December, 2022

Holographic thermal propagator for arbitrary scale dimensions

Borut<EMAIL_ADDRESS>and Adrián R.
<EMAIL_ADDRESS>
a J. Stefan Institute, 1000 Ljubljana, Slovenia

b Instituto de Física La Plata - CONICET, Diagonal 113 y 63, 1900 La Plata,
Argentina

c Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional
de La Plata

C.C. 67, 1900 La Plata, Argentina

###### Abstract

Using the AdS/CFT correspondence we model the behaviour of the two-point
correlator of an operator with arbitrary scale dimension $\Delta$ in arbitrary
spacetime dimension $d$ for small but non-zero temperature. The obtained
propagator coincides in the low temperature regime with the known result for
$d=4$ for large $\Delta$ at the order $T^{d}$ as well as with the $T^{d}$ and
$T^{2d}$ terms of the exact all order result for $d=2$. Furthermore, for
arbitrary $d$ we explicitly write down the expression for the order $T^{d}$ of
the propagator for arbitrary $\Delta$, and present a conjecture for the order
$T^{2d}$ in the large $\Delta$ limit.

###### Contents

  1. 1 Introduction
  2. 2 The propagator from the AdS/CFT approach
    1. 2.1 The $T=0$ case
    2. 2.2 The $T\neq 0$ case
  3. 3 First correction: $(T/k)^{d}$
  4. 4 Second correction: $(T/k)^{2d}$
    1. 4.1 $\nu\to-\nu$
  5. 5 Special $d$
    1. 5.1 $d=2$
    2. 5.2 $d=4$
  6. 6 Conclusions
  7. A Some useful identities
  8. B From $x$ to $k$ space

##  1 Introduction

Correlation functions of operators in conformal field theories are constrained
by symmetry. For example, the two-point correlation function in
$d$-dimensional Euclidean momentum space of a spin zero primary operator with
scale dimension $\Delta=d/2+\nu$ is given by

$G_{2}^{(0)}(k)\propto k^{2\nu}\quad,\quad
k^{2}=\omega^{2}+\vec{q}^{\,2}\quad,$ (1.1)

where we split the time and space part of the $d$-dimensional vector
$k^{\mu}=(\omega,\vec{q})$ for later use.

A bit less obvious is what happens when in such a theory we introduce a non-
zero temperature, i.e. in the spacetime $S_{1/T}^{1}\times\mathbb{R}^{d-1}$.
Conformal symmetry is now broken by $T\neq 0$ and correlators get, as we will
see, corrections of order ${\cal O}((T/k)^{d})$. There have been recently two
important developments in this direction. First, the problem has been attacked
in the limit of $\vec{q}=0$ for the absorption cross section by black branes
[1] or using the AdS/CFT correspondence [2, 3, 4] for large scale dimensions
$\Delta\gg 1$ [5, 6, 7]. This last limit was needed to be able to apply the
geodesic approximation in the background geometry of a black brane in AdS,
while the leading order has been given for general $\Delta$ recently in terms
of the ambient space formalism [8]. Second, by using the solution to the
connection problem of the Heun equation [9, 10] a complete analytic result for
the thermal two-point correlation function has been found recently in [11]
(with the generalisation to non-zero chemical potential in [12]).

These results are however not completely satisfactory. First, large scale
dimensions are not necessarily realistic, so results for smaller values are
welcome. Second, although the solution found in [11] is analytic, it is very
hard to make it work properly in the small temperature expansion of the black
brane case, mainly because an infinite number of terms contribute and the
resummation is far from obvious.

In this work we want to improve the state of the art calculating the first two
corrections (the $(T/k)^{d}$ and the $(T/k)^{2d}$ ones) to the zero
temperature propagator for a completely generic scale dimension $\Delta$ and
arbitrary spacetime dimension $d$. In some sense this paper is a continuation
of [13], although it uses a different approach. The method for calculating the
propagator we use in this paper is straightforward: we rewrite the usual
second order ordinary differential equation (ODE) for a perturbation in the
black brane AdS background as a matrix linear first order ODE, expand it in
powers of the temperature and solve it by explicit integration. When we will
not be able to explicitly and analytically compute the integrals involved, we
will evaluate them numerically. As we will show, our method turns out to be
tougher and tougher as the scale dimension increases, so it is particularly
useful especially for low $\Delta$ and so it is complementary to the large
$\Delta$ calculation of [5, 6, 7].

##  2 The propagator from the AdS/CFT approach

As we mentioned in the introduction we will model our CFT primary operator
with scale dimension $\Delta$ by a scalar field in the AdS black brane
background. The equation to solve in momentum space for the scalar field
perturbation

$\eta(z)=\int d^{d}x\,e^{-ix_{\mu}k^{\mu}}\phi(x,z)$ (2.1)

is thus (see for example [14] and references therein)

$\left(f(z/z_{h})\;z\;\frac{d}{dz}\;z\frac{d}{dz}-d\;z\;\frac{d}{dz}-z^{2}\;\left(k^{2}+\omega^{2}\;\frac{1-f(z/z_{h})}{f(z/z_{h})}\right)-\Delta(\Delta-d)\right)\,\eta(z)=0\quad,$
(2.2)

where

$f(x)\equiv 1-x^{d}$ (2.3)

with the AdS radius $L=1$ and the horizon radius (black brane position)
expressed in terms of the spacetime dimension $d$ and temperature $T$ through

$z_{h}=\frac{d}{4\pi T}\quad.$ (2.4)

Our choice of (Euclidean) boundary conditions for the two linearly independent
solutions $\eta_{\pm}(z)$ are

$\displaystyle z\rightarrow 0^{+}$ $\displaystyle:$
$\displaystyle\eta_{\pm}(z)\quad\longrightarrow z^{\frac{d}{2}\pm\nu}\quad,$
$\displaystyle z\rightarrow z_{h}$ $\displaystyle:$
$\displaystyle|\eta_{-}(z)|<\infty\quad,$ (2.5)

where

$\nu=\Delta-d/2$ (2.6)

is taken to be positive with non integer $2\nu$ all along the paper.

If we write

$\displaystyle\eta(z)=\frac{z^{\frac{d-1}{2}}}{f(z/z_{h})^{\frac{1}{2}}}\;h(z)\quad,$
(2.7)

then (2.2) implies that $h(z)$ obeys

$\left(\frac{d^{2}}{dz^{2}}-U(z)\right)\,h(z)=0\quad,$ (2.8)

where

$U(z)=-\frac{1}{4\,z^{2}}+\frac{d^{2}}{4}\,\frac{2\,f(z/z_{h})-1}{z^{2}\,f(z/z_{h})^{2}}+\frac{\Delta(\Delta-d)}{z^{2}\,f(z/z_{h})}+\frac{k^{2}}{f(z/z_{h})}+\omega^{2}\,\frac{1-f(z/z_{h})}{f(z/z_{h})^{2}}\quad.$
(2.9)

Let $h_{\pm}(z)$ be a basis of solutions to (2.8). The general solution can be
written as

$h(z)=C_{+}\,h_{+}(z)+C_{-}\,h_{-}(z)\quad,$ (2.10)

where $C_{\pm}$ are arbitrary constants. Taking into account the derivative of
(2.10), we can write,

$\left(\begin{array}[]{c}h(z)\\\ h^{\prime}(z)\end{array}\right)={\bf
w}\left(h_{+},h_{-};z\right)\;\left(\begin{array}[]{c}C_{+}\\\
C_{-}\end{array}\right)\quad,$ (2.11)

where

${\bf w}\left(f,g;z\right)\equiv\left(\begin{array}[]{cc}f(z)&g(z)\\\
f^{\prime}(z)&g^{\prime}(z)\end{array}\right)$ (2.12)

is the wronskian matrix of $f$ and $g$ at the point $z$. For any basis of
solutions of (2.8) its determinant, i.e. the wronskian, is a non-zero
constant. From (2.11), our problem reduces to the computation of the wronskian
matrix of a given basis. According to (2) the (Euclidean) boundary conditions
are

$\displaystyle z\rightarrow 0^{+}$ $\displaystyle:$ $\displaystyle
h_{\pm}(z)\quad\longrightarrow z^{\frac{1}{2}\pm\nu}\quad,$ $\displaystyle
z\rightarrow z_{h}$ $\displaystyle:$ $\displaystyle|h_{-}(z)|=0\quad.$ (2.13)

In this way, the wronskian results: $\det({\bf
w}\left(h_{+},h_{-};z\right))=-2\,\nu$.

We can recast the second order equation (2.8) as the following first order
system,

$\frac{d}{dz}\left(\begin{array}[]{c}h(z)\\\
h^{\prime}(z)\end{array}\right)={\bf A}(z)\;\left(\begin{array}[]{c}h(z)\\\
h^{\prime}(z)\end{array}\right)\quad,\quad{\bf
A}(z)\equiv\left(\begin{array}[]{cc}0&1\\\ U(z)&0\end{array}\right)\quad,$
(2.14)

or, for the wronskian matrix

$\frac{d}{dz}{\bf w}\left(h_{+},h_{-};z\right)={\bf A}(z)\;{\bf
w}\left(h_{+},h_{-};z\right)\quad.$ (2.15)

The potential ${\bf A}(z)$ can be split into a zero-temperature and a
temperature-dependent part as

${\bf A}(z)={\bf A}^{(0)}(z)+{\bf A}_{T}(z)\quad,$ (2.16)

where the $T=0$ matrix is

${\bf A}^{(0)}(z)=k^{2}\;\left(\begin{array}[]{cc}0&1\\\
1+\frac{\nu^{2}-\frac{1}{4}}{q^{2}}&0\end{array}\right)\quad,$ (2.17)

with

$q\equiv kz\quad,$ (2.18)

while that the temperature-dependent part can be expanded in powers of
$\left(T/k\right)^{d}$ as

${\bf A_{T}(z)}=k^{2}\;\sigma_{-}u_{T}(q)\quad,\quad
u_{T}(q)=\sum_{m\in\mathbb{N}^{+}}\frac{u^{(m)}(kz)}{(kz_{h})^{d\,m}}\quad,$
(2.19)

where

$\displaystyle\sigma_{-}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\quad,$ $\displaystyle u^{(m)}(q)$ $\displaystyle=$
$\displaystyle A_{m}\,q^{d\,m-2}+B_{m}\,q^{d\,m}\quad,$ $\displaystyle
A_{m}\equiv\nu^{2}-m\,\frac{d^{2}}{4}$ , $\displaystyle B_{m}\equiv
1+m\,(\omega/k)^{2}\quad.$ (2.20)

###  2.1 The $T=0$ case

At zero temperature (2.15) reads,

$\frac{d}{dz}{\bf w}\left(h^{(0)}_{+},h^{(0)}_{-};z\right)={\bf
A}^{(0)}(z)\;{\bf w}\left(h^{(0)}_{+},h^{(0)}_{-};z\right)\quad.$ (2.21)

The solutions are $z^{\frac{1}{2}}$ times Bessel functions of order $\nu$. A
summary of their properties and useful expansions is given in appendix (A).
The basis (2.13) is given by

$\displaystyle h^{(0)}_{+}(z)$ $\displaystyle=$
$\displaystyle\frac{2^{\nu}\,\Gamma(1+\nu)}{k^{\nu}}\;z^{\frac{1}{2}}\;I_{\nu}(q)\quad,$
(2.22) $\displaystyle h^{(0)}_{-}(z)$ $\displaystyle=$
$\displaystyle\frac{2^{1-\nu}\,k^{\nu}}{\Gamma(\nu)}\;z^{\frac{1}{2}}\;K_{\nu}(q)\quad.$
(2.23)

According to the AdS/CFT recipe the propagator is defined through the limit

$h_{-}(z)\quad\stackrel{{\scriptstyle z\rightarrow
0^{+}}}{{\longrightarrow}}\quad
z^{\frac{1}{2}-\nu}\;(1+\dots)+G_{2}(k)\;z^{\frac{1}{2}+\nu}\;(1+\dots)$
(2.24)

to be in the leading order

$G_{2}^{(0)}(k)=\frac{\Gamma(-\nu)}{\Gamma(\nu)}\;\left(\frac{k}{2}\right)^{2\,\nu}\quad.$
(2.25)

###  2.2 The $T\neq 0$ case

Going to the general case, if we make the ansatz for any $T$,

${\bf w}\left(h_{+},h_{-};z\right)={\bf
w}\left(h^{(0)}_{+},h^{(0)}_{-};z\right)\;\left({\bf 1}+{\bf
w}^{(T)}(q)\right)\quad,$ (2.26)

then from (2.15) and (2.21) we get for ${\bf w}^{(T)}(q)$

$\frac{d}{dq}{\bf w}^{(T)}(q)=q\;u_{T}(q)\;{\bf b}(q)\;\left({\bf 1}+{\bf
w}^{(T)}(q)\right)\quad,$ (2.27)

where $u_{T}(q)$ is given in (2.19) and the matrix ${\bf b}(q)$ is

$\displaystyle{\bf b}(q)$ $\displaystyle\equiv$ $\displaystyle z^{-1}\;{\bf
w}\left(h^{(0)}_{+},h^{(0)}_{-};z\right)^{-1}\;\sigma_{-}\;{\bf
w}\left(h^{(0)}_{+},h^{(0)}_{-};z\right)$ (2.30) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I_{\nu}(q)\,K_{\nu}(q)&\alpha\,K_{\nu}(q){}^{2}\\\
-\frac{1}{\alpha}\,I_{\nu}(q){}^{2}&-I_{\nu}(q)\,K_{\nu}(q)\end{array}\right)\quad,$

where

$\alpha\equiv\frac{2^{1-2\nu}\,k^{2\nu}}{\Gamma(\nu)\,\Gamma(1+\nu)}=-2\frac{\sin(\pi\nu)}{\pi}\,G_{2}^{(0)}(k)\quad.$
(2.31)

In order to solve (2.27) we expand

${\bf w}^{(T)}(q)=\sum_{m\in\mathbb{N}^{+}}\frac{{\bf
w}^{(m)}(q)}{(kz_{h})^{m\,d}}$ (2.32)

and get the following system of differential equations for the ${\bf
w}^{(m)}(q)$’s, $m>0$ (we define ${\bf w}^{(0)}=\mathbb{I}$)

$\frac{d}{dq}{\bf w}^{(m)}(q)=\sum_{k=0}^{m-1}\;q\;u^{(m-k)}(q)\;{\bf
b}(q)\;{\bf w}^{(k)}(q)\quad,$ (2.33)

to be solved iteratively, with the $u^{(m)}(q)$’s given in (2). The solution
is

${\bf w}^{(m)}(q)={\bf
w}_{0}^{(m)}+\sum_{k=0}^{m-1}\;\int^{q}dz\,z\;u^{(m-k)}(z)\;{\bf b}(z)\;{\bf
w}^{(k)}(z)\quad.$ (2.34)

From (2.26) and (2.12) we have

$\displaystyle h_{+}(z)$ $\displaystyle=$
$\displaystyle\left(1+w_{11}^{(T)}(q)\right)\;h^{(0)}_{+}(z)+w_{21}^{(T)}(q)\;h^{(0)}_{-}(z)\quad,$
(2.35) $\displaystyle h_{-}(z)$ $\displaystyle=$ $\displaystyle
w_{12}^{(T)}(q)\;h^{(0)}_{+}(z)+\left(1+w_{22}^{(T)}(q)\right)\;h^{(0)}_{-}(z)\quad.$
(2.37)

The integration constants present in the computation of the matrix elements
$w_{ij}^{(T)}(q)$ are fixed by (2.13), i.e. for $z\to 0$

$z\rightarrow 0^{+}\quad:\quad h_{\pm}(z)\quad\longrightarrow
z^{\frac{1}{2}\pm\nu}\quad.$ (2.38)

Since expanding everything around $T=0$ is equivalent to expanding everything
around $z_{h}=\infty$, we replace the last constraint of (2.13) with

$z\rightarrow\infty\quad:\quad h_{-}(z)\longrightarrow 0\quad.$ (2.39)

Near $z=0$,

$\displaystyle h_{+}(z)$ $\displaystyle=$
$\displaystyle\left(1+w_{11}^{(T)}(q)\right)\;z^{\frac{1}{2}+\nu}\,(1+{\cal
O}(q))$ (2.40) $\displaystyle+$ $\displaystyle
w_{21}^{(T)}(q)\;z^{\frac{1}{2}-\nu}\,\left(1+{\cal
O}(q)+G_{2}^{(0)}(k)\,z^{2\,\nu}\,(1+{\cal O}(q))\right)\quad,$ (2.41)
$\displaystyle h_{-}(z)$ $\displaystyle=$ $\displaystyle
w_{12}^{(T)}(q)\;z^{\frac{1}{2}+\nu}\,(1+{\cal O}(q))$ (2.42) $\displaystyle+$
$\displaystyle\left(1+w_{22}^{(T)}(q)\right)\,z^{\frac{1}{2}-\nu}\,\left(1+{\cal
O}(q)+G_{2}^{(0)}(k)\,z^{2\,\nu}\,(1+{\cal O}(q))\right)\quad.$ (2.43)

Conditions (2.38) are fulfilled if

$0=w_{11}^{(T)}(0)=w_{22}^{(T)}(0)=\left.q^{-2\,\nu}\,w_{21}^{(T)}(q)\right|_{q\rightarrow
0}=\left.q^{2\,\nu}\,w_{12}^{(T)}(q)\right|_{q\rightarrow 0}\quad.$ (2.44)

The first three conditions imply that $w_{11,21,22}^{(T)}(0)=0$, so they
determine the respective integration constants $w_{0\,ij}^{(T)}$ in (2.34).
Since in general up to a constant

$\displaystyle z\to 0$ $\displaystyle:$ $\displaystyle w_{ij}^{(m)}(z)\sim
z^{md+(i-j)2\nu}\quad,$ (2.45) $\displaystyle z\to\infty$ $\displaystyle:$
$\displaystyle w_{ij}^{(m)}(z)\sim z^{md+1-|i-j|}e^{(i-j)2z}\quad,$ (2.46)

all $(ij)$ except eventually $(12)$ are well behaved close to $z=0$. On the
other side, close to $z=\infty$ it is exactly the $(12)$ component which is
well behaved. We can thus rewrite (2.34) incorporating (2.38) as (let define
$w_{ij}^{(0)}=\delta_{ij}$)

$w_{ij}^{(m)}(q)=\sum_{k=0}^{m-1}\;\int_{0}^{q}dz\,z\;u^{(m-k)}(z)\;\left({\bf
b}(z)\;{\bf w}^{(k)}(z)\right)_{ij}$ (2.47)

for $(ij)=(11),(21),(22)$, while (2.39) implies

$w_{12}^{(m)}(q)=-\sum_{k=0}^{m-1}\;\int_{q}^{\infty}dz\,z\;u^{(m-k)}(z)\;\left({\bf
b}(z)\;{\bf w}^{(k)}(z)\right)_{12}\quad.$ (2.48)

Taking into account that the behavior of a solution is always of the form:
$a_{-}z^{\frac{1}{2}-\nu}\,(1+{\cal O}(z))+a_{+}z^{\frac{1}{2}+\nu}\,(1+{\cal
O}(z))$, we can write the following behaviour for $w_{12}^{(T)}(q)$ near
$q=0$,

$w_{12}^{(T)}(q)\quad\stackrel{{\scriptstyle q\rightarrow
0}}{{\longrightarrow}}\quad
G_{2}^{(0)}(k)\,g^{(T)}(k)+a\,q^{-2\,\nu+d}\,\left(1+{\cal O}(q)\right)\quad,$
(2.49)

where $g^{(T)}(k)$ is some free constant. Then the AdS/CFT recipe (2.24) gives
the propagator in the form

$G_{2}(k)=G_{2}^{(0)}(k)\,\left(1+g^{(T)}(k)\right)\quad.$ (2.50)

Similarly as ${\bf w}^{(T)}(k)$ in (2.32) also $g^{(T)}(k)$ gets expanded:

$g^{(T)}(k)=\sum_{m=1}^{\infty}\frac{g_{m}(\omega/k)}{(kz_{h})^{md}}\quad.$
(2.51)

Thus, to get the finite temperature correction to the propagator one needs to
extract the constant part of the expansion of $w_{12}^{(T)}(q)$ for $q\to 0$.
If the integral (2.48) converges, then the constant part of the expansion
(2.49) is simply

$G_{2}^{(0)}(k)g_{m}(\omega/k)=-\sum_{k=0}^{m-1}\;\int_{0}^{\infty}dz\,z\;u^{(m-k)}(z)\;\left({\bf
b}(z)\;{\bf w}^{(k)}(z)\right)_{12}\quad.$ (2.52)

However, for $2\nu-md>0$ the integral (2.48) diverges. In this case, to
extract the constant part, one needs first to split the integrand

$F(z)\equiv-\sum_{k=0}^{m-1}\,z\;u^{(m-k)}(z)\;\left({\bf b}(z)\;{\bf
w}^{(k)}(z)\right)_{12}\to\sum_{\alpha>0}\frac{F_{-\alpha}}{z^{\alpha+1}}+\sum_{\beta>0}F_{\beta}z^{\beta-1}$
(2.53)

as $z\to 0$, where $\alpha,\beta$ are in general non integer real numbers.
Notice that for finite $\nu$ there is always only a finite number of divergent
terms ($\alpha$). In this case (2.52) becomes

$G_{2}^{(0)}(k)g_{m}(\omega/k)=\int_{0}^{\infty}dz\,\left(F(z)-\sum_{\alpha>0}\frac{F_{-\alpha}}{z^{\alpha+1}}\right)\quad,$
(2.54)

which renders the integration at $z=0$ finite without changing the convergence
at $z=\infty$.

Of course, if one is able to compute analytically (2.48), then this is not
needed, and one just takes the constant term in the expansion around $q\to 0$
of the result as indicated in eq. (2.49). But apart from the $m=1$ case we
were unable to perform such an analytic integration, in which case the formula
(2.54) (which reduces to (2.52) for $2\nu-md<0$) is the one we will use in the
numerical evaluation.

##  3 First correction: $(T/k)^{d}$

At first order we obtain

${\bf
w}^{(1)}(q)=A_{1}\left(\begin{array}[]{cc}I_{d}[I_{\nu}\,K_{\nu};q,0]&\alpha\,I_{d}[K_{\nu}{}^{2};q,\infty]\\\
\\\
-\frac{1}{\alpha}\,I_{d}[I_{\nu}{}^{2};q,0]&-I_{d}[I_{\nu}\,K_{\nu};q,0]\end{array}\right)+\left((A_{1},d)\to(B_{1},d+2)\right)\quad,$
(3.1)

where we defined the integral

$I_{s}[f;q,q_{0}]\equiv\int_{q_{0}}^{q}\,dz\;z^{s-1}\;f(z)\quad.$ (3.2)

Although we will not use them, these integrals are explicitly computed for
completeness in Appendix A.

From (2.35) we have in our case

$\displaystyle h_{+}(z)$
$\displaystyle=\left(1+\frac{1}{(kz_{h})^{d}}\;w_{11}^{(1)}(q)\right)\;h^{(0)}_{+}(z)+\frac{1}{(kz_{h})^{d}}\;w_{21}^{(1)}(q)\;h^{(0)}_{-}(z)+{\cal
O}\left(\frac{1}{(kz_{h})^{2\,d}}\right)\quad,$ $\displaystyle h_{-}(z)$
$\displaystyle=\frac{1}{(kz_{h})^{d}}\;w_{12}^{(1)}(q)\;h^{(0)}_{+}(z)+\left(1+\frac{1}{(kz_{h})^{d}}\;w_{22}^{(1)}(q)\right)\;h^{(0)}_{-}(z)+{\cal
O}\left(\frac{1}{(kz_{h})^{2\,d}}\right)\quad.$ (3.3)

Using the definitions (3.1) we find the leading low $q$ expansion as

$w_{11}^{(1)}(q)\sim q^{d}\quad,\quad w_{22}^{(1)}(q)\sim q^{d}\quad,\quad
w_{21}^{(1)}(q)\sim q^{2\,\nu+d}\quad,$ (3.5)

while the only constant piece is in

$w_{12}^{(1)}(q)\sim
G_{2}^{(0)}(k)\left(A_{1}\alpha_{d}+B_{1}\alpha_{d+2}\right)+{\cal
O}\left(q^{-2\,\nu+d}\right)\quad,$ (3.6)

where we have defined

$\alpha_{s}=\frac{2\sin{(\pi\nu)}}{\pi}\int_{0}^{\infty}dzz^{s-1}K_{\nu}^{2}(z)\quad,$
(3.7)

which is taken at face value for $s-2\nu>0$, while for $s-2\nu<0$ it must be
understood in the sense of (2.54), i.e. with proper subtractions. However, in
both cases (we are interested in positive integer $s$ and $\nu>d/2$) the final
result is

$\alpha_{s}=\frac{\sqrt{\pi}}{2}\,\frac{\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{s+1}{2}\right)}\,\nu\frac{\Gamma(\frac{s}{2}+\nu)\Gamma(\frac{s}{2}-\nu)}{\Gamma(1+\nu)\Gamma(1-\nu)}\quad.$
(3.8)

The propagator (2.24) at this order is

$G_{2}(k)=G_{2}^{(0)}(k)\left(1+\frac{g_{1}(\omega/k)}{(kz_{h})^{d}}\right)\quad,$
(3.9)

with

$\displaystyle g_{1}(\omega/k)$ $\displaystyle=$ $\displaystyle
A_{1}\alpha_{d}+B_{1}\alpha_{d+2}$ (3.10) $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(\frac{d}{2}\right)}{\Gamma\left(\frac{d+3}{2}\right)}\frac{\Gamma\left(\frac{d}{2}+1+\nu\right)\Gamma\left(\frac{d}{2}+1-\nu\right)}{\Gamma\left(1+\nu\right)\Gamma\left(1-\nu\right)}\nu\left(d\,\left(\frac{\omega}{k}\right)^{2}-1\right)\quad,$

which agrees with [8]. For $d=4$ we get

$G_{2}(k)/G_{2}^{(0)}(k)-1=\left(\frac{\pi
T}{k}\right)^{4}\;\frac{2}{15}\;\nu\,(\nu^{2}-1)\,(\nu^{2}-4)\,\left(4\,\left(\frac{\omega}{k}\right)^{2}-1\right)\quad,$
(3.11)

which is the Fourier transform (B.13) of the result in [6] using
$\Delta=2+\nu$ (2.6). Notice that the result here is exactly the same as for
large $\Delta$. This peculiarity is not preserved at next order in the
temperature expansion.

Since the $d=2$ solution is known [15], for even $d$ there is an even easier
way to compute this leading order correction to the propagator, without
explicitly calculating the integral (3.7), which we now explicitly write as a
function of $\nu$:

$\alpha_{r}(\nu)=\frac{2}{\pi}\sin{(\pi\nu)}\int_{0}^{\infty}dzz^{r-1}K_{\nu}^{2}(z)\quad.$
(3.12)

Using

$K_{\nu}=-K_{\nu+1}^{\prime}-\frac{\nu+1}{z}K_{\nu+1}$ (3.13)

one can derive, applying repeatedly differentiation by parts

$\alpha_{r+2}(\nu)=\alpha_{r+2}(\nu+1)-\frac{r(r-2\nu-2)}{2}\alpha_{r}(\nu+1)\quad.$
(3.14)

Eq. (3.14) means that there is a relation for the $\alpha_{d+2}$ once
$\alpha_{d}$ is known. Since $\alpha_{2}$ is known exactly, $\alpha_{4}$ can
be calculated.

Take the ansatz333The constant term is missing because $g_{1}$ is odd under
$\nu\to-\nu$, see section 4.1, while higher orders $c_{n}$ for $n\geq d$ turn
out to be zero.

$\alpha_{4}(\nu)=\sum_{n=1}^{d-1}c_{n}\nu^{n}$ (3.15)

and plugging it into (3.14) we obtain ($\alpha_{2}(\nu)=\nu$)

$c_{3}\left((\nu+1)^{3}-\nu^{3}\right)+c_{2}\left((\nu+1)^{2}-\nu^{2}\right)+c_{1}\left((\nu+1)-\nu\right)=-2\nu(\nu+1)\quad.$
(3.16)

Equating the same powers of $\nu$ we get

$\alpha_{4}(\nu)=\frac{2}{3}\nu\left(1-\nu^{2}\right)\quad,$ (3.17)

which is the correct result got by direct integration of (2.31), which
Mathematica is able to do easily. Notice that here the recursion relation is
between coefficients in different dimensions but of the same order
(coefficients of $T^{d}$ vs. those of $T^{d+2}$ but not of $T^{2d}$ which are
of next order except for $d=2$). Unfortunately we did not succeed in computing
the next order term $T^{2d}$ using the same idea.

##  4 Second correction: $(T/k)^{2d}$

Going to the second order correction, taking into account (2.44) and (2.39),
we define

$w_{ij}^{(2)}(q)=\left\\{\begin{array}[]{lcc}\int^{q}_{0}dz\,z\;\left(u^{(2)}(z)\;b_{ij}(z)+u^{(1)}(z)\;b_{ik}(z)\;w_{kj}^{(1)}(z)\right)&,&\quad(ij)\neq(12)\\\
\\\
-\int^{\infty}_{q}dz\,z\;\left(u^{(2)}(z)\;b_{12}(z)+\;u^{(1)}(z)\;b_{1k}(z)\;w_{k2}^{(1)}(z)\right)&,&\quad(ij)=(12)\end{array}\right.$
(4.1)

With these definitions we assure that the conditions that define the basis
$h_{\pm}(z)$,

$\displaystyle w^{(2)}_{11}(0)$ $\displaystyle=$ $\displaystyle
w^{(2)}_{21}(0)=w^{(2)}_{22}(0)=0\quad,$ (4.2) $\displaystyle
w^{(2)}_{12}(\infty)$ $\displaystyle=$ $\displaystyle 0$ (4.3)

hold. To get the correction to the propagator, we focus on $w_{12}^{(2)}(q)$
and its limit

$w_{12}^{(2)}(q)\quad\stackrel{{\scriptstyle q\rightarrow
0}}{{\longrightarrow}}\quad
G_{2}^{(0)}(k)\,g_{2}(\omega/k)+a\,q^{-2\,\nu+2\,d}\,\left(1+{\cal
O}(q)\right)\quad.$ (4.4)

For $\nu<d$ the correction to the propagator is

$g_{2}(\omega/k)=A_{2}\alpha_{2d}+B_{2}\alpha_{2d+2}+A_{1}{}^{2}\,g_{dd}+B_{1}{}^{2}\,g_{d+2,d+2}+2A_{1}\,B_{1}\,g_{d,d+2}\quad,$
(4.5)

with

$g_{rs}=\frac{1}{2}\int_{0}^{\infty}dz\,\left(F_{rs}(z)+F_{sr}(z)\right)\quad,$
(4.6)

whose integrand is explicitly given by

$\frac{\pi}{2\sin{(\pi\nu)}}F_{rs}(z)=z^{r-1}\,I_{\nu}(z)\,K_{\nu}(z)\,I_{s}[K_{\nu}^{2};z,\infty]-z^{r-1}K^{2}_{\nu}(z)\,I_{s}[I_{\nu}K_{\nu};z,0]\quad.$
(4.7)

The behaviour of the integral can be read from the behaviour of the Bessel
functions, which main properties are summarised in appendix A. While for
$z\to\infty$ both terms in (4.7) are suppressed by $e^{-2\,z}$, for $z\to 0$
we have

$F_{rs}(z)\sim z^{-2\,\nu+r+s-1}\quad.$ (4.8)

So, the integral (4.6) converges if $2\,\nu<r+s$. Since we need
$r+s=2\,d,2\,d+2,2\,d+4$, for $\nu<d$ no substractions are needed, and the
contribution to the propagator at order $(T/k)^{2d}$, i.e. $g_{2}(\omega/k)$,
can be evaluated directly.

This procedure can be continued for larger $\nu$ using proper subtractions
(2.54). For example, for $\nu<d+1$ one gets the same formula (4.5) but now
with

$g_{dd}=\int_{0}^{\infty}dz\left(F_{dd}(z)-\frac{2^{2\nu}}{2\nu
d(d-2\nu)}\frac{\Gamma(\nu)}{\Gamma(-\nu)}z^{-2\nu+2d-1}\right)\quad,$ (4.9)

with all the rest the same.

More generally, for $2\nu>r+s$, the renormalised formula (4.6) looks like

$\displaystyle
g_{rs}=\frac{1}{2}\int_{0}^{\infty}dz\left[F_{rs}(z)-\sum_{n=0}^{\lfloor 2\nu-
r-s\rfloor}z^{-2\nu+r+s-1+n}\frac{\sqrt{\pi}\csc(\pi\nu)}{4\Gamma(n+1)\Gamma(n-\nu+1)}\times\right.$
$\displaystyle\left(\frac{\pi
4^{\nu}\Gamma(\nu+1)(-1)^{n}\,_{5}F_{4}\left(-n,\frac{1}{2}-\nu,-n-\nu,\frac{s}{2}-\nu,\nu-n;\frac{1}{2}-n,1-2\nu,1-\nu,\frac{s}{2}-\nu+1;1\right)}{\Gamma(n+\nu+1)\Gamma(\frac{1}{2}-n)(-2\nu)\Gamma(1-\nu)(\frac{s}{2}-\nu)}\right.$
$\displaystyle\left.\left.+\frac{\Gamma\left(n-\nu+\frac{1}{2}\right)\,_{5}F_{4}\left(\frac{1}{2},-n,\frac{s}{2},\nu-n,2\nu-n;\frac{s}{2}+1,1-\nu,\nu+1,-n+\nu+\frac{1}{2};1\right)}{\nu
s\Gamma(n-2\nu+1)}\right)\right]$ $\displaystyle+\left(r\leftrightarrow
s\right)\quad.$ (4.10)

Notice that both ${}_{5}F_{4}$ are actually finite sums.

###  4.1 $\nu\to-\nu$

A check of the results is the behaviour of the propagator for $\nu\to-\nu$
(or, equivalently, $\Delta\to d-\Delta$). In this case one should get the
inverse propagator:

$G_{2}(-\nu)=\frac{1}{G_{2}(\nu)}\quad.$ (4.11)

Since at this order

$G_{2}(\nu)=G_{2}^{(0)}(\nu)\left(1+\frac{g_{1}(\nu)}{(kz_{h})^{d}}+\frac{g_{2}(\nu)}{(kz_{h})^{2d}}\right)$
(4.12)

and since due to (2.25) indeed

$G_{2}^{(0)}(-\nu)=\frac{1}{G_{2}^{(0)}(\nu)}\quad,$ (4.13)

eq. (4.11) implies

$g_{1}(-\nu)=-g_{1}(\nu)\quad,$ (4.14)

which is easily satisfied by (3.10), and

$g_{2}(-\nu)=g_{1}^{2}(\nu)-g_{2}(\nu)\quad.$ (4.15)

To check this one explicitly, let us first rewrite (4.6) in terms of the
Wronskian

$g_{rs}=\frac{1}{2}\alpha_{r}\alpha_{s}-\frac{1}{4\nu}\int_{0}^{\infty}dz\,\left(W(F_{r},G_{s};z)+W(F_{s},G_{r};z)\right)$
(4.16)

and (see definitions in appendix A)

$\displaystyle F_{r}(z)$ $\displaystyle=$
$\displaystyle\frac{z^{r}}{r}\left(2f_{r}(z)+g_{r}^{(+)}(z)+g_{r}^{(-)}(z)\right)+2\nu\alpha_{r}\quad,$
(4.17) $\displaystyle G_{r}(z)$ $\displaystyle=$
$\displaystyle\frac{z^{r}}{4\nu
r}\left(g_{r}^{(+)}(z)-g_{r}^{(-)}(z)\right)\quad,$ (4.18)

$\displaystyle F_{r}^{\prime}(z)$ $\displaystyle=$
$\displaystyle-\frac{4\nu}{\pi}\sin{(\pi\nu)}z^{r-1}K_{\nu}^{2}(z)\quad,$
(4.19) $\displaystyle G_{r}^{\prime}(z)$ $\displaystyle=$
$\displaystyle\frac{z^{r-1}}{2}K_{\nu}(z)\left(I_{\nu}(z)+I_{-\nu}(z)\right)\quad.$
(4.20)

From (2.31), (A.26) and the above definitions it is straightforward to see
that under $\nu\to-\nu$ the quantity $\alpha_{r}$ is odd, while $A_{m}$,
$B_{m}$, $F_{r}$ and $G_{r}$ are even. Then in (4.15) the $A_{2}$, $B_{2}$ as
well as the integrals of Wronskians exactly drop out, while the
$\alpha_{r}\alpha_{s}/2$ terms in (4.16) take care of the $g_{1}^{2}$ term in
(4.15). The relation (4.11) is indeed correct up to order $T^{2d}$.

##  5 Special $d$

So far the only check we did is the analytic comparison with the large $\nu$
solution of [6]. Now we will specialise to the $d=2$ case which is exactly
known. We will compare our numerical solution with this exactly known result
and find agreement at second order $T^{4}$. Then we will pass on to $d=4$ and
predict the second order $T^{8}$ correction to the propagator.

###  5.1 $d=2$

Equation (5.8) of [15] in our euclidean setting is

$G(k)=C_{\Delta}\frac{\Gamma(z_{+}+\frac{\Delta}{2})}{\Gamma(z_{+}+1-\frac{\Delta}{2})}\,\frac{\Gamma(z_{-}+\frac{\Delta}{2})}{\Gamma(z_{-}+1-\frac{\Delta}{2})}\quad,$
(5.1)

where

$C_{\Delta}=\frac{\Gamma(-\nu)}{\Gamma(\nu)2^{2\Delta-2}}\left(\frac{2}{z_{h}}\right)^{2\Delta-2}$
(5.2)

is a normalisation constant, and

$z_{\pm}\equiv\frac{z_{h}}{2}\,\left(\omega\pm i\,q\right)\qquad,\qquad
z_{+}\,z_{-}=\frac{(kz_{h})^{2}}{4}\quad.$ (5.3)

We need the expansion for large $kz_{h}$, i.e. large $|z_{\pm}|$. To get it,
we use results in [16]. We have for large $|z|$

$\frac{\Gamma(z+a)}{\Gamma(z+b)}\sim
z^{a-b}\,\sum_{n\in\mathbb{N}}c_{n}(a,b)\,\frac{z^{-n}}{n!}\qquad{\rm
with}\qquad c_{n}(a,b)\equiv(-)^{n}\,(b-a)_{n}\,B_{n}^{(1+a-b)}(a)\quad,$
(5.4)

where $B_{n}^{(k)}(x)$ are generalized Bernoulli polynomials [16].

Applying to (5.1), taking into account that

$c_{2n}=(1-\Delta)_{2n}\,B_{2n}^{(\Delta)}\left(\Delta/2\right)\qquad,\qquad
c_{2n+1}=-(1-\Delta)_{2n+1}\,B_{2n+1}^{(\Delta)}\left(\Delta/2\right)=0\quad,$
(5.5)

we get the expansion

$\displaystyle G(k)$ $\displaystyle\sim$ $\displaystyle
C_{\Delta}\,(z_{+}\,z_{-})^{\Delta-1}\,\left(1+\frac{c_{2}}{2!\,z_{+}{}^{2}}+\frac{c_{4}}{4!\,z_{+}{}^{4}}+{\cal
O}(T^{6})\right)\,\left(1+\frac{c_{2}}{2!\,z_{-}{}^{2}}+\frac{c_{4}}{4!\,z_{-}{}^{4}}+{\cal
O}(T^{6})\right)$ (5.6) $\displaystyle=$ $\displaystyle
G^{(0)}(k)\,\left(1+\frac{c_{2}}{2\,z_{+}{}^{2}}+\frac{c_{2}}{2\,z_{-}{}^{2}}+\frac{c_{4}}{24\,z_{+}{}^{4}}+\frac{c_{2}{}^{2}}{4\,z_{+}{}^{2}\,z_{-}{}^{2}}+\frac{c_{4}}{24\,z_{-}{}^{4}}+{\cal
O}(T^{6})\right)\quad,$ (5.7)

where

$c_{2}=-\frac{1}{12}\,\prod_{l=0}^{2}(\Delta-l)\qquad,\qquad
c_{4}=\frac{1}{240}\,(5\,\Delta+2)\,\prod_{l=0}^{4}(\Delta-l)\quad.$ (5.8)

From (5.6) we find that the $T^{2}$-correction to the propagator

$\frac{g_{1}(\omega/k)}{(kz_{h})^{2}}=\left(\frac{2\pi
T}{k}\right)^{2}\,\frac{1}{3}\,\Delta(\Delta-1)(\Delta-2)\,(1-2\,(\omega/k)^{2})$
(5.9)

coincides with our result (3.10) at $d=2$, while the $T^{4}$-correction is

$\displaystyle\frac{g_{2}(\omega/k)}{(kz_{h})^{4}}$ $\displaystyle=$
$\displaystyle\left(\frac{2\pi
T}{k}\right)^{4}\,\frac{1}{90}\,\Delta(\Delta-1)(\Delta-2)$
$\displaystyle\times$
$\displaystyle\left(9\,\Delta\,(\Delta-2)-12+(5\,\Delta+2)\,(\Delta-3)\,(\Delta-4)\,(1-2\,(\omega/k)^{2})^{2}\right)\quad.$

Writing

$g_{2}(\omega/k)=\gamma_{0}+\gamma_{2}(\omega/k)^{2}+\gamma_{4}(\omega/k)^{4}$
(5.11)

and comparing $\gamma_{i}$ from the numerical evalulation of the integrals
(4.5) (with (4.9) instead of (4.6) for $2<\nu<5/2$) as functions of
$\nu\in\left[1,5/2\right]$ with the known analytic form (5.1) we get good
agreement, see figs. 1.

Figure 1:  Comparison of $\gamma_{i}$ defined in (5.11) between the exact
result (5.1) (continuous curves) and our numerical evaluation of. eq. (4.5)
(discrete points) as functions of $\nu\in\left[d/2,d+1/2\right]$ for the order
$T^{2d}$ correction to the Euclidean propagator in the $d=2$ case.

Since the expansion eq. (4.5) in [1] for the $D1/D5$-branes system agrees with
the full result [15] by identifying $\Delta=1+\nu$ and $q=0$, we obviously
agree with [1] too.

###  5.2 $d=4$

Here we first notice that our $T^{4}$ analytic result (3.11) completely agrees
with the first nontrivial term in [1] by identifying $\Delta=2+\nu=l+4$ and
$\vec{q}=\vec{0}$. Next we compare our $T^{8}$ numerical result (4.5) for
$g(1)$ with eq. (2.3) of [1] with the same identification as above, finding a
perfect numerical agreement as shown in Fig. 2.

Figure 2:  The function $g_{2}$ at $k=\omega$ for the $d=4$ Euclidean
propagator at order $T^{8}$ in our numerical evaluation of eq. (4.5) (discrete
points) as functions of $\nu$ compared to the eq. (2.3) of [1] (full line).

Reinvigorated by this matching at $d=4$ with the case $\omega/k=1$ and
$\nu=l+2$ of [1], and the matching in the previous section between the exactly
known $d=2$ result and our numerical evaluation of the second order
contribution to the Euclidean propagator as a function of $\nu$, which is
summarised in fig. 1, we go now to the arbitrary $\omega/k$ and $d=4$ case.
For the same $\gamma_{i}$, $i=0,2,4$ functions of $\nu$ we find the $T^{8}$
contributions in fig. 3.

Here the full $\nu$ dependent result is not known, but for large $\nu$ the
approximate values can be found in [5, 6] and is here taken from (B.26)444
Here there is an ambiguity of how to write the leading term, i.e.
$\propto\nu^{10}$ or $\propto\Delta^{10}$: they are equivalent at large $\nu$
but make a difference for small $\nu$. We choose a conservative approach and
wrote this contribution as a function of $\nu$, which is closer to our
numerical result.:

$g_{2}^{{\rm
large}\;\nu}(\omega/k)=\frac{2}{225}\nu^{10}\left(4\frac{\omega^{2}}{k^{2}}-1\right)^{2}\quad.$
(5.12)

It is interesting to know how far is the approximate result (5.12) from our
numerical evaluation. This is compared in fig. 4, which shows that indeed the
large $\nu$ approximation is not suitable for small $\nu$, which makes our
computation all the more important. On the other side, it should be stressed
here that our method is not suitable for large $\nu$ calculations: in fact we
need to explicitly subtract the divergent pieces and, as we have seen, their
number increases linearly with $\nu$. So the two methods, the geodesic
approximation used in [6, 7] for large $\nu$ and our method here, should be
considered as complementary rather than competitive.

Said that, we tried to analytically take the limit $\nu\to\infty$. Although we
were unable to prove it definitely, we have some indications to conjecture
that the large $\nu$ result for an arbitrary dimension $d$ is

$\displaystyle g_{2}^{{\rm large}\;\nu}(\omega/k)$ $\displaystyle=$
$\displaystyle\lim_{\nu\to\infty}\frac{1}{2}\left(A_{1}\alpha_{d}+B_{1}\alpha_{d+2}\right)^{2}$
(5.13) $\displaystyle=$
$\displaystyle\frac{2^{2d-5}\Gamma^{4}(d/2)}{(d+1)^{2}\Gamma^{2}(d)}\nu^{2d+2}\left(d\left(\frac{\omega}{k}\right)^{2}-1\right)^{2}\quad.$

This result coincides with (5.12) for $d=4$ and with the large $\nu$ limit of
(5.1) for $d=2$. The reason why the extreme large limit is maybe doable is
that one does not need to do it numerically, and analytical approximation of
Bessel functions at large $\nu$ can be used [17].

Notice that (5.13) confirms the exponentiation property of the propagator at
large $\Delta$ [6].

Figure 3:  The functions $\gamma_{i}$ defined in (5.11) for the $d=4$
Euclidean propagator at order $T^{8}$ in our numerical evaluation of eq. (4.5)
(discrete points) as functions of $\nu\in\left[d/2,d+1/2\right]$.

Figure 4:  Comparison of $\gamma_{i}$ defined in (5.11) for the $d=4$
Euclidean propagator at order $T^{8}$ between the approximate result valid for
large $\nu$ of [6] and our numerical evaluation of eq. (4.5) (discrete points)
as functions of $\nu\in\left[d/2,d+1/2\right]$.

##  6 Conclusions

We showed in this paper the way to calculate the euclidean thermal propagator
in a CFT using the AdS/CFT correspondence. Although the full formula for it
has been recently found [11], it is not suited for a small temperature
expansion. We found the first two corrections, the first analytical of order
$T^{d}$ in (3.10), and the second numerical of order $T^{2d}$ via (4.5) and
(4). The order $T^{d}$ agrees with [8] for general $d$ and with [5, 6] for
$d=4$ (summarised in (3.11)). The order $T^{2d}$ reproduces the well known
result [15] for $d=2$. This last check is summarised in fig. 1. We then
predicted the $T^{8}$ contributions in the $d=4$ case in fig. 3. Finally we
showed in fig. 4 explicitly that the large $\nu$ approximation does not
properly describe the small $\nu$ limit, a result which is not surprising but
had to be checked. The method presented here is thus appropriate for this
range of $\nu$, not described by the geodesic approximation.

Our approach can deal with general spacetime dimensions $d$. It is not tied to
the Heun equation as is the case of the analytical solution [11]. In fact in
$d>4$ apparently more singular points are present and so it is not clear if
the equation to be solved is still of the Heun type. In this paper we found
for arbitrary $d$ an explicit expression (3.10) for the propagator at order
$T^{d}$ for arbitrary $\nu$. At the order $T^{2d}$ we conjecture an analytic
formula (5.13), valid for large $\nu$.

The perturbative expansion in $T/k$, which we do in $d$-dimensional Euclidean
spacetime, seems to be compatible with the analytical continuation to
Minkowski spacetime. This is checked for example if we compare our Euclidean
result with the Minkowski all-order result in $d=2$.

Few words of caution are perhaps important at this point. As studied in [1],
the Heun equation has one potential problem and one real problem. The
potential one is connected to the uniform convergence of the asymptotic
expansion in the whole coordinate interval, also close to the singularities.
The authors of [1] use successfully the method of Langer and Olver [18, 19].
In our case this would roughly mean that the coefficients (2.47)-(2.48) need
to be finite, which we easily prove they are. More precisely, in the limit
$k=\omega$ we start from the same equation of [1]. The difference with them is
that we automatically find an expansion in positive integer powers of $T^{d}$,
while in [1] this turns out to be true only after some cancellations. For our
more general case of $k\neq\omega$ we do not know of any method equivalent to
Langer and Olver. Finding such a method is clearly beyond the scope of this
paper. We however check our results by comparing them to the known ones in the
literature and find perfect agreement whenever available. The second, real
problem, is the appearance in the solution of terms proportional to positive
powers of $\exp{(-1/T)}$. These terms, as in [1], are not available in our
method. We would like here to stress that these terms are certainly one of the
reasons why the full solution [11] mentioned above is not suited for a low $T$
expansion.

#### Acknowledgments

We thank Sašo Grozdanov for many illuminating discussions on subjects relating
this work, Vasil Avramov for pointing out and clarifying some aspects of
reference [9], Jorge Russo for very important remarks and Enrico Parisini,
Kostas Skenderis, Benjamin Withers and Kuo-Wei Huang for useful
correspondence. BB acknowledges the financial support from the Slovenian
Research Agency (research core funding No. P1-0035). AL acknowledges the
financial support from CONICET through PIP 02229 and PUE084, and from La Plata
University-11/X910. BB (AL) thanks the Physics Department of La Plata
University (Jožef Stefan Institute) for hospitality.

##  Appendix A Some useful identities

We first present some useful formulae regarding Bessel functions for indices
$\nu\notin\mathbb{Z}$. By using the expansions given in [20], we have for
$q\ll 1$,

$\displaystyle I_{\nu}(q)$ $\displaystyle=$
$\displaystyle=\frac{1}{\Gamma(1+\nu)}\,\left(\frac{q}{2}\right)^{\nu}\;_{0}{}F_{1}\left(1+\nu;\frac{q^{2}}{4}\right)\quad,$
(A.1) $\displaystyle K_{\nu}(q)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\nu)}{2}\left(\frac{q}{2}\right)^{-\nu}\;_{0}{}F_{1}\left(1-\nu;\frac{q^{2}}{4}\right)+\frac{\Gamma(-\nu)}{2}\;\left(\frac{q}{2}\right)^{\nu}\;_{0}{}F_{1}\left(1+\nu;\frac{q^{2}}{4}\right)\quad,$
(A.4)

and for $q\gg 1$ the asymptotic expansions is

$\displaystyle I_{\nu}(q)$ $\displaystyle\sim$
$\displaystyle\frac{e^{q}}{\sqrt{2\,\pi\,q}}\;_{2}{}F_{0}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\frac{1}{2\,q}\right)+i\,e^{i\,\pi\,\nu}\frac{e^{-q}}{\sqrt{2\,\pi\,q}}\;_{2}{}F_{0}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;-\frac{1}{2\,q}\right)\quad,$
(A.6) $\displaystyle K_{\nu}(q)$ $\displaystyle\sim$
$\displaystyle\sqrt{\frac{\pi}{2\,q}}\;e^{-q}\;_{2}{}F_{0}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;-\frac{1}{2\,q}\right)\quad.$
(A.9)

The elements of the matrix ${\bf b}(q)$ in (2.30) are defined by,

$\displaystyle 2\,\nu\,b_{11}(q)$ $\displaystyle\equiv$ $\displaystyle
2\,\nu\,I_{\nu}(q)\,K_{\nu}(q)$ (A.10) $\displaystyle=$
$\displaystyle{}_{1}F_{2}\left({\frac{1}{2}\atop
1+\nu,1-\nu};q^{2}\right)+\frac{\Gamma(-\nu)}{\Gamma(\nu)}\,\left(\frac{q}{2}\right)^{2\,\nu}\,{}_{1}F_{2}\left({\frac{1}{2}+\nu\atop
1+2\,\nu,1+\nu};q^{2}\right)\quad,$ (A.11) $\displaystyle b_{22}(q)$
$\displaystyle\equiv$ $\displaystyle-b_{11}(q)\quad,$ (A.13)
$\displaystyle\frac{4}{\alpha}\,b_{12}(q)$ $\displaystyle\equiv$
$\displaystyle
4\,K_{\nu}(q)^{2}=\Gamma(-\nu)^{2}\,\left(\frac{q}{2}\right)^{2\,\nu}\,{}_{1}F_{2}\left({\frac{1}{2}+\nu\atop
1+2\,\nu,1+\nu};q^{2}\right)$ (A.15) $\displaystyle+$
$\displaystyle\Gamma(\nu)\,\Gamma(-\nu)\,{}_{1}F_{2}\left({\frac{1}{2}\atop
1+\nu,1-\nu};q^{2}\right)+(\nu\rightarrow-\nu)\quad,$ (A.17)
$\displaystyle-\alpha\,b_{21}(q)$ $\displaystyle\equiv$ $\displaystyle
I_{\nu}(q)^{2}=\frac{1}{\Gamma(1+\nu)^{2}}\,\left(\frac{q}{2}\right)^{2\,\nu}\,{}_{1}F_{2}\left({\frac{1}{2}+\nu\atop
1+2\,\nu,1+\nu};q^{2}\right)\quad,$ (A.18)

where we have used (A.1), the notable product identity

${}_{0}F_{1}\left(a;z\right)\;{}_{0}F_{1}\left(b;z\right)={}_{2}F_{3}\left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop
a,b,a+b-1};4\,z\right)\quad,$ (A.19)

and the fact that

${}_{p+1}F_{q+1}\left({a_{1},\dots a_{p},c\atop b_{1},\dots
b_{q},c};z\right)={}_{p}F_{q}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};z\right)\quad.$ (A.20)

Using

$\int^{q}dz\;z^{s-1}\;{}_{p}F_{q}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};z^{2}\right)=\frac{q^{s}}{s}\;{}_{p+1}F_{q+1}\left({a_{1},\dots,a_{p},\frac{s}{2}\atop
b_{1},\dots,b_{q},1+\frac{s}{2}};q^{2}\right)\quad,$ (A.21)

the integrals (3.2) are (for positive $s$ )

$\displaystyle I_{s}[b_{11};q,0]$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\nu}\,\frac{q^{s}}{s}\left(f_{s}(q)+g^{(+)}_{s}(q)\right)\quad,$
(A.22) $\displaystyle I_{s}[b_{22};q,0]$ $\displaystyle=$
$\displaystyle-\frac{1}{2\,\nu}\,\frac{q^{s}}{s}\left(f_{s}(q)+g^{(+)}_{s}(q)\right)\quad,$
(A.23) $\displaystyle I_{s}[b_{12};q,\infty]$ $\displaystyle=$
$\displaystyle\,\frac{G_{2}^{(0)}(k)}{2\,\nu}\left(\frac{q^{s}}{s}\left(2\,f_{s}(q)+g^{(+)}_{s}(q)+g^{(-)}_{s}(q)\right)+2\nu\alpha_{s}\right)\quad,$
(A.24) $\displaystyle I_{s}[b_{21};q,0]$ $\displaystyle=$
$\displaystyle-\frac{1}{2\,\nu
G_{2}^{(0)}(k)}\frac{q^{s}}{s}g^{(+)}_{s}(q)\quad,$ (A.25)

where we defined the functions $f_{s}(q)$ and $g^{(\pm)}_{s}(q)$ as

$\displaystyle f_{s}(q)$
$\displaystyle\equiv{}_{2}F_{3}\left({\frac{s}{2},\frac{1}{2}\atop
1+\nu,1+\frac{s}{2},1-\nu};q^{2}\right)\quad,$ (A.26) $\displaystyle
g^{(\pm)}_{s}(q)$ $\displaystyle\equiv\gamma_{s}^{(\pm)}\;q^{\pm
2\,\nu}\;{}_{2}{}F_{3}\left({\frac{s}{2}\pm\nu,\frac{1}{2}\pm\nu\atop 1\pm
2\,\nu,1\pm\nu+\frac{s}{2},1\pm\nu};q^{2}\right)\quad,\quad\gamma_{s}^{(\pm)}\equiv\frac{2^{\mp
2\,\nu}\,s}{s\pm 2\,\nu}\;\frac{\Gamma(\mp\nu)}{\Gamma(\pm\nu)}\quad,$

and $\alpha_{s}$ in (2.31). Eq. (A.22) can be used to compute the elements of
the matrix (3.1)

$\int_{q_{0}}^{q}dz\,z\,u^{(m)}(z)\;{\bf b}(z)=A_{m}\,I_{d\,m}[{\bf
b};q,q_{0}]+B_{m}\,I_{d\,m+2}[{\bf b};q,q_{0}]\quad,\quad m=1,2,\dots\quad.$
(A.27)

##  Appendix B From $x$ to $k$ space

In reference [7], Rodríguez-Gómez and Russo computed the low temperature
expansion of the propagator up to second order in $T^{4}$, in four dimensions
and large operator dimension limit. Their result, given in coordinate space
$(\tau,\vec{x})$, can be written as

$G_{2}(x,\eta)=G_{2}^{(0)}(x)+G_{2}^{(1)}(x,\eta)+G_{2}^{(2)}(x,\eta)+\dots\quad,$
(B.1)

where $\;x\equiv\sqrt{\tau^{2}+\vec{x}^{2}}$, $\;\eta\equiv\frac{\tau}{x}$,
and

$\displaystyle G_{2}^{(0)}(x)$ $\displaystyle=$ $\displaystyle
x^{-2\,\Delta}\quad,$ (B.2) $\displaystyle G_{2}^{(1)}(x,\eta)$
$\displaystyle=$
$\displaystyle\frac{2\,\Delta}{15}\,\left(\frac{\pi\,T}{2}\right)^{4}\,C_{2}^{(1)}(\eta)\,x^{4-2\,\Delta}\quad,$
(B.3) $\displaystyle G_{2}^{(2)}(x,\eta)$ $\displaystyle=$
$\displaystyle\frac{2\,\Delta^{2}}{15^{2}}\,\left(\frac{\pi\,T}{2}\right)^{8}\,\left(C_{0}^{(1)}(\eta)+C_{2}^{(1)}(\eta)+C_{4}^{(1)}(\eta)\right)\,x^{8-2\,\Delta}\quad.$
(B.4)

Here $C_{m}^{(a)}(\eta)$ are Gegenbauer polynomials, the relevant ones are

$C_{0}^{(1)}(\eta)=1\quad,\quad C_{2}^{(1)}(\eta)=-1+4\,\eta^{2}\quad,\quad
C_{4}^{(1)}(\eta)=1-12\,\eta^{2}+16\,\eta^{4}\quad.$ (B.5)

To compare (B.1) with our results, we need (B.2) in momentum space
$(\omega,\vec{q})$. To this end, we first introduce the Fourier transform

$f_{2a}(k)\equiv\int d^{d}x\,e^{-i\,k\cdot
x}\,x^{-2\,a}=c_{2a}\,k^{2\,a-d}\quad,$ (B.6)

where $k\equiv\sqrt{\omega^{2}+\vec{q}^{2}}$. By using the representation

$x^{-2\,a}=\frac{1}{\Gamma(a)}\,\int_{0}^{\infty}dt\,t^{a-1}\,e^{-x^{2}\,t}\qquad,\qquad\Re(a),\Re(x^{2})>0\quad,$
(B.7)

we get

$\displaystyle c_{2a}$ $\displaystyle=$ $\displaystyle\int
d^{d}x\,e^{-i\,x_{\mu}{k^{\mu}}/k}\,x^{-2\,a}=\frac{1}{\Gamma(a)}\,\int_{0}^{\infty}dt\,t^{a-1}\,\int
d^{d}x\,e^{-x^{2}\,t-i\,x_{\mu}k^{\mu}/k}$ (B.8) $\displaystyle=$
$\displaystyle\frac{\pi^{\frac{d}{2}}}{\Gamma(a)}\,\int_{0}^{\infty}dt\,t^{a-\frac{d}{2}-1}\,e^{-\frac{1}{4\,t}}=(4\pi)^{\frac{d}{2}}\,\frac{\Gamma(\frac{d}{2}-a)}{2^{2\,a}\,\Gamma(a)}\quad.$
(B.9)

A useful relation that follows is (for $d=4$)

$\displaystyle\frac{2^{-2\,m}\,c_{2a}}{c_{2a+2m}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(2-a)}{\Gamma(2-a-m)}\,\frac{\Gamma(a+m)}{\Gamma(a)}=(-)^{m}\,(a-1)_{m}\,(a)_{m}$
(B.10) $\displaystyle=$
$\displaystyle(-)^{m}\,(a-1)\,(a-1+m)\,\prod_{l=1}^{m-1}(a-1+l)^{2}\quad.$
(B.11)

So, from (B.2) we get:

Zero order

$G_{2}^{(0)}(k)=f_{2\,\Delta}(k)=c_{2\Delta}\,k^{2\,\Delta-d}=\pi^{\frac{d}{2}}\,\frac{\Gamma(-\nu)}{\Gamma(\Delta)}\,\left(\frac{k}{2}\right)^{2\,\nu}\quad.$
(B.12)

Note the different normalization of (B.2) with respect to the one used in the
paper, equation (2.25).

First order

$\displaystyle G_{2}^{(1)}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{k^{4}}\,\frac{2\,\Delta\,k^{4}}{15\,2^{4}}\,\left(-f_{2\,\nu}(k)-4\,\partial_{\omega}^{2}f_{2\,\nu+2}(k)\right)$
(B.13) $\displaystyle=$
$\displaystyle\frac{1}{k^{4}}\,\frac{2\,\Delta\,k^{2\nu}}{15\,2^{4}}\,\left(-c_{2\,\nu}-8\,(\nu-1)\,c_{2\,\nu+2}\,\left(1+2\,(\nu-2)\,\left(\frac{\omega}{k}\right)^{2}\right)\right)$
(B.14) $\displaystyle=$
$\displaystyle\frac{G_{2}^{(0)}(k)}{k^{4}}\,\frac{2\,\Delta}{15\,2^{4}}\,\frac{c_{2\,\nu}}{c_{2\,\nu+4}}\,\left(-1-8\,(\nu-1)\,\frac{c_{2\,\nu+2}}{c_{2\,\nu}}\,\left(1+2\,(\nu-2)\,\left(\frac{\omega}{k}\right)^{2}\right)\right)$
(B.15) $\displaystyle=$
$\displaystyle\frac{G_{2}^{(0)}(k)}{k^{4}}\,\frac{2}{15}\,\nu\,(\nu^{2}-1)\,(\nu^{2}-4)\,C_{2}^{(1)}\left(\frac{\omega}{k}\right)\quad,$
(B.16)

where we have used (B.10) and the relation

$\int
d^{4}x\,e^{-i\,x_{\mu}k^{\mu}/x}\,x^{-2\,a}\,\eta^{m}=i^{m}\,c_{2a+m}\,k^{4-2a}\,\partial_{\omega}^{m}k^{2a-4+m}\quad.$
(B.17)

Second order

$\displaystyle G_{2}^{(2)}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{(kz_{h})^{8}}\,\frac{2\,\Delta^{2}\,k^{8}}{15^{2}\,2^{8}}\,\int
d^{4}x\,e^{-i\,k\cdot
x}\,x^{-2\,\Delta+8}\,\left(C_{0}^{(1)}(\eta)+C_{2}^{(1)}(\eta)+C_{4}^{(1)}(\eta)\right)$
(B.18) $\displaystyle=$
$\displaystyle\frac{G_{2}^{(0)}(k)}{(kz_{h})^{8}}\,\frac{2\,\Delta^{2}}{15^{2}\,2^{8}}\,\frac{c_{2\nu-4}}{c_{2\nu+4}}\sum_{l=0}^{2}I_{2\nu}^{(2l)}(\omega/k)\quad,$
(B.19)

where

$I_{2\nu}^{(2l)}(\omega/k)\equiv\frac{k^{8-2\nu}}{c_{2\nu-4}}\,\int
d^{4}x\,e^{-i\,k\cdot x}\,x^{-2\,\nu+4}\,C_{2\,l}^{(1)}(\eta)\quad.$ (B.20)

We get

$\displaystyle I_{2\nu}^{(0)}(x)$ $\displaystyle=$ $\displaystyle 1\quad,$
(B.21) $\displaystyle I_{2\,\nu}^{(2)}(x)$ $\displaystyle=$
$\displaystyle\frac{\nu-4}{\nu-2}\,C_{2}^{(1)}(x)\quad,$ (B.22) $\displaystyle
I_{2\,\nu}^{(4)}(x)$ $\displaystyle=$
$\displaystyle\frac{(\nu-4)\,(\nu-5)}{(\nu-1)\,(\nu-2)}\,C_{4}^{(1)}(x)\quad.$
(B.23)

From (B.18) we obtain

$\displaystyle G_{2}^{(2)}(k)$ $\displaystyle=$
$\displaystyle\frac{G_{2}^{(0)}(k)}{(kz_{h})^{8}}\,\frac{2}{15^{2}}\,(\nu+2)^{2}\,(\nu-3)\,(\nu+1)\,\nu^{2}\,(\nu-1)^{2}\,(\nu-2)^{2}\,\left(1+\frac{\nu-4}{\nu-2}\,C_{2}^{(1)}(\omega/k)\right.$
(B.24) $\displaystyle+$
$\displaystyle\left.\frac{(\nu-4)\,(\nu-5)}{(\nu-1)\,(\nu-2)}\,C_{4}^{(1)}(\omega/k)\right)\quad.$
(B.25)

For large $\nu$ we have

$\displaystyle\left.\frac{(kz_{h})^{8}}{G_{2}^{(0)}(k)}\,G_{2}^{(2)}(k)\right|_{\nu\rightarrow\infty}$
$\displaystyle=$
$\displaystyle\frac{2}{15^{2}}\,\nu^{10}\,\left(1+C_{2}^{(1)}(\omega/k)+C_{4}^{(1)}(\omega/k)\right)$
(B.26) $\displaystyle=$
$\displaystyle\frac{2}{15^{2}}\,\nu^{10}\,\left(4\,\left(\frac{\omega}{k}\right)^{2}-1\right)^{2}\quad.$

## References

  * [1] G. Policastro and A. Starinets, “On the Absorption by Near Extremal Black Branes,” Nucl. Phys. B 610 (2001), 117-143 [arXiv:hep-th/0104065 [hep-th]].
  * [2] J. M. Maldacena, “The Large $N$ Limit of Superconformal Field Theories and Supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231-252 [arXiv:hep-th/9711200 [hep-th]].
  * [3] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge Theory Correlators from Noncritical String Theory,” Phys. Lett. B 428 (1998), 105-114 [arXiv:hep-th/9802109 [hep-th]].
  * [4] E. Witten, “Anti-de Sitter Space and Holography,” Adv. Theor. Math. Phys. 2 (1998), 253-291 [arXiv:hep-th/9802150 [hep-th]].
  * [5] A. L. Fitzpatrick and K. W. Huang, “Universal Lowest-Twist in CFTs from Holography,” JHEP 08 (2019), 138 [arXiv:1903.05306 [hep-th]].
  * [6] D. Rodriguez-Gómez and J. G. Russo, “Correlation Functions in Finite Temperature CFT and Black Hole Singularities,” JHEP 06 (2021), 048 [arXiv:2102.11891 [hep-th]].
  * [7] D. Rodriguez-Gómez and J. G. Russo, “Thermal Correlation Functions in CFT and Factorization,” JHEP 11 (2021), 049 [arXiv:2105.13909 [hep-th]].
  * [8] E. Parisini, K. Skenderis and B. Withers, “An Embedding Formalism for CFTs in General States on Curved Backgrounds,” [arXiv:2209.09250 [hep-th]].
  * [9] B. Carneiro da Cunha and F. Novaes, “Kerr-De Sitter Greybody Factors via Isomonodromy,” Phys. Rev. D 93 (2016) no.2, 024045 [arXiv:1508.04046 [hep-th]].
  * [10] G. Bonelli, C. Iossa, D. P. Lichtig and A. Tanzini, “Irregular Liouville Correlators and Connection Formulae for Heun Functions,” [arXiv:2201.04491 [hep-th]].
  * [11] M. Dodelson, A. Grassi, C. Iossa, D. Panea Lichtig and A. Zhiboedov, “Holographic Thermal Correlators from Supersymmetric Instantons,” [arXiv:2206.07720 [hep-th]].
  * [12] A. Bhatta and T. Mandal, “Exact Thermal Correlators of Holographic $CFT$s,” [arXiv:2211.02449 [hep-th]].
  * [13] B. Bajc, A. R. Lugo and M. B. Sturla, “Spontaneous breaking of a discrete symmetry and holography,” JHEP 04 (2012), 119 [arXiv:1203.2636 [hep-th]].
  * [14] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large $N$ Field Theories, String Theory and Gravity,” Phys. Rept. 323 (2000), 183-386 [arXiv:hep-th/9905111 [hep-th]].
  * [15] S. Grozdanov, P. K. Kovtun, A. O. Starinets and P. Tadić, “The Complex Life of Hydrodynamic Modes,” JHEP 11 (2019), 097 [arXiv:1904.12862 [hep-th]].
  * [16] F. Tricomi and A. Erdélyi, “The asymptotic expansion of a ratio of gamma functions,” Pacific Journal of Mathematics Vol. 1 (1951), No. 1, 133-142.
  * [17] NIST Handbook of Mathematical Functions, Edited by Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark, National Institute of Standards and Technology, Maryland and University of Maryland, Chapter 10, https://dlmf.nist.gov/10.41.
  * [18] R. Langer, “On the asymptotic solutions of ordinary differential equations, with reference to the Stokes’ phenomenon about a singular point”, Trans. Amer. Math. Soc. 37 no. 3, 397–416 (1935).
  * [19] F. W. J. Olver, “The asymptotic solution of linear differential equations of the second order in a domain containing one transition point”, Philos. Trans. Roy. Soc. London. Ser. A. 249, 65–97 (1956), “Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter”, Philos. Trans. Roy. Soc. London. Ser. A 250 479–517 (1958).
  * [20] I.S. Gradshteyn and I.M. Ryzhik., “Table of integrals, series, and products,” Elsevier/Academic Press, Amsterdam (2007).