# The Page Curve for Reflected Entropy

Chris Akers  Thomas Faulkner  Simon Lin  and Pratik Rath

###### Abstract

We study the reflected entropy $S_{R}$ in the West Coast Model, a toy model of
black hole evaporation consisting of JT gravity coupled to end-of-the-world
branes. We demonstrate the validity of the holographic duality relating it to
the entanglement wedge cross section away from phase transitions. Further, we
analyze the important non-perturbative effects that smooth out the
discontinuity in the $S_{R}$ phase transition. By performing the gravitational
path integral, we obtain the reflected entanglement spectrum analytically. The
spectrum takes a simple form consisting of superselection sectors, which we
interpret as a direct sum of geometries, a disconnected one and a connected
one involving a closed universe. We find that area fluctuations of
$O(\sqrt{G_{N}})$ spread out the $S_{R}$ phase transition in the canonical
ensemble, analogous to the entanglement entropy phase transition. We also
consider a Renyi generalization of the reflected entropy and show that the
location of the phase transition varies as a function of the Renyi parameter.

##  1 Introduction

The black hole information problem has served as a beacon guiding us in the
quest to understand quantum gravity [1, 2, 3, 4]. Although a complete
resolution still eludes us and might require a better understanding of the UV-
complete theory of quantum gravity, significant progress has been made in
recent years simply by taking the gravitational path integral seriously [5,
6].

A commendable milestone in this endeavour is the calculation of the “Page
Curve” using the semiclassical theory [7, 8]. In fact, in the so-called West
Coast Model [7], a toy model of black hole evaporation consisting of Jackiw-
Teitelboim (JT) gravity coupled to end-of-the-world (ETW) branes, the detailed
curve including effects near the phase transition were computed.

Figure 1: The Lorentzian description of the state we consider in the West
Coast Model, a JT gravity black hole with an ETW brane. The ETW brane carries
two sub-flavours, denoted black and green, that are entangled (dashed,
coloured lines) with radiation systems $R_{1}$ and $R_{2}$ respectively. The
extremal surface is denoted in purple and the island that dominates after the
Page time is coloured gray.

Away from phase transitions, the Page curve can be computed using the quantum
extremal surface (QES) formula [9, 10, 11] which reads

$S(A)=\min_{\gamma_{A}}\frac{\mathrm{Area}(\gamma_{A})}{4G_{N}}+S_{\text{bulk}}(\Sigma_{A}),$
(1.1)

where $S(A)$ is the entanglement entropy of subsystem $A$. $\Sigma_{A}$ is a
partial Cauchy slice, whose domain of dependence is the entanglement wedge of
$A$, such that $\partial\Sigma_{A}=A\cup\gamma_{A}$. In the West Coast Model,
one considers a state with ETW branes carrying flavour indices that are
entangled with an auxiliary radiation system $R=R_{1}\cup R_{2}$.111The
division of $R$ into subsystems is a straightforward generalization, which
will be useful for our computation of reflected entropy. The model consists of
two parameters: the horizon area $S_{BH}$ and the number of ETW brane flavour
indices $k=k_{1}k_{2}$. Tuning these parameters simulates black hole
evaporation, and by applying Eq. (1.1) to $R$, one finds the Page curve to be

$S(R)=\min(S_{BH},\log k),$ (1.2)

where the candidate quantum extremal surfaces are the bifurcation surface and
the trivial surface respectively, as shown in Fig. (1).

Figure 2: A spatial slice of AdS with $A$ and $B$ chosen to be two intervals.
The figure depicts the entanglement wedge of $AB$ (gray), the entanglement
wedge of $C$ (green), the RT surface $\gamma_{AB}$ and the entanglement wedge
cross section $\Gamma_{A:B}$, which divides the entanglement wedge into
regions $a$ and $b$ homologous to $A$ and $B$ respectively.

While the entanglement entropy is essentially the unique measure for bipartite
entanglement in pure states, one would also like to understand the
multipartite entanglement structure of the radiation. Various distinct
measures of entanglement have been proposed for this purpose. Among these we
shall focus on the reflected entropy $S_{R}(A:B)$, which seems to be
particularly interesting for holography [12, 13].

Given a density matrix $\rho_{AB}$, the reflected entropy $S_{R}(A:B)$ is
defined by considering the canonically purified state
$\ket{\sqrt{\rho}}_{AA^{*}BB^{*}}$ in a doubled Hilbert space where $A^{*}$
and $B^{*}$ are the mirror copies of $A$ and $B$. The reflected entropy is
then defined as

$S_{R}(A:B)=S(AA^{*})_{\ket{\sqrt{\rho}}}.$ (1.3)

The reflected entropy is proposed to be holographically computed by the
entanglement wedge cross section [12] and corresponding quantum corrections
[14, 15], i.e.,

$S_{R}(A:B)=\min_{\Gamma_{A:B}}\left[2\frac{\mathrm{Area}(\Gamma_{A:B})}{4G_{N}}+S_{R,\text{bulk}}(a:b)\right],$
(1.4)

where $\Gamma_{A:B}$ is a surface dividing the entanglement wedge of $AB$ into
portions $a$ and $b$, which are homologous to $A$ and $B$ respectively (see
Fig. (2)). The reflected entropy has already proven useful in understanding
the requirement of tripartite entanglement in holographic states [13, 15].

Figure 3: The main result: (a) The reflected entanglement spectrum of
$\rho_{AA^{*}}$ in the canonically purified state $\ket{\sqrt{\rho_{AB}}}$ is
a mixture of two superselection sectors: a single pole of weight $p_{d}$
corresponding to the disconnected purification, and a mound of approximately
$k_{1}^{2}$ eigenvalues (assuming $k_{1}<k_{2}$) with weight $p_{c}$
corresponding to a connected purification involving a closed universe with the
entanglement wedge cross section denoted in orange. (b) The probability of the
connected purification $p_{c}$ as we vary $k$ across the Page transition. We
show analytic plots for the microcanonical and canonical ensemble. The latter
shows a spread in the phase transition of $O(\sqrt{G_{N}})$. We also show
plots for $p_{c}$ when considering the $(m,1)$-Renyi reflected entropy; these
undergo sharp transitions at $m$-dependent locations in the canonical
ensemble.

We analyze the reflected entropy between the radiation subsystems $R_{1}$ and
$R_{2}$. Simply applying Eq. (1.4) leads to

$S_{R}(R_{1}:R_{2})=\begin{cases}0&k<\exp(S_{BH})\\\ 2\min(\log k_{1},\log
k_{2})&k>\exp(S_{BH})\end{cases}$ (1.5)

This follows from the two possible saddles obtained upon canonical
purification: the disconnected geometry and the connected geometry depicted in
Fig. (3).

In this paper we compute the reflected entropy precisely, demonstrating the
validity of the holographic formula away from phase transitions. The fact that
the gravitational path integral can be performed exactly allows us to compute
the detailed behaviour of $S_{R}(R_{1}:R_{2})$ including corrections to Eq.
(1.5) near the phase transition. We term this behavior the ‘Page curve for
reflected entropy.’ This is the main result of this paper, summarized by Fig.
(3).

In fact, we compute the entire reflected entanglement spectrum, which takes a
very simple form: it consists of a mixture of two superselection sectors
corresponding to the disconnected and connected geometries shown in Fig. (3).
The probabilities $p_{d}$ and $p_{c}$ of the two sectors are computable
functions, that vary as we change $k$, leading to the phase transition in Eq.
(1.5). Moreover, we find corrections to Eq. (1.5) in a window of $\Delta\log
k=O(\frac{1}{\sqrt{G_{N}}})$ near the phase transition that arise from
fluctuations in the horizon area (equivalently thermal fluctuations, in this
model). This is analogous to a similar effect found in the entanglement
entropy Page curve [5, 16, 17], and we expect it to be a universal feature of
holographic reflected entropy.

### Overview

In Sec. (2), we set up the stage for our analysis. In Sec. (2.1), we review
the West Coast Model. In Sec. (2.2), we discuss basic aspects of the reflected
entropy in this model. The proposed holographic answer arises from a leading
saddle computation assuming replica symmetry. This calculation suffers from
various issues, which motivates us to analyze the problem in more detail.

In Sec. (3), we review the computation of the entanglement spectrum in the
West Coast Model. This serves as a warm-up for the resolvent trick which is
used to obtain the spectrum in Sec. (3.1). We then use the spectrum to compute
the Renyi entropies in this model in Sec. (3.2). Apart from being of interest
on their own, the results for the Renyi entropies serve as an input for the
reflected entropy calculation.

In Sec. (4), we analyze the reflected entanglement spectrum in the West Coast
Model. We describe the resolvent trick in Sec. (4.1) to obtain a Schwinger-
Dyson equation for the reflected entanglement spectrum. In Sec. (4.2), we
solve this equation to obtain the spectrum and use it to analyze the reflected
entropy. In Sec. (4.3), we go on to analyze a two-parameter Renyi
generalization of the reflected entropy, called the $(m,n)$-Renyi reflected
entropies.

In Sec. (5), we discuss these results and future directions.

Note: This paper provides a complementary analysis to Ref. [18] where we
analyze the reflected entropy phase transition in random tensor networks.

##  2 Preliminaries

In this section, we set up the background for our analysis. We will first
review the West Coast Model in Sec. (2.1). In Sec. (2.2), we review aspects of
the reflected entropy and discuss the proposed holographic dual.

###  2.1 West Coast Model

Figure 4: The Lorentzian description of the state we consider in the West
Coast Model, a JT gravity black hole with an ETW brane. The ETW brane carries
two sub-flavours, denoted black and green, that are entangled (dashed,
coloured lines) with radiation systems $R_{1}$ and $R_{2}$ respectively. The
extremal surface is denoted in purple and the island that dominates after Page
time is coloured gray.

The West Coast Model is a toy model that was used to derive the Page curve of
entanglement entropy. We briefly describe the model here, for more details
refer to Ref. [5]. The model consists of JT gravity coupled to ETW branes. The
action is given by

$\displaystyle I$ $\displaystyle=I_{JT}+\mu\int_{\text{brane}}ds$ (2.1)
$\displaystyle I_{JT}$
$\displaystyle=-\frac{S_{0}}{2\pi}\left[\frac{1}{2}\int_{\mathcal{M}}\sqrt{g}R+\int_{\partial\mathcal{M}}\sqrt{h}K\right]-\left[\frac{1}{2}\int_{\mathcal{M}}\sqrt{g}\phi(R+2)+\int_{\partial\mathcal{M}}\sqrt{h}\phi
K\right],$ (2.2)

where $S_{0}$ is the extremal entropy and $\mu$ is the mass of the brane. We
will take both the above parameters to be large in our analysis. The ETW
branes also possess a large number of flavours $k$, which we artificially
divide into two sub-flavour indices that number $k_{1}$ and $k_{2}$
respectively.

We consider a state where the ETW brane is entangled with two radiation bath
systems $R_{1}$ and $R_{2}$ as

$\ket{\Psi}=\frac{1}{\sqrt{k}}\sum_{i=1}^{k_{1}}\sum_{j=1}^{k_{2}}\ket{i}_{R_{1}}\ket{j}_{R_{2}}\ket{\psi_{ij}}_{B},$
(2.3)

where $\ket{\psi_{ij}}_{B}$ is the state of the black hole system $B$ with the
ETW brane chosen to be of sub-flavours $i$ and $j$ respectively. The state can
be prepared using a Euclidean path integral, and the Lorentzian description is
obtained by analytic continuation as shown in Fig. (4).

A diagrammatic description of the boundary conditions that compute the overlap
between two such states is

$\langle\psi_{i_{1}\,j_{1}}|\psi_{i_{2}\,j_{2}}\rangle=\
\includegraphics[scale={.7},valign={c}]{fig2/overlap.eps},$ (2.4)

where the red line represents an asymptotic AdS boundary, whereas the black
and green lines correspond to sub-flavour indices of the ETW brane.222Note
that we have changed the diagrammatic notation from Ref. [5]. In this paper,
we will consider two choices of boundary conditions imposed at the asymptotic
boundary: a) microcanonical, where we impose a fixed energy $E$, and b)
canonical, where we fix the renormalized length $\beta$ corresponding to the
inverse temperature. The computation in Eq. (2.4) is done by performing the
gravitational path integral over geometries consistent with these boundary
conditions, e.g.,

$\langle\psi_{i_{1}\,j_{1}}|\psi_{i_{2}\,j_{2}}\rangle=\
\includegraphics[scale={.7},valign={c}]{fig2/overlap_geometry.eps},$ (2.5)

where the ETW brane has a definite flavour and thus, makes the diagram
proportional to $\delta_{i_{1},i_{2}}\delta_{j_{1},j_{2}}$.

###  2.2 Reflected Entropy

The reflected entropy $S_{R}(A:B)$ is a function defined for a density matrix
$\rho_{AB}$ on a bipartite quantum system $AB$. One first considers the
_canonical purification_ of $\rho_{AB}$ on a doubled Hilbert space

$\ket{\sqrt{\rho_{AB}}}\in\mathrm{End}(\mathcal{H}_{A})\otimes\mathrm{End}(\mathcal{H}_{B})=(\mathcal{H}_{A}\otimes\mathcal{H}^{*}_{A})\otimes(\mathcal{H}_{B}\otimes\mathcal{H}^{*}_{B})~{},$
(2.6)

where the space of linear maps $\mathrm{End}(\mathcal{H}_{A})$ acting on
$\mathcal{H}_{A}$ itself forms a Hilbert space with inner product
$\braket{X}{Y}=\operatorname{tr}_{A}(X^{\dagger}Y)$. In general
$\mathrm{End}(\mathcal{H})$ is isomorphic to the doubled copy
$\mathcal{H}\otimes\mathcal{H}^{*}$. In other words, define
$\ket{\sqrt{\rho_{AB}}}$ by finding the unique positive matrix square root of
$\rho_{AB}$ and interpret the result as a state in
$\mathrm{End}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$. This procedure is a
generalization of the much more familiar procedure of purifying the thermal
density matrix by considering the thermofield double state [19].

The reflected entropy is then defined as [12]:

$S_{R}(A:B)=S(AA^{*})_{\sqrt{\rho_{AB}}}.$ (2.7)

In fact, it has a two parameter Renyi generalization based on the following
state:333This state is unnormalized as written. One can insert a factor of
$({\rm Tr}\rho_{AB}^{m})^{-1/2}$ to normalize it, though this is irrelevant in
the $m\rightarrow 1$ limit.

$\left|\psi^{(m)}\right>=\left|\rho_{AB}^{m/2}\right>.$ (2.8)

The Renyi generalization, which we call the $(m,n)$-Renyi reflected entropy,
is then given by

$S_{R}^{(m,n)}(A:B)=-\frac{1}{n-1}\ln{\rm
Tr}(\rho_{AA^{\star}}^{(m)})^{n}\,,\qquad\rho_{AA^{*}}^{(m)}=\frac{1}{{\rm
Tr}\rho_{AB}^{m}}{\rm
Tr}_{BB^{\star}}\left|\rho_{AB}^{m/2}\right>\left<\rho_{AB}^{m/2}\right|,$
(2.9)

for $m\geq 0,n\geq 0$.

In the West Coast Model, we are interested in calculating the reflected
entropy $S_{R}(R_{1}:R_{2})$. To motivate the proposed holographic answer, we
can first consider preparing the Renyi generalization of the canonically
purified state given in Eq. (2.8). For even integer $m$, this can be analyzed
using the replica trick. Consider the Euclidean path integral that computes
the norm of the state
$\langle\psi^{(m)}|\psi^{(m)}\rangle=\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$,
it is diagramatically represented by the boundary conditions: (e.g. $m=4$)

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})=\includegraphics[scale={.4},valign={c}]{fig2/can_bc.eps}.$
(2.10)

The boundary conditions have a dihedral symmetry composed of $\mathbb{Z}_{m}$
rotations and a $\mathbb{Z_{2}}$ reflection.

We now consider two natural saddle points that respect this replica symmetry
[12], and applying the saddle point approximation gives us

$\displaystyle\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$
$\displaystyle=\max\left[\includegraphics[scale={.4},valign={c}]{fig2/can_pur.eps}\right],$
(2.11) $\displaystyle=\max\left[kZ_{1}^{m},k^{m-1}Z_{m}\right],$ (2.12)

where $Z_{p}$ is the $p$-boundary partition function and the purple dot
indicates the location of the extremal surface. The disconnected saddle
dominates before the Page transition, i.e. $k\ll\exp(S_{0})$, while the
connected saddle dominates after the Page transition. Each of these saddles
has a $\mathbb{Z_{2}}$ symmetric slice which can be used to analytically
continue the Euclidean saddle to a Lorentzian solution, giving us two
candidate geometries corresponding to $\ket{\psi^{(m)}}$:

$\ket{\psi^{(m)}}=\includegraphics[scale={.4},valign={c}]{fig2/disconnected_CP.eps}\qquad\text{or}\qquad\includegraphics[scale={.4},valign={c}]{fig2/connected_CP.eps}.$
(2.13)

We refer to these geometries as the disconnected and connected purifications
respectively. Here, $m$ is simply a parameter that fixes the area of the
extremal surface and thus, the analytic continuation in $m$ is
straightforward. As $m\rightarrow 1$, these geometries can be interpreted as
arising from gluing together two copies of the entanglement wedge of
$R_{1}\cup R_{2}$ [20, 21].444See also Ref. [22] for a construction of the
canonical purification of the black hole system in a model of JT gravity
coupled to matter.

Within the respective phases, we can apply the QES formula once we know the
geometry corresponding to $\ket{\psi^{(m)}}$. For the disconnected saddle, the
QES is trivial and thus, the reflected entropy vanishes. On the other hand,
for the connected saddle we have two non-trivial candidate QESs represented by
the pink and orange cuts in Eq. (2.13). This leads to the proposed formula

$S_{R}(R_{1}:R_{2})=\begin{cases}0&k<\exp(S_{BH})\\\ 2\min(\log k_{1},\log
k_{2})&k>\exp(S_{BH})\end{cases}$ (2.14)

While this saddle point approximation is expected to work far away from the
Page transition, there are various issues with this proposal which will
require a more detailed analysis that we carry out in Sec. (4). In particular,
the saddle point calculation suffers from an order of limits issue when
analytically continuing $m,n\rightarrow 1$. Further, Eq. (2.14) has a
discontinuous jump at the phase transition which is resolved by non-
perturbative effects. Once these effects are included, we will find
significant corrections to this formula near the phase transition.

##  3 Entanglement Spectrum

Before we go to a more careful analysis of the reflected entropy, we first
discuss the simpler case of entanglement entropy. In Sec. (3.1), we first
review the resolvent trick that was used to compute the entanglement spectrum
in Ref. [5]. This serves as a warm-up for the analogous calculation for the
reflected spectrum. Further, we analyze the Renyi entropies in the West Coast
Model in Sec. (3.2) which will be useful for our later calculations.

###  3.1 Resolvent Trick

The technique we use here was presented in Ref. [5] (see also [23, 24, 25]).
This powerful approach enables us to write down a Schwinger-Dyson equation for
the resolvent of $\rho_{R_{1}R_{2}}$, which then gives full information about
the entanglement spectrum.

Consider the resolvent matrix $R_{ij}(\lambda)$ for the density matrix
$\rho_{R_{1}R_{2}}$ defined formally as

$\displaystyle
R_{ij}(\lambda)=\left(\frac{1}{\lambda-\rho_{R_{1}R_{2}}}\right)_{ij},$ (3.1)

where the $i,j$ indices run over both $R_{1}$ and $R_{2}$ labels and take
values from $1$ to $k=k_{1}k_{2}$. We then define the resolvent as the trace
$R(\lambda)=\operatorname{tr}R_{ij}(\lambda)$. From the resolvent, we can
obtain the density of eigenvalues $D(\lambda)$ using

$\displaystyle D(\lambda)$
$\displaystyle=-\frac{1}{\pi}\lim_{\epsilon\rightarrow
0}\text{Im}\,R(\lambda+i\epsilon).$ (3.2)

To evaluate $R(\lambda)$, we expand the matrix inverse around $\lambda=\infty$
as

$R(\lambda)=\frac{k}{\lambda}+\sum_{n=1}^{\infty}\frac{\operatorname{tr}(\rho_{R_{1}R_{2}}^{n})}{\lambda^{n+1}},$
(3.3)

sum the series and then analytically continue to $\lambda\in[0,1]$ on the real
axis where $D(\lambda)$ is non-zero. In terms of diagrams, this leads to the
boundary conditions:

$\includegraphics[width=375.95186pt]{fig2/resolvent1.eps},$ (3.4)

where each red line carries a power of $\frac{1}{kZ_{1}}$ for normalization.
These boundary conditions are then filled in with planar disk geometries since
crossing (higher genus) geometries are suppressed by powers of $\frac{1}{k}$
($e^{-S_{0}}$).

Now denote $F_{ij}(\lambda)$ to be the connected part of the resolvent,
defined by

(3.5)

and we define $F(\lambda)=\operatorname{tr}F_{ij}(\lambda)$. We can now write
down a Schwinger-Dyson equation for $R(\lambda)$ and $F(\lambda)$:

$\includegraphics[width=375.95186pt]{fig2/resolvent3.eps},$ (3.6)

where we note that it is self consistent to consider every term to be
proportional to $\delta_{ij}$, and thus
$F_{ij}(\lambda)=\frac{F(\lambda)}{k}\delta_{ij}$. After taking the trace of
Eq. (3.6), we obtain the following algebraic equation

$\displaystyle F(\lambda)$
$\displaystyle=\sum^{\infty}_{n=1}\frac{kZ_{n}R(\lambda)^{n-1}}{(kZ_{1})^{n}}.$
(3.7)

We could now consider boundary conditions that correspond to either the
microcanonical or canonical ensemble. The microcanonical case is identical to
a random state and is discussed in detail in Ref. [18],555In this model,
fixing the energy is the same as fixing the area of the horizon in the
semiclassical approximation. Thus, the calculations can be understood as
related to those done in random tensor networks via the connection to fixed-
area states [26, 27, 28] so our primary focus here will be on the canonical
ensemble. For this case, we can use an integral representation for the
$n$-boundary partition function $Z_{n}$ given by

$Z_{n}=\int_{0}^{\infty}ds\,\rho(s)y(s)^{n},$ (3.8)

where $\rho(s)$ is the density of states, and $y(s)$ is the Boltzmann weight
for the thermal spectrum given by

$\displaystyle\rho(s)$ $\displaystyle=\frac{e^{S_{0}}s\sinh{2\pi
s}}{2\pi^{2}}$ (3.9) $\displaystyle y(s)$ $\displaystyle=e^{-\frac{\beta
s^{2}}{2}}2^{1-2\mu}|\Gamma(\mu-\frac{1}{2}+is)|^{2}.$ (3.10)

It is convenient to define the normalized Boltzmann weight
$w(s)=\frac{y(s)}{Z_{1}}$. Using this representation in Eq. (3.7), we obtain

$F(\lambda)=\int_{0}^{\infty}ds\frac{\rho(s)w(s)}{k-w(s)R}.$ (3.11)

Finally, combining this with Eq. (3.5), we obtain an integral equation for
$R(\lambda)$:

$\displaystyle\lambda R=k+\int_{0}^{\infty}ds\frac{\rho(s)w(s)R}{k-w(s)R}$
(3.12)

Solving this equation will give us the entanglement spectrum and corresponding
entropies.

#### Approximate Entanglement Spectrum

Figure 5: (a) The thermal spectrum of the JT black hole. (b) The approximate
spectrum of $\rho_{R_{1}R_{2}}$ is a cutoff thermal spectrum which is obtained
by truncating the spectrum to the $k$ largest eigenvalues and shifting it by
$\lambda_{0}$ to make it normalized.

The resolvent equation was solved approximately in Ref. [5] to obtain an
approximate entanglement spectrum. It was shown that the spectrum of
$\rho_{R_{1}R_{2}}$ can be well approximated by a cutoff thermal spectrum (see
Fig. (5)), i.e.,

$D(\lambda)=\int_{0}^{s_{k}}ds\,\rho(s)\delta(\lambda-\lambda_{0}-w(s)),$
(3.13)

where the parameters $s_{k}$ and $\lambda_{0}$ can be computed by

$\displaystyle k$ $\displaystyle=\int_{0}^{s_{k}}ds\,\rho(s)$ (3.14)
$\displaystyle\lambda_{0}$
$\displaystyle=\frac{1}{k}\int_{s_{k}}^{\infty}\rho(s)w(s).$ (3.15)

For more details about the calculation, refer to Appendix F of Ref. [5]. Here,
we will simply use the result and discuss the interpretation of this spectrum.

Intuitively, the thermal spectrum of the black hole is divided into two
portions - a part that has already undergone the Page transition, and a part
that hasn’t. For states corresponding to sufficiently high energy in the
thermal spectrum, the Page transition has still not occurred since they
correspond to larger black holes. On the other hand, low energy states are
past the Page transition and in this part of the wavefunction, the radiation
system contains an island in the black hole interior. Thus, it is natural to
expect that details of the spectrum such as the probability distribution of
the horizon area should be observable. This natural ansatz is in fact a good
approximation and leads to a cutoff thermal spectrum where $s_{k}$ roughly
corresponds to the threshold dividing the pre-Page and post-Page parts of the
wavefunction. The post-Page portion of the spectrum is not normalized on its
own and thus, $\lambda_{0}$ is the additive constant contributed by the pre-
Page spectrum that shifts the spectrum of $\rho_{R_{1}R_{2}}$ to make it
appropriately normalized. Further, this ansatz captures the feature that the
rank of $\rho_{R_{1}R_{2}}$ cannot exceed $k$ even far past the Page
transition, since the dimension of the radiation Hilbert space is $k$.

###  3.2 Renyi Entropy

Given this approximate description of the spectrum, we will now analyze the
Renyi entropies in this model,

$S_{m}(\rho_{R_{1}R_{2}})=-\frac{1}{n-1}\log(\operatorname{tr}(\rho_{R_{1}R_{2}}^{m}))$
(3.16)

for different values of $m$. To simplify the analysis, we choose
$\mu\gg\frac{1}{\beta}$. In this limit, the $\mu$-dependent terms drop out
from $w(s)$. Further, we also work in the semiclassical limit
$\beta\rightarrow 0$.666Note that $\beta$ should be replaced with $\beta
G_{N}$ to restore explicit dependence on the Newton constant.

Using Eq. (3.13), we obtain

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})=\int_{0}^{s_{k}}\rho(s)(\lambda_{0}+w(s))^{m}.$
(3.17)

There are a few values of $k$ near the phase transition, equivalently
$s_{k}\approx\frac{\log k-S_{0}}{2\pi}$, that play an important role in this
calculation. First, the location of the peak of $\rho(s)w(s)^{m}$ is
$s_{m}=\frac{2\pi}{m\beta}$. Secondly, far from the phase transition the
calculation of Renyi entropies is dominated by either of the two
$\mathbb{Z}_{m}$ symmetric saddles. The location at which they exchange
dominance is denoted by $s_{k}^{(m)}$. Working in the saddle point
approximation, we obtain the condition

$\frac{1}{k^{m-1}}=\exp\left[(1-m)S_{0}+\frac{2\pi^{2}}{\beta}(\frac{1}{m}-m)\right],$
(3.18)

which leads to $s_{k}^{(m)}\approx\frac{\pi}{\beta}(1+\frac{1}{m})$. Lastly,
from Eq. (3.15), we see that $\lambda_{0}$ has a phase transition at
$s_{k}=s_{1}$.

With these ingredients in hand, we can now analyze Eq. (3.17). An important
feature that appears in this model is that different Renyi entropies have
transitions at different values of $k$. We will analyze them separately for
$m>1$, $m=1$ and $m<1$ since they have qualitatively different behaviour.

(a)

(b)

Figure 6: (a) For $m>1$, the Renyi entropy (denoted by the green line) has a
sharp transition at $s_{k}=s_{k}^{(m)}$ where the two replica symmetric saddle
contributions (denoted by blue and red dotted lines) exchange dominance. Phase
transitions in the two terms in Eq. (3.19) happen at $s_{k}=s_{1},s_{m}$
(depicted with gray dashed lines) and thus don’t affect the Renyi entropies.
(b) For $m<1$, the Renyi entropy (denoted by the green line) has a transition
in the range $s_{k}\in[s_{1},s_{m}]$ whereas the two naive replica symmetric
saddles (denoted by blue and red dotted lines) exchange dominance at
$s_{k}=s_{k}^{(m)}$. Thus, the phase transition is over a large window of size
$O(\frac{1}{\beta})$ and the Renyi entropy has $O(\frac{1}{\beta})$
corrections compared to a naive analytic continuation of the replica symmetric
saddles.

#### $m>1$

For $m>1$, we have $s_{m}<s_{k}^{(m)}<s_{1}$. This implies that
$\lambda_{0}\approx\frac{1}{k}$ for $s_{k}\approx s_{k}^{(m)}$. As a simple
approximation, we can compute Eq. (3.17) by

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})\approx\max\left[\int_{0}^{s_{k}}ds\,\rho(s)\lambda_{0}^{m},\int_{0}^{s_{k}}ds\,\rho(s)w(s)^{m}\right].$
(3.19)

Since at the level of the saddle point approximation, this comparison is
identical to Eq. (3.18), we see that
$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$ has a phase transition at
$s_{k}=s_{k}^{(m)}$. More so, the second term in Eq. (3.19) is the integral of
a Gaussian with a width $\sigma_{m}=\frac{1}{\sqrt{m\beta}}$. Since
$s_{k}^{(m)}-s_{m}\gg\sigma_{m}$, the second term is almost unchanging in this
region. Thus, we see that for $m>1$,
$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$ has a rather sharp phase
transition. The Renyi entropy takes the form

$S_{m}(\rho_{R_{1}R_{2}})=\min\left[\log
k,S_{0}+\frac{2\pi^{2}}{\beta}(1+\frac{1}{m})\right],$ (3.20)

as shown in Fig. (6).

#### $m=1$

The case $m=1$ corresponds to the entanglement entropy which was analyzed in
detail in Ref. [5]. In this case, $s_{m}=s_{k}^{(m)}=s_{1}$ and thus, there
isn’t a sharp transition. Instead, one finds enhanced corrections of
$O(\frac{1}{\sqrt{\beta}})$ in a window of size $\Delta\log
k=O(\frac{1}{\sqrt{\beta}})$ near $s=s_{1}$. Again, we can use the approximate
spectrum in Eq. (3.13) to obtain

$S(\rho_{R_{1}R_{2}})\approx-\int_{0}^{s_{k}}\rho(s)(\lambda_{0}+w(s))\log(\lambda_{0}+w(s)).$
(3.21)

#### $m<1$

For $m<1$, we instead have $s_{m}>s_{k}^{(m)}>s_{1}$. Thus, we see that the
phase transition in $\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$ starts
happening at $s_{k}=s_{1}$ when $\lambda_{0}$ starts decaying. This is
qualitatively different from the $m>1$ case and will require a more careful
analysis.777Similar results have been discussed in Ref. [25]. We are very
grateful to Sean Mcbride and Wayne Weng for discussions related to this topic.

In order to compute Eq. (3.17), we can use the following inequality on the
integrand:

$\max[\lambda_{0}^{m},w(s)^{m}]\leq\left(\lambda_{0}+w(s)\right)^{m}\leq\lambda_{0}^{m}+w(s)^{m}.$
(3.22)

Upon performing the integral for $s_{k}=s_{1}-O(\frac{1}{\beta})$, since
$\lambda_{0}\gg w(s)$, we obtain

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})\approx
k\lambda_{0}^{m}\approx\frac{1}{k^{m-1}},$ (3.23)

since both the upper and lower bounds take the same value at leading order. In
the regime $s_{1}+O(\frac{1}{\beta})<s_{k}<s_{m}-O(\frac{1}{\beta})$, both the
terms are comparable. Thus, we obtain

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})\approx\\#\exp\left[2\pi
s_{k}-\frac{m\beta s_{k}^{2}}{2}+(1-m)S_{0}-\frac{2\pi^{2}m}{\beta}\right],$
(3.24)

where there is a potential $O(1)$ multiplicative uncertainty. Similarly in the
regime $s_{k}=s_{m}+O(\frac{1}{\beta})$, the $w(s)$ term dominates over the
$\lambda_{0}$ term giving us

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})\approx\exp\left[(1-m)S_{0}+\frac{2\pi^{2}}{\beta}\left(\frac{1}{m}-m\right)\right],$
(3.25)

where we have used the saddle point approximation. Combining these results, we
obtain

$S_{m}(\rho_{R_{1}R_{2}})\approx\begin{cases}\log k&s_{k}<s_{1}\\\ \frac{2\pi
s_{k}-\frac{m\beta
s_{k}^{2}}{2}+(1-m)S_{0}-\frac{2\pi^{2}}{m\beta}}{1-m}&s_{1}<s_{k}<s_{m}\\\
S_{0}+\frac{2\pi^{2}}{\beta}\left(1+\frac{1}{m}\right)&s_{k}>s_{m}\end{cases},$
(3.26)

where we have ignored subleading corrections at $s_{k}=s_{1}$ and
$s_{k}=s_{m}$. The result is sketched in Fig. (6), along with the naive
expectation from analytically continuing the results of the $\mathbb{Z}_{m}$
symmetric saddles a la Lewkowycz-Maldacena [29, 30]. We see that there are in
fact large $O(\frac{1}{\beta})$ corrections to the naive holographic answer
for Renyi index $m<1$. This is consistent with generic expectations based on
chaotic behaviour [31, 32, 16]. As shown, the transition happens over a
parametrically large window of $\Delta\log k\approx O(\frac{1}{\beta})$.

Before moving on, we analyze the problem a little more carefully in a window
$s_{k}=s_{1}\pm O(\frac{1}{\sqrt{\beta}})$ since this will be relevant for the
reflected entropy transition. We can use the inequality in Eq. (3.22) and
perform the integrals, which are simple Gaussian integrals. In fact, we will
include one-loop effects to find the answer to greater accuracy. Computing the
relevant quantities, we obtain

$\displaystyle\begin{split}Z_{1}&\simeq\frac{e^{S_{0}+2\pi^{2}/\beta}}{\sqrt{2\pi\beta^{3}}}\\\
k&\simeq e^{S_{0}}\frac{s_{k}e^{2\pi s_{k}}}{8\pi^{3}}\\\
\lambda_{0}&\simeq\frac{\text{erfc}(\frac{\beta
s_{k}-2\pi}{\sqrt{2\beta}})}{2k}\\\
\int_{0}^{s_{1}}\rho(s)w(s)^{m}&\simeq\frac{\beta^{3m/2-1}e^{4\pi^{2}(1-m)/\beta}}{(1-m)(2\pi)^{m/2-2}},\end{split}$
(3.27)

where $\text{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-x^{2}}dx$ is
the error function. Thus, integrating Eq. (3.22) with the measure $\rho(s)$
and using Eq. (3.27), we obtain

$\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})\approx k\lambda_{0}^{m},$ (3.28)

where we have used the fact that the integral of $\rho(s)w(s)^{m}$ is
suppressed by powers of $\beta$ in the regime of interest.

##  4 Reflected Entanglement Spectrum

We now use a generalization of the resolvent trick in Sec. (4.1) to analyze
the reflected entanglement spectrum. Solving the Schwinger-Dyson equation
leads us to the spectrum and reflected entropy in Sec. (4.2). Finally, we
analyze the $(m,n)$-Renyi reflected entropies in Sec. (4.3).

###  4.1 Resolvent Trick

Our goal here is to find the resolvent for the reduced density matrix
$\rho_{R_{1}R_{1}^{*}}$ obtained from the Renyi generalization of the
canonically purified state $\ket{\psi^{(m)}}$ defined in Eq. (2.8), i.e.,

$R_{m}(\lambda)=\frac{k_{1}^{2}}{\lambda}+\sum^{\infty}_{n=1}\frac{\operatorname{tr}(\rho^{(m)}_{R_{1}R_{1}^{*}})^{n}}{\lambda^{n+1}}$
(4.1)

where the integer moments of the (normalized) density matrix can be computed
using the replica trick on $mn$ copies of the system [12, 18].

In order to do so, we first set up a slightly more general problem.888This
problem is analyzed in more detail in Ref. [18], here we attempt to be brief
and focus on the main results. Consider the following $2\times 2$ “matrix” of
resolvents:

$\mathbb{R}(\lambda)=\sum_{k=0}^{\infty}\lambda^{-1-k/2}\begin{pmatrix}0&(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma^{\dagger}}\\\
(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}&0\end{pmatrix}^{k},$ (4.2)

where $\rho\rightarrow\rho^{\Gamma}$ is an involution defined to take a linear
operator $\rho$ on $\mathcal{H}_{R_{1}R_{2}}$ to a linear operator from
$\mathcal{H}_{R_{1}R_{1}^{\star}}\rightarrow\mathcal{H}_{R_{2}R_{2}^{\star}}$.
It is defined by re-arranging the incoming/outgoing legs in the obvious (and
canonical/basis independent) way. Each insertion of
$(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}$ (and
$(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma^{\dagger}}$) has $m/2$ replicas of each of
the bra and ket of the original state $\ket{\psi}$. These involuted density
operators are related to the density matrices in the canonically purified
state by

$(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma^{\dagger}}(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}=\rho^{(m)}_{R_{1}R_{1}^{\star}}\,,\qquad(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma^{\dagger}}=\rho^{(m)}_{R_{2}R_{2}^{\star}}.$
(4.3)

Diagramatically this is given by the following boundary conditions

$(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}=\includegraphics[scale={.4},valign={c}]{fig2/offdiagonal.eps},$
(4.4)

where we have chosen $m=6$ for example. As before, any calculation using the
gravitational path integral in this model will amount to a sum over geometries
of different topology connecting the various asymptotic boundaries, e.g, in
computing Eq. (4.4) we get

$(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}=\sum_{\text{topologies}}\includegraphics[scale={.5},valign={c}]{fig2/filling_gamma.eps}.$
(4.5)

For convenience, we will use a shorthand representation

(4.6)

From these results we can infer an alternative expression for matrix
$\mathbb{R}(\lambda)$:

$\mathbb{R}(\lambda)=\sum_{n=0}^{\infty}\lambda^{-1-n}\begin{pmatrix}\rho_{R_{1}R_{1}^{\star}}^{n}&0\\\
0&\rho_{R_{2}R_{2}^{\star}}^{n}\end{pmatrix}+\sum_{n=1}^{\infty}\lambda^{-1/2-n}\begin{pmatrix}0&\rho_{R_{1}R_{1}^{\star}}^{n-1}(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma^{\dagger}}\\\
\rho_{R_{2}R_{2}^{\star}}^{n-1}(\rho^{m/2}_{R_{1}R_{2}})^{\Gamma}&0\end{pmatrix},$
(4.7)

which diagrammatically looks like

$\includegraphics[width=398.06111pt]{diagram-matrix}.$ (4.8)

The resolvent for system $R_{1}\cup R_{1}^{*}$ in the state
$\ket{\rho^{m/2}_{R_{1}\,R_{2}}}$ is given by

$R(\lambda)={\rm Tr}_{R_{1}R_{1}^{\star}}(\mathbb{R}_{11}(\lambda))$ (4.9)

Figure 7: Example geometries that give rise to the projector or the identity.
All possible contributions are either proportional to the identity operator
$1_{AA^{*}},1_{BB^{*}}$, or one of the four projectors
$\ket{\epsilon_{A,B}}\bra{\epsilon_{A,B}}$.

A single copy of $(\rho^{(m/2)}_{AB})^{\Gamma}$ is referred to as a _puddle_
and given two contiguous sets of puddles, there are two classes of diagrams,
either _disconnected_ or _connected_. As before, consider the connected part
of $\mathbb{R}$ that we call $\mathbb{F}$. This corresponds to a sum over the
sub-diagrams over all connected contractions . We have

$\displaystyle\mathbb{R}(\lambda)=\frac{1}{\lambda}+\frac{\mathbb{F}}{\lambda^{2}}+\frac{\mathbb{F}^{2}}{\lambda^{3}}+\ldots=\frac{1}{\lambda-\mathbb{F}}$
(4.10)

A useful fact is that all diagrams in $\mathbb{F}$ and $\mathbb{R}$ take the
following form: (see Fig. (7))

$\displaystyle\mathbb{F}(\lambda)$
$\displaystyle=\begin{pmatrix}G_{11}(1_{R_{1}R_{1}^{\star}}-e_{R_{1}})+F_{11}e_{R_{1}}&F_{12}\left|\epsilon_{R_{1}}\right>\left<\epsilon_{R_{2}}\right|\\\
F_{21}\left|\epsilon_{R_{2}}\right>\left<\epsilon_{R_{1}}\right|&G_{22}(1_{R_{2}R_{2}^{\star}}-e_{R_{2}})+F_{22}e_{R_{2}}\end{pmatrix},$
(4.11) $\displaystyle\mathbb{R}(\lambda)$
$\displaystyle=\begin{pmatrix}S_{11}(1_{R_{1}R_{1}^{\star}}-e_{R_{1}})+R_{11}e_{R_{1}}&R_{12}\left|\epsilon_{R_{1}}\right>\left<\epsilon_{R_{2}}\right|\\\
R_{21}\left|\epsilon_{R_{2}}\right>\left<\epsilon_{R_{1}}\right|&S_{22}(1_{R_{2}R_{2}^{\star}}-e_{R_{2}})+R_{22}e_{R_{2}}\end{pmatrix},$
(4.12)

where $\ket{\epsilon_{R_{1}}}=\frac{1}{\sqrt{k_{1}}}\ket{1_{R_{1}}}$ is the
maximally mixed state on $R_{1}R_{1}^{*}$ (same for $\ket{\epsilon_{R_{2}}}$),
$e_{R_{1}}=\left|\epsilon_{R_{1}}\right>\left<\epsilon_{R_{1}}\right|$ and
$e_{R_{2}}=\left|\epsilon_{R_{2}}\right>\left<\epsilon_{R_{2}}\right|$ are
normalized minimal projectors and $F$ is a $2\times 2$ matrix of scalars (not
to be confused with $\mathbb{F}$).

In order to write down a Schwinger-Dyson equation for $\mathbb{F}(\lambda)$,
we can organize the different diagrammatic contributions to $\mathbb{F}$ by
the number of $\mathbb{R}$ insertions, labelled $k-1$, where $k$ is the number
of puddles in each diagram:

(4.13)

where the shorthand conn. indicates connected contractions. Thus, combining
Eq. (4.13) and Eq. (4.10), we now have self-consistency equations that we can
solve to obtain $\mathbb{R}$ and $\mathbb{F}$, order by order in $k$.

###  4.2 Spectrum and Reflected Entropy

The solution to the Schwinger-Dyson equation is identical to that obtained in
Ref. [18] for a random tripartite state. Thus, we only discuss the results
here. The results follow from a conjecture that the important contributions to
the matrix $\mathbb{F}$ come from the lowest two orders in $k$, i.e.
$\mathbb{F}\simeq\mathbb{F}^{(1)}+\mathbb{F}^{(2)}$, in the limit we are
considering. Evidence for this conjecture is provided in Ref. [18], while here
we simply utilize the conjecture and obtain physically sensible results.

Since the solution to the Schwinger-Dyson equation is obtained by considering
$k=1$ and $k=2$, the two parameters that determine the spectrum turn out to be
given by (e.g. for $m=6$)

$\displaystyle D_{m}$
$\displaystyle=\frac{\sqrt{\lambda}}{\sqrt{k_{1}k_{2}}}\,\,\includegraphics[scale={.5},valign={c}]{fig2/dm1.eps}=\frac{\sqrt{\lambda}}{\sqrt{k_{1}k_{2}}}\,\sum_{\text{topologies}}\,\,\includegraphics[scale={.5},valign={c}]{fig2/dm.eps},$
(4.14) $\displaystyle B_{m}$
$\displaystyle=\frac{\sqrt{\lambda}}{\sqrt{k_{1}k_{2}}}\,\,\includegraphics[scale={.5},valign={c}]{fig2/bm1.eps}=\frac{\sqrt{\lambda}}{\sqrt{k_{1}k_{2}}}\,\sum_{\begin{subarray}{c}\text{connected}\\\
\text{topologies}\end{subarray}}\,\,\includegraphics[scale={.5},valign={c}]{fig2/bm.eps},$
(4.15)

where connected topologies are the ones which have geometries joining at least
two asymptotic boundaries on either side of the vertical dashed line. We
define the normalized quantities $\hat{B}_{m}=\frac{B_{m}}{Z_{1}}$ and
$\hat{D}_{m}=\frac{D_{m}}{\sqrt{Z_{1}}}$. Further, these quantities are
related by a sum rule:

$\widehat{D}_{m}^{2}+\widehat{B}_{m}=1$ (4.16)

The solution to the resolvent is then given by

$R(\lambda)\approx\frac{(\chi_{A}\chi_{B})^{2}}{2\widehat{B}_{m}\lambda}\left(\lambda+\widehat{B}_{m}\left(\frac{1}{\chi^{2}_{B}}-\frac{1}{\chi^{2}_{A}}\right)-\sqrt{(\lambda-\lambda_{+})(\lambda-\lambda_{-})}\right)+\frac{1}{\lambda-\widehat{D}^{2}_{m}}$
(4.17)

with

$\lambda_{\pm}=\frac{\widehat{B}_{m}}{(k_{1}k_{2})^{2}}\left(\sqrt{k_{1}^{2}-1}\pm\sqrt{k_{2}^{2}-1}\right)^{2}$
(4.18)

The spectrum obtained from Eq. (4.17) takes a simple form (see Fig. (8)), with
two superselection sectors: a single pole of weight $p_{d}=\hat{D}_{m}^{2}$
and a mound of eigenvalues with weight $p_{c}=\hat{B}_{m}$. The mound is a
Marchenko-Pastur (MP) distribution with support between
$\lambda_{\pm}\simeq\widehat{B}_{m}(k_{1}^{-1}\pm k_{2}^{-1})^{2}$ and
$\min(k_{1}^{2}-1,k_{2}^{2}-1)$ eigenvalues. I.e., the spectrum is given by

$\displaystyle D(\lambda)=\frac{(k_{1}k_{2})^{2}}{2\pi
p_{c}\lambda}\sqrt{(\lambda_{+}-\lambda)(\lambda-\lambda_{-})}+\delta(\lambda)(k_{1}^{2}-k_{2}^{2})\theta(k_{1}^{2}-k_{2}^{2})+\delta(\lambda-
p_{d}),$ (4.19)

where the sum rule then takes the form:

$p_{c}+p_{d}=1,$ (4.20)

and thus we see that these quantities act like two “classical probabilities”.

Figure 8: The spectrum obtained from the resolvent Eq. (4.17). We take
$k_{1}<k_{2}$ here so the MP distribution has no zero eigenvalues. Figure 9:
Plot of $p_{c}(q)\approx\frac{S_{R}(R_{1}:R_{2})}{2\min(S(R_{1}),S(R_{2}))}$
for $m=1$ as a function of $q=\frac{\beta s_{k}}{2\pi}$, for $\beta=1$, in the
canonical ensemble (blue) and microcanonical ensemble (red). The phase
transition in the canonical ensemble is spread out over a region of size
$\Delta s=O(\frac{1}{\sqrt{\beta}})$ due to thermal fluctuations.

Using Eq. (4.14), we obtain

$\displaystyle p_{d}$
$\displaystyle=\frac{(\operatorname{tr}(\rho_{R_{1}R_{2}}^{m/2}))^{2}}{k\operatorname{tr}{\rho_{R_{1}R_{2}}^{m}}}$
(4.21) $\displaystyle p_{c}$ $\displaystyle=1-p_{d}.$ (4.22)

we interpret these superselection sectors as corresponding to the two
different geometries shown in Fig. (10), a disconnected and connected
purification respectively. Thus, the reflected spectrum depends on the Renyi
entropies of $\rho_{R_{1}R_{2}}$, which we computed in Sec. (3.2). For $m=1$,
the physically most interesting case, we have from Eq. (4.21) and Eq. (3.28),

$p_{d}\approx k\lambda_{0}\approx\int_{s_{k}}^{\infty}ds\,\rho(s)w(s),$ (4.23)

which can be interpreted as the Pre-Page probability as discussed earlier.
Consequently, the connected probability $p_{c}$ is interpreted as the Post-
Page probability. We plot this result and compare the phase transitions in the
canonical and microcanonical ensemble in Fig. (9). Due to thermal
fluctuations, the phase transition is spread out over a range of parameter
space of size $\Delta s=O(\frac{1}{\sqrt{\beta}})$.

(a)

(b)

Figure 10: (a) The canonical purification in the disconnected phase has
radiation systems purifying themselves. (b) The canonical purification in the
connected phase has a state of radiation systems entangled via a closed
universe. The two candidate RT surfaces are denoted orange and pink.

The reflected entropy is given by (assuming $k_{1}<k_{2}$)

$\displaystyle S_{R}(R_{1}:R_{2})=\begin{cases}0&k<\exp(S_{BH})\\\ 2\min(\log
k_{1},\log k_{2})&k>\exp(S_{BH}).\end{cases}$ (4.24)

This is the main result of the paper. The reflected entropy is given by:

$\displaystyle S_{R}(R_{1}:R_{2})\approx p_{c}(2\min(S(R_{1}),S(R_{2}))),$
(4.25)

where we generally have the upper bound $S_{R}(R_{1}:R_{2})\leq
2\min(S(R_{1}),S(R_{2}))$. Thus, Fig. (9) serves as an approximate plot of
$S_{R}(R_{1}:R_{2})$ normalized by the upper bound.

###  4.3 Renyi Reflected Entropy

Having discussed the reflected entropy, we now analyze the $(m,n)$-Renyi
reflected entropies. Since the spectrum is an analytic function of $m$, the
Renyi entropies can be computed by using the sum of the moments of the two
sectors in the spectrum

$S_{n}(R_{1}R_{1}^{*})=\frac{1}{1-n}\ln\left(p^{n}_{d}+k_{1}^{2(1-n)}r^{-2}C_{n}(r^{2})p^{n}_{c}\right)$
(4.26)

where $r=k_{1}^{2}/k_{2}^{2}$ and $C_{n}(q)$ is the $q$-Catalan number defined
by

$C_{n}(q)=\sum^{n}_{k=0}q^{k}N(n,k),$ (4.27)

where $N(n,k)$ is the Narayana number. Notably, the probabilities are the only
source of $m$-dependence. Thus, we need to analyze the behaviour of $p_{d}(m)$
as we cross the Page transition using the analysis in Sec. (3.2).

Figure 11: Analytic plot of $p_{c}(m,q)$ for $m=1.5$ (yellow), $m=3$ (orange)
and $m=100$ (green) and numerical plot for $m=0.7$ (thin, brown) as a function
of $q=\frac{\beta s_{k}}{2\pi}$ for $\beta=1$. The locations at which the
$(m,1)$ Renyi reflected entropies undergo a transition changes as a function
of $m$ for $m>1$.

For $m\geq 2$, we obtain

$p_{d}\approx\begin{cases}1&s_{k}<s_{k}^{(m)}\\\
\frac{\exp\left[(m-1)S_{0}+(m-\frac{1}{m})\frac{2\pi^{2}}{\beta}\right]}{k^{m-1}}&s_{k}^{(m)}<s_{k}<s_{k}^{(m/2)}\\\
\frac{\exp\left[S_{0}+\frac{6\pi^{2}}{m\beta}\right]}{k}&s_{k}^{(m/2)}<s_{k}.\end{cases}$
(4.28)

There are two continuous transitions in $p_{d}$ and hence, also in the
$(m,n)$-Renyi reflected entropy. However, the second transition is almost
invisible since $p_{d}$ is already non-perturbatively small at
$s_{k}=s_{k}^{(m/2)}$ and the reflected entropy is dominated by the connected
entanglement wedge.

For $m\in(1,2]$, the phase transition in $p_{d}$ is largely determined by the
phase transition in $\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$ which occurs at
$s_{k}=s_{k}^{(m)}<s_{1}$. Thus, we obtain

$p_{d}\approx\begin{cases}1&s_{k}<s_{k}^{(m)}\\\
\frac{\exp\left[(m-1)S_{0}+(m-\frac{1}{m})\frac{2\pi^{2}}{\beta}\right]}{k^{m-1}}&s_{k}^{(m)}<s_{k}<s_{1}\\\
O(e^{-\frac{1}{\beta}})&s_{1}<s_{k},\end{cases}$ (4.29)

where the value in the last line is difficult to compute analytically,
although it is non-perturbatively suppressed and has a negligible effect on
the reflected entropy. We plot the behavior for $p_{d}(m)$ for different
values of $m>1$ in Fig. (11).

Finally, for the case $m<1$, there is a multiplicative uncertainty in the
calculation of $\operatorname{tr}(\rho_{R_{1}R_{2}}^{m})$ which makes it
difficult to compute $p_{d}(q)$ analytically. We provide a numerical plot
instead in Fig. (11). Qualitatively, the transition happens over a large
region in parameter space corresponding to $s_{k}\in[s_{1},s_{m}]$.

To summarize, the phase transition in the $(m,n)$-Renyi reflected entropy
comes from a shift in dominance between the two terms in Eq. (4.26). As
$n\rightarrow 1$, the transition is dictated by the phase transition in
$p_{d}$. This transition is qualitatively different for $m>1$ and $m<1$. For
$m>1$, it is a sharp transition at $s_{k}=s_{k}^{(m)}$ and for $m<1$, it is
spread out over $s_{k}\in[s_{1},s_{m}]$.

##  5 Summary and Discussion

In this paper we have analyzed the canonical purification and reflected
entropy in the West Coast Model. We found that the holographic proposal is
satisfied in the expected regimes: the canonical purification is related to a
doubled spacetime, and the reflected entropy is related to the entanglement
wedge cross section. Furthermore, by summing over the contributions from all
relevant saddles, we have understood how the geometric picture evolves as we
cross an entanglement phase transition. We now discuss certain interesting
aspects of our calculation and potential directions for future research.

###  5.1 Relation to Petz Map

Consider a code subspace of bulk states encoded in the boundary Hilbert space.
The Petz map provides an explicit realization in this setting of a
reconstruction that maps a given bulk operator to a boundary operator [33, 34,
5]. In the West Coast Model, one can consider bulk operators that live on the
ETW brane. Before the Page transition, they are reconstructable by the black
hole $B$, whereas after the Page transition they are reconstructable by the
radiation system $R$. More so, in Ref. [5], they computed the probability of
the Petz map acting on the radiation to succeed. It turns out to be given by
$p_{c}$, precisely the probability that showed up in our calculation of the
reflected spectrum (regardless of the choice of how to split $R$ into
$R_{1}R_{2}$). This is true in both the microcanonical ensemble and the
canonical ensemble (see Fig. (9) for our $p_{c}$, which matches the Petz
recovery probability computed in Ref. [5]).

This hints at a connection between the reflected entropy and the Petz map
success probability. From one point of view, this seems completely sensible
within the paradigm of entanglement wedge reconstruction. The reflected
entropy $S_{R}(R_{1}:R_{2})$ is a sharp diagnostic of whether the entanglement
wedge is connected, and only when it’s connected does the entanglement wedge
of $R_{1}R_{2}$ include the black hole interior. Indeed, the probability
$p_{c}$ is simply the probability of measuring the canonical purification of
$R_{1}R_{2}$ in the connected phase. It will be interesting to investigate a
possible deeper connection between the reflected entropy and reconstruction
maps.

###  5.2 Area Fluctuations

Here, we considered a holographic state in which the horizon area for $B$ had
area fluctuations. However, in order to do the calculation, we needed to
consider a maximally entangled state between $R$ and the ETW brane flavours.
In the ER=EPR story, this can be interpreted as considering a three boundary
wormhole with area fluctuations for the horizon of $B$ but fixed area boundary
conditions for the horizons of $R_{1}$ and $R_{2}$ respectively, see Figure
12. It would be interesting in the future to understand how the area
fluctuations of $R_{1}$ and $R_{2}$ affect the reflected entropy. Presumably,
it will affect the spectrum on $R_{1}R_{1}^{*}$ by including the effects of
thermal fluctuations into the corresponding superselection sectors. One may
attempt to analyze this using the techniques of the equilibrium approximation
[35, 36].

Figure 12: The three-boundary wormhole corresponding to our current setup.

## Acknowledgments

We would like to thank Xi Dong, Jonah Kudler-Flam, Don Marolf, Sean McBride
and Wayne Weng for useful discussions. PR is supported in part by a grant from
the Simons Foundation, and by funds from UCSB. CA is supported by the Simons
foundation as a member of the It from Qubit collaboration. This material is
based upon work supported by the Air Force Office of Scientific Research under
award number FA9550-19-1-0360. TF is supported in part by Department of Energy
under award number DE-SC0019183.

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[1]Patrick JANOT

1]CERN, 1 Esplanade des Particules, 1217 Meyrin, Switzerland 2]LPNHE,
IN2P3-CNRS, 4 place Jussieu, 75005 Paris, France 3]University of Geneva, 24
rue du Général-Dufour, 1211 Geneva, Switzerland *]e-mail:
<EMAIL_ADDRESS>
# The carbon footprint of proposed $\rm e^{+}e^{-}$ Higgs factories

Alain BLONDEL  [  [  [  [

###### Abstract

The energy consumption of an ${\rm e^{+}e^{-}}$ Higgs factory in operation
will be everything but negligible. Future Higgs boson studies may therefore
have a significant environmental impact. This short note proposes ways to
estimate the environmental footprint during the operation of all the Higgs
factory projects that can credibly operate immediately after the end of LHC,
namely the projects for three linear colliders (CLIC, operating at
$\sqrt{s}=380$ GeV; and ILC and $\rm C^{3}$, operating at $\sqrt{s}=250$ GeV)
and two circular colliders (CEPC and FCC-ee, operating at $\sqrt{s}=240$ GeV).
The projected carbon footprint varies by a factor hundred depending on the
Higgs factory considered.

###### keywords:

Higgs, Factory, Carbon, Footprint

##  1 Introduction

In January 2020, the European Strategy Group concluded that an electron-
positron Higgs factory is the highest-priority next collider, thus giving to
CERN the mandate to prepare for a Higgs factory, followed by a future hadron
collider with sensitivity to energy scales an order of magnitude higher than
those of the LHC CERN-ESU-015 . The CERN Council endorsed this vision in June
2020. In its June 2021 session, the CERN Council approved FCC-FS-plans:202106
; FCC-FS-org:202106  and funded CERNmtp  the FCC technical and financial
feasibility study, with focus on the first step, i.e., the tunnel and the
first stage machine FCC-ee. This additional step somehow clarified the path
for Europe towards the intermediate future.

In July 2022, the Snowmass’21 effort held its last public meeting in Seattle
Snowmass21 , and the Energy Frontier working group came to similar general
conclusions Reina , pending the P5 recommendations and the subsequent DoE
decision: after the immediate future (HL-LHC), the intermediate future is an
$\rm e^{+}e^{-}$ Higgs factory, based on either a linear (ILC in Japan
ILCInternationalDevelopmentTeam:2022izu , $\rm C^{3}$ in the US Dasu:2022nux ,
CLIC at CERN Brunner:2022usy ) or a circular (FCC-ee at CERN Bernardi:2022hny
, CEPC in China Gao:2022lew ) collider, and the long-term future is a multi-
TeV ($\mu\mu$ and/or pp) collider.

The difficult choice for the highest-priority electron-positron Higgs factory
between three linear colliders and two circular colliders, situated at four
different possible locations, may appear subjective, and even a matter of
taste. Indeed, the extensive discussions of the scientific pros and cons
throughout the European Strategy and Snomwass studies concluded that a
substantial part of the Higgs physics programmes at linear and circular $\rm
e^{+}e^{-}$ colliders are similar. In addition, significant domains of physics
exist, in which each of these two types of machines is unique and
irreplaceable Blondel:2019ykp :

  * •

For linear colliders, the ability to explore lepton collisions above 365 GeV
is unique. This is particularly interesting for the searches for new particles
in the gaps possibly left by hadron colliders, and may prove essential should
a new particle be discovered at the LHC in the suitable energy regime. In
addition, centre-of-mass energies above 1 Tev also give the opportunity of a
first (often called “direct”) measurement of the Higgs self-coupling via
double Higgs production, with an expected precision of about 10%.

  * •

For circular colliders, the high luminosity and the multiple detectors, the
exquisite energy calibration at the Z, WW, ZH and $\rm t\bar{t}$ energies, and
the possibility of monochromatisation at $\sqrt{s}=m_{\rm H}$, are building
blocks of a unique program. The Z factory with several trillions of Z
produced, offers unique opportunities for a multitude of electroweak and QCD
measurements, flavour (b, c, $\tau$) physics, as well as direct searches for
SM symmetry violations and feebly coupled particles. The possibility to
observe the $s$-channel Higgs production leads to a unique chance of measuring
the Yukawa coupling of the electron. A circular collider also remains the most
pragmatic, safest, and most effective way towards a 100 TeV pp exploration
machine.

Surely, the members of the linear (circular) Higgs factory efforts are keen on
putting forward that, should a circular (linear) factory go ahead without a
complementary linear (circular) collider, considerable physics opportunities
might be lost.

In any case, the acceptance of the next facility by the governments and the
public will be highly facilitated if it can be shown that environmental
concerns have taken an important place in the choice. In our times of
measurable climate change due to a global warming of unprecedented rapidity,
resulting from the emission of greenhouse gases, the most important criterion
is that the scientific outcome for a given carbon footprint be maximised.

##  2 Carbon footprint per physics outcome

The Higgs factory actors have made no mistake about it: they regularly
communicate about their potential energy savings while in operation with
better technology choices and sustained technology R&D, as well as about the
recycling and recovery of as much as possible of the energy that cannot be
saved in the first place (e.g., under the form of domestic heating, etc.).
Several propositions can be found in the literature on this important topic,
which do not immediately help towards a resolution of the dispute. Three
striking examples of these subtle communication attempts can be found below.

######  Proposition 1.

The estimated electrical power for the linear Higgs factories is in general
lower than circular machines forum .

######  Proposition 2.

The ILC is certainly the less “energy hungry” of all future projects thanks,
in particular, to its superconducting radiofrequency (RF) technology
ILCnewsline .

######  Proposition 3.

The expected AC site power consumption for Higgs production, calibrated to
performance, is probably the lowest for the FCC-ee, compared with other
proposed future projects Benedikt:2020ejr .

To reconcile the three propositions and draw conclusions as to the
environmental footprint of each collider, it is necessary to go through the
actual numbers. The third row of Table 1, which displays the instantaneous
power consumption estimates ($P$) of the five Higgs factory projects (CLIC at
380 GeV, ILC/$\rm C^{3}$ at 250 GeV, and CEPC/FCC-ee at 240 GeV) as reported
in Ref. Roser:2022sht , validates the first proposition. The annual energy
consumption in operation $E$, approximated by $E=PT(1+\epsilon)/2\epsilon$,
where $T$ is the time spent in collisions, $\epsilon$ is the operational
efficiency, and assuming that the power during downtime operation is half that
in collision mode, is displayed in the last row. Setting aside the recent CLIC
performance update Brunner:2022usy , the second proposition is validated.

Table 1: For each of the Higgs factory projects (1st row): Centre-of-mass energy (2nd row); Instantaneous wall-plug power forum  (3rd row); Assumed annual operational time and operational efficiency Bordry:2018gri ; Bambade:2019fyw ; Blondel:2019yqr ; CEPCStudyGroup:2018rmc ; CEPCPhysicsStudyGroup:2022uwl  (4th and 5th rows); Inferred annual energy consumption in operation (last row). Higgs factory | CLIC | ILC | $\rm C^{3}$ | CEPC | FCC-ee
---|---|---|---|---|---
$\sqrt{s}$ (GeV) | 380 | 250 | 250 | 240 | 240
Instantaneous power $P$ (MW) | 110 | 140 | 150 | 340 | 290
Annual collision time $T$ ($10^{7}$ s) | 1.20 | 1.60 | 1.60 | 1.30 | 1.08
Operational efficiency $\epsilon$ (%) | 75 | 75 | 75 | 60 | 75
Annual energy consumption $E$ (TWh) | 0.4 | 0.7 | 0.8 | 1.6 | 1.0
|  |  |  |  |

As this point, it is important to ask whether or not the instantaneous power
and the annual energy consumption are relevant measures for the environmental
impact of a given factory. Maybe not! Take for example a car factory that
produces 1000 (electric) cars per year, consuming energy $E_{1}$, and a second
factory producing 10,000 such cars every year with twice the total energy
$E_{2}(=2E_{1})$. If an environment-aware firm wants to buy 5,000 cars, they
will naturally choose the second factory, which can produce these cars with
five times less energy, not to mention the additional bonus that the totality
of the cars will be delivered ten times quicker. Therefore, one concludes that
to meet the client’s demands of a given number of cars, the latter factory is
five times more environmental-friendly than the former, in spite of its twice
larger annual energy consumption. The same is true for a Higgs factory: given
that the physics outcome depends on the total number of Higgs bosons produced,
a Higgs factory must be judged by its ability to produce Higgs bosons, and in
particular by the energy needed to produce each Higgs boson, if the
environmental impact is of any concern.

The total number of Higgs bosons produced and the time needed to produce them,
as expected in the current operation model of each of the five Higgs
factories, is shown in Table 2. The last row displays the energy consumption
per Higgs boson produced, as obtained from these figures and the last row of
Table 1. The FCC-ee is, by a large factor, the most energy-efficient Higgs
factory project, with 3 MWh/Higgs: this observation validates the third
proposition.

Table 2: For each of the Higgs factory projects (1st row): Total number of Higgs boson produced with the baseline values for the centre-of-mass energy, beam longitudinal polarisation and integrated luminosity (2nd row); Running time in the current operation model (3rd row); Energy consumption per Higgs boson (4th row).  Higgs factory | CLIC | ILC | $\rm C^{3}$ | CEPC | FCC-ee
---|---|---|---|---|---
Running time as a Higgs factory (year) | 8 | 11.5 | 11.5 | 10 | 3
Total number of Higgs bosons produced ($10^{6}$) | 0.25 | 0.5 | 0.5 | 4 | 1
Energy consumption per Higgs boson (MWh) | 14 | 17 | 18 | 4.1 | 3.0
|  |  |  |  |

The baseline FCC-ee operation model has so far included only two interaction
points (IP), but the collider layout was recently modified to be compatible
with the operation of up to four detectors Bernardi:2022hny . The total number
of Higgs bosons expected to be produced at these four IPs is expected to be up
to 1.7 times larger than that produced at two IPs, thus reducing the energy
consumption per Higgs boson to less than 1.8 MWh/Higgs. As a Higgs factory,
FCC-ee with four IPs would therefore be one order-of-magnitude more energy-
efficient than ILC or $\bf C^{3}$, for example.

While the energy efficiency expressed in MWh/Higgs is a closer evaluation of
the environmental footprint of each collider than the instantaneous power or
the annual energy consumption, it still does not give the full and accurate
picture of the environmental impact, dominated by the contribution to global
warming resulting from greenhouse gas emission, which itself strongly depends
on where and when the energy is produced. Today, about 80% of the energy
produced in France originates from nuclear plants, and is therefore mostly
carbon-free. As a consequence, a MegaWatt-hour used at CERN to power a
collider has a much smaller carbon footprint than a MegaWatt-hour used in the
US, in Japan, or in China.

The carbon intensity for electricity production (in kg $\rm CO_{2}$ eq. / MWh)
by the different countries can be obtained from Ref. CO2 , at least for Japan,
France and US Illinois, as shown in Fig. 1 for 2021. Data from China were not
made available to this application at the time of writing this note, and the
specific location of CEPC is not known, but average data over the whole China
in 2021 can be found at https://lowcarbonpower.org/map-gCO2eq-kWh-iea, still
allowing a rough footprint estimate to be made. These data are displayed in
the second row of Table 3 for each of the five $\rm e^{+}e^{-}$ Higgs factory
projects, under the assumptions that CLIC or FCC-ee would be operated at CERN,
ILC at KEK, $\rm C^{3}$ at FNAL, and CEPC from somewhere in China. The carbon
footprint of each Higgs boson produced is the product of the energy
consumption per Higgs boson from Table 2 by the carbon intensity of the energy
used by each Higgs factory, and is displayed in the last row of Table 3.

Table 3: For each of the Higgs factory projects (1st row): Assumed collider location (2nd row); Carbon intensity of electricity production in 2021 at each location, except for China where the carbon intensity is averaged over the whole 2021 (3rd row); Corresponding carbon footprint of each Higgs boson produced (last row). Higgs factory | CLIC | ILC | $\rm C^{3}$ | CEPC | FCC-ee
---|---|---|---|---|---
Operated from | CERN | KEK | FNAL | China | CERN
| Carbon intensity
---
(kg $\rm CO_{2}$ eq. / MWh)
56 | 565 | 381 | 546 | 56
| Carbon footprint per Higgs boson
---
(t $\rm CO_{2}$ eq.)
0.8 | 9.4 | 6.8 | 2.2 | 0.17
|  |  |  |  |

Figure 1: Carbon intensity of electricity production around the world in 2021,
in $\rm kgCO_{2}$ equivalent / MWh, obtained from Ref. CO2 . The carbon
footprints of the energy produced in Illinois, Japan, and France, and used
respectively at FNAL, KEK, and CERN, are highlighted.

Should it operate today, the Higgs factory with the least environmental
impact, i.e., with the smallest carbon footprint for a given physics outcome,
would therefore be FCC-ee operating at CERN, a factor 5 ahead of the next
contender (CLIC). With 4 IPs, the FCC-ee carbon footprint would further reduce
to 0.1 t $\rm CO_{2}$ eq. for each Higgs boson produced, almost two orders of
magnitude better than ILC operating at KEK.

These estimates will need to be revised regularly, as the energy carbonation
levels might change and hopefully improve in developed/developing countries by
the end of HL-LHC. For example, over the 2005-2019 period, the carbon
intensity of the French economy fell by 32% while GDP increased by 17%. France
plans to further reduce the carbon intensity of its electricity production by
another 40% by the end of the decade (and by a factor of almost three by 2050)
France , by further developing its nuclear fleet and by increasing the
production of renewable energies. A 450 B$ Climate Bill has just been adopted
by US Senate (at the beginning of August 2022) with a similar target for the
decade to come. As of 23 July 2022, only 6 out of 17 major nuclear power
plants operate in Japan Japan , while the 11 others are pending restart.
Together with the development of renewable energies in Japan, this may allow
the ILC figures to reduce as well in similar proportions. Finally, the carbon
intensity of China’s electricity production was already reduced by 15% between
2011 and 2021 China , and this reduction is expected to amplify after 2030, to
comply with the 2015 Paris agreements.

##  3 Conclusion

The discussions of the relative scientific merits of linear and circular Higgs
factories (CLIC, ILC, $\rm C^{3}$, CEPC, and FCC-ee) have been extensive
throughout the European Strategy Update and the Snowmass studies, with the
various communities having agreed on standard candles of performance
comparison. Given the increasing relevance of the environmental impact of
future colliders, common meaningful criteria for the comparison between Higgs
factories, which differ from those that have been tabulated so far (e.g.,
instantaneous power, annual energy consumption), have been proposed in this
note, relating the scientific output to the energy consumption, or to the
carbon footprint. These criteria exhibit very significant differences between
the five contenders and might therefore be a convincing tie-breaker in these
times of global warming.

  1. 1.

Because of the much larger instantaneous luminosity and the possibility of
operating four detectors, circular Higgs factories are found to be
significantly less energy hungry to produce Higgs bosons – which is the
raison-d’être of a Higgs factory – than their linear counterparts.
Specifically, the energy consumed by FCC-ee to produce one Higgs boson is
expected to be an order of magnitude smaller than by any of the proposed
linear Higgs factories.

  2. 2.

Because the energy produced in France is mostly carbon-free (unlike in US,
Japan, or China today), the carbon footprint of the FCC-ee operating as a
Higgs factory with 4 IPs at CERN is typically two orders of magnitude smaller
than that of ILC/$\rm C^{3}$, a factor 20 smaller than that of CEPC, and
almost one order of magnitude smaller than that of CLIC, for the same
scientific outcome.

In short, and as illustrated in Fig. 2, FCC-ee is – by very large factors –
the least disruptive in terms of environmental impact during operation, among
the ${\rm e^{+}e^{-}}$ candidate Higgs factories aimed at operating by the end
of HL-LHC.

Figure 2: Energy consumption (top) and carbon footprint (bottom) for the five
Higgs factory projects (CLIC at 380 GeV, ILC/$\rm C^{3}$ at 250 GeV, and
CEPC/FCC-ee at 240 GeV), per Higgs boson produced, i.e., for an equivalent
physics outcome. In these plots, FCC-ee is assumed to operate only two
detectors. With four detectors, the FCC-ee estimators would be divided by a
factor 1.7.

Certainly, the numbers presented in this note are not devoid of uncertainty:
the efficiency of real machines, i.e., the fraction of the available facility
power that can be used; and the fraction of the time this power can be used;
are somewhat arbitrary. All the above “predictions” must therefore be read
with some caution at the 10-20% level and perhaps more. Nevertheless the
differences are well beyond these uncertainties. Meanwhile, common
sustainability studies and R&D activities are being pursued both in the
laboratory and in industry to further improve the energy efficiency of these
colliders, along the four lines of sobriety, efficiency, energy management and
waste heat recovery.

Supplementary information

  * •

In this note, most of the operation parameters (e.g., annual operation time,
operation efficiency, total running time, total number of Higgs bosons
produced) for $\rm C^{3}$ have been assumed to be identical to the
corresponding ILC published numbers.

  * •

It is often argued that the physics outcome of a Higgs factory does not depend
solely on the number of Higgs bosons produced, but that longitudinal beam
polarization (foreseen at linear colliders) is also to be taken into account
in the energy-efficiency determination. Table 4 displays the projected
statistical precision of the Higgs coupling measurements made with one million
Higgs bosons, produced at $\sqrt{s}=250$ GeV either with or without
longitudinal polarization, as extrapolated from Table XIX of Ref.
Bambade:2019fyw .

Table 4: Projected statistical uncertainties (in %) on the Higgs boson couplings with one million Higgs bosons produced in the $\rm e^{+}e^{-}\to ZH$ process at $\sqrt{s}=250$ GeV, obtained with the EFT method without (2nd row) and with (3rd row) longitudinal beam polarisation, as straightforwardly extrapolated from Table XIX of Ref. Bambade:2019fyw . Higgs Coupling | ZZ/WW | bb | cc | gg | $\tau\tau$
---|---|---|---|---|---
| Precision w/o long. polarization (%)
---
0.41 | 0.72 | 1.2 | 1.1 | 0.81
| Precision w/ long. polarization (%)
---
0.36 | 0.71 | 1.3 | 1.2 | 0.79
|  |  |  |  |

It can be inferred from Table 4 that, while longitudinal beam polarization may
increase the $\rm e^{+}e^{-}\to ZH$ production cross section – by a factor
already included in all estimates made so far in this note – the projected
uncertainties (i.e., the physics outcome) with a given number of Higgs bosons
are mostly independent of whether beams are longitudinally polarized or not.
The small excursions (within $\pm 10\%$) observed with the EFT method entirely
disappear when the couplings are determined in the $\kappa$ framework
Blondel:2019yqr .

  * •

In this note, the annual energy consumption of each Higgs factory includes
only the collider consumption while in operation (including downtime).An
alternative footprint estimate could be obtained by including the energy spent
in beam commissioning, machine-development periods, technical stops, and
shutdown periods. Such an exercise has been done, e.g., for FCC-ee in Ref.
Burnet  and for CLIC in Ref. Brunner:2022usy , which increases the annual
energy consumption from 1.0 to 1.52 TWh, and from 0.4 to 0.6 TWh,
respectively. Because similar factors would apply to the other Higgs factories
as well, the conclusions of this note are largely unchanged with this
alternative estimate.

  * •

The current status of sustainability studies and related R&D activities to
further improve the energy efficiency and environmental impact of linear
colliders at CERN, FNAL, or KEK, can be found in Ref. Benny . Many of the
solutions presented therein are considered for circular colliders as well.

  * •

All the facilities mentioned in this note require the construction of a new
tunnel, which adds to the carbon footprint from running the collider. (Two
facilities, ILC and CEPC, would start from a green field, and thus require in
addition the construction of a brand-new international particle physics
centre.) Because the FCC-ee footprint in operation is so much smaller than
that of the other contenders, it is important to make sure that the tunnel
construction does not saturate the total carbon footprint. Several estimates
exist in the literature for the 91-km-long FCC tunnel, should conventional
construction methods be used. An experimental approach from existing road
tunnel construction is followed in Ref. RODRIGUEZ2021103704 , extrapolating to
250-300 kt $\rm CO_{2}$ for the FCC tunnel, assuming that the footprint scales
with the tunnel length and the square of its diameter. In a more recent
contribution to Snowmass’21 Bloom:2022gux , a footprint of 221 kt $\rm CO_{2}$
is predicted from a bottom-up calculation driven by the tunnel parameters.
Both estimates are consistent, and are comparable with the 170 kt $\rm CO_{2}$
eq. from the three years of FCC-ee running at $\sqrt{s}=240$ GeV as a Higgs
factory. Importantly, the tunnel construction footprint is also significantly
smaller than the $\simeq 1.25$ Mt $\rm CO_{2}$ eq. expected from the 15 years
of the FCC-ee programme Burnet . The same tunnel may then serve a 100 TeV
hadron collider (and/or potentially a multi-TeV muon collider) for several
additional decades.

Acknowledgments

Special thanks are due to Panagiotis Charitos, Marcin Chrząszcz, Mike
Koratzinos, Joachim Kopp, Michelangelo Mangano, Bill Murray, Vladimir
Shiltsev, Steinar Stapnes, and Frank Zimmermann for their useful comments and
suggestions during the preparation of this note. This work is partially funded
from the European Union’s Horizon 2020 research and innovation programme under
grant agreement No. 951754.

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