aainstitutetext: Departamento de Matemática, Escola de Tecnologias e
Arquitetura,
ISCTE - Instituto Universitário de Lisboa
and Instituto de Telecomunicações,
Av. das Forças Armadas, 1649-026 Lisboa, Portugalbbinstitutetext:  Centro de
Física da Universidade de Coimbra,
Rua Larga, 3004-516 Coimbra, Portugal

# Asymptotic quasinormal modes of string-theoretical $d$-dimensional black
holes

Filipe Moura b João Rodrigues<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the quasinormal frequencies of $d$-dimensional spherically
symmetric black holes with leading string $\alpha^{\prime}$ corrections for
tensorial gravitational perturbations in the highly damped regime. We solve
perturbatively the master differential equation and we compute the monodromies
of the master perturbation variable (analytically continued to the complex
plane) in different contours, in order to obtain the quasinormal mode spectra.
We proceed analogously for the quasinormal modes of test scalar fields.
Differently than in Einstein gravity, we obtain distinct results for the two
cases.

###### Keywords:

Black holes, Black holes in string theory

††arxiv: 2105.02616

##  1 Introduction

Black hole quasinormal modes (QNMs) are associated to perturbations that can
be related either to the black hole metric (gravitational perturbations) or to
external fields. They represent resonances in black hole scattering problems
that are purely ingoing at the event horizon and purely outgoing at infinity.
These boundary conditions imply that the associated frequencies must be
complex. The quasinormal ringing frequencies carry unique information about
parameters of the black hole in the ringdown phase resulting from a black hole
collision; they can be directly measured by the gravitational wave detectors.
This feature turns QNMs into preferential probes for testing theories of
gravity beyond Einstein, since the ringing frequencies represent a universal
part of the gravitational wave signals. These frequencies do not depend on
what drives the perturbations: they are given exclusively in terms of
intrinsic physical quantities of the black hole such as mass, charge or spin,
and eventually (beyond Einstein gravity) some other parameters of the theory.
With the advent of gravitational wave astronomy, therefore, interest in the
study of black hole QNMs has raised. Literature concerning QNMs is vast and
rich; many excellent references can be provided, such as the seminal works
Konoplya:2011qq ; Konoplya:2003ii ; Konoplya:2002zu ; Berti:2009kk . More
recent works have extended the calculation of QNMs in several different black
hole solutions Rincon:2018sgd ; Panotopoulos:2017hns ; Destounis:2018utr ;
Destounis:2020pjk ; Panotopoulos:2018hua .

Most of the times, QNMs have to be computed numerically. Nonetheless,
different analytical methods have been developed in order to compute QNMs in
some limiting cases. One of such cases is the asymptotic (highly damped)
limit. Quasinormal frequencies in that limit have been computed in Motl:2003cd
; Birmingham:2003rf ; Andersson:2003fh  for $d$ dimensional asymptotically
flat spherically symmetric black holes. These results have been extended to
rotating black holes in $d=4$ Musiri:2003ed  and to nonasymptotically flat
black holes: in $d=4$ in Cardoso:2004up  and, for $d>4$, in Musiri:2003rs ;
Natario:2004jd . Numerical studies have confirmed the results of these
calculations, and have also been extended to other solutions Nollert:1993zz ;
Cardoso:2003cj ; Berti:2003zu ; Berti:2003jh .

From what we mentioned, it is relevant to extend these results to theories
beyond Einstein gravity, namely theories with higher derivatives. There are
several recent works computing QNMs of black holes with higher derivative
corrections coming from different theories, namely Blazquez-Salcedo:2016enn ;
Cano:2020cao ; Pierini:2021jxd ; Moura:2021eln . Concerning specifically the
asymptotic limit, numerical results have been obtained in Daghigh:2006xg  for
vectorial perturbations of $d$-dimensional spherically symmetric black holes
in Gauss-Bonnet gravity.

In this article we will consider $d$-dimensional spherically symmetric black
holes with leading string-theoretical $\alpha^{\prime}$ corrections, and
analytically compute their quasinormal frequencies in the asymptotic (highly
damped) limit corresponding to tensorial gravitational perturbations and test
scalar fields. The article is organized as follows. In section 2 we will
review the tensorial gravitational perturbations of spherically symmetric
black holes in $d$ dimensions. We then concentrate on the effective action of
bosonic/heterotic superstrings with leading $\alpha^{\prime}$ corrections and
on a $d$-dimensional black hole solution of its field equations. We write down
the master equation and the respective potential corresponding to these
perturbations. Given this information, in section 3 we compute the quasinormal
spectrum in the highly damped limit corresponding to these perturbations. We
solve the master differential equation perturbatively, considering the radial
variable $r$ (or more precisely the tortoise coordinate) to be complex. We
consider two different (but homotopic) contours in the complex $r$ plane,
associating to each of them one of the quasinormal modes’ boundary conditions.
By computing the monodromies of the master perturbation variable along each of
these contours and equating such monodromies, we are able to obtain a
condition that we can solve for the black hole quasinormal frequencies. In
section 4 we proceed analogously with test scalar fields in the background of
the same black hole, and we compute the respective quasinormal spectrum in the
highly damped limit. In the end we discuss and compare our results in both
cases.

##  2 String-corrected spherically symmetric black holes and their tensorial
perturbations

A general static spherically symmetric metric in $d$ dimensions can always be
cast in the form

$ds^{2}=-f(r)\ dt^{2}+f^{-1}(r)\ dr^{2}+r^{2}d\Omega^{2}_{d-2}.$ (1)

General tensors of rank at least 2 on the $(d-2)$-sphere $\mathbb{S}^{d-2}$
can be uniquely decomposed in their tensorial, vectorial and scalar
components. That is the case of general perturbations $h_{\mu\nu}=\delta
g_{\mu\nu}$ of a $d-$dimensional spherically symmetric metric like (1). We
have then scalar, vectorial and (for $d>4$) tensorial gravitational
perturbations.

Each type of perturbation is described in terms of a master variable. In
Einstein gravity, each of these master variables obeys a second order
differential equation (“master equation”) with a potential that depends on the
kind of perturbation one considers ik03a .

Tensorial perturbations are expressed in terms of the eigentensors
$\mathcal{T}_{ij}$ of $D^{2}$, with $D$ being the covariant derivative on the
$(d-2)$-sphere $\mathbb{S}^{d-2}$:

$\left(D^{k}D_{k}+\ell\left(\ell+d-3\right)-2\right)\mathcal{T}_{ij}\left(\theta^{k}\right)=0,\,\,\ell=2,3,4,\ldots$

Here $i,j,k=1,\ldots,d-2$ represent coordinates of $\mathbb{S}^{d-2}$ and
$\ell$ the multipole number. $\mathcal{T}_{ij}$ also satisfy

$D^{i}\mathcal{T}_{ij}=0,\,g^{ij}\mathcal{}T_{ij}=0.$

Specifically, tensorial gravitational perturbations of the metric (1) are
given in terms of a function $H_{T}(r,t)$ by

$\displaystyle h_{ir},\,h_{it},\,h_{rr},\,h_{rt},\,h_{tt}=0,$ $\displaystyle
h_{ij}=2r^{2}H_{T}(r,t)\mathcal{T}_{ij}.$ (2)

Assuming an oscillatory time dependence, from the perturbation function
$H_{T}(r,t)$ we can define a master variable $\psi(r)$ through

$H_{T}(r,t)=\frac{\mbox{e}^{i\omega t}}{\kappa(r)}\psi(r),$ (3)

with $\kappa(r)=r^{\frac{d-2}{2}}.$ In terms of the tortoise coordinate $x$
for the metric (1) defined by

$dx=\frac{dr}{f(r)},$ (4)

the master variable $\psi(r)$ satisfies a second order differential equation
with a potential, the “master equation”, given by

$\frac{d^{2}\psi}{d\,x^{2}}+\omega^{2}\psi=V\left[f(r)\right]\psi$ (5)

with the potential $V\left[f(r)\right]$ in this case being given by the
minimal potential

$V_{\textsf{min}}[f(r)]=f(r)\left(\frac{\ell\left(\ell+d-3\right)}{r^{2}}+\frac{\left(d-2\right)\left(d-4\right)f(r)}{4r^{2}}+\frac{\left(d-2\right)f^{\prime}(r)}{2r}\right).$
(6)

In the presence of higher order corrections in the lagrangian, one can still
have spherically symmetric black holes of the form (1), but the master
equation obeyed by each perturbation variable is expected to change.
Concretely, we will consider the following $d$–dimensional effective action
with string $\alpha^{\prime}$ corrections:

$\frac{1}{16\pi
G}\int\sqrt{-g}\left(\mathcal{R}-\frac{4}{d-2}\left(\partial^{\mu}\phi\right)\partial_{\mu}\phi+\mbox{e}^{\frac{4}{d-2}\phi}\frac{\lambda}{2}\
\mathcal{R}^{\mu\nu\rho\sigma}\mathcal{R}_{\mu\nu\rho\sigma}\right)\mbox{d}^{d}x.$
(7)

This is the effective action of bosonic and heterotic string theories, to
first order in the inverse string tension $\alpha^{\prime}$, with
$\lambda=\frac{\alpha^{\prime}}{2},\frac{\alpha^{\prime}}{4}$, respectively.
111Type II superstring theories do not have $\alpha^{\prime}$ corrections to
this order. In both cases, since we are only interested in purely
gravitational corrections, we can consistently set all other bosonic and
fermionic fields present in the string spectrum to zero except for the dilaton
field $\phi$.

Tensorial metric perturbations in the presence of these leading string
$\alpha^{\prime}$ corrections for a spherically symmetric metric of the form
(1) are also defined through (2). The master variable $\psi$ is also defined
through (3), with

$\kappa(r)=\frac{1}{\sqrt{f}}\exp\left(\int\frac{\frac{f^{\prime}}{f}+\frac{d-2}{r}+\frac{4}{r^{3}}(d-4)\lambda(1-f)-\frac{4}{r^{2}}\lambda
f^{\prime}-\frac{2}{rf}\lambda f^{\prime 2}}{2-\frac{4}{r}\lambda
f^{\prime}}dr\right).$

In Moura:2006pz ; Moura:2012fq  it has been shown that, perturbing the field
equations resulting from this action, for tensorial perturbations of the
metric (1) one also obtains a second order master equation like (5). The
corresponding potential, as expected, is an $\alpha^{\prime}$-corrected
version of the minimal potential in (6) given by

$\displaystyle V_{\textsf{T}}[f(r)]$ $\displaystyle=$ $\displaystyle\lambda\
\frac{f(r)}{r^{2}}\left[\left(\frac{2\ell\left(\ell+d-3\right)}{r}+\frac{\left(d-4\right)\left(d-5\right)f(r)}{r}+\left(d-4\right)f^{\prime}(r)\right)\left(2\frac{1-f(r)}{r}+f^{\prime}(r)\right)\right.$
(8) $\displaystyle+$
$\displaystyle\left.\Big{(}4(d-3)-(5d-16)f(r)\Big{)}\frac{f^{\prime}(r)}{r}-4\left(f^{\prime}(r)\right)^{2}+\left(d-4\right)f(r)f^{\prime\prime}(r)\right]+V_{\textsf{min}}[f(r)].$

Spherically symmetric $d-$dimensional black hole solutions with leading
$\alpha^{\prime}$ corrections have been obtained in cmp89 ; Moura:2009it .
Specifically concerning the action (7), a solution of the respective field
equations is of the form (1), with

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
f_{0}(r)\left(1+\frac{\lambda}{R_{H}^{2}}\delta f(r)\right),$ (9)
$\displaystyle f_{0}(r)$ $\displaystyle=$ $\displaystyle
1-\frac{R_{H}^{d-3}}{r^{d-3}},$ (10) $\displaystyle\delta f(r)$
$\displaystyle=$ $\displaystyle-\frac{(d-3)(d-4)}{2}\
\frac{R^{d-3}_{H}}{r^{d-3}}\
\frac{1-\frac{R_{H}^{d-1}}{r^{d-1}}}{1-\frac{R^{d-3}_{H}}{r^{d-3}}}.$ (11)

The only horizon of this metric occurs at the same radius $r=R_{H}$ of the
Tangherlini solution, which is the metric with $f(r)=f_{0}(r)$ obtained in the
Einstein limit $\lambda=0$.

This black hole solution has been obtained by Callan, Myers and Perry in cmp89
, where some of its properties have been studied. For our purposes, it is
enough to quote here the explicit expression for its temperature, given by

$T_{\mathcal{H}}=\frac{f^{\prime}(R_{H})}{4\pi}=\frac{f_{0}^{\prime}(R_{H})}{4\pi}\left(1+\frac{\lambda}{R_{H}^{2}}\delta
f(R_{H})\right)=\frac{d-3}{4\pi
R_{H}}\left(1-\frac{\left(d-1\right)\left(d-4\right)}{2}\
\frac{\lambda}{R_{H}^{2}}\right).$ (12)

Throughout this article we will use for the perturbative expansion the small
dimensionless parameter

$\lambda^{\prime}=\frac{\lambda}{R_{H}^{2}}=\lambda\frac{16\pi^{2}}{(d-3)^{2}}T_{\mathcal{H}}^{2}.$
(13)

The stability of this solution under tensorial gravitational perturbations has
been studied in Moura:2006pz , and the spectra of quasinormal modes
corresponding to such perturbations and also of test scalar fields in the
eikonal limit has been obtained in Moura:2021eln . In this article we will
compute the same spectra of quasinormal modes for both cases in the highly
damped regime.

##  3 Asymptotic quasinormal modes of tensorial gravitational perturbations to
the Callan-Myers-Perry black hole

###  3.1 Quasinormal modes, boundary conditions and the monodromy method

Quasinormal modes of tensorial gravitational perturbations are solutions to
the corresponding master equation (5) subject to the boundary conditions

$\displaystyle\psi\propto e^{-i\omega x}\hskip 3.0pt,\hskip 3.0ptr\to+\infty;$
(14) $\displaystyle\psi\propto e^{i\omega x}\hskip 3.0pt,\hskip 3.0ptr\to
R_{H}.$ (15)

In terms of the tortoise coordinate $x$, these boundary conditions are written
as

$\displaystyle\psi\propto e^{-i\omega x}\hskip 3.0pt,\hskip 3.0ptx\to+\infty;$
(16) $\displaystyle\psi\propto e^{i\omega x}\hskip 3.0pt,\hskip
3.0ptx\to-\infty.$ (17)

It should be clear that any method devised to compute quasinormal frequencies
will have to make use of these defining boundary conditions. Because
quasinormal frequencies are complex and $r$ and $x$ are real, gathering
information of these boundary conditions amounts to distinguish between an
exponentially vanishing and an exponentially growing term. Clearly, a
numerical approach will face problems with such task. Moreover, any kind of
analytical approximate approach will also fail to some extent. Indeed, many
lower order terms of the approximation are needed in order to make sense of
the exponentially decreasing term, otherwise this term might be much smaller
than the approximation error and consequently needs to be disregarded.

An elegant solution to half of this issue emerges from the moment we allow $r$
(and $x$) to take complex values and consequently assume an analytic
continuation of functions of $r$ to the complex plane. Indeed, if one takes
the contour $\mathop{\rm Im}\left(\omega x\right)=0$ in the complex $r$ plane,
then $|e^{\pm i\omega x}|=1,$ the asymptotic behavior of $e^{\pm i\omega x}$
is always oscillatory and there will be no problems with exponentially growing
versus exponentially vanishing terms. Thus if one considers the Stokes lines
$\mathop{\rm Im}\left(\omega x\right)=0$, imposing the boundary condition (16)
in the complex $r$ plane no longer poses a challenge to an approximate
analytical method.

Near the event horizon, we can distinguish the two exponential terms by
computing the respective monodromies around it. As we will see in the next
section, these monodromies are non trivial because $x$ has a branch point in
the event horizon. As it turns out, the boundary condition (17) can be set as
a monodromy condition.

The monodromy method we will be using was introduced in Motl:2003cd . In order
to apply it one needs two contours, and the solutions to the master equation
in the respective regions, to impose the appropriate boundary conditions. The
general idea goes as follows:

  * •

We pick two closed homotopic contours on the complex $r$-plane. Both these
contours enclose only the physical horizon: none of them encloses the origin
of the complex $r$-plane nor any fictitious horizon.

  * •

One of these contours, which we designate as the big contour, seeks to encode
information of the boundary condition (16) on the monodromy of $\psi$
associated with a full loop around it.

  * •

The other contour, which we designate as the small contour, seeks to encode
information of the boundary condition (17) on the monodromy of $\psi$
associated with a full loop around it.

  * •

As both contours are homotopic, the monodromy theorem asserts that the
respective monodromies must be the same. Thus, equating them hopefully yields
a restriction on the values of the quasinormal frequencies $\omega$, from the
complex plane to an infinite but countable subset.

We restrict our analysis in this article to the highly damped regime of
quasinormal modes defined by the condition

$\mathop{\rm Im}\left(\omega\right)\gg\mathop{\rm Re}\left(\omega\right).$
(18)

This condition is equivalent to $\omega$ being approximately imaginary. The
definition of a Stokes line comes thus as

$\mathop{\rm Im}\left(\omega x\right)=0\Rightarrow\mathop{\rm
Re}\left(x\right)=0.$ (19)

###  3.2 Choice of coordinates

####  3.2.1 Behavior close to the origin

The tortoise coordinate (4) for the metric (9) we are working with is given,
up to an integration constant $C_{X}$, in terms of the Gauss hypergeometric
function ${}_{2}F_{1}$ by

$\displaystyle x$ $\displaystyle=$
${}_{2}F_{1}\left(1,-\frac{1}{d-3};1-\frac{1}{d-3};\left(\frac{R_{H}}{r}\right)^{d-3}\right)r-\left[\left(\frac{R_{H}}{r}\right)^{d-3}\frac{{}_{2}F_{1}\left(2,\frac{d-4}{d-3};1+\frac{d-4}{d-3};\left(\frac{R_{H}}{r}\right)^{d-3}\right)}{d-4}\right.$
(20) $\displaystyle-$
$\displaystyle\left.\left(\frac{R_{H}}{r}\right)^{2d-4}\frac{{}_{2}F_{1}\left(2,\frac{2d-5}{d-3};1+\frac{2d-5}{d-3};\left(\frac{R_{H}}{r}\right)^{d-3}\right)}{2d-5}\right]\frac{(d-3)(d-4)}{2}\frac{\lambda}{R_{H}^{2}}r+C_{X}.$

Close to the origin, this coordinate $x$ can be approximated as

$\displaystyle x$ $\displaystyle\sim$
$\displaystyle-\frac{1}{d-2}\frac{r^{d-2}}{R_{H}^{d-3}}+\frac{(d-3)(d-4)}{2}\frac{\lambda}{r}$
(21) $\displaystyle+$ $\displaystyle\mathrm{e}^{\frac{\pi
i}{d-3}}\frac{\pi}{(d-3)\sin\left(\frac{\pi}{d-3}\right)}R_{H}+\frac{d-4}{d-3}\frac{\pi}{\sin\left(\frac{\pi}{d-3}\right)}\frac{\lambda}{2R_{H}}\left(\mathrm{e}^{\frac{\pi
i}{d-3}}-(d-2)\mathrm{e}^{\frac{-\pi i}{d-3}}\right)+C_{X}.$

We see that, associated to the $\alpha^{\prime}$ correction, there is a
singularity in the coordinate $x$ at $r=0$. Because of such singularity, close
to the origin the Stokes lines $\mathop{\rm Re}(x)=0$ are very difficult to
handle. Since the analysis of these lines is crucial for our calculation, we
must find an alternative coordinate in order to avoid that singular behavior
close to the origin.

We have chosen to rather take the tortoise coordinate $z$ corresponding to the
Tangherlini solution: since there are no $\alpha^{\prime}$ corrections
associated to it, we thought that was the most sensible choice, considering
the perturbative analysis we will make. Such coordinate is given simply by

$dz=\frac{dr}{f_{0}(r)}$ (22)

with $f_{0}(r)$ given by (10). After integration one gets simply the
$\lambda=0$ part of (20):

$z=_{2}F_{1}\left(1,-\frac{1}{d-3};1-\frac{1}{d-3};\left(\frac{R_{H}}{r}\right)^{d-3}\right)r+C_{Z}$
(23)

for some integration constant $C_{Z}$.

Both coordinates $x,z$ are defined up to the constants $C_{X},C_{Z}$. We may
choose $C_{Z}$ in (23) in such a way that, close to the origin, $z$ can be
approximated simply as

$z\sim-\frac{1}{d-2}\frac{r^{d-2}}{R_{H}^{d-3}}.$ (24)

With this choice of $C_{Z}$, $z$ in (23) can also be written as Motl:2003cd

$z=r+\frac{1}{2}\sum_{n=0}^{d-4}\frac{\exp\left(\frac{2\pi
in}{d-3}\right)}{d-3}\log\left(1-\frac{r}{R_{H}}\exp\left(-\frac{2\pi
in}{d-3}\right)\right).$ (25)

We see that $z(r)$ is a multivalued function (just like $x(r)$); indeed, from
(22) we see that each zero of $f_{0}(r)$ is a branch point. There are $d-3$
zeros of $f_{0}(r)$:

$f_{0}(r)=0\Leftrightarrow R_{n}=R_{H}\exp\left(\frac{2\pi
in}{d-3}\right),\,n=0,1,\ldots,d-4.$ (26)

Only the solution $R_{0}=R_{H}$ corresponds to a physical horizon; the other
$d-4$ solutions are “fictitious” horizons.

####  3.2.2 Behavior at infinity

At spatial infinity one can easily notice that $f(r)\sim f_{0}(r)\sim 1$, and
therefore from (20), (23) we get, in this limit, $x\sim r+C_{X}$, $z\sim
r+C_{Z}$ and

$z\sim x+C_{Z}-C_{X},$ (27)

i.e. in spatial infinity the coordinates $r$, $x$ and $z$ are the same up to
constants. The corresponding boundary condition in this limit, equivalent to
(16), is then written as

$\psi\propto e^{-i\omega z}\hskip 3.0pt,\hskip 3.0ptz\to+\infty.$ (28)

In this limit we then have

$\frac{dz}{dx}\sim 1,\,\frac{d^{2}z}{dx^{2}}\sim 0;$ (29)

therefore, in this region the master equation describing the perturbations can
be written with respect to the variable $z$ exactly in the same way as it is
written with respect to $x$, i.e. (5). Also from $f(r)\sim f_{0}(r)\sim 1$ we
see that $V_{\textsf{min}}[f(r)]$ given by (6), and more generally
$V_{\textsf{T}}[f(r)]$ given by (8), vanish. This way, (5) is written in this
region simply as

$\frac{d^{2}\psi}{dz^{2}}+\omega^{2}\psi=0,$

which is compatible with the required boundary condition (28).

####  3.2.3 The topology of Stokes lines

Since we are using the standard tortoise coordinate $z$ of the non corrected
$d$-dimensional Tangherlini black hole spacetime, we will have the same
structure of Stokes lines. As we saw in (24), close to the origin of the
complex $r$-plane, the leading term of $z$ is given by
$z\sim-\frac{1}{R_{H}^{d-3}}\frac{r^{d-2}}{d-2}.$ If we parameterize $r$ as
$r=\rho e^{i\theta}$ for $\rho\in\mathbb{R}^{+}_{0}$ and $\theta\in[0,2\pi[$,
we know that

$\mathop{\rm Re}\left(r^{d-2}\right)=\rho^{d-2}\cos\left((d-2)\theta\right).$
(30)

Equating the expression above to zero yields

$\theta=\frac{\pi}{d-2}\left(n+\frac{1}{2}\right)$ (31)

for $1\leq n\leq 2d-4$. This means that in general we will have $2d-4$ Stokes
lines emerging from the origin of the complex $r$-plane, all equally
distributed and separated by an angle of $\frac{\pi}{d-2}$. As argued in
Natario:2004jd  we know that two of such lines are bounded, forming angles of
$\pm\frac{\pi}{2(d-2)}$ with the real axis at the origin, and forming a loop
around the real physical horizon $r=R_{H}$. The next two adjacent Stokes lines
are unbounded, going towards complex infinity and forming angles of
$\pm\frac{3\pi}{2(d-2)}$ with the real axis at the origin. Between these two
unbounded Stokes lines there will be no “fictitious” horizons, as these are
all located in a circumference of radius $R_{H}$ and the ones that are
adjacent to the real horizon form angles of $\pm\frac{2\pi}{d-3}$ with the
real axis at the origin. Only for $d\geq 8$ there will be fictitious horizons
with $\mathop{\rm Re}(r)>0$, but as we can see in figure 1 these horizons lie
outside the domain contained between the two unbounded Stokes lines we
mentioned.

(a) $d=5$

(b) $d=6$

(c) $d=7$

(d) $d=8$

(e) $d=9$

(f) $d=10$

Figure 1: Numerical plot of the Stokes lines topology for different
dimensions. The horizontal axis stands for $\mathop{\rm Re}(r/R_{H})$ and the
vertical axis stands for $\mathop{\rm Im}(r/R_{H})$. We denoted the positions
of the physical horizon $R_{H}$ and of the fictitious horizons by red dots.

###  3.3 The master equation and its perturbative solution

We want to recast the differential equation (5) with respect to the variable
$z$. We may write the second order derivative in (5) as

$\frac{d^{2}\psi}{dx^{2}}=\frac{d^{2}\psi}{dz^{2}}\left(\frac{dz}{dx}\right)^{2}+\frac{d\psi}{dz}\frac{d^{2}z}{dx^{2}}.$
(32)

From (4), (9) and (22) we get

$\frac{dz}{dx}=1+\frac{\lambda}{R_{H}^{2}}\delta f(r);$ (33)

using the definition (4) and differentiating again (33), we obtain

$\frac{d^{2}z}{dx^{2}}=f\frac{d}{dr}\left(\frac{dz}{dx}\right)=\frac{\lambda}{R_{H}^{2}}f(r)\left(\delta
f(r)\right)^{\prime}.$ (34)

This way, (5) is written in terms of $z$ as

$\left(\frac{dz}{dx}\right)^{2}\frac{d^{2}\psi}{dz^{2}}+\frac{d^{2}z}{dx^{2}}\frac{d\psi}{dz}+\left(\omega^{2}-V\right)\psi=0,$
(35)

with $\frac{dz}{dx}$ and $\frac{d^{2}z}{dx^{2}}$ given as functions of $r$ by
(33) and (34), respectively.

Replacing (33) and (34), we rewrite (35) as

$\left[1+\lambda^{\prime}\delta
f(r)\right]^{2}\frac{d^{2}\psi}{dz^{2}}+\lambda^{\prime}f(r)\left[\delta
f(r)\right]^{\prime}\frac{d\psi}{dz}+\left(\omega^{2}-V\right)\psi=0,$ (36)

In order to solve the differential equation (36) in different regions of the
complex $r$-plane, we apply standard perturbation theory. This way we expand,
to first order in $\lambda^{\prime}$, the perturbation function $\psi$ and the
potential $V$:

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\psi_{0}+\lambda^{\prime}\psi_{1},$ (37) $\displaystyle V(r)$
$\displaystyle=$ $\displaystyle V_{0}(r)+\lambda^{\prime}V_{1}(r).$ (38)

$V(r)$ is the full potential $V_{\textsf{T}}[f(r)]$ given by (8). The
$\lambda^{\prime}=0$ part is given by $V_{0}(r)=V_{\textsf{min}}[f_{0}(r)]$,
with $V_{\textsf{min}}$ given by (6): it is the classical (uncorrected)
potential evaluated with the uncorrected metric function (10). All the
$\lambda^{\prime}$ corrections appear in $V_{1}(r)$: those that are implicit
in $V_{\textsf{min}}[f(r)]$, from evaluating $V_{\textsf{min}}$ with a
$\lambda^{\prime}$-corrected function, and those that are explicit in (8).

Replacing the above expansions in (36) and expanding again in
$\lambda^{\prime}$, by separately considering the terms of order zero and
first order in $\lambda^{\prime}$ we obtain two separate differential
equations, a homogeneous and a nonhomogeneous one:

$\displaystyle\frac{d^{2}\psi_{0}}{dz^{2}}+\left(\omega^{2}-V_{0}\right)\psi_{0}$
$\displaystyle=$ $\displaystyle 0,$ (39)
$\displaystyle\frac{d^{2}\psi_{1}}{dz^{2}}+\left(\omega^{2}-V_{0}\right)\psi_{1}$
$\displaystyle=$ $\displaystyle\xi,$ (40)

with the function $\xi$ given by

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\xi_{1}+\xi_{2}+\xi_{3},$
(41) $\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle-2\delta
f(r)\left(\frac{d^{2}\psi_{0}}{dz^{2}}\right),$ (42) $\displaystyle\xi_{2}$
$\displaystyle=$ $\displaystyle-f(r)\left[\delta
f(r)\right]^{\prime}\left(\frac{d\psi_{0}}{dz}\right),$ (43)
$\displaystyle\xi_{3}$ $\displaystyle=$ $\displaystyle V_{1}(r)\psi_{0}.$ (44)

The boundary conditions at spatial infinity of the differential equations (39)
and (40) will be a simple extension of the boundary condition (28):
$\psi_{0}(z)\sim e^{-i\omega z},\,\psi_{1}(z)\sim e^{-i\omega z},\hskip
3.0ptz\to+\infty.$

###  3.4 The big contour

In order to build the big contour, we use the properties we found when
studying the topology of the Stokes lines. More precisely, for every dimension
$d$, there will be two Stokes lines, emerging from the origin of the complex
$r$-plane, encircling the event horizon $R_{H}$. Furthermore, these lines are
followed, counterclockwise and clockwise, by two unbounded Stokes lines,
asymptotically parallel to the imaginary axis. The big contour will follow
these unbounded Stokes lines, reaching the condition $|r|\to+\infty$ twice.
There, the contour abandons the Stokes lines and follows a large arc shaped
path enclosing it.

Overall, the big contour is well represented as depicted in figure 2. Looking
at this figure, we notice the proportions may not be right. However, the
topology of the big contour is well represented by the blue dashed line for
every dimension $d\geq 5$.

Figure 2: Schematic depiction of the big contour, as the blue dashed line. The
Stokes lines are depicted as red curves. Naturally, not all Stokes lines are
depicted. Furthermore, we marked by $D$ and $U$ the regions where the boundary
condition (28) may be imposed.

The boundary condition (28) is to be imposed in the regions marked by $D$ or
$U$. Here, we choose the region $D$ to impose it. Furthermore, we choose to
follow the contour in the clockwise direction.

In the surroundings of $r=0$, the big contour is represented as depicted in
figure 3. From equation (31) we conclude that the (small) arc shaped portion
of the big contour in this region, depicted in this figure, sweeps an angle of
$\frac{3\pi}{d-2}$.

Figure 3: Schematic representation of the small arc-shaped portion of the big
contour, in the surroundings of $r=0$, as the blue dashed line. The Stokes
lines are represented by red curves. Naturally, not all Stokes lines are
depicted.

###  3.5 Solution in a neighborhood of the origin and asymptotic behavior

We will now compute the solutions to the differential equations (39) and (40)
in a neighborhood of the origin of the complex $r-$plane. We then proceed to
study the asymptotic behavior of these solutions in the portion of the big
contour that follows the Stokes lines extending to complex infinity.

We start by solving (39), the differential equation for $\psi_{0}$. This is
precisely the same equation describing the same quasinormal modes in Einstein
gravity. The resolution of this equation has been studied in detail in
Motl:2003cd ; Natario:2004jd ; we will review the main steps.

In an arbitrarily small neighborhood of the origin, the leading term of the
potential $V_{0}(r)$ is given by

$V_{0}(r)\sim-\left(\frac{d-2}{2}\right)^{2}\frac{R_{H}^{2d-6}}{r^{2d-4}}.$
(45)

Replacing $z$ given by (24) in $V_{0}(r)$, we obtain for the leading term
$V_{0}(z)\sim-\frac{1}{4z^{2}}$. We can then write the differential equation
(39), in this neighborhood, as

$\frac{d^{2}\psi_{0}}{dz^{2}}+\left(\omega^{2}-\frac{j^{2}-1}{4z^{2}}\right)\psi_{0}=0$
(46)

for $j=0$. Following the procedure of Motl:2003cd ; Natario:2004jd  we will
consider the general solution, for arbitrary $j$, of the above differential
equation, and at the end take the limit $j\rightarrow 0$. Such solution is
given by

$\psi_{0}(z)=A_{+}\sqrt{2\pi}\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)+A_{-}\sqrt{2\pi}\sqrt{\omega z}J_{-\frac{j}{2}}(\omega z),$ (47)

where $J_{\pm\frac{j}{2}}(\omega z)$ are Bessel functions of the first kind
and $A_{+},A_{-}$ arbitrary constants.

Bessel functions verify the asymptotic condition (for $|\omega z|\gg 1$)

$\displaystyle\sqrt{2\pi}\sqrt{\omega z}J_{\pm\frac{j}{2}}(\omega z)\sim
2\cos\left(\omega z-\alpha_{\pm}\right),$
$\displaystyle\alpha_{\pm}=\frac{\pi}{4}\left(1\pm j\right).$ (48)

The condition $|\omega z|\gg 1$ holds in the portion of the big contour
following the Stokes lines everywhere except near the origin. Therefore, in
this portion of the contour extending to complex infinity $\psi_{0}$ will be a
linear combination of plane waves:

$\displaystyle\psi_{0}(z)$ $\displaystyle\sim$ $\displaystyle
2A_{+}\cos\left(\omega z-\alpha_{+}\right)+2A_{-}\cos\left(\omega
z-\alpha_{-}\right)$ (49) $\displaystyle=$
$\displaystyle\left(A_{+}e^{-i\alpha_{+}}+A_{-}e^{-i\alpha_{-}}\right)e^{i\omega
z}+\left(A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}\right)e^{-i\omega z}.$

Imposing the boundary condition (28) to $\psi_{0}$, we get a linear system of
equations for the coefficients $A_{+},A_{-}$ (with $c\in\mathbb{C}$):

$\begin{cases}A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}=c,\\\
A_{+}e^{-i\alpha_{+}}+A_{-}e^{-i\alpha_{-}}=0,\end{cases}$ (50)

which we can solve:

$\begin{cases}A_{-}=\frac{ce^{-i\alpha_{+}}}{2i\sin(\alpha_{-}-\alpha_{+})},\\\
A_{+}=-\frac{ce^{-i\alpha_{-}}}{2i\sin(\alpha_{-}-\alpha_{+})}.\end{cases}$
(51)

We now turn to the solution of the differential equation (40), the
nonhomogeneous differential equation for $\psi_{1}$. In order to obtain a
particular solution to it we use the method of variation of parameters to
write

$\psi_{1}(z)=2\pi\sqrt{\omega z}J_{-\frac{j}{2}}(\omega z)\int\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\frac{\xi(z)}{W}dz-2\pi\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\int\sqrt{\omega z}J_{-\frac{j}{2}}(\omega
z)\frac{\xi(z)}{W}dz,$ (52)

where $W$ denotes the wronskian of the basis of solutions of (46)

$W=\frac{d}{dz}\left(\sqrt{2\pi}J_{-\frac{j}{2}}(\omega
z)\right)\sqrt{2\pi}J_{\frac{j}{2}}-\frac{d}{dz}\left(\sqrt{2\pi}J_{\frac{j}{2}}(\omega
z)\right)\sqrt{2\pi}J_{-\frac{j}{2}}=-4\omega\sin\left(\frac{\pi
j}{2}\right).$ (53)

The extra function $\xi$ depends on $\psi_{0}$, given by (47), and its
derivatives, given by

$\displaystyle\frac{d\psi_{0}}{dz}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2}}\sqrt{\frac{\omega}{z}}\left(A_{+}J_{\frac{j}{2}}(z\omega)+A_{+}\omega
zJ_{\frac{j}{2}-1}(z\omega)-A_{+}\omega zJ_{\frac{j}{2}+1}(z\omega)\right)$
(54) $\displaystyle+$
$\displaystyle\sqrt{\frac{\pi}{2}}\sqrt{\frac{\omega}{z}}\left(A_{-}J_{-\frac{j}{2}}(z\omega)+A_{-}\omega
zJ_{-\frac{j}{2}-1}(z\omega)-A_{-}\omega zJ_{1-\frac{j}{2}}(z\omega)\right),$
$\displaystyle\frac{d^{2}\psi_{0}}{dz^{2}}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{8}}\sqrt{\omega}z^{-\frac{3}{2}}\left(j^{2}-4\omega^{2}z^{2}-1\right)\left(A_{+}J_{\frac{j}{2}}(z\omega)+A_{-}J_{-\frac{j}{2}}(z\omega)\right).$
(55)

From (41) we see that $\xi$ also depends on three other quantities, whose
leading terms in an arbitrarily small neighborhood around the origin are given
by

$\displaystyle V_{1}(r)$ $\displaystyle\sim$
$\displaystyle\frac{1}{4}(d-4)(d(d-5)(2d-7)-22)\frac{R_{H}^{3d-7}}{r^{3d-5}},$
(56) $\displaystyle f(r)\left[\delta f(r)\right]^{\prime}$ $\displaystyle\sim$
$\displaystyle-\frac{(d-4)(d-3)(d-1)}{2}\frac{R_{H}^{2d-4}}{r^{2d-3}},$ (57)
$\displaystyle 2\delta f(r)$ $\displaystyle\sim$
$\displaystyle-(d-3)(d-4)\left(\frac{R_{H}}{r}\right)^{d-1}.$ (58)

Expressed in terms of $z$ through (24), the same leading terms are given by

$\displaystyle 2\delta f(r)$ $\displaystyle\sim$
$\displaystyle\Upsilon_{1}\left(\frac{R_{H}}{z}\right)^{-\rho-2},$ (59)
$\displaystyle f(r)\left[\delta f(r)\right]^{\prime}$ $\displaystyle\sim$
$\displaystyle\Upsilon_{2}z^{\rho+1}R_{H}^{-\rho-2},$ (60) $\displaystyle
V_{1}(r)$ $\displaystyle\sim$
$\displaystyle\Upsilon_{3}z^{\rho}R_{H}^{-\rho-2},$ (61)

with the definitions

$\displaystyle\rho$ $\displaystyle\equiv$ $\displaystyle-2-\frac{d-1}{d-2},$
(62) $\displaystyle\Upsilon_{1}$ $\displaystyle\equiv$
$\displaystyle(-1)^{\rho+2}(d-2)^{\rho+2}(d-4)(d-3),$ (63)
$\displaystyle\Upsilon_{2}$ $\displaystyle\equiv$
$\displaystyle(-1)^{\rho+1}(d-2)^{\rho+1}\frac{(d-4)(d-3)(d-1)}{2},$ (64)
$\displaystyle\Upsilon_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4}(-1)^{\rho}(d-2)^{\rho}(d-4)((d-5)d(2d-7)-22).$ (65)

Analogously to (41), we decompose the particular solution to the
nonhomogeneous differential equation (40) into a sum of three terms, each one
corresponding to one term of $\xi$:

$\displaystyle\psi_{1}(z)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\phi_{i}(z),$ $\displaystyle\phi_{i}(z)$
$\displaystyle=$ $\displaystyle 2\pi\sqrt{\omega z}J_{-\frac{j}{2}}(\omega
z)\int\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\frac{\xi_{i}(z)}{W}dz-2\pi\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\int\sqrt{\omega z}J_{-\frac{j}{2}}(\omega z)\frac{\xi_{i}(z)}{W}dz.$ (66)

In appendix A we obtain explicit expressions for the functions
$\phi_{1}(z),\phi_{2}(z),\phi_{3}(z)$. Based on these results, we can write
the asymptotic behavior of $\psi_{1}$ in the limit $|\omega z|\gg 1$ as

$\psi_{1}(z)\sim\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right)e^{i\omega
z}+\left(\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}\right)e^{-i\omega
z},$ (67)

with the definition

$\Lambda_{I}^{\pm}(d,j,\omega,R_{H})=\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}\sum_{k=1}^{3}\Theta_{k}^{\pm}(d,j,1,1)$
(68)

and the coefficients $\Theta_{1,2,3}^{\pm}(d,j,\omega,R_{H})$ (also depending
on the constants $A_{+},A_{-}$ from (47)) given at the end of appendix A.
These coefficients all share the same overall factor
$\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}$. We will discuss this issue in
section 3.9.

We see that the above asymptotic behavior (67) is not compatible with the
boundary condition (28). In order to fix this we use the fact that we can
always add a solution $\psi^{*}$ of the homogeneous equation associated with
the differential equation (40) to the _particular_ solution (52). We use this
fact to redefine $\psi_{1}$ as

$\psi_{1}\to\psi_{1}+\psi^{*}.$ (69)

The general solution of such homogeneous equation $\psi^{*}$ is, analogously
to (47), given by

$\psi^{*}(z)=C_{+}\sqrt{2\pi}\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)+C_{-}\sqrt{2\pi}\sqrt{\omega z}J_{-\frac{j}{2}}(\omega z)$ (70)

for some $C_{+},C_{-}\in\mathbb{C}$. The asymptotic behavior of $\psi^{*}$ is
also analogous to the one of $\psi_{0}$:

$\psi^{*}(z)\sim\left(C_{+}e^{-i\alpha_{+}}+C_{-}e^{-i\alpha_{-}}\right)e^{i\omega
z}+\left(C_{+}e^{i\alpha_{+}}+C_{-}e^{i\alpha_{-}}\right)e^{-i\omega z}.$ (71)

We require $C_{+},C_{-}$ to be solutions of the following linear system of
equations (with $c\in\mathbb{C}$):

$\begin{cases}C_{+}e^{i\alpha_{+}}+C_{-}e^{i\alpha_{-}}=c,\\\
C_{+}e^{-i\alpha_{+}}+C_{-}e^{-i\alpha_{-}}=-\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right).\end{cases}$
(72)

The second equation in (72) is simply the condition for the $e^{i\omega z}$
term in (71) to cancel the $e^{i\omega z}$ term in (67) with the redefinition
(69). Similarly to this redefinition, we may decompose the coefficients
$C_{\pm}$ in (70) as

$C_{\pm}=A_{\pm}+B_{\pm},$

with the coefficients $A_{\pm}$ satisfying (50) and $B_{\pm}$ satisfying

$\begin{cases}B_{+}e^{i\alpha_{+}}+B_{-}e^{i\alpha_{-}}=0,\\\
B_{+}e^{-i\alpha_{+}}+B_{-}e^{-i\alpha_{-}}=-\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right),\end{cases}$
(73)

whose solution is given by

$\begin{cases}B_{-}=\frac{e^{i\alpha_{+}}\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right)}{2i\sin\left(\alpha_{-}-\alpha_{+}\right)},\\\
B_{+}=-\frac{e^{i\alpha_{-}}\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right)}{2i\sin\left(\alpha_{-}-\alpha_{+}\right)}.\end{cases}$
(74)

One can immediately see that, by adding the systems of equations (50) and
(73), one obtains (72). This means one can actually decompose $\psi^{*}$ in
(70) into two parts. The coefficients of the first part are $A_{\pm}$
satisfying (50); having in mind (37), the first part can actually be absorbed
by $\psi_{0}$. Therefore, without loss of generality one can define $\psi^{*}$
as in (70), but with coefficients $B_{\pm}$ satisfying (73) (i.e. just by the
second part of the splitting).

We get finally
$\psi_{1}(z)\sim\left(\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}\right)e^{-i\omega
z}$ (for the function after the redefinition (69)) and, from (37), (49), (50),
we obtain asymptotically as $|\omega z|\gg 1$

$\begin{split}\psi(z)\sim\left(A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}\right)e^{-i\omega
z}+\lambda^{\prime}\left(\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}\right)e^{-i\omega
z}\\\
=\left(A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}\right)\left(1+\lambda^{\prime}\left(\frac{\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}\right)\right)e^{-i\omega
z}.\end{split}$ (75)

###  3.6 Computation of the total monodromy in the big contour

We now proceed with the analysis of the behavior of $\psi$ around the big
contour described in section 3.4 and depicted in figure 2.

In the contour we consider, we start from region $D$ in the lower part of the
complex _r_ -plane (close to $r=-i\infty$ for $d=5$ and in the fourth quadrant
for $d>5$), where $\psi(z)$ is given by (75). We follow the Stokes line
towards the origin. In a neighborhood of the origin, we make a
$\frac{3\pi}{d-2}$ rotation around the origin; we present the detailed study
of the behavior of $\psi=\psi_{0}+\lambda^{\prime}\psi_{1}$ under such
rotation in appendix B. We then follow again a portion of the contour
coincident with a Stokes line and all the way to the upper portion of the
contour back to complex infinity (region $U$), where the asymptotic values of
$\psi_{0},\psi_{1}$ are given by (127) and (133) respectively. We notice that
both asymptotic values contain terms proportional to $e^{i\omega z}$ and to
$e^{-i\omega z}$. This is to be expected from the WKB approximation.

Now we want to see how $\psi$ behaves in the large arc shaped portion of the
big contour. We notice that this portion of the contour no longer coincides
with a Stokes line of the WKB approximation of the master equation (39).
Because of the condition $\mathop{\rm Im}(\omega z)\gg 0,$ the term
proportional to $e^{i\omega z}$ gives only an exponentially small contribution
to $\psi$ in this region of the complex $r-$plane; therefore, as we abandon
$U$ and start following this portion of the big contour this term can be
modified by the small correction terms arising from the WKB approximation that
have been neglected in the plane wave approximation. This way, we cannot trust
the coefficient multiplying $e^{i\omega z}$, as we cannot expect it to stay
the same after closing the big contour. However, even not knowing the
coefficient of this term, we know it is exponentially small in the large arc,
and can be discarded there. Thus near $D$, after closing the big contour we
will then have (from (127) and (133), with coefficients
$\Lambda_{F}^{\pm}(d,j,\omega,R_{H})$ given by (132), and discarding the
$e^{i\omega z}$ term)

$\psi(z)\sim\left(A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}\right)e^{-i\omega
z}+\lambda^{\prime}\left(\Lambda_{F}^{+}e^{5i\alpha_{+}}+\Lambda_{F}^{-}e^{5i\alpha_{-}}\right)e^{-i\omega
z}.$ (76)

This way, we use the plane wave approximation to obtain the coefficient
multiplying $e^{-i\omega z}$ in $\psi$, and we assume that the corresponding
monodromy is equivalent to the monodromy of $\psi$, i.e. that the monodromy of
$\psi$ is multiplicative. This assumption will be fully justified when we
compute the monodromy of $\psi$ in the small contour, in section 3.7.

Before computing the total monodromy of $\psi$, we must consider one further
detail. As we mentioned in section 3.2, as a function $z(r)$ has a branch
point in each zero of $f_{0}(r)$: the real and the “fictitious” horizons (26).
From the discussion in section 3.2.3, the big contour does not enclose any of
these “fictitious” horizons but, since it encloses the real horizon, a full
loop around it is bound to cross a branch cut somewhere. Thus the expression
(76) above is written with respect to a variable $z$, defined on a branch, in
the Riemann surface of $z$, different from the branch in which the variable
$z$ used in (75) is defined. In order to relate these two variables we simply
need to compute the monodromy of $z$ associated with one clockwise loop around
the event horizon $r=R_{H}$.

Close to the horizon, from (22) we write (ignoring an irrelevant integration
constant)

$z=\int\frac{dr}{f_{0}(r)}\approx\frac{1}{f^{\prime}_{0}(R_{H})}\ln\left(1-\frac{r}{R_{H}}\right)=\frac{R_{H}}{d-3}\ln\left(1-\frac{r}{R_{H}}\right).$
(77)

From the parametrization $1-\frac{r}{R_{H}}=\rho e^{i\theta}$ for $\rho\geq 0$
and $0\leq\theta<2\pi$, we can rewrite the expansion (77) near $r=R_{H}$ as
$z(r)\sim\frac{R_{H}}{d-3}\left(\ln\left(\rho\right)+i\theta\right).$
Following a full clockwise loop around $r=R_{H}$ is equivalent to letting
$\theta$ run from $2\pi$ to $0$. Therefore, the monodromy of $z$, associated
with a full clockwise loop around $r=R_{H}$, is

$m:=-2\pi i\frac{R_{H}}{d-3}.$ (78)

Using this monodromy, we can relate the previously mentioned $z$ variables by
redefining the one used in (76) as $z\mapsto z+m.$ We can then consider the
monodromy of $e^{-i\omega z}$ around the big contour and rewrite (76) near $D$
as

$\psi(z)\sim\left(A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}\right)e^{-i\omega
m}e^{-i\omega
z}\left[1+\lambda^{\prime}\left(\frac{\Lambda_{F}^{+}e^{5i\alpha_{+}}+\Lambda_{F}^{-}e^{5i\alpha_{-}}}{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}\right)\right].$
(79)

Comparing (75) and (79), we can finally write the final monodromy of $\psi$
around the big contour as

$\mathcal{M}_{1}=\left(\frac{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}\right)e^{-i\omega
m}\left(1+\lambda^{\prime}\delta\mathcal{M}_{1}\right)$ (80)

where, to first order in $\lambda^{\prime}$,

$\delta\mathcal{M}_{1}=\frac{\Lambda_{F}^{+}e^{5i\alpha_{+}}+\Lambda_{F}^{-}e^{5i\alpha_{-}}}{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}-\frac{\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{+}e^{i\alpha_{-}}}.$
(81)

We recall that $\alpha_{\pm}$ are given by (48); $A_{\pm}$ are given by (51);
$\Lambda_{I}^{\pm}(d,j,\omega,R_{H})$ are given by (68) with coefficients
$\Theta_{1,2,3}^{\pm}(d,j,\omega,R_{H})$ defined in appendix A;
$\Lambda_{F}^{\pm}(d,j,\omega,R_{H})$ are given by (132) with coefficients
$B_{\pm}$ given by (74) and $\Xi_{1,2,3}^{\pm}(d,j,\omega,R_{H})$ defined in
appendix B.

One can check that, taking the limit $j\to 0$,

$\frac{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}=-3.$
(82)

Also in the limit $j\to 0$, and after a lot of algebraic manipulations, we get
for (81)

$\displaystyle\delta\mathcal{M}_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{R_{H}\omega}{d-2}\right)^{\frac{d-1}{d-2}}\mathrm{e}^{-\frac{2\pi
i}{d-2}}\Pi_{\textsf{T}}(d),$ (83) $\displaystyle\Pi_{\textsf{T}}(d)$
$\displaystyle:=$
$\displaystyle\frac{2}{3}\frac{(d-4)\left(d(d-5)+2\right)}{d-1}\sqrt{\pi}\sin\left(\frac{\pi}{d-2}\right)\frac{\Gamma\left(\frac{1}{2(d-2)}\right)\Gamma\left(\frac{d-3}{2(d-2)}\right)}{\left[\Gamma\left(\frac{d-1}{2(d-2)}\right)\right]^{2}}.$

This way, we can rewrite (80) as

$\mathcal{M}_{1}=-3e^{-i\omega
m}\left(1+\lambda^{\prime}\delta\mathcal{M}_{1}\right).$ (84)

###  3.7 Computation of the monodromy in the small contour

Compared to the big one, the small contour is remarkably simple. Indeed, we
build an arbitrarily small closed contour around the event horizon $R_{H}$.
Such contour can be represented as the dashed orange contour in figure 4.

Figure 4: Schematic depiction of the small and big contours as the orange and
blue dashed lines respectively. The orange contour is to be interpreted as
arbitrarily close to $R_{H}$. The Stokes lines are depicted by red curves.
Naturally, not all Stokes lines are depicted.

In this contour, we don’t need to solve perturbatively the master equation
(5). Indeed, since the contour is arbitrarily small, we can simply solve (5)
expanded in a neighborhood of the event horizon. From $f(R_{H})=0$ we see that
$V_{\textsf{min}}[f(r)]$ given by (6), and more generally
$V_{\textsf{T}}[f(r)]$ given by (8), vanish at the horizon. This way, (5) is
written in this region simply as

$\frac{d^{2}\psi}{dx^{2}}+\omega^{2}\psi=0,$ (85)

with general solution given by $\psi(x)=C_{+}e^{i\omega x}+C_{-}e^{-i\omega
x}$ for some constants $C_{+},C_{-}$. Compatibility with the boundary
condition (17) requires the restriction $C_{-}=0$; therefore we may write

$\psi(x)\sim C_{+}e^{i\omega x}$ (86)

in the small contour.

Close to the horizon, from (9) we expand to first order in $\lambda$
$f(r)\approx f^{\prime}_{0}(R_{H})\left(1+\frac{\lambda}{R_{H}^{2}}\delta
f(R_{H})\right)(r-R_{H})$, from which we get (using (77) and also ignoring an
irrelevant integration constant)

$x(r)=\int\frac{dr}{f(r)}\approx\frac{1}{f^{\prime}_{0}(R_{H})}\left(1-\frac{\lambda}{R_{H}^{2}}\delta
f(R_{H})\right)\ln\left(1-\frac{r}{R_{H}}\right)\approx\left(1-\frac{\lambda}{R_{H}^{2}}\delta
f(R_{H})\right)z(r)$ (87)

in the small contour. Thus, analogously to $z$, the variable $x$ has a branch
point at the event horizon $R_{H}$. The monodromy $n$ of $x$, associated with
a full clockwise loop around the small contour, is related to the equivalent
monodromy $m$ of $z$, given by (78), as

$n:=\left(1-\frac{\lambda}{R_{H}^{2}}\delta
f(R_{H})\right)m=\left(1+\frac{\lambda^{\prime}}{2}(d-4)(d-1)\right)m.$ (88)

From (86), the monodromy of $\psi$ associated with a full clockwise loop
around the small contour is multiplicative and given by

$\mathcal{M}_{2}:=e^{i\omega n}.$ (89)

The monodromy theorem tells us that this monodromy of $\psi$ is the same for
every contour that is homotopic to the small contour, namely any larger
contour around the horizon $R_{H}$, where $\psi$ is given by a combination of
$e^{-i\omega z}$ and $e^{i\omega z}$. Thus the monodromy of (86), i.e. the
monodromy of $e^{i\omega x}$, is indeed the monodromy of $\psi$. The
multiplicativity of this monodromy is therefore a general property. The
assumption of multiplicativity of the monodromy around the big contour is
indeed justified, and so is its calculation in section 3.6.

###  3.8 Equating monodromies

Now, we want to relate the monodromies $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$. To this end, we start by noticing that the big contour is
homotopic to the small one. This is so, because one can continuously deform
the big contour into the small one. Thus, using the monodromy theorem, we know
that the monodromies of $\psi$, associated with the full clockwise loops
around the big and the small contour, are the same. Hence, the equation

$\mathcal{M}_{1}=\mathcal{M}_{2}$ (90)

must hold. From (84) and (89) we can rewrite the equation above, to first
order in $\lambda^{\prime}$, as

$-3=e^{i\omega(m+n)}(1-\lambda^{\prime}\delta\mathcal{M}_{1}).$ (91)

Taking the logarithm on both sides of (91) we get, to first order in
$\lambda^{\prime}$,

$\ln(-3)-i\omega(m+n)+\lambda^{\prime}\delta\mathcal{M}_{1}=0.$ (92)

From (78) and (88), we can write

$m+n=-\left(2+\lambda^{\prime}\frac{(d-4)(d-1)}{2}\right)\frac{2\pi
i}{d-3}R_{H}.$ (93)

Using (12) we can rewrite $m+n$, to first order in $\lambda^{\prime}$, as a
function of the black hole temperature $T_{\mathcal{H}}$:

$m+n=-\frac{i}{T_{\mathcal{H}}}\left(1-\lambda^{\prime}\frac{(d-1)(d-4)}{4}\right).$
(94)

Similarly, from (12) we can also write, to first order in $\lambda^{\prime}$,

$\lambda^{\prime}\left(\frac{R_{H}\omega}{d-2}\right)^{\frac{d-1}{d-2}}=\lambda^{\prime}\left[\frac{d-3}{d-2}\frac{\omega}{4\pi
T_{\mathcal{H}}}\right]^{\frac{d-1}{d-2}}.$ (95)

Taking (94) for $m+n$, (83) for $\delta\mathcal{M}_{1}$ and using (95), we can
rewrite (92) as

$\ln(3)+(2k+1)\pi
i=\frac{\omega}{T_{\mathcal{H}}}\left(1-\lambda^{\prime}\frac{(d-1)(d-4)}{4}-\lambda^{\prime}\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\left[\frac{\omega}{4\pi
T_{\mathcal{H}}}\right]^{\frac{1}{d-2}}\frac{\Pi_{\textsf{T}}(d)}{4\pi}\mathrm{e}^{-\frac{2}{d-2}\pi
i}\right)$ (96)

for $k\in\mathbb{N}$. Here we take $k>0$ in order to get $\mathop{\rm
Im}(\omega)>0$ in the Einstein limit $\lambda^{\prime}\to 0$, and also
considering condition (18). We obtain this way a transcendental equation that
restricts the possible values of the quasinormal frequencies $\omega$ from the
complex plane to an infinite but countable (for every value of the mode number
$k$) subset. In the limit of Einstein gravity, we recover the results of
Natario:2004jd , as we should:

$\left.\frac{\omega}{T_{\mathcal{H}}}\right|_{\alpha^{\prime}=0}=\ln(3)+(2k+1)\pi
i,\,k\in\mathbb{N}.$ (97)

Since we are working perturbatively in $\lambda^{\prime}$, we can simply
consider this value of $\frac{\omega}{T_{\mathcal{H}}}$ for the
$\lambda^{\prime}$ correction in (96) and solve this equation for
$\frac{\omega}{T_{\mathcal{H}}}$, obtaining

$\frac{\omega}{T_{\mathcal{H}}}=\left(\ln(3)+(2k+1)\pi
i\right)\left(1+\lambda^{\prime}\frac{(d-1)(d-4)}{4}+\lambda^{\prime}\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\left[\frac{\ln(3)+(2k+1)\pi
i}{4\pi}\right]^{\frac{1}{d-2}}\frac{\Pi_{\textsf{T}}(d)}{4\pi}\mathrm{e}^{-\frac{2}{d-2}\pi
i}\right).$ (98)

With this replacement, from (98) it is not easy to evaluate the real and
imaginary parts of $\omega$. One should keep in mind that we have been working
in the highly damped limit (18), which allows us to consider
$\left.\frac{\omega}{T_{\mathcal{H}}}\right|_{\alpha^{\prime}=0}\approx(2k+1)\pi
i$ in the $\lambda^{\prime}$ correction in (98). We also make the dependence
on the temperature more explicit by writing
$\lambda^{\prime}=\lambda\frac{16\pi^{2}}{(d-3)^{2}}T_{\mathcal{H}}^{2}$,
according to (13). This way we obtain for the $\alpha^{\prime}$-corrected
asymptotic quasinormal spectra:

$\frac{\omega}{T_{\mathcal{H}}}=\left(\ln(3)+(2k+1)\pi
i\right)\left[1+\lambda\frac{16\pi^{2}}{(d-3)^{2}}T_{\mathcal{H}}^{2}\left(\frac{(d-1)(d-4)}{4}+\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\left[\frac{2k+1}{4}\right]^{\frac{1}{d-2}}\frac{\Pi_{\textsf{T}}(d)}{4\pi}\,\mathrm{e}^{-\frac{3\pi
i}{2(d-2)}}\right)\right],$ (99)

with $k\in\mathbb{N}$ and $\Pi_{\textsf{T}}(d)$ given by (83).

Written in this form, for each value of $d$ one can evaluate the real and
imaginary parts of the frequencies $\omega$. We notice that because of the
$\mathrm{e}^{-\frac{3\pi i}{2(d-2)}}$ term the $\alpha^{\prime}$ correction is
complex, which means it will affect both the real and the imaginary parts of
$\omega$. The real part of the asymptotic limit of
$\frac{\omega}{T_{\mathcal{H}}}$, therefore, is no longer equal to the
universal value $\ln(3)$: because of the $\alpha^{\prime}$-correction, it now
also depends on the spacetime dimension $d$ and on the mode number $k$.

It is interesting to study the magnitudes of the different contributions to
the $\lambda^{\prime}$ correction in (99). For that purpose, we have evaluated
$\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\frac{\Pi_{\textsf{T}}(d)}{4\pi}$
numerically for the relevant values of $d$. This factor grows monotonically
with $d$, varying from approximately 0.19 (corresponding to $d=5$) to
approximately 13.5 (corresponding to $d=10$). Just for comparison,
$\frac{(d-1)(d-4)}{4}$ varies between 1 and $13.5$ on the same range. For
values of $k$ that are not very large (although verifying the condition
$(2k+1)\pi\gg\ln(3)$ defining the highly damped limit (18)), the two terms in
the $\lambda^{\prime}$ correction in (99) are of comparable orders of
magnitude. For these “intermediate” values of $k$ one must consider those two
terms and, therefore, (99) cannot be simplified.

For fixed values of $\omega$, we notice our results may loose precision as $d$
increases. Indeed, as we discussed in appendix A, while computing the
monodromy of $\psi$ around the big contour we made an approximation in the
asymptotic expansion (116) by discarding terms of order at most
$\left(\frac{\omega}{T_{\mathcal{H}}}\right)^{-\frac{1}{d-2}}.$ This is
acceptable for the theoretical asymptotic limit, in which we assume that
$|\omega|\to+\infty$. However, one must take care, when applying our results
for very large, but not arbitrary so, values of $|\omega|$. Indeed, suppose
one uses our transcendental equations to compute values of $\omega$ such that

$\Big{|}\frac{\omega}{T_{\mathcal{H}}}\Big{|}\sim 10^{8}.$ (100)

Then, those values were computed assuming that $10^{-\frac{8}{d-2}}$ is
negligible. However, taking $d=5,7,9$ yields

$10^{-\frac{8}{3}}\approx 0.002,\hskip 20.0pt10^{-\frac{8}{5}}\approx
0.03,\hskip 20.0pt10^{-\frac{8}{7}}\approx 0.07,$ (101)

respectively. As we can see, the approximation grows worst as we increase $d$.
Thus, our results for progressively higher dimensions only apply to
progressively higher values of $\Big{|}\frac{\omega}{T_{\mathcal{H}}}\Big{|}$.

###  3.9 The asymptotic limit

The decomposition of $\omega$ into its real and imaginary parts is simpler to
obtain if one takes the asymptotic limit (i.e. the limit of large $k$) in
(99).

One should keep in mind that we have been assuming in our calculations the
highly damped limit (18). Assuming the result of Einstein gravity (97), the
highly damped limit (18) is simply equivalent to having $(2k+1)\pi\gg\ln(3)$;
a very large value of $k$ implies then the highly damped limit, but the
converse does not need to be true. One may now ask the question: considering
the $\alpha^{\prime}$ correction to $\omega$, does the large $k$ limit still
imply the highly damped limit (18)?

Throughout this work we have been taking a perturbative expansion in $\lambda$
(or $\alpha^{\prime}$). With such expansion, we have taken only terms of first
order in $\lambda$, because that was the order considered in the lagrangian
(7): terms of higher order are meaningless unless they had been included in
the lagrangian since the beginning, changing the field equations and their
solutions like (9). These are only approximate and subject to even higher
order corrections in $\lambda$, and the same is true for the perturbation
potential $V_{\textsf{T}}[f(r)]$ given by (8). But once the order of
perturbation is fixed and $V_{\textsf{T}}[f(r)]$ is given, the master equation
(5) is also fixed and can be solved exactly, like we did. The only reason we
took the expansions (20) of the tortoise coordinate and (37), (38) of the
perturbation function and potential was the fact that the $\lambda-$corrected
master differential equation is nonhomogeneous, and the corresponding
homogeneous equation is precisely the $\lambda=0$ master equation. The
motivation for considering (37), (38) was not a perturbative expansion, but
the (exact) method of variation of parameters to solve nonhomogeneous
differential equations. This way, our solution to the master equation is exact
to first order in $\lambda$, and so are the mathematical consequences that we
can extract from it.

In general, a perturbative expansion means that a higher order term is
expected to be negligible when compared to a lower order term, the reason
being that the perturbative parameter is supposed to be small. That is
certainly the case for our parameter $\lambda$. But that does not mean at all
that $\lambda$ can be taken as arbitrarily small, and therefore it may happen
that a higher order term of a perturbative expansion in $\lambda$ is not
negligible if such term is made arbitrarily large.

But from equation (125) in appendix A and from (67) and (68) we see that
$\lambda^{\prime}\psi_{1}$ has an overall factor
$\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}$ in the asymptotic limit when
$|\omega z|\gg 1$. Such overall factor is also present in the $\lambda$
corrections $\delta\mathcal{M}_{1}$ to the monodromy $\mathcal{M}_{1}$ in (83)
and to the result for $\frac{\omega}{T_{\mathcal{H}}}$ in eq. (96) (in this
case after having $R_{H}$ expressed in terms of $T_{\mathcal{H}}$ in (95)).
One expects the perturbative parameter $\lambda^{\prime}$ to be small, but if
the absolute value $|\omega|$ is very large none of these $\lambda^{\prime}$
corrections to $\psi$ and $\mathcal{M}_{1}$ is negligible, and nor is the one
of $\omega$ itself. This is what happens in the asymptotic limit, with very
large mode number $k$ and very large $\mathop{\rm Im}(\omega)$. One may argue
that $\lambda^{\prime}$ is a small parameter, and therefore the larger the
order in $\lambda^{\prime}$, the smaller the (relative) magnitude of the
$\lambda^{\prime}$ correction, even if their (absolute) magnitude is large.
That argument may be true in our concrete discussion for large but finite $k$,
but not in the asymptotic limit of very large $k$, say $k\to\infty$. For such
limit the first order $\lambda^{\prime}$ correction becomes too large, as one
can see from (98) or from (99), and it cannot be neglected in any
circumstance, contrarily to what one expects from a perturbative correction.
Presumably the same is true for higher order $\lambda^{\prime}$ corrections,
which means in this limit the perturbative expansion breaks down. Therefore,
strictly speaking, the asymptotic limit cannot be inferred from our result.
The condition (18) we requested for the validity of our method should not be
understood as “arbitrarily large $\mathop{\rm Im}(\omega)$ (or $k$)”:
$\mathop{\rm Im}(\omega)$ and $k$ can be very large, but not so large to break
down the perturbative expansion. Other works computing asymptotic quasinormal
modes with higher derivative corrections, such as Daghigh:2006xg , require a
similar interpretation for (18).

We suppose now that $k$ is large enough so that we can neglect the whole term
multiplied by $\ln 3$ and also the term $\frac{(d-1)(d-4)}{4}$ in the
$\lambda^{\prime}$ correction in (99). If that is the case, we obtain then the
simpler result that we were looking for in the beginning of this section:

$\frac{\omega}{T_{\mathcal{H}}}=\ln(3)+(2k+1)\pi
i+\lambda\left(\frac{4\pi}{d-3}\right)^{2}T_{\mathcal{H}}^{2}\left[\frac{d-3}{d-2}\frac{(2k+1)}{4}\right]^{\frac{d-1}{d-2}}\Pi_{\textsf{T}}(d)\,\mathrm{e}^{\frac{d-5}{2(d-2)}\pi
i}$ (102)

Similarly to (99), because of the $\mathrm{e}^{\frac{d-5}{2(d-2)}\pi i}$ term
the $\alpha^{\prime}$ correction is complex, which means it will affect both
the real and the imaginary parts of $\omega$. We have evaluated this term
numerically for the relevant values of $d$. While
$\cos\left(\frac{d-5}{2(d-2)}\pi\right)$ varies between 1 (corresponding to
$d=5$) and approximately 0.56 (corresponding to $d=10$),
$\sin\left(\frac{d-5}{2(d-2)}\pi\right)$ varies on the same range from 0 to
approximately 0.83. For $d=5$, therefore, in this limit only the real part of
$\omega$ gets an $\alpha^{\prime}$ correction. For other relevant values of
$d$, (first order) $\alpha^{\prime}$ corrections to the real and the imaginary
parts of $\omega$ are both positive and have similar orders of magnitude in
this limit (for $d=8$ they are actually equal).

##  4 Asymptotic quasinormal modes of test scalar fields

A minimally coupled test scalar field propagating in the background of a black
hole of the form (1) can be expanded as

$\Phi(t,r,\theta)=e^{i\omega
t}\sum_{\ell}\Phi_{\ell,\omega}(r)Y_{\ell}(\theta)\,,$ (103)

where $\omega$ is the wave frequency, $\ell$ is the angular quantum number
associated with the polar angle $\theta$ and $Y_{\ell}(\theta)$ are the usual
spherical harmonics defined over the $(d-2)$ unit sphere $\mathbb{S}^{d-2}$.
Each component $\Phi_{\ell,\omega}(r)$ obeys a field equation like (5), with
precisely the same potential of the tensorial perturbations in Einstein
gravity, given hep-th/0206084  by $V_{\textsf{min}}[f_{0}(r)]$ in (6), with
$f_{0}(r)$ given by (10). In the presence of $\alpha^{\prime}$ corrections
this field equation remains the same (since the scalar field is minimally
coupled); the only effect of the $\alpha^{\prime}$ corrections in it is
indirect, through the metric. The potential corresponding to such equation is
now given by $V_{\textsf{min}}[f(r)]$, with $f(r)$ corresponding to the
$\alpha^{\prime}$-corrected metric (in our case given by (9)).

Naturally, if the field equation is the same, so is the spectrum of
quasinormal modes. In Einstein gravity, therefore, test scalar fields and
tensorial gravitational perturbations share the same spectra of quasinormal
modes. For an $\alpha^{\prime}$-corrected metric, because of the difference in
the potentials that is no longer true: the two spectra are indeed different.
In Moura:2021eln  we verified this fact by computing the quasinormal modes in
the eikonal limit for both cases.

In this section we address the calculation of the quasinormal modes
corresponding to test scalar fields in the highly damped regime, like we
previous did for tensorial gravitational perturbations, in the background of
the $\alpha^{\prime}$-corrected black hole given by (9). The calculation is
totally analogous to the one described in section 3; we just have to consider
the effects of changing the potential.

This change in the potential does not affect the monodromy $\mathcal{M}_{2}$
around the small contour, since close to the horizon the general potential
$V_{\textsf{T}}[f(r)]$ vanishes, and so does the potential
$V_{\textsf{min}}[f(r)]$ we are now considering. The same is true at infinity.

The result affected by the change in the potential, as one could expect, is
the monodromy $\mathcal{M}_{1}$ around the big contour. The procedure we took
in subsections 3.3 to 3.6 to compute this monodromy, with a perturbative
solution to the master equation, remains valid; it is actually totally
analogous. The only change is in the expansion (38) of the potential:
$V_{0}(r)$ remains given by $V_{\textsf{min}}[f_{0}(r)]$, but because $V(r)$
is now different, so is $V_{1}(r)$.

Close to the origin, $V_{1}(r)$ is given by (61), but with $\Upsilon_{3}$ in
(65) replaced by the new value

$\Upsilon_{3}\mapsto\Upsilon_{3}=\frac{1}{4}(-1)^{\rho}(d-2)^{\rho+1}(d-4)(d-3)(2d-3).$
(104)

This change in the value of $\Upsilon_{3}$ implies a change in the values of
the coefficients
$\Omega_{3}^{\pm}(d,j,\omega,R_{H}),\,\Theta_{3}^{\pm}(d,j,\omega,R_{H})$ and
$\Xi_{3}^{\pm}(d,j,\omega,R_{H})$, and those changes will affect the values of
$\Lambda_{I}^{\pm}(d,j,\omega,R_{H})$, $\Lambda_{F}^{\pm}(d,j,\omega,R_{H})$,
and consequently of the $\lambda$ correction $\delta\mathcal{M}_{1}$ in the
monodromy $\mathcal{M}_{1}$ around the big contour. This correction is now
given by

$\displaystyle\delta\mathcal{M}_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{R_{H}\omega}{d-2}\right)^{\frac{d-1}{d-2}}\mathrm{e}^{-\frac{2\pi
i}{d-2}}\Pi_{\textsf{S}}(d),$ (105) $\displaystyle\Pi_{\textsf{S}}(d)$
$\displaystyle:=$
$\displaystyle\frac{8}{3}\frac{(d-4)(d-3)(d-2)}{d-1}\frac{\pi^{2}}{2^{\frac{1}{d-2}}}\sin\left(\frac{\pi}{2(d-2)}\right)\frac{\Gamma\left(\frac{1}{d-2}\right)}{\left[\Gamma\left(\frac{d-1}{2d-4}\right)\right]^{4}}.$

This expression has exactly the same form as the one previously obtained in
(83), just with $\Pi_{\textsf{T}}(d)$ replaced by $\Pi_{\textsf{S}}(d)$. All
the results we previously obtained for the tensorial perturbations remain
therefore valid for the test scalar field, just replacing
$\Pi_{\textsf{T}}(d)$ by $\Pi_{\textsf{S}}(d)$. For the quasinormal
frequencies we have then

$\frac{\omega}{T_{\mathcal{H}}}=\left(\ln(3)+(2k+1)\pi
i\right)\left[1+\lambda\frac{16\pi^{2}}{(d-3)^{2}}T_{\mathcal{H}}^{2}\left(\frac{(d-1)(d-4)}{4}+\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\left[\frac{2k+1}{4}\right]^{\frac{1}{d-2}}\frac{\Pi_{\textsf{S}}(d)}{4\pi}\,\mathrm{e}^{-\frac{3\pi
i}{2(d-2)}}\right)\right],$ (106)

with $k\in\mathbb{N}$ and $\Pi_{\textsf{S}}(d)$ given by (105).

We have evaluated
$\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\frac{\Pi_{\textsf{S}}(d)}{4\pi}$
numerically for the relevant values of $d$. This factor grows monotonically
with $d$, varying from approximately 0.58 (corresponding to $d=5$) to
approximately 14.48 (corresponding to $d=10$). These values are of the same
order of magnitude of those we obtained for the frequencies corresponding to
tensorial perturbations (with $\Pi_{\textsf{T}}(d)$) in the discussion
following (99). This means that all the conclusions we got in such discussion
corresponding to tensorial perturbations (namely in section 3.9) are also
valid for test scalar fields. In particular the equivalent to the asymptotic
limit (102) is also valid in this case in the same conditions it was obtained:

$\frac{\omega}{T_{\mathcal{H}}}=\ln(3)+(2k+1)\pi
i+\lambda\left(\frac{4\pi}{d-3}\right)^{2}T_{\mathcal{H}}^{2}\left[\frac{d-3}{d-2}\frac{(2k+1)}{4}\right]^{\frac{d-1}{d-2}}\Pi_{\textsf{S}}(d)\,\mathrm{e}^{\frac{d-5}{2(d-2)}\pi
i}.$ (107)

##  5 Conclusions and future directions

In this work we have computed analytically the quasinormal frequencies in the
highly damped limit corresponding to tensorial gravitational perturbations and
scalar test fields for the simplest case of a $d-$dimensional spherically
symmetric black hole solution with leading string $\alpha^{\prime}$
corrections given by the Callan-Myers-Perry black hole (9). In both cases, we
have obtained conditions that restrict the quasinormal frequencies $\omega$ to
an infinite but countable set.

$\alpha^{\prime}$ corrections are expressed through a transcendental (in
$\omega$) term in those conditions, that we may solve perturbatively in
$\omega$, obtaining a closed form of the final result. In the Einstein limit,
both the real and the imaginary parts of the quasinormal frequencies are
independent of the spacetime dimension $d$, but the $\alpha^{\prime}$
corrections we obtained depend strongly on $d$. We checked that the magnitude
of these $\alpha^{\prime}$ corrections increases with $d$.

The $\alpha^{\prime}$ corrections we obtained grow with the mode number $k$,
which means the allowed values of $k$ should be large (in order to verify the
highly damped limit we have assumed), but cannot be taken arbitrarily large,
or the perturbative expansion will break down. Taking arbitrarily large values
of $k$ does not have an immediate practical relevance; we considered that
asymptotic limit only as a partial analysis. Assuming the higher order in
$\alpha^{\prime}$ corrections also to grow with the mode number $k$, a
complete analysis of the asymptotic limit should include the complete
contributions from all such orders.

Since the value of $k$ is large but not arbitrarily large, corrections in
powers of $1/k$ to the highly damped quasinormal frequencies could also be
included. Corrections of that type have been studied in Musiri:2003bv ;
Cardoso:2003vt , and those results should be considered together with the ones
we obtain in this article.

In the highly damped limit, the $\alpha^{\prime}$ correction terms we obtained
are different for quasinormal frequencies of tensorial gravitational
perturbations and quasinormal frequencies of scalar test fields. Therefore,
under the stringy correction considered, the highly damped limits of the
associated quasinormal spectra are different. This situation is similar to the
eikonal limit, where we also found different results for quasinormal
frequencies of tensorial gravitational perturbations and scalar test fields
Moura:2021eln . However, in Einstein gravity all quasinormal frequencies (for
tensorial, vectorial and scalar gravitational perturbations and for scalar
test fields as well) are equal in the eikonal and in the highly damped limits.
Stringy $\alpha^{\prime}$ corrections, therefore, allow us to distinguish
between the quasinormal spectra of tensorial gravitational perturbations and
scalar test fields in the eikonal and highly damped limits. It would be very
interesting to check if vectorial and scalar gravitational perturbations also
exhibit a similar behaviour in the stringy solution we considered. That is,
should the quasinormal spectra associated with these perturbations be affected
under the stringy correction as well and, if so, should they differ from the
spectra associated with the remaining perturbations in the eikonal and highly
damped limits?

In order to answer these questions, one must study vectorial and scalar
gravitational perturbations in the Callan-Myers-Perry black hole and then
obtain their quasinormal spectra in the highly damped and eikonal limits.
These are topics for future works.

##### Acknowledgements

This work has been supported by Fundação para a Ciência e a Tecnologia under
contracts IT (UIDB/50008/2020 and UIDP/50008/2020) and project CERN/FIS-
PAR/0023/2019.

##  Appendix A The functions $\phi_{1},\phi_{2},\phi_{3}$

We start by writing down explicit expressions for the functions
$\phi_{1}(z),\phi_{2}(z),\phi_{3}(z)$ defined in (66), with $\rho$ defined in
(62). $\phi_{1}(z)$ is given by

$\displaystyle\phi_{1}(z)$ $\displaystyle=$
$\displaystyle\left(\Omega_{1}^{-}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{-\frac{j}{2}}(\omega
z)\right)\left(j^{2}-4\omega^{2}z^{2}-1\right)z^{\rho}dz$ (108)
$\displaystyle+$
$\displaystyle\left(\Omega_{1}^{+}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega
z)\right)\left(j^{2}-4\omega^{2}z^{2}-1\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{1}^{-}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)\left(j^{2}-4\omega^{2}z^{2}-1\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{1}^{+}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)\left(j^{2}-4\omega^{2}z^{2}-1\right)z^{\rho}dz,$

with $\Upsilon_{1}$ defined in (63) and

$\Omega^{\pm}_{1}(d,j,\omega,R_{H})=\pm\frac{\pi\Upsilon_{1}R_{H}^{\frac{d-1}{d-2}}}{8\omega}\csc\left(\frac{\pi
j}{2}\right).$ (109)

$\phi_{2}(z)$ is given by

$\displaystyle\phi_{2}(z)$ $\displaystyle=$
$\displaystyle\left(\Omega_{2}^{-}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{-\frac{j}{2}}(\omega
z)\right)z^{\rho}dz$ (110) $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{+}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{-}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{+}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{-}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}-1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{+}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}-1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{-}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{\frac{j}{2}-1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{2}^{+}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{\frac{j}{2}-1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle-$
$\displaystyle\left(\Omega_{2}^{-}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}+1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle-$
$\displaystyle\left(\Omega_{2}^{+}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}+1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle-$
$\displaystyle\left(\Omega_{2}^{-}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{\frac{j}{2}+1}(\omega z)\right)\omega z^{\rho+1}dz$ $\displaystyle-$
$\displaystyle\left(\Omega_{2}^{+}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{\frac{j}{2}+1}(\omega z)\right)\omega z^{\rho+1}dz,$

with $\Upsilon_{2}$ defined in (64) and

$\Omega^{\pm}_{2}(d,j,\omega,R_{H})=\pm\frac{\pi\Upsilon_{2}R_{H}^{\frac{d-1}{d-2}}}{4\omega}\csc\left(\frac{\pi
j}{2}\right).$ (111)

Finally $\phi_{3}(z)$ is given by

$\displaystyle\phi_{3}(z)$ $\displaystyle=$
$\displaystyle\left(\Omega_{3}^{-}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{-\frac{j}{2}}(\omega
z)\right)z^{\rho}dz$ (112) $\displaystyle+$
$\displaystyle\left(\Omega_{3}^{+}A_{-}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{3}^{-}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)z^{\rho}dz$ $\displaystyle+$
$\displaystyle\left(\Omega_{3}^{+}A_{+}\right)\left(\sqrt{2\pi}\sqrt{\omega
z}J_{\frac{j}{2}}(\omega z)\right)\int\left(\sqrt{\omega
z}J_{-\frac{j}{2}}(\omega z)\right)\left(\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)\right)z^{\rho}dz,$

with $\Upsilon_{3}$ defined in (65) and

$\Omega_{3}^{\pm}(d,j,\omega,R_{H})=\pm\frac{\pi\Upsilon_{3}R_{H}^{\frac{d-1}{d-2}}}{2\omega}\csc\left(\frac{\pi
j}{2}\right).$ (113)

In order to evaluate these functions, we need to study the following class of
indefinite integrals:

$\mathcal{P}_{mnk}\left(x\right)=\int x^{k}J_{m}(x)J_{n}(x)dx,$ (114)

for $m,n\in\mathbb{R}$ and $k<0$. These integrals evaluate to

$\mathcal{P}_{mnk}\left(x\right)=\frac{2^{-m-n}x^{k+m+n+1}\,_{3}F_{4}\left(\frac{m+n+1}{2},\frac{m+n+2}{2},\frac{k+m+n+1}{2};m+1,\frac{k+m+n+3}{2},n+1,m+n+1;-x^{2}\right)}{\Gamma(m+1)\Gamma(n+1)(k+m+n+1)},$
(115)

where

$\,{}_{3}F_{4}\left(a,b,c;d,e,f,g;w\right)=\sum_{n=0}^{+\infty}\frac{\Gamma(a+n)\Gamma(b+n)\Gamma(c+n)\Gamma(d)\Gamma(e)\Gamma(f)\Gamma(g)}{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(d+n)\Gamma(e+n)\Gamma(f+n)\Gamma(g+n)}\frac{w^{n}}{n!}$

is a generalized hypergeometric function. This expression has the following
asymptotic behavior for $|x|\gg 1$, considering that $k<0$:

$\mathcal{P}_{mnk}\left(x\right)\sim\mathcal{H}(m,n,k)+\mathcal{O}(x^{k}),$
(116)

where

$\mathcal{H}(m,n,k):=\frac{\Gamma\left(\frac{1}{2}-\frac{k}{2}\right)\Gamma\left(-\frac{k}{2}\right)\Gamma\left(\frac{k}{2}+\frac{m}{2}+\frac{n}{2}+\frac{1}{2}\right)}{2\sqrt{\pi}\Gamma\left(-\frac{k}{2}+\frac{m}{2}-\frac{n}{2}+\frac{1}{2}\right)\Gamma\left(-\frac{k}{2}+\frac{n}{2}-\frac{m}{2}+\frac{1}{2}\right)\Gamma\left(-\frac{k}{2}+\frac{m}{2}+\frac{n}{2}+\frac{1}{2}\right)}.$
(117)

Finally, as a first approximation we can use the above asymptotic expansion
together with $(\ref{alfa})$ to obtain the asymptotic behavior of
$\phi_{1}(z),\phi_{2}(z),\phi_{3}(z)$ for $|\omega z|\gg 1$:

$\phi_{p}(z)\sim\left(\Theta_{p}^{+}e^{i\alpha_{+}}+\Theta_{p}^{-}e^{i\alpha_{-}}\right)e^{-i\omega
z}+\left(\Theta_{p}^{+}e^{-i\alpha_{+}}+\Theta_{p}^{-}e^{-i\alpha_{-}}\right)e^{i\omega
z},\,p=1,2,3.$ (118)

The coefficients $\Theta_{1,2,3}^{\pm}$ are given in terms of
$\Omega_{1,2,3}^{\pm}$ and of the asymptotic coefficients $\mathcal{H}$
defined in (117) by

$\begin{split}\Theta_{1}^{+}(d,j,\omega,R_{H}):=\Omega^{+}_{1}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\left(j^{2}-1\right)\\\
-4\Omega^{+}_{1}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+3\right)+A_{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+3\right)\right),\end{split}$
(119)

$\begin{split}\Theta_{1}^{-}(d,j,\omega,R_{H}):=\Omega^{-}_{1}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\left(j^{2}-1\right)\\\
-4\Omega^{-}_{1}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+3\right)+A_{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+3\right)\right),\end{split}$
(120)

$\begin{split}\Theta^{+}_{2}(d,j,\omega,R_{H}):=\Omega^{+}_{2}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\\\
+\Omega^{+}_{2}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2}-1,\rho+2\right)+A_{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2}-1,\rho+2\right)\right)\\\
-\Omega_{2}^{+}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},1-\frac{j}{2},\rho+2\right)+A^{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2}+1,\rho+2\right)\right),\end{split}$
(121)

$\begin{split}\Theta^{-}_{2}(d,j,\omega,R_{H}):=\Omega^{-}_{2}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\\\
+\Omega^{-}_{2}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2}-1,\rho+2\right)+A_{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2}-1,\rho+2\right)\right)\\\
-\Omega_{2}^{-}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},1-\frac{j}{2},\rho+2\right)+A^{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2}+1,\rho+2\right)\right),\end{split}$
(122)

$\Theta_{3}^{+}(d,j,\omega,R_{H}):=\Omega^{+}_{3}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right),$
(123)

$\Theta_{3}^{-}(d,j,\omega,R_{H}):=\Omega^{-}_{3}\omega^{-\rho-1}\left(A_{-}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right).$
(124)

After looking at these coefficients, we notice that discarding terms of order
$\mathcal{O}(x^{k})$ in the asymptotic expansion (116) amounted to discard
terms proportional to at most $\omega^{\rho+3}=\omega^{-\frac{1}{d-2}}$.

These coefficients share the same overall factor
$\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}$: they all obey the relation

$\Theta_{1,2,3}^{\pm}(d,j,\omega,R_{H})=\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}\Theta_{1,2,3}^{\pm}(d,j,1,1).$
(125)

We notice that, since we took $k<0$ in the definition (117),
$\Theta_{1}^{\pm}$ are defined only for $\rho<-3$.

##  Appendix B Monodromy of $\psi$ near the origin under a $3\pi$ rotation in
$z$-plane

Here we compute the monodromy of $\psi(z)$ under a $3\pi$ rotation around the
origin of the complex $z$-plane, or analogously a $\frac{3\pi}{d-2}$ rotation
around the origin in the complex $r$-plane. Under such rotation, the monodromy
of the Bessel functions is

$\sqrt{\pi e^{3\pi i}z}J_{\pm\frac{j}{2}}(\omega e^{3\pi
i}z)=e^{6i\alpha_{\pm}}\sqrt{\pi z}J_{\pm\frac{j}{2}}(\omega z).$ (126)

It is easy to obtain the monodromy of $\psi_{0}$ from (47) and (126). The
asymptotic behavior of $\psi_{0}$, which was originally given by (49), after
the rotation is given by

$\psi_{0}(z)\sim\left(A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}\right)e^{-i\omega
z}+\left(A_{+}e^{7i\alpha_{+}}+A_{-}e^{7i\alpha_{-}}\right)e^{i\omega z}.$
(127)

In order to obtain the monodromy of $\psi_{1}$ after the same rotation, we
have to study the functions $\phi_{1}(z),\phi_{2}(z),\phi_{3}(z)$ given
respectively by (112), (110), (108) and the indefinite integral
$\mathcal{P}_{mnk}(z)$ given by (114). From the result for this indefinite
integral obtained at (115), we see that we can write

$\mathcal{P}_{mnk}(z)=z^{k+m+n+1}\mathcal{A}(z).$ (128)

where $\mathcal{A}$ is an even and analytic function of $z$ near the origin.
Therefore, after a $3\pi$ rotation around the origin of the complex $z$-plane,
we can associate to $\mathcal{P}_{mnk}(z)$ the monodromy

$\mathcal{P}_{mnk}(z)\to
e^{3i\pi\left(k+m+n+1\right)}z^{k+m+n+1}\mathcal{A}(z)=e^{3i\pi\left(m+n+k+1\right)}\mathcal{P}_{mnk}(z).$
(129)

We recall that asymptotically $\mathcal{P}_{mnk}(z)$ behaves as
$\mathcal{H}_{mnk}$ given by (117), and $\phi_{1},\phi_{2},\phi_{3}$ behave as
(118).

After completing the $3\pi$ rotation around the origin of the complex
$z$-plane, and taking into consideration the monodromy of the Bessel functions
(126), $\phi_{1}(z),\phi_{2}(z),\phi_{3}(z)$ have the following asymptotic
expansions for $|\omega z|\gg 1$:

$\phi_{p}(z)\sim\left(\Xi_{p}^{+}e^{5i\alpha_{+}}+\Xi_{p}^{-}e^{5i\alpha_{-}}\right)e^{-i\omega
z}+\left(\Xi_{p}^{+}e^{7i\alpha_{+}}+\Xi_{p}^{-}e^{7i\alpha_{-}}\right)e^{i\omega
z},\,p=1,2,3.$ (130)

where, using the coefficients $\Omega_{1,2,3}^{\pm}$ defined in appendix (A),
we further define the coefficients

$\displaystyle\Xi_{1}^{+}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{+}_{1}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2-j)}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3\pi
i(\rho+2)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\left(j^{2}-1\right)$
$\displaystyle-$ $\displaystyle
4\Omega^{+}_{1}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+4-j)}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+3\right)+A_{+}e^{3\pi
i(\rho+4)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+3\right)\right),$
$\displaystyle\Xi_{1}^{-}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{-}_{1}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2)}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3\pi
i(\rho+2+j)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right)\left(j^{2}-1\right)$
$\displaystyle-$ $\displaystyle
4\Omega^{-}_{1}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+4)}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+3\right)+A_{+}e^{3\pi
i(j+\rho+4)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+3\right)\right),$
$\displaystyle\Xi^{+}_{2}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{+}_{2}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2-j)}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3\pi
i(\rho+2)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right)$
$\displaystyle+$
$\displaystyle\Omega^{+}_{2}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2-j)}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2}-1,\rho+2\right)+A_{+}e^{3\pi
i(\rho+2)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2}-1,\rho+2\right)\right)$
$\displaystyle-$
$\displaystyle\Omega_{2}^{+}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+4-j)}\mathcal{H}\left(-\frac{j}{2},1-\frac{j}{2},\rho+2\right)+A_{+}e^{3\pi
i(\rho+4)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2}+1,\rho+2\right)\right),$
$\displaystyle\Xi^{-}_{2}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{-}_{2}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2)}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3\pi
i(j+\rho+2)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right)$
$\displaystyle+$
$\displaystyle\Omega^{-}_{2}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+2)}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2}-1,\rho+2\right)+A_{+}e^{3\pi
i(j+\rho+2)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2}-1,\rho+2\right)\right)$
$\displaystyle-$
$\displaystyle\Omega_{2}^{-}\omega^{-\rho-1}\left(A_{-}e^{3\pi
i(\rho+4)}\mathcal{H}\left(\frac{j}{2},1-\frac{j}{2},\rho+2\right)+A_{+}e^{3\pi
i(\rho+4+j)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2}+1,\rho+2\right)\right),$
$\displaystyle\Xi_{3}^{+}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{+}_{3}\omega^{-\rho-1}\left(A_{-}e^{3i\pi(\rho-j+2)}\mathcal{H}\left(-\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3\pi
i(\rho+2)}\mathcal{H}\left(-\frac{j}{2},\frac{j}{2},\rho+1\right)\right),$
$\displaystyle\Xi_{3}^{-}(d,j,\omega,R_{H})$ $\displaystyle=$
$\displaystyle\Omega^{-}_{3}\omega^{-\rho-1}\left(A_{-}e^{3i\pi\left(\rho+2\right)}\mathcal{H}\left(\frac{j}{2},-\frac{j}{2},\rho+1\right)+A_{+}e^{3i\pi\left(\rho+2+j\right)}\mathcal{H}\left(\frac{j}{2},\frac{j}{2},\rho+1\right)\right).$
(131)

We must consider the redefinition (69) and the contribution of $\psi^{*}$,
whose asymptotic behavior after the $3\pi$ rotation is similar to the one of
$\psi_{0}$, given by (127), but with the coefficients $A_{\pm}$ replaced by
$B_{\pm}$ given by (74). This way we are led to the definition

$\Lambda^{\pm}_{F}(d,j,\omega,R_{H})=\left(R_{H}\omega\right)^{\frac{d-1}{d-2}}\sum_{k=1}^{3}\Xi_{k}^{\pm}(d,j,1,1)+B_{\pm}.$
(132)

Combining (130), (132) and the asymptotic behavior of $\psi^{*}$, we get for
the asymptotic behavior of $\psi_{1}$ after a $3\pi$ rotation around the
origin of the complex $z$-plane

$\psi_{1}(z)\sim\left(\Lambda_{F}^{+}e^{5i\alpha_{+}}+\Lambda_{F}^{-}e^{5i\alpha_{-}}\right)e^{-i\omega
z}+\left(\Lambda_{F}^{+}e^{7i\alpha_{+}}+\Lambda_{F}^{-}e^{7i\alpha_{-}}\right)e^{i\omega
z}.$ (133)

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