would soften the corrections to the $\rho$ parameter. Finally
modifying the RS metric near the IR boundary can also help in reduction of the
strong electroweak precision constraints Falkowski:2008fz ; Cabrer .
Allowing fermions to propagate in the bulk has interesting implications for
flavor physics. The bulk profiles of the fermion fields are determined by
their bulk masses in a manner similar to Arkani-hamed and Schmaltz mechanism
in ADD models ArkaniSch . In the RS model, however, the warped geometry
facilitates the so-called ‘automatic’ localization of fermionsHisano . The
profiles are also no longer gaussian, but are exponentially suppressed. It has
been proposed that RS could be a theory of flavour, where the fermion mass
hierarchy can be explained in terms of a few $\mathcal{O}$(1) parameters. This
is analogous to the popular Froggatt-Nielsen (FN) models FN ; Babu in four
dimensions. While in the FN model, it is the gauge and the heavy fermion
sector which determine the hierarchies in the Yukawa couplings, in the RS
case, it is the geometry of the bulk. The role of the FN charges can be played
by the five dimensional Dirac masses for the bulk fermions. The expectation is
that by taking $\mathcal{O}(1)$ bulk mass parameters as well as Yukawa
couplings, one would be able to explain the large hierarchies in the quark and
leptonic mass spectrum. While this is true in general for quarks and charged
lepton masses, as we will see subsequently, in case of neutrino masses, the
situation is a bit more involved.
Flavour violation in the hadronic sector has been explored by various authors
AgasheSoni ; Huber1 ; cedric , a recent comprehensive analysis can be found in
neubert1 ; neubert2 . In the present work, we are interested in studying
neutrino masses and mixing angles within the RS context. One method of
generating neutrino masses in the RS model would be to allow only the right
handed neutrino to propagate in the bulk, while the SM particles are confined
to the IR brane. This leads to a higher dimensional seesaw mechanism gross .
However, unlike the case of ADD models, here only the lightest KK modes
participate in the seesaw mechanism. Furthermore, lepton flavour violating
decay rates are extremely large in this case pushing the lightest KK mode to
be heavier than $m_{\text{kk}}\gtrsim 25~{}\text{TeV}$ Kitano . Neutrino mass
models have also been explored in the alternative scheme where all the
fermionic fields are allowed to propagate in the bulk. In the present work, we
will concentrate on this set up and study the neutrino mass phenomenology and
lepton flavor violation gross ; Kitano ; Huber4 ; Huber3 ; Huber2 ; Agashe ;
Fitzpatrick ; Chen ; AgasheSundrum . We have assumed Higgs to be localized on
the IR brane. Fermion mass fits in scenarios with Higgs also propagating in
the bulk have been considered in Huber1 ; Archer .
In this RS set-up (fermions in the bulk, Higgs localized on IR brane )
neutrino mass models can be divided broadly into Dirac mass models or Majorana
mass models. In the case of Majorana fermions, the number of possibilities is
more than one. In the present work we discuss three cases in detail (a) The
higher dimensional LH LH operator (b) the Dirac neutrino case and finally (c)
Majorana neutrinos with bulk seesaw terms. In these models, typically two sets
of parameters determine the charged lepton masses and neutrino masses and
mixing angles. These are the afore mentioned set of bulk Dirac masses for the
fermions and then the $\mathcal{O}(1)$ parameters containing the Yukawa
couplings. In each of these cases, we have numerically minimized a $\chi^{2}$
function containing the model parameters and the leptonic masses and mixing
data, to determine the ‘best fit’ regions of the parameter space. The Yukawa
couplings are varied from $0.1$ to $4$ whereas the ranges for the bulk
parameters are judiciously chosen to be as wide as possible.
We found that in the (a) higher dimensional LHLH operator case, the bulk mass
parameters of the charged singlets are required to be negative and extremely
large. This gets reflected into an extremely hierarchal Kaluza-Klein mass
spectrum of the first KK states of the SM fermions. In fact, the best fit
regions are those with Standard Model charged singlets being completely
composite333This interpretation is based on the AdS/CFT correspondence.. On
the other hand, if one considers Dirac neutrinos, it is quite possible to fit
the data naturally with the bulk Dirac masses within reasonable ranges without
any large hierarchies. Both hierarchal and inverse hierarchal neutrino mass
schemes can be fit in this case though it is much more difficult to find
regions which satisfy inverse hierarchal neutrino mass relations compared to
normal hierarchy. The bulk equations of motion in the presence of a Majorana
mass term are coupled and more complicated than the Dirac or LHLH case. We
have solved them numerically and given example points where data can be fit
easily either the inverted or the normal hierarchy scheme. We have not
conducted an extended numerical scan of the parameter space for the bulk
Majorana case.
Fitting neutrino masses in any of the above models in RS set up potentially
leads to large lepton flavor violation. A detailed analysis was presented in
Agashe , where the authors discussed the implications of flavor physics in the
lepton sector with both the brane localized and the bulk Higgs. Neutrinos were
assumed to be of Dirac nature. They observed that with a bulk Higgs, the
branching fraction for the process $\mu\rightarrow e\gamma$ requires a KK mass
scale of around $\sim 20$ TeV to keep it below the present experimental
limits. Similar comments were made in AgasheSundrum on how the higher
dimensional operator case is not conductive for suppressing process like
$\mu\rightarrow eee$, especially when the KK mass is low. Higgs was allowed to
propagate in the bulk in this work. In the present work, we revisited the
flavor constraints for all the three cases, concentrating on the best fit
regions in the LHLH and the Dirac case. For the LHLH case, the couplings of SM
fermions to KK gauge bosons are universal in the best fit region, leading to
no apparent constraint, at least at the leading order from the tree level
flavor violating decays. However, there are large Yukawa couplings in this
model which make it unattractive from perturbation theory point of view. The
best fit region of the Dirac case is strongly constrained from tree level
decays as well as loop induced decays like $\mu\to e+\gamma$. In the brane
localized Higgs scenario we are considering here, the limits from dipole
processes are cut-off dependent. But, for cut-off values close to the first KK
mass scale, the limits are comparably much stronger. For the bulk Majorana
case too, the points we have considered display strong constraints from
leptonic flavor violation and are ruled out. One would thus need ways to
circumvent these strong limits from lepton flavor violation.
We explored Minimal Flavour Violation (MFV) ansatz implemented in the RS
scenario to evade the flavour constraints in the Dirac and Majorana cases
Fitzpatrick ; perez . We provide example symmetry groups where the flavor
violating constraints can be removed for both the Dirac and the Majorana
cases.
The paper is organized as follows. In section (II), we discuss lepton mass
fits in three models of neutrino mass generation, the higher dimensional LHLH
operator, the Dirac case and the bulk Majorana mass terms case spread over
three subsections. In section (III), we discuss the lepton flavor violating
constraints for the three cases of neutrino masses. In section (IV) we discuss
the minimal flavor violating schemes for the Dirac and Majorana cases and show
example points where flavor violating constraints are alleviated. We close
with a summary and outlook in the final section V.
## II Lepton Mass Fits
The observed neutrino and charged lepton data is fit to the set of theory
parameters which determine the charged lepton and neutrino mass matrices
through a $\chi^{2}$ minimization. Thus the observables correspond to three
charged lepton masses, three mixing angles and two (neutrino) mass squared
differences, while, the bulk mass parameters and Yukawa couplings form the set
of theory parameters. The number of theory parameters varies from model to
model, as discussed in the following sub-sections. We have chosen the
following central values for the observables pdg ; valle :
Table 1: Experimental Data masses | mass-squared | mixing angles
---|---|---
(MeV ) | ($\text{eV}^{2}$) |
$m_{e}=0.51^{+0.0000007}_{-0.0000007}$ | $\Delta m^{2}_{12}=7.59^{+0.20}_{-0.21}\times 10^{-5}$ | $\theta_{12}=0.59^{+0.02}_{-0.015}$
$m_{\mu}=105.6^{+0.000003}_{-0.000003}$ | $\Delta m_{23}^{2}=2.43^{+0.13}_{-0.13}\times 10^{-3}$ | $\theta_{23}=0.79^{+0.12}_{-0.12}$
$m_{\tau}=1776^{+0.00016}_{-0.00016}$ | | $\theta_{13}=0.154^{+0.016}_{-0.016}$
We use the standard $\chi^{2}$ definition for N observables given by
$\chi^{2}=\sum_{i=1}^{N}\left(\frac{y_{i}^{exp}-y_{i}^{theory}}{\sigma_{i}}\right)^{2}$
(1)
where, $y_{i}^{theory}$ is the value of the $i^{th}$ observable predicted by
the model and $y_{i}^{exp}$ is its corresponding experimental number measured
with a uncertainty of $\sigma_{i}$. Since, the values of the charged lepton
are measured to a very high accuracy, it is difficult to fit masses to such
high accuracy. Thus, we incorporate up to $\sim 1.5\%$ errors in the masses of
charged leptons444This approach is very similar to fermion mass fitting in GUT
theories. See for example, GUT1 ; GUT2 .. The $\chi^{2}$ relevant to our study
is
$\displaystyle\chi^{2}=\frac{(\theta_{sol}-0.59)^{2}}{(0.02)^{2}}+\frac{(\theta_{atm}-0.79)^{2}}{(0.12)^{2}}+\frac{(\theta_{13}-0.154)^{2}}{(0.02)^{2}}+\frac{(\Delta
m^{2}_{sol}-7.59\times 10^{-23})^{2}}{(0.2\times 10^{-23})^{2}}$
$\displaystyle+\frac{(\Delta m^{2}_{atm}-2.43\times 10^{-21})^{2}}{(0.2\times
10^{-21})^{2}}+\frac{(m_{e}-0.00051)^{2}}{(0.00001)^{2}}+\frac{(m_{\mu}-0.1056)^{2}}{(0.0001)^{2}}+\frac{(m_{\tau}-1.77)^{2}}{(0.02)^{2}}$
(2)
As mentioned above, the fermion masses (and mass squared differences) and
mixing angles appearing in Eq.(2) are functions of bulk parameters. The
minimization was performed using MINUIT minuit . For a given scan, MINUIT
looks for a local minima for the $\chi^{2}$ around a certain input guess value
of the bulk masses and Yukawa parameters. This scan is repeated by randomly
varying the guess values and in the process of looking for a global minima.
### II.1 The $LHLH$ operator
In the absence of detailed specification of the mechanism which generates
neutrino masses, one can always write an effective higher dimensional operator
at the weak scale to account for non-zero neutrino masses. In the Standard
Model, this operator is simply the $(LHLH)/\Lambda$ operator , where $\Lambda$
is the high scale at which neutrino masses are generated. In the Randall
Sundrum model one can write a similar operator for non-zero neutrino masses.
The model has been earlier studied in Huber1 ; Huber3 . The 5D action for the
RS model with the Higgs localized on the IR brane is given by
$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{\text{kin}}+S_{\text{Yuk}}$ $\displaystyle S_{kin}$ $\displaystyle=$
$\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}~{}\left(~{}\bar{L}(i\not{D}-m_{L})L+\bar{E}(i\not{D}-m_{E})E~{}\right)$
$\displaystyle S_{\text{Yuk}}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}\left(\frac{\mathbf{\kappa}}{\Lambda^{(5)}}LHLH~{}+~{}Y_{E}\bar{L}EH\right)\delta(y-\pi
R)$ (3)
where $\Lambda^{(5)}\sim 2.2\times 10^{18}$ GeV is the fundamental five
dimensional reduced Planck scale and
$D_{M}=\partial_{M}+\Omega_{M}+\frac{ig_{5}}{2}\tau^{a}W_{M}^{a}(x,y)+\frac{ig^{\prime}}{2}Q_{Y}B_{M}(x,y)$
(4)
with $\Omega_{M}=(-k/2e^{-ky}\gamma_{\mu}\gamma^{5},0)$ being the spin
connection and $Q_{Y}$ is the hypercharge. $M$ is the five dimensional Lorentz
index. $R$ is the compactification radius and $\kappa$ and $Y_{E}$ are the
coupling of the neutrino mass operator and the Yukawa coupling for the charged
leptons respectively. They are three dimensional matrices in flavour space and
we have suppressed the generation indices in writing the above equation. $L$
and $E$ are the 5D fermionic fields which transform as doublets and singlets
respectively under the Standard Model $\text{SU(2)}_{\text{W}}$ gauge group
with the covariant derivative given by Eq.(4) acting accordingly. $m_{L}$ and
$m_{E}$ are five dimensional Dirac masses of the $L$ and $E$ fields. As we
will see below, after Kaluza-Klein decomposition, these masses determine the
profiles of the zero and higher KK modes in the extra dimension. Since the
effective operator is suppressed by the 5D Planck mass, one can imagine that
the neutrino masses are as a result of some fundamental lepton number
violation beyond the 5D Planck scale.
|
---|---
|
Figure 1: Regions in $c_{i}$ for the LHLH case which give best fit to lepton
masses and mixing. The graphs in the upper row shows the region of parameter
space for the bulk masses for doublets which fits small neutrino masses.
Neutrino masses are assumed to have normal hierarchy in this analysis. The
graphs in the lower row shows the region for the bulk masses for the charged
singlets $c_{E_{i}}$. We have used log scale for $c_{E_{i}}$
The left and right components of the $L$ and $E$ fields have different $Z_{2}$
properties. These are chosen such that the $Z_{2}$ even zero modes correspond
to the SM fields. We assign the following $Z_{2}$ parity for the $L_{l,r}$ and
the $E_{l,r}$ fields, where the subscript $(l,r)$ correspond to the left and
right handed components of $L$ and $E$ 555The $\gamma_{5}$ required to define
the left and right components remains the same as the four dimensional case..
$\displaystyle Z_{2}(y)L_{l}(x,y)\rightarrow L_{l}(x,y)$ , $\displaystyle
Z_{2}(y)L_{r}(x,y)\rightarrow-L_{r}(x,y)$ $\displaystyle
Z_{2}(y)E_{r}(x,y)\rightarrow E_{r}(x,y)$ , $\displaystyle
Z_{2}(y)E_{l}(x,y)\rightarrow-E_{l}(x,y),$
where $Z_{2}(y):y\rightarrow-y$. The 5D fields can be expanded in terms of the
KK modes, with the expansion given by gross ; AgasheSoni
$\displaystyle L_{l}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}L_{l}^{(n)}(x)f^{(n)}_{L}(y)$ ; $\displaystyle
L_{r}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}L_{r}^{(n)}(x)\chi^{(n)}_{L}(y)$ $\displaystyle
E_{r}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}E_{r}^{(n)}(x)f^{(n)}_{E}(y)$ ; $\displaystyle
E_{l}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}E_{l}^{(n)}(x)\chi^{(n)}_{E}(y)$ (5)
where the exponential factor is chosen such that the fields are canonically
normalized. The profiles $f_{L,E}$ and $\chi_{L,E}$ are determined by :
$\displaystyle(\partial_{y}+c_{L}\sigma^{\prime})f_{L,E}^{(n)}(y)=m^{(n)}e^{\sigma(y)}\chi^{(n)}_{L,E}(y)$
$\displaystyle(-\partial_{y}+c_{L}\sigma^{\prime})\chi_{L,E}^{(n)}(y)=m^{(n)}e^{\sigma(y)}f_{L,E}^{(n)}(y)$
(6)
where the 5D masses $m_{L,E}$ are written in terms of the fundamental scale as
$m_{L,E}=c_{L,E}\sigma^{\prime}$ and $\sigma^{\prime}=\partial_{y}\sigma=k$.
The following orthonormality conditions are used for the profiles $f_{L,E}$
and $\chi_{L,E}$ to arrive at Eq.(II.1)
${1\over\sqrt{2\pi R}}\int_{-\pi R}^{\pi
R}~{}dy~{}e^{\sigma}\chi^{(n)}_{L,E}(y)\chi^{(m)}_{L,E}(y)={1\over\sqrt{2\pi
R}}\int_{-\pi R}^{\pi
R}~{}dy~{}e^{\sigma}f^{(n)}_{L,E}(y)f^{(m)}_{L,E}(y)=\delta^{nm}$ (7)
The above equations decouple for the zero mode solutions where $m^{(n)}=0.$
The solution for the $Z_{2}$ even part, $f_{L}(y)$ is given as
$f_{L}^{(0)}(y)=N_{0}(c_{L})e^{-c_{L}\sigma^{\prime}y}\;\;\;\;;\;\;\;N_{0}(c_{L})=\sqrt{\pi
R}\sqrt{\frac{(1-2c_{L})k}{e^{(1-2c_{L})k\pi R}-1}}$ (8)
$N_{0}$ being the normalization constant. The solution is the same for profile
of $E$, $f_{E}(y)$, with $c_{L}$ replaced by $c_{E}$. The bulk wave functions
are exponentials which peak towards the UV (IR) for $c>1/2$ ($c<1/2$) as can
be seen from Eq.(8). Typically, particles lighter in mass like leptons require
$c>1/2$ whereas heavier particles like top quark is localized much closer to
the IR brane with $c<1/2$. For the charged leptons and the neutrino masses one
would expect all the corresponding $c_{i}$ to be $>1/2$. The KK expansions
(II.1) are put into the Yukawa part of the action Eq.(II.1) leading to
$\displaystyle S_{\text{Yuk}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\int_{0}^{\pi R}dy\frac{1}{\pi
R}\sum_{n,m}~{}\left(~{}Y_{E}\bar{L}^{(n)}(x)f_{L}^{(n)}(y)E^{(m)}(x)f_{E}^{(m)}(y)e^{kR\pi}H\right.$
(9) $\displaystyle+$
$\displaystyle\left.\frac{\kappa}{\Lambda^{(5)}}f_{L}^{(n)}(y)f_{L}^{(m)}(y)L^{(n)}L^{(m)}HHe^{2kR\pi}~{}\right)\delta(y-\pi
R),$
where we have used $H\rightarrow e^{kR\pi}H$ to canonically normalize the
Higgs field and suppressed the subscripts $(l,r)$ for the $Z_{2}$ even fields.
The odd fields are neglected as they are removed from the boundary as a
consequence of the $Z_{2}$ symmetry. The charged lepton mass matrix and the
neutrino mass matrix are determined when the zero modes of the fields are
taken. The charged lepton mass matrix, corresponding to the $L^{(0)}E^{(0)}H$
operator in the action is given by
$\displaystyle{\mathcal{M}}^{(0,0)}_{e}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}\tilde{Y}_{E}+\mathcal{O}\Big{(}f_{L}^{(0)}(\pi
R)\frac{v^{3}}{M_{KK}^{2}}f_{E}^{(0)}(\pi R)\Big{)}$
$\displaystyle\tilde{Y}_{E}$ $\displaystyle=$ $\displaystyle{Y_{E}\over
R\pi}~{}N_{0}(c_{L})N_{0}(c_{E})~{}e^{(1-c_{L}-c_{E})kR\pi},$ (10)
| |
---|---|---
| |
| |
Figure 2: The distribution of electron Yukawa couplings ($Y_{E}^{\prime}$)
which give a ‘good fit’ to the charged fermion mass data in the LH LH operator
case. Neutrinos are assumed to follow normal hierarchy in this analysis. The
binning is done with an interval of 0.2
where the matrix $\tilde{Y}_{E}$ can be considered equivalent to the 4D
dimensionless Yukawa couplings. The neutrino mass matrix defined as the co-
efficient of the $L^{(0)}L^{(0)}HH$ operator in the action, is given as
$\displaystyle{\mathcal{M}}^{(0,0)}_{\nu_{ij}}$ $\displaystyle=$
$\displaystyle\tilde{\kappa}_{ij}\frac{v^{2}}{2\Lambda^{(5)}}+\mathcal{O}\left(\frac{1}{M_{KK}}\left(\frac{f_{L}^{(0)}(\pi
R)v^{2}}{\Lambda^{(5)}}\right)^{2}\right)$ $\displaystyle\tilde{\kappa}_{ij}$
$\displaystyle=$ $\displaystyle\kappa_{ij}~{}e^{2kR\pi}f_{L_{i}}(\pi
R)f_{L_{j}}(\pi R)={\kappa_{ij}\over
R\pi}~{}N_{0}(c_{L_{i}})N_{0}(c_{L_{j}})~{}e^{(2-c_{L_{i}}-c_{L_{j}})kR\pi},$
(11)
where $i,j$ are generation indices and $M_{KK}$ is the typical mass of higher
KK fermions. The corrections are from higher order KK modes and can be
neglected. Before fitting the mass matrices, we introduce new $\mathcal{O}(1)$
Yukawa parameters entering the mass matrices, which are defined as
$Y_{E}^{\prime}=2kY_{E}\;\;\;;\;\;\kappa^{\prime}=2k\kappa$ (12)
In terms of these new Yukawa parameters, the mass matrices are explicitly
given as
$\displaystyle({\mathcal{M}}^{(0,0)}_{e})_{ij}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}({Y}_{E}^{\prime})_{ij}e^{(1-c_{L}-c_{E})kR\pi}\sqrt{\frac{(0.5-c_{L_{i}})}{e^{(1-2c_{L_{i}})\pi
kR}-1}}\sqrt{\frac{(0.5-c_{E_{j}})}{e^{(1-2c_{E_{j}})\pi kR}-1}},$
$\displaystyle({\mathcal{M}}^{(0,0)}_{\nu})_{ij}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2\Lambda^{(5)}}(\kappa^{\prime})_{ij}e^{(2-c_{L_{i}}-c_{L_{j}})kR\pi}\sqrt{\frac{(0.5-c_{L_{i}})}{e^{(1-2c_{L_{i}})\pi
kR}-1}}\sqrt{\frac{(0.5-c_{L_{j}})}{e^{(1-2c_{L_{j}})\pi kR}-1}}$ (13)
The matrices are diagonalised as
$U_{eL}^{\dagger}\mathcal{M}^{(0,0)}_{e}U_{eR}=\text{Diag}[\\{m_{e},m_{\mu},m_{\tau}\\}]$
and
$U_{\nu}\mathcal{M}^{(0,0)}_{\nu}U_{\nu}^{T}=\text{Diag}[\\{m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}}\\}]$
and $U_{PMNS}=U_{\nu}^{\dagger}U_{eL}$. The eigenvalues of the charged lepton
mass matrix and the mass squared differences of the neutrino mass matrix and
the $U_{PMNS}$ mixing angles are fit to the data as per Table 1. In this case,
there are three $c_{L_{i}}$ and three $c_{E_{i}}$ and fifteen Yukawa
parameters fitting three charged lepton masses, three angles and two mass
squared differences. Given the dependence of the leptonic mass matrices on the
Yukawa parameters, we have chosen them strictly to be of $\mathcal{O}(1)$
nature. By this we mean, they are varied roughly between -4 and 4.
Furthermore, in order to avoid regions where the Yukawa parameters are
unnaturally close to zero, we put a lower bound on the Yukawas such that $|Y|$
lies between $\sim$ 0.08 and 4.
| |
---|---|---
| |
Figure 3: The distribution of neutrino Yukawa couplings ($\kappa^{\prime}$)
which give a ‘good fit’ to the fermion mass data in the LH LH operator case.
Neutrinos are assumed to follow normal hierarchy in this analysis. The binning
is done with an interval of 0.2.
Since the charged leptons and neutrinos have relatively light mass spectrum
compared to heavy quarks, one would have expected that varying $c_{L}$ and
$c_{E}$ between $1/2$ and $1$ would be sufficient to fit the data. However, in
the present context such values for $c_{E}$ will not satisfy the data. This is
because the neutrino mass matrix depends only on $c_{L_{i}}$ and requiring the
neutrino masses to be of the $\mathcal{O}(10^{-1})\text{eV}$ automatically
sets $c_{L_{i}}$ to be around $0.9$, close to the UV brane. The charged lepton
mass matrix, which in turn is determined by both $c_{L_{i}}$ and $c_{E_{i}}$
should off-set the effect of $c_{L_{i}}$ and increase the effective 4D Yukawa
coupling by pushing it towards the IR brane. This can only be achieved by
taking large and negative values666 One way to avoid large negative c
parameters would be to consider very large O(1) Yukawa parameters. The
required Yukawa couplings are in the range $\sim O(10^{3}-10^{4})$ to make any
connection with data. of the $c_{E_{i}}$. The range for the scan of the
$c_{L,E}$ has been judiciously chosen between 0.82 and 1.0 for bulk doublets
and $-5\times 10^{7}<c_{E_{1}}<-0.2$, $-10^{8}<c_{E_{2}}<-8000$ and
$-10^{9}<c_{E_{3}}<-9000$ for first, second and third generation charged
singlets respectively. A larger democratic range does not change the results
significantly.
All the parameters, the fifteen Yukawa couplings and the six $c_{L,E}$
parameters are varied so as to minimize the function in Eq.(2). The points
which give a $\chi^{2}$ between 1 and 8 are considered to give a ‘good fit’ to
the data. In Fig.[1] we present the regions in $c_{L_{1,2,3}}$ and
$c_{E_{1,2,3}}$ which have minimum $\chi^{2}$ assuming normal hierarchy for
neutrino masses. It is important to remember that Yukawa couplings are also
varied in obtaining this range in the $c_{L,E}$ parameter space. From the
figures we see that the strong constraint of neutrino masses limits the
$c_{L_{i}}$ to be within a limited range. On the other hand, $c_{E}$ seem to
have much larger ranges spanning orders of magnitudes. In particular,
$c_{E_{1}}$ is virtually unconstrained from $\mathcal{O}(-1)$ to
$\mathcal{O}(-10^{6})$. This is an artifact of the unconstrained lightest
neutrino mass, $m_{\nu_{1}}$. $c_{E_{2}}$ and $c_{E_{3}}$ have lesser freedom
as they are constrained by the mass squared differences. The allowed ranges in
the $c_{L,E}$ which satisfy the minimum $\chi^{2}$ requirement are summarized
in Table 2.
Table 2: Allowed range for the bulk parameters with minimum $\chi^{2}$. Neutrino masses have normal hierarchy. Range of first KK scale of the doublet(singlet) $M^{(1)}_{L}$($M^{(1)}_{E}$) corresponding to the bulk mass parameter is also give. parameter | range | range of $M^{(1)}_{L}$ (TeV) | parameter | range | range of $M^{(1)}_{E}$(TeV)
---|---|---|---|---|---
$c_{L_{1}}$ | 0.87-0.995 | 1.49-1.59 | $c_{E_{1}}$ | $-10.0$ to $-5.0\times 10^{6}$ | 7.9-$3.9\times 10^{6}$
$c_{L_{2}}$ | 0.86-0.98 | 1.48-1.58 | $c_{E_{2}}$ | $-1.0\times 10^{4}$ to $-1.2\times 10^{8}$ | $7.9\times 10^{3}$-$9.5\times 10^{7}$
$c_{L_{3}}$ | 0.84-0.92 | 1.47-1.53 | $c_{E_{3}}$ | $-7.0\times 10^{5}$ to $-1\times 10^{9}$ | $5.5\times 10^{5}$ $7.9\times 10^{8}$
It would be interesting to see distribution of the Yukawa couplings
$Y^{\prime}_{E}$ and $\kappa^{\prime}$ for the ‘best fit’ regions of the
parameter space. The distributions are presented in Figs.[2] and [3]. For most
of the $Y^{\prime}_{E}$ parameters, there is peaking at the two ends of the
range chosen, around 0.2 and 3.8. The exception is the lower $2\times 2$ block
of the Yukawa matrix, for which there seems to be a flatter profile for the
upper row parameters $(Y^{\prime}_{E})_{22}$ and $(Y^{\prime}_{E})_{23}$ and a
progressively increasing distribution for the second row parameters.
For almost all of the $Y^{\prime}_{E}$ parameters, peaking seems to be
happening at high values $\sim 3.8$, except for $(Y^{\prime}_{E})_{22}$. There
are also second peaks at very low values $\sim 0.2$ for some of the
parameters. Distributions in $\kappa^{\prime}$ on other hand, show peak at
very large value $\sim 3.8$ for the first two generation couplings and very
low values $\sim 0.4$ for $\kappa^{\prime}_{33}$ and $\kappa^{\prime}_{23}$.
With the exception of peaks, there is an underlying though highly subdued,
‘anarchical’ nature in the distribution of $Y^{\prime}_{E}$ Yukawa
couplings777Anarchy in the Yukawa distributions does not necessarily mean
anarchical structure in the mass matrix.. Thus, for a given choice of
$\mathcal{O}$(1) Yukawa couplings within our chosen range (-4 to 4), it seems
to be possible to find $c$ values which can fit the data well 888Increasing
the scan range for the $\mathcal{O}$(1) Yukawa couplings from -10 to 10 does
not change the gross features of the distributions much. For example,
$Y^{\prime}_{E}$ are peaked near the end points, showing that the lepton
masses in this case prefer large or small Yukawa couplings. The
$\kappa^{\prime}$ distribution has the same features scaled now to to 0 to 10
from 0 to 4. The ranges of the $c_{L,E}$ do not change significantly.. From
the allowed parameter space, we have randomly chosen two sample points, which
we call Point A and Point B, and we provided the corresponding observables in
Table 3. The corresponding Yukawa couplings are given in Eqs. (14) and (15).
Table 3: Sample points with corresponding fits of observables for Normal Hierarchy in LHLH case with $\mathcal{O}(1)$ Yukawas. The masses are in GeV Point | A | B
---|---|---
$\chi^{2}$ | 2.07 | 5.5
$c_{L_{1}}$ | 0.9755 | 0.903
$c_{L_{2}}$ | 0.9162 | 0.93
$c_{L_{3}}$ | 0.87 | 0.8443
$c_{E_{1}}$ | -692416.99 | -17.35
$c_{E_{2}}$ | -2647794.18 | -946125.13
$c_{E_{3}}$ | -80717122.21 | -47941542.53
$m_{e}$ | $5.07\times 10^{-4}$ | $5.08\times 10^{-4}$
$m_{\mu}$ | 0.1056 | 0.1056
$m_{\tau}$ | 1.767 | 1.771
$\theta_{12}$ | 0.58 | 0.589
$\theta_{23}$ | 0.68 | 0.743
$\theta_{13}$ | .168 | 0.163
$\delta m_{sol}^{2}$ | $7.49\times 10^{-23}$ | $7.48\times 10^{-23}$
$\delta m_{atm}^{2}$ | $2.47\times 10^{-21}$ | $1.99\times 10^{-21}$
Yukawa coupling matrices for Point A:
$Y^{\prime}_{E}=\begin{bmatrix}0.5023&1.9546&3.9730\\\ 3.2482&2.9629&2.7742\\\
2.6865&2.0383&1.2369\end{bmatrix}\;\;\;;\;\;\;\kappa^{\prime}=\begin{bmatrix}3.8933&3.9717&3.9818\\\
3.9717&-2.6660&-1.1409\\\ 3.9818&-1.1409&1.4597\end{bmatrix}$ (14)
Yukawa coupling matrices for Point B
$Y^{\prime}_{E}=\begin{bmatrix}3.0571&0.6316&0.8978\\\ 1.4085&0.9952&3.5597\\\
0.7971&0.9579&0.5539\end{bmatrix}\;\;\;;\;\;\;\kappa^{\prime}=\begin{bmatrix}0.2315&-3.8320&0.3490\\\
-3.8320&-0.6632&-1.1287\\\ 0.3490&-1.1287&0.0802\end{bmatrix}$ (15)
In Appendix A we have presented our results assuming neutrinos have an inverse
hierarchical mass ordering. We find very few points which satisfy the data in
this case. This is because inverted hierarchical spectrum requires two masses
at the atmospheric neutrino scale with their mass difference satisfying
$\Delta m^{2}_{sol}$. Thus the results are very sensitive to the
$\mathcal{O}$(1) Yukawa parameters. For a fixed Yukawa, however it is easy to
find points. More discussion is present in Appendix A.
The analysis presented so far has been purely phenomenological. Let us digress
from the fermion fits for a moment to discuss about the large negative $c$
parameters. Such large negative values for the bulk mass parameters are in
conflict with the 5D cutoff scale $k$. We have neglected this conflict in
fitting the data where we have considered them to be purely phenomenological
parameters which can take any value999We prefer to keep the $k$ (and also the
radius R) value fixed by noting that only the charged singlets required large
negative $c$ values. In case we shift the 5D cut-off scale to $|c|k$ keeping
$k$ fixed, the corresponding IR would shift to $c\Lambda_{IR}$, thus spoiling
the solution to the hierarchy problem in this scenario.. In terms of the bulk
wave-functions the large negative $c$ values would mean that the zero mode
wave-function $f^{(0)}\gg 1$, which is not the case when we choose the $c$
parameters between 0 and 1.
It is preferable to understand the large negative $c$ values in terms of
localization on the IR brane. The limit $c\rightarrow-\infty$ corresponds to
the case where the fermions are completely localized on the IR singular
pointGherghetta:2003he . In the limit $c\rightarrow-\infty$,
$f^{(0)}\rightarrow\infty$ indicating full overlap of the bulk wave-function
with the brane. The value of the $c$ parameters also affects the masses of the
KK modes. These masses are determined from Eqs.(II.1) by considering
$m_{n}\neq 0$ and choosing appropriate boundary conditions for the 5D fields.
The resultant differential equation has solution in terms of Bessel’s function
which describe bulk wave-functions of the KK modes whereas the masses are
given in terms of the zeros of the Bessel functiongher . The order of the
Bessel function is roughly given by $|c|$ for large values of c. In the
asymptotic limit the first KK mode has mass $\thickapprox|c|ke^{-kR\pi}$. Thus
we see that the phenomenologically relevant first KK mode mass also grows as
$\sim c\Lambda_{IR}$, where $\Lambda_{IR}\sim TeV$, the IR cutoff. The masses
of the first KK modes are presented in the Table[2]. The bulk wave-function of
the KK mode tend to zero as $|c|\rightarrow\infty$.
One might wonder if such large negative values of the $c_{E_{i}}$ parameters
would have some implications in terms of the AdS/CFT
correspondencerandallporrati ; gherghetta . The CFT interpretation for the
bulk scalars has been studied in batellgherghetta1 ; gherghetta and for bulk
fermions in continopomarol . The best fit $c_{L,E}$ parameters of LLHH case
given in Table 2 leads to an unusual situation where the left handed leptons
are almost completely elementary while the right handed singlets are
completely composite. This can be easily verified using the ‘holographic
basis’ of batellgherghetta2 . The composite component of the $c_{L}$ is
proportional to $e^{-(c_{L}-0.5)kR\pi}$, which goes to 0 when $c_{L}\to 0.99$.
Thus, the zero modes for the doublets are elementary. For the $c_{E}$ fields
however, the elementary component for the zero mode is given as
$\sqrt{(c_{E}-0.5)(c_{E}+1.5)}e^{-|1.5+c_{E}|kR\pi}$. Thus we see that the
zero mode for the charged singlets have a vanishing elementary component and
are completely composite fields. The effective 4-D Yukawa coupling of the zero
mode to the KK modes, is given as $Y^{\prime}_{E}\sqrt{(0.5-c_{E})}$. A
problematic feature of these models is that this coupling enters the non-
perturbative regime for $c_{E}$ large and negative. This non perturbative
coupling appears for all including the first KK mode, which is
phenomenologically relevant. This, non-perturbative feature is restricted to
the Yukawa coupling. The gauge coupling on the other hand do not face this
problem. In fact as we shall see later (Section V, Figure[13]), the coupling
strength of the zero mode fermions to gauge KK modes quickly approaches the
coupling of the brane localized fermions to gauge bosons for relatively
moderate values of $|c|$ parameter.
|
---|---
Figure 4: The figures above correspond to the case in which neutrinos are of
Dirac type. The points in the above figures correspond to a $\chi^{2}$ between
1 and 8. The plot represents the parameter space for the bulk masses of the
doublets. This case corresponds to the normal hierarchial case.
### II.2 Dirac Neutrinos
Dirac neutrino mass models in the RS setting have been extensively studied in
the literature Agashe . In AgasheSundrum , the authors talked about the
difficulty of fitting neutrino masses and mixing angles in the same scenario
as quarks. Their argument drew inspiration from the fact that neutrino mixing
angles are anarchic in nature. To address this issue they had a bulk Higgs,
with the profile ’sufficiently peaked’ near the IR brane and introduced a
‘switching behaviour’ to fit the both charged fermion and the neutrino masses
and mixing angles. We, on other hand, approach this problem in the same way as
we have done in the LHLH case of the previous section. We look for regions in
the parameter space of the bulk masses which give ‘good’ fits for a reasonable
choice of $\mathcal{O}$(1) Yukawa couplings. The particle spectrum of the
Standard Model is extended by adding singlet right handed neutrino. Global
lepton number is assumed to be conserved. It can be violated by quantum
gravity effects which manifest at the 5D Planck scale. However, for most of
the present analysis, we require lepton number violation present to be highly
suppressed.
The bulk and Yukawa actions in Eq.(II.1) now take the form:
$\displaystyle S_{kin}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int
dy\sqrt{-g}\left(~{}~{}\bar{L}(i\not{D}-m_{L})L+\bar{E}(i\not{D}-m_{E})E+\bar{N}(i\not{D}-m_{N})N~{}~{}\right)$
$\displaystyle S_{yuk}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}\left(~{}~{}Y_{N}\bar{L}NH+~{}Y_{E}\bar{L}EH~{}\right)\delta(y-\pi
R),$ (16)
|
---|---
Figure 5: The plot represents the parameter space for the bulk masses of
charged singlets.
where $N$ stands for the 5D right handed neutrino fields. The rest of the
parameters carry the same meaning as in the previous section. The components
of the $N$ field are assigned the same $Z_{2}$ properties as the $E$ field. We
expand the $N$ fields as
$\displaystyle N_{r}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}N_{r}^{(n)}(x)f^{(n)}_{N}(y)$ ; $\displaystyle
N_{l}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}N_{l}^{(n)}(x)\chi^{(n)}_{N}(y)$ (17)
Using Eq.(17) and Eq.(II.1) one can derive the equations of motion and
solutions similar to Eq.(8) for the profiles of $N$ fields. Substituting them,
the zero mode mass matrices for the charged lepton and neutrinos take the
form:
$\displaystyle{\mathcal{M}}_{e}^{(0,0)}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}\tilde{Y}_{E}\;;\;\tilde{Y}_{E}={Y_{E}\over
R\pi}~{}N_{0}(c_{L})N_{0}(c_{E})~{}e^{(1-c_{L}-c_{E})kR\pi}$
$\displaystyle{\mathcal{M}}_{\nu}^{(0,0)}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}\tilde{Y}_{N}\;;\;\tilde{Y}_{N}={Y_{N}\over
R\pi}~{}N_{0}(c_{L})N_{0}(c_{N})~{}e^{(1-c_{L}-c_{N})kR\pi},$ (18)
where we have neglected corrections from higher KK modes. As before, we
perform a scan over the parameter space of the bulk fermion masses and order
one Yukawa parameters to minimize the $\chi^{2}$ in Eq.(2) for the masses and
mixing angles. To specify the parameters which are scanned, it is useful to
look at the explicit form of the mass matrices equivalent to those of
Eq.(II.1):
$\displaystyle({\mathcal{M}}^{(0,0)}_{e})_{ij}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}({Y}_{E}^{\prime})_{ij}e^{(1-c_{L_{i}}-c_{E_{j}})kR\pi}\sqrt{\frac{(0.5-c_{L_{i}})}{e^{(1-2c_{L_{i}})\pi
kR}-1}}\sqrt{\frac{(0.5-c_{E_{j}})}{e^{(1-2c_{E_{j}})\pi kR}-1}}$
$\displaystyle({\mathcal{M}}^{(0,0)}_{\nu})_{ij}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}({Y}_{N}^{\prime})_{ij}e^{(1-c_{L_{i}}-c_{N_{j}})kR\pi}\sqrt{\frac{(0.5-c_{L_{i}})}{e^{(1-2c_{L_{i}})\pi
kR}-1}}\sqrt{\frac{(0.5-c_{N_{j}})}{e^{(1-2c_{N_{j}})\pi kR}-1}},$ (19)
where $Y^{\prime}_{E,N}=2kY_{E,N}$. Each of the $c_{i}$ parameters
($i=\\{L,N,E\\}$) which are three in number are varied along with eighteen
$\mathcal{O}(1)$ Yukawa parameters, i.e, a total of 27 parameters are varied
to fit the data and minimize the $\chi^{2}$. The $c$ parameters are varied as
follows: The doublets ($c_{L_{i}}$) and the the charged singlets are varied
between 0.02 and 1, while the neutral singlets are varied between between 1
and 1.9. The order one Yukawa couplings, $Y^{\prime}_{E,N}$, are varied
randomly between -4 and 4 with a lower bound $|Y|\gtrsim 0.08$. We consider
all the regions of the $c_{i}$ parameter space where the $\chi^{2}$ is between
1 and 8 as a ‘good’ fit region. In Figs.[4,5,6] we present regions in the
$c_{i}$ parameter space which give ‘good’ fit to the leptonic mass and mixing
angles. A summary of these regions is presented in Table (4).
Table 4: Allowed ranges of bulk parameters with normal hierarchy of neutrino masses. The range of first KK scale corresponding to the range of c values is also given. parameter | range | $M_{L}^{(1)}$ TeV | parameter | range | $M_{E}^{(1)}$ TeV | parameter | range | $M_{\nu}^{(1)}$ TeV
---|---|---|---|---|---|---|---|---
$c_{L_{1}}$ | 0.05-0.76 | 0.839-1.4 | $c_{E_{1}}$ | 0.2-0.88 | 0.959-1.5 | $c_{N_{1}}$ | 1.1-1.9 | 1.67-2.31
$c_{L_{2}}$ | 0.05-0.72 | 0.839-1.37 | $c_{E_{2}}$ | 0.05-0.73 | 0.839-1.38 | $c_{N_{2}}$ | 1.1-1.9 | 1.67-2.31
$c_{L_{3}}$ | 0.05-0.64 | 0.839-1.31 | $c_{E_{3}}$ | 0.05-0.64 | 0.839-1.31 | $c_{N_{3}}$ | 1.1-1.9 | 1.67-2.31
|
---|---
Figure 6: The plot represents the parameter space for the bulk masses of the
neutrino singlets.
The Dirac neutrino mass matrix in the RS model seems to fit the data more
naturally compared to the $LHLH$ discussed in the previous subsection. A large
section of the points fall in the regime $c_{i}$ $>1/2$ indicating that they
are localized closer to the UV brane. The distributions of the Yukawa
couplings in the ‘good fit’ region, presented in Figs.(7,8) show that most of
them peak in the last bins for all the Yukawas at ($3.8-4.0$). A secondary
peak can also been seen at $(0.2-0.4)$ bin for some of the $Y^{\prime}_{N}$
parameters. Electron Yukawa couplings on the other hand do not seem to show
any such secondary peak. In this case too the distribution of the
$\mathcal{O}$(1) Yukawa couplings displays an underlying anarchic nature
especially for the $Y^{\prime}_{E}$. This will prove useful in our analysis of
Minimal Flavour violation where the $\mathcal{O}$(1) Yukawa couplings and the
bulk mass matrices need to be simultaneously diagonalizable. In Table(5), we
presented two sample points. Point A has all the $c_{i}>1/2$ where as Point B
has $c_{E_{2}},c_{E_{3}}~{}<1/2$. The corresponding Yukawa couplings are given
in Eqs.(20,21).
As before we use the holographic basis to comment on the partial compositeness
of the bulk fermions. The zero modes of singlet right handed neutrinos are
dominantly elementary, with almost zero component of compositeness. The
composite component for the zero modes of the doublets and the charged
singlets becomes smaller as the corresponding c values becomes greater than
0.5. Essentially they have partially composite nature.
Table 5: Sample points with corresponding fits of observables for Normal Hierarchy in Dirac case with O(1) Yukawas. The masses are in GeV Parameter | Point A | Point B
---|---|---
$\chi^{2}$ | 0.28 | 0.39
$c_{L_{1}}$ | 0.6263 | 0.7166
$c_{L_{2}}$ | 0.5932 | 0.6382
$c_{L_{3}}$ | 0.5293 | 0.6126
$c_{E_{1}}$ | 0.6704 | 0.5911
$c_{E_{2}}$ | 0.5541 | 0.1939
$c_{E_{3}}$ | 0.5131 | 0.2647
$c_{N_{1}}$ | 1.2233 | 1.2791
$c_{N_{2}}$ | 1.2692 | 1.1215
$c_{N_{3}}$ | 1.2948 | 1.2343
$m_{e}$ | $5.09\times 10^{-4}$ | $5.09\times 10^{-4}$
$m_{\mu}$ | 0.1055 | 0.1055
$m_{\tau}$ | 1.77 | 1.77
$\theta_{12}$ | 0.59 | 0.589
$\theta_{23}$ | 0.80 | 0.792
$\theta_{13}$ | 0.153 | 0.153
$\delta m_{sol}^{2}$ | $7.49\times 10^{-23}$ | $7.49\times 10^{-23}$
$\delta m_{atm}^{2}$ | $2.39\times 10^{-21}$ | $2.40\times 10^{-21}$
Yukawa Coupling Matrix for Point A:
$Y^{\prime}_{E}=\begin{bmatrix}3.9502&-1.6538&0.5889\\\
-0.7276&-2.0054&-3.9004\\\
-1.4061&1.4756&1.5318\end{bmatrix}\;\;\;;\;\;\;Y_{N}^{\prime}=\begin{bmatrix}-3.8918&-3.9447&-3.8380\\\
-2.6439&2.5796&3.9962\\\ -0.9223&-1.3577&0.6417\par\end{bmatrix}$ (20)
Yukawa Coupling Matrix for Point B:
$Y^{\prime}_{E}=\begin{bmatrix}3.3847&1.8639&-1.3814\\\
-1.8107&-0.7219&-0.9499\\\
-2.5435&-1.0497&-3.3588\end{bmatrix}\;\;\;;\;\;\;Y_{N}^{\prime}=\begin{bmatrix}2.4435&-1.8006&-1.9575\\\
0.4198&-3.1594&3.5905\\\ -0.2505&1.3172&2.1521\end{bmatrix}$ (21)
| |
---|---|---
| |
| |
Figure 7: The distribution of electron Yukawa couplings ($Y_{E}^{\prime}$)
which give a ‘good fit’ to the fermion mass data in the Dirac case. Neutrinos
are assumed to follow normal hierarchy in this analysis. The binning is done
with an interval of 0.2
### II.3 Bulk Majorana mass term
Singlet neutrinos typically accommodate Majorana mass terms in addition to the
Dirac mass terms. These bare mass terms which break lepton number at a very
high scale play an essential role in the standard four dimensional seesaw
mechanism to generate light neutrino masses. The seesaw mechanism with bulk
Majorana mass terms has been first considered in Huber2 . There have been
other works which have considered brane localised Majorana mass terms
Goldberger:2002pc ; Nomura:2003du ; Gherghetta:2003he ; perez . Our analysis
follows the work of Huber2 and extends it by computing the numerical
solutions. The part of the action which contains the singlet right handed
neutrinos is given by
$S_{N}=\int d^{4}x\int
dy\sqrt{-g}\big{(}m_{M}\bar{N}N^{c}+m_{D}\bar{N}N+\delta(y-\pi
R)Y_{N}\bar{L}\tilde{H}N\big{)}$ (22)
where $N^{c}=C_{5}\bar{N}^{T}$ with $C_{5}$ being the five-dimensional charge
conjugation matrix101010$C_{5}$ is taken to be $C_{4}$. and $m_{M}=c_{M}k$,
with $k$ being the reduced Planck scale111111Majorana mass terms does not have
the same interpretation in the bulk as in 4D.. The bulk Dirac mass for the
right handed neutrino is parametrized as $m_{D}=c_{N}k$. As before we consider
all the mass parameters to be real. The bulk singlet fields N have the
following KK expansions:
$\displaystyle N_{L}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}N_{L}^{(n)}(x)g^{(n)}_{L}(y)$ ; $\displaystyle
N_{R}(x,y)=\sum_{n=0}^{\infty}{1\over\sqrt{\pi
R}}e^{2\sigma(y)}N_{R}^{(n)}(x)g^{(n)}_{R}(y),$
where $g_{L}$ and $g_{R}$ are profiles of the singlet neutrinos in the bulk.
They follow the following orthonormal conditions
$\frac{1}{2\pi R}\int_{-\pi R}^{\pi
R}~{}dy~{}e^{\sigma}\Big{(}g_{L}^{(n)}g_{L}^{(m)}+g_{R}^{(n)}g_{R}^{(m)}\Big{)}=\delta^{(n,m)}$
(24)
Using this, the eigenvalues equations for the $g_{L,R}$ fields become Huber2
$\displaystyle(\partial_{y}+m_{D})g_{L}^{(n)}(y)=m_{n}e^{\sigma}g_{R}^{(n)}(y)-m_{M}g_{R}^{(n)}(y)$
$\displaystyle(-\partial_{y}+m_{D})g_{R}^{(n)}(y)=m_{n}e^{\sigma}g_{L}^{(n)}(y)-m_{M}g_{L}^{(n)}(y)$
(25)
where we have assumed the five dimensional wave functions to be real.
| |
---|---|---
| |
| |
Figure 8: The distribution of neutrino Yukawa couplings ($Y_{N}^{\prime}$)
which give a ‘good fit’ to the fermion mass data in the Dirac case. Neutrinos
are assumed to follow normal hierarchy in this analysis. The binning is done
with an interval of 0.2
Unlike the Dirac and higher dimensional LHLH term cases, the present system of
equations, in Eq.(25) are not consistent with a zero mode solution $m_{n}=0$
for $m_{D}\neq 0$. This is because the zero mode solutions, $\propto
e^{\sqrt{c_{N}^{2}-c_{M}^{2}}\sigma}$ do not satisfy either Dirichlet or the
more general $(\partial_{y}+m_{d})g_{L}(y)=0$ boundary condition. Thus in the
following analysis, we will consider the first KK mode not to be the zero mode
but $m_{n}=m_{(1)}$. Furthermore, Eq.(25) does not have simple analytical
solutions, though numerical solutions exist. We have obtained the numerical
solutions of $g_{L,R}$ by solving the second order equations derived from
Eq.(25). The equation for the $Z_{2}$ even part takes the form:
$g_{L}^{\prime\prime}(y)-\frac{m_{n}kRe^{kRy}}{m_{n}e^{kRy}-c_{M}k}g_{L}^{\prime}(y)-\left(\frac{c_{N}m_{n}e^{kRy}k^{2}}{m_{n}e^{kRy}-c_{M}k}+c_{N}^{2}k^{2}-\left(m_{n}e^{kRy}-c_{M}k\right)^{2}\right)R^{2}g_{L}(y)=0$
(26)
The second order equation for the $Z_{2}$ odd part $g_{R}$ is given as
$g_{R}^{\prime\prime}(y)-\frac{m_{n}kRe^{kRy}}{m_{n}e^{kRy}-c_{M}k}g_{R}^{\prime}(y)-\left(\frac{-c_{N}m_{n}e^{kRy}k^{2}}{m_{n}e^{kRy}-c_{M}k}+c_{N}^{2}k^{2}-(m_{n}e^{kRy}-c_{M}k)^{2}\right)R^{2}g_{R}(y)=0,$
(27)
where we have used the notation $m_{D}=c_{N}k$ and $M_{M}=c_{M}k$ introduced
earlier. The primes on $g_{L}(y)$ and $g_{R}(y)$ indicate derivatives on the
profiles. For a given choice of $c_{N}$ and $c_{M}$ one would expect to
numerically find solutions using the above equations for $g_{L,R}$ as long as
they satisfy two conditions: (i) $m_{(1)}$ is also fixed such that the
boundary conditions are satisfied consistently (ii) There are no singularities
in coefficients of the differential equations in the interval $[0,\pi R]$.
This second condition requires that for unique solutions, only those values of
$c_{M}$ and $m_{n}$ are allowed for which $m_{n}e^{\sigma}-m_{M}$ is non zero.
Note that this condition is always true when $c_{M}$ is negative. For positive
$c_{M}$, the allowed region is shown in Fig.[9], where all the shaded region
has $m_{n}e^{\sigma}-m_{M}$ non zero. As can be seen from the figure, as
$c_{M}$ increases, the KK mass scale also increases.
Figure 9: Region of $c_{M}$-$m_{1}$ parameter space, for positive $c_{M}$ for
which the coefficients of the differential equation in Eq.(26) are analytic in
the interval $[0,\pi R]$
In Fig.[10] we show solutions to Eq.(26) for a fixed value of $c_{N}=0.58$.
$c_{M}$ is varied from 0.55 to 1. From the figure, it is clear as the profile
becomes oscillatory as $c_{M}$ becomes greater than $c_{N}$. In fact the
solutions are sinusoidal for $c_{M}$=1 and $c_{N}$=0. We now address the
question of fitting the lepton masses and mixing.
|
---|---
|
Figure 10: The Figure shows the form of the profile for solution to Eq.(26)
for a fixed bulk dirac mass of 0.58 for the right handed neutrinos. We see
that the profile becomes oscillatory as $c_{M}$ becomes greater than $c_{N}$.
The charged lepton mass matrix has the same form as in earlier sections
$m_{l}^{(0,0)}=\frac{v}{\sqrt{2}}\tilde{Y}_{E}+O(\frac{v^{2}}{M_{KK}^{2}})\;;\;\tilde{Y}_{E}={Y_{E}\over
R\pi}~{}N_{0}(c_{L})N_{0}(c_{E})~{}e^{(1-c_{L}-c_{E})kR\pi}.$ (28)
Choosing $g_{L}^{(1)}$ to be the $Z_{2}$ even profile for the right handed
neutrino, the Dirac mass matrix takes the form
$m^{(0,1)}_{D}={Y_{N}\over R\pi}N_{0}(c_{L})e^{(1-c_{L})kR\pi}g_{L}^{(1)}(\pi
R)$ (29)
where $g_{L}^{(1)}(y)$ is the solution to Eq.(26). The singlet Majorana mass
matrix is determined in the flavor space by the choice of $c_{N}$ and $c_{M}$
for each of the generations. For simplicity, for the present analysis, we take
all of them equal $c_{N_{i}}=c_{N}$ and $c_{M_{i}}=c_{M}$ for all the three
generations121212This can be achieved by imposing an $O(3)$ symmetry on the
$N$ fields.. With this, singlet neutrino mass matrix becomes proportional to
the unit matrix $M_{R}=\textbf{1}~{}m_{(1)}$. The light neutrino mass matrix
now takes the see-saw form given by
$m_{\nu}^{(0,0)}=m_{D}^{(0,1)}\frac{1}{M_{R}}{{m_{D}^{(0,1)}}^{T}}+\mathcal{O}\left({\left(m_{D}^{(0,k)}\right)^{2}\over
m_{(k)}}\right)$ (30)
where higher order corrections are from higher KK states. To fit the neutrino
masses and mixing angles we neglect higher order corrections as before.
Defining $Y^{\prime}_{N}=2kY_{N}$, we have
$m_{\nu}^{(0,0)}=Y^{\prime}_{N}e^{(1-c_{L})kR\pi}g_{L}(\pi
R)(M_{R}^{-1})Y^{\prime}_{N}e^{(1-c_{L})kR\pi}g_{L}(\pi R)$ (31)
In Table (6), we present two sample points one for inverted hierarchy and
another for normal hierarchy, which fit the neutrino masses and mixing angles
as well as charged lepton masses with the accuracy we have specified in
section II. Both these examples131313These solutions require that the profiles
of the $N$ fields have very small values on the UV brane. have $c_{M}<c_{N}$.
The corresponding Yukawa coupling matrices are presented in Eqs. (32,33).
Table 6: Sample points with corresponding fits of observables for Normal and Inverted Hierarchy schemes in Bulk Majorana case with O(1) Yukawas. The masses are in GeV Parameter | Normal | Inverted
---|---|---
$M_{kk}$ | 161.4 | 161.4
$c_{M_{i}}$ | 0.55 | 0.55
$g_{L}^{(1)}(\pi R)$ | $3\times 10^{-13}$ | $1.2\times 10^{-12}$
$c_{L_{1}}$ | 0.58 | 0.59
$c_{L_{2}}$ | 0.56 | 0.57
$c_{L_{3}}$ | 0.55 | 0.55
$c_{E_{1}}$ | 0.735 | 0.735
$c_{E_{2}}$ | 0.5755 | 0.575
$c_{E_{3}}$ | 0.501 | 0.501
$c_{N_{i}}$ | 0.58 | 0.58
$m_{e}$ | $5.09\times 10^{-4}$ | $5.08\times 10^{-4}$
$m_{\mu}$ | 0.1055 | 0.1055
$m_{\tau}$ | 1.77 | 1.774
$\theta_{12}$ | 0.58 | 0.58
$\theta_{23}$ | 0.80 | 0.8
$\theta_{13}$ | 0.13 | 0.13
$\Delta m_{sol}^{2}$ | $7.8\times 10^{-23}$ | $7.8\times 10^{-23}$
$\Delta m_{atm}^{2}$ | $2.4\times 10^{-21}$ | $2.4\times 10^{-21}$
Yukawa parameters for inverted hierarchy
$Y^{\prime}_{N}=\begin{bmatrix}2.73&1.81&.108\\\ -0.83&0.975&.328\\\
0.327&-0.679&.182\end{bmatrix}\;\;Y^{\prime}_{E}=\begin{bmatrix}3.44&-0.41&.87\\\
0.62&1.583&0.332\\\ 2.74&0.55&2.33\end{bmatrix}$ (32)
Yukawa parameters for normal hierarchy
$Y^{\prime}_{N}=\begin{bmatrix}2.56&1.69&1.26\\\ -0.795&0.927&3.89\\\
0.414&-0.859&2.86\end{bmatrix}\;\;Y^{\prime}_{E}=\begin{bmatrix}2.825&-0.41&.87\\\
0.62&1.2008&0.332\\\ 2.74&0.55&2.31\end{bmatrix}$ (33)
### II.4 Brane localized Majorana mass term
Following our discussion with a bulk Majorana mass term, there could be
special cases where the Majorana mass term could be localized on either
boundary. In this case the bulk profiles for the right handed singlets $N_{i}$
remain unchanged. The eigenvalue equations are same as in Eq.(II.1).
#### II.4.1 UV localized mass term
The case with UV localized Majorana mass term was studied in Huber2 ; perez .
The action in this case is given as
$S_{N}=\int d^{4}x\int
dy~{}\sqrt{-g}~{}\big{(}\delta(y)\bar{N}N^{c}+m_{D}\bar{N}N+\delta(y-\pi
R)Y_{N}\bar{L}\tilde{H}N\big{)}$ (34)
where we have expressed $m_{M}=\delta(y)$. Substituting the KK expansions from
Eq.(17), the effective 4-D neutrino mass matrix, in the basis
$\chi^{T}=\\{\nu_{L}^{(0)},N_{R}^{(0)},N_{R}^{(1)},N_{L}^{(1)}\\}$ takes the
form
$\mathcal{L}_{m}=-{1\over
2}\chi^{T}M_{N}\chi\;\;\;;\;\;\;\;M_{N}=\begin{bmatrix}0&\mathcal{M}^{(0,0)}_{\nu}&\mathcal{M}^{(0,1)}_{\nu}&0\\\
\mathcal{M}^{(0,0)}_{\nu}&M^{Maj}_{\nu^{(0,0)}}&M^{Maj}_{\nu^{(0,1)}}&0\\\
\mathcal{M}^{(0,1)}_{\nu}&M^{Maj}_{\nu^{(0,1)}}&M^{Maj}_{\nu^{(1,1)}}&M_{KK}\\\
0&0&M_{KK}&0\end{bmatrix}$ (35)
where $\mathcal{M}^{(0,0)}_{\nu}$ is defined in Eq.(II.2). Let
$f^{(1)}_{N}(0)$ denote the value of the profile of the first KK mode of N at
the UV brane i.e, y=0 and $f_{N}(0)$, defined in Eq.(8), is the zero mode
profile of N evaluated at y=0. The individual elements of Eq.(35) are then
defined as: $\mathcal{M}^{(0,1)}_{\nu}=\frac{v}{\sqrt{2}}\frac{1}{\sqrt{\pi
R}}f_{N}(\pi R)Y_{N}^{\prime}$; $M^{Maj}_{\nu^{(0,0)}}=\frac{1}{\pi
R}f_{N}^{2}(0)$ ;$M^{Maj}_{\nu^{(0,1)}}=\frac{1}{\pi
R}f^{(1)}_{N}(0)f_{N}(0)$; $M^{Maj}_{\nu^{(1,1)}}=\frac{1}{\pi
R}f^{(1)}_{N}(0)f^{(1)}_{N}(0)$ and $M_{KK}$ is the KK mass of first KK mode
of N. The small neutrino masses can be fit by choosing $c_{N}\sim 0.32$ for
which $M^{Maj}_{\nu^{(0,0)}}\sim 10^{14}\text{GeV}$. The charged leptons are
fit by choosing $c_{L,E}>0.5.$ This scenario along with flavour implications
has been extensively dealt in perez .
#### II.4.2 Pure Majorana Case
An interesting sub case of the Bulk Majorana term would be the situation where
$m_{D}=c_{N}k=0$. As we have seen from the discussion in the previous section,
in such a case, the profile equations become oscillatory. The eigenvalue
equations now take the form:
$\displaystyle\partial_{y}g_{L}^{(n)}(y)$ $\displaystyle=$ $\displaystyle
m_{n}e^{\sigma}g_{R}^{(n)}(y)-m_{M}g_{R}^{(n)}(y)$
$\displaystyle-\partial_{y}g_{R}^{(n)}(y)$ $\displaystyle=$ $\displaystyle
m_{n}e^{\sigma}g_{R}^{(n)}(y)-m_{M}g_{L}^{(n)}(y)$ (36)
Contrary to the Dirac+ Majorana case of the previous section, the above set of
equations allow solutions for zero modes, $m_{0}=0$. The solutions are given
as
$\displaystyle g_{L}(y)$ $\displaystyle=$ $\displaystyle
N\cos(\frac{m_{n}e^{\sigma}}{k}-m_{M}y)$ $\displaystyle g_{R}(y)$
$\displaystyle=$ $\displaystyle N\sin(\frac{m_{n}e^{\sigma}}{k}-m_{M}y),$ (37)
where $N$ is the normalization factor given by $N=\sqrt{\pi
Rk}e^{-0.5\sigma(\pi R)}$. These solutions are consistent with the boundary
conditions. The neutrino mass matrix has a specific structure in this case, as
there are contributions from the first KK mode, which might be important. In
the basis, $\chi^{T}=\\{\nu_{L}^{(0)},N^{(0)},N^{(1)}\\}$ the mass matrix
takes the form
$\mathcal{L}_{m}=-{1\over
2}\chi^{T}\mathcal{M}\chi\;\;\;;\;\;\;\;\mathcal{M}=\left(\begin{array}[]{ccc}0&m_{D}^{(0,0)}&m_{D}^{(0,1)}\\\
m_{D}^{(0,0)}&0&0\\\ m_{D}^{(0,1)}&0&m_{(1)}\end{array}\right)$ (38)
From the above, we see that at the zeroth level, light neutrino and singlet
neutrinos form a pseudo-Dirac structure, leading to maximal mixing between
these two states. For the three flavor states, we would have three light
states which are sterile. We have not pursued the phenomenology of this model
further.
## III Lepton Flavor Violation
We now study lepton flavor violating constraints on the three neutrino mass
models considered in the present work. Lepton flavor violation within the RS
framework has been studied in detail in Agashe . The localization of the
fermions in the bulk at different places leads to non-zero flavour mixing
between the zero mode SM fermions and higher KK states, which contribute to
flavor violating processes both at the tree and the loop level. The tree level
flavor violating decay modes of the form $l_{i}\to l_{j}l_{k}l_{k}$ are due to
non-universal overlap of the zero mode fermions with the Z-boson KK modes. At
the 1-loop level, penguin graphs contribute to rare decays like $l_{j}\to
l_{i}+\gamma$. The SM states mix with their heavier KK states on the IR brane,
and thus may give rise to significant contributions to dipole processes in
particular. The present LFV limits are very strong and are listed in
Table[LABEL:lfv-tab]
Table 7: Present Experimental Bounds on LFV Processes Process | Experiment | Present upper bound
---|---|---
BR$(\mu\rightarrow e\,\gamma)$ | MEG meg11 ; DeGerone:2011fg | $2.4\times 10^{-12}$
BR$(\mu\rightarrow e\,e\,e)$ | MEG meg11 ; DeGerone:2011fg | $1.0\times 10^{-12}$
CR$(\mu\rightarrow e\,{\rm in}\,{\bf Ti})$ | SINDRUM-II Wintz:1996va | $6.1\times 10^{-13}$
BR$(\tau\rightarrow\mu\,\gamma)$ | BABAR/Belle :2009tk | $4.4\times 10^{-8}$
BR$(\tau\rightarrow e\,\gamma)$ | BABAR/Belle :2009tk | $3.3\times 10^{-8}$
BR$(\tau\rightarrow\mu\,\mu\,\mu)$ | BABAR/Belle :2009tk | $2.0\times 10^{-8}$
BR$(\tau\rightarrow e\,e\,e)$ | BABAR/Belle :2009tk | $2.6\times 10^{-8}$
In this section we calculate the Branching fractions for the leptonic FCNC.
The effective 4-D lagrangian describing $l\rightarrow l^{\prime}$ process is
given by Agashe
$\displaystyle-\mathcal{L_{{\rm eff}}}$ $\displaystyle=$ $\displaystyle
A_{R}(q^{2})\frac{1}{2m_{\mu}}\bar{e}_{R}\sigma^{\mu\nu}F_{\mu\nu}\mu_{L}+A_{L}(q^{2})\frac{1}{2m_{\mu}}\bar{e}_{L}\sigma^{\mu\nu}F_{\mu\nu}\mu_{R}$
(39)
$\displaystyle+\frac{4G_{F}}{\sqrt{2}}\left[a_{3}(\bar{e}_{R}\gamma^{\mu}\mu_{R})(\bar{e}_{R}\gamma_{\mu}e_{R})+a_{4}(\bar{e}_{L}\gamma^{\mu}\mu_{L})(\bar{e}_{L}\gamma_{\mu}e_{L})\right.$
$\displaystyle+\left.a_{5}(\bar{e}_{R}\gamma^{\mu}\mu_{R})(\bar{e}_{L}\gamma_{\mu}e_{L})+a_{6}(\bar{e}_{L}\gamma^{\mu}\mu_{L})(\bar{e}_{R}\gamma_{\mu}e_{R})\right]+{\rm
h.c.}$
### III.1 Tree level decays
The breaking of the electroweak symmetry at the IR brane mixes the zero mode
gauge boson with the higher modes. To parametrize this mixing, let ($Z^{(0)}$,
$Z^{(1)}$) and (${Z^{\prime}}^{(0)}$ ${Z^{\prime}}^{(1)}$) denote the gauge
boson states before and after diagonalisation of the gauge boson mass matrix
respectively. Assuming only one KK mode for simplicity, they are related as
Agashe
$\displaystyle{Z^{\prime}}^{(0)}=Z^{(0)}+\sqrt{2kR\pi}\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}Z^{(1)}$
$\displaystyle{Z^{\prime}}^{(1)}=Z^{(1)}-\sqrt{2kR\pi}\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}Z^{(0)}$
where $M_{Z^{(1)}}$ is the mass of first KK excitation of the Z boson. Owing
to its flat profile the $Z^{(0)}$ couples universally to all three
generations. However, the coupling of ${Z}^{(1)}$, whose profile is peaked
near the IR brane, is generation dependent. This coupling depends on the
localization of the fermions along the extra-dimension thus giving rise to
non-universality. Let $\eta^{T}=$ {$e_{M}$,$\mu_{M}$, $\tau_{M}$} be vector of
fermions in the mass basis. Let $a_{ij}^{(1)}$ be a $3\times 3$ matrix which
denotes the coupling of SM fermions in the mass basis to ${Z^{\prime}}^{(1)}$.
It is given as
$a^{(1)ij}_{L,R}=g_{L,R}~{}\bar{\eta}_{L,R}.D_{L,R}^{\dagger}.\begin{bmatrix}I_{e}&0&0\\\
0&I_{\mu}&0\\\
0&0&I_{\tau}\end{bmatrix}.D_{L,R}.\eta_{L,R}~{}\not{{Z^{\prime}}}^{(1)}$ (41)
where $g_{L,R}$ is the SM coupling, $D_{L,R}$ are $3\times 3$ unitary matrices
for rotating the zero mode (SM) fermions from the flavour basis to the mass
basis. I is the overlap of the profiles of two zero mode fermions and first KK
gauge boson. It is given by
$I(c)=\frac{1}{\pi R}\int_{0}^{\pi
R}dye^{\sigma(y)}(f_{i}^{(0)}(y,c))^{2}\xi^{(1)}(y)$ (42)
$\xi^{(1)}(y)$ denotes the profile of the first KK gauge boson. It is plotted
as a function of a generic bulk mass parameter c in Fig.[13]. As we can see
from this figure, the overlap function $I(c)$ becomes universal for $c>0.5$
and for $c\lesssim-15$. The off diagonal elements of $a_{ij}^{(1)}$ represent
the flavour violating couplings. The contribution to $l_{i}\to
l_{j}l_{k}l_{k}$ from direct $Z^{(1)}$ exchange is suppressed compared to that
of ${Z}^{(0)}$. The contributions to the coefficients $a^{ij}_{3,.,6}$ in
Eq.(39) due to the flavour violating coupling of ${Z}^{(0)}$ as well as direct
${Z}^{(1)}$ exchange are given as
$\displaystyle a^{ij}_{3}$ $\displaystyle=$
$\displaystyle-2g_{R}(\sqrt{2kR\pi}-I_{j})\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}a^{(1)ij}_{R}$
(43) $\displaystyle a^{ij}_{4}$ $\displaystyle=$
$\displaystyle-2g_{L}(\sqrt{2kR\pi}-I_{j})\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}a^{(1)ij}_{L}$
$\displaystyle a^{ij}_{5}$ $\displaystyle=$
$\displaystyle-2g_{L}(\sqrt{2kR\pi}-I_{j})\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}a^{(1)ij}_{R}$
$\displaystyle a^{ij}_{6}$ $\displaystyle=$
$\displaystyle-2g_{R}(\sqrt{2kR\pi}-I_{j})\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}a^{(1)ij}_{L}$
The Branching fractions for the tree level decays are given as Agashe
$\displaystyle BR(\mu\rightarrow 3e)$ $\displaystyle=$ $\displaystyle
2\left(|a^{\mu e}_{3}|^{2}+|a^{\mu e}_{4}|^{2}\right)+|a^{\mu
e}_{5}|^{2}+|a^{\mu e}_{6}|^{2}$ $\displaystyle BR(\tau\rightarrow 3\mu)$
$\displaystyle=$
$\displaystyle\left\\{2\left(|a^{\tau\mu}_{3}|^{2}+|a^{\tau\mu}_{4}|^{2}\right)+|a^{\tau\mu}_{5}|^{2}+|a^{\tau\mu}_{6}|^{2}\right\\}BR(\tau\to
e\nu\nu)$ $\displaystyle BR(\tau\rightarrow 3e)$ $\displaystyle=$
$\displaystyle\left\\{2\left(|a^{\tau e}_{3}|^{2}+|a^{\tau
e}_{4}|^{2}\right)+|a^{\tau e}_{5}|^{2}+|a^{\tau
e}_{6}|^{2}\right\\}BR(\tau\to e\nu\nu)$ $\displaystyle BR(\tau\rightarrow\mu
ee)$ $\displaystyle=$
$\displaystyle\left\\{|a^{\tau\mu}_{3}|^{2}+|a^{\tau\mu}_{4}|^{2}+|a^{\tau\mu}_{5}|^{2}+|a^{\tau\mu}_{6}|^{2}\right\\}BR(\tau\to
e\nu\nu)$ $\displaystyle BR(\tau\rightarrow e\mu\mu)$ $\displaystyle=$
$\displaystyle\left\\{|a^{\tau e}_{3}|^{2}+|a^{\tau e}_{4}|^{2}+|a^{\tau
e}_{5}|^{2}+|a^{\tau e}_{6}|^{2}\right\\}BR(\tau\to e\nu\nu).$ (44)
Figure 11: Tree level contribution to $\mu\rightarrow eee$ due to exchange of
${Z^{\prime}}^{(1)}$. The effective ${Z}^{(0)}$ contribution is proportional
to this graph.
Similarly, the relevant quantities for $\mu\to e$ conversion in Ti are given
as:
$\displaystyle a_{L,R}^{\mu e}$ $\displaystyle=$
$\displaystyle-\sqrt{2kR\pi}\frac{m_{Z}^{2}}{M_{Z^{(1)}}^{2}}a^{(1)\mu
e}_{L,R}$ $\displaystyle\hskip 22.76228ptBR(\mu\rightarrow e)~{}\text{ in
Nuclei}$ $\displaystyle=$
$\displaystyle\frac{2p_{e}F_{p}^{2}E_{e}G_{F}^{2}m_{\mu}^{3}\alpha^{3}Z^{4}_{eff}Q_{N}^{2}}{\pi^{2}Z\Gamma_{capt}}[|a_{R}^{\mu
e}|^{2}+|a_{L}^{\mu e}|^{2}]$ (45)
where $p_{e}\sim E_{e}\sim m_{\mu}$. $G_{F}$ is the Fermi constant, $\alpha$
is the electromagnetic coupling. The most stringent constraint for $\mu-e$
conversion comes from Titanium ($Ti^{48}_{22}$). Atomic constants are defined
as: $Q_{N}=v^{u}(2Z+N)+v^{d}(2N+Z)$ with N being the neutron number,
$Z_{eff}=17.61$, form factor $F_{p}=0.55$, $\Gamma_{capt}=2.6\times 10^{6}$
$s^{-1}$ for Titanium Chang .
### III.2 Dipole Transition $l_{j}\rightarrow l_{i}\gamma$
The dominant graph is due to scalar exchange in the loop. One of them is due
to Higgs exchange as shown in Fig.[12]. The amplitude for this process is
given as
$M_{j\rightarrow
i\gamma}=\sum_{n,m}\int\frac{d^{4}k}{(2\pi)^{4}}\bar{u}_{i}(p^{\prime}){\Lambda^{i}_{L}}^{\dagger}{Y^{\prime}_{E}}\frac{\hat{p}^{\prime}+M_{n}}{\hat{p}^{\prime
2}-M_{n}^{2}}e\gamma^{\mu}\frac{\hat{p}+M_{n}}{\hat{p}^{2}-M_{n}^{2}}v{Y^{\prime}_{E}}^{\dagger}\frac{\hat{p}+M_{m}}{\hat{p}^{2}-M_{m}^{2}}Y^{\prime}_{E}\Lambda^{j}_{R}u_{i}(p)\frac{1}{k^{2}-m_{H}^{2}}\epsilon_{\mu}$
(46)
where $\hat{p}=p-k$, $\hat{p}^{\prime}=p^{\prime}-k$ and $q=p-p^{\prime}$.
$\Lambda^{i}_{L,R}=F_{L,R}^{i}D_{L,R}$ and $M_{n}$ denotes the mass of the
$n^{th}$ mode KK fermion. $F_{L,E}$ is a function of bulk masses which are
taken to be diagonal in the flavour space. It is given as
$F_{L,R}=\begin{bmatrix}f_{c_{L_{1}},c_{E_{1}}}(\pi R)&0&0\\\
0&f_{c_{L_{2}},c_{E_{2}}}(\pi R)&0\\\ 0&0&f_{c_{L_{3}},c_{E_{3}}}(\pi
R)\end{bmatrix}$ (47)
Figure 12: Higgs mediated $j\rightarrow i\gamma$. The dot represents the mass
insertion. Flavour indices have been suppressed in the internal charged KK
lines. (L,R) represents the KK modes corresponding to the left and right
chiral zero modes.
The amplitude for Eq.(46) can be rewritten as
$M(j\rightarrow
i\gamma)=(eD_{L}^{\dagger}F_{L}Y^{\prime}_{E}{Y^{\prime}_{E}}^{\dagger}vY^{\prime}_{E}F_{R}D_{R})_{ij}J(\hat{p},\hat{p}^{\prime},q)$
(48)
The expression $J(\hat{p},\hat{p}^{\prime},q)$ is the momentum integral in
Eq.(46). It is log divergent owing to a double-independent sum over two KK
modes. We regularise it using a cutoff of $\Lambda\sim 4\pi M^{(1)}_{kk}\sim
15$ TeV. The other dominant contribution is due to Fig.[19] is discussed in
Appendix[B]. The Branching fraction for the dipole decays $l_{j}\rightarrow
l_{i}\gamma$ is given as
$BR(l_{j}\rightarrow
l_{i}\gamma)=\frac{12\pi^{2}}{(G_{F}m_{j}^{2})^{2}}(A_{L}^{2}+A_{R}^{2})$ (49)
where the coefficient due to Figs.[12,19] is given as
$A_{L}=2\frac{em_{j}}{16\pi^{2}}\frac{1}{M_{KK}^{2}}\frac{v}{\sqrt{2}}D_{L}^{\dagger}F_{L}(Y^{\prime}_{N}{Y^{\prime}_{N}}^{\dagger}+Y^{\prime}_{E}{Y^{\prime}_{E}}^{\dagger})Y_{E}F_{R}D_{R}$
(50)
and $A_{R}=A_{L}^{\dagger}$. The other dipole contributions are discussed in
Appendix[B]. We now proceed to discuss the LFV rates for the mass models
discussed in Section[II]. The quantities, like the KK masses of fermions, the
rotation matrices $D_{L,R}$ etc. which determine the LFV rates are functions
of the bulk mass parameters. We compute these quantities for each point of the
best fit parameter space obtained earlier for the LHLH and the Dirac case and
use it to constrain the parameter space from flavour violation.
### III.3 LHLH Case
The contributions to trilepton decays from graphs like Fig.[11] are highly
suppressed in the parameter space of interest. This is because the couplings
of the zero mode fermions to the KK gauge boson become universal for the
fermions sufficiently localized towards IR and UV branes, as can be seen in
Fig.[13]. However, there could be other potentially large contributions. This
comes from the large mixing between zero mode charged singlet states and the
first KK modes of the lepton doublets; the corresponding Yukawa coupling is
very large due to the large negative $c_{E}$ values. Example of such a graph
is shown in Fig.[14]. Exact value of the contribution, of course depends on
the values of $D_{L,R}$ and other parameters. We have not considered these
graphs in the present work. We note that for a fairly degenerate bulk doublet
masses, ($c_{L_{i}}$), the combination of the matrices which enter in these
graphs are aligned with the zero mode mass matrix for charged leptons. The
best parameter space does contain such regions where all the $c_{L_{i}}$ are
degenerate. We found several examples of that kind. Another potential problem
with the highly localized IR charged singlets, is the shift in the universal
coupling constant $g_{R}$. This could effect $Z\rightarrow ll$ branching
fractions. Models with custodial symmetries or very heavy KK gauge bosons
could avoid this problem. We have not addressed this issue here.
Finally, contribution to $l_{j}\rightarrow l_{i}\gamma$ due to loop diagrams
of the form in Fig.[20] are heavily suppressed owing to the heavy KK mass
scales corresponding to the charged singlets. The corresponding masses are in
shown in Table[2]. Additionally, the large effective 4-D Yukawa couplings of
the charged singlets to the KK modes make it difficult to apply techniques of
perturbation theory to calculate graphs like those in Fig.[12,19].
Figure 13: Coupling of two zero mode fermions to $Z_{1}$ as a function of bulk
mass parameter Gher. Figure 14: Additional tree level contribution to
$\mu\rightarrow eee$. For a fairly degenerate bulk doublet mass in LHLH case
this contribution is negligible. For the Dirac case this graph receives wave
function suppression in addition to the KK scale suppression.
### III.4 Constraints on Dirac Neutrinos
The Dirac case gives a good fit to the leptonic data for a reasonable choice
of $\mathcal{O}$(1) parameters. However, the parameter space is strongly
constrained from flavour considerations. In the parameter space of interest
the dominant contribution to tree-level decays comes from Fig.[11]. The
parameter space of the bulk doublets and charged singlets consistent with tree
level contribution is shown in Fig.[15]. The lightest $M_{Z^{(1)}}$ mass
required to satisfy all constraints from tree-level processes $\sim
1.9~{}\text{TeV}$. Fig.[15] shows the points within the best fit parameter
space consistent with all constraints from tree-level processes. As can be
seen from the figure, very few points pass the constraints. The black point is
allowed for a KK gauge boson scale of $1.9$ TeV, where as the green points are
for mass of 3 TeV.
|
---|---
|
Figure 15: The black dot and the green region represent the parameter space
permitted by tree-level constraints for a KK gauge boson scale of 1920 and
3000 GeV respectively
The constraints from dipole processes are far more severe. Corresponding to
the $c_{L,E}$ values in the best fit parameter space, the mass of the first KK
excitation of the leptons varied from approximately 850 GeV to 1400 GeV as
presented in Table (4). We found no points which satisfied the constraints
from $\mu\rightarrow e\gamma$, $\tau\rightarrow e\gamma$ and
$\tau\rightarrow\mu\gamma$ simultaneously. The constraint from $\mu\rightarrow
e\gamma$ was most severe and required a KK fermion mass scale
$\mathcal{O}$(10) TeV to suppress it below the experimental limit given in
Table [LABEL:lfv-tab].
### III.5 Constraints on scenarios with bulk Majorana mass
The tree-level decays only constrain the parameter space of the bulk doublets
and charged singlets as seen in Fig.[15]. Since, the charged lepton mass
fitting is independent of any right handed neutrino parameter, the constraints
coming from tree-level decays in the Dirac case are applicable in this case as
well.
The contribution to dipole decays of the form $l_{j}\rightarrow l_{i}\gamma$
due to charged Higgs shown in Fig.[19] is small. This is because, as shown in
Table[6], $g_{L}^{(1)}(\pi R)$ is required to be small to fit neutrino masses.
Thus, the dominant contribution to dipole decays in this case is due to Higgs
exchange diagram shown in Fig.[12]. They are calculated for the both the
normal and inverted hierarchy cases presented earlier and are given in
Table[8]. The branching fractions are evaluated for $M_{KK}\sim 1250$ GeV
which is the first KK scale of the doublet.
Table 8: BR for dipole decays for the case with bulk Majorana mass Hierarchy | BR($\mu\rightarrow e\gamma$) | BR($\tau\rightarrow\mu\gamma$) | BR($\tau\rightarrow e\gamma$)
---|---|---|---
Inverted | $2.4\times 10^{-5}$ | $1.9\times 10^{-5}$ | $7.6\times 10^{-6}$
Normal | $1.4\times 10^{-5}$ | $3.4\times 10^{-5}$ | $1.3\times 10^{-5}$
## IV Minimal Flavor Violation(MFV)
From the discussion above it is clear that lepton flavor violating constraints
are strong on RS models with fermions localized in bulk and Higgs localized on
the IR brane. In the Dirac and the Bulk Seesaw case flavor violation rules out
most of the ‘best fit’ parameter space. One option to evade these bounds would
be to increase the scale of KK masses. As we have seen in the LHLH case, the
fits indicate to the highly hierarchal spectrum with KK masses of the
$\mathcal{O}(10^{2})$ TeV for the singlet charged leptons, the flavor
violating amplitudes are highly suppressed and thus do not put severe
constraint on the model. However, the Dirac and the Majorana cases whose best
fit regions have lighter KK spectrum would essentially be ruled out. The
misalignment between the Yukawa coupling matrix and bulk mass terms which
determine the profile is the cause of the large flavor violating transitions
leading to strong restrictions on these models. In a4delaunay the authors
imposed discrete symmetries to constrain Flavour Changing Neutral Currents
(FCNC). In this work we adopt the Minimal Flavour violation ansatz which
reduces the misalignment by demanding an alignment between the Yukawa matrices
and the bulk parameters.
The ansatz of Minimal Flavour violation was first proposed for the hadronic
sector mfv1 . It proposes that new physics adds no new flavor structures and
thus entire flavor structure in Nature is determined by the Standard Model
Yukawa couplings. In the leptonic sector, MFV in not uniquely defined due to
the possibility of the seesaw mechanism. Several schemes of leptonic minimal
flavor violation are possible cirigliano .
The proposal to use the MFV hypothesis in RS was first introduced in
Fitzpatrick in the quark sector. There were subsequent extensions in the
leptonic sector by perez ; Chen . The MFV ansatz assumes that the Yukawa
couplings are the only sources of flavor violation. In the RS setting this
would require that the bulk mass terms should now be expressed in terms of the
Yukawa couplings Fitzpatrick . The exact expression would depend on the
particle content and the flavor symmetry assumed.
### IV.1 Dirac Neutrino Case
In the presence of right handed neutrinos the flavour group is
$SU(3)_{L}\times SU(3)_{E}\times SU(3)_{N}$; the lepton number is conserved.
The $Y_{E}$ transforms as $Y_{E}\rightarrow(3,\bar{3},1)$ and $Y_{N}$
transforms as $Y_{N}\rightarrow(3,1,\bar{3})$. The Yukawa couplings are
aligned with the five dimensional bulk mass matrices. The bulk masses can be
expressed in terms of the Yukawas as
$c_{L}=a_{1}I+a_{2}{Y^{\prime}}_{E}Y^{\prime\dagger}_{E}+a_{3}Y^{\prime}_{N}Y^{\prime\dagger}_{N}\;\;\;\;\;c_{E}=bY^{\prime\dagger}_{E}Y^{\prime}_{E}\;\;\;\;\;c_{N}=cY^{\prime\dagger}_{N}Y^{\prime}_{N}$
(51)
where a,b,c $\in\Re$ and $Y^{\prime}_{E,N}$ are as defined earlier as
$Y^{\prime}_{E,N}=2kY_{E,N}$. Owing to the flavor symmetry we work in a basis
in which $Y^{\prime}_{E}$ is diagonal. We then rotate $Y^{\prime}_{N}$ by the
PMNS matrix i.e, writing $Y^{\prime}_{N}\rightarrow
V_{PMNS}\text{Diag}(Y^{\prime}_{N})$ where the
$\text{Diag}(Y^{\prime}_{N})=\text{Diag}(0.709,0.709,0.75)$. The $c_{L}$ value
chosen is $0.5802$ for all three generations. The $c_{N}$ values chosen are
respectively 1.17241, 1.172, 1.311 respectively. The bulk singlet mass
parameters are $c_{E}=(0.7477,0.58059,0.401)$
The simplest Yukawa combination transforming as (8,1,1) under the flavour
group is given as
$\Delta=Y^{\prime}_{N}Y^{\prime\dagger}_{N}$ (52)
Thus the BR for $\mu\rightarrow e\gamma$, which is the most constrained is
given as perez
$BR(\mu\rightarrow e\gamma)=4\times
10^{-8}~{}(Y^{\prime}_{N}Y^{\prime\dagger}_{N})^{2}_{12}~{}\Big{(}\frac{3\text{TeV}}{M_{KK}}\Big{)}^{4}$
(53)
$Y^{\prime}_{N}=\begin{bmatrix}0.586033&0.383951&0.115044\\\
-0.335962&0.370429&0.53165\\\ 0.215349&-0.466953&0.516346\end{bmatrix}$ (54)
The (1,2) element of $\Delta$ which is responsible for $\mu\rightarrow
e\gamma$ is 0.006 which gives a contribution of $1.44\times 10^{-12}$, for a
fermion KK mass of around 3 TeV.
### IV.2 Bulk Majorana mass term
Owing to the presence of a bulk Majorana mass term, we choose the flavour
group for the lagrangian in Eq.(22) is $SU(3)_{L}\times SU(3)_{E}\times
O(3)_{N}$. $Y_{E}$ transforms as $Y_{E}\rightarrow(3,\bar{3},1)$ and $Y_{N}$
transforms as $Y_{N}\rightarrow(3,1,3)$. The bulk Majorana term $\bar{N}^{c}N$
transforms as $(1,1,6)$ under this flavour group. In terms of the
dimensionless Yukawa couplings, $Y^{\prime}_{E,N}$ the bulk mass parameters
can be expressed as
$c_{L}=a_{1}I+a_{2}{Y^{\prime}}_{E}Y^{\prime\dagger}_{E}+a_{3}Y^{\prime}_{N}Y^{\prime
T}_{N}\;\;\;\;\;c_{E}=1+bY^{\prime\dagger}_{E}Y^{\prime}_{E}\;\;\;\;\;c_{N}=1+cY^{\prime
T}_{N}Y^{\prime}_{N}\;\;\;\;\;c_{M}=dI_{3\times 3}$ (55)
where a,b,c,d $\in\Re$. $c_{M}=0.55$ and $c_{N}=0.58$ are chosen for the right
handed neutrino bulk mass parameters. The value of profiles for the singlets
are chosen appropriately at the boundary so as to fit the neutrino data using
the $\mathcal{O}$(1) Yukawa couplings. As before we work in a basis in which
$Y^{\prime}_{E}$ is diagonal. In this basis
$Y^{\prime}_{N}=V_{PMNS}\text{Diag}(Y^{\prime}_{N})$. This removes the
dominant contribution to dipole decays due to the Higgs exchange in Fig.[12].
The contribution due to Fig.[19] is very small owing to wavefunction
suppression of the singlet neutrinos. Thus, we see that the MFV ansatz is
successful in suppressing FCNC’s for both the Dirac and the bulk Majorana
case.
## V Summary and Outlook
Understanding neutrino masses and mixing is an important aspect of most
physics beyond the Standard Model frameworks. The Randall-Sundrum setup while
solving the hierarchy problem could also form a natural setting to explain
flavour structure of the Standard Model Yukawa couplings. The quark sector has
already been explored in this context in detail. While there have been several
analysis in the leptonic sector, in the present work we have tried to explore
the same in a comprehensive manner, filling the gaps wherever we found it
necessary. Our aim had been to determine quantitavely the parameter space of
both the $\mathcal{O}(1)$ (dimensionless) Yukawa couplings as well as the bulk
mass parameters which can give good fits to the leptonic data.
We have concentrated on the RS setup with the Higgs field localized on the IR
boundary. We have considered three cases of neutrino mass models (a) The LH LH
higher dimensional operator (b) The Dirac case and the (c) Majorana case. The
LHLH fits require large negative c-parameters which reflect the composite
nature of the charged singlets. There is some parameter space in this case
where the flavor constraints are weak. However, the model has very large
effective 4-D Yukawa couplings between the zero mode SM fermions and the KK
fermions, which makes it unattractive from perturbation theory point of view.
We have also presented the distributions of the Yukawa couplings in the best
fit region. Most of the individual Yukawa couplings are concentrated on the
higher side of the $\mathcal{O}(1)$ range we have chosen. The Dirac and
Majorana cases offer large parameter space without the need of large
hierarchies in the $c$ parameters. We have also presented the distribution of
the Yukawa couplings in the Dirac case. We could not find strong correlations
between the Yukawa couplings and the $c$-parameters. There are strong
constraints from the lepton flavor violating rare processes. These can be
circumvented by a suitable choice of Yukawa couplings and c-parameters guided
by the MFV ansatz. The Majorana case, in particular allows for several classes
of MFV schemes, which will be explored in an upcoming publication ourrs2 .
While we restricted ourselves to the Higgs located on the IR brane, it can
also be allowed to propagate in the bulk. Lepton flavor violating amplitudes
however are now cut-off independent, which makes the computations more
predictive. But with the Higgs boson in the bulk one has to invoke other
scenarios like supersymmetry to solve the hierarchy problem.
Acknowledgments
We thank Bhavik Kodrani for important and interesting inputs. We appreciate D.
Chowdhury and R. Garani’s help with the numerics. We also thank V.S. Mummidi
for carefully reading the manuscript. SKV acknowledges support from DST
Ramanujam fellowship SR/S2/RJN-25/2008 of Government of India.
## Appendix A Inverted Mass fits
We present the results of the scan performed for inverted hierarchy for both
the LLHH and the Dirac case. In the case for the normal hierarchy it was
easier to find c values and order one Yukawa entries which satisfied all
constraints. However, the choice of these parameters which fits the data in
the inverted case is very subtle. This is because one requires two large mass
eigenvalues in the inverted case which must satisfy the $\Delta m^{2}_{sol}$
constraint. This requires a very careful choice of order one Yukawa
parameters. The parameter space for c values does not differ much between the
normal and the inverted case. For the case of inverted hierarchy, we choose
points which satisfy $0<\chi^{2}<10$. For the Dirac case we performed a scan
only for $c>0.5$.
(A) LHLH case
Table 9: Sample points for Inverted Hierarchy in LHLH case with O(1) Yukawas. The masses are in GeV Point | A | B
---|---|---
$\chi^{2}$ | 7.48 | 6.61
$c_{L_{1}}$ | 0.8967 | 0.9162
$c_{L_{2}}$ | 0.8983 | 0.8920
$c_{L_{3}}$ | 0.8913 | 0.8945
$c_{E_{1}}$ | -3758.1502 | -2099.8993
$c_{E_{2}}$ | -6005847.4955 | -552577.8188
$c_{E_{3}}$ | -32730342.0982 | -23953472.2265
$m_{e}$ | $5.11\times 10^{-4}$ | $5.09\times 10^{-4}$
$m_{\mu}$ | 0.1056 | 0.1056
$m_{\tau}$ | 1.775 | 1.755
$\theta_{12}$ | 0.584 | 0.55
$\theta_{23}$ | 0.829 | 0.875
$\theta_{13}$ | 0.148 | 0.160
$\delta m_{sol}^{2}$ | $7.49\times 10^{-23}$ | $7.46\times 10^{-23}$
$\delta m_{atm}^{2}$ | $1.90\times 10^{-21}$ | $2.7\times 10^{-21}$
Yukawa for Point A
$Y^{\prime}_{E}=\begin{bmatrix}0.8249&0.8516&1.1111\\\ 1.3600&1.5956&1.8402\\\
3.5831&3.5664&2.9092\end{bmatrix}\;\;\;;\;\;\;\kappa^{\prime}=\begin{bmatrix}-3.5528&2.6612&1.4503\\\
2.6612&3.8149&1.2903\\\ 1.4503&1.2903&-0.6682\end{bmatrix}$ (56)
Yukawa for Point B
$Y^{\prime}_{E}=\begin{bmatrix}2.5874&0.5123&3.6064\\\ 3.9696&2.4876&1.9903\\\
3.8604&1.1438&3.9712\end{bmatrix}\;\;\;;\;\;\;\kappa^{\prime}=\begin{bmatrix}-3.6860&-3.6778&3.9987\\\
-3.6778&2.1362&3.3252\\\ 3.9987&3.3252&-0.8497\end{bmatrix}$ (57)
(B) Dirac Case
Table 10: Sample points for Inverted Hierarchy in Dirac case with O(1) Yukawas. The masses are in GeV Parameter | Point A | Point B
---|---|---
$\chi^{2}$ | 0.30 | 8.04
$c_{L_{1}}$ | 0.5565 | 0.51
$c_{L_{2}}$ | 0.5556 | 0.5316
$c_{L_{3}}$ | 0.5433 | 0.5012
$c_{E_{1}}$ | 0.7681 | 0.8092
$c_{E_{2}}$ | 0.6186 | 0.6498
$c_{E_{3}}$ | 0.5044 | 0.5674
$c_{N_{1}}$ | 1.2450 | 1.2765
$c_{N_{2}}$ | 1.2421 | 1.2755
$c_{N_{3}}$ | 1.2546 | 1.2941
$m_{e}$ | $5.1\times 10^{-4}$ | $5.08\times 10^{-4}$
$m_{\mu}$ | 0.1055 | 0.1055
$m_{\tau}$ | 1.769 | 1.81
$\theta_{12}$ | 0.59 | 0.59
$\theta_{23}$ | 0.80 | 0.72
$\theta_{13}$ | 0.155 | 0.152
$\delta m_{sol}^{2}$ | $7.49\times 10^{-23}$ | $7.48\times 10^{-23}$
$\delta m_{atm}^{2}$ | $2.40\times 10^{-21}$ | $2.16\times 10^{-21}$
Yukawa for Point A
$Y^{\prime}_{E}=\begin{bmatrix}2.2645&2.7691&0.4272\\\
1.0499&-3.6695&-1.0818\\\
-2.2402&-0.5400&-1.9176\end{bmatrix}\;\;\;;\;\;\;Y_{N}^{\prime}=\begin{bmatrix}-0.2202&-2.3054&1.5602\\\
3.4794&-2.2140&0.2302\\\ -2.0676&-1.7529&0.7888\end{bmatrix}$ (58)
Yukawa for Point B
$Y^{\prime}_{E}=\begin{bmatrix}-3.7916&-0.3960&-2.5573\\\
1.2699&-2.3757&3.2167\\\
-3.5010&3.4430&2.8224\end{bmatrix}\;\;\;;\;\;\;Y_{N}^{\prime}=\begin{bmatrix}-3.9443&-0.9714&0.1848\\\
-2.5788&0.2609&3.3684\\\ 0.5020&-3.0268&-3.1765\end{bmatrix}$ (59)
|
---|---
Figure 16: The plot represents the parameter space for the bulk masses of charged doublets for inverted hierarchy |
---|---
Figure 17: The plot represents the parameter space for the bulk masses of charged singlets for inverted hierarchy |
---|---
Figure 18: The plot represents the parameter space for the bulk masses of
neutrino singlets for inverted hierarchy
## Appendix B Amplitudes for dipole transitions
In this section we review the other potential contributions to the dipole
processes $i\rightarrow j\gamma$
a) Internal flip in neutrino KK line in Dirac Case
This contribution is absent for the LHLH case as it involves neutral internal
KK lines corresponding to the right handed neutrino. In the unitary gauge the
charged Higgs is nothing but the longitudnal component of the W boson.
Figure 19: “Charged” Higgs mediated $j\rightarrow i\gamma$. The dot represents
the mass insertion. Flavour indices have been suppressed in the internal
neutral KK lines. (L,R) represents the KK modes corresponding to the left and
right chiral zero modes respectively.
This displays a similar divergence to Fig.[12] owing to the presence of double
KK sum.
$M_{j\rightarrow
i\gamma}=(F_{L}Y^{\prime}_{N}{Y^{\prime}_{N}}^{\dagger}e\frac{v}{\sqrt{2}}Y^{\prime}_{E}F_{E})_{ij}\int\sum_{n,m}\frac{d^{4}k}{(2\pi)^{4}}\bar{u}_{i}(p^{\prime})(2k^{\mu}-q^{\mu})\frac{(\hat{p}^{\prime}+M_{n})}{\hat{p}^{\prime
2}-M_{N}^{2})}\frac{\hat{p}+M_{n}}{\hat{p}^{2}-M_{m}^{2}}\frac{1}{k^{2}-m_{H}^{2}}\frac{1}{(k-q)^{2}-m_{H}^{2}}u_{j}(p)$
(60)
b) Gauge contribution
Additional contributions arise due to KK gauge bosons in the loop as shown in
Fig.[20]
Figure 20: Contribution to the dipole graph due exchange of KK gauge bosons
and charged KK fermion lines.
The amplitude for Fig.[20] is given as
$M_{j\rightarrow
i\gamma}=(A^{0,n,l}\frac{v}{\sqrt{2}}{Y^{\prime}_{E}}A^{0,m,l})_{ij}\sum_{n,m}\int\frac{d^{4}k}{(2\pi)^{4}}\bar{u}_{i}(p^{\prime})\frac{\hat{p}^{\prime}+M_{n}}{\hat{p}^{\prime
2}-M_{n}^{2}}e\gamma^{\mu}\frac{\hat{p}+M_{n}}{\hat{p}^{2}-M_{n}^{2}}\frac{\hat{p}^{\prime}+M_{m}}{\hat{p}^{\prime
2}-M_{m}^{2}}u_{j}(p)\frac{1}{k^{2}-m_{H}^{2}}$ (61)
where $A^{0,n,l}$ represents the coupling of zero mode fermion to $n^{th}$
mode fermion and $l^{th}$ mode gauge boson. The contribution from this sector
is suppressed in both the Dirac and LHLH case in the parameter space under
consideration.
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|
arxiv-papers
| 2012-06-20T04:42:45 |
2024-09-04T02:49:31.980984
|
{
"license": "Public Domain",
"authors": "Abhishek M. Iyer and Sudhir K. Vempati",
"submitter": "Sudhir Vempati",
"url": "https://arxiv.org/abs/1206.4383"
}
|
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