# Solar axions in large extra dimensions

Mar Bastero-Gil  Cyprien Beaufort  Daniel Santos

###### Abstract

The axion could be used as a probe for extra dimensions. In large extra
dimensions, besides the QCD axion one obtains an infinite tower of massive
Kaluza-Klein (KK) states. We describe the processes of KK axions production in
the Sun via the axion-photon coupling, $g_{a\gamma\gamma}$, and we derive the
number density of KK axions that get trapped into the solar gravitational
field and then accumulate over cosmic times. The large multiplicity of states,
as well as their masses in the keV-range, deeply alter the phenomenology of
the axion. This scenario leads us to propose the presence of KK axions as an
interpretation of the non-thermal distribution of the solar X-rays. In this
work, we dedicate special attention on the astrophysical and cosmological
bounds that apply to the model. In particular, we show how the KK axions may
escape the EBL limit that constrains standard ALPs in the same mass range.
Present searches for KK axions make use of the decay channel,
$a\rightarrow\gamma\gamma$, for which we revise the event rate; our value lies
orders of magnitude below the rate usually quoted in the literature. This
major conclusion stems from recent measurements of the luminosity of the quiet
Sun which acts as an irreducible limit. The revised model remains a viable and
an attractive explanation for multiple astrophysical observations, and we
propose several approaches to search for solar KK axions in the near future.

##  1 Introduction

In theories with large extra dimensions, the fundamental scale of quantum
gravity can be lowered near the TeV-scale without violating experimental
constraints [1, 2]. Such a framework naturally solves the mass hierarchy
problem and it can be embedded in string theory [3, 4, 5] while preserving
gauge coupling unification [6, 7]. This approach offers new models describing,
among others, inflation [8, 9], the baryon asymmetry [10] or the proton
stability [11]. In the original scenario, called ADD from the names of the
founders, the Standard Model (SM) particles are constrained to live on a
$4$-dimensional Minkowski brane whereas singlets under the SM gauge group can
propagate into the bulk of dimension $(4+n)$. The weakness of gravity at large
distances is explained by the propagation of gravitons in the bulk. The $n$
extra dimensions are compactified and large compared to the electroweak scale,
up to the micrometer range, which allows for experimental tests of the theory.

The ADD’s claim that the electroweak scale is the only fundamental scale
seems, at first glance, incompatible with the usual need for intermediate
scales to address phenomenological issues as the neutrino oscillations or the
strong CP problem. However, the potential propagations of right-handed
neutrinos and axions in the bulk give rise to a higher-dimensional seesaw
mechanism without a heavy mass scale [12, 13], and to an experimentally
allowed breaking of the Peccei-Quinn (PQ) symmetry near the electroweak scale
that preserves axion invisibility [14, 15, 16, 2].

In the standard picture, the axion is a hypothetical pseudoscalar that arises
from the breaking of a global $U(1)$ symmetry called the PQ symmetry [17, 18,
19]. When the axion relaxes to its minimum of the potential, it dynamically
compensates the CP-violating term of QCD and consequently solves the strong CP
problem. The scale of the PQ symmetry breaking, $f_{PQ}$, is assumed to be in
the range $f_{PQ}\sim(10^{9}-10^{12})\,\mathrm{GeV}$ which makes the axion a
stable particle, weakly coupled to the SM particles and with a mass
$m_{PQ}\sim\mathcal{O}(10\,\mathrm{\mu eV})$. The axion is hence a viable non-
baryonic candidate for Dark Matter [20, 21, 22] even if this interesting
feature was not the initial purpose of its prediction.

The situation drastically changes in the framework of large extra dimensions.
In the simplest scenario, the $n$ extra dimensions are toroidally compactified
and we assume axions are free to propagate in $\delta\leq n$ extra dimensions.
Due to the periodicity of the extra dimensions, the axion field in the bulk
can be Fourier expanded into an infinite number of modes. From our
4-dimensional brane, the axion is seen as an infinite superposition of massive
states, called a Kaluza-Klein (KK) tower, and there is one KK tower for each
of the $\delta$ extra dimensions in which the axion propagates. In other
words, observed from our 4-dimensional brane, the standard QCD axion appears
as complemented by $\delta$ infinite towers of massive KK states arising from
the new phase space available in the bulk.

The presence of the KK towers and the extra dimensions deeply affect the
physics of the axion. First, the coupling of the KK axions remains
parametrized by $f_{PQ}$ whereas the overall scale of the PQ symmetry
breaking, $\bar{f}_{PQ}$, is now suppressed by a volume-renormalization
factor. While the experimentally constrained scale $f_{PQ}$ is unchanged,
$\bar{f}_{PQ}$ can lie near the electroweak scale. Second, only the ground
state of the KK tower, $a_{0}$, transforms under a PQ transformation [15]. The
strong CP problem is still solved in extra large dimensions and $a_{0}$ can be
identified as the standard QCD axion. Third, KK axions would be produced with
masses in the keV-range. Each KK mode can decay into 2 photons with lifetimes
that can be of the order of the age of the universe, which leads to important
consequences on astrophysics and cosmology, but also on detection since the
channel $a\rightarrow\gamma\gamma$ has been proposed for experimental searches
[16, 23]. Finally, the large number of states strongly enhances any phenomenon
mediated by the axions.

KK axion modes up to the kinematic limit would be produced in astrophysical
objects. The Sun appears as a particularly interesting source due to its
proximity and its relatively well-known characteristics. Moreover, part of the
KK axions produced by the Sun are sufficiently non-relativistic to be trapped
into the solar gravitational field. These orbiting axions accumulate over
cosmic times and consequently increase the present KK axion flux in a detector
on Earth by about 5 orders of magnitude compared to the direct flux of axions
emitted by the Sun. The decay of these trapped KK axions into photons acts as
a new X-rays source in the solar system and has been proposed as an
explanation to the coronal heating problem [23] and to the non-thermal
distribution of the solar X-rays measurements [24].

Searches for solar KK axions have started using axion helioscopes [25, 26] or
via the decay into photons [27, 28, 29, 30, 31]. As far as we know, there is
only a single paper describing the model of trapped KK axions [23] which sets
the basis for experimental searches, and this pioneer paper contains few
mathematical details. The present work aims to entirely revise the solar KK
axion model, to provide analytical expressions for phenomenology, and to
update the experimental constraints that apply on the model with recent
measurements. We expect to shine light on how KK axions can be used as probe
for extra dimensions.

The paper is organized as follows. In Section 2 we describe the physics of the
axions in large extra dimensions. Section 3 is dedicated to processes of KK
axions production in the Sun in order to determine the solar KK axions flux.
The case of gravitational trapped KK axions is discussed in Section 4 where we
derive an expression for the number density of KK axions, which plays a major
role for experimental searches. In Section 5, we revise the constraints that
apply on the model and we discuss a possible explanation of the non-thermal
distribution of the solar X-rays measurements. Finally, we conclude by
studying some strategies of detection in Section 6. In particular, we update
the decay event rate ($a\rightarrow\gamma\gamma$) in a detector on Earth
showing that this channel of detection, the most investigated in experiments,
seems inaccessible with present technologies, except if one can reject almost
perfectly the background. We anyway propose other approaches to search for
solar KK axions.

##  2 Axions in large extra dimensions

Let first briefly describe the standard axions before discussing the higher-
dimensional case. Axions are pseudo Goldstone bosons associated to the
breaking of a global $U(1)_{PQ}$ symmetry, as proposed by Peccei and Quinn in
order to solve the strong CP problem [32]. For the moment we limit our study
to the minimal scenario of hadronic models in which axions do not couple to
fermions at tree-level. The effective axion Lagrangian of interest for us is
given by:

$\mathcal{L}_{eff}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{2}\big{(}\partial_{\mu}a\big{)}^{2}\,-\,\frac{1}{2}m_{PQ}^{2}a^{2}\,+\,\frac{g_{a\gamma\gamma}}{4}a\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,,$
(2.1)

in which $a$ is the axion, $F^{\mu\nu}$ is the electromagnetic field tensor,
$\tilde{F}_{\mu\nu}$ is its dual, and $m_{PQ}$ is the axion mass that can be
expressed as [33]:

$m_{PQ}=5.70(7)\,\mu\mathrm{eV}\leavevmode\nobreak\
\bigg{(}\frac{${10}^{12}$\,\mathrm{GeV}}{f_{PQ}}\bigg{)}\,.$ (2.2)

We call $f_{PQ}$ the scale of the $U(1)_{PQ}$ symmetry breaking and we define
the effective axion-photon coupling $g_{a\gamma\gamma}$ as:

$g_{a\gamma\gamma}\equiv\frac{\xi\alpha_{em}}{2\pi f_{PQ}}\,,$ (2.3)

where $\xi$ is a model-dependent constant of order unity and $\alpha_{em}$ is
the fine-structure constant. The scale of symmetry breaking, $f_{PQ}$, is
constrained from astrophysical (lower limit) [34, 35] and cosmological (upper
limit) considerations [20, 21, 22, 36], with the usually quoted bounds:

$10^{9}\,\mathrm{GeV}\lesssim f_{PQ}\lesssim 10^{12}\,\mathrm{GeV}\,.$ (2.4)

Finally, we present the expression for the lifetime of the axion:

$\tau_{a\rightarrow\gamma\gamma}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{64\pi}{g_{a\gamma\gamma}^{2}m_{PQ}^{3}}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ 1.5\times
10^{45}\,\mathrm{days}\leavevmode\nobreak\
\bigg{(}\frac{10^{-15}\,\mathrm{GeV^{-1}}}{g_{a\gamma\gamma}}\bigg{)}^{2}\,\bigg{(}\frac{10^{-5}\,\mathrm{eV}}{m_{PQ}}\bigg{)}^{3}\,.$
(2.5)

From the previous expressions we see that the axion is feebly coupled to
Standard Model particles, that it is a light particle and that its lifetime in
the allowed range given by Eq.(2.4) is much larger than the age of the
universe, rendering the axion effectively stable. This consideration rules out
the possibility of detecting the axion through the decay channel
$a\rightarrow\gamma\gamma$.

We will now discuss the higher-dimensional case and the consequences for axion
physics. Let consider $n$ large extra spatial dimensions to be compactified on
a $\mathbb{Z}_{2}$ orbifold with an orbifold action
$\mathbf{y}\rightarrow-\mathbf{y}$. For simplicity and conceptual reasons we
consider all the extra dimensions to have the same compactification radius
$R$. In this scenario, the graviton is free to propagate in all dimensions and
the fundamental gravitational scale, $M_{\ast}$, can be much lower than the
Planck scale, $M_{P}$, due to the extra dimensions volume suppression. They
are related to each other via the expression [1]:

$M_{P}=\big{(}2\pi RM_{\ast}\big{)}^{n/2}\,M_{\ast}\,.$ (2.6)

The original motivation of such a scenario was to keep the fundamental scale
$M_{\ast}$ in the TeV range in order to solve the hierarchy problem of the SM.
While the particles of the SM are constrained to live on a $4$-dimensional
Minkowski brane, any singlets under the SM symmetry may propagate in
$\delta\leq n$ dimensions. This can be the case for the axion and, similarly
to the gravitational case, the $4$-dimensional axion scale $f_{PQ}$ can be
much larger than the fundamental axion scale $\bar{f}_{PQ}$ of the overall
theory:

$f_{PQ}=\big{(}2\pi RM_{\ast}\big{)}^{\delta/2}\,\bar{f}_{PQ}\,.$ (2.7)

This relation tells us that the fundamental scale $\bar{f}_{PQ}$ could be very
low, perhaps in the TeV range, while letting the experimentally constrained
4-dimensional scale $f_{PQ}$ in the range of Eq.(2.4). It is also tempting to
keep the fundamental scale $M_{\ast}\sim\mathcal{O}(1-100\,\mathrm{TeV})$ in
order to solve the mass hierarchy problem of the Standard Model. Without
introducing any other fundamental besides $M_{\ast}$ and the compactification
radius $R$, i.e. taking $\bar{f}_{PQ}=M_{\ast}$, we see that if the axion
propagates in the same number of extra dimensions than gravity, the
4-dimensional axion scale $f_{PQ}$ must be at the Planck scale and thus
disagree with the experimental constraints of Eq.(2.4).

The size of the extra dimensions, described by $R$, influences many
observables and is consequently constrained from multiple measurements. A
considerable review of such limits can be found in [37], we list in Table 1
the most striking bounds on $R$ obtained from tests of Newton law, colliders
and astrophysical considerations.

$n$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$R_{max}\,[\mathrm{keV}^{-1}]$ | Pendulum | $2.2\times 10^{5}$ | / | / | / | / | /
Colliders | / | $2.5\times 10^{4}$ | $6.2\times 10^{-1}$ | $3.1\times 10^{-3}$ | $1.2\times 10^{-4}$ | $1.5\times 10^{-5}$
SN 1987A | $2.5\times 10^{12}$ | $4.9\times 10^{3}$ | $5.8$ | $1.9\times 10^{-1}$ | $2.5\times 10^{-2}$ | $6.1\times 10^{-3}$
Neutron star | $2.2\times 10^{5}$ | $7.9\times 10^{-1}$ | $1.3\times 10^{-2}$ | $1.7\times 10^{-3}$ | $5.1\times 10^{-4}$ | $2.2\times 10^{-4}$
Table 1: Limits on $R$ from torsion-pendulum experiments [38], from colliders
(CMS) [39] and from the duration of the neutrino signal of the supernova SN
1987A and neutron-star excess heat [40].

The results quoted as "neutron stars" set the most stringent limits on $R$ for
$\delta<5$. They are obtained from the hypothetical $\gamma$ rays production
due to the decay of KK gravitons trapped in a neutron star gravitational
field. However, we suppose here the existence of KK axions, and it has been
shown that KK gravitons would mainly decay into KK axions if they exist [14],
process that would alter the derived limits. Plus, the branching ratio of KK
graviton decay into photons could be drastically lowered if another brane
exists as pointed out by the founders of the ADD model [2]. For those reasons,
we do not consider the limits on $R$ derived from neutron stars in this work.

With those elements in mind, and following the approach of [2, 14, 15, 16],
let us have a look to the physics of the axion in large extra dimensions as
seen from our 4-dimensional brane. Let denote
$\mathbf{y}=(y_{1},y_{2},\dots,y_{\delta})$ the coordinates of the axion in
each extra dimensions $\delta\leq n$ it propagates so that the coordinates of
the axion can be written $x^{M}\equiv(x^{\mu},\mathbf{y})$ where $M$ is a
spacetime index that runs over the $(4+\delta)$ dimensions. The
$(4+\delta)$-action is given by:

$\mathcal{S}_{4+\delta}\leavevmode\nobreak\ =\leavevmode\nobreak\ \int
d^{4}x\,d^{\delta}\mathbf{y}\,\bigg{\\{}\frac{1}{2}M_{\ast}^{\delta}(\partial_{M}a)(\partial^{M}a)\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{\xi\alpha_{em}}{8\pi\bar{f}_{PQ}}\,a\,F_{\mu\nu}\tilde{F}^{\mu\nu}\,\delta^{(\delta)}(\mathbf{y})\,\bigg{\\}}\,.$
(2.8)

We now need to compactify the extra dimensions in order to get the effective
4-dimensional theory. Since the axion field is compactified on a
$\mathbb{Z}_{2}$ orbifold, one has:

$a(x^{\mu},\mathbf{y})=a(x^{\mu},\mathbf{y}+2\pi R)\hskip
14.22636pt\mathrm{and}\hskip
14.22636pta(x^{\mu},\mathbf{y})=a(x^{\mu},-\mathbf{y})\,,$ (2.9)

and therefore one can Fourier expand the field without loss of generality:

$a(x^{\mu},\mathbf{y})=\sum_{\mathbf{n}=0}^{\infty}\leavevmode\nobreak\
a_{\mathbf{n}}(x^{\mu})\,\cos\Big{(}\frac{\mathbf{n}\,\mathbf{y}}{R}\Big{)}\,,$
(2.10)

yielding an infinite superposition of modes; this procedure is called a
Kaluza-Klein (KK) decomposition. In the equation,
$a_{\mathbf{n}}(x^{\mu})\in\mathbb{R}$ are the KK modes,
$\mathbf{n}=(n_{1},n_{2},\dots,n_{\delta})$ is a $\delta$-dimensional vector
that labels the KK modes, and the sum runs over all extra dimensions
$\sum^{\infty}_{\mathbf{n}=0}\equiv\sum^{\infty}_{n_{1}=0}\,\sum^{\infty}_{n_{2}=0}\dots\sum^{\infty}_{n_{\delta}=0}$.
The next step is to plug this KK decomposition into the $(4+\delta)$-action
and to integrate over the $\delta$ extra dimensions to get the effective
4-dimensional Lagrangian of the theory that describes the axion as seen from
our brane:

$\mathcal{L}_{eff}^{4D}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{2}\,\sum^{\infty}_{\mathbf{n}=0}\big{(}\partial_{\mu}a_{\mathbf{n}}\big{)}\big{(}\partial^{\mu}a_{\mathbf{n}}\big{)}\leavevmode\nobreak\
-\leavevmode\nobreak\ \frac{1}{2}\,m_{PQ}^{2}a_{0}^{2}\leavevmode\nobreak\
-\leavevmode\nobreak\ \frac{1}{2}\,\sum^{\infty}_{\mathbf{n\neq
0}}\,\frac{\mathbf{n}^{2}}{R^{2}}\,a_{\mathbf{n}}\leavevmode\nobreak\
+\leavevmode\nobreak\ \frac{\xi\alpha_{em}}{8\pi
f_{PQ}}\,\Bigg{(}\sum^{\infty}_{\mathbf{n}=0}\,r_{\mathbf{n}}a_{\mathbf{n}}\Bigg{)}\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,,$
(2.11)

with $r_{0}=1$ and $r_{\mathbf{n}\neq 0}=\sqrt{2}$ which are rescaling
coefficients to ensure that the KK modes $a_{\mathbf{n}}$ have canonically
normalized kinetic-energy terms [15].

A few comments are needed at this stage. First, a single axion propagating in
the bulk will be seen from our 4-dimensional brane as a sum of $\delta$
infinite superpositions of massive states with same quantum numbers, the
overall superposition being called a Kaluza-Klein tower. The mass of each KK
mode depends on the value of the momentum of the axion along the extra
dimensions, this momentum being quantized due the compactification of the
extra dimensions. The KK axion masses are given by [15]:

$m_{a_{0}}=\min\big{(}m_{PQ}\,,\,\frac{1}{2R}\big{)}\hskip
14.22636pt\mathrm{and}\hskip
14.22636ptm_{a_{\mathbf{n}}}\simeq\frac{|\mathbf{n}|}{R}=\frac{\sqrt{n_{1}^{2}+n_{2}^{2}+\dots+n_{\delta}^{2}}}{R}\,.$
(2.12)

In the phenomenologically interesting situations that we will consider later
we have $m_{PQ}\ll 1/R$, so the fundamental mode of the KK tower is equal to
the 4-dimensional Peccei-Quinn mass, $m_{a_{0}}=m_{PQ}$. It means that one can
let $m_{PQ}\sim\mathrm{10\,\mu eV}$ as in the standard case while having
larger masses in the KK tower. We also see that the mass splitting of the KK
tower is $\sim 1/R$. The differential mode multiplicity can reach very large
values and is given by:

$\frac{dN}{dm}=\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\,R^{\delta}\,m^{(\delta-1)}\,.$
(2.13)

Second, it can be shown that a higher-dimensional Peccei-Quinn mechanism
exists and that only the fundamental mode $a_{0}$ transforms under the
$U(1)_{PQ}$ symmetry [15]. In other words, the strong CP problem is still
solved when the axion propagates in the bulk and one can identify the lightest
KK mode $a_{0}$ to the standard QCD axion. Third, the last term of the
effective Lagrangian Eq.(2.11) tells us that each individual KK mode couples
to the electromagnetic tensor with the same coupling than in the 4-dimensional
case, $\frac{1}{4}g_{a\gamma\gamma}$ defined in Eq.(2.3), up to a factor
$r_{\mathbf{n}}$ that we will ignore for the rest of the paper for simplicity.
The lifetime of each individual KK mode is now given by:

$\tau\big{(}a_{\mathbf{n}}\rightarrow\gamma\gamma\big{)}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\
\frac{64\pi}{g_{a\gamma\gamma}^{2}m_{a_{\mathbf{n}}}^{3}}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ 4.9\times 10^{17}\,\mathrm{s}\leavevmode\nobreak\
\bigg{(}\frac{10^{-13}\,\mathrm{GeV^{-1}}}{g_{a\gamma\gamma}}\bigg{)}^{2}\,\bigg{(}\frac{30\,\mathrm{keV}}{m_{a_{\mathbf{n}}}}\bigg{)}^{3}\,.$
(2.14)

We can now touch the attractiveness of such a framework. The axion was first
proposed to solve the strong CP problem and it has received considerable
interest for more than four decades, both from theoretical and experimental
aspects [41, 42]. In theories with large extra dimensions, the axion remains a
compelling solution of the strong CP problem and a viable dark matter
candidate, but its phenomenology changes. Instead of having one light
particle, we must now consider an infinite number of massive states (the
Kaluza-Klein axions), forming a continuum for large enough extra dimensions,
with a large multiplicity of states.

As in the standard case, KK axions would be produced in astrophysical objects.
We will show that the Sun would produced KK axions with masses distributed in
the $(1-30)\,\mathrm{keV}$ range. With such masses, the lifetime of KK axions
could be of the order of the age of the universe and the photons obtained from
its decay $a_{\mathbf{n}}\rightarrow\gamma\gamma$ would have impact on
cosmology and astrophysics, and would open a new channel for experimental
searches. Moreover, as we will demonstrate, part of the KK axions produced by
the Sun would be trapped in its gravitational field.

##  3 Production of KK axions in the Sun

In hadronic models, KK axions are produced in the Sun via three dominant
processes. The Primakoff process $\gamma+Ze\rightarrow Ze+a$ converts a photon
into an axion in the electrostatic field of a nucleus or an electron. The
second process is the coalescence of two photons $\gamma\gamma\rightarrow a$,
which is kinematically suppressed for small masses (as for the QCD axion) but
turns to be significant for producing KK axions with masses in the keV-range.
In the thermal environment of the Sun, photons acquire a thermal "mass" and a
longitudinal component. This phenomenon opens up a third production process
from the decay of a transverse plasmon $\gamma_{T}\rightarrow\gamma_{L}+a$. It
is worth to mention at this stage that the phenomenology of solar KK axions is
almost exclusively driven by the process of photon coalescence since it
produces non-relativistic KK axions that are susceptible to be trapped in the
gravitational field of the Sun, as we will see in Section 4.

###  3.1 Primakoff process

The Primakoff transition rate for KK axions produced in the Sun is derived in
[43, 16]

$\Gamma_{\gamma\rightarrow a}^{\mathrm{Primakoff}}\leavevmode\nobreak\
=\leavevmode\nobreak\
\frac{g_{a\gamma\gamma}^{2}T\kappa^{2}}{32\pi^{2}}\leavevmode\nobreak\
\frac{|\mathbf{k}|}{\omega}\int
d\Omega\frac{|\mathbf{k}\times\mathbf{p}|^{2}}{(\mathbf{k}-\mathbf{p})^{2}((\mathbf{k}-\mathbf{p})^{2}+\kappa^{2})}\,,$
(3.1)

where $T$ is the temperature at production, $\mathbf{p}$ is the axion
momentum, $\mathbf{k}$ the photon one, and $\omega$ the photon energy;
$\kappa$ is the Debye-Hückel screening scale, given in general by
$\kappa^{2}=(4\pi\alpha/T)\sum_{j}Z_{j}^{2}n_{j}$, where $n_{j}$ is the number
density of charged particles. For small momenta, it can be expressed as:

$\Gamma_{\gamma\rightarrow a}^{\mathrm{Primakoff}}\leavevmode\nobreak\
=\leavevmode\nobreak\
\frac{g_{a\gamma\gamma}^{2}T\kappa^{2}}{32\pi}\bigg{(}\frac{8p^{2}}{3(\kappa^{2}+m^{2})}+\mathcal{O}(p^{4})\bigg{)}\,,$
(3.2)

where $p=|\mathbf{p}|$ and $m$ is the axion mass. The differential axion flux
at Earth, integrating over a standard solar model, is given by

$\Phi^{\mathrm{Primakoff}}_{a}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{4\pi d_{ST}^{2}}\int_{\rm
Sun}d^{3}\mathbf{r}\Gamma_{\gamma\rightarrow
a}^{\mathrm{Primakoff}}\leavevmode\nobreak\
\frac{E^{2}}{\pi^{2}}\leavevmode\nobreak\ f_{a}^{eq}(E)\,,$ (3.3)

where $d_{ST}$ is the Earth-Sun distance, $E$ the axion energy, and
$f_{a}^{eq}(E)=(e^{E/T}-1)^{-1}$ is the Bose-Einstein axion distribution
function. The authors of [16] arrived at the expression for the differential
axion flux at Earth:

$\Phi^{\mathrm{Primakoff}}_{a}\leavevmode\nobreak\ =\leavevmode\nobreak\
\Big{(}4.20\times 10^{10}\leavevmode\nobreak\
\mathrm{cm^{-2}\,s^{-1}\,keV^{-1}}\Big{)}\leavevmode\nobreak\
g_{10}^{2}\,\frac{E\,p^{2}}{e^{E/1.1}-0.7}\big{(}1+0.02m\big{)}\,,$ (3.4)

where $g_{10}=g_{a\gamma\gamma}\times 10^{10}\,\mathrm{GeV^{2}}$ is the
dimensionless coupling to photon, and $E$, $p$, and $m$ are given in keV.

###  3.2 Photon coalescence

Let us review the production of axions from the photon coalescence process
$\gamma\gamma\rightarrow a$ in a thermal environment as such of the Sun. We
start with the Boltzmann equation for the axion distribution function,
$f_{a}$, including only production from the inverse decay [44]:

$\displaystyle\frac{df_{a}(E)}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{2E}\int\frac{d^{3}k_{1}}{(2\pi)^{3}2\omega_{1}}\frac{d^{3}k_{2}}{(2\pi)^{3}2\omega_{2}}(2\pi)^{4}\delta^{4}(k_{1}+k_{2}-p)|M_{\gamma}|^{2}f_{1}^{eq}(\omega_{1})f_{2}^{eq}(\omega_{2})(1+f_{a}(E))$
(3.5) $\displaystyle\simeq$ $\displaystyle f_{a}^{eq}(E)\leavevmode\nobreak\
\frac{1}{2E}\int\frac{d^{3}k_{1}}{(2\pi)^{3}2\omega_{1}}\frac{d^{3}k_{2}}{(2\pi)^{3}2\omega_{2}}(2\pi)^{4}\delta^{4}(k_{1}+k_{2}-p)|M_{\gamma}|^{2}(1+f_{1}^{eq}(\omega_{1})+f_{2}^{eq}(\omega_{2}))$
(3.6) $\displaystyle=$ $\displaystyle\Gamma_{\gamma\gamma\rightarrow
a}^{\rm{Coal.}}(T)f_{a}^{eq}(E)\,,$ (3.7)

where $E$ is the axion energy, $p$ its momentum, and $\omega_{i}$ are the
photon energies, with $E=\omega_{1}+\omega_{2}$. In the the second line we
have used:

$f_{1}^{eq}(\omega_{1})f_{2}^{eq}(\omega_{2})(1+f_{a}^{eq}(E))=f_{a}^{eq}(E)\cdot(1+f_{1}^{eq}(\omega_{1})+f_{2}^{eq}(\omega_{2}))\,,$
(3.8)

for the Bose-Einstein distribution functions. Performing the momentum
integrals, the thermal inverse decay rate, $\Gamma_{\gamma\gamma\rightarrow
a}^{\rm{Coal.}}(T)$, is given by [44, 45]:

$\Gamma_{\gamma\gamma\rightarrow
a}^{\rm{Coal.}}(T)=\Gamma_{a\gamma\gamma}\frac{m^{2}-4\omega_{P}^{2}}{m^{2}}\left(\frac{m}{E}\right)\left(1+\frac{2T}{p}\ln\frac{1-e^{-(E+p)/2T}}{1-e^{-(E-p)/2T}}\right)\,,$
(3.9)

where $\Gamma_{a\gamma\gamma}$ is the standard axion decay rate into photons
at $T=0$, and $\omega_{P}^{2}$ the photon thermal "mass" that can be expressed
as:

$\omega_{P}^{2}=\frac{4\pi\alpha n_{e}}{m_{e}}\,,$ (3.10)

$n_{e}$ being the electron number density and $m_{e}$ its mass. Finally, for
the number of axions per unit volume, per unit time, and per unit energy, we
make use of $dp/dE=E/p$ to obtain:

$\frac{dN_{a}^{\rm{Coal.}}}{dE}=\Gamma_{\gamma\gamma\rightarrow
a}^{\rm{Coal.}}(T)\,\frac{pE}{2\pi^{2}}\,f_{a}^{eq}(E)\,.$ (3.11)

We will apply approximations for later analytical calculations in order to
simplify the expressions, as done in [16]. Neglecting the photon thermal
"mass" and the temperature corrections, Eq.(3.9) simplifies into:

$\Gamma_{\gamma\gamma\rightarrow a}^{\rm{Coal.}}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ \Gamma_{a\gamma\gamma}\frac{m}{E}\,.$ (3.12)

Integrating over a standard solar model (in this work we use the Saclay solar
model [46]), we obtain the differential flux at Earth for axions produced by
the coalescence of two photons:

$\displaystyle\Phi^{\rm{Coal.}}_{a}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4\pi d_{ST}^{2}}\int
d^{3}r\frac{dN^{\rm{Coal.}}_{a}}{dE}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{R_{\odot}^{3}}{4\pi
d_{ST}^{2}}4\pi\Gamma_{a\gamma\gamma}\frac{pm}{2\pi^{2}}\int_{0}^{1}d\bar{r}\bar{r}^{2}f_{a}^{eq}(E)$
(3.13) $\displaystyle=$ $\displaystyle\Big{(}7.45\times
10^{11}\leavevmode\nobreak\
\mathrm{cm^{-2}\,s^{-1}\,keV^{-1}}\Big{)}\leavevmode\nobreak\
g_{10}^{2}\,m^{4}\,p\int_{0}^{1}d\bar{r}\bar{r}^{2}f_{a}^{eq}(E)\,,$ (3.14)

where $g_{10}$ is the dimensionless coupling described before, and $m$ and $p$
are given in keV. Still, when performing the integral, this expression
slightly differs from the one derived by Di Lella et al. [16]

$\Phi_{a}\simeq\Big{(}1.68\times 10^{9}\leavevmode\nobreak\
\mathrm{cm^{-2}\,s^{-1}\,keV^{-1}}\Big{)}\leavevmode\nobreak\
g_{10}^{2}\,m^{4}p\,\bigg{(}{\frac{10}{0.2+E^{2}}+1+6\times
10^{-4}E^{3}}\bigg{)}e^{-E}\,.$ (3.15)

Figure 1: Energy dependency of the coalescence rate. We plot $f_{C}(E)\,e^{E}$
where $f_{C}(E)=\int_{0}^{1}d\bar{r}\bar{r}^{2}f_{a}^{eq}(E)$. The solid black
line is obtained with a Maxwell-Boltzmann distribution, and the solid red line
with a Bose-Einstein distribution. The dotted purple line shows the
approximation derived in [16].

For comparison, Fig. 1 shows the energy dependency of the coalescence rate in
Eq. (3.14) and (3.15). For example, in the energy range of interest,
$E\sim\mathcal{O}(1-10)\,\mathrm{keV}$, the approximations applied by Di Lella
et al. would overestimate the flux of photons by about 25% after integrating
in the case $\delta=1$.

###  3.3 Plasmon decay

In the thermal environment of the Sun, the excitations of the electromagnetic
field follow non-trivial dispersion relations. These excitations, either
longitudinal or transverse, are called "plasmons". The presence of plasmons in
the solar interior opens up the possibility of producing KK axions by decay of
a transverse plasmon, $\gamma_{T}\rightarrow\gamma_{L}+a$, since this process
becomes kinematically allowed. In a thermal plasma, transverse and
longitudinal photons have different dispersion relations:

$\displaystyle\omega_{T}^{2}$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\Big{(}1+\frac{k_{T}^{2}}{\omega_{P}^{2}+k_{T}^{2}}\frac{T}{m_{e}}\Big{)}+k_{T}^{2}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ k_{T}^{2}+\omega_{P}^{2}\,,$ (3.16)
$\displaystyle\omega_{L}^{2}$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\Big{(}1+3\frac{k_{L}^{2}}{\omega_{P}^{2}}\frac{T}{m_{e}}\Big{)}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ \omega_{P}^{2}\,,$ (3.17)

where $k_{T,\,L}$ are the transverse and longitudinal 3-momenta, and
$\omega_{P}$ is the plasma frequency defined previously. In the Sun,
$T\lesssim 1.3$ keV and $T/m_{e}\lesssim 0.0025$, so the dispersions relations
can be well approximated by the second equality in Eqs. (3.16) and (3.17).

We will compute the rate of production of axions with momentum $p$ and energy
$E$ from:

$\frac{dN_{a}^{\rm{Decay}}}{dE}=\Gamma_{\gamma_{T}\rightarrow
a\gamma_{L}}^{\rm{Decay}}(T)\frac{pE}{2\pi^{2}}f_{a}^{eq}(E)\,,$ (3.18)

where

$\Gamma_{\gamma_{T}\rightarrow
a\gamma_{L}}^{\rm{Decay}}(T)=\frac{1}{2E}\int\frac{d^{3}k_{T}}{(2\pi)^{3}2\omega_{T}}\frac{d^{3}k_{L}}{(2\pi)^{3}2\omega_{L}}(2\pi)^{4}\delta^{4}(k_{T}-k_{L}-p)|M_{\gamma_{T}\rightarrow
a\gamma_{L}}|^{2}(1+f^{eq}(\omega_{T}))\,,$ (3.19)

and the transition amplitude is given by [47]:

$|M_{\gamma_{T}\rightarrow
a\gamma_{L}}|^{2}=g^{2}_{a\gamma\gamma}\frac{|(\mathbf{e}_{T}\times\mathbf{k}_{T})\cdot\mathbf{k}_{L}|^{2}}{k_{L}^{2}}\frac{\omega_{P}^{4}}{k_{L}^{2}+\omega_{P}^{2}}\,,$
(3.20)

in which $\mathbf{e}_{T}$ is a polarization vector such that
$\mathbf{e}_{T}\cdot\mathbf{k}_{T}=0$.

The details of the derivation of the plasmon decay rate are presented in
Appendix A. As it will be discussed in section 4, we are interested in non-
relativistic KK axions so we restrict our analysis to the limit $p/k_{T}\ll
1$, for which we get:

$\Gamma_{\gamma_{T}\rightarrow
a\gamma_{L}}^{\rm{Decay}}\simeq\frac{g^{2}_{a\gamma\gamma}}{24\pi}\,\frac{\omega_{P}^{4}\,k_{T}}{k_{T}^{2}+\omega_{P}^{2}}\,\Big{(}\frac{p}{E}\Big{)}^{2}\,.$
(3.21)

We now have all the elements to give the expression for the differential flux
at Earth for KK non-relativistic axions produced by plasmon decay:

$\Phi^{\rm{Decay}}_{a}\simeq\Big{(}2.0\times 10^{12}\leavevmode\nobreak\
\mathrm{cm^{-2}\,s^{-1}\,keV^{-1}}\Big{)}\leavevmode\nobreak\ \
g_{10}^{2}\,\int_{0}^{1}d\bar{r}\bar{r}^{2}\,\frac{\omega_{P}^{4}\,k_{T}}{k_{T}^{2}+\omega_{P}^{2}}\,\frac{p^{3}}{E}\,f_{a}^{eq}(E)\,.$
(3.22)

The differential fluxes at Earth for Primakoff, Coalescence and Plasmon decay
are shown in Fig. 2, for different values of the axion mass as indicated in
the plot. The Primakoff differential rate has been computed with the
transition rate in Eq. (3.1), including the thermal photon mass and using the
approximation $\kappa\simeq 7T$ [16] for the Debye-Hückel screening scale; for
Coalescence we use Eq. (3.14), and that of plasmon decay has been computed
with the full thermal decay rate given in Eq. (A.6). We have plotted the
fluxes with respect to the axion momentum instead of energy to stress their
behavior at low momentum. Primakoff is the main production channel for axions
upto $m\sim O(10)$ keV and $p\sim E$. However, it will be suppressed with
respect to Coalescence in the limit of small momentum $p\ll E$. Same happens
with the plasmon decay channel: in both cases the rate is proportional to
$p^{2}$ which suppresses the production of non-relativistic KK axions.

Figure 2: Differential flux at Earth for individual axions versus the axion
momentum, for different production channels: black lines for Primakoff, red
lines for Coalescence, and green ones for Plasmon decay. We have taken
$g_{10}=10^{-2}$, and different values of the axion mass as indicated in the
plot.

We could also consider the production due to the conversion of thermal photons
in the presence of the large-scale solar magnetic field [48, 49, 50]. However,
given the profile of the solar magnetic field, the conversion is mainly
efficient near the resonance for $m\sim O(10-100)$ eV [50]. Because we are
interested in keV axions, this channel for us will be subdominant when
compared to coalescence or Primakoff.

##  4 Gravitationally trapped KK axions

Part of the KK axions produced in the Sun will be sufficiently massive and
non-relativistic to be trapped in the solar gravitational field. Those KK
axions will start orbiting the Sun and they will accumulate over the Sun
history. The large astrophysical times involved in this picture would result
in a significant amount of trapped KK axions which overpasses by several
orders of magnitude the amount of KK axions instantaneously emitted by the
Sun. This scenario of trapped KK axions was first proposed by DiLella and
Zioutas [23] to explain some astrophysical issues as the heating of the solar
corona, but also as a window for a possible detection of KK axions in a Time
Projection Chamber (TPC). Recently, the phenomenological implications of such
"stellar basins" of massive particles has been studied in more details
yielding to striking consequences on direct and indirect detection [51].

The article from DiLella and Zioutas about trapped KK axions [23] pioneered
the domain and its results (obtained by simulations for a given set of
parameters) have been used as reference for all searches for KK axions [28,
30, 31]. This article is almost 20 years old and, as far as we know, no
additional theoretical work has been conducted about trapped KK axions. This
paper aims to cover this issue by deriving analytically some phenomenological
quantities and by updating the astrophysical constraints on the model with
recent measurements.

###  4.1 Number density of KK axions

The number density of trapped KK axions as a function of the distance to the
Sun represents a crucial quantity for the determination of the event rate in a
detector and also for some astrophysical constraints as presented later in
Section 5. Since the Primakoff process and the plasmon decay are suppressed
for low-momenta, it can be shown [23] that the amount of trapped KK axions is
about 3 orders of magnitude larger for the production via the coalescence of
two photons than for the other processes. For this reason, we only consider
the coalescence of two photons in the following calculations.

The details of the calculations of the number density are presented in
Appendix B. The derivation starts by looking at the gravitational potential
inside the Sun to determine the trajectories of the trapped KK axions. For
instance, we show that in order to reach the Earth, the kinetic energy per
unit mass of particles produced at Sun surface must be in the range
$[0.995363,0.995374]$ in units of $GM/R_{\odot}$. The next step is to
integrate the Boltzmann equation over momenta for which the trajectories are
bounded at a distance $r$ to the Sun. Finally, we integrate over the KK tower
and time, taking into account the potential decay of the axions. We arrive to
the following expression for the number density of trapped KK axions:

$\displaystyle n_{KK}^{(T)}(t,r)$ $\displaystyle=$
$\displaystyle\Big{(}2.2\times 10^{14}{\rm
cm}^{-3}\Big{)}\,g_{10}^{2}\,\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\int
dm\,(Rm)^{\delta}m^{5}\,\frac{1-e^{-t_{\odot}\Gamma_{a\gamma\gamma}}}{t_{\odot}\Gamma_{a\gamma\gamma}}$
(4.1)
$\displaystyle\times\,\int_{0}^{1}\bar{r}_{0}^{2}d\bar{r}_{0}\,\frac{1}{e^{E/T}-1}\,\frac{m}{E}\,I_{v}[\bar{r},\bar{r}_{0}]\,,$

where $g_{10}=g_{a\gamma\gamma}\times 10^{10}\,\mathrm{GeV^{-1}}$, $\bar{r}$
is the distance to the Sun normalized by the solar radius, $m$, $R^{-1}$, $T$
and $E$ are given in keV, and the product $t_{\odot}\Gamma_{a\gamma\gamma}$
must be dimensionless; the second integral must be performed over a solar
model. Finally, $I_{v}[\bar{r},\bar{r}_{0}]$ is the integral over velocities
(in the plane of the trapped orbit) of the probability density for an axion of
energy $E$ to be at radius $\bar{r}$. In Appendix B we give two expressions
for the quantity $I_{v}[\bar{r},\bar{r}_{0}]$ depending on which expression we
use for the probability density:

$\displaystyle I_{v}^{0}(\bar{r}_{0},\bar{r})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\bar{r}^{4}}\sqrt{2\bigg{(}\bar{\Phi}_{G}(\bar{r}_{0})-\frac{1}{\bar{r}}\bigg{)}}\,,$
(4.2) $\displaystyle I_{v}^{1}(\bar{r}_{0},\bar{r})$ $\displaystyle=$
$\displaystyle\int_{\bar{v}_{min}}^{\bar{v}_{max}}d\bar{v}\bar{v}^{2}\frac{1}{10\pi^{2}}\frac{\big{(}2\bar{\Phi}_{G}(\bar{r}_{0})-\bar{v}^{2}\big{)}^{7/2}}{\Big{(}\bar{v}^{2}-2\big{(}\bar{\Phi}_{G}(\bar{r}_{0})-\frac{1}{\bar{r}}\big{)}\Big{)}^{1/2}}\,,$
(4.3)

in which $\bar{\Phi}_{G}(\bar{r}_{0})$ is the gravitational potential
normalized by its value at Sun surface, and $\bar{v}$ is the velocity
normalized by $(GM/R_{\odot})^{1/2}$.

The number density behaves as $1/r^{4}$ as pointed out in [23]. For
$\delta=2$, $g_{10}=9.2\times 10^{-4}$ and $R=10^{3}\,\mathrm{keV^{-1}}$ (set
of parameters used in [23]) we obtain

$\displaystyle n_{KK}^{(T)}(t_{\odot},R_{\odot})$ $\displaystyle\simeq$
$\displaystyle 0.15\times 10^{16}\,{\rm cm}^{-3}\,,$ (4.4) $\displaystyle
n_{KK}^{(T)}(t_{\odot},1\,\rm{A.U.})$ $\displaystyle\simeq$ $\displaystyle
0.79\times 10^{6}\,{\rm cm}^{-3}\,,$ (4.5)

which is about 50 times lower than the number density at Earth obtained in
[23]. Possible explanations of such a difference are difficult to point out
since very few details are given in the article about how the simulation
works. In Figure 3 we show the trapped number density for different choices of
parameters, and we have compared the number density of trapped KK axions
computed with the different probability densities (solid and dashed lines):
there is only a factor of 2 difference.

Figure 3: Number density of trapped KK axions for model parameters as
indicated in the plot versus the radial distance in units of the solar radius.
We have taken $R_{1}=2\times 10^{5}$, $R_{2}=10^{3}$, $R_{3}=1$, all given in
keV-1. Solid lines are obtained with (4.2), while dashed lines correspond to
(4.3). The dotted line indicates the value obtained in [23] for
$n_{KK}^{(T)}(t_{\odot},1\,\rm{A.U.})$ and $\delta=2$, $g_{10}=9.2\times
10^{-4}$.

###  4.2 Monte Carlo simulation

Since the number density plays a crucial role for phenomenology, we aim to
cross-check the analytic calculations with a Monte Carlo (MC) simulation, both
approaches using a different philosophy.

The backbone of the simulation is to produce a given amount of KK axions in
the Sun and to follow the trajectory of each of them by solving the Equations
of Motion (EoM). In this way, one knows the position of each KK axion at any
time and one can compute the number density at a distance $r$. The number of
KK axions of mass $m$ produced by the Sun evolves in time as:

$\frac{dN_{a}(t,m)}{dt}=-\Gamma_{a\gamma\gamma}(m)\,N_{a}(t,m)+P_{a}(m)\,,$
(4.6)

where $P_{a}(m)$ is the solar production rate. It implies that, at present
time $t=t_{\odot}$ and summing over the mode multiplicity, one obtains:

$N_{a}\big{(}\delta,R,g_{10},t_{\odot}\big{)}=\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\,R^{\delta}\int_{0}^{\infty}dm\,m^{(\delta-1)}\,\frac{P_{a}(m)}{\Gamma_{a\gamma\gamma}(m)}\,\bigg{(}1-e^{-t_{\odot}\Gamma_{a\gamma\gamma}(m)}\bigg{)}\,.$
(4.7)

We have already seen that the coalescence mechanism is the main source of
trapped axions. The solar axion production, per energy and time unit, through
the coalescence process is given by:

$P_{a}(m,E)=\frac{g_{10}^{2}}{32\pi^{2}}\,m^{4}\,\sqrt{E^{2}-m^{2}}\,\int_{Sun}\,dr\,\frac{r^{2}}{e^{E/T(r)}-1}\,,$
(4.8)

in which the integration is over a solar model. In this work we use the Saclay
solar model [46]. The present total number of KK axions susceptible of being
trapped is then given by

$\begin{split}N_{a}\big{(}\delta,R,g_{10},t_{\odot}\big{)}=\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\,&R^{\delta}\int_{0}^{\infty}dm\,\frac{m^{(\delta-1)}}{\Gamma_{a\gamma\gamma}(m)}\,\bigg{(}1-e^{-t_{\odot}\Gamma_{a\gamma\gamma}(m)}\bigg{)}\\\
&\times\int_{m}^{\infty}\,dE\,\frac{g_{10}^{2}}{32\pi^{2}}\,m^{4}\,\sqrt{E^{2}-m^{2}}\,\int_{Sun}\,dr\,\frac{r^{2}}{e^{E/T(r)}-1}\,.\end{split}$
(4.9)

The principle of the MC simulation is to generate $n_{MC}$ KK axions of mass
$m$ and energy $E$, to follow them in their trajectories around the Sun, and
to count the proportion that will be located in a $1\,\mathrm{cm^{3}}$ box at
distance $r$ to the Sun. Since the solar axion production is isotropic this
box can be located at any position in the $(\phi,\theta)$ plane. We then scale
$n_{MC}$ according to Eq.(4.9) to get the present number density of KK axions.

The randomness of the MC comes from the choice of the initial conditions in
the EoM solver: $r_{0}$ is taken randomly in the distribution of Eq.(4.8);
$\theta_{0}$ and $\phi_{0}$ are randomly chosen in a way to construct an
isotropic angular distribution; and the duration of the integration,
$t_{rand}$, is taken randomly in a power law distribution such that $t_{rand}$
is large compared to the orbit period.

A comparison between the MC simulation and the analytical derivation is
presented in Figure 4. For $r<40\,R_{\odot}$ the two approaches differ by less
than 20%. For larger $r$ the statistics of the MC becomes too low and we
observe significant statistical fluctuations. This 20% difference is small and
is likely due to numerical uncertainties and approximations made in both
approaches. The MC simulation agree on the order of magnitude obtained by the
analytical derivation and thus it cross-checks the calculations.

Figure 4: Comparison of the numbers density obtained from the MC simulation
and from the analytical derivation. We have taken $\delta=2$,
$R=10^{3}\,\mathrm{keV^{-1}}$ and $g_{10}=9.2\times 10^{-4}$, which are the
parameters cited in [23].

The code of the MC simualtions is provided on a public repository
111https://github.com/BeaufortC/KKaxionDensity.

##  5 Constraining the solar KK axion model

The existence of extra dimensions and of KK axions would alter some
astrophysical and cosmological considerations. To be viable, the solar KK
axions model must be in agreement with experimental constraints. In this
section we review the constraints applying on the model in order to determine
the parameter space in which KK axions should be searched for.

###  5.1 Solar luminosity from KK axion

The first constraint that we will consider is the one that puts the most
stringent limits on the solar KK axion model. Paradoxically, it is also the
one that brings the most interest in searching for KK axions to possibly
explain the non-thermal X-ray emission of the Sun. As explained in Section 4,
part of the KK axions gets trapped in the solar gravitational field and
accumulates. A KK axion eventually decays into two photons and contributes to
the solar luminosity. This solar luminosity due to trapped KK axions should
not exceed the observed solar luminosity, requirement that sets constraints on
the parameters of the model.

The maximal value for the axion-photon coupling obtained by DiLella and
Zioutas using ASCA/SIS X-ray data [23] has been reduced by more than one order
of magnitude few years later using measurements of the quiet Sun from RHESSI
[52, 53]. While the solar axion luminosity is stable during short time
periods, the standard solar luminosity depends on the Sun’s activity. For this
reason, we must compare our predictions to measurements of quiet Sun when the
solar X-ray emissions are not related to flares, sunspots or active regions.
In this work, we refer to solar X-rays measurements in the range
$(1-6)\,\mathrm{keV}$ from the SphinX spectrophotometer [54, 55] during a
period of deep solar minimum in 2009 known for its extremely low solar
activity.

The SphinX spectrometer is orbiting the Earth and it has a conical field of
view (FoV) of $2^{\circ}$. We want to determine the flux of photons coming
from KK axion decays that enters the FoV of SphinX. Considering for simplicity
that the detector covers a cone of aperture $\alpha_{0}$, we should integrate
the number density of trapped KK axions over the solid angle of the detector,
$d\Omega_{D}$, and over a radial distance $D$ from the detector (i.e., the
Earth). Here we follow the approach in references [56, 57, 58, 59]. In these
references, the flux of photons due to decaying dark matter (number of photons
per unit time and unit surface) is given by:

$F_{DM}=\Gamma_{DM}\frac{E_{\gamma}}{m_{DM}}\int_{FoV}\frac{\rho_{DM}(r)}{4\pi|{\bf
r_{D}}+{\bf r}|^{2}}d^{3}{\bf r}\,,$ (5.1)

where $\Gamma_{DM}$ is the decay rate into photons, $r_{D}$ is the distance
between the Sun and the Earth. The integral is done within the FoV of the
detector. For example, for a small FoV, and DM decaying into two photons with
$E_{\gamma}=m_{DM}/2$, one has:

$F_{DM}[\alpha]\simeq\Gamma_{DM}\frac{\Omega_{fov}}{8\pi}\int\rho_{DM}(r)dz\,,$
(5.2)

where the integral is the column integral along the line of sight $z$,
$r=r_{D}^{2}+z^{2}-2r_{D}z\cos\alpha$, and $\alpha$ is the angle between $D$
and $r_{D}$ centered on the Earth.

In our case, the number of trapped KK axions within the detector FoV is given
by (see Appendix C for the details of the derivation):

$\displaystyle N_{a}^{T,\,FoV}(m)$ $\displaystyle=$
$\displaystyle\int_{FoV}d{\bf r}^{3}\frac{r^{2}}{4\pi D^{2}}n_{a}^{T}(r,m)\,$
(5.3) $\displaystyle\simeq$
$\displaystyle\frac{R_{\odot}n_{a}^{T}(R_{\odot},m)}{2\bar{r}_{D}^{3}}\left(2\bar{r}_{D}-\frac{2}{3\bar{r}_{D}}-\frac{1}{2}(1-\frac{\alpha_{0}-\pi}{\tan\alpha_{0}})\right)\,,$

and the flux of photons due to the decay of trapped KK axions and that enters
the FoV of the detector by:

$\frac{dF}{dE_{\gamma}}=2\frac{dF}{dm}=2\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}(Rm)^{\delta}m^{-1}\Gamma_{a\gamma\gamma}N_{a}^{T,\,FoV}(m)\,.$
(5.4)

The spectral photon flux in a SphinX-like detector from the decay of trapped
KK axions is presented in Figure 5, for $\delta=1,\,2$ and different values of
the extra dimension radius222Notice that strictly speaking the curves for
$R=10^{-1}\,\rm{keV}^{-1}$ should start at $E=5$ keV, given that the lightest
state in the KK tower would have a mass $m_{1}=10$ keV, but we include the
full curves for comparison.. Remaining below the SphinX measurements implies
stringent limits on the maximal value of the axion-photon coupling. The set of
parameters proposed by DiLella and Zioutas [23] is now ruled out.

---
Figure 5: The spectral photon flux from trapped KK axions as observed from
Earth and the flux measured by SphinX. The top panel shows the case $\delta=1$
whereas $\delta=2$ is presented on the bottom one. Model parameters are
indicated in the plots and they have been adjusted to remain below the SphinX
measurements. The black dashed line on the bottom panel represents the photon
flux obtained using the parameters from [23].

Measurements of the X-ray spectrum of the quiet Sun in Figure 5 does not
follow an isothermal distribution as expected from the element abundances.
Significant deviations from the theoretical spectrum obtained with the CHIANTI
database are observed for energies above $2.5\,\mathrm{keV}$ [55]. Similar
deviations are observed by RHESSI [53]. This could indicate the presence of
non-thermal components (for instance via particle acceleration in solar
microflares as measured by NuSTAR [60]), but it could also be explained by
trapped KK axions as initially proposed by Zioutas et al. [24]. In Figure 5,
one can see that the predictions of the photon flux from KK axions decay
follow similar tendencies than the spectrum measured by SphinX, in particular
for the case $\delta=1$. Measurements of the solar X-rays spectrum do not only
constraints the KK axions model, they contribute to motivate it as a promising
explanation for the non-thermal behavior of the solar X-rays emission.

Moreover, the decay of trapped KK axions would irradiate the Sun and was
proposed [23] as an explanation for the heating of the solar corona. This
hypothesis is supported by the abrupt temperature changes in the solar corona
that share similar tendencies with the temperature fast increase in the Earth
atmosphere under X-rays irradiation from the Sun [61]. Such an explanation of
the 80-year old solar corona problem remains plausible and is consistent with
simulations of stellar atmosphere irradiation from a hotter X-rays source
[62].

###  5.2 Solar energy loss

KK axions production in the solar core would result in energy loss and would
consequently modify the helioseismological sound-speed profiles and the fluxes
of neutrinos emitted by the Sun. Helioseismology has implied that the solar
axion production would be indistinguishable from standard solar models as long
as $L_{a}^{D}<0.2\,L_{\odot}$ [63], value that has been used to establish the
solar KK axion model [16, 23]. Solar neutrino fluxes have restrained to
$L_{a}^{D}<0.1\,L_{\odot}$ [64] and a study combining helioseismology and
neutrino emissions has set the limit $L_{a}^{D}<0.03\,L_{\odot}$ [65]. The
solar axion luminosity referred here is the direct luminosity from axions
produced inside the Sun, which we distinguish from the trapped axion
luminosity thanks to the superscript $D$ ($L_{a}^{D}$).

The direct solar KK axion luminosity have been derived in [16],

$L_{a}^{D}=A\,L_{\odot}\,g_{10}^{2}\,\bigg{(}\frac{R}{\mathrm{keV}^{-1}}\bigg{)}^{\delta}\,,$
(5.5)

with $A=0.018$ for $\delta=1$, $A=0.19$ for $\delta=2$, and $A=2.1$ for
$\delta=3$. We remind that $g_{10}=g_{a\gamma\gamma}\times
10^{10}\,\mathrm{GeV^{-1}}$. Using the limit $L_{a}^{D}<0.03\,L_{\odot}$ we
obtain the constraint from solar energy loss:

$g_{10}\,\bigg{(}\frac{R}{\mathrm{keV}^{-1}}\bigg{)}^{\delta/2}\leavevmode\nobreak\
<\leavevmode\nobreak\ \begin{cases}1.3&\mathrm{for\leavevmode\nobreak\
\delta=1}\,,\\\ 0.40&\mathrm{for\leavevmode\nobreak\ \delta=2}\,,\\\
0.12&\mathrm{for\leavevmode\nobreak\ \delta=3}\,.\\\ \end{cases}$ (5.6)

###  5.3 Extragalactic background light

The last constraint we will revise relies on the Extragalactic Background
Light (EBL). Axions would be produced in the early Universe and in
astrophysical objects, resulting in a large number of axions all around the
Universe. The decay of an axion into two photons would then contribute to the
luminosity of the Universe. In particular, measurements of the EBL put
stringent limits on Axion-Like Particles (ALP) in the keV-range. One can
compute the axion contribution to the EBL by matching the axion density with
present observed DM density and taking into account the expansion of the
Universe; then, if an axion decays, one must consider the redshift of the
photons and the eventual absorption on atoms of the intergalactic medium. In
this work, we extend the procedure of [45] to the case of KK axions. We
consider the photoionization process on hydrogen and we neglect other
processes as the Compton scattering or the pair production on atoms which do
not play a role in the energy range considered [66].

We need to compare the EBL measurements with the intensity of the photons flux
from the KK axions decay denoted by $I_{E}$, multiplied by the energy to
recover usual units. It is given by:

$EI_{E}=\int_{2E_{0}}^{\infty}dm\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}R^{\delta}m^{(\delta-1)}\frac{3E_{0}\rho(z_{0},m)}{4\pi
m}xe^{-x}e^{-\kappa_{P}(E_{0})}\,,$ (5.7)

where the subindex “0” denotes values today,
$x=\Gamma_{a\gamma\gamma}t_{0}(2E_{0}/m)^{3/2}$, $\rho(z_{0},m)$ is the energy
density today, and $\kappa_{P}(E_{0})$ is the absorption factor due to
photoionization process:

$\displaystyle\kappa_{P}(z,E)$ $\displaystyle=$
$\displaystyle\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})(1+z^{\prime})}n_{H}(z^{\prime})\sigma_{P}(E)\,,$
(5.8) $\displaystyle\sigma_{P}$ $\displaystyle=$ $\displaystyle 4.57\times
10^{-20}\left(\frac{10\,\rm{keV}}{E}\right)^{7/2}\,{\rm cm}^{2}\,,$ (5.9)

where $n_{H}$ is the hydrogen number density, $\sigma_{P}$ the hydrogen
photoionization cross-section, $H$ the Hubble parameter and $z$ the redshift.
In Eq.(5.9), we approximate the photoionization cross-section from [67] by a
power-law.

The main difference with standard ALPs lies in the superposition of states in
the KK tower. To determine the EBL contribution due to KK axions, one must
consider how the energy is distributed among the KK modes, which is given by
the spectral function $\rho(z_{0},m)$. This distribution depends on the
production mechanism but also on mode-mixing: when the system starts
oscillating, the energy gets redistributed among the whole KK tower. The
system starts oscillating during the phase transition at which the axion gets
a mass and, following the approach of [68], one can parametrize the time
dependent mass like:

$m(t)=\frac{m}{2}\left(1+{\rm
erf}\big{(}\frac{1}{\sqrt{2\delta_{G}}}\ln\frac{t}{t_{G}}\big{)}\right)\,,$
(5.10)

where $m$ is the late time axion mass, $m(t_{G})=m/2$, and $\delta_{G}$
controls how fast the transition takes place. In general, increasing
$\delta_{G}$ suppresses the energy density in the KK tower; while increasing
$mt_{G}$ leads to opposite effect, the energy density tends to be more
democratically distributed. The authors of [68] define the parameter:

$\eta_{KK}=1-{\rm max}_{\lambda}\frac{\rho_{\lambda}}{\rho}\,,$ (5.11)

where the subindex $\lambda$ refers to the mass eigenstate, and $\rho$ is the
total energy density. Small values of $\eta_{KK}$ indicate that most of the
energy density is stored practically in one mode.

The next step is to determine the mass distribution function $\rho(z_{0},m)$.
The derivation of this function should follow from integrating the EoM of the
coupled systems, but such a detailed study is beyond the scope of this work.
Instead, we proceed by making a plausible ansatz for this function by
considering a Gaussian of width $\Delta_{K}$:

$\displaystyle\rho(z_{0},m)$ $\displaystyle=$
$\displaystyle\bar{\rho}_{KK}f_{G}(\bar{m},\Delta_{K})\,,$ (5.12)
$\displaystyle f_{G}(\bar{m},\Delta_{K})$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi\Delta_{K}}}exp(-(\bar{m})^{2}/\Delta_{K})\,,$
(5.13)

where $\bar{m}=m/m_{1}$, $m_{1}=1/R$, and $\bar{\rho}_{KK}$ is a parameter
with units of energy density. We choose to normalize the function
$f_{G}(\bar{m},\Delta_{K})$ to have unit area, such that:

$\rho_{KK}\leavevmode\nobreak\ =\leavevmode\nobreak\
\bar{\rho}_{KK}\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\int_{1}^{\infty}d\bar{m}{\bar{m}}^{(\delta-1)}f_{G}(\bar{m},\Delta_{K})\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ \bar{\rho}_{KK}I_{\delta}^{\infty}\,.$ (5.14)

We still need to relate the value of $\bar{\rho}_{KK}$ with the DM energy
density today. Note that within the KK tower most of the modes will have
lifetimes smaller than the age of the universe and will not contribute to the
present DM energy density. The heaviest stable mode is given by:

$m_{DM}=\left(\frac{1.327\times
10^{-4}}{g_{a\gamma\gamma}^{2}\tau_{u}}\right)^{1/3}=\bigg{(}1.46\times
10^{-2}\,\rm{keV}\bigg{)}\,g_{10}^{-2/3}\,,$ (5.15)

and the fraction of KK modes that contribute to the DM energy density is:

$\rho_{KK}^{DM}\leavevmode\nobreak\ =\leavevmode\nobreak\
\bar{\rho}_{KK}\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\int_{1}^{\bar{m}_{DM}}d\bar{m}{\bar{m}}^{(\delta-1)}f_{G}(\bar{m},\Delta_{K})\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ \bar{\rho}_{KK}I_{\delta}^{\bar{m}_{DM}}\,.$ (5.16)

Finally, using the present value of the DM energy density, we obtain:

$\displaystyle EI_{E}$ $\displaystyle=$ $\displaystyle\bigg{(}1.28916\times
10^{9}{\rm nW}\leavevmode\nobreak\ {\rm m}^{-2}\leavevmode\nobreak\ {\rm
str}^{-1}\bigg{)}$ $\displaystyle\times$
$\displaystyle\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\,g^{2}_{10}\,\frac{\eta_{KK}}{(1-\eta_{KK})I_{\delta}^{\infty}+\eta_{KK}I_{\delta}^{\bar{m}_{DM}}}\,\left(\frac{E_{0}}{m_{1}}\right)^{5/2}\left(\frac{m_{1}}{\rm{keV}}\right)^{3}\,F_{A\delta}\bigg{\\{}\frac{E_{0}}{m_{1}},g_{10},m_{1}\bigg{\\}}\,,$
$\displaystyle
F_{A\delta}\bigg{\\{}\frac{E_{0}}{m_{1}},g_{10},m_{1}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\int_{\bar{m}_{min}}^{\infty}d\bar{m}f_{G}(\bar{m},\Delta_{K})\bar{m}^{(\delta-1/2)}e^{-C_{x}\bar{m}^{3/2}}\,e^{-\kappa_{P}(E_{0})}\,,$
(5.17) $\displaystyle C_{x}$ $\displaystyle\equiv$ $\displaystyle 92\times
g_{10}^{2}\left(\frac{E_{0}}{m_{1}}\right)^{3/2}\left(\frac{m_{1}}{\rm{keV}}\right)^{3}\,.$
(5.18)

Each KK mode cannot contribute to the spectrum with energies $E_{0}$ larger
than half their mass, and therefore $\bar{m}_{min}=2E_{0}/m_{1}$ when
$2E_{0}\geq m_{1}$ and one otherwise.

We are now ready to compare the expected light from KK axions decay to the EBL
measurements. We approximately parametrize the upper limit of EBL published in
[69] and [70] by the formula:

$EI_{E}^{max}\simeq 0.20412\times\lambda[\mathring{A}]^{0.47317}\,{\rm
nW\leavevmode\nobreak\ m}^{-2}\leavevmode\nobreak\ {\rm str}^{-1}\,.$ (5.19)

---
Figure 6: Top panel: spectral intensity function from the decay of the KK
tower for a set of parameters indicated in the plot. Solid lines correspond to
$g_{a\gamma\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$ and dashed lines are obtained
for $g_{a\gamma\gamma}=10^{-13}\,\mathrm{GeV^{-1}}$. The purple dotted line
$EI_{E}^{\rm{max}}$ corresponds to the experimental limit given by Eq.(5.19).
Bottom panel: maximum allowed value of $\eta_{KK}$ in order to have
$EI_{E}\leq EI_{E}^{max}$.

In Figure 6 are presented the EBL predictions from our model for several sets
of parameters (top panel) and $\delta=2$ as an example. On the bottom panel we
have plotted the maximum allowed value for $\eta_{KK}$ to guaranty that the
contribution remains below the observational constraint as a function of the
compactification radius $R$. In general, the smaller the width of the
Gaussian, the easier to fulfil the EBL constraint since the energy density
left in the tower diminishes. However, this trend is reverse when increasing
$g_{a\gamma\gamma}$ (and therefore their decay rate into photons) and
$R\lesssim 1\,\rm{keV}^{-1}$, as shown on the bottom panel in Figure 6.
Nevertheless, while the EBL puts stringent limits in the case of standard ALPs
in the energy range considered [45], KK axions can easily escape this
constraint, depending on how the energy is distributed among the KK tower.

The revision and the update of the constraints that apply on the solar KK
axion model, as detailed in this section, brings us to the determination of
possible sets of parameters. Some of them are proposed in Table 2. For
instance, we see that the case $\delta=1$, $R=10^{3}\,\mathrm{keV}$, and
$g_{10}=9.8\times 10^{-4}$ fulfils the constraints and enables to have the
fundamental gravitational scale, $M_{\ast}$, as well as the overall scale of
the PQ symmetry breaking, $\bar{f}_{PQ}$, near the electroweak scale.

$(\delta\,,\,n)$ | $R\leavevmode\nobreak\ [\rm{keV^{-1}}]$ | $g_{10}$ | $\eta_{KK}$ | $\Delta_{K}$ | $M_{\ast}\leavevmode\nobreak\ [\mathrm{TeV}]$ | $f_{PQ}\leavevmode\nobreak\ [\mathrm{GeV}]$ | $\bar{f}_{PQ}\leavevmode\nobreak\ [\mathrm{GeV}]$
---|---|---|---|---|---|---|---
$(1\,,\,2)$ | $5\times 10^{-1}$ | $7.3\times 10^{-3}$ | $8\times 10^{-6}$ | 1 | $2.0\times 10^{3}$ | $1.6\times 10^{9}$ | $6.4\times 10^{2}$
$(1\,,\,2)$ | $10^{3}$ | $9.8\times 10^{-4}$ | – | – | $4.4\times 10^{1}$ | $1.2\times 10^{10}$ | $7.1\times 10^{2}$
$(2\,,\,2)$ | $5\times 10^{-1}$ | $3.2\times 10^{-3}$ | $3\times 10^{-5}$ | 1 | $2.0\times 10^{3}$ | $3.6\times 10^{9}$ | $5.9\times 10^{-4}$
$(2\,,\,3)$ | $5\times 10^{-1}$ | $3.2\times 10^{-3}$ | $3\times 10^{-5}$ | 1 | $5.4$ | $3.6\times 10^{9}$ | $2.1\times 10^{-1}$
$(3\,,\,3)$ | $5\times 10^{-1}$ | $1.7\times 10^{-3}$ | $10^{-4}$ | 1 | $5.4$ | $6.8\times 10^{9}$ | $3.1\times 10^{-6}$
Table 2: Possible sets of parameters allowed experimentally and the
corresponding quantities defined in Section 2.

##  6 Experimental searches for KK axion

Searching for standard axion represents a considerable challenge due to its
feeble coupling to the particles of the Standard Model. As already stressed
out, the situation changes drastically when considering large extra dimensions
in which the axion can propagate. In this framework, the large multiplicity of
the KK modes and their masses in the keV-range open a window for original
detection strategies. This new point of view on the phenomenology could lead
to a shift in the way of searching for axion, and the KK axion could be used
as a probe for extra dimensions.

###  6.1 Event rate of the KK axion decay in a detector on Earth

The first approach proposed to search for KK axions was the decay channel,
$a\rightarrow\gamma\gamma$, since the decay could happen in a detector on
Earth, the event rate depending not on the type of detector but only on its
volume. The signature of the decay of a KK axion is the emission of two
photons back-to-back, each of them carrying the same energy $E_{\gamma}=m/2$
since the gravitationally trapped axions are non-relativistic. In a Time
Projection Chamber (TPC) with a 3D track reconstruction, as the MIMAC detector
[71] for instance, such a signature is almost unique, so it would lead to a
very efficient background discrimination. Searching for KK axions via the
detection of the photons emitted by the decay has been proposed in [16, 27,
28, 29] and was used to set experimental constraints on solar KK axions [30,
31].

The differential decay rate of trapped KK axions into photons is given by:

$\frac{dR}{dm}\leavevmode\nobreak\ =\leavevmode\nobreak\
\Gamma_{a\gamma\gamma}\,\frac{dn_{KK}^{(T)}}{dm}\leavevmode\nobreak\
=\leavevmode\nobreak\ \Big{(}6.51\times
10^{-12}\,\mathrm{day^{-1}}\Big{)}\,g_{10}^{2}\,\bigg{(}\frac{m}{\mathrm{keV}}\bigg{)}^{3}\,\frac{dn_{KK}^{(T)}}{dm}\,,$
(6.1)

and is represented in Figure 7. The KK axion mass ranges from roughly
$(1-30)\,\rm{keV}$ with an expected peak at $m=2E_{\gamma}\simeq O(10)$ for
each set of parameters.

Figure 7: Differential decay rate of trapped KK axions into photons in a
detector on Earth for the sets of parameters given in Table 2.

Integrating Eq.(6.1) over the mass range we finally obtain the even rate. For
the set of parameters in Table 2 and Figure 7, the event rate in a detector on
Earth gives $R\simeq 3\times 10^{-7}\,\mathrm{day^{-1}\cdot m^{-3}}$. The
revision of the number density and the updated constraints from solar
measurements have strongly reduced the event rate of $R\simeq
0.21\,\mathrm{day^{-1}\cdot m^{-3}}$ that was obtained from the DiLella and
Zioutas model [28].

This event rate forces to have a huge TPC-active volume requiring many
observables to cope with the huge volume background. The high rejection power
needed could be foreseen built on an upgraded MIMAC-type micro-TPC with 3D
track reconstruction at few keV energies [72, 73].

A last comment about other axion production channels. So far we have
restricted our study to hadronic axion models, even though considering a non-
zero axion-electron coupling, $g_{ae}$, offers additional mechanisms to
produce KK axions in the Sun at a lower mass. The solar photon flux due to the
electron-axion coupling will be proportional to
$g_{a\gamma\gamma}^{2}g^{2}_{ae}$, while that due to coalescence is
proportional to $g_{a\gamma\gamma}^{4}$. And the same coupling dependence
holds for the detection rate. So let us call $F_{\rm{SphinX}}$ the measured
solar photon flux, and $F^{\gamma}=g_{a\gamma\gamma}^{4}\bar{F}^{\gamma}$ and
$F^{e}=g_{a\gamma\gamma}^{2}g^{2}_{ae}\bar{F}^{e}$ the flux due to coalescence
and axion-electron production respectively, such that:

$g_{a\gamma\gamma}^{2}(g_{a\gamma\gamma}^{2}\bar{F}^{\gamma}+g^{2}_{ae}\bar{F}^{e})\leq
F_{\rm{SphinX}}\,.$ (6.2)

Similarly denoting by $R_{\gamma,\,e}$ the event rate from axion-photon or
axion-electron coupling respectively, and using the "bar" notation to
normalize quantities from values at unit coupling, one can then show that:

$R=g_{a\gamma\gamma}^{2}(g_{a\gamma\gamma}^{2}\bar{R}_{\gamma}+g^{2}_{ae}\bar{R}_{e})=g_{a\gamma\gamma}^{4}\bar{R}_{\gamma}+(\frac{F_{\rm{SphinX}}}{\bar{F}^{e}}-g_{a\gamma\gamma}^{4}\frac{\bar{F}^{\gamma}}{\bar{F}^{e}})\bar{R}_{e}\,.$
(6.3)

Eq.(6.3) tells us that considering the axion-electron coupling to produce KK
axions via other processes in the Sun would only have minor influence on the
constraint on $g_{a\gamma\gamma}$ imposed by measurements of the solar X-rays
spectrum. In other words, any detector searching for KK axions via the
coupling to photons will be limited by the constraint on $g_{a\gamma\gamma}$
from solar luminosity regardless of the production mechanisms.

###  6.2 Other strategies of detection

A promising approach to search for axions lies on the use of a magnetic field
to convert an axion into a photon via the reverse Primakoff process. Such a
detection strategy can be adapted to the case of KK axions. The CERN Axion
Solar Telescope (CAST) has already set limits on KK axions in the frame of two
additional dimensions [25, 26]. Since axion helioscopes are improving their
sensitivities, dedicated searches for KK axions via the reverse Primakoff
process could give interesting results. In a similar way, the experiments of
"light shining through a wall" use the Primakoff effect in the aim of
detecting the photon regeneration process $\gamma\rightarrow
a\rightarrow\gamma$. Adapting the design of such experiments to the framework
of large extra dimensions could open a detection channel for KK axions.

The search for KK axions can also be performed via indirect processes. In
Section 5, we have shown examples of how the phenomenology of the Sun or of
the Extragalactic Background Light could be affected by the possible existence
of KK axions. Burst of axions are expected in supernovae leading to a photon
flux after axion decay or after axion-photon conversion. Oscillations between
axions and photons are also expected, which would modify the photon fluxes
from distant sources. Revisiting X-rays and $\gamma$-rays measurements in the
framework of KK axions instead of standard ALP could lead to interesting
results. We have shown that the trapped KK axions density evolves as
$1/r^{4}$, with $r$ the distance to the Sun. Measurements of the solar photon
flux at multiple distances to the Sun could reveal this $1/r^{4}$ behavior.
For instance, the STIX telescope [74] placed on Solar Orbiter will soon
measure hard X-rays at distances to the Sun down to $0.22\,A.U.$.

Finally, we have restrained our work to hadronic axion models in which the
axions do not couple to fermions at tree level. Our study could be later
extended by considering the axion-electron coupling, $g_{ae}$, that offers
additional production mechanisms in the Sun [75] as well as other channels for
detection. The main projects of dark matter direct detection have started to
search for standard ALP since, for a non-zero axion-electron coupling, axions
could induce events of the electron-recoils class [76, 77]. Next generation of
axion helioscopes, as the IAXO [78] for instance, could also improve their
sensibility if axions couple to electrons. Once again, the existence of large
extra dimensions would alter the phenomenology of the axions, and detection of
KK axions via the axion-electron coupling appears as conceivable.

##  7 Conclusion

The scenario of axions propagating in large extra dimensions addresses several
theoretical issues $-$ as the hierarchy problem, the strong CP problem, or the
nature of dark matter $-$ while offering a new scenario for explaining
astrophysical observations. In this work we have entirely revised the solar KK
axion model in which the particles are trapped in the gravitational field of
the Sun. We here corroborate some results of the paper that pioneered the
model [23]: (1) the number density of trapped KK axions evolves as $1/r^{4}$;
(2) photon coalescence is the dominant production mechanism of trapped KK
axions in the Sun; (3) the decay event rate $a\rightarrow\gamma\gamma$ in a
detector on Earth is mainly constrained from measurements of the solar
luminosity.

The revision proposed in this paper aimed to provide analytical expressions of
phenomenologically interesting quantities relevant for experimental searches
of KK axions, and mainly to update the constraints with recent measurements.
We have computed the fluxes of KK axions produced in the Sun via the dominant
processes in the case of hadronic models in which axions do not couple to
fermions at tree level. The derivation of the number density of trapped KK
axions represents the central element of the model. We obtain a density
lowered by $\sim\mathcal{O}(50)$ compared with the previous estimation and we
have cross-checked our result by an independent MC code provided on a public
repository.

The properties of ALP in the keV-range are usually constrained by the
measurement of the Extragalactic Background Light. However, depending on how
the energy is distributed among the KK tower, the KK axions can escape this
cosmological bound. The most stringent constraint on the model comes from the
X-rays measurements of the quiet Sun. This limit on the decay event rate in a
detector cannot be avoided: the amount of KK axions decaying on Earth is
correlated to the amount of KK axions that decay in between the Earth and the
Sun and which contribute to the observed solar luminosity. We show that any
strategy of detection making use of the axion-photon coupling,
$g_{a\gamma\gamma}$, will experience this astrophysical bound, even when
considering production mechanisms via the axion-electron coupling. We derive a
decay event rate of about $3\times 10^{-7}\cdot\mathrm{day^{-1}\cdot m^{-3}}$
which lies $6$ orders of magnitude below previous estimations [23, 28], with
major consequences on current experiments searching for KK axions via this
channel.

The X-rays measurements of the quiet Sun strongly limit the decay event rate
in a detector though they also bring additional motivation for searching for
KK axions. We have shown that the non-thermal component of the X-rays
distribution could possibly be explained by the decay of trapped KK axions.
This appealing hypothesis should be further investigated, for instance by
studying the physics of KK axions during solar microflares or by upgrading our
analysis with upcoming STIX measurements at $0.22\,\mathrm{A.U.}$. Further
work could also extend the study of KK axions to non-hadronic models that
offer additional production mechanisms in the Sun [75] and open other channels
for detection. The axion-electron coupling could play an interesting role in
solar microflares.

Searches for KK axions via the decay in a detector on Earth appears as very
challenging for present TPCs. However, complementary strategies of detection,
from direct or indirect approaches, could be used in the near future to search
for KK axions. The framework of large extra dimensions offers new
possibilities for the physics of the axion. Current experiments searching for
ALP could be revisited and extended to this scenario. The axion could then be
used as a probe for extra dimensions.

## Acknowledgments

We wish to thank Barbara and Janusz Sylwester who kindly shared with us the
SphinX measurements of solar X-rays and who answered with details our
questions about the working principles of the SphinX detector. MBG would like
to thank the LPSC (UGA, CNRS, Grenoble INP) for kind hospitality. The work of
MBG has been partially supported by MICINN
(PID2019-105943GB-I00/AEI/10.13039/501100011033) and “Junta de Andalucía"
grants P18-FR-4314 and A-FQM-211-UGR18.

##  Appendix A Plasmon decay rate

To compute the rate of KK axion production from the plasmon decay
$\gamma_{T}\rightarrow a+\gamma_{L}$ we need the thermal decay rate:

$\Gamma_{\gamma_{T}\rightarrow
a\gamma_{L}}(T)=\frac{1}{2E}\int\frac{d^{3}k_{T}}{(2\pi)^{3}2\omega_{T}}\frac{d^{3}k_{L}}{(2\pi)^{3}2\omega_{L}}(2\pi)^{4}\delta^{4}(k_{T}-k_{L}-p)\big{|}M_{\gamma_{T}\rightarrow
a\gamma_{L}}\big{|}^{2}(1+f^{eq}(\omega_{T}))\,,$ (A.1)

where [47]:

$\big{|}M_{\gamma_{T}\rightarrow
a\gamma_{L}}\big{|}^{2}=g_{a\gamma\gamma}^{2}\frac{\Big{|}(\mathbf{e}_{T}\times\mathbf{k}_{T})\cdot\mathbf{k}_{L}\Big{|}^{2}}{k_{L}^{2}}\frac{\omega_{P}^{2}}{k_{L}^{2}+\omega_{P}^{2}}\omega_{L}^{2}=g_{a\gamma\gamma}^{2}\frac{|\mathbf{k}_{T}\times\mathbf{k}_{L}|^{2}}{2k_{L}^{2}}\frac{\omega_{P}^{2}}{k_{L}^{2}+\omega_{P}^{2}}\omega_{L}^{2}\,,$
(A.2)

and
$\delta^{4}(k_{T}-k_{L}-p)=\delta^{3}(\mathbf{k}_{T}-\mathbf{k}_{L}-\mathbf{p})\delta(\omega_{T}-\omega_{L}-E)$
enforced energy and momentum conservation:

$\displaystyle E$ $\displaystyle=$ $\displaystyle\omega_{T}-\omega_{L}\,,$
(A.3) $\displaystyle\mathbf{p}$ $\displaystyle=$
$\displaystyle\mathbf{k}_{T}-\mathbf{k}_{L}\,,$ (A.4) $\displaystyle
k_{L}^{2}$ $\displaystyle=$ $\displaystyle
k_{T}^{2}+p^{2}-2k_{T}p\cos\theta_{T}\,.$ (A.5)

Neglecting the thermal correction for the time being (i.e., taking
$f^{eq}(\omega_{T})\ll 1$), we have:

$\displaystyle 2E\Gamma_{\gamma_{T}\rightarrow a\gamma_{L}}(T)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\cdot\frac{1}{4\omega_{L}\omega_{T}}\int
d^{3}k_{T}\delta(\omega_{T}-\omega_{L}-E)\,\big{|}M_{\gamma_{T}\rightarrow
a\gamma_{L}}\big{|}^{2}$ (A.6) $\displaystyle=$
$\displaystyle\frac{g^{2}_{a\gamma\gamma}}{2\pi^{2}}\cdot\frac{\omega_{L}^{2}}{4\omega_{L}\omega_{T}}\int
d\Omega_{T}\frac{k^{2}_{T}}{|dW/dk_{T}|}\cdot\frac{\omega_{P}^{2}}{k_{L}^{2}+\omega_{P}^{2}}\cdot\frac{k_{T}^{2}p^{2}(1-\cos^{2}\theta_{T})}{k_{L}^{2}}$
$\displaystyle=$
$\displaystyle\frac{g^{2}_{a\gamma\gamma}}{32\pi}\cdot\frac{\omega_{L}^{2}\omega_{P}^{2}k_{T}^{2}}{\omega_{T}p}\int
d\cos\theta_{T}\frac{1-\cos\theta_{T}^{2}}{(A-\cos\theta_{T})(B-\cos\theta_{T})(C-\cos\theta_{T})}\,,$

where in the second line we have:

$\displaystyle W$ $\displaystyle=$ $\displaystyle\omega_{T}-\omega_{L}-E\,,$
(A.7) $\displaystyle\frac{dW}{dk_{T}}$ $\displaystyle=$
$\displaystyle\frac{k_{T}}{\omega_{T}}-\frac{k_{L}}{\omega_{L}}\frac{dk_{L}}{dk_{T}}=\frac{p}{\omega_{L}}\left(\frac{E}{\omega_{T}}\cdot\frac{k_{T}}{p}-\cos\theta_{T}\right)\,,$
(A.8)

and in the last line in (A.6) we have defined:

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{E}{\omega_{T}}\cdot\frac{k_{T}}{p}\,,$ (A.9)
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{k_{T}}{p}+\frac{p}{k_{T}})\,,$ (A.10)
$\displaystyle C$ $\displaystyle=$ $\displaystyle
B+\frac{\omega_{P}^{2}}{2k_{T}p}\,.$ (A.11)

For slow-moving axions, in the limit $p/k_{T}\ll 1$ we have:

$2E\Gamma_{\gamma_{T}\rightarrow
a\gamma_{L}}\simeq\frac{g^{2}_{a\gamma\gamma}}{16\pi}\frac{\omega_{P}^{2}\omega_{L}^{2}p^{2}}{k_{T}E}\cdot\frac{k_{T}^{2}}{k_{T}^{2}+\omega_{P}^{2}}\int\int
d\cos\theta_{T}(1-\cos\theta_{T}^{2})\simeq\frac{g^{2}_{a\gamma\gamma}}{12\pi}\frac{\omega_{P}^{2}\omega_{L}^{2}p^{2}}{k_{T}E}\cdot\frac{k_{T}^{2}}{k_{T}^{2}+\omega_{P}^{2}}\,,$
(A.12)

and therefore in this limit:

$\frac{dN_{a}}{dE}\simeq\frac{f_{a}^{eq}(E)}{2\pi^{2}}\frac{g^{2}_{a\gamma\gamma}}{24\pi}\frac{\omega_{P}^{4}k_{T}}{k_{T}^{2}+\omega_{P}^{2}}\cdot\frac{p^{3}}{E}\,,$
(A.13)

where we have replaced $\omega_{L}^{2}\simeq\omega_{P}^{2}$.

##  Appendix B Derivation of the number density

We start by deriving the gravitational potential inside the Sun from the
Poisson equation:

$\displaystyle\Phi_{G}(r<R_{\odot})$ $\displaystyle=$ $\displaystyle-4\pi
G\left(\frac{1}{r}\int_{0}^{r}{r^{\prime}}^{2}\rho(r^{\prime})dr^{\prime}+\int_{r}^{R}r^{\prime}\rho(r^{\prime})dr^{\prime}\right)$
(B.1) $\displaystyle=$
$\displaystyle-\frac{GM}{R_{\odot}}\left(1+\int_{\bar{r}}^{1}\frac{\bar{M}}{\bar{r}^{2}}\right)d\bar{r}=-\frac{GM}{R_{\odot}}\bar{\Phi}_{G}(\bar{r})\,,$

where in the second line “bar” quantities are normalized by their Sun value.
Bounded trajectories follow from the condition $v_{r}=\dot{r}=0$, which can be
written as:

$v_{r,0}^{2}+r_{0}^{2}{\dot{\phi}_{0}}^{2}(1-\bar{r}_{0}^{2}\bar{u}_{r}^{2})=\frac{2GM}{R_{\odot}}(\bar{\Phi}_{G}(\bar{r}_{0})-\bar{u}_{r})\,,$
(B.2)

where the subindex “0” refers to the initial condition for radius and
velocities inside the Sun, and $\bar{u}_{r}=R_{\odot}/r$. In terms of the
initial kinetic energy per unit mass,
$\bar{E}_{K}=(v_{r,0}^{2}+r_{0}^{2}\dot{\phi}_{0}^{2})/2$, and using
$a^{2}=v_{r,0}^{2}/(2\bar{E}_{K})$, we have:

$\bar{E}_{K}(\bar{r}_{0})=\frac{GM}{R_{\odot}}\cdot\frac{\bar{\Phi}_{G}(\bar{r}_{0})-\bar{u}_{r}}{1-\bar{r}_{0}^{2}(1-a^{2})\bar{u}_{r}^{2}/2}\,.$
(B.3)

Bounded orbits at a radial distance $r$ are obtained for particles produced
inside the Sun with kinetic energy in the range $(E_{K}^{min},E_{K}^{max})$
depending on the initial angular momentum. To reach the Earth, for particles
produced at the Sun surface, their kinetic energy must be in the range
[0.995363, 0.995374] in units of $GM/R_{\odot}=2v_{esc}^{2}$; for those
produced at half the Sun radius we have instead $E_{K}\in[1.952736,1.95274]$.

In order to get the number density of trapped KK axions, we start again with
the Boltzmann equation for axions produced inside the Sun due to photon
coalescence. We neglect the other production processes since they are
suppressed for low momenta (c.f. Section 3):

$\frac{dn_{a}}{dt}=\frac{1}{(2\pi)^{3}}\int
d^{3}p\Gamma_{a\gamma}f_{a}^{eq}(E)\,,$ (B.4)

where for the rate of production we use the simplification of Eq.(3.12):

$\Gamma_{a\gamma}\simeq\Gamma_{a\gamma\gamma}\frac{m}{E}\,.$ (B.5)

Restricting the momentum integral to those values which give bounded orbits at
a distance $r$ we obtain the number density of trapped KK axions. They will
have small velocities and behave as non-relativistic particles, and therefore:

$d^{3}p=dp_{x}dp_{y}dp_{z}\simeq m^{3}v^{2}dvd\phi d\cos\theta\,.$ (B.6)

Due to isotropy, we can perform the integral along $\theta$ and work in the
$XY$ plane:

$d^{3}p=2m^{3}v^{2}dvd\phi=2m^{3}\frac{v^{2}}{\sqrt{v^{2}-v_{\phi}^{2}}}dvdv_{\phi}\,.$
(B.7)

We can now perform the integral in velocities with the constraint for bounded
trajectories Eq.(B.2), which gives the required velocity at production at a
radius $r_{0}$ inside the Sun to get to a distance $r$ from the center of the
Sun. Integrating over the solar volume we finally obtain the number density of
trapped KK axions $n_{a}^{(T)}$ per unit time:

$\frac{dn_{a}^{(T)}}{dt}=\frac{m^{3}}{(2\pi)^{3}}4\pi
R_{\odot}^{3}\int_{0}^{1}\bar{r}_{0}^{2}d\bar{r}_{0}\int_{0}^{\infty}dv\int_{0}^{v}dv_{\phi}\frac{4v^{2}}{\sqrt{v^{2}-v_{\phi}^{2}}}\cdot
N_{\delta}\,\delta(f_{T}(v,v_{\phi}))\cdot\Gamma_{a\gamma}(T)f_{a}^{eq}(E)\,,$
(B.8)

where

$f_{T}(v,v_{\phi})=v_{r}^{2}+v_{\phi}^{2}(1-\bar{r}_{0}^{2}\bar{u}_{r}^{2})-\frac{2GM}{R_{\odot}}(\bar{\Phi}_{G}(\bar{r}_{0})-\bar{u}_{r})\,.$
(B.9)

The function $N_{\delta}\delta(f_{T})$ is the probability density for a
particle of energy $E(v,v_{\phi})$ to be at radius r, such that

$\int d^{3}rN_{\delta}\delta(f_{T})=1\,.$ (B.10)

Let us consider for the time being only radial trajectories to simplify the
analyses, i.e., $v_{\phi}=0$. The constraint Eq.(B.2) simplifies and the
maximum radial distance of the bounded trajectory is given by:

$\bar{r}_{B}=\frac{2}{2\bar{\Phi}_{G}(\bar{r}_{0})-\bar{v}^{2}}\,,$ (B.11)

where $\bar{v}^{2}=v^{2}/(GM/R_{\odot})$, $\bar{r}_{B}=r_{B}/R_{\odot}$, and
the normalization constant of the probability density is:

$N_{\delta}=\frac{1}{32\pi
R_{\odot}^{3}}(2\bar{\Phi}_{G}(\bar{r}_{0})-\bar{v}^{2})^{4}\,.$ (B.12)

We can now perform the integral over velocities to get the number of trapped
axions:

$I_{v}(\bar{r}_{0},\bar{r})=R_{\odot}^{3}\left(\frac{GM}{R_{\odot}}\right)^{3/2}\int_{\bar{v}_{min}}^{\bar{v}_{max}}d\bar{v}\bar{v}^{2}N_{\delta}\delta(f_{T})=\frac{1}{2\pi\bar{r}^{4}}\sqrt{B(\bar{r},\bar{r}_{0})}\,,$
(B.13)

where we have defined:

$B(\bar{r},\bar{r}_{0})=2(\bar{\Phi}_{G}(r_{0})-\bar{u}_{r})\,.$ (B.14)

The number density per unit time of trapped KKaxions is then:

$\frac{dn_{a}^{(T)}}{dt}=\frac{m^{3}}{4\pi^{3}\bar{r}^{4}}\int_{0}^{1}\bar{r}_{0}^{2}d\bar{r}_{0}\sqrt{B(\bar{r},\bar{r}_{0})}\cdot\Gamma_{a\gamma}(T)\cdot
f_{a}^{eq}(E)\,,$ (B.15)

where $E\simeq
m\big{(}1+\frac{GM}{2R_{\odot}}\sqrt{B(\bar{r},\bar{r}_{0})}\big{)}$. This
gives the $1/r^{4}$ behavior as in [23].

Following [51], the probability density is inversely proportional to the
radial velocity, so instead of a delta function to implement the constraint of
bounded trajectories, one must use:

$P(\bar{r},\bar{r}_{0})=\frac{N_{P}}{\sqrt{\bar{v}^{2}-B(\bar{r},\bar{r}_{0})}}\,,$
(B.16)

where again imposing:

$4\pi R_{\odot}^{3}\int d\bar{r}\bar{r}^{2}P(\bar{r},\bar{r}_{0})=1\,,$ (B.17)

one gets:

$R_{\odot}^{3}N_{P}=\frac{1}{10\pi^{2}}(2\bar{\Phi}_{G}(\bar{r}_{0})-\bar{v}^{2})^{7/2}\,.$
(B.18)

Because trapped axions are non-relativistic, one can approximate $E\simeq m$,
and as before perform the velocity integral:

$I_{v}(\bar{r}_{0},\bar{r})=\int_{\bar{v}_{min}}^{\bar{v}_{max}}d\bar{v}\bar{v}^{2}\frac{1}{10\pi^{2}}\frac{(2\bar{\Phi}_{G}(\bar{r}_{0})-\bar{v}^{2})^{7/2}}{(\bar{v}^{2}-B(\bar{r},\bar{r}_{0}))^{1/2}}\,,$
(B.19)

which can be done numerically and again gives the $1/r^{4}$ behavior for the
trapped axions.

The last step is to sum over the KK-tower, and integrate over time with the
decay term from the Boltzmann equation. For a KK tower of massive axions, the
lifetime of some modes can be smaller than the Sun age, so one must include
production and decay at the same time in the Boltzmann equation. Taking this
into account and summing over the tower of KK axions, we obtain for the number
density of trapped axions333To recover the number density of axions in cm-3,
we recall that $GM/R_{\odot}$ is $GM/R_{\odot}/c^{2}=2.11119\times 10^{-6}$,
and with all other quantities in units of energy, i.e. keV, we get $n_{a}$ in
units of $\rm{keV}^{3}$. Dividing by $(\hbar c)^{3}$ we convert to cm-3.:

$\displaystyle n_{KK}^{(T)}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2\pi^{2}}\cdot\left(\frac{GM}{R_{\odot}}\right)^{3/2}\cdot\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}R^{\delta}\cdot\frac{g_{a\gamma\gamma}^{2}}{64\pi}\int
dm\frac{m^{(\delta+5)}}{\Gamma_{a\gamma\gamma}}(1-e^{-t_{\odot}\Gamma_{a\gamma\gamma}})$
(B.20) $\displaystyle\times\int_{0}^{1}\bar{r}_{0}^{2}d\bar{r}_{0}\cdot
f_{a}^{eq}(m)\frac{m}{E}I_{v}[\bar{r},\bar{r}_{0}]\,,$

where the function $I_{v}$ is given either by Eq.(B.13) or Eq.(B.19) depending
on the probability density considered.

Finally, collecting all factors:

$n_{KK}^{(T)}(t,r)\simeq 2.2\times 10^{14}{\rm cm}^{-3}\cdot g_{10}^{2}\cdot
I[\bar{r},\delta]\,,$ (B.21)

where $g_{10}=g_{a\gamma\gamma}\times 10^{10}$ GeV-1, and the integral is
given by444When doing the integral, $\Gamma_{a\gamma\gamma}t_{\odot}$ is
dimensionless, i.e., if $t_{\odot}$ is given in $s$ then
$\Gamma_{a\gamma\gamma}$ is given in $s^{-1}$. :

$I[\bar{r},\delta]=\frac{2\pi^{\delta/2}}{\Gamma[\delta/2]}\int
dm(Rm)^{\delta}m^{5}\cdot\frac{1-e^{-t_{\odot}\Gamma_{a\gamma\gamma}}}{\Gamma_{a\gamma\gamma}t_{\odot}}\times\int_{0}^{1}\bar{r}_{0}^{2}d\bar{r}_{0}\cdot
f_{a}^{eq}(m)\frac{m}{E}\cdot I_{v}[\bar{r},\bar{r}_{0}]\,,$ (B.22)

where $m$ the axion mass, $R^{-1}$, $T$ and $E$ are given in keV.

##  Appendix C KK axions within the FoV of SphinX

Given the number density of trapped KK axions:

$n_{a}^{T}(r,m)\simeq\frac{n_{a}^{T}(R_{\odot},m)}{\bar{r}^{4}}\,,$ (C.1)

the number of trapped KK axions within the detector FoV is given by:

$\displaystyle N_{a}^{T}(m)$ $\displaystyle=$ $\displaystyle\int_{FoV}d{\bf
r}^{3}\frac{r^{2}}{4\pi
D^{2}}n_{a}^{T}(r,m)=\frac{2\pi}{4\pi}\int_{0}^{\infty}d\leavevmode\nobreak\
D\int^{1}_{\cos\alpha_{0}}d\cos\alpha\leavevmode\nobreak\ n_{a}^{T}(r,m)$
(C.2) $\displaystyle\simeq$
$\displaystyle\frac{R_{\odot}n_{a}^{T}(R_{\odot},m)}{2\bar{r}_{D}^{3}}\int_{0}^{\infty}dx\int_{y_{0}}^{1}dy\frac{1}{(x^{2}+1-2xy)^{2}}\,,$

where we perform the integral from the “Earth” reference system with
$y=\cos\alpha$, $x=\bar{D}/\bar{r}_{D}$, and consider the Sun as a point
source. However, we have to subtract the Sun contribution when $\bar{r}<1$;
i.e., we have to take the limits of integration such that:

$x^{2}-2xy+1>\frac{1}{\bar{r}_{D}^{2}}\,.$ (C.3)

The roots of this equation are given by:

$x_{\pm}(y)=y\pm\sqrt{y^{2}-\Delta^{2}}\,,$ (C.4)

where $\Delta^{2}=1-1/\bar{r}_{D}^{2}$, $x_{\pm}(1)=1\pm\sqrt{1-\Delta^{2}}$
and $x_{\pm}(\Delta)=\Delta$. Therefore, we have now the integral:

$I^{E}=\int_{0}^{\infty}dx\int_{y_{0}}^{1}dy\frac{1}{(x^{2}+1-2xy)^{2}}-\int_{\Delta}^{1}dy\int_{x_{-}}^{x_{+}}dx\frac{1}{(x^{2}+1-2xy)^{2}}=I^{E}_{All}-I^{E}_{Sun}\,.$
(C.5)

Doing first the integral in “$x$”, we have:

$\displaystyle I_{All}^{E}[y]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\frac{1}{(x^{2}+1-2xy)^{2}}$ (C.6)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\pi/2}{(1-y^{2})^{3/2}}+\frac{y}{1-y^{2}}+\frac{1}{(1-y^{2})^{3/2}}\arctan{\frac{y}{\sqrt{1-y^{2}}}}\right)\,,$

and therefore:

$\displaystyle I^{E}_{All}$ $\displaystyle=$
$\displaystyle\int_{y_{0}}^{1}dyI_{All}^{E}[y]=\frac{1}{2}\left(\frac{y}{\sqrt{1-y^{2}}}\cdot\frac{\pi}{2}+\frac{y}{\sqrt{1-y^{2}}}\cdot\arctan{\frac{y}{\sqrt{1-y^{2}}}}\right)_{y_{0}}^{1}$
(C.7) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\underset{{\alpha\to
0}}{\rm
limit}\frac{\pi}{\tan\alpha}-\frac{\pi}{\tan\alpha_{0}}-1+\frac{\alpha_{0}}{\tan\alpha_{0}}\right)\,.$

Similarly, for the Sun contribution we have:

$\displaystyle I_{Sun}^{E}[y]$ $\displaystyle=$
$\displaystyle\int_{r_{+}}^{r_{-}}dx\frac{1}{(x^{2}+1-2xy)^{2}}=\frac{\sqrt{y^{2}-\Delta^{2}}}{(1-\Delta^{2})(1-y^{2})}+\frac{1}{(1-y^{2})^{3/2}}\arctan{\frac{\sqrt{y^{2}-\Delta^{2}}}{\sqrt{1-y^{2}}}}$
(C.8) $\displaystyle I_{Sun}^{E}$ $\displaystyle=$
$\displaystyle\int_{\Delta}^{1}dyI_{Sun}^{E}[y]=\left(\frac{y}{\sqrt{1-y^{2}}}\arctan{\frac{\sqrt{y^{2}-\Delta^{2}}}{\sqrt{1-y^{2}}}}-\frac{\Delta^{2}}{1-\Delta^{2}}\ln(y+\sqrt{y^{2}-\Delta^{2}})\right)_{\Delta}^{1}$
(C.9) $\displaystyle=$ $\displaystyle\underset{{\alpha\to 0}}{\rm
limit}\frac{\pi/2}{\tan\alpha}-\frac{1}{\sqrt{1-\Delta^{2}}}-\frac{\Delta^{2}}{1-\Delta^{2}}\ln\frac{1+\sqrt{1-\Delta^{2}}}{\Delta}\,.$

The divergence at the Sun center (when $\alpha\rightarrow 0$) cancels out, and
finally we get:

$\displaystyle N_{a}^{T}(m)$ $\displaystyle=$
$\displaystyle\frac{R_{\odot}n_{a}^{T}(R_{v},m)}{2\bar{r}_{D}^{3}}\left(\bar{r}_{D}+(\bar{r}_{D}^{2}-1)\ln\frac{1+\bar{r}_{D}}{\sqrt{\bar{r}_{D}^{2}-1}}-(1-\frac{(\alpha_{0}-\pi)}{\tan\alpha_{0}})\right)$
$\displaystyle\underset{\bar{r}_{D}\gg 1}{\simeq}$
$\displaystyle\frac{R_{\odot}n_{a}^{T}(R_{\odot},m)}{2\bar{r}_{D}^{3}}\left(2\bar{r}_{D}-\frac{2}{3\bar{r}_{D}}-\frac{1}{2}(1-\frac{\alpha_{0}-\pi}{\tan\alpha_{0}})\right)\,.$

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