aainstitutetext: Department of Physics, University of California,
Santa Barbara, CA 93106, USAbbinstitutetext: Kavli Institute for Theoretical
Physics, University of California,
Santa Barbara, CA 93106, USAccinstitutetext: Center for Cosmology and Particle
Physics, Department of Physics, New York University,
New York, NY 10003, USAddinstitutetext: Center for the Fundamental Laws of
Nature, Harvard University,
Cambridge, MA 02138, USAeeinstitutetext: Department of Physics, Cornell
University,
Ithaca, NY 14853, USA

# Holographic entanglement from the UV to the IR

Xi Dong<EMAIL_ADDRESS>a,b,c Grant N. Remmen<EMAIL_ADDRESS>a,d
Diandian Wang<EMAIL_ADDRESS>a,e Wayne W. Weng
<EMAIL_ADDRESS>a and Chih-Hung Wu<EMAIL_ADDRESS>
###### Abstract

In AdS/CFT, observables on the boundary are invariant under renormalization
group (RG) flow in the bulk. In this paper, we study holographic entanglement
entropy under bulk RG flow and find that it is indeed invariant. We focus on
tree-level RG flow, where massive fields in a UV theory are integrated out to
give the IR theory. We explicitly show that in several simple examples,
holographic entanglement entropy calculated in the UV theory agrees with that
calculated in the IR theory. Moreover, we give an argument for this agreement
to hold for general tree-level RG flow. Along the way, we generalize the
replica method of calculating holographic entanglement entropy to bulk
theories that include matter fields with nonzero spin.

##  1 Introduction

Holographic entanglement entropy has played a significant role in
understanding the emergence of spacetime in AdS/CFT, a duality between a bulk
gravitational theory in anti-de Sitter space (AdS) and a boundary conformal
field theory (CFT). At leading order in the large-$N$ expansion, the
entanglement entropy of a subregion in the CFT is related to a geometric
quantity in the bulk. For Einstein gravity, this quantity is simply the area
of an extremal surface, given by the Ryu-Takayanagi (RT) formula Ryu:2006bv ;
Ryu:2006ef ; Hubeny:2007xt :

$S=\operatorname*{ext}_{\gamma}\frac{2\pi}{\kappa^{2}}\int\mathrm{d}^{d}y\,\sqrt{h},\qquad\kappa^{2}=8\pi
G_{N},$ (1)

where the extremization is over all candidate codimension-two surfaces
$\gamma$ with induced metric $h$ and satisfying certain homology constraints.
This formula was derived by Lewkowycz and Maldacena (LM) Lewkowycz:2013nqa
using the gravitational path integral.

For more general bulk gravity theories, such as higher-derivative gravity, the
boundary entanglement entropy is instead given by Dong:2013qoa ; Camps:2013zua
; Dong:2017xht

$S=\operatorname*{ext}_{\gamma}A_{\text{gen}}[\gamma],$ (2)

where $A_{\text{gen}}$ is an entropy functional evaluated on $\gamma$,
generalizing the area in (1). This formula can be derived via a generalization
of the replica method of LM to arbitrary bulk theories.111Nevertheless, we
will continue to refer to this generalization as the LM method. For Einstein
gravity, $A_{\text{gen}}$ reduces to the area of the surface (1) divided by
$4G_{N}$. For the case of $f(\text{Riemann})$ theories, the corresponding
entropy functional was derived in Ref. Dong:2013qoa . For instance, when the
bulk theory includes a Ricci-squared correction, ${\cal L}=(R+\lambda
R_{\mu\nu}R^{\mu\nu})/2\kappa^{2}$, $A_{\text{gen}}$ is given by the integral
of the quantity
$1+\lambda(R_{a}^{\;\;a}-K_{a}K^{a}/2)+{\mathcal{O}}(\lambda^{2})$ over the
extremizing surface, where $K_{a}$ is the trace of the extrinsic curvature
along the directions (or two-dimensional space) orthogonal to
$\gamma$.222Throughout, we will use $a,b,\ldots$ for directions in the normal
bundle to the codimension-two surface $\gamma$, while we use $i,j,\ldots$ for
directions within the surface and Greek letters $\mu,\nu,\dots$ for full
spacetime Lorentz indices. In general, $A_{\text{gen}}$ is a local functional
determined from the action of the bulk theory. We will work to leading order
in the gravitational constant $G_{N}$ and therefore ignore quantum corrections
from bulk fields, which are generally subleading.

Since the holographic entanglement entropy formula relates a quantity on the
boundary to a quantity in the bulk, this raises the question of what happens
if there are different bulk descriptions of the same boundary theory. For the
holographic duality to be consistent, the different bulk descriptions must
give the same result for the boundary quantity, i.e.,

$S=\operatorname*{ext}_{\gamma}A_{\text{gen},1}[\gamma;\Phi_{1}]=\operatorname*{ext}_{\gamma}A_{\text{gen},2}[\gamma;\Phi_{2}].$
(3)

Here $\Phi_{1}$ and $\Phi_{2}$ represent schematically two distinct sets of
dynamical fields (including the metric) that are present in the two bulk
descriptions of the theory and evaluated on-shell in $A_{\text{gen}}$, whereas
$A_{\text{gen},1}$ and $A_{\text{gen},2}$ are the corresponding holographic
entropy functionals. The descriptions encoded by $\Phi_{1,2}$ could represent
any reorganization of the bulk path integral, e.g., two different field
redefinitions of the bulk action or a repackaging in terms of auxiliary
fields.

Figure 1: Commutative diagram for the holographic entanglement entropy
calculation from the UV to the IR under bulk RG flow. The invariance of the
entropy is demonstrated by first computing it in a tree-level UV extension
with massive nonminimally coupled matter; upon integrating out the massive
fields, the IR entropy agrees with that computed directly from the IR EFT.

One general situation where two bulk theories can have the same boundary
description occurs in a bulk renormalization group (RG) flow, i.e., where the
actions of $\Phi_{1,2}$ represent the same bulk theory evaluated at different
energy scales. The boundary CFT is not sensitive to the scale at which the
bulk effective field theory (EFT) is defined.333This is different from the
boundary RG flow (which holographically corresponds to changing a radial
cutoff in the bulk). For this reason, Eq. (3) specializes to

$S=\operatorname*{ext}_{\gamma}A_{\text{gen},\text{UV}}[\gamma;g_{\rm
UV},\Phi]=\operatorname*{ext}_{\gamma}A_{\text{gen},\text{IR}}[\gamma;g_{\rm
IR}],$ (4)

where we focus on situations in which the IR theory is obtained by integrating
out massive matter fields $\Phi$ in the UV theory,
${\mathcal{L}}_{\text{UV}}\left[g_{\rm
UV},\Phi\right]\to{\mathcal{L}}_{\text{IR}}\left[g_{\rm IR}\right]$, and
$A_{\rm gen,IR}$, $A_{\rm gen,UV}$ are the holographic entropy functionals in
the IR and UV theory, respectively, with $g,\Phi$ evaluated on-
shell.444Furthermore, we will find in explicit examples of gravitational EFTs
and their tree-level UV extensions that a stronger version of Eq. (4) holds,
equating $A_{\text{gen},{\rm IR}}[g_{\text{IR}}]$ with $\lim_{{\rm
UV}\rightarrow{\rm IR}}A_{\text{gen},{\rm UV}}[g_{\text{UV}},\Phi]$ at the
functional level—i.e., at the level of the entropy functional before
extremization—where the ${\rm UV}\rightarrow{\rm IR}$ limit denotes imposition
of the equations of motion for $\Phi$ at low energies. Here, $g_{\text{UV}}$
is a metric solution in the UV theory, and $g_{\text{IR}}$ is the
corresponding one in the IR. Requiring Eq. (4) to hold is a nontrivial
consistency check for the holographic entropy formulas.

Our main goal is to show that the UV entropy flowed to the IR matches the
entropy computed in the IR theory, as summarized in Fig. 1 as a commutative
diagram. We do this by first demonstrating the matching in a few examples,
namely certain UV extensions of four-derivative gravity, and then providing a
general argument for all tree-level UV extensions of gravitational EFTs.

Throughout, we work in the formalism of gravitational EFTs, where the leading
corrections to the Einstein-Hilbert action are given by the four-derivative
curvature-squared terms $R^{2}$, $R_{\mu\nu}R^{\mu\nu}$, and
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. To study the RG flow in these
theories, we introduce one-particle UV extensions, where the higher-derivative
terms are generated at the tree level by integrating out massive fields
nonminimally coupled to the curvature.555For example, in four dimensions one
could couple a scalar $\phi$ of mass $m$ via the interaction term $M_{\rm
Pl}\phi R$, generating $M_{\rm Pl}^{2}R^{2}/m^{2}$ at low energies. In this
way, the introduction of $\phi$ allows the four-derivative term to be replaced
with one of lower mass dimension, while the scale at which the EFT breaks down
is raised from $m$ to $M_{\rm Pl}$. As a rule, we will refer to such theories
as UV extensions rather than UV completions; what happens above their regime
of validity will not affect our analysis.

We emphasize that while we were initially inspired by the conjecture of
Susskind and Uglum Susskind:1994sm  which states that the generalized
entropy—defined as the sum of $A_{\text{gen}}$ and the bulk entropy
$S_{\text{bulk}}$—is invariant under RG flow, the statement that we want to
make here is only about the behavior of $A_{\text{gen}}$ itself. Indeed, as
most of our discussion involves tree-level effects, we will confine our
consideration to $A_{\rm gen}$ throughout. It would be interesting to
investigate how our results extend to the full generalized entropy beyond the
tree level.

The remainder of the paper is organized as follows. We start by specifying the
tree-level extensions of four-derivative gravity in Sec. 2, and show that the
UV entropies match the entropies derived from the IR theory under the tree-
level RG flow. The toolkit needed for deriving their entropy functionals is
described in Sec. 3, with the details of the derivation presented in App. A.
In Sec. 4, we give a general argument for the entropy matching in tree-level
UV extensions. We summarize our results and discuss certain future directions
in Sec. 5. In App. B, we study holographic entanglement entropy under field
redefinitions in two simple examples.

##  2 Tree-level UV extensions and their entropy functionals

We now study holographic entanglement entropy under bulk RG flow in several
simple examples. A physically motivated way to achieve such an RG flow is via
a UV extension of the higher-derivative interactions where the equations of
motion are second order in the UV.666Throughout, we use the phrase “UV
extensions” rather than “UV completions” to denote integrating in massive
fields that render the equations of motion second order and for which, upon
integrating out these fields at the tree level, we obtain our desired EFTs.
That is, we are replacing interaction terms of high mass dimension with ones
of lower mass dimension. In the case of massive scalars, this will raise the
UV cutoff of the theory, while for massive higher-spin fields the situation is
more subtle Arkani-Hamed:2002bjr ; Cheung:2019cwi . In all cases, we will be
agnostic about the UV completion above the scale of the massive fields, as it
will not play a role in our analysis. We will therefore consider various tree-
level UV extensions of the curvature-squared corrections $R^{2}$,
$R_{\mu\nu}R^{\mu\nu}$, and $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.777We
include $R^{2}$ and $R_{\mu\nu}R^{\mu\nu}$, even though IR EFTs with solely
these corrections are equivalent to vacuum Einstein gravity under field
redefinition, since we will find that working through the formalism
demonstrating invariance of the holographic entropy under RG flow is already
enlightening in these examples. We also discuss such field redefinitions in
App. B.

Studying tree-level UV extensions is both well-motivated and tractable. Tree-
level completions in gravity arise naturally from a string-theoretic
perspective; the nonzero effective curvature-squared terms that appear in
(heterotic, type I, and bosonic) string theory Gross:1986mw ; Tseytlin:1995bi
; Zwiebach:1985uq  are generated in the IR via a tower of massive states
exchanged at the tree level between scattering gravitons. Moreover, an
$\hbar$-counting argument implies that graviton scattering in pure Einstein
gravity must be UV-completed at the tree level in any weakly-coupled
completion CheungRemmen  (as opposed to a loop-level completion, as in the
case of the Euler-Heisenberg Lagrangian), so considering higher-derivative
gravitational terms that are themselves generated at the tree level may be
well-motivated. In an amplitude context, it has been shown that the Källén-
Lehmann spectral representation describing the integrating out of massive
degrees of freedom, even at the loop level, can be viewed as a sum over tree-
level composite states Arkani-Hamed:2020blm .

As in Refs. CheungRemmen ; Cheung:2018cwt ; Cheung:2019cwi , we therefore
integrate in new states at the tree level as a UV extension of the original
EFT.888This is similar to the generation of a $(\partial\pi)^{4}$ term via
$h(\partial\pi)^{2}$ in a linear sigma model of a Goldstone $\pi$ and Higgs
field $h$ Adams:2006sv . This allows us to access features of the RG evolution
of the entropy calculation in toy models within the arena of quantum field
theory and semiclassical gravity while sidestepping the full UV completion of
gravity itself.

We now turn to the construction of our three example theories for generating
the curvature-squared terms, using massive fields of spin zero, two, and four.

###  2.1 Massive scalar extension of $R^{2}$

To generate $R^{2}$, we consider a UV theory of a massive scalar nonminimally
coupled to gravity, reminiscent of a dilaton in a model with broken
supersymmetry. The bulk action is

${\cal S}=\int{\rm
d}^{d+2}x\,\sqrt{-g}\left[\frac{1}{2\kappa^{2}}R+\frac{\alpha}{\kappa}\phi
R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}\right],$ (5)

where $\alpha$ is a dimensionless constant. The $\phi$ equation of motion for
this theory is

$\Box\phi-m^{2}\phi+\alpha\kappa^{-1}R=0.$ (6)

At low energies, we find the solution

$\phi=\frac{\alpha R}{\kappa m^{2}}$ (7)

up to higher order terms in $1/m^{2}$. We now integrate out $\phi$ at the tree
level, plugging (7) into (5) and finding the low energy EFT

${\mathcal{L}}_{{\rm
EFT}}=\frac{1}{2\kappa^{2}}R+\frac{\alpha^{2}}{2\kappa^{2}m^{2}}R^{2},$ (8)

dropping terms going like $(\alpha^{2}/m^{2})\times{\mathcal{O}}(1/m^{2})$.
The entropy functional for the UV theory (5) is derived in App. A.1 and is
given by

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int\mathrm{d}^{d}y\sqrt{h}\,\Big{(}1+2\alpha\kappa\phi\Big{)}.$
(9)

We note that this equation is exact to all orders in $\alpha$ and $1/m^{2}$.
After RG flow to the IR—that is, taking the large-$m$ limit while keeping
$\alpha/m$ fixed—the entropy functional $A_{\text{gen}}$ becomes

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int\mathrm{d}^{d}y\sqrt{h}\,\left(1+\frac{2\alpha^{2}}{m^{2}}R\right).$
(10)

This expression is again exact in $\alpha$, but has been truncated to leading
order in the $1/m^{2}$ expansion, since we have flowed to the IR. This result
matches the entropy formula for the IR theory (8), computed using Eq. (29) and
following the prescription to be reviewed in Sec. 3. Thus, we have
demonstrated entropy matching in this example.

###  2.2 Massive spin-two extension of $R_{\mu\nu}R^{\mu\nu}$

To generate the $R_{\mu\nu}R^{\mu\nu}$ term, we consider a UV theory that
couples the Ricci tensor to a massive spin-two field $\phi_{\mu\nu}$. As noted
in footnote 6, this is not in general a full UV completion, and massive
higher-spin states famously can lead to worse UV divergences or ghosts in the
action unless we carefully choose the kinetic, mass, and interaction terms.
Nonetheless, such considerations will not impact our purposes here, where we
are interested in treating $\phi_{\mu\nu}$ effectively as a massive auxiliary
field, providing us with a system that is equivalent to
$R+R_{\mu\nu}R^{\mu\nu}$ gravity in the IR, while still possessing a second-
order equation of motion that will make the entropy calculation qualitatively
different. As such, we will not require $\phi_{\mu\nu}$ to be a canonical
(i.e., Fierz-Pauli) massive spin-two state Hinterbichler:2011tt .

Thus, we consider the UV action

${\cal S}=\int{\rm
d}^{d+2}x\sqrt{-g}\left(\frac{1}{2\kappa^{2}}R+\frac{\alpha}{\kappa}\phi_{\mu\nu}R^{\mu\nu}-\frac{1}{2}\nabla_{\alpha}\phi_{\mu\nu}\nabla^{\alpha}\phi^{\mu\nu}-\frac{1}{2}m^{2}\phi_{\mu\nu}\phi^{\mu\nu}\right).$
(11)

The equation of motion for $\phi_{\mu\nu}$ is

$\Box\phi_{\mu\nu}-m^{2}\phi_{\mu\nu}+\alpha\kappa^{-1}R_{\mu\nu}=0,$ (12)

which at low energies $\nabla/m\ll 1$ leads to an effective Lagrangian with a
Ricci-squared term,

${\mathcal{L}}_{{\rm
EFT}}=\frac{1}{2\kappa^{2}}R+\frac{\alpha^{2}}{2\kappa^{2}m^{2}}R_{\mu\nu}R^{\mu\nu},$
(13)

again dropping terms going like
$(\alpha^{2}/m^{2})\times{\mathcal{O}}(1/m^{2})$. The UV entropy formula is
derived in App. A.2 at leading order in a $1/m^{2}$ expansion, keeping
$\alpha/m$ fixed. The result is

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+\alpha\kappa\phi_{a}^{a}-\frac{\alpha^{2}}{2m^{2}}K_{a}K^{a}\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right).$
(14)

After RG flow to the IR (which, in this case, amounts to the replacement
$\phi_{\mu\nu}=\alpha R_{\mu\nu}/\kappa m^{2}$), $A_{\text{gen}}$ becomes

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+\frac{\alpha^{2}}{m^{2}}\left(R_{a}^{a}{}-\frac{1}{2}K_{a}{}K^{a}{}\right)\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right),$
(15)

which agrees with the IR entropy formula derived from Eq. (29).

###  2.3 Massive spin-four extension of
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$

To complete our consideration of gravitational EFTs at four-derivative order,
we now turn to the Riemann-squared term. Consideration of Riemann-squared is
in particular necessary if we are to consider Gauss-Bonnet gravity and is
nontrivial because, unlike $R^{2}$ and $R_{\mu\nu}R^{\mu\nu}$, it cannot be
related to Einstein gravity via a field redefinition. Using the tree-level
generation of the Gauss-Bonnet term in the IR EFT of various string theories
Gross:1986mw ; Tseytlin:1995bi ; Zwiebach:1985uq  as motivation, we will
follow Ref. CheungRemmen  and consider a toy model in which the Riemann tensor
is coupled to a single massive quantum field, providing a spin-four-like999We
say “spin-four-like” since a canonical spin-four field is totally symmetric on
its four Lorentz indices, which is incompatible with the symmetries of the
Riemann tensor. Also, as in the case of the spin-two field, we will not be
using a canonical kinetic or mass term, possibly leading to ghosts or
tachyons, though our results will be insensitive to these subtleties.
generalization of the spin-zero and -two cases considered previously.

Suppose that in the UV we have a massive field $\phi_{\mu\nu\rho\sigma}$
possessing all of the index symmetries of the Riemann tensor,
$\phi_{\mu\nu\rho\sigma}=-\phi_{\nu\mu\rho\sigma}=-\phi_{\mu\nu\sigma\rho}=\phi_{\rho\sigma\mu\nu}$
as well as $\phi_{\mu[\nu\rho\sigma]}=0$, with action

${\cal S}\,{=}\\!\int\\!{\rm
d}^{d{+}2}x\sqrt{{-}g}\left(\frac{1}{2\kappa^{2}}R{+}\frac{\alpha}{\kappa}\phi_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}{-}\frac{1}{2}\nabla_{\alpha}\phi_{\mu\nu\rho\sigma}\nabla^{\alpha}\phi^{\mu\nu\rho\sigma}{-}\frac{1}{2}m^{2}\phi_{\mu\nu\rho\sigma}\phi^{\mu\nu\rho\sigma}\\!\right)\\!.$
(16)

The equation of motion for $\phi_{\mu\nu\rho\sigma}$ is

$\Box\phi_{\mu\nu\rho\sigma}-m^{2}\phi_{\mu\nu\rho\sigma}+\alpha\kappa^{-1}R_{\mu\nu\rho\sigma}=0,$
(17)

where, again in the regime $\nabla/m\ll 1$, we have a low-energy effective
Lagrangian with a Riemann-squared term,

${\mathcal{L}}_{{\rm
EFT}}=\frac{1}{2\kappa^{2}}R+\frac{\alpha^{2}}{2\kappa^{2}m^{2}}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(18)

up to terms that go like $(\alpha^{2}/m^{2})\times{\mathcal{O}}(1/m^{2})$. The
UV entropy formula is derived in App. A.3, again at leading order in a
$1/m^{2}$ expansion, keeping $\alpha/m$ fixed. The result is

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+2\alpha\kappa\phi^{ab}{}_{ab}-\frac{2\alpha^{2}}{m^{2}}K_{aij}{}K^{aij}{}\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right).$
(19)

After RG flow to the IR (using the replacement $\phi_{\mu\nu\rho\sigma}=\alpha
R_{\mu\nu\rho\sigma}/\kappa m^{2}$), $A_{\text{gen}}$ becomes

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+\frac{2\alpha^{2}}{m^{2}}\left(R^{ab}{}_{ab}-K_{aij}K^{aij}\right)\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right),$
(20)

which agrees with the IR entropy formula using Eq. (29).

##  3 Holographic entropy toolkit

In this section, we describe how to derive the entropy functionals presented
in the previous section. We will begin with a review of the LM method
Lewkowycz:2013nqa , and then explain the generalizations needed for it to work
in our examples. We will apply this method to the example theories discussed
in Sec. 2, but since the details of the calculation are complicated, we
relegate the full calculation to App. A.

###  3.1 Review of Lewkowycz-Maldacena

Consider a spatial subregion $R$ in a quantum field theory and a density
matrix $\rho_{R}$ on $R$. The entanglement entropy of the state is defined as

$S=-\operatorname{Tr}\left(\rho_{R}\log\rho_{R}\right).$ (21)

This is often computed by taking the $n\to 1$ limit of the Rényi entropy,

$S_{n}=-\frac{1}{n-1}\log\operatorname{Tr}\left(\rho^{n}_{R}\right),$ (22)

a procedure referred to as the replica trick. For integer $n\geq 2$, the Rényi
entropy may be computed by

$S_{n}=-\frac{1}{n-1}\log\left[\frac{Z_{n}}{(Z_{1})^{n}}\right],$ (23)

where $Z_{n}$ is the partition function of the quantum field theory on an
$n$-fold cover. This partition function is often computed via a Euclidean path
integral that prepares $n$ copies of the state $\rho_{R}$.

We now apply this replica trick to holographic field theories, following the
LM method. In that context, we use the holographic dictionary to translate Eq.
(23) into a bulk gravitational path integral,

$Z_{n}=\int{\mathcal{D}}\Phi\,e^{-I\left[\Phi\right]},$ (24)

integrating over all bulk fields (including the metric) with boundary
conditions set by the $n$-fold cover. Here, $I$ is the Euclidean gravitational
action in the bulk. In the limit $G_{N}\to 0$, this gravitational path
integral can be computed by a saddle-point approximation
$Z_{n}=e^{-I\left[B_{n}\right]}$ evaluated on the dominant saddle $B_{n}$. If
we further assume that $B_{n}$ preserves the boundary $\mathbb{Z}_{n}$ replica
symmetry, then the action can be computed by a $\mathbb{Z}_{n}$ quotient,
$I[\hat{B}_{n}]=I[B_{n}]/n$, thought of as the on-shell action of the orbifold
spacetime $\hat{B}_{n}=B_{n}/\mathbb{Z}_{n}$. Applying this prescription to
Eq. (23), we find the formula for the Rényi entropy,

$S_{n}=\frac{n}{n-1}\left(I[\hat{B}_{n}]-I[\hat{B}_{1}]\right).$ (25)

The entanglement entropy is then obtained by taking the $n\to 1$ limit,
yielding

$\displaystyle S=\left.\partial_{n}I[\hat{B}_{n}]\right|_{n=1}.$ (26)

To compute this quantity, we now use what is known as the regularized cone
method Lewkowycz:2013nqa . We write

$\displaystyle S=\lim_{n\to
1}\frac{1}{n-1}\left(I[\hat{B}_{n}]-I[\widetilde{B}_{n}^{(a)}]+I[\widetilde{B}_{n}^{(a)}]-I[\hat{B}_{1}]\right),$
(27)

where we have subtracted and added the action of a new geometry,
$\widetilde{B}_{n}^{(a)}$, called the regularized cone, parameterized by a
regulator $a$ and defined so that $\lim_{a\to
0}\widetilde{B}_{n}^{(a)}=\hat{B}_{n}$ and $\lim_{n\to
1}\widetilde{B}_{n}^{(a)}=\hat{B}_{1}$ (independent of $a$); see Fig. 2. The
last two terms in Eq. (27) then cancel in the limit because $\hat{B}_{1}$ is
on-shell. So the final expression is

$S=\partial_{n}(I_{0}-I_{a})\Big{|}_{n=1},$ (28)

where we have defined the shorthand $I_{a}\equiv I[\widetilde{B}_{n}^{(a)}]$
and $I_{0}\equiv I[\hat{B}_{n}]$; note that $I_{0}$ can be obtained from
$I_{a}$ by setting $a=0$. Evaluating the variation in Eq. (28) ultimately
leads to Eq. (2), where the entropy can be expressed as the extremum of an
entropy functional $A_{\text{gen}}$ that generally takes the form of an
integral of a local quantity (e.g., a local geometric invariant)
Lewkowycz:2013nqa ; Lewkowycz_2018 ; upcoming .

Figure 2: Pictorial representation of the regularized cone method, as detailed
in Eq. (27). The leftmost picture represents the singular cone configuration
$\hat{B}_{n}$, while the middle two pictures are its regularized version
$\widetilde{B}^{(a)}_{n}$. The rightmost picture is the solution $\hat{B}_{1}$
at $n=1$.

The entropy functional has been derived using this method in the presence of
higher-derivative terms. For example, for ${\mathcal{L}}=f(\text{Riemann})$
gravity, it was derived in Ref. Dong:2013qoa :

$\displaystyle A_{\text{gen}}=$
$\displaystyle\;2\pi\int\mathrm{d}^{d}y\sqrt{h}\bigg{\\{}-\frac{\partial{\mathcal{L}}}{\partial
R_{\mu\rho\nu\sigma}}\varepsilon_{\mu\rho}\varepsilon_{\nu\sigma}-\frac{\partial^{2}{\mathcal{L}}}{\partial
R_{\mu_{1}\rho_{1}\nu_{1}\sigma_{1}}\partial
R_{\mu_{2}\rho_{2}\nu_{2}\sigma_{2}}}2K_{\lambda_{1}\rho_{1}\sigma_{1}}K_{\lambda_{2}\rho_{2}\sigma_{2}}\times$
(29)
$\displaystyle\;\;\;{\times}\left[\left(n_{\mu_{1}\mu_{2}}n_{\nu_{1}\nu_{2}}{+}\varepsilon_{\mu_{1}\mu_{2}}\varepsilon_{\nu_{1}\nu_{2}}\right)n^{\lambda_{1}\lambda_{2}}{-}\left(n_{\mu_{1}\mu_{2}}\varepsilon_{\nu_{1}\nu_{2}}{+}\varepsilon_{\mu_{1}\mu_{2}}n_{\nu_{1}\nu_{2}}\right)\varepsilon^{\lambda_{1}\lambda_{2}}\right]{+}\cdots\bigg{\\}},$

where ${\mathcal{L}}$ is the Lagrangian in Lorentzian signature,
$K_{\lambda\rho\sigma}$ is the extrinsic curvature along the $x^{\lambda}$
direction, $\varepsilon_{\mu\nu}$ is the Levi-Civita tensor in the two normal
directions (orthogonal to the surface), and $n_{\mu\nu}$ is the induced metric
in the normal directions. The $\cdots$ represent terms involving higher orders
in derivatives along the two normal directions. When evaluated in the examples
that we studied in Sec. 2, this expression precisely gives the IR entropies
(10), (15), and (20).

###  3.2 Expansions in conical geometries

We would now like to apply the LM method to our theories of interest and
calculate the UV entropies. Before doing so, several comments are in order.
First, the derivation of the formula (29) was known to suffer to some extent
from ambiguities called the splitting problem Dong:2013qoa ; Miao:2014nxa ;
Dong:2017xht . One aspect of this problem can be seen from the requirement
that the desired expression for $A_{\text{gen}}$ be a functional of quantities
evaluated on a candidate codimension-two surface in the original spacetime
$\hat{B}_{1}$, not the singular cone $\hat{B}_{n}$. To satisfy this
requirement and resolve the ambiguities, we need to use certain equations of
motion to express quantities in the singular cone in terms of $n=1$ quantities
uniquely.

A framework for solving the splitting problem concretely was provided in Ref.
Dong:2019piw . It provided a prescription for a systematic _conical expansion_
of the metric, i.e., an expansion of the metric around the conical singularity
when $n\neq 1$. To begin, write the metric for the singular cone as in Ref.
Dong:2019piw :

$ds^{2}=dzd\bar{z}+T\frac{(\bar{z}dz-
zd\bar{z})^{2}}{z\bar{z}}+h_{ij}dy^{i}dy^{j}+2iU_{j}dy^{j}(\bar{z}dz-
zd\bar{z}),$ (30)

where the coordinates $z,{\bar{z}}$ satisfy the periodicity condition $z\sim
z\,e^{2\pi i/n}$. For any $X\in\\{T,U_{i},h_{ij}\\}$, expand $X$ in coupling
constants $\lambda_{1},\lambda_{2},\dots,\lambda_{k}$ as

$X=\sum_{r_{1},\dots,r_{k}=0}^{\infty}\lambda_{1}^{r_{1}}\cdots\lambda_{k}^{r_{k}}X^{(\vec{r})}.$
(31)

Additionally, the expansion coefficients $X^{(\vec{r})}$ themselves have the
following expansions in $z$ and ${\bar{z}}$,

$\displaystyle T^{(\vec{r})}$ $\displaystyle=\sum_{p,q=0,\,s=-r\atop
pq>0\text{ or
}s>0}^{\infty}T_{pqs}^{(\vec{r})}\,z^{np}\bar{z}^{nq}(z\bar{z})^{s},$ (32)
$\displaystyle U_{i}^{(\vec{r})}$ $\displaystyle=\sum_{p,q=0,\,s=-r\atop
pq>0\text{ or }s\geq
0}^{\infty}U_{i,pqs}^{(\vec{r})}\,z^{np}\bar{z}^{nq}(z\bar{z})^{s},$
$\displaystyle h_{ij}^{(\vec{r})}$ $\displaystyle=\sum_{p,q=0,\,s=-r\atop
pq>0\text{ or }s\geq
0}^{\infty}h_{ij,pqs}^{(\vec{r})}\,z^{np}\bar{z}^{nq}(z\bar{z})^{s},$

where the coefficients $X^{(\vec{r})}_{pqs}$ are functions of $y^{i}$ only. We
will refer to the expressions in Eq. (32) as conical expansions.

Our goal here is to generalize the above prescription to theories involving
additional matter fields, especially those nonminimally coupled to the metric.
For a component $X$ of a general tensorial field (e.g.,
$X=\phi_{zi{\bar{z}}j}$ or $X=R_{z{\bar{z}}z{\bar{z}}}$), we first define an
expansion in terms of the coupling constants $\lambda_{1},\lambda_{2},\dots$
as in Eq. (31). As in Eq. (32), the coefficients $X^{(\vec{r})}$ themselves
have a conical expansion,

$X^{(\vec{r})}=\left(\frac{z}{{\bar{z}}}\right)^{l/2}\sum_{p,q,s}X^{(\vec{r})}_{pqs}\;z^{np}{\bar{z}}^{nq}(z{\bar{z}})^{s},$
(33)

where the angular momentum $l$ is defined to be the number of upper $z$
indices minus that of upper ${\bar{z}}$ indices for the field component $X$
(each lower $z$ is considered as an upper ${\bar{z}}$, and vice versa). The
ranges of the indices $p,q,s$ depend on the specific theory and field
component being considered.

To find the entropy functional, we define a regularized metric given by

$ds^{2}=e^{\epsilon A}\left[dzd{\bar{z}}+\widetilde{T}\frac{({\bar{z}}dz-
zd{\bar{z}})^{2}}{z{\bar{z}}}+2i\widetilde{U}_{j}dy^{j}({\bar{z}}dz-
zd{\bar{z}})\right]+\widetilde{h}_{ij}dy^{i}dy^{j},$ (34)

where $\widetilde{T}$, $\widetilde{U}_{j}$, and $\widetilde{h}_{ij}$ are
appropriately regularized versions of $T$, $U_{j}$, and $h_{ij}$, and
$\epsilon\equiv 1-1/n$. When matter fields are present, we also regularize
them in a similar way. Here $A$ is a suitable regulating function of
$z,{\bar{z}}$ and is parameterized by the same regulator $a$ that appeared in
Eq. (27). The entropy functional is found by evaluating Eq. (28) in this
regularized field configuration. In App. A, we show how to apply the above
procedure to the examples in Sec. 2.

##  4 General argument for entropy matching

In the examples we have considered in Sec. 2, we found that the calculation of
holographic entanglement entropy always commutes with RG flow. Specifically,
whether we compute the entropy in the UV theory using the method of Sec. 3 and
then flow to the IR, or in the IR EFT directly using Eq. (29), we obtain the
same result. This relationship is depicted in the commutative diagram shown in
Fig. 1.

We now give a formal argument for this entropy matching in general
theories.101010It is worth noting that in the examples of Sec. 2, matching
works at the level of the entropy functional $A_{\text{gen}}$ (before
extremization). This is sufficient (although not necessary) for the entropy
value (obtained after extremization) to match. In comparison, our general
argument works at the level of the entropy value directly. We will show this
result by proving that the Rényi entropies at integer Rényi index $n\geq 2$
satisfy this property. Our claim for the entanglement entropy then follows
from the analytic continuation to $n=1$.

To be concrete, consider a general IR EFT with a tree-level UV extension.
Recall from (23) that the Rényi entropy with integer $n\geq 2$ is computed
from replicated partition functions. Define $Z_{\rm UV}$ and $Z_{\rm IR}$ as
the replicated partition functions of the UV extension and the IR EFT,
respectively. By construction, the IR EFT is obtained by integrating out the
UV degrees of freedom,

$\int{\cal D}g\,e^{-I_{\rm IR}[g]}=\int{\cal D}g\,{\cal D}\phi\,e^{-I_{\rm
UV}[g,\phi]},$ (35)

where we schematically write $\phi$ for any arbitrary UV fields, $g$ for the
metric (and any other IR fields), and $I$ for the Euclidean action for each
theory. Thus, given well-defined boundary conditions for the metric (including
the integer $n$ in the replica method), the partition functions are formally
equal, $Z_{\rm IR}=Z_{\rm UV}$. Hence, the Rényi entropies are equal.
Therefore, their analytic continuation in $n$, and in particular the
entanglement entropy, match between the UV and the IR, thus demonstrating our
desired conclusion.

We can understand this result in more detail by investigating what happens
near $n=1$, and more explicitly in terms of substituting in the equations of
motion for the massive fields $\phi$. The path integral for $\phi$ is
dominated by its saddle point that solves the equations of motion (e.g., of
the schematic form $(\Box-m^{2})\phi\sim R$). Writing the solution
collectively as $\bar{\phi}[g]$, we find at the tree level

$\int{\cal D}g\,e^{-I_{\rm IR}[g]}=\int{\cal D}g\,e^{-I_{\rm
UV}[g,\bar{\phi}[g]]}.$ (36)

Now, both $I_{\rm IR}[g]$ and $I_{\rm UV}[g,\bar{\phi}[g]]$ in general contain
an infinite tower of higher-curvature terms, which we can expand as a series
in $\nabla/m$. Retaining only the leading order in this series corresponds to
the EFT requirement that all curvature scales are larger than the Compton
wavelength of any UV states. In this case, $\bar{\phi}[g]$ reduces to a local
functional of $g$. The path integral over $g$ is dominated by its saddle
point, putting the metric on-shell, $g\rightarrow\bar{g}$. We therefore find

$I_{\rm IR}[\bar{g}(n)]=I_{\rm UV}[\bar{g}(n),\bar{\phi}(n)],$ (37)

where we explicitly write the argument $n$ to emphasize that the solutions are
functions of the Rényi index $n$ defining the boundary conditions. Assuming
that we can analytically continue in $n$ so that we can act with
$\lim_{n\rightarrow 1}\partial_{n}$ on Eq. (37), we obtain the matching of the
von Neumann entropy: on the left-hand side, it is the entropy computed
directly in the IR EFT, while on the right-hand side, it is the entropy
computed in the UV theory, which we RG-flow to the IR by coarse-graining the
metric and field configuration on length scales much larger than $1/m$.

As we noted previously, in our example theories of Sec. 2 we made a stronger
observation, namely that the entropy functionals in fact match (before
extremization). This can be understood by adding appropriate sources, as we
will discuss in Sec. 5.

##  5 Discussion

In this paper, we have studied the matching of entanglement entropies between
bulk gravitational effective theories and their UV extensions. In particular,
through several example EFTs, we have demonstrated that the entropy computed
directly in the IR theory matches that obtained by computing the entropy in
the UV and flowing to the IR. We then gave a general argument based on the
bulk gravitational path integral for why this matching should occur in any
tree-level UV extension.

Along the way, we have generalized the LM method for deriving the holographic
entropy formula to allow for certain matter fields. We expect this procedure
to work for more general higher-derivative theories coupled to matter.

We now conclude with some open questions.

#### Entropy functional matching:

The argument in Sec. 4 holds at the level of the entropy value $S$. On the
other hand, from the examples in Sec. 2, we expect the matching to hold at the
level of the entropy functional $A_{\text{gen}}$, before imposing the
extremality condition. Thus, it would be interesting to generalize the
argument in Sec. 4 to one that works at the level of the entropy functional.
One way of doing so is to allow the codimension-two surface to move from its
extremal location to an arbitrary one. This can be accomplished by introducing
a source term $T_{\mu\nu}\delta g^{\mu\nu}$ into the action, as in Ref.
Dong:2017xht . Here, $T_{\mu\nu}$ is a background stress tensor giving a
source for the metric fluctuation $\delta g_{\mu\nu}$, and it is chosen
appropriately to allow us to open a conical defect along a general, not
necessarily extremal, codimension-two surface. We expect this to upgrade the
argument in Sec. 4 to one showing a stronger matching at the level of the
entropy functional, although we leave the details to future work.111111This
functional-level matching between the UV and the IR is quite remarkable, as it
need not hold in other contexts such as field redefinitions. In particular, as
shown in App. B, under field redefinitions the entropy values match, but not
the entropy functionals in general.

#### Finite-$\boldsymbol{m}$ entropy calculation:

For the spin-two and spin-four examples that we considered, in order to solve
the equations of motion in the conical expansion, we found it convenient to
perform a $1/m^{2}$ expansion in the UV. It would be interesting to see if it
is possible to do a finite-$m$ calculation as in the scalar case.

#### Quantum corrections:

In a more general context, we expect the full generalized entropy
Faulkner:2013ana ; Engelhardt:2014gca , rather than the value of $A_{\rm gen}$
alone, to match between the UV and IR as in Fig. 1.121212See Ref. Kabat:1995jq
for an example using an $O(N)$ invariant linear $\sigma$ model. It would be
interesting to extend our results to incorporate quantum corrections from bulk
fields, especially to cases where the bulk matter entropy contribution is
significant. Further, it would be illuminating to extend our argument to loop-
level extensions of IR EFTs. We leave such investigations of quantum effects
to future work.

###### Acknowledgements.

We thank Don Marolf, Pratik Rath, and Zi-Yue Wang for interesting discussions,
and especially Jiuci Xu for collaboration on related topics. This material is
based upon work supported by the Air Force Office of Scientific Research under
Award Number FA9550-19-1-0360. This material is also based upon work supported
by the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, under Award Numbers DE-SC0023275 and DE-SC0011702. X.D. and W.W.W.
were supported in part by funds from the University of California. G.N.R. is
supported by the James Arthur Postdoctoral Fellowship at New York University,
and was supported at the Kavli Institute for Theoretical Physics by the Simons
Foundation (Grant No. 216179) and the National Science Foundation (Grant No.
NSF PHY-1748958) and at the University of California, Santa Barbara by the
Fundamental Physics Fellowship. D.W. is supported by NSF grant PHY2107939.
C-H.W. was supported in part by the Ministry of Education, Taiwan.

##  Appendix A Derivation of the entropy functionals

In this appendix, we compute the holographic entanglement entropy functional
for each of the UV extensions given in Sec. 2 using the replica method
described in Sec. 3.

Recall that in Sec. 3, the metric of the singular cone $\hat{B}_{n}$ with
opening angle $2\pi/n$ can be written as in Eq. (30), which we repeat here for
convenience:

$ds^{2}=dzd{\bar{z}}+T\frac{({\bar{z}}dz-
zd{\bar{z}})^{2}}{z{\bar{z}}}+2iU_{j}dy^{j}({\bar{z}}dz-
zd{\bar{z}})+h_{ij}dy^{i}dy^{j},$ (38)

where the coordinates $z,{\bar{z}}$ satisfy the periodicity condition $z\sim
z\,e^{2\pi i/n}$. The regularized version of Eq. (38) is given in Eq. (34),
which we reproduce here as well:

$ds^{2}=e^{\epsilon A}\left[dzd{\bar{z}}+\widetilde{T}\frac{({\bar{z}}dz-
zd{\bar{z}})^{2}}{z{\bar{z}}}+2i\widetilde{U}_{j}dy^{j}({\bar{z}}dz-
zd{\bar{z}})\right]+\widetilde{h}_{ij}dy^{i}dy^{j},$ (39)

where as before $\epsilon=1-1/n$, and $A=A(a,z,{\bar{z}})$ is an appropriate
regulating function that depends on the regulator $a$. A simple example that
is sufficient for our calculation is

$A=\log\left(\frac{z^{n}{\bar{z}}^{n}}{z^{n}{\bar{z}}^{n}+a^{2n}}\right).$
(40)

Recall that $\widetilde{T}$, $\widetilde{U}_{j}$, and $\widetilde{h}_{ij}$,
which appear in Eq. (39), are appropriately regularized versions of $T$,
$U_{j}$, and $h_{ij}$. We denote the entire regularized metric defined by Eq.
(39) as $\widetilde{g}_{\mu\nu}$. More generally, we write any regularized
quantity with a tilde; for example, any matter fields $\phi$ that are present
in the theory are regularized to $\widetilde{\phi}$, and the regularized
version of the Ricci scalar $R$ is written as $\widetilde{R}$. As in (33), a
field component $X$ (such as a component of $\phi$ or $R$) in the singular
cone has a conical expansion with coefficients $X_{pqs}$, and its regularized
version $\widetilde{X}$ has a similar expansion with coefficients
$\widetilde{X}_{pqs}$.

Recall that the entropy $S$ can be calculated using Eq. (28) by evaluating the
action difference between the singular and regularized cone to linear order in
$\epsilon$. Here we would like to derive the entropy functional
$A_{\text{gen}}$ whose extremum gives $S$. It can be shown that
$A_{\text{gen}}$ is given by an equation similar to Eq. (28):

$A_{\text{gen}}=\partial_{\epsilon}(I_{0}-I_{a})\Big{|}_{\epsilon=0},$ (41)

where $I_{0}$ now denotes the action of a singular cone in which we only
impose a suitable subset of equations of motion (EOMs) upcoming , and $I_{a}$
is an appropriately regularized version. Basically, not imposing all EOMs
allows us to open a conical defect along a generic, not necessarily extremal,
codimension-two surface, whereas imposing some of the EOMs allows us to solve
quantities in the conical geometry in terms of quantities in the $n=1$
solution. Therefore, we will first evaluate Eq. (41) as a function of conical
expansion coefficients $X_{pqs}$ and then rewrite them in terms of standard
Taylor expansion coefficients $X_{pq}$ in the $n=1$ solution:

$X\big{|}_{n=1}=\left(\frac{z}{{\bar{z}}}\right)^{l/2}\sum_{p,q=0}^{\infty}X_{pq}\;z^{p}{\bar{z}}^{q},$
(42)

by using the imposed EOMs and the continuity condition at $n=1$:

$X_{pq}=\sum_{s}X_{p-s,q-s,s}\Big{|}_{n=1}.$ (43)

Here (and in the calculation below), we use commas to separate the indices of
conical coefficients whenever necessary to avoid confusion. We will refer to
$X_{pq}$ as the $n=1$ coefficients.

In principle, one can evaluate Eq. (41) explicitly,

$I_{0}-I_{a}=\int\left(L_{0}-L_{a}\right),$ (44)

integrating over the entire bulk, where $L_{0}$ and $L_{a}$ are the
Lagrangians (containing a factor of $\sqrt{g}$) evaluated on the singular and
regularized cones, respectively.131313We could denote $I_{a}$ and $L_{a}$ as
$\widetilde{I}$ and $\widetilde{L}$, respectively, but for these two
quantities we choose to use the current notation to make the $a$-dependence
explicit. However, it can be shown upcoming  that Eq. (44) reduces to

$I_{0}-I_{a}=\int_{|z|<a}L_{0}-\int
L_{a}^{\mathrm{I}}+{\mathcal{O}}\big{(}\epsilon^{2}\big{)},$ (45)

and it is simpler to evaluate the right-hand side of this equation. Here
$L_{a}^{\mathrm{I}}$ denotes the sum of a suitable subset of terms in $L_{a}$
that are called “Type I” terms upcoming . Roughly speaking, these terms
contain a factor of $\partial_{z}\partial_{\bar{z}}A$ or its derivatives
(after appropriate integration by parts), and they contribute to Eq. (45) as
regularized delta functions of $z,{\bar{z}}$. The precise definition of Type I
terms is provided in Ref. upcoming . For simplicity, in this appendix we will
use Eq. (45) to calculate entropy functionals. However, one can always
reproduce these results by using Eq. (44) directly, although the calculation
would be more complicated.

###  A.1 Massive scalar extension of $R^{2}$

Consider first the spin-zero theory (5) with Euclidean action

$I=\int{\rm
d}^{d+2}x\,\sqrt{g}\left[-\frac{1}{2\kappa^{2}}R-\frac{\alpha}{\kappa}\phi
R+\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}\phi^{2}\right].$ (46)

Since this theory has a scalar field, we need to prescribe an expansion for it
in the singular cone similar to what we have done for the metric. As in the
case of the metric (32), let us write the conical expansion for $\phi$ as

$\phi=\sum_{p,q,s=0}^{\infty}\phi_{pqs}\,z^{np}{\bar{z}}^{nq}(z{\bar{z}})^{s}.$
(47)

Due to the simplicity of this theory, we do not have to expand in $\alpha$ or
$1/m^{2}$, so we will effectively work with finite $\alpha$ and $m$. In
particular, the $s$ index in (47) and the conical expansion of the metric is
always nonnegative.

We now evaluate $I_{0}-I_{a}$ using Eq. (45). The only nonzero contribution
comes from the $\widetilde{R}_{z{\bar{z}}{\bar{z}}z}$ component in regularized
Lagrangian $L_{a}$:

$L_{a}\supset
2\left(\sqrt{g}\right)_{000}\left(g^{z{\bar{z}}}_{000}\right)^{2}\partial_{z}\partial_{\bar{z}}\left(g_{z{\bar{z}},000}e^{\epsilon
A}\right)\left(\frac{1}{2\kappa^{2}}+\frac{\alpha}{\kappa}\phi_{000}\right)(1+{\mathcal{O}}(\epsilon)).$
(48)

Here, we have collected additional factors of $e^{\epsilon A}$ into
$(1+{\mathcal{O}}(\epsilon))$ because the entropy functional is not sensitive
to anything beyond the linear order in $\epsilon$ in the action. From now on,
we will sometimes omit writing $(1+{\mathcal{O}}(\epsilon))$ in the
regularized Lagrangian, with the caveat above understood. Using Eq. (45), we
obtain the corresponding contribution to $I_{0}-I_{a}$ by integrating the
$\partial_{z}\partial_{\bar{z}}A$ term in Eq. (48):

$I_{0}-I_{a}\supset
4\pi\epsilon\int\mathrm{d}^{d}y\,(\sqrt{h})_{000}\left(\frac{1}{2\kappa^{2}}+\frac{\alpha}{\kappa}\phi_{000}\right)+{\mathcal{O}}\big{(}\epsilon^{2}\big{)}.$
(49)

Using Eq. (41) and the continuity condition at $n=1$

$\phi(z=0,{\bar{z}}=0)\big{|}_{n=1}=\phi_{00}=\sum_{p=0}^{\infty}\phi_{p,p,-p}=\phi_{000},$
(50)

the resulting entropy functional is given by

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int\mathrm{d}^{d}y\sqrt{h}\,\big{(}1+2\alpha\kappa\phi\big{)},$
(51)

which is precisely Eq. (9). This formula is exact in the mass parameter $m$,
i.e., we are not doing any large-mass expansion, in contrast to the spin-two
and spin-four cases discussed in the next two subsections.

###  A.2 Massive spin-two extension of $R_{\mu\nu}R^{\mu\nu}$

In this subsection, we derive the entropy functional for the spin-two theory
given in Eq. (11), with Euclidean action

$I=\int{\rm
d}^{d+2}x\sqrt{g}\left(-\frac{1}{2\kappa^{2}}R-\frac{\alpha}{\kappa}\phi_{\mu\nu}R^{\mu\nu}+\frac{1}{2}\nabla_{\alpha}\phi_{\mu\nu}\nabla^{\alpha}\phi^{\mu\nu}+\frac{1}{2}m^{2}\phi_{\mu\nu}\phi^{\mu\nu}\right).$
(52)

We work perturbatively in both $\alpha$ and $1/m^{2}$. In other words, we
specialize Eqs. (31) and (33) to

$X=\left(\frac{z}{{\bar{z}}}\right)^{l/2}\sum_{r=0}^{\infty}\sum_{t=0}^{\infty}\sum_{p,q,s}X^{(r)\langle
t\rangle}_{pqs}\;\alpha^{r}m^{-2t}z^{np}{\bar{z}}^{nq}(z{\bar{z}})^{s}.$ (53)

Here we have used a single $(r)$ index rather than $(\vec{r})$ as it
corresponds to one coupling constant $\alpha$ in this example. The angle
brackets $\langle\,\cdot\,\rangle$ label the order in a large-mass expansion
as powers of $1/m^{2}$.

Let us first analyze the equation of motion for $\phi_{\mu\nu}$ (which we
impose):

$\Phi_{\mu\nu}\equiv\Box\phi_{\mu\nu}-m^{2}\phi_{\mu\nu}+\alpha\kappa^{-1}R_{\mu\nu}=0.$
(54)

Working at ${\mathcal{O}}(\alpha^{0})$, the equation at leading order in
$1/m^{2}$ is

$\Phi^{(0)\langle-1\rangle}_{\mu\nu}=-\phi^{(0)\langle 0\rangle}_{\mu\nu}=0.$
(55)

At the next order in $1/m^{2}$, we have

$\Phi^{(0)\langle
0\rangle}_{\mu\nu}=\left(\Box\phi_{\mu\nu}\right)^{(0)\langle
0\rangle}-\phi^{(0)\langle 1\rangle}_{\mu\nu}=0.$ (56)

Since $\phi_{\mu\nu}^{(0)\langle 0\rangle}=0$ from the previous order, this
equation sets $\phi^{(0)\langle 1\rangle}_{\mu\nu}=0$. Repeating this
procedure at each order in the large-mass expansion, we have

$\phi^{(0)}_{\mu\nu}=0$ (57)

to all orders in $1/m^{2}$.

Moving to ${\mathcal{O}}(\alpha)$, the matter equation of motion (54) at
leading order in $1/m^{2}$ reads

$\Phi^{(1)\langle-1\rangle}_{\mu\nu}=-\phi^{(1)\langle 0\rangle}_{\mu\nu}=0.$
(58)

The next order in $1/m^{2}$ gives

$\Phi^{(1)\langle
0\rangle}_{\mu\nu}=\left(\Box\phi_{\mu\nu}\right)^{(1)\langle
0\rangle}-\phi^{(1)\langle
1\rangle}_{\mu\nu}+\frac{1}{\kappa}R_{\mu\nu}^{(0)\langle 0\rangle}=0,$ (59)

where the first term is zero according to Eq. (58), so

$\phi^{(1)\langle
1\rangle}_{\mu\nu,pqs}=\frac{1}{\kappa}R_{\mu\nu,pqs}^{(0)\langle 0\rangle}.$
(60)

We now evaluate $I_{0}-I_{a}$ using Eq. (45). First, we work at
${\mathcal{O}}(\alpha^{0})$. Note that even though setting $\alpha=0$ in the
Lagrangian gives Einstein gravity with matter, after using Eq. (57) the
Lagrangian is simply

$L^{(0)}=-\frac{1}{2\kappa^{2}}(\sqrt{g})^{(0)}R^{(0)},$ (61)

which is the same as in pure Einstein gravity without matter. Similarly, the
regularized version is

$L_{a}^{(0)}=-\frac{1}{2\kappa^{2}}\left(\sqrt{\widetilde{g}}\right)^{(0)}{\widetilde{R}}^{(0)}.$
(62)

Since the $s$ index of $\widetilde{R}^{(0)}_{pqs}$ starts at $-1$, the only
important contribution comes from the following term (in
$\widetilde{R}_{z{\bar{z}}{\bar{z}}z}$):

$L_{a}^{(0)}\supset\frac{1}{\kappa^{2}}(\sqrt{g})^{(0)}_{000}\left(g^{z{\bar{z}},(0)}_{000}\right)^{2}\partial_{z}\partial_{\bar{z}}\left(g^{(0)}_{z{\bar{z}},000}e^{\epsilon
A}\right).$ (63)

This contribution integrates to

$(I_{0}-I_{a})^{(0)}\supset\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,\big{(}\sqrt{h}\big{)}^{(0)}_{000}+{\mathcal{O}}(\epsilon^{2})=\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,\big{(}\sqrt{h}\big{)}^{(0)}_{00}+{\mathcal{O}}(\epsilon^{2}),$
(64)

where we have rewritten the conical coefficient
$\big{(}\sqrt{h}\big{)}^{(0)}_{000}$ in terms of the $n=1$ coefficient
$\big{(}\sqrt{h}\big{)}^{(0)}_{00}$. We identify the contribution of Eq. (64)
as the zeroth-order area term in the series expansion in $\alpha$.

Next, at ${\mathcal{O}}(\alpha)$, we find

$\displaystyle L_{a}^{(1)}$
$\displaystyle=\left(\sqrt{\widetilde{g}}\right)^{(1)}\widetilde{{\mathcal{L}}}^{(0)}$
(65)
$\displaystyle\;\;+\\!\left(\sqrt{\widetilde{g}}\right)^{\\!(0)}\\!\\!\bigg{[}{-}\,\frac{1}{2\kappa^{2}}\widetilde{R}^{(1)}{-}\,\frac{1}{\kappa}\widetilde{R}^{(0)}_{\mu\nu}\widetilde{\phi}^{\mu\nu(0)}{+}\,(\widetilde{\nabla}_{\rho}\widetilde{\phi}_{\mu\nu})^{(0)}(\widetilde{\nabla}^{\rho}\widetilde{\phi}^{\mu\nu})^{(1)}{+}\,m^{2}\widetilde{\phi}_{\mu\nu}^{(0)}\widetilde{\phi}^{\mu\nu(1)}\bigg{]}$
$\displaystyle=\left(\sqrt{\widetilde{g}}\right)^{(1)}\left(-\frac{1}{2\kappa^{2}}\widetilde{R}^{(0)}\right)+\left(\sqrt{\widetilde{g}}\right)^{(0)}\left(-\frac{1}{2\kappa^{2}}\widetilde{R}^{(1)}\right),$

where we have used Eq. (57) to set $\phi_{\mu\nu}^{(0)}$ and its regularized
version to zero. This is still just Einstein gravity, whose entropy functional
is just the area, so we can write down its contribution directly:

$(I_{0}-I_{a})^{(1)}=\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,\left[\sum_{p=0}^{\infty}(\sqrt{h})^{(1)}_{p,p,-p}\right]+{\mathcal{O}}(\epsilon^{2})=\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,(\sqrt{h})^{(1)}_{00}+{\mathcal{O}}(\epsilon^{2}).$
(66)

Note that the above contribution comes solely from the first term on the last
line of Eq. (65), while the second term there turns out not to contribute, as
can be checked explicitly.

At the next order ${\mathcal{O}}(\alpha^{2})$, we have

$\displaystyle L_{a}^{(2)}$
$\displaystyle=\Big{(}\sqrt{\widetilde{g}}\Big{)}^{(2)}\left[-\frac{1}{2\kappa^{2}}\widetilde{R}^{(0)}\right]+\Big{(}\sqrt{\widetilde{g}}\Big{)}^{(1)}\left[-\frac{1}{2\kappa^{2}}\widetilde{R}^{(1)}\right]$
(67)
$\displaystyle\;\;+\\!\Big{(}\sqrt{\widetilde{g}}\Big{)}^{\\!(0)}\\!\\!\left[{-}\frac{1}{2\kappa^{2}}\widetilde{R}^{(2)}{-}\frac{1}{\kappa}\widetilde{R}_{\mu\nu}^{(0)}\widetilde{\phi}^{\mu\nu}{}^{(1)}{+}\frac{1}{2}(\widetilde{\nabla}_{\rho}\widetilde{\phi}_{\mu\nu})^{(1)}(\widetilde{\nabla}^{\rho}\widetilde{\phi}^{\mu\nu})^{(1)}{+}\frac{m^{2}}{2}\widetilde{\phi}_{\mu\nu}^{(1)}\widetilde{\phi}^{\mu\nu}{}^{(1)}\\!\right]\\!.$

The first three terms are from the Einstein-Hilbert action, so their
contributions are known. In particular, the second and third terms turn out
not to contribute, and the first term simply yields

$(I_{0}-I_{a})^{(2)}\supset\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,\left[\sum_{p=0}^{\infty}(\sqrt{h})^{(2)}_{p,p,-p}\right]+{\mathcal{O}}(\epsilon^{2})=\frac{2\pi\epsilon}{\kappa^{2}}\int\mathrm{d}^{d}y\,(\sqrt{h})^{(2)}_{00}+{\mathcal{O}}(\epsilon^{2}).$
(68)

This is similar to what we have seen at order ${\mathcal{O}}(\alpha)$.

Let us now focus on the last three terms in (67) and work order by order in
$1/m^{2}$. At the zeroth order in $1/m^{2}$, we find

$L_{a}^{(2)\langle
0\rangle}{\supset}\\!\left(\\!\\!\sqrt{\widetilde{g}}\right)^{\\!(0)\langle
0\rangle}\\!\\!\left[\\!{-}\frac{1}{\kappa}\widetilde{R}_{\mu\nu}^{(0)\langle
0\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle
0\rangle}\\!{+}\frac{1}{2}(\widetilde{\nabla}_{\rho}\widetilde{\phi}_{\mu\nu})^{(1)\langle
0\rangle}\\!(\widetilde{\nabla}^{\rho}\widetilde{\phi}^{\mu\nu})^{(1)\langle
0\rangle}\\!{+}\widetilde{\phi}_{\mu\nu}^{(1)\langle
1\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle 0\rangle}\\!\right]\\!.$ (69)

This is set to zero by (58). At first order in $1/m^{2}$, we have

$\displaystyle L_{a}^{(2)\langle 1\rangle}$
$\displaystyle\supset\\!(\sqrt{\widetilde{g}})^{(0)\langle
0\rangle}\\!\\!\left[\\!{-}\frac{1}{\kappa}\widetilde{R}_{\mu\nu}^{(0)\langle
0\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle
1\rangle}\\!{+}(\widetilde{\nabla}_{\rho}\widetilde{\phi}_{\mu\nu})^{(1)\langle
0\rangle}\\!(\widetilde{\nabla}^{\rho}\widetilde{\phi}^{\mu\nu})^{(1)\langle
1\rangle}\\!{+}\frac{1}{2}\widetilde{\phi}_{\mu\nu}^{(1)\langle
1\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle 1\rangle}\\!\right]$ (70)
$\displaystyle=\\!(\sqrt{\widetilde{g}})^{(0)\langle
0\rangle}\\!\\!\left[\\!{-}\frac{1}{\kappa}\widetilde{R}_{\mu\nu}^{(0)\langle
0\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle
1\rangle}+\frac{1}{2}\widetilde{\phi}_{\mu\nu}^{(1)\langle
1\rangle}\widetilde{\phi}^{\mu\nu}{}^{(1)\langle 1\rangle}\right].$

We now work out the contribution of Eq. (70) to $I_{0}-I_{a}$ according to Eq.
(45). First, focus on the first term in the last line of Eq. (70), which
contributes in two ways. The first way is through the following Type I terms
in $L_{a}$ coming from $(\mu,\nu)=(z,{\bar{z}})$ and $({\bar{z}},z)$:

$\displaystyle L_{a}^{(2)\langle
1\rangle}\supset\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\frac{1}{\kappa}g^{z{\bar{z}},(0)\langle
0\rangle}_{000}\partial_{z}\partial_{\bar{z}}\left(g^{(0)\langle
0\rangle}_{z{\bar{z}},000}e^{\epsilon
A}\right)2\left[\phi^{z{\bar{z}}}{}^{(1)\langle
1\rangle}_{000}+\phi^{z{\bar{z}}}{}^{(1)\langle 1\rangle}_{11-1}\right],$ (71)

which integrates to

$\displaystyle(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\int\mathrm{d}^{d}y\,\frac{2\pi\epsilon}{\kappa}(\sqrt{h})^{(0)\langle
0\rangle}_{000}\,\left[\phi^{z{\bar{z}}}{}^{(1)\langle
1\rangle}_{000}+\phi^{z{\bar{z}}}{}^{(1)\langle
1\rangle}_{11-1}\right]+{\mathcal{O}}(\epsilon^{2}).$ (72)

Here $\phi^{z{\bar{z}}}{}^{(1)\langle 1\rangle}_{11-1}$ could be written more
clearly as $\phi^{z{\bar{z}}}{}^{(1)\langle 1\rangle}_{1,1,-1}$, but we will
use the former for simplicity. The second way for the first term in Eq. (70)
to contribute is through the following non-Type I terms in $L_{0}$:

$\displaystyle L_{0}^{(2)\langle
1\rangle}\supset\frac{1}{\kappa}\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\bigg{[}$ $\displaystyle\,\frac{1}{2}h^{ij}{}^{(0)\langle
0\rangle}_{000}\partial_{z}\partial_{z}\left(h_{ij}{}^{(0)\langle
0\rangle}_{100}z^{n}\right)\phi^{zz}{}^{(1)\langle
1\rangle}_{01-1}{\bar{z}}^{n-2}$ (73) $\displaystyle+$
$\displaystyle\,\frac{1}{2}h^{ij}{}^{(0)\langle
0\rangle}_{000}\partial_{\bar{z}}\partial_{\bar{z}}\left(h_{ij}{}^{(0)\langle
0\rangle}_{010}{\bar{z}}^{n}\right)\phi^{{\bar{z}}{\bar{z}}}{}^{(1)\langle
1\rangle}_{10-1}z^{n-2}\bigg{]},$

which is integrated within $|z|<a$ according to Eq. (45) and gives

$\displaystyle(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{2\pi}{4\kappa}h^{ij}{}^{(0)\langle
0\rangle}_{000}h_{ij}{}^{(0)\langle 0\rangle}_{100}\phi^{zz}{}^{(1)\langle
1\rangle}_{01-1}+{\rm c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (74)

This expression is not manifestly ${\mathcal{O}}(\epsilon)$, but becomes so
after we rewrite the spin-two field. Using the matter equations of motion
(60), Eq. (74) becomes

$\displaystyle(I_{0}-I_{a})^{(2)\langle 1\rangle}$
$\displaystyle\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{2\pi}{4\kappa^{2}}h^{ij}{}^{(0)\langle
0\rangle}_{000}h_{ij}{}^{(0)\langle 0\rangle}_{100}R^{zz}{}^{(0)\langle
0\rangle}_{01-1}+{\rm c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (75)

Recall the definition $K_{zij}\equiv\frac{1}{2}\partial_{z}h_{ij}$. Using

$\displaystyle K_{z,00}{}^{(0)\langle
0\rangle}=\frac{1}{2}h^{ij}{}^{(0)\langle 0\rangle}_{00}h_{ij}{}^{(0)\langle
0\rangle}_{10}=\frac{1}{2}h^{ij}{}^{(0)\langle
0\rangle}_{000}h_{ij}{}^{(0)\langle 0\rangle}_{100}+{\mathcal{O}}(\epsilon),$
(76)

and

$\displaystyle R^{zz}{}^{(0)\langle 0\rangle}_{01-1}$
$\displaystyle=\,g^{z{\bar{z}}}{}^{(0)\langle
0\rangle}_{000}g^{z{\bar{z}}}{}^{(0)\langle
0\rangle}_{000}\left[-\frac{1}{2}n(n-1)\right]h_{ij}{}^{(0)\langle
0\rangle}_{010}h^{ij}{}^{(0)\langle 0\rangle}_{000}$ (77)
$\displaystyle=-2n(n-1)h_{ij}{}^{(0)\langle
0\rangle}_{010}h^{ij}{}^{(0)\langle 0\rangle}_{000}$ $\displaystyle=-4\epsilon
K_{{\bar{z}},00}^{(0)\langle 0\rangle}+{\mathcal{O}}(\epsilon^{2}),$

Eq. (75) becomes

$(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}-\frac{4\pi\epsilon}{\kappa^{2}}K_{z,00}{}^{(0)\langle
0\rangle}K_{{\bar{z}},00}^{(0)\langle 0\rangle}+{\rm
c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (78)

Now we move on to the second term in the last line of Eq. (70). It only
contributes through the following non-Type I terms in $L_{0}$,

$\displaystyle\left(L_{0}\right)^{(2)\langle 1\rangle}\supset$
$\displaystyle\,\frac{1}{2}\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\left[\phi_{zz}{}^{(1)\langle
1\rangle}_{10-1}z^{n-2}\phi^{zz}{}^{(1)\langle
1\rangle}_{01-1}{\bar{z}}^{n-2}+\phi_{{\bar{z}}{\bar{z}}}{}^{(1)\langle
1\rangle}_{01-1}{\bar{z}}^{n-2}\phi^{{\bar{z}}{\bar{z}}}{}^{(1)\langle
1\rangle}_{10-1}z^{n-2}\right]\\!,$ (79)

which is again integrated according to Eq. (45) and gives

$\displaystyle(I_{0}-I_{a})^{(2)\langle
1\rangle}{\supset}\\!\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{2\pi}{4\epsilon}\phi_{zz}{}^{(1)\langle
1\rangle}_{10-1}\phi^{zz}{}^{(1)\langle 1\rangle}_{01-1}{+}{\rm
c.c.}\right]{+}{\mathcal{O}}(\epsilon^{2}).$ (80)

Again using (60), this expression becomes

$\displaystyle(I_{0}-I_{a})^{(2)\langle 1\rangle}$
$\displaystyle{\supset}\\!\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{2\pi}{4\kappa^{2}\epsilon}R_{zz}{}^{(0)\langle
0\rangle}_{10-1}R^{zz}{}^{(0)\langle 0\rangle}_{01-1}{+}{\rm
c.c.}\\!\right]\\!{+}\,{\mathcal{O}}(\epsilon^{2})$ (81)
$\displaystyle{=}\\!\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{2\pi\epsilon}{\kappa^{2}}K_{z}{}^{(0)\langle
0\rangle}_{,00}K_{\bar{z}}{}^{(0)\langle 0\rangle}_{,00}\times
2\right]{+}\,{\mathcal{O}}(\epsilon^{2}).$

We note that Eqs. (72), (78), and (81) are the only contributions of the last
three terms in (67) to $I_{0}-I_{a}$ at ${\mathcal{O}}(\alpha^{2}/m^{2})$, as
can be verified explicitly. Combining them gives

$\displaystyle(I_{0}\,{-}\,I_{a})^{(2)\langle 1\rangle}$ (82)
$\displaystyle\supset\,2\pi\epsilon\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{1}{\kappa}\left(\phi^{z{\bar{z}}}{}_{000}^{(1)\langle
1\rangle}{+}\phi^{z{\bar{z}}}{}_{11-1}^{(1)\langle
1\rangle}\right)-\frac{2}{\kappa^{2}}K_{z,00}{}^{(0)\langle
0\rangle}K_{{\bar{z}},00}^{(0)\langle
0\rangle}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$

Adding this to the contributions at order $\alpha^{0}$ given in Eq. (64), at
order $\alpha$ given in Eq. (66), and at order $\alpha^{2}$ given in Eq. (68),
we use Eq. (41) to find the entropy functional

$A_{\text{gen}}\,{=}\,\frac{2\pi}{\kappa^{2}}\int\\!\mathrm{d}^{d}y\sqrt{h}\left[1\,{+}\,\frac{\alpha^{2}}{m^{2}}\left(\kappa\phi^{z{\bar{z}}}{}^{(1)\langle
1\rangle}{-}{2}K_{z}{}^{(0)\langle 0\rangle}K_{{\bar{z}}}{}^{(0)\langle
0\rangle}\right)\right]\\!{+}\,{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right)$
(83)

up to higher order terms in $1/m^{2}$. Here we have used the continuity
condition at $n=1$

$\phi_{\mu\nu}(z=0,{\bar{z}}=0)\big{|}_{n=1}=\phi_{\mu\nu,00}=\sum_{p=0}^{\infty}\phi_{\mu\nu,p,p,-p}.$
(84)

Writing Eq. (83) covariantly in $z$, ${\bar{z}}$ and using
$\phi^{a}_{a}{}^{(0)}=0$ from Eq. (57), we find

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+\alpha\kappa\phi_{a}^{a}{}-\frac{\alpha^{2}}{2m^{2}}K_{a}{}K^{a}{}\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right),$
(85)

which is precisely Eq. (14).

###  A.3 Massive spin-four extension of
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$

Finally, we consider the spin-four theory (16) with Euclidean action

$I\,{=}\\!\int{\rm
d}^{d+2}x\sqrt{g}\left(\\!{-}\frac{1}{2\kappa^{2}}R{-}\frac{\alpha}{\kappa}\phi_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}{+}\frac{1}{2}\nabla_{\alpha}\phi_{\mu\nu\rho\sigma}\nabla^{\alpha}\phi^{\mu\nu\rho\sigma}{+}\frac{1}{2}m^{2}\phi_{\mu\nu\rho\sigma}\phi^{\mu\nu\rho\sigma}\\!\right)\\!,$
(86)

and we will again work perturbatively in both $\alpha$ and $1/m^{2}$.

Repeating the order-by-order analysis that we performed for the spin-two
field’s equation of motion, we find analogously that, to all orders in the
$1/m^{2}$ expansion,

$\phi^{(0)}_{\mu\nu\rho\sigma}=0.$ (87)

We also find

$\displaystyle\phi^{(1)\langle 0\rangle}_{\mu\nu\rho\sigma,pqs}$
$\displaystyle=0,$ (88) $\displaystyle\phi^{(1)\langle
1\rangle}_{\mu\nu\rho\sigma,pqs}$
$\displaystyle=\frac{1}{\kappa}R_{\mu\nu\rho\sigma,pqs}^{(0)\langle
0\rangle}.$

We now analyze the action difference $I_{0}-I_{a}$ to linear order in
$\epsilon$. First, note that at ${\mathcal{O}}(\alpha^{0})$ the action is
exactly the same as that of the spin-two theory with $\alpha$ set to zero, so
we obtain Eq. (64) again without having to repeat the calculation. In other
words, the leading-order result is just the area term of Einstein gravity.

At ${\mathcal{O}}(\alpha)$, using Eq. (87), we can write down the analogue of
Eq. (65):

$L_{a}^{(1)}=\left(\sqrt{\widetilde{g}}\right)^{(1)}\left(-\frac{1}{2\kappa^{2}}\widetilde{R}^{(0)}\right)+\left(\sqrt{\widetilde{g}}\right)^{(0)}\left(-\frac{1}{2\kappa^{2}}\widetilde{R}^{(1)}\right).$
(89)

The second term turns out not to contribute (as in the spin-two case), and the
first term gives Eq. (66) again.

At ${\mathcal{O}}(\alpha^{2})$, we have

$\displaystyle L_{a}^{(2)}$
$\displaystyle=\;\;\Big{(}\sqrt{\widetilde{g}}\Big{)}^{(2)}\left[-\frac{1}{2\kappa^{2}}\widetilde{R}^{(0)}\right]+\Big{(}\sqrt{\widetilde{g}}\Big{)}^{(1)}\left[-\frac{1}{2\kappa^{2}}\widetilde{R}^{(1)}\right]$
(90)
$\displaystyle\;\;\;+\Big{(}\sqrt{\widetilde{g}}\Big{)}^{(0)}\bigg{[}-\frac{1}{2\kappa^{2}}\widetilde{R}^{(2)}-\frac{1}{\kappa}\widetilde{R}_{\mu\nu\rho\sigma}^{(0)}\widetilde{\phi}^{\mu\nu\rho\sigma}{}^{(1)}$
$\displaystyle\qquad\qquad\;\;\;\;\;\;\,\,+\frac{1}{2}(\widetilde{\nabla}_{\alpha}\widetilde{\phi}_{\mu\nu\rho\sigma})^{(1)}(\widetilde{\nabla}^{\alpha}\widetilde{\phi}^{\mu\nu\rho\sigma})^{(1)}+\frac{m^{2}}{2}\widetilde{\phi}_{\mu\nu\rho\sigma}^{(1)}\widetilde{\phi}^{\mu\nu\rho\sigma}{}^{(1)}\bigg{]}.$

The first three terms give Eq. (68) again. Now we focus on the remaining three
terms and work order by order in $1/m^{2}$. At ${\mathcal{O}}(m^{0})$, they
vanish after using the matter equations of motion. At
${\mathcal{O}}(1/m^{2})$, the remaining three terms give

$L_{a}^{(2)\langle 1\rangle}\supset(\sqrt{\widetilde{g}})^{(0)\langle
0\rangle}\left[-\frac{1}{\kappa}\widetilde{R}_{\mu\nu\rho\sigma}^{(0)\langle
0\rangle}\widetilde{\phi}^{\mu\nu\rho\sigma}{}^{(1)\langle
1\rangle}+\frac{1}{2}\widetilde{\phi}_{\mu\nu\rho\sigma}^{(1)\langle
1\rangle}\widetilde{\phi}^{\mu\nu\rho\sigma}{}^{(1)\langle 1\rangle}\right].$
(91)

The first term contributes through Type I terms

$\displaystyle(L_{a})^{(2)\langle
1\rangle}\supset\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\left[\frac{1}{\kappa}\partial_{z}\partial_{\bar{z}}\left(g^{(0)\langle
0\rangle}_{z{\bar{z}},000}e^{\epsilon
A}\right)4\left(\phi^{z{\bar{z}}{\bar{z}}z}{}^{(1)\langle
1\rangle}_{000}+\phi^{z{\bar{z}}{\bar{z}}z}{}^{(1)\langle
1\rangle}_{11-1}\right)\right],$ (92)

where the factor of 4 comes from symmetries of $\phi^{\mu\nu\rho\sigma}$, and
the above equation integrates to

$\displaystyle(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\frac{2\pi\epsilon}{\kappa}\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)}_{000}\,\left(\phi^{z{\bar{z}}{\bar{z}}z}{}^{(1)\langle
1\rangle}_{000}+\phi^{z{\bar{z}}{\bar{z}}z}{}^{(1)\langle
1\rangle}_{11-1}\right)+{\mathcal{O}}(\epsilon^{2}).$ (93)

The first term in Eq. (91) also contributes through non-Type I terms

$\displaystyle\left(L_{0}\right)^{(2)\langle
1\rangle}\supset\,\frac{1}{\kappa}\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\bigg{[}$
$\displaystyle\,\frac{1}{2}\partial_{z}\partial_{z}\left(h_{ij}{}^{(0)\langle
0\rangle}_{100}z^{n}\right)4\phi^{zizj}{}^{(1)\langle
1\rangle}_{01-1}{\bar{z}}^{n-2}$ (94) $\displaystyle+$
$\displaystyle\,\frac{1}{2}\partial_{\bar{z}}\partial_{\bar{z}}\left(h_{ij}{}^{(0)\langle
0\rangle}_{010}{\bar{z}}^{n}\right)4\phi^{{\bar{z}}i{\bar{z}}j}{}^{(1)\langle
1\rangle}_{10-1}z^{n-2}\bigg{]},$

which is integrated according to Eq. (45) and gives

$\displaystyle(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{2\pi}{\kappa}h_{ij}{}^{(0)\langle
0\rangle}_{100}\phi^{zizj}{}^{(1)\langle 1\rangle}_{01-1}+{\rm
c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (95)

As before, we proceed by rewriting the spin-four field. Using the matter
equations of motion (88), Eq. (95) becomes

$\displaystyle(I_{0}-I_{a})^{(2)\langle 1\rangle}$
$\displaystyle\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{2\pi}{\kappa^{2}}h_{ij}{}^{(0)\langle
0\rangle}_{100}R^{zizj}{}^{(0)\langle 0\rangle}_{01-1}+{\rm
c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (96)

Using

$\displaystyle K_{zij,00}^{(0)\langle
0\rangle}=\frac{1}{2}h_{ij,10}^{(0)\langle
0\rangle}=\frac{1}{2}h_{ij,100}^{(0)\langle 0\rangle}+{\mathcal{O}}(\epsilon)$
(97)

and

$\displaystyle R^{zizj}{}^{(0)\langle 0\rangle}_{01-1}$
$\displaystyle=\,g^{z{\bar{z}}}{}^{(0)\langle
0\rangle}_{000}g^{z{\bar{z}}}{}^{(0)\langle
0\rangle}_{000}\left(-\frac{1}{2}n(n-1)\right)h_{kl}{}^{(0)\langle
0\rangle}_{010}h^{ik}{}^{(0)\langle 0\rangle}_{000}h^{jl}{}^{(0)\langle
0\rangle}_{000}$ (98) $\displaystyle=-2n(n-1)h_{kl}{}^{(0)\langle
0\rangle}_{010}h^{ik}{}^{(0)\langle 0\rangle}_{000}h^{jl}{}^{(0)\langle
0\rangle}_{000}$ $\displaystyle=-4\epsilon
K_{{\bar{z}}}^{ij}{}_{00}^{(0)\langle 0\rangle}+{\mathcal{O}}(\epsilon^{2}),$

Eq. (96) becomes

$(I_{0}-I_{a})^{(2)\langle
1\rangle}\supset\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}-\frac{16\pi\epsilon}{\kappa^{2}}K_{zij}{}_{00}^{(0)\langle
0\rangle}K_{{\bar{z}}}^{ij}{}^{(0)\langle 0\rangle}_{00}+{\rm
c.c.}\bigg{]}+{\mathcal{O}}(\epsilon^{2}).$ (99)

Moving on to the second term in Eq. (91), we find that it only contributes
through the non-Type I terms

$\displaystyle\left(L_{0}\right)^{(2)\langle
1\rangle}\supset\frac{1}{2}\left(\sqrt{g}\right)^{(0)\langle
0\rangle}_{000}\,4\left(\phi_{zizj}{}^{(1)\langle
1\rangle}_{10-1}\phi^{zizj}{}^{(1)\langle
1\rangle}_{01-1}+\phi_{{\bar{z}}i{\bar{z}}j}{}^{(1)\langle
1\rangle}_{01-1}\phi^{{\bar{z}}i{\bar{z}}j}{}^{(1)\langle
1\rangle}_{10-1}\right)(z{\bar{z}})^{n-2},$ (100)

which gives

$\displaystyle(I_{0}\,{-}\,I_{a})^{(2)\langle
1\rangle}{\supset}\\!\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{2\pi}{\epsilon}\phi_{zizj}{}^{(1)\langle
1\rangle}_{10-1}\phi^{zizj}{}^{(1)\langle 1\rangle}_{01-1}{+}{\rm
c.c.}\right]\\!{+}\,{\mathcal{O}}(\epsilon^{2}).$ (101)

Again using Eq. (88), this expression becomes

$\displaystyle(I_{0}-I_{a})^{(2)\langle 1\rangle}$
$\displaystyle{\supset}\\!\int\\!\\!\mathrm{d}^{d}y(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{2\pi}{\kappa^{2}\epsilon}R_{zizj}{}^{(0)\langle
0\rangle}_{10-1}R^{zizj}{}^{(0)\langle 0\rangle}_{01-1}{+}{\rm
c.c.}\\!\right]\\!{+}\,{\mathcal{O}}(\epsilon^{2}).$ (102)
$\displaystyle{=}\\!\int\\!\\!\mathrm{d}^{d}y(\sqrt{h})^{(0)\langle
0\rangle}_{000}\left[\frac{8\pi\epsilon}{\kappa^{2}}K_{zij}{}^{(0)\langle
0\rangle}_{00}K_{{\bar{z}}}^{ij}{}^{(0)\langle 0\rangle}_{00}\times
2\right]+{\mathcal{O}}(\epsilon^{2}).$

Again, at this order in $\alpha$ and $1/m^{2}$, we have no further
contributions. Combining the results from Eqs. (93), (99), and (102), we have

$\displaystyle(I_{0}-I_{a})^{(2)\langle 1\rangle}\supset$ (103)
$\displaystyle\,2\pi\epsilon\int\mathrm{d}^{d}y\,(\sqrt{h})^{(0)\langle
0\rangle}_{000}\bigg{[}\frac{1}{\kappa}\left(\phi^{z{\bar{z}}{\bar{z}}z}{}_{000}^{(1)\langle
1\rangle}+\phi^{z{\bar{z}}{\bar{z}}z}{}_{11-1}^{(1)\langle
1\rangle}\right)-\frac{8}{\kappa^{2}}K_{zij}{}_{00}^{(0)\langle
0\rangle}K_{{\bar{z}}}^{ij}{}_{00}^{(0)\langle 0\rangle}\,\bigg{]}$
$\displaystyle+{\mathcal{O}}(\epsilon^{2}).$

Adding this result to the area contributions given in Eqs. (64), (66), and
(68), we find

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int\mathrm{d}^{d}y\sqrt{h}\left[1+\frac{\alpha^{2}}{m^{2}}\left(\kappa\phi^{z{\bar{z}}{\bar{z}}z}{}^{(1)\langle
1\rangle}-8K_{zij}{}^{(0)\langle 0\rangle}K_{{\bar{z}}}^{ij}{}^{(0)\langle
0\rangle}\right)\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right)$
(104)

up to higher order terms in $1/m^{2}$. Writing this covariantly in $z$,
${\bar{z}}$ and using $\phi^{ab}{}_{ab}{}^{(0)}=0$, we find

$A_{\text{gen}}=\frac{2\pi}{\kappa^{2}}\int{\rm
d}^{d}y\sqrt{h}\left[1+2\alpha\kappa\phi^{ab}{}_{ab}-\frac{2\alpha^{2}}{m^{2}}K_{aij}{}K^{aij}{}\right]+{\mathcal{O}}\left(\frac{\alpha^{4}}{m^{4}}\right),$
(105)

which is precisely Eq. (19).

##  Appendix B Holographic entanglement entropy under field redefinition

A particular representation of an EFT Lagrangian, in terms of higher-dimension
interactions and their associated Wilson coefficients, is only well-defined—in
the sense of being uniquely specified given IR physics—modulo field
redefinition. For example, Einstein gravity ${\mathcal{L}}\sim R$ can be field
redefined to ${\mathcal{L}}\sim
R+\lambda_{1}R^{2}+\lambda_{2}R^{\mu\nu}R_{\mu\nu}$ (to leading order in the
Wilson coefficients $\lambda_{1},\lambda_{2}$), so the two theories are in
fact equivalent.141414Note that the $R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$
term is not related to Einstein gravity via a field redefintion. Hence, we do
not consider it in this appendix. A priori, there might be some tension: if
two bulk theories are equivalent, the dual boundary theory should be the same,
and therefore the boundary entropy should be independent of which bulk
description we use; on the other hand, the two equivalent theories do have
different Lagrangians leading to potentially different entropy functionals,
and one might question whether they have the same extremal value.

In this appendix, we will see that the two representations of the theory do in
fact give the same entanglement entropy, despite apparent dissimilarities. We
will demonstrate this in two simple examples. The key ingredients in the
resolution are the equations of motion and the extremality condition.

###  B.1 From Einstein to $R^{2}$

To demonstrate this equivalence with an example, consider the Euclidean action

$I_{\diamond}=\frac{1}{2}\int\mathrm{d}^{D}x\sqrt{g_{\diamond}}\left(-R_{\diamond}+2\Lambda+g_{\diamond}^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right)$
(106)

where the subscript $\diamond$ denotes the quantities before redefinition and
$D=d+2$ is the bulk spacetime dimension. We have included a scalar field
$\phi$ to make the example more nontrivial. The entropy functional is simply
the area

$A_{\text{gen},\diamond}=2\pi\int\mathrm{d}^{d}y\sqrt{h_{\diamond}}.$ (107)

Einstein’s equation is given by the vanishing of

$\displaystyle E_{\diamond}^{\mu\nu}$
$\displaystyle\equiv\frac{2}{\sqrt{g_{\diamond}}}\frac{\delta I}{\delta
g_{\diamond\mu\nu}}=R_{\diamond}^{\mu\nu}-\frac{1}{2}R_{\diamond}g_{\diamond}^{\mu\nu}+\frac{1}{2}\left(2\Lambda+\nabla_{\mu}\phi\nabla^{\mu}\phi\right)g_{\diamond}^{\mu\nu}-\nabla^{\mu}\phi\nabla^{\nu}\phi,$
(108)

where it is understood that the index on $\nabla^{\mu}$ is raised with
$g_{\diamond}$.

Now let us perform a small field redefinition,

$g_{\diamond\mu\nu}=(1+\lambda R)g_{\mu\nu},\quad\delta g_{\mu\nu}\equiv
g_{\diamond\mu\nu}-g_{\mu\nu}=\lambda Rg_{\mu\nu}$ (109)

and work to linear order in $\lambda$. The action becomes

$\displaystyle I$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{D}x\sqrt{g}\left[-R+2\Lambda+(\nabla\phi)^{2}\right]+\frac{1}{2}\int\mathrm{d}^{D}x\sqrt{g}E^{\mu\nu}\delta
g_{\mu\nu}$ (110)
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{D}x\sqrt{g}\left[-R+2\Lambda+(\nabla\phi)^{2}+\lambda\left(-\frac{d}{2}R^{2}+(d+2)\Lambda
R+\frac{d}{2}(\nabla\phi)^{2}R\right)\right],$

up to boundary terms that are not important for our purposes. Using the method
in Sec. 3, we find the entropy functional $A_{\text{gen}}$ for this new
action:

$A_{\text{gen}}=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left[1+\lambda\left(dR-(d+2)\Lambda-\frac{d}{2}(\nabla\phi)^{2}\right)\right].$
(111)

On the other hand, if we simply apply the field redefinition (109) to the
entropy functional (107), we find

$A_{\text{gen}}^{\prime}=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left(1+\frac{d}{2}\lambda
R\right),$ (112)

which appears to disagree with (111).

To resolve this discrepancy, we need to use the equations of motion (108).
Specifically, we only need the trace,

$E_{\mu}^{\mu}=-\frac{d}{2}R+(d+2)\Lambda+\frac{d}{2}(\nabla\phi)^{2}=0.$
(113)

We then have

$A_{\text{gen}}=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left(1+\frac{d}{2}\lambda
R\right),$ (114)

so the expressions agree: $A^{\prime}_{\text{gen}}=A_{\text{gen}}$.

In this example, we have seen that the equations of motion alone are enough to
equate two apparently different expressions for the entropy functional. Next,
we consider an example where this is not the case.

###  B.2 From Einstein to $R_{\mu\nu}R^{\mu\nu}$

Consider again Einstein gravity with a cosmological constant (106), but this
time for simplicity without the matter field $\phi$. To generate a Ricci-
squared term, we take our field redefinition to be

$g_{\diamond\mu\nu}=g_{\mu\nu}+\lambda R_{\mu\nu}.$ (115)

Then the action becomes

$\displaystyle
I=\frac{1}{2}\int\mathrm{d}^{D}x\sqrt{g}\left[-R+2\Lambda+\lambda\left(\Lambda
R-\frac{1}{2}R^{2}+R_{\mu\nu}R^{\mu\nu}\right)\right].$ (116)

Applying Eq. (29) to this new action, we find the entropy functional to linear
order in $\lambda$:

$A_{\text{gen}}=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left[1+\lambda\left(-\Lambda+R-R^{a}_{a}+\frac{1}{2}K^{a}K_{a}\right)\right].$
(117)

Applying the field redefinition (115) directly to the entropy functional
(107), on the other hand, yields

$A_{\text{gen}}^{\prime}=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left(1+\frac{\lambda}{2}R^{i}_{i}\right).$
(118)

To check whether this agrees with (117), we again use the equations of motion.
Taking the trace of the metric equation of motion in all directions, in the
$a$-type directions, and in the $i$-type directions gives, respectively,

$\displaystyle R=\frac{2(d+2)\Lambda}{d},\qquad
R^{a}_{a}=\frac{4\Lambda}{d},\qquad R^{i}_{i}=2\Lambda.$ (119)

Substituting these equations into Eqs. (117) and (118), we obtain

$\displaystyle A_{\text{gen}}$
$\displaystyle=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left[1+\lambda\left(\Lambda+\frac{1}{2}K^{a}K_{a}\right)\right],$
(120) $\displaystyle A_{\text{gen}}^{\prime}$
$\displaystyle=2\pi\int\mathrm{d}^{d}y\sqrt{h}\left(1+\lambda\Lambda\right).$

The difference between $A_{\text{gen}}$ and $A_{\text{gen}}^{\prime}$ is now
given by a term proportional to $K^{a}K_{a}$. This is where we need to use the
extremality condition at zeroth order in $\lambda$, $K_{a}=0$, so the extra
term $K^{a}K_{a}$ vanishes (up to higher order terms in $\lambda$). Thus, we
have shown that even though the entropy functional of the field-redefined
theory generally does not match the field-redefined entropy functional, their
extremal values (i.e., the entropy values) are indeed the same.

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