††institutetext: International Centre for Theoretical Sciences (ICTS-TIFR),
Tata Institute of Fundamental Research,
Shivakote, Hesaraghatta, Bangalore 560089, India.

# Proper time to singularity and thermal correlators

Kaustubh Singhi<EMAIL_ADDRESS>
###### Abstract

We study certain higher point thermal correlators of heavy and light scalar
primaries in a holographic CFT. Assuming simple self-interactions and
couplings of the scalars in the bulk theory, we show that the thermal
correlators contain a signature of the proper time to singularity from the
horizon. The key idea is to use WKB approximations for the propagators and
evaluate the bulk integrals using saddle point method.

##  1 Introduction

Given the success of the AdS/CFT correspondence in correctly computing CFT
correlation functions from entirely bulk supergravity calculations, it is
natural to ask which features of the AdS bulk are present in the boundary CFT.
In particular, the semiclassical bulk theory permits various kinds of black
hole solutions, and so one can ask whether (and how) the geometrical features
of black holes are encoded in the dual CFT.

A reason that this question is not straightforward to answer is that the
statement of the duality is best stated in the Euclidean theories. In presence
of a black hole, the bulk Euclidean theory lives only on the exterior of the
black hole. In this case, the correspondence relates Euclidean CFT correlators
to those of fields living in the exterior geometry.

Regardless of the fact that the Euclidean bulk only consists of exterior
geometry, the metric is still obtained from continuation of the Lorentzian
one. So, it is often expected that geometrical features such as the black hole
interior and singularity, present in the full Lorentzian geometry, have to be
obtained by some analytic continuation of the Euclidean CFT correlators. This
connection has been explored in the early days of AdS/CFT Kraus:2002iv ;
Fidkowski:2003nf ; Festuccia:2005pi  and recently in Horowitz:2023ury ;
Ceplak:2024bja . The key result in these papers is a signature of the black
hole singularity in the analytically continued Euclidean correlators.

In contrast, Grinberg and Maldacena Grinberg:2020fdj  have argued that in the
presence of a higher derivative coupling of a heavy scalar, such as Weyl
tensor squared, thermal one point function of the dual CFT primary contains a
signature of the proper time to singularity from the horizon. The bulk
calculation is performed in the large mass (WKB) limit, where the scalar wave
equation can be solved using geodesics Louko_2000 . In this limit, some bulk
integrals can be performed using saddle point approximation, and so the
calculation reduces to finding the location of saddle points in the bulk
geometry.

Grinberg and Maldacena’s proposal has renewed interest in thermal correlators
in holographic CFTs and signatures of the black hole singularity. David and
Kumar David:2022nfn  have strengthened their claim by performing exact
calculations in different black hole backgrounds while also considering a
Gauss-Bonnet coupling of the scalar instead of Weyl squared one. They have
also tried addressing the problem from the CFT side David:2022nfn ;
David:2023uya . To probe richer physics than that available from the one point
function, there have been several attempts to generalize the claim to higher
point functions Rodriguez-Gomez:2021pfh ; Rodriguez-Gomez:2021mkk ;
Georgiou:2022ekc ; Georgiou:2023xpg . These attempts focus on using geodesics
to study the leading structure of thermal two and three point functions of CFT
primaries of large conformal dimension.

In this paper, we look for geometrical features of the bulk in more general
higher point thermal correlators of CFT primaries of large and small conformal
dimensions. To this end, we introduce heavy and light bulk scalar fields in a
black brane background and consider bulk interactions that will source higher
point functions of the dual operators. We argue that to obtain the information
of proper time to singularity in thermal correlators, it is not necessary to
couple the heavy scalar to higher derivative curvature terms — even simple
self-interaction terms of the heavy scalar, or interactions of the heavy and
light scalar are sufficient to give this signature of singularity. Our
emphasis in this paper is on keeping the first two terms in the WKB expansion
and then directly performing the bulk integrals using saddle point method.
This is in contrast to earlier attempts where the focus has been to replace
the calculation with some geodesic computations.

The organization of the paper is as follows. In sections 2 and 3, we provide
the requisite background and obtain the WKB expressions for the scalar
propagators required to compute thermal correlators. To my knowledge, with the
exception of an appendix in Grinberg:2020fdj , previous computations have
featured only the leading order WKB solution, i.e. the geodesic approximation,
for the boundary-bulk propagators. Since we require both boundary-bulk and
bulk-bulk propagators at next-to-leading order in WKB, we spend some time in
deriving these. In sections 4 and 5, we calculate appropriate higher point
thermal correlators of CFT primaries and show that these too contain the
information of proper radial time to singularity. Contributions to thermal
correlators from Witten diagrams containing bulk-bulk propagators are
considered in section 6 along with arguments for the appearance of the time to
singularity in these contributions as well. Finally, we conclude with some
discussions on our results.

##  2 Setup

Consider a $d+1$-dimensional black brane AdS geometry with heavy and light
probe scalar fields, $\phi$ and $\chi$, respectively. These scalars are dual
to conformal primaries $\mathcal{O}_{\phi}$ and $\mathcal{O}_{\chi}$ in a
thermal state of the holographic CFT. Using the AdS/CFT dictionary (refer
Freedman_Van_Proeyen_2012  for details on AdS/CFT in the supergravity limit),
the relation between the conformal dimensions of $\mathcal{O}_{\phi}$ and
$\mathcal{O}_{\chi}$ to the masses, $m_{\phi}$ and $m_{\chi}$, of bulk scalars
$\phi$ and $\chi$, respectively, is given by111The AdS length scale has been
set to unity.

$\displaystyle\Delta_{\phi}$
$\displaystyle=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m_{\phi}^{2}},$ (1)
$\displaystyle\Delta_{\chi}$
$\displaystyle=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m_{\chi}^{2}}.$ (2)

By taking $m_{\phi}$ large, we are studying correlators of CFT primaries of
large conformal dimension.

In Grinberg:2020fdj , Grinberg and Maldacena considered a Weyl squared
coupling of the heavy scalar in the bulk (as in (32)). From the point of view
of AdS/CFT, the Weyl squared coupling sources the thermal one point function
of the dual primary operator. In the probe limit, it is simply given by a bulk
integral of the boundary-bulk $\phi$ propagator weighted by the value of the
Weyl squared term (see Figure 5). Using WKB and saddle point methods, it can
be argued that the thermal one point function of $\mathcal{O}_{\phi}$ takes
the following form

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}}_{\beta}\sim\exp[-m_{\phi}(\ell_{h}+i\tau_{s})].$
(3)

In this expression, $\ell_{h}$ is the regularized distance between the black
hole horizon and the boundary, and $\tau_{s}$ is the proper time a radially
infalling geodesic observer takes to reach the singularity from the horizon.
The above form has been verified by exact computation as well. Calculation of
the thermal one point function has been reviewed in appendix B.

A form similar to (3) has been conjectured for a variety of symmetric black
hole solutions. In the case of black holes with an inner horizon, instead of
$i\tau_{s}$, the one point function features the proper radial time between
the two horizons ($i\tau_{in}$) and the proper radial distance between the
inner horizon and the singularity ($\ell_{s}$). David and Kumar David:2022nfn
have verified by exact computation this conjecture in the case of a charged
black hole in a particular large $d$ (spacetime dimension) limit. They have
also obtained similar results for the thermal one point function by
considering a Gauss-Bonnet coupling instead of Weyl squared.

In this paper, we establish that to see a geometrical feature such as the time
to singularity from the thermal CFT, it is _not_ necessary to introduce a bulk
coupling of the scalar with higher derivative curvature terms. In fact, even
correlators sourced by simple self-interaction terms or scalar exchange
couplings contain the signature of time to singularity.

We begin by considering two classes of bulk couplings. Firstly, we take a
self-interaction term of the form $(\phi)^{n}$ for $n\geq 3$. In the dual CFT,
the $n$-point function of the primary operator $\mathcal{O}_{\phi}$ gets a
tree level contribution from this term (see Figure 1). Secondly, we consider a
coupling between the heavy and light scalars. We take a ‘heavy-light’
interaction of the kind $(\phi)^{N}(\chi)^{M}$, for positive integers $N,M$.
Such a coupling sources correlators of the schematic kind
$(\mathcal{O}_{\phi})^{N}(\mathcal{O}_{\chi})^{M}$ (see Figure 2).

The Euclidean bulk action is written as

$\displaystyle S[\phi,\chi]=S_{\rm free}[\phi]+S_{\rm free}[\chi]+S_{\rm
int}[\phi,\chi].$ (4)

The free part is simply the Klein-Gordon action

$\displaystyle S_{\rm free}[\phi]$
$\displaystyle=\frac{1}{2}\int{d^{d+1}x}\sqrt{g}\left[\partial_{\mu}\phi\partial^{\mu}\phi+m_{\phi}^{2}\phi^{2}\right]$
(5) $\displaystyle S_{\rm free}[\chi]$
$\displaystyle=\frac{1}{2}\int{d^{d+1}x}\sqrt{g}\left[\partial_{\mu}\chi\partial^{\mu}\chi+m_{\chi}^{2}\chi^{2}\right],$
(6)

and the interaction, as just discussed, is chosen to be

$\displaystyle S_{\rm
int}[\phi,\chi]=\int{d^{d+1}x}\sqrt{g}\left[\frac{\lambda_{n}}{n!}(\phi)^{n}+\frac{\kappa_{N,M}}{N!M!}(\phi)^{N}(\chi)^{M}\right].$
(7)

To avoid cumbersome expressions, factors of $16\pi G$ have been suppressed.
Setting the AdS length scale to 1, the bulk Euclidean metric for the black
brane is given by

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\left(f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+d\vec{x}^{2}\right)\quad;\quad
f(z)=1-\frac{z^{d}}{z_{h}^{d}}.$ (8)

The Euclidean bulk geometry consists only of the exterior of the black hole,
i.e. of the region $0<z\leq z_{h}$, and all bulk integrals run over this range
only.

Thermal correlators in the CFT can be evaluated holographically by computing
integrals over bulk propagators. For example, the leading contribution from
the self-interaction in (7) to the thermal $n$-point function of
$\mathcal{O}_{\phi}$ is given by the following bulk integral

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}=\lambda_{n}\int{dz}\sqrt{g(z)}G_{k_{1}}^{\phi}(z)\ldots
G_{k_{n}}^{\phi}(z).$ (9)

Similarly, the heavy-light interaction sources the following thermal
correlator

$\displaystyle\begin{split}\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}&\\\
&=\kappa_{N,M}\int{dz}\sqrt{g(z)}G_{k_{1}}^{\phi}(z)\ldots
G_{k_{N}}^{\phi}(z)G_{q_{1}}^{\chi}(z)\ldots G_{q_{M}}^{\chi}(z).\end{split}$
(10)

In the above expressions, the momentum space boundary-bulk propagators for the
fields $\phi$ and $\chi$ are denoted, respectively, by $G_{k}^{\phi}(z)$ and
$G_{q}^{\chi}(z)$.222Since the black brane background has translation
isometries in all non-radial directions, momentum space fields and correlators
are related to position space ones by a simple Fourier transform. A factor of
$(2\pi)^{d}$ and an overall momentum conserving delta function have been
suppressed.

##  3 Preliminaries

In this section, we obtain the approximate expressions for the boundary-bulk
and bulk-bulk propagators of the $\phi$ field in the large mass (WKB) limit.
These WKB expressions will then be used to compute integrals of the kind (9)
and (10). The bulk-bulk propagators will be of use for the computations in
section 6. Explicit expressions for $\chi$-propagators won’t be required in
the saddle point computations we perform.

We introduce momentum $k\equiv(\omega,\vec{k})$ corresponding to the
translational isometries of the black brane backgrounds. Then, only the radial
part of the wave equation remains to be solved.333A similar statement can be
made for other spherically symmetric black hole backgrounds. Now, there is a
choice to be made on which our approximate WKB solutions _will_ depend:
whether to scale the momentum with the mass or not.

The limit in which we work throughout this paper is where we take the mass
$m_{\phi}$ to be large while keeping the momentum fixed. In this case, the
‘leading’ order WKB solutions do not get any momentum dependence. In
particular, as we will see below, the turning point is still at the horizon in
this limit. Since we compute only leading order exponential dependence on
$m_{\phi}$ of the correlators in question, it suffices to keep only the next-
to-leading order WKB solutions.

The other limit, in which the momentum is scaled with a factor of $m_{\phi}$
and then the large mass limit is taken, relates the WKB solutions to bulk
geodesics. In this case, the turning point lies in the exterior region, so the
saddle arguments to approximate the bulk integrals have to be made much more
carefully. Even after restricting to a specific dimension and momentum values,
it is not at all clear that the CFT correlators in this limit will give
results similar to the ones obtained in the other limit. An analysis is
performed in Rodriguez-Gomez:2021mkk  to compute thermal three point function
by reducing the WKB solution to geodesics carrying generic momentum.

We emphasize that the relation between the WKB ansatz, like the one obtained
in (13), and the geodesic distances is only for the first term. The successive
terms in the WKB expansion have to be obtained by solving the wave equation.
Furthermore, if the turning point lies in the domain of the bulk integrals, as
it does in the second limit, then a prescription has to be adopted for fixing
the WKB solutions. This limit will be studied at a later time. Of course, it
is perfectly reasonable to just study low modes in the large mass limit.
Hence, we proceed with the first case in which the momentum is left untouched
while taking the large mass limit.

###  3.1 WKB approximations

We want to get approximate solutions to the Klein-Gordon equation for the
heavy scalar, $\phi$, in the black brane background:

$\displaystyle\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}\partial^{\mu}\phi)-m_{\phi}^{2}\phi=0,$
(11)

where $g_{\mu\nu}$ is the Euclidean metric (8). After going to Fourier space,
the radial equation reads as

$\displaystyle
z^{d+1}\partial_{z}(z^{-d+1}f(z)\partial_{z}\phi_{k}(z))-\left(\frac{\omega^{2}}{f(z)}z^{2}+\absolutevalue*{\vec{k}}^{2}z^{2}+m_{\phi}^{2}\right)\phi_{k}(z)=0$
(12)

with $k\equiv(\omega,\vec{k})$.

Now, we find WKB solutions to (12) in the large mass limit by taking the
ansatz $\phi_{k}(z)\propto\exp(m_{\phi}S(z))$, where $S(z)$ admits an
expansion in $\frac{1}{m_{\phi}}$. Since we are _not_ scaling the momentum by
$m_{\phi}$, the leading order WKB solution reads the same as in the case of
vanishing momentum.444From hereon, by leading order WKB solution we mean the
first two terms in the $\frac{1}{m_{\phi}}$ expansion of $S(z)$. The momentum
contributes to $S(z)$ at order $\displaystyle\frac{1}{m_{\phi}^{2}}$.

Substituting the WKB ansatz into the equation, we find the solutions for
$S(z)$. They are conveniently written in the variable
$w=\frac{z^{d}}{z_{h}^{d}}$

$\displaystyle S(z)\equiv
S(w)=\pm\ell_{w}+\frac{1}{4m_{\phi}}\log(\frac{w^{2}}{1-w})+O\left(\frac{1}{m_{\phi}^{2}}\right),$
(13)

where the $\pm$ sign corresponds to normalizable and non-normalizable
solutions, respectively. $\ell_{w}$ is the regularized radial geodesic
distance from the boundary to the bulk point $w$. It is evaluated by the
integral

$\displaystyle\ell_{w}=\frac{1}{d}\lim_{w_{c}\to
0}\left(\int_{w_{c}}^{w}{\frac{dw}{w\sqrt{1-w}}}+\log
w_{c}\right)=\ell_{h}-\frac{2}{d}\tanh^{-1}\sqrt{1-w}.$ (14)

The identification $\ell_{h}\equiv\frac{2}{d}\log 2$ has been made for the
regularized distance to the horizon.

Some comments are in order concerning the validity of WKB approximation for
the solutions. The second term in (13) gives the prefactor for the exponent.
Because of fractional powers of $w$ and $1-w$, it appears as if both the
solutions have branch points at _both_ $w=0,1$. But this is not quite correct,
since the expansion itself will break down at the turning point, which in this
case is at the horizon, $w=1$. This can be seen more explicitly by solving the
Klein-Gordon equation (12) exactly for $\omega,\vec{k}=0$. The normalizable
and non-normalizable solutions are respectively given by

$\displaystyle\phi_{0}^{\rm norm}(w)$
$\displaystyle=w^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi};2h_{\phi};w),$ (15)
$\displaystyle\phi_{0}^{\rm nn}(w)$
$\displaystyle=w^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi};1;1-w),$ (16)

where we have introduced the parameter $\displaystyle
h_{\phi}\equiv\frac{\Delta_{\phi}}{d}$.

It is clear from (15) that the normalizable solution has a branch point at
$w=0$ because of the $w^{h}$ term and a branch point at $w=1$ due to the
hypergeometric. In contrast, from (16) it can be seen that the non-
normalizable solution only has a branch point at $w=0$. This means that in the
WKB solutions, _part_ of the branch point at $w=1$ is indeed a spurious
artefact of the solution being invalid near the turning point.

It may be possible to account for this spurious feature by performing an
‘exact WKB’ computation kawai2005algebraic , or even by solving the equation
near the turning point and matching it against the WKB solution. But for the
calculations in this paper, it is not necessary to do so. In the saddle point
analysis that follows, we will simply restrict to using the WKB solutions in
the region away from the turning point so that the approximation still holds.
We hence only consider saddle points away from the turning point.

The large mass expression for the solutions at vanishing momentum can also be
got by using an integral representation for the hypergeometric and then
evaluating the integral in saddle point method. As a consistency check, we
give an alternative derivation of the WKB expression of the boundary-bulk
propagator (proportional to the non-normalizable solution) in appendix A.

Now that we have the solutions, we need expressions for the boundary-bulk and
bulk-bulk propagators. Since we are only interested in correctly producing the
exponent in the saddle point approximation, we can suppress some overall
factors and numerical constants.

The boundary-bulk propagator is simply proportional to the non-normalizable
solution. In the WKB limit taken to obtain (13), the boundary-bulk propagator
is simply given as

$\displaystyle
G_{k}^{\phi}(w)\sim\sqrt{\frac{w}{\sqrt{1-w}}}e^{-m_{\phi}\ell_{w}}\left(1+O\left(\frac{1}{m_{\phi}}\right)\right),$
(17)

where $\ell_{w}$ is given in (14). The normalization of the boundary-bulk
propagator is obtained by the factor of the horizon distance $\ell_{h}$.

Given the two independent solutions in (13), the bulk-bulk propagator can be
constructed in the usual way using the Heaviside function

$\displaystyle\mathcal{G}_{k}^{\phi}(w_{1};w_{2})\sim\sqrt{\frac{w_{1}w_{2}}{\sqrt{1-w_{1}}\sqrt{1-w_{2}}}}\left[e^{-m_{\phi}\ell_{w_{1}}}e^{m_{\phi}\ell_{w_{2}}}\Theta(w_{1}>w_{2})+e^{-m_{\phi}\ell_{w_{2}}}e^{m_{\phi}\ell_{w_{1}}}\Theta(w_{1}<w_{2})\right].$
(18)

Since $\ell_{w}$ is a monotonically increasing function of $w$ as we move from
the boundary to the horizon, the bulk-bulk propagator can be written more
succinctly as

$\displaystyle\mathcal{G}_{k}^{\phi}(w_{1};w_{2})\sim\sqrt{\frac{w_{1}w_{2}}{\sqrt{1-w_{1}}\sqrt{1-w_{2}}}}e^{-m_{\phi}\absolutevalue{\ell_{w_{1}}-\ell_{w_{2}}}}\left(1+O\left(\frac{1}{m_{\phi}}\right)\right).$
(19)

The prefactors in (17)-(19) coming from the leading order WKB solution are
often ignored. A reason is that these prefactors ‘shift’ the saddle points in
a way that only subleading contributions get affected. But we keep these
factors along to fix the saddle locations. This will become clear when we
explicitly evaluate the integrals for the correlation functions using saddle
point method.

##  4 $n$-point function
$\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}$

We begin by computing the thermal $n$-point function of a CFT primary,
$\mathcal{O}_{\phi}$, of large conformal dimension, $\Delta_{\phi}$. Such a
correlator is sourced holographically by an $n$-point bulk self-interaction of
the dual heavy scalar, $\phi$, as introduced in the action (7). The Witten
diagram for the thermal correlator is illustrated in Figure 1, and the
corresponding bulk integral over the Euclidean black brane geometry is given
by (9). Shifting to the coordinate $w=\frac{z^{d}}{z_{h}^{d}}$ and plugging in
the measure, the integral reads

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}\sim\lambda_{n}\int_{0}^{1}{\frac{dw}{w^{2}}}G_{k_{1}}^{\phi}(w)\ldots
G_{k_{n}}^{\phi}(w).$ (20)

Recall that we are looking for signatures of the black hole singularity in the
higher point thermal correlators. In the large mass limit, the propagators get
linked to geodesics through the WKB solutions. We are interested in extracting
the exponent in the large mass limit of the resulting integrals. Hence, we
will not keep track of overall factors.

$w=0$$G_{k_{1}}^{\phi}$$G_{k_{2}}^{\phi}$$G_{k_{n}}^{\phi}$$\bm{+}$$w$ Figure
1: Witten diagram for the $n$-point function of $\mathcal{O}_{\phi}$ being
sourced holographically by a self-interaction term of the dual scalar. The
interaction vertex has been marked and the bulk point is integrated over the
exterior region. The $\phi$ propagators are denoted by solid lines. Momenta
carried by the propagators are indicated and are small in the sense that they
_do not_ scale by the mass of the heavy scalar.

Substituting the WKB expression (17) for the boundary bulk propagator, and
bringing all integrand factors in the exponent, the bulk integral reads

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}\sim\lambda_{n}\int_{0}^{1}{dw}\exp[-nm_{\phi}\ell_{w}+\frac{n}{4}\log(\frac{w^{2}}{1-w})-\log
w^{2}].$ (21)

The leading behaviour of the integral in (21) can be computed easily in saddle
point method. The saddle equation reads

$\displaystyle-
nm_{\phi}\evaluated{\partial_{w}\ell_{w}}_{w^{*}}+\frac{n}{4}\left(\frac{2}{w^{*}}+\frac{1}{1-w^{*}}\right)-\frac{2}{w^{*}}=0,$
(22)

which, after putting in (14), simplifies to

$\displaystyle-\frac{nm_{\phi}}{d}\frac{1}{w^{*}\sqrt{1-w^{*}}}+\frac{2n-8-(n-8)w^{*}}{4w^{*}(1-w^{*})}=0.$
(23)

At large $m_{\phi}$, the leading behaviour of the saddle location is easily
obtained as555There is another solution near $w=1$ of equation (23), but as
discussed earlier, the WKB approximation is not valid here. Hence, this saddle
should be ignored to avoid spurious results.

$\displaystyle
w^{*}\approx-\left(\frac{4n}{n-8}\frac{m_{\phi}}{d}\right)^{2}.$ (24)

Now, to obtain the leading exponential behaviour of the integral in (21), we
simply need to evaluate the integrand at the saddle point. Since the saddle
(24) lies at infinity, we run into an ambiguity in defining the regularized
length $\ell_{w}$ at this point. One way to resolve this ambiguity is by
giving $m_{\phi}$ a negative imaginary part, so that for large $m_{\phi}$, the
expression (14) yields

$\displaystyle\ell_{w^{*}}\approx\ell_{h}+i\tau_{s},$ (25)

where we have made the identification $\tau_{s}=\frac{\pi}{d}$ for the radial
proper time to singularity.

Thus, the $n$-point function evaluates to the following form

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}\sim\lambda_{n}\exp[-nm_{\phi}(\ell_{h}+i\tau_{s})]\times\text{powers
of }m_{\phi}.$ (26)

##  5 Heavy-light correlator
$\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}$

Now, we consider the effect of an exchange interaction between a heavy and a
light scalar in the bulk. A coupling between these scalars holographically
sources a higher point correlator of the dual primaries. The Witten diagram
for such a contribution to the thermal correlator is illustrated in Figure 2.
The corresponding bulk integral (10) is performed over the exterior black
brane geometry. Plugging in the measure, the integral is easily written in the
$w$ variable as follows

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}\sim\kappa_{N,M}\int_{0}^{1}{\frac{dw}{w^{2}}}G_{k_{1}}^{\phi}(w)\ldots
G_{k_{N}}^{\phi}(w)\mathcal{A}_{M}(w),$ (27)

where we have clubbed the $\chi$ propagators into

$\displaystyle\mathcal{A}_{M}(w)\equiv G_{q_{1}}^{\chi}(w)\ldots
G_{q_{M}}^{\chi}(w).$ (28)

$w=0$$G_{k_{1}}^{\phi}$$G_{q_{M}}^{\chi}$$\bm{+}$$w$ Figure 2: Witten diagram
for the heavy-light correlator of primaries $\mathcal{O}_{\phi}$ and
$\mathcal{O}_{\chi}$ being sourced holographically by an exchange coupling
between the dual scalars. The interaction vertex has been marked and the bulk
point is integrated over the exterior region. The $\phi$ propagators are
denoted by solid lines while the $\chi$ propagators are denoted by dashed
lines. Momenta carried by the propagators are indicated and are small in the
sense that they _do not_ scale by the mass of the heavy scalar.

Substituting the WKB form for the $\phi$ propagators, the integral in (27)
reads

$\displaystyle\begin{split}\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}&\\\
&\sim\kappa_{N,M}\int_{0}^{1}{dw}\mathcal{A}_{M}(w)\exp[-Nm_{\phi}\ell_{w}+\frac{N}{4}\log(\frac{w^{2}}{1-w})-\log
w^{2}].\end{split}$ (29)

The saddle equation for only the exponent in the above integrand will give the
leading order behaviour of the integral in the large mass limit. Notice the
fact that the exponent in (29) is same as the one in (21) with $n$ replaced by
$N$. Of course, this is not a coincidence but simply because the exponent came
entirely from the $\phi$ propagator factors. Nonetheless, it should be noted
that this occurrence is because we chose not to scale the momenta while
deriving the WKB expressions. If instead the momenta were scaled, then the
$\chi$ propagators will depend on $m_{\phi}$ and hence contribute to the
exponent.

The location of the saddle is now simply obtained by replacing $n$ with $N$ in
the previously obtained solution (24)

$\displaystyle w^{*}=-\left(\frac{4N}{N-8}\frac{m_{\phi}}{d}\right)^{2}.$ (30)

Thus, the leading behaviour of the heavy-light thermal correlator is given as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}\sim\kappa_{N,M}\mathcal{A}_{M}(w^{*})\exp[-Nm_{\phi}(\ell_{h}+i\tau_{s})].$
(31)

As $m_{\phi}$ is taken to be large, $w^{*}$ (given by (30)) is also large and
hence the factor $\mathcal{A}_{M}(w^{*})$ can be evaluated by using the
asymptotics of the boundary $\chi$ propagator. We suspect that the
contribution will not be large, but in case it does modify the exponent, then
the shift in the saddle from the $\chi$ propagators should be taken into
account.

##  6 Higher point correlators and bulk-bulk propagators

The purpose of this section is to discuss higher point thermal correlators
which are sourced holographically by Witten diagrams involving bulk-bulk
propagators of the heavy scalar. We still look for a signature of time to
singularity in higher point thermal correlators while using WKB and saddle
point techniques.

The novelty of the calculations and discussion presented can be summarized
here itself. Integrals involving bulk-bulk propagators are notorious to
evaluate since they typically partition the region of integration. Despite
this, we argue for approximating the integral in each region using saddle
point method and then summing the contribution. In the correlators we study,
the leading behaviour of the integral in each region produces the same
exponential factor containing the time to singularity, and hence summing these
contributions gives the same leading behaviour for the full correlator.

To make things more concrete, in addition to the scalar interactions in (7),
we add to the action a Weyl squared coupling of the heavy scalar

$\displaystyle S_{\rm W}[\phi]=\alpha\int{d^{d+1}x}\sqrt{g}\,\phi W^{2}.$ (32)

In the black brane background, the Weyl tensor squared is given by

$\displaystyle W^{2}=d(d-1)^{2}(d-2)w^{2}.$ (33)

This means that for $d\geq 3$, we can form tree level Witten diagrams using a
vertex each of (7) and (32) (see for example Figures 3 and 4). These
contribute, respectively, to corresponding higher point thermal correlators at
order $\alpha\lambda_{N}$ and $\alpha\kappa_{N,M}$.

Since we work in the probe limit, the Weyl tensor squared in (32) acts as a
bulk source for the $\phi$ field, providing a $w^{2}$ weight for bulk
integrals.666Actually, the $w^{2}$ weight from Weyl tensor squared just
cancels against the $\frac{1}{w^{2}}$ coming from the measure. Furthermore,
the momentum conserving delta function at the interaction vertex forces the
attached propagator to be at vanishing momentum.

###  6.1
$\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n-1})}_{\beta}$

If we take an $n$-point self-interaction vertex of the heavy scalar along with
a vertex of the Weyl squared coupling, then we can source the thermal
$n-1$-point function of the dual primary. The Witten diagram is illustrated in
Figure 3.

$w=0$$G_{k_{1}}^{\phi}$$G_{k_{2}}^{\phi}$$G_{k_{n-1}}^{\phi}$$\bm{+}$$w_{1}$$\mathcal{G}_{0}^{\phi}$$\bm{\times}$$W^{2}(w_{2})$
Figure 3: Witten diagram for the $n-1$-point function of $\mathcal{O}_{\phi}$
being sourced holographically by the self-interaction and the Weyl squared
coupling of the dual scalar. The two bulk interaction vertices have been
marked separately and both bulk points are integrated over the entire exterior
region. The $\phi$ propagators are denoted by solid lines. The momenta carried
by the propagators are indicated and are small in the sense that they _do not_
scale by the mass of the heavy scalar.

The bulk integral for the thermal correlator reads as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n-1})}_{\beta}\sim\alpha\lambda_{n}\int{\frac{dw_{1}dw_{2}}{w_{1}^{2}}}G_{k_{1}}^{\phi}(w_{1})\ldots
G_{k_{n-1}}^{\phi}(w_{1})\mathcal{G}_{0}^{\phi}(w_{1};w_{2}).$ (34)

Now, to get the leading exponential behaviour in large $m_{\phi}$ we simply
need to evaluate the value of the integrand at the saddle location. Hence, we
rewrite the integral as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n-1})}_{\beta}\sim\alpha\lambda_{n}\int{dw_{1}dw_{2}}\,\mathcal{I},$
(35)

where the integrand $\mathcal{I}$ is composed from the propagators and the
measure in (34). Using the WKB forms (17) and (19) for the propagators, the
integrand can be expressed as

$\displaystyle\mathcal{I}=\exp[\mathcal{F}(w_{1},w_{2})]$ (36)

with the function $\mathcal{F}(w_{1},w_{2})$ defined by

$\displaystyle\mathcal{F}=-(n-1)m_{\phi}\ell_{w_{1}}-m_{\phi}\absolutevalue{\ell_{w_{1}}-\ell_{w_{2}}}+\frac{1}{4}\log(\frac{w_{1}^{2n-8}w_{2}^{2}}{(1-w_{1})^{n}(1-w_{2})}).$
(37)

As we mentioned at the start of the section, the factor of
$\absolutevalue{\ell_{w_{1}}-\ell_{w_{2}}}$ coming from the bulk-bulk
propagator partitions the region of integration into $w_{1}>w_{2}$ and
$w_{1}<w_{2}$. In each of these regions, respectively, the replacement
$\absolutevalue{\ell_{w_{1}}-\ell_{w_{2}}}=\ell_{w_{1}}-\ell_{w_{2}}$ and
$\absolutevalue{\ell_{w_{1}}-\ell_{w_{2}}}=\ell_{w_{2}}-\ell_{w_{1}}$ has to
be made. Thus, the full integrand can be decomposed into two integrands

$\displaystyle\mathcal{I}=\mathcal{I}_{a}+\mathcal{I}_{b},$ (38)
$\displaystyle\mathcal{I}_{a}=\exp[\mathcal{F}_{a}(w_{1},w_{2})],\quad\mathcal{I}_{b}=\exp[\mathcal{F}_{b}(w_{1},w_{2})],$
(39)

where the functions $\mathcal{F}_{a}$ and $\mathcal{F}_{b}$ are defined by

$\displaystyle\mathcal{F}_{a}(w_{1},w_{2})$
$\displaystyle=-nm_{\phi}\ell_{w_{1}}+m_{\phi}\ell_{w_{2}}+\frac{1}{4}\log(\frac{w_{1}^{2n-8}w_{2}^{2}}{(1-w_{1})^{n}(1-w_{2})}),$
(40) $\displaystyle\mathcal{F}_{b}(w_{1},w_{2})$
$\displaystyle=-(n-2)m_{\phi}\ell_{w_{1}}-m_{\phi}\ell_{w_{2}}+\frac{1}{4}\log(\frac{w_{1}^{2n-8}w_{2}^{2}}{(1-w_{1})^{n}(1-w_{2})}).$
(41)

Note that since each of the integrands $\mathcal{I}_{a}$ and $\mathcal{I}_{b}$
has a large parameter ($m_{\phi}$) in the exponent, we can evaluate the
corresponding integrals using saddle point.777The case $n=2$ is a little
different since there is no large parameter accompanying a function of $w_{1}$
in (41). Nonetheless, the saddle in $w_{2}$ reproduces the correct expected
factor of time to singularity. Denoting the location of saddle points of
$\mathcal{F}_{a}(w_{1},w_{2})$ and $\mathcal{F}_{b}(w_{1},w_{2})$,
respectively, by ($w_{1,a}^{*},w_{2,a}^{*}$) and ($w_{1,b}^{*},w_{2,b}^{*}$),
we proceed to solve the equations

$\displaystyle\evaluated{\partial_{w_{1}}\mathcal{F}_{a}(w_{1},w_{2})}_{w_{1,a}^{*},w_{2,a}^{*}}=0=\evaluated{\partial_{w_{2}}\mathcal{F}_{a}(w_{1},w_{2})}_{w_{1,a}^{*},w_{2,a}^{*}},$
(42)
$\displaystyle\evaluated{\partial_{w_{1}}\mathcal{F}_{b}(w_{1},w_{2})}_{w_{1,b}^{*},w_{2,b}^{*}}=0=\evaluated{\partial_{w_{2}}\mathcal{F}_{b}(w_{1},w_{2})}_{w_{1,b}^{*},w_{2,b}^{*}}.$
(43)

It is clear from the expressions for $\mathcal{F}_{a}$ and $\mathcal{F}_{b}$
that the saddle equations for $w_{1}$ and $w_{2}$ decouples for each of them.
Substituting for $\mathcal{F}_{a}$ and $\mathcal{F}_{b}$ and using (14) for
the regularized lengths, the saddle equations reduce to

$\displaystyle\begin{split}-\frac{nm_{\phi}}{d}\frac{1}{w_{1,a}^{*}\sqrt{1-w_{1,a}^{*}}}+\frac{2n-8-(n-8)w_{1,a}^{*}}{4w_{1,a}^{*}(1-w_{1,a}^{*})}&=0,\\\
\frac{m_{\phi}}{d}\frac{1}{w_{2,a}^{*}\sqrt{1-w_{2,a}^{*}}}+\frac{2-w_{2,a}^{*}}{4w_{2,a}^{*}(1-w_{2,a}^{*})}&=0,\end{split}$
(44)
$\displaystyle\begin{split}-\frac{(n-2)m_{\phi}}{d}\frac{1}{w_{1,b}^{*}\sqrt{1-w_{1,b}^{*}}}+\frac{2n-8-(n-8)w_{1,b}^{*}}{4w_{1,b}^{*}(1-w_{1,b}^{*})}&=0,\\\
-\frac{m_{\phi}}{d}\frac{1}{w_{2,b}^{*}\sqrt{1-w_{2,b}^{*}}}+\frac{2-w_{2,b}^{*}}{4w_{2,b}^{*}(1-w_{2,b}^{*})}&=0.\end{split}$
(45)

Notice the fact that the equation for $w_{1,a}^{*}$ is exactly the same as the
equation for $w^{*}$ in (23). This is simply because dependence of
$\mathcal{F}_{a}$ on $w_{1}$ is same as dependence of the exponent in (21) on
$w$. The dependence of $\mathcal{F}_{b}$ on $w_{1}$ is a little different,
thus modifying the saddle equation (and solution) for $w_{1,b}^{*}$ a bit.
Also notice that the saddle equations for $w_{2,a}^{*}$ and $w_{2,b}^{*}$ only
differ by the sign of the $m_{\phi}$ term. This means that the solution is the
same for both in the cut plane but differs in the covering space.

The solutions at large $m_{\phi}$ of (44) and (45) are given as

$\displaystyle
w_{1,a}^{*}\approx-\left(\frac{4n}{n-8}\frac{m_{\phi}}{d}\right)^{2},\quad
w_{2,a}^{*}\approx-\frac{16m_{\phi}^{2}}{d^{2}},$ (46) $\displaystyle
w_{1,b}^{*}\approx-\left(\frac{4(n-2)}{n-8}\frac{m_{\phi}}{d}\right)^{2},\quad
w_{2,b}^{*}\approx-\frac{16m_{\phi}^{2}}{d^{2}}.$ (47)

Now, we simply need to evaluate the integrands $\mathcal{I}_{a}$ and
$\mathcal{I}_{b}$ at these saddle values. Using the same prescription as
before, we evaluate the regularized lengths at the saddles as

$\displaystyle\ell_{w_{1,a}^{*}}=\ell_{w_{2,a}^{*}}=\ell_{w_{1,b}^{*}}=\ell_{w_{2,b}^{*}}=\ell_{h}+i\frac{\pi}{d}+O\left(\frac{1}{m_{\phi}}\right).$
(48)

so that each of the integrands start of as

$\displaystyle\mathcal{I}_{a}\sim\exp[-(n-1)m_{\phi}(\ell_{h}+i\tau_{s})],$
(49)
$\displaystyle\mathcal{I}_{b}\sim\exp[-(n-1)m_{\phi}(\ell_{h}+i\tau_{s})].$
(50)

Thus, we conclude that the leading behaviour of the contribution coming from
Witten diagram in Figure 3 to the thermal $n-1$-point function of
$\mathcal{O}_{\phi}$ is given as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{n})}_{\beta}\sim\alpha\lambda_{n}\exp[-(n-1)m_{\phi}(\ell_{h}+i\tau_{s})].$
(51)

###  6.2
$\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N-1})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}$

Similar to the previous computation, we now consider a contribution to a
heavy-light thermal correlator coming from the Witten diagram formed from a
heavy-light vertex and a Weyl squared interaction vertex. The diagram is shown
in Figure 4.

$w=0$$G_{k_{1}}^{\phi}$$G_{q_{M}}^{\chi}$$\bm{+}$$w_{1}$$\mathcal{G}_{0}^{\phi}$$\bm{\times}$$W^{2}(w_{2})$
Figure 4: Witten diagram for heavy-light correlator being sourced
holographically by bulk couplings between the dual fields and the Weyl squared
coupling of the heavy scalar. The two bulk interaction vertices have been
marked separately and both the bulk points are integrated over the entire
exterior region. Solid lines denote propagators for the heavy scalar, $\phi$,
and dashed ones are for the light scalar, $\chi$. The momenta carried by the
propagators are indicated and are small in the sense that they _do not_ scale
by the mass of the heavy scalar.

The bulk integral for the contribution reads as

$\displaystyle\begin{split}\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N-1})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}&\\\
&\sim\alpha\kappa_{N,M}\int{\frac{dw_{1}dw_{2}}{w_{1}^{2}}}G_{k_{1}}^{\phi}(w_{1})\ldots
G_{k_{N-1}}^{\phi}(w_{1})\mathcal{G}_{0}^{\phi}(w_{1};w_{2})\mathcal{A}_{M}(w_{1}),\end{split}$
(52)

where $\mathcal{A}_{M}(w)$ is the same product of propagators defined in (28).
The product of $\phi$ propagators is the same as that appearing in the
integrand of (37) with the replacement of $n$ with $N$. Since the factor of
$\mathcal{A}_{M}(w_{1})$ does not contribute any terms with the large
parameter $m_{\phi}$, the leading behaviour from the saddle point analysis can
be obtained by considering the remaining integrand only. Hence, the results of
the previous subsection can be borrowed.

Thus, we deduce that the contribution to the thermal correlator (52) is
approximated to leading order as

$\displaystyle\begin{split}\expectationvalue{\mathcal{O}_{\phi}(k_{1})\cdots\mathcal{O}_{\phi}(k_{N-1})\mathcal{O}_{\chi}(q_{1})\cdots\mathcal{O}_{\chi}(q_{M})}_{\beta}&\\\
&\sim\alpha\kappa_{N,M}\exp[-(N-1)m_{\phi}(\ell_{h}+i\tau_{s})]\left(c_{a}\mathcal{A}_{M}(w_{1,a}^{*})+c_{b}\mathcal{A}_{M}(w_{1,b}^{*})\right).\end{split}$
(53)

Here, $w_{1,a}^{*}$ and $w_{1,b}^{*}$ are obtained, respectively, from (46)
and (47) by replacing $n$ with $N$. $c_{a}$ and $c_{b}$ are the coefficients
coming from performing the integral after the partitioning due to the bulk-
bulk propagator. These will, at most, contain powers of $m_{\phi}$.

##  7 Conclusion

We have argued in this paper that for a thermal CFT admitting a holographic
description of probe fields in a black brane background, certain higher point
thermal correlators may contain the information of the black hole singularity.
We consider the bulk theory with a self-interaction term of a heavy scalar and
a coupling between a heavy and light scalar.

The interactions source thermal correlators of the dual CFT primaries, which
are given by bulk integrals of propagators over the Euclidean geometry. In the
limit of large conformal dimension, or equivalently large mass of the heavy
scalar, these integrals can be evaluated using WKB forms of the propagators
and saddle point method. We have shown that the saddle contribution contains
geometrical factors such as the proper radial time to singularity from the
horizon. The explicit form of the thermal correlators is given by (26) and
(31). The use of WKB and saddle point approximations is itself justified by
its success in matching with the exact calculation of the thermal one point
function being sourced by a bulk coupling of the scalar to the Weyl tensor
squared Grinberg:2020fdj . This has been reviewed in our notation in appendix
B.

An important conclusion that we emphasize is that to obtain a signature of
singularity in thermal CFT correlators, it was not necessary to include
couplings of dual scalars to higher derivative curvature terms. Simple self-
interaction terms also work to give information of the time to singularity.
The reason is that the saddle of the bulk integrals lies at $\infty$ (in the
$w$ variable), and then the WKB expressions have the proper length to the bulk
vertex in the exponent, thus giving a factor of the time to singularity. Such
computations _do not_ necessitate factors coming from curvature terms.

We have also considered contributions to thermal correlators coming from
Witten diagrams containing bulk-bulk propagators. Bulk-bulk propagators
partition the region of integration making the analysis a bit involved.
Nonetheless, it is argued that each integral can be approximated to leading
order using saddle point method. Since each of them contains the same exponent
containing the time to singularity, the full contribution of the Witten
diagram also has the same form.

The temperature of the CFT is inversely proportional to the horizon radius,
$T\propto z_{h}$. So, by restoring factors of $z_{h}$ in the propagators and
the measure, we can recover temperature dependence of the correlators. For
instance, the overall temperature dependence in (26) of the $n$-point function
of $\mathcal{O}_{\phi}$ can be fixed to $(T)^{n(\Delta_{\phi}-d)+d}$, since a
factor of $z_{h}^{d-\Delta_{\phi}}$ comes from each propagator insertion (see
for example David:2022nfn  for fixing normalization of the propagators), and
$z_{h}^{-d}$ comes from the measure of the bulk integral. Similarly, the
temperature dependence of the heavy-light correlator in (31) can be fixed as
$T^{N(\Delta_{\phi}-d)+M(\Delta_{\chi}-d)+d}$. If we evaluate the next order
terms in the WKB expansion of the propagator, both momentum dependence and
factors of $z_{h}$ will appear. Thus, the terms subleading in
$\displaystyle\frac{1}{m_{\phi}}$ have a different temperature dependence than
the leading term.

A significant advantage of the WKB approach over just using the geodesic
approximations, as in Rodriguez-Gomez:2021pfh ; Rodriguez-Gomez:2021mkk ;
Georgiou:2022ekc ; Georgiou:2023xpg , to get the form of the propagators is
that the corrections coming from the momentum and the temperature dependence
can be computed systematically even at small values of the momentum. At large
momentum (scaled by $m_{\phi}$), the expressions for the propagators are more
complicated and it is not clear whether the saddle analysis will give
information about the singularity. Nonetheless, the WKB expressions can be
corrected and are amenable for computing bulk integrals.

It is evident from David and Kumar’s analysis David:2022nfn  that the
structure of the thermal one point function in (3) holds for more general
(‘spherically’ symmetric) black holes. So, one might ask whether our results
for the higher point thermal correlators also carry forward to these cases. In
highly isometric black hole solutions, the wave equation reduces to an ODE in
a single (radial) variable, with the coefficients containing parameters
corresponding to the isometries (analogue of $k=(\omega,\vec{k})$). In such a
case, it is generically possible to use the WKB ansatz to find a solution in a
$\displaystyle\frac{1}{m_{\phi}}$ expansion. If in the large $m_{\phi}$ limit,
the other parameters are not scaled, then we expect the saddle arguments to go
through. The precise location of the saddle in the radial variable may, of
course, differ. But as $m_{\phi}$ is taken to $\infty$, the saddle would also
go to $\infty$ and produce the factor of time to singularity, or for black
holes with inner horizon, the factor of time to inner horizon from the outer
one and space-like distance to singularity from the inner horizon.

Although our computation of the thermal correlators is holographic and relies
on the supergravity approximation of the bulk theory, it is general enough
that simple bulk interaction terms make the correlators vary exponentially
with the dimension of the fields. It is interesting to consider whether this
behaviour is preserved when the ’t Hooft coupling is not large, i.e. when
supergravity is no longer a good approximation. In fact, it may be the case
that such a variation of thermal correlators of heavy fields is a general
feature in weakly coupled QFTs as well. One can then rely on perturbative or
non-perturbative methods at weak coupling for verifying a feature like this
directly from the QFT side. This should be a less formidable task than trying
to verifying this feature in strongly coupled CFTs as attempted in
David:2023uya .

There are several ways for improving upon the analysis done here. To start
with, obtaining the form of the corrections coming from the higher order WKB
terms containing factors of the momentum will give us more information about
the dynamics in the thermal CFT. In particular, it may be possible to connect
the factor of time to singularity appearing as a ‘phase’ in the higher point
thermal correlators with diffusive phenomena in the CFT. On the other hand,
the method followed here may be helpful in explicitly performing bulk
calculations in black hole backgrounds which do not have spherical symmetry,
for example rotating black holes.888I thank Justin David and Srijan Kumar for
suggesting this possibility. It will be nice if the method presented in this
paper can be generalized to thermal correlation functions of primaries
carrying large momentum (scaling with the conformal dimension) to see whether
these contain geometrical features of the bulk geometry, and in particular, of
the black hole.

###### Acknowledgements.

I express my gratitude towards my mentor, R. Loganayagam, for helping me
improve upon the draft and painfully going over a few calculations with me. I
would like to thank Justin David and Srijan Kumar for illuminating
discussions. I also thank Gautam Mandal, Godwin Martin, Suvrat Raju, Ashoke
Sen, Shivam Sharma, Omkar Shetye, Shridhar Vinayak, Avi Wadhwa, and the rest
of the ICTS string group for various interactions resulting in an improvement
of this work. I acknowledge support of the Department of Atomic Energy,
Government of India, under project no. RTI4001.

##  Appendix A WKB expression of $G_{0}^{\phi}$

We provide an alternative derivation for the WKB expression (17) of the
boundary-bulk propagator $G_{0}^{\phi}(w)$ at vanishing momentum. The exact
solution is given as (suppressing overall factors that do not contribute to
the exponent in large $h_{\phi}$ limit) David:2022nfn

$\displaystyle
G_{0}^{\phi}(w)=\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}w^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi};1;1-w).$
(54)

We make use of the following integral representation for the hypergeometric
(equation 9.112 in tableofintegrals )

$\displaystyle{}_{2}F_{1}(p,p+n;n+1;z^{2})=z^{-n}\frac{\Gamma(p)n!}{\Gamma(p+n)}\int_{0}^{2\pi}{\frac{dt}{2\pi}}\frac{\cos
nt}{(1-2z\cos t+z^{2})^{p}},$ (55)

where $n$ is a whole number and $p\neq 0,-1,-2,\ldots$. The hypergeometric
appearing in (54) is obtained by putting $n=0$, $p=h_{\phi}$ and
$z=\sqrt{1-w}$. The expression for the propagator then reads

$\displaystyle
G_{0}^{\phi}(w)=\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}w^{h_{\phi}}\int_{0}^{2\pi}{\frac{dt}{2\pi}}\frac{1}{\left(2-w-2\sqrt{1-w}\cos
t\right)^{h_{\phi}}}.$ (56)

For large $h_{\phi}$, the $t$ integral can be performed using saddle point. It
is easy to see that saddles lie at solutions of $\sin t=0$, i.e. at $t=n\pi$
for integers $n$. Furthermore, the steepest descent contours for even $n$ lie
parallel to the real axis and for odd they lie parallel to the imaginary axis
(for $z>0$). Thus, for the integration contour in (56), we pick up the
contributions around $t=0$ and $t=2\pi$. The two contributions are equal due
to symmetry about $t=\pi$, and thus we can expand the integrand about $t=0$ to
quadratic order, leaving us with the following Gaussian integral

$\displaystyle
G_{0}^{\phi}(w)\approx\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}\left(\frac{w}{(1-\sqrt{1-w})^{2}}\right)^{h_{\phi}}\int_{-\infty}^{\infty}{\frac{dt}{2\pi}}\exp[-h_{\phi}\frac{\sqrt{1-w}}{(1-\sqrt{1-w})^{2}}t^{2}].$
(57)

Performing the $t$ integration and using
$\frac{w}{1-\sqrt{1-w}}=1+\sqrt{1-w}$, we get the following expression

$\displaystyle G_{0}^{\phi}(w)\approx\frac{1}{\sqrt{4\pi
h_{\phi}}}\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}\sqrt{\frac{(1-\sqrt{1-w})^{2}}{\sqrt{1-w}}}\exp[h_{\phi}\log(\frac{1+\sqrt{1-w}}{1-\sqrt{1-w}})].$
(58)

The relation between $h_{\phi}$ and the mass of the scalar is
$h_{\phi}=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{m_{\phi}^{2}}{d^{2}}}$, which
for large $m_{\phi}$ simplifies to
$h_{\phi}=\frac{m_{\phi}}{d}+\frac{1}{2}+O(\frac{1}{m_{\phi}})$. Taking a
large $m_{\phi}$ limit of (58) yields

$\displaystyle G_{0}^{\phi}(w)\sim
4^{-\frac{m_{\phi}}{d}}\sqrt{\frac{w}{\sqrt{1-w}}}\exp[\frac{m_{\phi}}{d}\log(\frac{1+\sqrt{1-w}}{1-\sqrt{1-w}})],$
(59)

where we have suppressed $\frac{1}{m_{\phi}}$ corrections, overall numerical
factors, and powers of $m_{\phi}$. Since $\ell_{h}=\frac{2}{d}\log 2$ and
$\tanh^{-1}\sqrt{1-w}=\frac{1}{2}\log(\frac{1+\sqrt{1-w}}{1-\sqrt{1-w}})$, we
get the geodesic distance in the exponent of the propagator, thus reproducing
the first two terms in the WKB expansion of the boundary-bulk propagator

$\displaystyle
G_{0}^{\phi}(w)\sim\sqrt{\frac{w}{\sqrt{1-w}}}e^{-m_{\phi}\ell_{w}}.$ (60)

##  Appendix B One point Function

In this appendix, we use the notation and expressions developed in section 3
for reviewing the calculation of the thermal one point function (3), first
performed in Grinberg:2020fdj . We consider one point function being sourced
by the coupling of the bulk heavy scalar to Weyl tensor squared, as in (32).
The Witten diagram for this contribution is shown in Figure 5. The computation
at large values of $m_{\phi}$, or equivalently $h_{\phi}$, can be performed in
two ways — by exactly performing the integral and then taking a limit, or by
using the WKB expression and evaluating the integral using saddle point
method. The match between the two engenders support for the use of WKB and
saddle point method for bulk integrals.

The contribution of the Witten diagram reads as follows

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(0)}_{\beta}=\alpha\int_{0}^{1}{\frac{dw}{w^{2}}}G_{0}^{\phi}(w)W^{2}(w),$
(61)

which reduces to the following integral after substituting the value (33) for
the Weyl tensor squared.

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(0)}_{\beta}\sim\alpha\int_{0}^{1}{dw}G_{0}^{\phi}(w).$
(62)

The one point function is evaluated at vanishing momentum because of the
translational isometries.

$w=0$$G_{0}^{\phi}$$\bm{\times}$$W^{2}(w)$ Figure 5: Witten diagram for
thermal CFT one point function of a primary being sourced holographically by a
coupling to Weyl squared term. The interaction vertex has been marked and the
bulk point is integrated over the entire exterior region. Boundary-bulk
propagator of the dual scalar is denoted by a solid line.

###  B.1 Exact calculation

Since the propagator in (61) is at vanishing momentum, we can use the exact
form given in (54). Substituting this yields the following integral

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(0)}_{\beta}\sim\alpha\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}\int_{0}^{1}{dw}\,w^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi};1;1-w).$
(63)

This integral is divergent for $\real(h_{\phi})>2$ but can be performed
exactly otherwise. It is an example of the integral formula in equation
7.511-4 in tableofintegrals , which reads

$\displaystyle\int_{0}^{1}{dx}x^{\gamma-1}(1-x)^{\rho-1}{}_{2}F_{1}(\alpha,\beta;\gamma;x)=\frac{\Gamma(\gamma)\Gamma(\rho)\Gamma(\gamma+\rho-\alpha-\beta)}{\Gamma(\gamma+\rho-\alpha)\Gamma(\gamma+\rho-\beta)}.$
(64)

Taking the integration variable as $x=1-w$ and other parameters to be
$\alpha=\beta=h_{\phi}$, $\gamma=1$, and $\rho=h_{\phi}+1$ produces the
integral in (63). Thus, we can take the result of (64) for $\real(h_{\phi})<2$
and analytically continue the answer for large $h_{\phi}$ Freedman:1998tz .
Then, the result of the integration is given as

$\displaystyle\int_{0}^{1}{dw}\,w^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi};1;1-w)=\Gamma(h_{\phi}+1)\Gamma(2-h_{\phi})=h_{\phi}(1-h_{\phi})\frac{\pi}{\sin\pi
h_{\phi}},$ (65)

where in the last equality we have made use of the identities
$\Gamma(x+1)=x\Gamma(x)$ and
$\displaystyle\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin\pi x}$.

The one point function is now given as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(0)}_{\beta}\sim\alpha\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}\frac{\pi
h_{\phi}(1-h_{\phi})}{\sin\pi h_{\phi}}.$ (66)

We see that the one point function is a well defined meromorphic function of
the conformal dimension of the primary (and hence the mass of the bulk field).
The poles at $h_{\phi}=n$ ($\Delta_{\phi}=nd$) for integers $n>2$ correspond
to mixing of $\mathcal{O}_{\phi}$ with powers of the CFT stress tensor $T^{n}$
(see Berenstein:2022nlj  and Grinberg:2020fdj  for further discussion).

We want to study the large $h_{\phi}$, or equivalently, large $m_{\phi}$ limit
of the one point function. The Gamma functions, at leading order, produce the
factor of horizon distance in the exponent

$\displaystyle\frac{\Gamma(h_{\phi})^{2}}{\Gamma(2h_{\phi})}\sim
2^{-2h_{\phi}}\sim e^{-m_{\phi}\ell_{h}}.$ (67)

To extract a limit out of the $\sin$ factor in the denominator of (66), a
negative imaginary part is given to $h_{\phi}$. At leading order, we can
approximate $\displaystyle\frac{1}{\sin\pi h_{\phi}}\sim 2e^{-i\pi
h_{\phi}}\sim 2e^{-im_{\phi}\tau_{s}}$, where we have identified the factor of
proper radial time to singularity. Combining this with (67) reproduces the
exponential behaviour claimed earlier

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}(0)}_{\beta}\sim\alpha\exp[-m_{\phi}(\ell_{h}+i\tau_{s})].$
(68)

###  B.2 Saddle point method

Our aim here is to reproduce the form (68) for the thermal one point function
using the WKB expression for the boundary-bulk propagator at vanishing
momentum and then approximating the bulk integral using saddle point method.
Using the WKB form of the propagator in (17), the integral (62) for the
thermal one point function is written as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}}_{\beta}\sim\alpha\int_{0}^{1}{dw}\exp[-m_{\phi}\ell_{w}+\frac{1}{4}\log(\frac{w^{2}}{1-w})].$
(69)

The saddle equation for the integrand reads

$\displaystyle-\frac{m_{\phi}}{d}\frac{1}{w^{*}\sqrt{1-w^{*}}}+\frac{2-w^{*}}{4w^{*}(1-w^{*})}=0,$
(70)

which is solved at large $m_{\phi}$ to give

$\displaystyle w^{*}\approx-\left(\frac{4m_{\phi}}{d}\right)^{2}.$ (71)

As $m_{\phi}$ is taken the be large, the saddle lies at $\infty$. Giving
$m_{\phi}$ a negative imaginary part defines the proper length at the saddle
as

$\displaystyle\ell_{w^{*}}\approx\ell_{h}+i\tau_{s}.$ (72)

So, at leading order, the exponential contribution to the one point function
is read off from the saddle point answer as

$\displaystyle\expectationvalue{\mathcal{O}_{\phi}}_{\beta}\sim\alpha\exp[-m_{\phi}\ell_{w^{*}}]\sim\alpha\exp[-m_{\phi}(\ell_{h}+i\tau_{s})].$
(73)

It matches exactly with the form obtained from the exact computation of the
bulk integral.

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