We explore entanglement entropy of a cap-like region for a generic quantum field
theory residing in the Bunch-Davies vacuum on de Sitter space. Entanglement
entropy in our setup is identical with the thermal entropy in the static patch of de
Sitter, and we derive a simple relation between the vacuum expectation value of the
energy-momentum tensor trace and the RG flow of entanglement entropy. In
particular, renormalization of the bare couplings and logarithmic divergence of the
entanglement entropy are interrelated in our setup. We confirm our findings by
recovering known universal contributions for a free field theory deformed by a
mass operator as well as obtain correct universal behaviour at the fixed
points. Simple examples of entanglement entropy flows are elaborated in
Article funded by SCOAP3
$, the (divergent) constant $c'$ drops out of connected correlators. } \be R\frac{d S_{\mt{EE}}}{d R} = {2\pi\over d} \, R{d\over dR} \({C\over R^d}\) \int_A \xi\cdot n = { \Omega_d R^{d+1} \over d} \,{d\over dR} \({C\over R^d}\)\,, \labell{dSdR} \ee where $\Omega_d$ is the surface area of a unit $d$-dimensional sphere, \be \Omega_d={2\pi^{d+1\over 2}\over \Gamma\({d+1\over 2}\)}~. \ee It worth mentioning that \reef{dSdR} relates finite quantities on both sides of the equation since $C$ is finite by definition. In particular, this formula eliminates the power law divergent terms of EE such as the well known `area law'. Of course, there is nothing bad about absence of these terms since they are scheme dependent and therefore vanish under appropriate choice of the regularization scheme (\eg dimensional regularization). However, the logarithmic divergence of EE is always retained, and its coefficient is scheme independent. This observation may cast doubts on possible applications of \reef{dSdR} for a QFT in even dimensions, where the universal EE is associated with a logarithmic divergence. In section~\ref{free} we elaborate on simple examples which clarify this subtlety. We show that the universal EE in our setup is directly related to the logarithmic divergences of the bare parameters defining the theory, and the standard renormalization results in a finite EE with the universal data being encoded in the coefficient of the (finite) logarithmic running. Throughout the paper we apply dimensional regularization scheme, and therefore non-universal divergences of EE will be absent from our calculations. ]]>
$,\footnote{When deriving this relation it should be noted that the difference between $\lambda\frac{\partial C}{\partial\lambda}$ and $\lambda\frac{\partial S}{\partial\lambda}$ stems from the fact that $\left<\frac{\del T_{\mu\nu}}{\del \lambda}\right>\sim g_{\mu\nu}\left
2$. Such divergences are scheme dependent, and therefore one can choose a particular scheme where it vanishes, \eg in dimensional regularization one gets $\chi^{2-d}\to 0$ for $d<2$, and therefore analytic continuation to higher $d$ also vanishes.} we have \be \langle \phi^2 \rangle = {\Gamma(1-d/2)\Gamma(\lambda)\Gamma(d-1-\lambda)\over \pi (4\pi)^{d/2} R^{d-2}} \sin\bigg({\pi\over 2}(d-2\lambda)\bigg) ~. \labell{vacphi2} \ee In particular, $\langle \phi^2 \rangle$ is finite for odd $d$, whereas for even $d$ it exhibits a simple pole which corresponds to the logarithmic divergence. We treat these cases separately. \paragraph{\boldmath Odd $d$.} For odd $d$, we have \bea C_\phi&=& {\pi (mR)^2 \coth(\pi\sqrt{m^2R^2-1/4})\over \sqrt{m^2R^2-1/4}} { (-)^{d-1 \over 2} \over (4\pi)^{d/2} \Gamma\({d\over 2}\) } \prod_{j=1}^{d-1\over 2} \( (d/2-1/2-j)^2-1/4+m^2R^2\) \non &=&{\pi (mR)^2 \coth(\pi\sqrt{m^2R^2-1/4})\over \sqrt{m^2R^2-1/4}} { (-)^{d-1 \over 2} \over (4\pi)^{d/2} \Gamma\({d\over 2}\) } \\ && \times \Bigg( (mR)^{d-1}+{(d-1)(d^2-5d+3)\over 24}(mR)^{d-3} +\ldots+{\pi\over 2\,\Gamma\({2-d\over 2}\)\Gamma\({4-d\over 2}\)}\Bigg)~. \nonumber \eea Appearance of the hyperbolic function in both the $C_\phi$ and the entanglement entropy after using \reef{dSdR} is not surprising. Indeed, such functions are the direct associates of any thermal computation, whereas the state in de Sitter space has thermal interpretation. To get rid of the thermal effects, let us consider the behaviour of entanglement entropy in the IR limit, $mR\gg 1$. In this regime curvature corrections are negligibly small, while thermal effects are exponentially suppressed, and we get \begin{flalign} C_\phi &\underset{\mt{$mR\gg 1$}}{=} { (-)^{d-1 \over 2} \pi \over (4\pi)^{d/2} \Gamma\({d\over 2}\) } \( (mR)^d+{d(d-2)(d-4)\over 24}(mR)^{d-2} +\ldots\)\\ R{d S_\mt{univ}^\mt{scalar}\over dR} &\underset{\mt{$mR\gg 1$}}{=} {(d-2)(d-4)\over 24(d-1)}{ (-)^{d+1 \over 2} \pi \over (4\pi)^{d-2\over 2} \Gamma\({d\over 2}\) } A_\Sigma m^{d-2} + \ldots \end{flalign} where ellipsis encapsulate curvature corrections to the leading order term also known as universal `area law'. As shown in appendix~\ref{freeflat}, this term is identical to the universal `area law' of entanglement entropy of a half space for a conformally coupled scalar field residing in the Minkowski vacuum. This result is a consequence of the fact that any surface and any background are locally flat. \paragraph{Renormalized Entanglement Entropy in 3D.} For a vacuum state in 3-dimensional flat space QFT, the general pattern for EE of a disk is \begin{eqnarray} S_\mt{EE} = c_1{R\over \delta} - c_0 \,, \end{eqnarray} where $R$ is the radius of the disk and $\delta$ is a UV cut off. At the fixed point, $c_0$ and $c_1$ are some constants,\footnote{At the IR fixed point there could be additional terms, which are remnants of the RG flow~\cite{Casini-ml-2014yca}, \eg the universal `area law' $\sim\, mR$ for massive QFTs~\cite{Hertzberg-ml-2010uv,Huerta-ml-2011qi,Lewkowycz-ml-2012qr}. However, the characteristic (relevant) scale of such terms is very large in the deep IR, and therefore in this regime they are not really distinguishable from the $R/\delta$ term. Hence, we do not write them out. } whereas outside the fixed point $c_0$ is a function of $R$ and various scales characterizing a given QFT. Obviously, $c_1$ is scheme dependent while $c_0$ is universal, therefore the authors of~\cite{Liu-ml-2012eea,Casini-ml-2012ei,Liu-ml-2013una} defined the so-called renormalized entanglement entropy (REE) to isolate the universal contribution,\footnote{See~\cite{Liu-ml-2012eea,Liu-ml-2013una} for definition of REE in general dimension.} \begin{eqnarray} \mathcal{S}_3 \equiv R\frac{dS_\mt{EE}}{dR} - S_\mt{EE} \equiv RS_\mt{EE}'-S_\mt{EE}~. \end{eqnarray} Using strong sub-additivity of EE~\cite{Lieb-ml-1973cp}, Casini and Huerta proved that $S_\mt{EE}'' \leq 0$ for a disk in flat space. Hence, $\mathcal{S}_3' = RS_\mt{EE}''\leq 0$, which clearly indicates that REE has a monotonic RG flow in flat space, and~\cite{Casini-ml-2012ei} \begin{eqnarray} \Delta c_0 = c_0^\mt{UV} - c_0^\mt{IR} = - \int_0^\infty dR \,R \, S_\mt{EE}''\geq 0 ~. \label{Fthm} \end{eqnarray} Of course, the RG flow itself is ambiguous but the fixed points satisfy an F-theorem~\cite{Jafferis-ml-2011zi,Klebanov-ml-2011gs}. Furthermore, conformal symmetry relates the REE at the fixed points in flat space to the universal EE in our setup~\cite{Casini-ml-2011kv}, hence it is worthwhile to explore REE on a sphere outside the fixed points. For a conformally coupled scalar field, $S''_\mt{EE}$ can be readily evaluated based on \reef{dSdR}, \reef{Cphi} and \reef{vacphi2}. Figure~\ref{fig:S''forscalarin3d} shows the corresponding plot, and it can be seen that $S''_\mt{EE}$ \emph{changes} sign. A similar issue is discussed in~\cite{Klebanov-ml-2011gs} for the free energy on a sphere, where the authors argue that additional subtractions are necessary because of emergence of the cosmological constant term in the IR limit. Although REE is not monotonic, other choices of subtraction scheme might result in a monotonic flow as was suggested for the free energy in~\cite{Klebanov-ml-2011gs}. Moreover, REE in our setup exhibits stationarity at the fixed points, \ie $S''_\mt{EE}\to0$ as $mR\rightarrow0,\,\infty$. This result should be contrasted with numerical studies in~\cite{Klebanov-ml-2012va}, where it has been shown that REE for a disk in flat space is not stationary in the massless limit. Similar considerations for the free energy showed that although the IR divergent free energy obeys stationarity, the existence of a subtraction scheme that maintains stationarity, monotonicity and analyticity is not obvious~\cite{Klebanov-ml-2011gs}. We have calculated $\Delta c_0\sim 0.0638$ which agrees with the expected result from the free energy calculation of a conformally coupled massless scalar in the UV and an empty theory in the IR, in spite of thermal effects mentioned above. \begin{figure} \centering \includegraphics[width=0.7\linewidth]{"Sforscalarin3d"} ]]>
2$.\footnote{The special case $d=2$ can be treated similarly. In this case the entangling surface consists of two disjoint points $\theta'=\pm{\pi\over 2}$, and the integral over $\theta'$ in \reef{TK} should be extended from $-{\pi\over 2}$ to ${\pi\over 2}$ in order to cover all of $S^2$, see \reef{coor}.} Finally, integrating the right hand side of \reef{TK} by parts twice, yields \bea \langle T(x) K \rangle_\mt{con}&=& {2\pi(d-1)\over \Omega_{d-1}} \int_{\mu_\phi}^\infty d\mu \bigg(\xi_c\cR + {d\over R^2} + \mu^2 \bigg) \rho_0(\mu) G_{\mu}^{(2)}\bigg(\tau,\theta,\theta'={\pi\over 2}\bigg)\,, \labell{TK2} \eea where $G_{\mu}^{(2)}(\tau,\theta,\theta'={\pi\over 2})$ is independent of $\tau$ since $\theta'={\pi\over 2}$, and the system exhibits rotational symmetry in the transverse space. As expected, $\langle T(x) K \rangle$ is just a function of $\theta$ which parametrizes the geodesic distance from the entangling surface in the transverse space. Another useful result follows directly from \reef{green} \be \int d^dx \, \sqrt{g(x)} G_\mu(x,y)={1\over \xi_c\cR+\mu^2 }~. \labell{intGreen} \ee Indeed, the integral on the left hand side is convergent since the Green's function is everywhere regular on a sphere except at $x=y$, where it diverges as $\sigma^{2-d}$. However, this divergence is balanced by the integration measure which behaves as $\sigma^{d-1}$. Furthermore, $S^d$ is a maximally symmetric space, and therefore the result of integration is given by some constant independent of $y$. In particular, \reef{intGreen} follows from the following identities \be (\xi_c\cR+\mu^2) \int d^dx \, \sqrt{g(x)} G_\mu(x,y)=(-\nabla^2+\xi_c\cR+\mu^2) \int d^dx \, \sqrt{g(x)} G_\mu(x,y)=1\,, \ee where the last equality rests on \reef{green}. Substituting now \reef{TK2} into \reef{flow2}, and using \reef{G2}, \reef{intGreen} to integrate $G^{(2)}_\mu$over $x$, yields \be R {dS_\mt{EE} \over dR} = - {(d-1) \Gamma\big({d\over2}\big)\over \pi^{d-2\over 2}} \mathcal{A}_\Sigma\int_{\mu_\phi}^\infty d\mu \(1 + {d\over R^2(\xi_c\cR+ \mu^2)} \) \rho_0(\mu) \,, \labell{Sprime} \ee where $\mathcal{A}_\Sigma$ is the area of the entangling surface. The integral on the right hand side of \reef{Sprime} is convergent at the lower bound since sphere introduces a natural IR cut off, but it does not necessarily converges at the upper bound. Indeed, in the UV limit curvature corrections are irrelevant, and convergence of the integral is essentially the same as in Minkowski space. Note also that in the limit $\mu^2\cR\gg1$ we recover the result of~\cite{Casini-ml-2014yca}. Furthermore, if the integral on the right hand side converges, then positivity of the spectral function guarantees $R S_\mt{EE}' \leq 0$ along the RG flow. However, it certainly does not ensure \reef{Fthm} since finite EE at the IR fixed point, $S_\mt{EE}^\mt{IR}$, includes terms which are part of the UV physics. These terms should be subtracted to isolate the universal contribution, $c_0^\mt{IR}$. We elaborate details of this point in the discussion section. ]]>