10

YITP-21-23

# Axion Cloud Decay due to the Axion-photon Conversion
with Background Magnetic Fields

Chul-Moon Yoo<EMAIL_ADDRESS>Division of Particle and
Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan  Atsushi Naruko  Center for
Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto
University, Kyoto 606-8502, Japan  Yusuke Sakurai  Division of Particle and
Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan  Keitaro Takahashi  Kumamoto
University, Graduate School of Science and Technology, Kumamoto, 860-8555,
Japan  International Research Organization for Advanced Science and
Technology, Kumamoto University, Kumamoto 860-8555, Japan  National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Yohsuke Takamori  National Institute of Technology (KOSEN), Wakayama College,
Gobo, Wakayama 644-0023, Japan  Daisuke Yamauchi  Faculty of Engineering,
Kanagawa University, Kanagawa-ku, Yokohama-shi, Kanagawa, 221-8686, Japan

###### Abstract

We consider an axion cloud around a black hole with background magnetic
fields. We calculate the decay rate of the axion cloud due to the axion-photon
conversion associated with the axion-photon coupling. For simplicity, we
consider the situation where the axion configuration is dominated by a
solution for the eigenvalue equation equivalent to that for the Hydrogen atom,
and the coupling term can be evaluated by a successive perturbation method.
For the monopole background, we find the decay rate of the axion cloud is
given by $\sim q^{2}\kappa^{2}(GM)^{5}\mu^{8}$, where $\mu$, $M$, $G$,
$\kappa$ and $q$ are the axion mass, black hole mass, gravitational constant,
coupling constant of the axion-photon coupling and monopole charge,
respectively. For the uniform background magnetic field, we obtain the decay
rate of the axion cloud $\sim B_{0}^{2}\kappa^{2}(GM)^{7}\mu^{6}$, where
$B_{0}$ is the magnetic field strength. Applying our formula to the central
black hole in our galaxy, we find that the value of the decay rate for the
case of the uniform magnetic field is comparable to the growth rate of the
superradiant instability with $\kappa\sim 10^{-12}{\rm GeV^{-1}}$, $B_{0}\sim
10^{3}{\rm G}$ and $\mu\sim 10^{-18}{\rm eV}$. The ratio is $10^{5}$ times
larger for the monopole magnetic field with the same values of the parameters.

##  I Introduction

Recently, axion-like particles(ALPs) have been attracting much interest in
cosmology, astrophysics and particle physics. The axion was originally
proposed as a solution of the strong CP problem in quantum chromodynamics
(QCD) Peccei:1977hh ; Peccei:1977ur ; Weinberg:1977ma ; Wilczek:1977pj , and
the possibility of ALPs in string theory was pointed out in late years
Svrcek:2006yi ; Arvanitaki:2009fg . ALPs including the QCD axion are
attractive candidates for dark matter Preskill:1982cy ; Dine:1982ah ;
Abbott:1982af . Various experimental and observational searches for ALPs are
ongoing, and the constraints are actively updated (see, e.g. Zyla:2020zbs  for
a summary of constraints).

One interesting phenomenon associated with the existence of an ALP is
formation of the axion cloud surrounding a black hole Arvanitaki:2010sy . For
a massive bosonic field around a rotating black hole, superradiant instability
takes placeZouros:1979iw ; Detweiler:1980uk ; Brito:2015oca , and the specific
modes of the massive field exponentially grow with time. The growing time
scale for the massive scalar field is analytically estimated in Refs.
Zouros:1979iw ; Detweiler:1980uk  under certain approximations and numerically
calculated in Ref. Dolan:2007mj . Possible observable phenomena and those
relevance have been discussed in a number of papers Cardoso:2011xi ;
Yoshino:2012kn ; Yoshino:2013ofa ; Brito:2014wla ; Yoshino:2015nsa ;
Arvanitaki:2014wva ; Brito:2017zvb ; Baumann:2018vus ; Zhang:2018kib ;
Zhang:2019eid ; Baumann:2019ztm ; Ding:2020bnl ; Omiya:2020vji . If the
Compton wavelength of an ALP is similar to the gravitational radius of a
rotating black hole, the angular momentum of the black hole may be efficiently
extracted and exhausted by the superradiant instability within the cosmic
history. Therefore the existence of a rotating black hole may give a
constraint on the existence of an ALP in the corresponding mass range
Stott:2020gjj . However, once we consider more general situations, there may
be an efficient decaying process which spoils the persistent growth of the
cloud. For instance, the existence of a binary compact object may dissolve the
cloud due to the level transitions Takahashi . In this paper, we focus on the
dissipative effect through the coupling with the electro-magnetic field.

An ALP may have a coupling with the electro-magnetic field as the Chern-Simons
coupling. Many axion search experiments rely on this coupling and gave
constraints on the magnitude of the coupling constant Anastassopoulos:2017ftl
; Boutan:2018uoc ; Ouellet:2018beu ; Calore:2020tjw ; Salemi:2021gck . The
birefringence due to this coupling term under the existence of the ALP is also
actively discussed Carroll:1989vb ; Carroll:1991zs ; Harari:1992ea ;
Ivanov:2018byi ; Fujita:2018zaj ; Liu:2019brz ; Fedderke:2019ajk ;
Caputo:2019tms ; Chen:2019fsq ; Yuan:2020xui ; Basu:2020gsy . Another
characteristic phenomenon is the axion-photon conversion with a background
magnetic field Maiani:1986md ; Raffelt:1987im , that is, the axion field can
be converted to electro-magnetic waves through the coupling with the
background magnetic field. There is a number of related papers to the axion-
photon conversion Sikivie:1983ip ; Sikivie:1985yu ; Ahonen:1995ky ;
Pshirkov:2007st ; Espriu:2011vj ; Espriu:2014lma ; Huang:2018lxq ;
Hook:2018iia ; Hertzberg:2018zte ; Arza:2018dcy ; Masaki:2019ggg ;
Carenza:2019vzg ; Arza:2020eik ; Leroy:2019ghm ; Buckley:2020fmh . In this
paper, we discuss the effect of the axion-photon conversion on the
growth/decay rate of the axion cloud.

Let us consider a rotating black hole surrounded by a magnetic field and an
ALP cloud. While the ALP has unstable bound states without the electro-
magnetic field, if the ALP has the coupling with the electro-magnetic field,
the axion-photon conversion takes place and the photon can dissipate to
infinity (see also Ref. Boskovic:2018lkj  for effects of a magnetic field on
an axion cloud). The decay due to the axion-photon conversion and its
phenomenological implications are also considered in Ref. Blas:2020nbs .
Therefore we expect the competing effects of the superradiant growth and the
decay due to the axion-photon conversion.111 The effect of a magnetic field on
the axion field through the modification of the background geometry is
discussed in Ref. Brito:2014nja , and the charged scalar field in a magnetized
black hole is discussed in Ref. Turimov:2019afv . However, as far as we know,
the effect of the axion-photon conversion to the stability of the axion cloud
has not been reported yet.  As a first step, we consider the case in which the
typical size of the cloud is much larger than the gravitational radius of the
black hole. Then the spin of the black hole and the absorption through the
horizon can be neglected, that is, the black hole can be treated as a
gravitational source of the point mass. We calculate the decay rate due to the
axion-photon conversion for the dominant mode of the superradiant instability
assuming the axion-photon coupling term is perturbatively small. We note that,
although the spin of the black hole is essential for the superradiant
instability, as will be shown in the text, the effect of the axion-photon
conversion can be estimated independently of the spin of the black hole once
the dominant mode of the axion cloud is specified. The magnetic field
configuration is assumed to be the monopole in Sec. III and uniform
Wald:1974np  in Sec. IV. Finally we compare the decay rate to the growth rate
of the superradiant instability in Sec. V.

Throughout this paper, we use the natural units in which both the speed of
light and the reduced Planck constant are unity, $c=\hbar=1$, and the
gravitational constant is denoted by $G$.

##  II Equations of motion, background and perturbations

Let us consider the axion-electro-magnetic system given by the following
action:

$S=\int\sqrt{-g}\mathrm{d}^{4}x\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\kappa\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-\frac{1}{2}\mu^{2}\phi^{2}\right),$
(1)

where

$\tilde{F}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\lambda\rho}F_{\lambda\rho}$
(2)

with $\varepsilon$ being the Levi-Civita tensor, and we neglected the non-
linear self-interaction of the axion field. From the variation with respect to
the axion field, we obtain the following equation of motion for the axion:

$\left(\nabla_{\mu}\nabla^{\mu}-\mu^{2}\right)\phi=\frac{1}{4}\kappa
F_{\mu\nu}\tilde{F}^{\mu\nu}.$ (3)

The equations of motion for the gauge field are given by

$\nabla_{\mu}F^{\mu\nu}=-\kappa\tilde{F}^{\mu\nu}\nabla_{\mu}\phi,$ (4)

where we have used the following identity:

$\nabla_{\mu}\tilde{F}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\lambda}\nabla_{\mu}F_{\rho\lambda}=0.$
(5)

For the background geometry, we consider the Schwarzschild metric with the
mass $M$ given by

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}),$
(6)

where $f(r)=1-2GM/r$.

In the following sections, we will consider perturbations on the two types of
background magnetic field configurations satisfying Eq. (4) with $\phi=0$.
That is, the axion $\phi$ and the gauge fields $A^{\rm tot}_{\mu}$ will given
in the following form:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\delta\phi,$ (7)
$\displaystyle A^{\rm tot}_{\mu}$ $\displaystyle=$ $\displaystyle A^{\rm
bg}_{\mu}+\delta A_{\mu},$ (8)

where $A^{\rm bg}$ is the background gauge field satisfying Eq. (4) with
$\phi=0$, and $\delta\phi$ and $\delta A_{\mu}$ are perturbations. We will
consider the equations of motion for $\delta\phi$ and $\delta A_{\mu}$ at the
linear order.

In the form of the vector spherical harmonics Zerilli:1974ai  and the Fourier
mode expansion with the frequency $\omega$, we can expand the axion field and
each component of the gauge field as

$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle\sum_{lm}\Phi_{lm}Y_{lm}{\rm e}^{\mathrm{i}\omega t},$ (9)
$\displaystyle\delta A_{t}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\sum_{lm}A^{a}_{lm}Y_{lm}{\rm e}^{\mathrm{i}\omega
t},$ (10) $\displaystyle\delta A_{r}$ $\displaystyle=$
$\displaystyle\sum_{lm}A^{b}_{lm}Y_{lm}{\rm e}^{\mathrm{i}\omega t},$ (11)
$\displaystyle\delta A_{\theta}$ $\displaystyle=$
$\displaystyle\sum_{lm}\frac{1}{\sqrt{l(l+1)}}\left(A^{c}_{lm}\partial_{\theta}Y_{lm}+A^{d}_{lm}\frac{1}{\sin\theta}\partial_{\varphi}Y_{lm}\right){\rm
e}^{\mathrm{i}\omega t},$ (12) $\displaystyle\delta A_{\varphi}$
$\displaystyle=$
$\displaystyle\sum_{lm}\frac{1}{\sqrt{l(l+1)}}\left(A^{c}_{lm}\partial_{\varphi}Y_{lm}-A^{d}_{lm}\sin\theta\partial_{\theta}Y_{lm}\right){\rm
e}^{\mathrm{i}\omega t},$ (13)

where, $Y_{lm}$ is the spherical harmonic function of degree $l$ and order
$m$, and $\Phi_{lm}$, $A^{a}_{lm}$, $A^{b}_{lm}$, $A^{c}_{lm}$ and
$A^{d}_{lm}$ are functions of $r$. The signs of $A^{a}_{lm}$, $A^{b}_{lm}$ and
$A^{c}_{lm}$ change as $(-1)^{l}$ for the parity transformation
$(\theta,\varphi)\rightarrow(-\theta,\varphi+\pi)$ and the sign of
$A^{d}_{lm}$ changes as $(-1)^{l+1}$, and they are called even and odd parity
modes, respectively.

##  III Analysis with the monopole magnetic field

###  III.1 perturbation equations

First, as the simplest case, let us consider the following perturbation around
the monopole magnetic field:

$\displaystyle A^{\rm tot}_{\mu}$ $\displaystyle=$ $\displaystyle
q(1-\cos\theta)(\mathrm{d}\varphi)_{\mu}+\delta A_{\mu}.$ (14)

Since the monopole field does not violate the spherical symmetry, the odd
parity modes and the even parity modes are decoupled with each other, and only
the even parity modes can be coupled with the axion field equation. Since each
mode labelled by $lm$ decouples from the other modes, we just focus on the
single mode specified by $l$ and $m$, and omit the labels $l$ and $m$ for
notational simplicity in this section.

We fix the gauge as $A^{c}=0$ Zerilli:1974ai . The equations of motion for the
gauge field are given by

$\displaystyle l(l+1)A^{a}-f\partial_{r}(r^{2}\partial_{r}A^{a})-\omega
f\partial_{r}(r^{2}A^{b})$ $\displaystyle=$
$\displaystyle-\mathrm{i}fq\kappa\partial_{r}\Phi,$ (15) $\displaystyle-
fl(l+1)A^{b}+\omega r^{2}(\partial_{r}A^{a}+\omega A^{b})$ $\displaystyle=$
$\displaystyle\mathrm{i}q\kappa\omega\Phi,$ (16) $\displaystyle
f\partial_{r}(fA^{b})-\omega A^{a}$ $\displaystyle=$ $\displaystyle 0.$ (17)

Substituting Eq. (17) into Eq. (16), we find

$-fl(l+1)A^{b}+r^{2}\partial_{r}\left[f\partial_{r}(fA^{b})\right]+r^{2}\omega^{2}A^{b}=\mathrm{i}q\kappa\omega\Phi.$
(18)

Eq. (15) is redundant. The axion field equation is given by

$\frac{f}{r^{2}}\partial_{r}\left(r^{2}f\partial_{r}\Phi\right)+\omega^{2}\Phi-f\mu^{2}\Phi-f\frac{l(l+1)}{r^{2}}\Phi=\frac{1}{4}\kappa
fF_{\mu\nu}\tilde{F}^{\mu\nu},$ (19)

where $F_{\mu\nu}\tilde{F}^{\mu\nu}$ is given by

$F_{\mu\nu}\tilde{F}^{\mu\nu}=-\frac{4\mathrm{i}q}{r^{2}}\left(\partial_{r}A^{a}+\omega
A^{b}\right)=-\mathrm{i}\frac{4q}{\omega
r^{2}}\left[\partial_{r}(f\partial_{r}(fA^{b}))+\omega^{2}A^{b}\right].$ (20)

Therefore we obtain the two coupled equations (18) and (19) as master
equations.

###  III.2 Approximate Analysis in analogy with the Hydrogen atom

In order to analytically evaluate the equations, we impose the following
conditions:

$\displaystyle 1/(GM)\gg\omega\sim\mu\gg 1/r,$ (21) $\displaystyle q\kappa\ll
r,$ (22)

where $r$ denotes the typical length scale of the system. The last condition
indicates that the coupling terms are relatively small and can be evaluated by
using a successive approximation. Under the condition (21), the master
equations (18) and (19) can be reduced to

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}A^{b}+\omega^{2}A^{b}-\frac{l(l+1)}{r^{2}}A^{b}$
$\displaystyle=$ $\displaystyle\mathrm{i}\frac{\omega q\kappa}{r^{2}}\Phi,$
(23)
$\displaystyle\frac{1}{r^{2}}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{\mathrm{d}}{\mathrm{d}r}\Phi\right)+\left(\omega^{2}-\mu^{2}+\frac{2GM}{r}\mu^{2}-\frac{l(l+1)}{r^{2}}\right)\Phi$
$\displaystyle=$ $\displaystyle-\mathrm{i}\frac{q\kappa}{\omega
r^{2}}\left[\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}A^{b}+\omega^{2}A^{b}\right].$
(24)

Combining these two equations, we can rewrite Eq. (24) as

$\frac{1}{r^{2}}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{\mathrm{d}}{\mathrm{d}r}\Phi\right)+\left(\omega^{2}-\mu^{2}+\frac{2GM\mu^{2}}{r}-\frac{l(l+1)}{r^{2}}\right)\Phi=\frac{q^{2}\kappa^{2}}{r^{4}}\Phi-\mathrm{i}\frac{q\kappa
l(l+1)}{\omega r^{4}}A^{b}.$ (25)

Hereafter we successively evaluate Eqs. (23) and (25) setting the lowest order
equation as

$\frac{\mathrm{d}^{2}\Phi_{0}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\Phi_{0}}{\mathrm{d}r}+\left(\omega_{0}^{2}-\mu^{2}\right)\Phi_{0}+\frac{2GM\mu^{2}}{r}\Phi_{0}-\frac{l(l+1)}{r^{2}}\Phi_{0}=0.$
(26)

This equation is equivalent to the equation for the Hydrogen atom. The radius
$a_{0}$ corresponding to the Bohr radius is given by

$a_{0}=1/(GM\mu^{2}).$ (27)

This radius gives the typical radius of the system. Let us focus on the
$l=m=1$ and $n=2$ mode since it has the largest growth rate in the context of
the superradiant instability, where $n$ is the principal quantum number. The
solution for $l=1$ and $n=2$ mode is given by

$\Phi_{0}=\omega_{0}\left(\frac{1}{2a_{0}\omega_{0}}\right)^{3/2}\frac{r}{\sqrt{3}a_{0}}\exp\left(-\frac{r}{2a_{0}}\right),$
(28)

and the frequency is given by

$\omega_{0}^{2}-\mu^{2}=-\frac{1}{4a_{0}^{2}},$ (29)

where we have normalized $\Phi_{0}$ as
$\omega_{0}\int\mathrm{d}rr^{2}\Phi_{0}^{2}=1$ keeping the dimension of
$\Phi_{0}$ as the mass dimension one.

As for Eq. (23), using the variable $\mathcal{A}:=A^{b}/(\omega_{0}r)$, we
obtain

$\frac{\mathrm{d}^{2}\mathcal{A}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}r}+\omega_{0}^{2}\mathcal{A}-\frac{l(l+1)}{r^{2}}\mathcal{A}=\mathrm{i}\frac{q\kappa}{r^{3}}\Phi_{0},$
(30)

where we neglected the higher order contributions, and replaced $\Phi$ and
$\omega$ by $\Phi_{0}$ and $\omega_{0}$, respectively. By using the
dimensionless variable $x:=\omega r$, this equation can be rewritten as

$\frac{\mathrm{d}}{\mathrm{d}x}\left(x^{2}\frac{\mathrm{d}}{\mathrm{d}x}\mathcal{A}\right)+x^{2}\mathcal{A}-l(l+1)\mathcal{A}=\mathrm{i}q\kappa\omega_{0}x^{-1}\Phi_{0}.$
(31)

The homogeneous solution can be described by the spherical Bessel functions
$j_{l}(x)$ and $n_{l}(x)$. The solution which is regular at the origin is
given by $j_{l}$. We note that the absorption through the black hole horizon
has been neglected under the assumption (21). For the boundary condition at
infinity, since there is no outer potential barrier, we should take the
outgoing boundary condition. Thus the behavior at infinity should be given by
$\sim{\rm e}^{-ix}$. This boundary condition can be satisfied by the second
spherical Hankel function defined by $h_{l}(x):=j_{l}-\mathrm{i}n_{l}$. Then
the appropriate Green’s function can be written as

$\mathcal{G}(x,\xi)=\Theta(\xi-x)j_{l}(x)h_{l}(\xi)/W+\Theta(x-\xi)j_{l}(\xi)h_{l}(x)/W,$
(32)

where $\Theta$ is the Heviside’s step function and
$W=x^{2}[\partial_{x}j_{l}(x)h_{l}(x)-j_{l}(x)\partial_{x}h_{l}(x))]$. For a
given $\Phi_{0}$, the solution of $\mathcal{A}$ can be given by

$\mathcal{A}(x)=-\mathrm{i}q\kappa\omega_{0}\int^{\infty}_{0}\mathrm{d}\xi\mathcal{G}(x,\xi)\xi^{-1}\Phi_{0}(\xi).$
(33)

For $l=1$, we have

$j_{1}(x)=\frac{\sin x}{x^{2}}-\frac{\cos
x}{x},~{}h_{1}(x)=\frac{\mathrm{i}-x}{x^{2}}{\rm e}^{-\mathrm{i}x}.$ (34)

Then $W=\mathrm{i}$ and the real part of $\mathcal{A}$ is given by

${\rm
Re}\mathcal{A}(x)=-q\kappa\omega_{0}j_{1}(x)\int^{\infty}_{0}\mathrm{d}\xi
j_{1}(\xi)\xi^{-1}\Phi_{0}(\xi).$ (35)

The 1-st order equation for the axion field is given by

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}r}+\left(\omega_{0}^{2}-\mu^{2}\right)\Phi_{1}+2\omega_{0}\omega_{1}\Phi_{0}+\frac{2}{a_{0}r}\Phi_{1}-\frac{l(l+1)}{r^{2}}\Phi_{1}=\frac{q^{2}\kappa^{2}}{r^{4}}\Phi_{0}-\mathrm{i}\frac{q\kappa
l(l+1)}{r^{3}}\mathcal{A}.$ (36)

Multiplying $r^{2}\Phi_{0}$ and integrating by parts, we obtain

$2\omega_{0}\omega_{1}\int^{\infty}_{0}\mathrm{d}rr^{2}\Phi_{0}^{2}=q^{2}\kappa^{2}\int^{\infty}_{0}\mathrm{d}r\frac{\Phi_{0}^{2}}{r^{2}}-\mathrm{i}q\kappa
l(l+1)\int^{\infty}_{0}\mathrm{d}r\frac{\Phi_{0}\mathcal{A}}{r}.$ (37)

Then the imaginary part of $\omega_{1}$ can be calculated as

$\displaystyle{\rm Im}\omega_{1}$ $\displaystyle=$
$\displaystyle-q\kappa\int^{\infty}_{0}\mathrm{d}\xi\frac{\Phi_{0}(\xi)}{\xi}{\rm
Re}\mathcal{A}(\xi)=q^{2}\kappa^{2}\omega_{0}\left(\int^{\infty}_{0}\mathrm{d}\xi\frac{\Phi_{0}(\xi)}{\xi}j_{1}(\xi)\right)^{2}$
(38) $\displaystyle=$
$\displaystyle\frac{q^{2}\kappa^{2}}{96a_{0}^{3}}\frac{\left(2a_{0}\omega_{0}-{\rm
ArcTan}(2a_{0}\omega_{0})\right)^{2}}{a_{0}^{4}\omega_{0}^{4}}\sim\frac{1}{24}\frac{q^{2}\kappa^{2}}{a_{0}^{2}}\frac{1}{a_{0}^{3}\omega_{0}^{3}}\omega_{0}\sim\frac{1}{24}\mu^{2}q^{2}\kappa^{2}(GM\mu)^{5}\mu,$
(39)

where we estimated the value of $\omega_{0}$ as $\omega_{0}\sim\mu$ in the
last expression. Therefore the imaginary part is positive, and the amplitude
of $\phi$ decays.

##  IV Analysis with the uniform magnetic field

###  IV.1 background and perturbation equations

Let us consider the magnetic field in the Schwarzschild geometry given by
Wald:1974np

$F=B_{0}r\sin^{2}\theta\mathrm{d}r\wedge\mathrm{d}\varphi+B_{0}r^{2}\sin\theta\cos\theta\mathrm{d}\theta\wedge\mathrm{d}\varphi,$
(40)

where $F$ is the two-form field strength. By using the radial coordinate
$\rho:=r\sin\theta$ of the cylindrical coordinates, we obtain

$F=B_{0}r\sin\theta\mathrm{d}\rho\wedge\mathrm{d}\varphi=B_{0}e_{\rho}\wedge
e_{\varphi}.$ (41)

Therefore the field strength describes the uniform magnetic field along the
$z$-direction. The vector potential is given by

$A_{\mu}^{\rm
bg}=\frac{1}{2}B_{0}r^{2}\sin^{2}\theta(\mathrm{d}\varphi)_{\mu}.$ (42)

This field configuration satisfies the field equations with $\phi=0$. Let us
consider the following perturbation:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\delta\phi,$ (43)
$\displaystyle A^{\rm tot}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}B_{0}r^{2}\sin^{2}\theta(\mathrm{d}\varphi)_{\mu}+\delta
A_{\mu}.$ (44)

Hereafter we will take the gauge $A^{c}_{lm}=0$ Zerilli:1974ai  as in the
previous section.

Since the background magnetic field violates the spherical symmetry, multiple
modes are coupled with each other. Therefore the scalar field equation is
given by Eq. (19) with the summation symbol, that is,

$\sum_{lm}\left\\{\left[\frac{f}{r^{2}}\partial_{r}\left(r^{2}f\partial_{r}\Phi_{lm}\right)+\omega^{2}\Phi_{lm}-f\mu^{2}\Phi_{lm}-f\frac{l(l+1)}{r^{2}}\Phi_{lm}\right]Y_{lm}\right\\}=\frac{1}{4}\kappa
fF_{\mu\nu}\tilde{F}^{\mu\nu},$ (45)

where

$\displaystyle F_{\mu\nu}\tilde{F}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4B_{0}}{r}\left[r\cos\theta\left(\partial_{r}\delta
A_{t}-\partial_{t}\delta A_{r}\right)+\sin\theta\left(-\partial_{\theta}\delta
A_{t}+\partial_{t}\delta A_{\theta}\right)\right]$ (46) $\displaystyle=$
$\displaystyle\frac{4\mathrm{i}B_{0}}{r}\sum_{lm}\Biggl{[}-r\cos\theta\left(\partial_{r}A^{a}_{lm}+\omega
A^{b}_{lm}\right)Y_{lm}$ (48) $\displaystyle\hskip 85.35826pt+\sin\theta
A^{a}_{lm}\partial_{\theta}Y_{lm}+\frac{\omega}{\sqrt{l(l+1)}}A^{d}_{lm}\partial_{\varphi}Y_{lm}\Biggr{]}.$

The gauge field equations are given by

$\displaystyle\sum_{l,m}\left\\{\left[f\partial_{r}(r^{2}\partial_{r}A^{a}_{lm})+\omega
f\partial_{r}(r^{2}A^{b}_{lm})-l(l+1)A^{a}_{lm}\right]Y_{lm}\right\\}$ (49)
$\displaystyle\hskip 142.26378pt=\mathrm{i}B_{0}\kappa
rf\sum_{l,m}\left(r\partial_{r}\Phi_{lm}\cos\theta
Y_{lm}-\Phi_{lm}\sin\theta\partial_{\theta}Y_{lm}\right),$ (50)
$\displaystyle\sum_{l,m}\left\\{\left[\omega
r^{2}\partial_{r}A^{a}_{lm}+\omega^{2}r^{2}A^{b}_{lm}-l(l+1)fA^{b}_{lm}\right]Y_{lm}\right\\}$
(51) $\displaystyle\hskip 142.26378pt=\mathrm{i}B_{0}\kappa\omega
r^{2}\cos\theta\sum_{l,m}\Phi_{lm}Y_{lm},~{}~{}$ (52)
$\displaystyle\sum_{l,m}\Bigl{\\{}r^{2}\partial_{\theta}Y_{lm}\left[\omega
A^{a}_{lm}-f\partial_{r}(fA^{b}_{lm})\right]$ (53) $\displaystyle\hskip
42.67912pt+\frac{\partial_{\varphi}Y_{lm}}{\sqrt{l(l+1)}\sin\theta}\left[r^{2}f\partial_{r}(f\partial_{r}A^{d}_{lm})+\omega^{2}r^{2}A^{d}_{lm}-l(l+1)fA^{d}_{lm}\right]\Bigr{\\}}$
(54) $\displaystyle\hskip 142.26378pt=-\mathrm{i}B_{0}\kappa\omega
r^{3}f\sin\theta\sum_{lm}\Phi_{lm}Y_{lm},$ (55)
$\displaystyle\sum_{l,m}\Biggl{\\{}r^{2}\partial_{\varphi}Y_{lm}\left[\omega
A^{a}_{lm}-f\partial_{r}(fA^{b}_{lm})\right]$ (56) $\displaystyle\hskip
42.67912pt-\frac{\sin\theta\partial_{\theta}Y_{lm}}{\sqrt{l(l+1)}}\left[r^{2}f\partial_{r}(f\partial_{r}A^{d}_{lm})+\omega^{2}r^{2}A^{d}_{lm}-l(l+1)fA^{d}_{lm}\right]\Biggr{\\}}=0.$
(57)

Combining Eq. (55) and Eq. (57), we obtain the following equation for
$A^{d}_{lm}$:

$\sum_{l,m}\left\\{\sqrt{l(l+1)}\left[r^{2}f\partial_{r}(f\partial_{r}A^{d}_{lm})+\omega^{2}r^{2}A^{d}_{lm}-l(l+1)fA^{d}_{lm}\right]Y_{lm}\right\\}=\mathrm{i}B_{0}\kappa\omega
r^{3}f\sum_{lm}\Phi_{lm}Y_{lm}.$ (58)

We can obtain the other independent equation from Eq. (55) and Eq. (57) as
follows:

$\sum_{l,m}\left\\{l(l+1)\left[\omega
A^{a}_{lm}-f\partial_{r}(fA^{b}_{lm})\right]Y_{lm}\right\\}=\mathrm{i}B_{0}\kappa\omega
r\sum_{lm}\Phi_{lm}\left(2\cos\theta
Y_{lm}+\sin\theta\partial_{\theta}Y_{lm}\right).$ (59)

Therefore $A^{a}_{lm}$ and $A^{b}_{lm}$ are given by the coupled equations
(52) and (59).

###  IV.2 Approximate Analysis under the condition (21)

Let us impose the condition (21) and

$\kappa B_{0}\ll 1/a_{0}$ (60)

so that we can consider the right-hand side of Eq. (45) is perturbatively
small. Then we can assume that the dominant part of the axion field is given
by the $n=2$, $l=1$ and $m=\pm 1$ mode of the solution for Eq. (26) as in the
previous section, where we have assumed that the uniform magnetic field is
parallel ($m=1$) or anti-parallel ($m=-1$) to the dominant mode. In the
context of the superradiant instability, the magnetic field is parallel (anti-
parallel) to the angular momentum of the black hole for $m=1$ ($m=-1$). The
dominant part of the axion field can be explicitly written as

$\delta\phi_{0}=\Phi_{0}(r){\rm e}^{\mathrm{i}\omega t}Y_{1~{}\pm
1}(\theta,\phi)=\mp\sqrt{\frac{3}{8\pi}}\sin\theta{\rm e}^{\mathrm{i}\omega
t\pm\mathrm{i}\varphi}\Phi_{0}(r),$ (61)

where $\Phi_{0}$ is given in Eq. (28).

From Eq. (58), we find that $A^{d}_{1~{}\pm 1}$ is induced by $\delta\phi_{0}$
through the following equation:

$r^{2}f\partial_{r}(f\partial_{r}A^{d}_{1~{}\pm
1})+\omega^{2}r^{2}A^{d}_{1~{}\pm 1}-2fA^{d}_{1~{}\pm
1}=\mp\frac{1}{\sqrt{2}}B_{0}\kappa\omega r^{3}f\Phi_{0}.$ (62)

From Eqs. (52) and (59), having the expression

$Y_{2~{}\pm 1}=\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta{\rm
e}^{\pm\mathrm{i}\varphi},$ (63)

we find that only $A^{a,b}_{2~{}\pm 1}$ modes can be induced by
$\delta\phi_{0}$. Then, eliminating $A^{a}_{2~{}\pm 1}$ from Eqs. (59) and
(52) with $l=2$ and $m=\pm 1$, we obtain

$-6fA^{b}_{2~{}\pm 1}+r^{2}\partial_{r}(f\partial_{r}(fA_{2~{}\pm
1}^{b}))+r^{2}\omega^{2}A^{b}_{2~{}\pm
1}=\frac{\mathrm{i}}{2\sqrt{5}}B_{0}\kappa\omega
r^{2}\left[2\Phi_{0}-\partial_{r}(rf\Phi_{0})\right],$ (64)

and $A^{a}_{2~{}\pm 1}$ is given by

$A^{a}_{2~{}\pm 1}=\frac{f}{\omega}\partial_{r}(fA^{b}_{2~{}\pm
1})+\frac{\mathrm{i}}{2\sqrt{5}}B_{0}\kappa fr\Phi_{0}.$ (65)

Introducing $\mathcal{B}_{\pm}:=A^{b}_{2~{}\pm 1}/(\omega_{0}r)$ and
$\mathcal{D}_{\pm}:=A^{d}_{1~{}\pm 1}/r$, and assuming the condition (21), we
obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(x^{2}\frac{\mathrm{d}}{\mathrm{d}x}\mathcal{B}_{\pm}\right)+x^{2}\mathcal{B}_{\pm}-6\mathcal{B}_{\pm}=\mathrm{i}\frac{B_{0}\kappa}{2\sqrt{5}\omega_{0}}x\left(\Phi_{0}-x\frac{\mathrm{d}}{\mathrm{d}x}\Phi_{0}\right),$
(66)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(x^{2}\frac{\mathrm{d}}{\mathrm{d}x}\mathcal{D}_{\pm}\right)+x^{2}\mathcal{D}_{\pm}-2\mathcal{D}_{\pm}=\mp\frac{B_{0}\kappa}{\sqrt{2}\omega_{0}}x^{2}\Phi_{0}$
(67)

from Eqs. (62) and (64). Eqs. (66) and (67) can be solved by using the Green’s
function (32). As in the case of Eq. (35), the functions ${\rm
Re}\mathcal{B}_{\pm}$ and ${\rm Im}\mathcal{D}_{\pm}$ can be expressed as

$\displaystyle{\rm Re}\mathcal{B}_{\pm}(x)$ $\displaystyle=$
$\displaystyle\frac{B_{0}\kappa}{2\sqrt{5}\omega_{0}}j_{2}(x)\int^{\infty}_{0}\mathrm{d}\xi\xi
j_{2}(\xi)\left(\xi\partial_{\xi}\Phi_{0}(\xi)-\Phi_{0}(\xi)\right),$ (68)
$\displaystyle{\rm Im}\mathcal{D}_{\pm}(x)$ $\displaystyle=$
$\displaystyle\mp\frac{B_{0}\kappa}{\sqrt{2}\omega_{0}}j_{1}(x)\int^{\infty}_{0}\mathrm{d}\xi\xi^{2}j_{1}(\xi)\Phi_{0}(\xi).$
(69)

For $F_{\mu\nu}\tilde{F}^{\mu\nu}$, we obtain

$\displaystyle F_{\mu\nu}\tilde{F}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4\mathrm{i}B_{0}}{r}\Biggl{[}-r\left(\partial_{r}A^{a}_{2~{}\pm
1}+\omega A^{b}_{2~{}\pm 1}\right)\cos\theta Y_{2~{}\pm 1}+A^{a}_{2~{}\pm
1}\sin\theta\partial_{\theta}Y_{2~{}\pm
1}\pm\frac{\mathrm{i}\omega}{\sqrt{2}}A^{d}_{1~{}\pm 1}Y_{1~{}\pm 1}\Biggr{]}$
(70) $\displaystyle=$ $\displaystyle
4\mathrm{i}B_{0}\Biggl{\\{}\left[-\frac{1}{\sqrt{5}}\left(\partial_{r}A^{a}_{2~{}\pm
1}+\omega A^{b}_{2~{}\pm 1}+\frac{3}{r}A^{a}_{2~{}\pm
1}\right)\pm\frac{\mathrm{i}}{\sqrt{2}r}\omega A^{d}_{1~{}\pm
1}\right]Y_{1~{}\pm 1}$ (72) $\displaystyle\hskip
28.45274pt+\sqrt{\frac{8}{35}}\left(-\partial_{r}A^{a}_{2~{}\pm 1}-\omega
A^{b}_{2~{}\pm 1}+\frac{2}{r}A^{a}_{2~{}\pm 1}\right)Y_{3~{}\pm
1}\Biggr{\\}}.$

Taking the same approximation (21), from the axion field equation (45) with
$l=1$, we obtain the equation for $\Phi_{1}$ and $\omega_{1}$ as follows:

$\displaystyle\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}r}+\left(\omega_{0}^{2}-\mu^{2}\right)\Phi_{1}+2\omega_{0}\omega_{1}\Phi_{0}+\frac{2}{a_{0}r}\Phi_{1}-\frac{2}{r^{2}}\Phi_{1}$
(73) $\displaystyle\hskip 56.9055pt=\mathrm{i}\kappa
B_{0}\left[-\frac{1}{\sqrt{5}}\left(\partial_{r}A^{a}_{2~{}\pm
1}+\omega_{0}A^{b}_{2~{}\pm 1}+\frac{3}{r}A^{a}_{2~{}\pm
1}\right)\pm\frac{\mathrm{i}}{\sqrt{2}r}\omega_{0}A^{d}_{1~{}\pm 1}\right].$
(74)

Multiplying $r^{2}\Phi_{0}$ and integrating by parts, we obtain

$2\omega_{0}\omega_{1}\int^{\infty}_{0}\mathrm{d}rr^{2}\Phi_{0}^{2}=\mathrm{i}\kappa
B_{0}\int^{\infty}_{0}\mathrm{d}rr^{2}\Phi_{0}\left[-\frac{1}{\sqrt{5}}\left(\partial_{r}A^{a}_{2~{}\pm
1}+\omega_{0}A^{b}_{2~{}\pm 1}+\frac{3}{r}A^{a}_{2~{}\pm
1}\right)\pm\frac{\mathrm{i}}{\sqrt{2}r}\omega_{0}A^{d}_{1~{}\pm 1}\right].$
(75)

Focusing on the imaginary part of $\omega_{1}$ and integrating by parts, we
obtain

${\rm Im}\omega_{1}=\frac{\kappa
B_{0}}{2\omega_{0}^{2}}\int^{\infty}_{0}\mathrm{d}x\left[\frac{3}{\sqrt{5}}x\left(x\partial_{x}\Phi_{0}-\Phi_{0}\right){\rm
Re}\mathcal{B}_{\pm}\mp\frac{1}{\sqrt{2}}x^{2}\Phi_{0}{\rm
Im}\mathcal{D}_{\pm}\right].$ (76)

Then ${\rm Im}\omega_{1}$ is given by

$\displaystyle{\rm Im}\omega_{1}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}B_{0}^{2}}{4\omega_{0}^{3}}\left[\frac{3}{5}\left(\int^{\infty}_{0}\mathrm{d}\xi\xi
j_{2}(\xi)\left(\xi\partial_{\xi}\Phi_{0}(\xi)-\Phi_{0}(\xi)\right)\right)^{2}+\left(\int^{\infty}_{0}\mathrm{d}\xi\xi^{2}j_{1}(\xi)\Phi_{0}(\xi)\right)^{2}\right]$
(77) $\displaystyle=$
$\displaystyle\frac{2^{14}}{15}a_{0}^{2}\kappa^{2}B_{0}^{2}\frac{a_{0}^{3}\omega_{0}^{3}}{(1+4a_{0}^{2}\omega_{0}^{2})^{6}}\omega_{0}\sim\frac{4}{15}a_{0}^{2}\kappa^{2}B_{0}^{2}\frac{1}{a_{0}^{9}\omega_{0}^{9}}\omega_{0}\sim\frac{4}{15}\frac{\kappa^{2}B_{0}^{2}}{\mu^{2}}(GM\mu)^{7}\mu.$
(78)

##  V Comparison with the growth rate of the superradiant instability

Let us compare the decay rates obtained in Sec. III and Sec. IV with the
growth rate $\omega_{\rm sr}$ of the superradiant instability given in Refs.
Detweiler:1980uk ; Pani:2012bp  as follows:

$\omega_{\rm sr}\sim\frac{1}{48}(GM\mu)^{8}\mu\sim 2\times 10^{-17}{\rm
s}^{-1}\left(\frac{\mu}{10^{-18}{\rm eV}}\right)^{9}\left(\frac{M}{4\times
10^{6}M_{\odot}}\right)^{8}.$ (79)

The ratio between $\omega_{\rm sr}$ and ${\rm Im}\omega_{1}$ is given by

$\displaystyle\left(\frac{{\rm Im}\omega_{1}}{\omega_{\rm sr}}\right)_{\rm
mono}$ $\displaystyle\sim$ $\displaystyle
2\left(\frac{\kappa^{2}q^{2}}{a_{0}^{2}}\right)\times(GM\mu)^{-5},$ (80)
$\displaystyle\left(\frac{{\rm Im}\omega_{1}}{\omega_{\rm sr}}\right)_{\rm
uni}$ $\displaystyle\sim$
$\displaystyle\frac{64}{5}(a_{0}^{2}\kappa^{2}B_{0}^{2})\times(GM\mu)$ (81)

for the monopole magnetic field and the uniform magnetic field, respectively.
We note that we have assumed $\kappa q/a_{0}\ll 1$, $a_{0}\kappa B_{0}\ll 1$
and $GM\mu\ll 1$ in our analytic evaluation. Within the relevant parameter
region of our approximation, the ratio can be much larger than unity if the
axion mass is sufficiently smaller than $1/(GM)$ for the monopole magnetic
field. On the other hand, the ratio cannot be much larger than unity for the
uniform magnetic field within the relevant parameter region of our
approximation. Nevertheless, it would be reasonable to expect that a similar
decaying process exists even if $a_{0}\kappa B_{0}\sim 1$ and/or $GM\mu\sim
1$.

Applying the formulae (80) and (81) to the black hole at the center of our
galaxy, we find

$\displaystyle\left(\frac{{\rm Im}\omega_{1}}{\omega_{\rm sr}}\right)_{\rm
mono}\sim 2\mu^{2}\kappa^{2}q^{2}(GM\mu)^{-3}$ (82) $\displaystyle\hskip
28.45274pt\sim 3\times 10^{8}\left(\frac{\kappa}{10^{-12}{\rm
GeV}^{-1}}\right)^{2}\left(\frac{q\mu^{4}(GM)^{2}}{10^{3}{\rm
G}}\right)^{2}\left(\frac{\mu}{10^{-18}{\rm
eV}}\right)^{-9}\left(\frac{M}{4\times 10^{6}M_{\odot}}\right)^{-7},$ (83)
$\displaystyle\left(\frac{{\rm Im}\omega_{1}}{\omega_{\rm sr}}\right)_{\rm
uni}\sim\frac{64}{5}\frac{\kappa^{2}B_{0}^{2}}{\mu^{2}}\frac{1}{GM\mu}$ (84)
$\displaystyle\hskip 28.45274pt\sim 2\left(\frac{\kappa}{10^{-12}{\rm
GeV}^{-1}}\right)^{2}\left(\frac{B_{0}}{10^{3}{\rm
G}}\right)^{2}\left(\frac{\mu}{10^{-18}{\rm
eV}}\right)^{-3}\left(\frac{M}{4\times 10^{6}M_{\odot}}\right)^{-1},$ (85)

where we referred to Ref. Berg:2016ese  for the value of $\kappa$.

##  VI summary and discussion

We derived the decay rate of the axion cloud around a black hole considering
the axion-photon coupling with background magnetic fields. Although the axion
field has unstable bound states around a rotating black hole without the
electro-magnetic fields, the axion-photon coupling together with a background
magnetic field induces the axion-photon conversion, and the photon can
dissipate to infinity. We obtained analytic expression of the decay rate for
the monopole and uniform magnetic field configurations under the approximation
for which the axion configuration is dominated by the solution of the equation
equivalent to that of the Hydrogen atom. For the monopole background, we found
the decay rate ${\rm
Im}~{}\omega_{1}\sim\mu^{2}q^{2}\kappa^{2}(GM\mu)^{5}\mu/24$, where $\kappa$,
$q$, $\mu$ and $M$ are the axion-photon coupling constant, magnetic charge,
axion mass and black hole mass, respectively. For the uniform background
magnetic field, we obtain the decay rate as
$\sim\kappa^{2}B_{0}^{2}\mu^{6}(GM)^{7}$, where $B_{0}$ is the magnetic field
strength. Since the two expressions for the monopole background and the
uniform magnetic field are different from each other, the decay rate may
significantly depend on the configuration of the background magnetic field.

When we discuss an axion cloud around a black hole, the decay rate should be
sufficiently small compared to the growth rate of the superradiant instability
and the inverse of the relevant time scale. Let us consider the central black
hole of our galaxy, Sgr A* as a specific example. The ambient environments of
Sgr A* will be probed with S-stars Amorim:2019hwp  and possible pulsars
orbiting around Sgr A* which are expected to be discovered by SKA Liu:2011ae .
Applying our formulae, we found that the magnitude of the decay rate due to
the axion-photon conversion can be comparable to that of the growth rate of
the superradiant instability for the central black hole in our galaxy.
Therefore Sgr A* is one of notable systems which could be relevant to the
phenomenon we described in this paper.

It should be noted that, in the analysis, we have imposed several conditions
to perform the analytic evaluation. Moreover, the frequency of the photon,
which is comparable to the axion mass scale in the present case, would be much
lower than the plasma frequency in the black hole environment, and the photon
emission would be prohibited. Nevertheless, one would expect that an
alternative wave mode such as Alfvén waves may play a similar role to the
photon and a similar decay process may generally exist. Obviously, in order to
clarify the decay process, we need to investigate more realistic situations
possibly realized in a black hole system. These are our future issues.

## Acknowledgements

We thank Jiro Soda for helpful comments and his excellent lecture on axion
physics, which was publicly open and remotely provided as an intensive lecture
course at Rikkyo University. We also thank Takahiro Tanaka for helpful
comments. This work was supported by JSPS KAKENHI Grant Numbers JP19H01895
(C.Y.), JP20H05850 (C.Y.), JP20H05853 (C.Y.), 20H05852 (A.N.), JP19H01891
(A.N. and D.Y.), 15H05896 (K.T.), 16H05999 (K.T.), 17H01110 (K.T.), 20H00180
(K.T.) and 17K14304 (D.Y.), Bilateral Joint Research Projects of JSPS (K.T.),
and the ISM Cooperative Research Program [2020-ISMCRP-2017] (K.T.).

## References

  * (1) R. D. Peccei and H. R. Quinn,  Phys. Rev. Lett. 38, 1440 (1977), CP Conservation in the Presence of Instantons.
  * (2) R. D. Peccei and H. R. Quinn,  Phys. Rev. D 16, 1791 (1977), Constraints Imposed by CP Conservation in the Presence of Instantons.
  * (3) S. Weinberg,  Phys. Rev. Lett. 40, 223 (1978), A New Light Boson?
  * (4) F. Wilczek,  Phys. Rev. Lett. 40, 279 (1978), Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons.
  * (5) P. Svrcek and E. Witten,  JHEP 06, 051 (2006), arXiv:hep-th/0605206, Axions In String Theory.
  * (6) A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell,  Phys. Rev. D 81, 123530 (2010), arXiv:0905.4720, String Axiverse.
  * (7) J. Preskill, M. B. Wise, and F. Wilczek,  Phys. Lett. B 120, 127 (1983), Cosmology of the Invisible Axion.
  * (8) M. Dine and W. Fischler,  Phys. Lett. B 120, 137 (1983), The Not So Harmless Axion.
  * (9) L. F. Abbott and P. Sikivie,  Phys. Lett. B 120, 133 (1983), A Cosmological Bound on the Invisible Axion.
  * (10) Particle Data Group, P. A. Zyla et al.,  PTEP 2020, 083C01 (2020), Review of Particle Physics.
  * (11) A. Arvanitaki and S. Dubovsky,  Phys. Rev. D 83, 044026 (2011), arXiv:1004.3558, Exploring the String Axiverse with Precision Black Hole Physics.
  * (12) T. J. M. Zouros and D. M. Eardley,  Annals Phys. 118, 139 (1979),  Instabilities of Massive Scalar Perturbations of a Rotating Black Hole .
  * (13) S. L. Detweiler,  Phys. Rev. D 22, 2323 (1980),  Klein-Gordon Equation and Rotating Black Holes.
  * (14) R. Brito, V. Cardoso, and P. Pani,  Lect. Notes Phys. 906, pp.1 (2015), arXiv:1501.06570, Superradiance: New Frontiers in Black Hole Physics.
  * (15) S. R. Dolan,  Phys. Rev. D 76, 084001 (2007), arXiv:0705.2880, Instability of the massive Klein-Gordon field on the Kerr spacetime.
  * (16) V. Cardoso, S. Chakrabarti, P. Pani, E. Berti, and L. Gualtieri,  Phys. Rev. Lett. 107, 241101 (2011), arXiv:1109.6021, Floating and sinking: The Imprint of massive scalars around rotating black holes.
  * (17) H. Yoshino and H. Kodama,  Prog. Theor. Phys. 128, 153 (2012), arXiv:1203.5070, Bosenova collapse of axion cloud around a rotating black hole.
  * (18) H. Yoshino and H. Kodama,  PTEP 2014, 043E02 (2014), arXiv:1312.2326, Gravitational radiation from an axion cloud around a black hole: Superradiant phase.
  * (19) R. Brito, V. Cardoso, and P. Pani,  Class. Quant. Grav. 32, 134001 (2015), arXiv:1411.0686, Black holes as particle detectors: evolution of superradiant instabilities.
  * (20) H. Yoshino and H. Kodama,  Class. Quant. Grav. 32, 214001 (2015), arXiv:1505.00714, The bosenova and axiverse.
  * (21) A. Arvanitaki, M. Baryakhtar, and X. Huang,  Phys. Rev. D 91, 084011 (2015), arXiv:1411.2263, Discovering the QCD Axion with Black Holes and Gravitational Waves.
  * (22) R. Brito et al.,  Phys. Rev. D 96, 064050 (2017), arXiv:1706.06311, Gravitational wave searches for ultralight bosons with LIGO and LISA.
  * (23) D. Baumann, H. S. Chia, and R. A. Porto,  Phys. Rev. D 99, 044001 (2019), arXiv:1804.03208, Probing Ultralight Bosons with Binary Black Holes.
  * (24) J. Zhang and H. Yang,  Phys. Rev. D 99, 064018 (2019), arXiv:1808.02905, Gravitational floating orbits around hairy black holes.
  * (25) J. Zhang and H. Yang,  Phys. Rev. D 101, 043020 (2020), arXiv:1907.13582, Dynamic Signatures of Black Hole Binaries with Superradiant Clouds.
  * (26) D. Baumann, H. S. Chia, R. A. Porto, and J. Stout,  Phys. Rev. D 101, 083019 (2020), arXiv:1912.04932, Gravitational Collider Physics.
  * (27) Q. Ding, X. Tong, and Y. Wang,  Astrophys. J. 908, 78 (2021), arXiv:2009.11106, Gravitational Collider Physics via Pulsar-Black Hole Binaries.
  * (28) H. Omiya, T. Takahashi, and T. Tanaka,  (2020), arXiv:2012.03473, Renormalization group analysis of superradiant growth of self-interacting axion cloud.
  * (29) M. J. Stott,  (2020), arXiv:2009.07206, Ultralight Bosonic Field Mass Bounds from Astrophysical Black Hole Spin.
  * (30) T. Takahashi, H. Omiya, and T. Tanaka,  Private communication.
  * (31) CAST, V. Anastassopoulos et al.,  Nature Phys. 13, 584 (2017), arXiv:1705.02290, New CAST Limit on the Axion-Photon Interaction.
  * (32) ADMX, C. Boutan et al.,  Phys. Rev. Lett. 121, 261302 (2018), arXiv:1901.00920, Piezoelectrically Tuned Multimode Cavity Search for Axion Dark Matter.
  * (33) J. L. Ouellet et al.,  Phys. Rev. Lett. 122, 121802 (2019), arXiv:1810.12257, First Results from ABRACADABRA-10 cm: A Search for Sub-$\mu$eV Axion Dark Matter.
  * (34) F. Calore, P. Carenza, M. Giannotti, J. Jaeckel, and A. Mirizzi,  Phys. Rev. D 102, 123005 (2020), arXiv:2008.11741, Bounds on axionlike particles from the diffuse supernova flux.
  * (35) C. P. Salemi et al.,  (2021), arXiv:2102.06722, The search for low-mass axion dark matter with ABRACADABRA-10cm.
  * (36) S. M. Carroll, G. B. Field, and R. Jackiw,  Phys. Rev. D 41, 1231 (1990), Limits on a Lorentz and Parity Violating Modification of Electrodynamics.
  * (37) S. M. Carroll and G. B. Field,  Phys. Rev. D 43, 3789 (1991), The Einstein equivalence principle and the polarization of radio galaxies.
  * (38) D. Harari and P. Sikivie,  Phys. Lett. B 289, 67 (1992), Effects of a Nambu-Goldstone boson on the polarization of radio galaxies and the cosmic microwave background.
  * (39) M. M. Ivanov et al.,  JCAP 02, 059 (2019), arXiv:1811.10997, Constraining the photon coupling of ultra-light dark-matter axion-like particles by polarization variations of parsec-scale jets in active galaxies.
  * (40) T. Fujita, R. Tazaki, and K. Toma,  Phys. Rev. Lett. 122, 191101 (2019), arXiv:1811.03525, Hunting Axion Dark Matter with Protoplanetary Disk Polarimetry.
  * (41) T. Liu, G. Smoot, and Y. Zhao,  Phys. Rev. D 101, 063012 (2020), arXiv:1901.10981, Detecting axionlike dark matter with linearly polarized pulsar light.
  * (42) M. A. Fedderke, P. W. Graham, and S. Rajendran,  Phys. Rev. D 100, 015040 (2019), arXiv:1903.02666, Axion Dark Matter Detection with CMB Polarization.
  * (43) A. Caputo et al.,  Phys. Rev. D 100, 063515 (2019), arXiv:1902.02695, Constraints on millicharged dark matter and axionlike particles from timing of radio waves.
  * (44) Y. Chen, J. Shu, X. Xue, Q. Yuan, and Y. Zhao,  Phys. Rev. Lett. 124, 061102 (2020), arXiv:1905.02213, Probing Axions with Event Horizon Telescope Polarimetric Measurements.
  * (45) G.-W. Yuan et al.,  (2020), arXiv:2008.13662, Testing the ALP-photon coupling with polarization measurements of Sagittarius A*.
  * (46) A. Basu, J. Goswami, D. J. Schwarz, and Y. Urakawa,  (2020), arXiv:2007.01440, Searching for axion-like particles under strong gravitational lenses.
  * (47) L. Maiani, R. Petronzio, and E. Zavattini,  Phys. Lett. B 175, 359 (1986), Effects of Nearly Massless, Spin Zero Particles on Light Propagation in a Magnetic Field.
  * (48) G. Raffelt and L. Stodolsky,  Phys. Rev. D 37, 1237 (1988), Mixing of the Photon with Low Mass Particles.
  * (49) P. Sikivie,  Phys. Rev. Lett. 51, 1415 (1983), Experimental Tests of the Invisible Axion,  [Erratum: Phys.Rev.Lett. 52, 695 (1984)].
  * (50) P. Sikivie,  Phys. Rev. D 32, 2988 (1985), Detection Rates for ’Invisible’ Axion Searches,  [Erratum: Phys.Rev.D 36, 974 (1987)].
  * (51) J. Ahonen, K. Enqvist, and G. Raffelt,  Phys. Lett. B 366, 224 (1996), arXiv:hep-ph/9510211, The Paradox of axions surviving primordial magnetic fields.
  * (52) M. S. Pshirkov and S. B. Popov,  J. Exp. Theor. Phys. 108, 384 (2009), arXiv:0711.1264, Conversion of Dark matter axions to photons in magnetospheres of neutron stars.
  * (53) D. Espriu and A. Renau,  Phys. Rev. D 85, 025010 (2012), arXiv:1106.1662, Photon propagation in a cold axion background with and without magnetic field.
  * (54) D. Espriu and A. Renau,  Int. J. Mod. Phys. A 30, 1550099 (2015), arXiv:1401.0663, Photons in a cold axion background and strong magnetic fields: polarimetric consequences.
  * (55) F. P. Huang, K. Kadota, T. Sekiguchi, and H. Tashiro,  Phys. Rev. D 97, 123001 (2018), arXiv:1803.08230, Radio telescope search for the resonant conversion of cold dark matter axions from the magnetized astrophysical sources.
  * (56) A. Hook, Y. Kahn, B. R. Safdi, and Z. Sun,  Phys. Rev. Lett. 121, 241102 (2018), arXiv:1804.03145, Radio Signals from Axion Dark Matter Conversion in Neutron Star Magnetospheres.
  * (57) M. P. Hertzberg and E. D. Schiappacasse,  JCAP 11, 004 (2018), arXiv:1805.00430, Dark Matter Axion Clump Resonance of Photons.
  * (58) A. Arza,  Eur. Phys. J. C 79, 250 (2019), arXiv:1810.03722, Photon enhancement in a homogeneous axion dark matter background.
  * (59) E. Masaki, A. Aoki, and J. Soda,  Phys. Rev. D 101, 043505 (2020), arXiv:1909.11470, Stability of Axion Dark Matter-Photon Conversion.
  * (60) P. Carenza, A. Mirizzi, and G. Sigl,  Phys. Rev. D 101, 103016 (2020), arXiv:1911.07838, Dynamical evolution of axion condensates under stimulated decays into photons.
  * (61) A. Arza, T. Schwetz, and E. Todarello,  JCAP 10, 013 (2020), arXiv:2004.01669, How to suppress exponential growth—on the parametric resonance of photons in an axion background.
  * (62) M. Leroy, M. Chianese, T. D. P. Edwards, and C. Weniger,  Phys. Rev. D 101, 123003 (2020), arXiv:1912.08815, Radio Signal of Axion-Photon Conversion in Neutron Stars: A Ray Tracing Analysis.
  * (63) J. H. Buckley, P. S. B. Dev, F. Ferrer, and F. P. Huang,  Phys. Rev. D 103, 043015 (2021), arXiv:2004.06486, Fast radio bursts from axion stars moving through pulsar magnetospheres.
  * (64) M. Boskovic, R. Brito, V. Cardoso, T. Ikeda, and H. Witek,  Phys. Rev. D 99, 035006 (2019), arXiv:1811.04945, Axionic instabilities and new black hole solutions.
  * (65) D. Blas and S. J. Witte,  Phys. Rev. D 102, 103018 (2020), arXiv:2009.10074, Imprints of Axion Superradiance in the CMB.
  * (66) R. Brito, V. Cardoso, and P. Pani,  Phys. Rev. D 89, 104045 (2014), arXiv:1405.2098, Superradiant instability of black holes immersed in a magnetic field.
  * (67) B. Turimov, B. Toshmatov, B. Ahmedov, and Z. Stuchlík,  Phys. Rev. D 100, 084038 (2019), arXiv:1910.00939, Quasinormal modes of magnetized black hole.
  * (68) R. M. Wald,  Phys. Rev. D 10, 1680 (1974), Black hole in a uniform magnetic field.
  * (69) F. J. Zerilli,  Phys. Rev. D 9, 860 (1974), Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry.
  * (70) P. Pani, V. Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi,  Phys. Rev. D 86, 104017 (2012), arXiv:1209.0773, Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric.
  * (71) M. Berg et al.,  Astrophys. J. 847, 101 (2017), arXiv:1605.01043, Constraints on Axion-Like Particles from X-ray Observations of NGC1275.
  * (72) GRAVITY, A. Amorim et al.,  Mon. Not. Roy. Astron. Soc. 489, 4606 (2019), arXiv:1908.06681, Scalar field effects on the orbit of S2 star.
  * (73) K. Liu, N. Wex, M. Kramer, J. M. Cordes, and T. J. W. Lazio,  Astrophys. J. 747, 1 (2012), arXiv:1112.2151, Prospects for Probing the Spacetime of Sgr A* with Pulsars.