# Calculated event rates for Axion Detection via Atomic and Nuclear Processes

John D. Vergados  University of Ioannina, Ioannina, Gr 451 10, Greece and
Center for Axion and Precision Physics Research, IBS, Daejeon 34051, Republic
of Korea  Paraskevi C. Divari  Department of Physical Sciences and
Applications, Hellenic Military Academy, Vari 16673, Attica, Greece  Hiroyasu
Ejiri  Research Center for Nuclear Physics, Osaka University, Osaka 567-0047,
Japan

###### Abstract

The possibility of detection of 5.5 MeV and 14.4 keV solar axions by observing
axion induced nuclear and atomic transitions is investigated. The presence of
nuclear transitions between spin orbit partners can be manifested by the
subsequent de-excitation via gamma ray emissions. The transition rates can
also be studied in the context of radiative axion absorption by a nucleus. The
elementary interaction is obtained in the context of the axion-quark couplings
predicted by existing axion models. Then these couplings will be transformed
to the nucleon level utilizing reasonable existing models, which lead to
effective transition operators. Using these operators we calculate the needed
nuclear matrix elements employing wave functions obtained in the context of
the nuclear shell model. With these ingredients we discuss possibilities of
experimental observation of the axion-induced nuclear gamma-rays. In the
second part we will examine the axion induced production of X-rays (axion-
photon conversion) or ionization from deeply bound electron orbits. In this
case the axion electron coupling is predicted by existing axion models, no
renormalization is needed. The experimental signal is the observation of
directly produced electrons and/or the emission of hard X-rays and Auger
electrons, following the de-excitation of the final atom. Critical discussion
is made on the experimental feasibility of detecting the solar axions by using
multi-ton scale NaI detectors.

Axion detection, nuclear excitations, WIMP, event rate, solar axions, MeV
energy axions

###### pacs:

93.35.+d 98.35.Gi 21.60.Cs

##  I Introduction

In the standard model there is a source of CP violation from the phase in the
Kobayashi-Maskawa mixing matrix. This, however, is not large enough to explain
the baryon asymmetry observed in nature. Another source is the phase in the
interaction between gluons ($\theta$-parameter), naively expected to be of
order unity. The non observation, however, of elementary electron dipole
moment, set limits its value to be $\theta\leq 10^{-9}$. This has been known
as the strong CP problem. A solution to this problem has been the P-Q (Peccei-
Quinn) mechanism. In extensions of the Standard Model (S-M), e.g. two Higgs
doublets, the Lagrangian has a global P-Q chiral symmetry $U_{PQ}(1)$, which
is spontaneously broken, generating a Goldstone boson, the axion (a). In fact
the axion has been proposed a long time ago as a solution to the strong CP
problem Peccei and Quinn (1977) resulting to a pseudo Goldstone Boson Weinberg
(1978); Wilczek (1978). The two most widely cited models of invisible axions
are the KSVZ (Kim, Shifman, Vainshtein and Zakharov) or hadronic axion models
Kim (1979); Shifman et al. (1980) and the DFSZ (Dine,Fischler,Srednicki and
Zhitnitskij) or GUT axion model Dine et al. (1981); Zhitnisky (1980). This
also led to the interesting scenario of the axion being a candidate for dark
matter in the universe Abbott and Sikivie (1983); Dine and Fischler (1983);
Preskill et al. (1983) and it can be searched for by real experiments Asztalos
et al. (2010); Duffy et al. (2005); Wagner et al. (2010); Irastorza and Garca
(2012). For a recent review see Shokair et al. . The relevant phenomenology
has also been reviewed Marsh (2016).

Axions could be generated mainly by Bremsstrahlung processes and Compton
scattering Kaplan (1985), originally proposed for star cooling via axion
emission, see e.g. Turner (1997) and a more recent work Carenza et al. (2019)
for Bremsstrahlung processes and Compton scattering studied a long time ago
Borisov and Grishina (1996) and more recently Derbin et al. (2011a). Also
generated via the Primakoff conversion of photons to axions in the presence of
a magnetic field Primakoff (1951). All these yield a continuous axion
spectrum, the Primakoff conversion with an average energy around 4 keV Zioutas
et al. (2005). They can also be generated by nuclear transitions, which
involve monochromatic axions, e.g. 14.4 keV in the case of the de-excitation
of 57Fe* and 5.5 MeV in the case $p+d\rightarrow^{3}_{2}$He*. Axions produced
in nuclear processes are mono-energetic because their energies correspond to
the transition energies to specific nuclear states. These solar axions can be
emitted and escape from the solar core due to the very weak interaction
between the axion and matter.

In the Intensity Frontier of physics a search for new light particles,
including axions among others, is growing. Generally speaking axions can be
produced in an environment of intense photon production. The axion sources
considered so far are cosmological or in the sun. Such environment on earth
can be the core of a nuclear reactor, see e.g. the recent works Sierra et al.
(2021); deNiverville et al. (2021). It is, of course, well known that
neutrinos were first detected in reactors and the most accurate neutrino
oscillation parameters were determined using reactor neutrinos. It is amusing
to think that history may repeat itself, i.e. reactors may have the potential
to produce axion like particles (ALPs) of sufficient intensity through
Primakoff conversion, Compton-like processes, nuclear transitions as well as
dark photons. The (ALPs) can subsequently interact with the material of a
nearby detector via the standard processes, i.e. the inverse Primakoff effect,
the inverse Compton-like scatterings and via axio-electric absorption. It is
claimed Sierra et al. (2021); deNiverville et al. (2021), that reactor- based
neutrino experiments have a high potential to test ALP-photon couplings and
masses.

Searches for solar axions have been carried out with various experimental
techniques: magnetic helioscopes Andriamonje et al. (2009); Arik et al.
(2014), low temperature bolometers Alessandria et al. (2013) and thin foil
nuclear targets Namba (2007). CUORE(Cryogenic Underground Observatory for Rare
Events) Arnaboldi et al. (2004); CUO ; Artusa et al. (2015); Ald ; Cuore
Collaboration (2013) as well as the Majorana Demonstrator Abgrall et al.
(2016) which are designed to search for neutrinoless double beta decay
($0\nu\beta\beta$) using very low background detectors are used for the axion
searches. The Ge detectors in GERDA Agostini et al. (2013) can also be used to
search for dark matter weakly interacting massive particles (WIMPs) and axion
searches Li et al. (2016); F.T. Avignone III (2009); Aalseth et al. (2011).

As we have mentioned high energy axions can be produced in the sun, e.g. in
the keV range, by the de-excitation of 57Fe* F.T. Avignone III et al. (2018)
or even at the MeV range Bellini et al. (2012a); Andriamonje et al. (2010);
Bhusal et al. (2020) via the reaction :

$p+d\rightarrow^{3}_{2}\mbox{He}+a(5.5)\mbox{MeV together with the observed
}p+d\rightarrow^{3}_{2}\mbox{He}+\gamma(5.5)\mbox{MeV}$ (1)

Both solar axions mentioned above can be exploited for axion detection. Those
in the energy range of Eq. (1) are suitable for the study of axion induced
nuclear transitions in the laboratory, while the keV axions for atomic
processes.

To this end in section II we will discuss the particle model needed in
understanding these processes. In section III we discuss how one obtains a set
of axion-nucleon couplings $g_{aN}$, going from the quark to the nucleon
level, needed in constructing the effective nuclear transition operators. In
section IV we will briefly discus the experimental issues involved in the
detection of axions in large detectors involving materials like NaI. In
section V we will derive the cross sections needed for the rates: A) in the
case of spin induced nuclear transitions, B) the radiative axion absorption by
nuclei and C) axion induced weak process induced by axion absorption. Then in
section VI we will briefly discuss the nuclear model needed to evaluate the
cross sections for cases A) and B) above for the Na and I nuclei. In section
VII we will present the obtained nuclear matrix elements for the nuclei of
interest and we will exhibit our results for the relevant cross sections and
the obtained event rates.
After that we will utilize the Fe∗-57 14.4 keV source, which is ideal to study
axion induced atomic processes. Thus, after determining the axion-electron
coupling from the Borexiono experiment in section VIII, we will calculate in
sections IX and X the cross sections for axion induced atomic transitions and
ionization from deeply bound orbits. In this case one expects good
experimental signatures, in addition of observing the primary electrons one
expects signals of hard X-rays and Auger electrons, following the atomic de-
excitation.

##  II The particle model

We remind the reader that the axion, $a$, is a pseudoscalar particle with a
coupling to fermions.
i) The axion coupling to the electron can be described by a Lagrangian of the
form:

${\cal L}=\frac{g_{e}}{f_{a}}i\partial_{\mu}a\bar{\psi}_{e}({\bf
p}^{\prime},s)\gamma^{\mu}\gamma_{5}\psi_{e}({\bf p},s)$ (2)

where $g_{e}$ is a coupling constant and $f_{a}$ a scale parameter with the
dimension of energy. $g_{e}$ can be calculated in various axion models. For an
axion with mass $m_{a}$ it easy to show that in the non relativistic limit:

  * •

the time component $\mu=0$ is given by:

${\cal
L}=\langle\phi|\Omega|\phi\rangle,\quad\Omega=\frac{g_{e}m_{a}}{2f_{a}}\frac{\mbox{\boldmath$\sigma$}.{\bf
q}}{m_{e}},\quad{\bf q=p^{\prime}-p}$ (3)

which is negligible for $m_{a}<<m_{e}$.

  * •

The space component, $\mu\neq 0$,

${\cal
L}_{aee}=\langle\phi|\Omega|\phi\rangle,\quad\Omega=\frac{g_{e}}{2f_{a}}\mbox{\boldmath$\sigma$}.{\bf
q},\quad{\bf q=p^{\prime}-p}$ (4)

where ${\bf p}$ and ${\bf p}^{\prime}$ are the initial and final electron
momenta, $f_{a}$ the axion decay constant and $\sigma$ the spin of the
electron.

ii) The axion coupling to the quarks is similarly given by:

${\cal L}=\frac{g_{q}}{f_{a}}i\partial_{\mu}a\bar{\psi}_{q}({\bf
p}^{\prime},s)\gamma^{\mu}\gamma_{5}\psi_{q}({\bf p},s),$ (5)

where $g_{q}$ is the coupling constant. The space component, $\mu\neq 0$, in
the non relativistic limit is given by

${\cal
L}=\langle\phi|\Omega|\phi\rangle,\quad\Omega=\frac{g_{aq}}{2f_{a}}\mbox{\boldmath$\sigma$}.{\bf
q},$ (6)

with $\sigma$ the Pauli matrices and $\phi$ the quark wave function and ${\bf
q}$ is the axion momentum.

We will concentrate on the last term involving the operator $g_{aq}\sigma$.
The quantities $g_{aq}$ can be evaluated at various axion models.

$\mbox{me}_{q}=\langle
q|\left(g_{aq}^{0}+g_{as}+g_{aq}^{3}\tau_{3}\right)\sigma|q\rangle,$ (7)

where we have ignored the contribution of heavier quarks and, thus, the
isovector and isoscalar components become

$g_{aq}^{3}=\frac{1}{2}(g_{au}-g_{ad}),\quad
g_{aq}^{0}=\frac{1}{2}(g_{au}+g_{ad}).$ (8)

Some authors use the notation $c_{q}$ instead of $g_{aq}$.

The scale parameter $f_{a}$ has been related to the axion mass $m_{a}$ via the
relation:

$\frac{1}{f_{a}}=\frac{m_{a}}{\Lambda^{2}_{QCD}},$ (9)

with $\Lambda_{QCD}=218\mbox{ MeV}$. For the determination of $\Lambda_{QCD}$
see Particle Data Group Patrignani et al. (2016). More recently, however, a
model independent relation has been provided Cor

$m_{a}f_{a}=5.691(51)\times 10^{3}\mbox{MeV}^{2}$ (10)

We will employ the expression

$m_{a}f_{a}\approx 6000\mbox{MeV}^{2}$ (11)

Taking $f_{a}$ to be $10^{10}$MeV then $m_{a}=0.6$eV.

##  III The effective axion nucleon couplings

In this section we follow the procedure discussed in a previous work F.T.
Avignone III et al. (2018) and references therein. For the benefit of the
reader we are going to give a summary here.

In going from the quark to the nucleon level one can follow a procedure
analogous for the determination of the nucleon spin from that of the quarks
Bali et al. (2012); Li and Thomas (2016). The matrix element at the nucleon
level can be written as:

$\mbox{me}_{N}=\langle N|\left(g_{aq}^{0}(\delta_{0}-\Delta s)+g_{as}\Delta
s+g_{aq}^{3}\tau_{3}\delta_{1}\right)\sigma|N\rangle,$ (12)

where

$\delta_{0}=(\Delta u+\Delta d+\Delta s),\,\delta_{1}=(\Delta u-\Delta d),$
(13)

$\delta_{0}-\Delta s=\Delta u+\Delta d$. The quantities $\Delta u,\,\Delta d$
and $\Delta s$ have been obtain previously F.T. Avignone III et al. (2018).
Alternatively these quantities can be expressed in terms of the quantities $D$
and $F$ defined by Ellis JEL

$\delta_{1}=F+D,\,\delta_{0}=3F-D+3\Delta s.$

The quantity $\delta_{1}$ is essentially fixed by the axial current to be
approximately 1.27 of the vector current. No such constraint exists for the
isoscalar part, so for that we have to rely on models. For the quantities $D$
and $F$ one can use experimental information JEL . Thus, e.g., from hyperon
beta decays and flavor SU(3) symmetry one gets

$\frac{3F-D}{\sqrt{3}}=0.34\pm 0.02.$

Different approach has been suggested by Kaplan Kaplan (1985), based on the
DFSZ Dine and Fischler (1983); Zhitnisky (1980) axion, which is analyzed in
some detail in ref. F.T. Avignone III et al. (2018).

On the other hand measurements of $\nu p$ and $\nu\bar{p}$ elastic scattering
of the recent MicroBooNE experiment Mic  indicate that $\Delta s=\pm 0.036\pm
0.003$. This is also discussed in F.T. Avignone III et al. (2018). It is,
however, concluded F.T. Avignone III et al. (2018) that the relevant couplings
cannot be extracted from experiment in the presence of both proton and neutron
components in the nuclear wave functions.

So we have to rely on the above models. The obtained results are found in
Tables 2 and 3 in ref. F.T. Avignone III et al. (2018). From these the
favorite choice is

$\displaystyle C_{p}$ $\displaystyle=$ $\displaystyle
0.0663,C_{n}=0.0663\mbox{ for }\tan{\beta}=1\mbox{ MODEL E},$ $\displaystyle
C_{p}$ $\displaystyle=$ $\displaystyle 0.2712,C_{n}=-0.1248\mbox{ for
}\tan{\beta}=10\mbox{ MODEL F}.$ (14)

where $C_{p}$ and $C_{n}$ are hadronic couplings for protons and neutrons
respectively and $\tan{\beta}$ is the ratio of the vacuum expectation values
of the two Higgs doublets (see ref. F.T. Avignone III et al. (2018)).
Theoretical preference is given to the higher value of $\tan{\beta}$. It also
yields an isovector coupling larger than the isoscalar, which is sometimes
useful in obtaining the ratio of axion to photon production rates (the photon
production being dominated by the isovector transition).

The effective axion-nucleon coupling can be cast in the form:

$H_{aNN}=\frac{1}{2f_{a}}\left(g_{aN}^{0}+g_{aN}^{3}\tau_{3}\right)\mbox{\boldmath$\sigma$}\cdot{\bf
q},\,g_{aN}^{0}=\frac{1}{2}\left(C_{p}+C_{n}\right),\,g_{aN}^{3}=\frac{1}{2}\left(C_{p}-C_{n}\right)$
(15)

Before proceeding further with the evaluation of the nuclear and atomic matrix
elements of interest we would like to consider the experimental situation in
detecting the resulting signals using a large NaI detector, already developed
and in use for rare event detection.

##  IV Experimental aspects of the axion search by NaI

We will now briefly present some remarks on experimental aspects of the axion
searches by nuclear interactions. The 5.5 MeV solar axion has been well
investigated by using the Borexino detector with 278 tons of
$\mbox{C}_{9}\mbox{H}_{12}$ scintillation detector. Among the interactions
studied are the axion-electron/photon interaction. Then, the upper limits for
the product of the axion flux and the cross section is around $4.5\times
10^{-3}$SNU, much smaller, by almost 3 orders of magnitude, than the rates for
8B neutrinos without oscillation. Note the axion energy and the 8B neutrino
energy are similar. The limits on the axion couplings are derived as
$|g^{3}_{aN}\times g_{aE}|\leq 5.5\times 10^{-13}$. It is noted that the
coupling $|g^{3}_{aN}|^{2}$ is studied from the axion nuclear interaction.
Experimental signals from the axion-induced photo-production are ultra rare
5.5 MeV $\gamma$-rays. The signal rate $N_{S}$ per t y, with t being the
target nuclear mass in units of tonne and y being year, is given as

$N_{S}=600NA^{-1}$ (16)

where $N$ is the signal rate per $10^{27}$ nuclei and $A$ is the mass number.
So we get $N_{S}=0.5$ and $N_{S}=0.005$ for $N=0.1$ and 0.001 in the case of
$A=127$ for Iodine nuclei. We note the signal rates are just of the same order
of magnitude as the rates of the neutrinoless double beta decays (DBDs),
respectively, for the inverted-mass and normal-mass hierarchies (IH and NH )
for typical DBD nuclei. The signal energy of 5.5. MeV is similar to that of
the DBD, i.e. of around 3 MeV. Then one may have to use multi-ton or even
multi-hundred ton scale detectors as used for the IH and NH neutrino-mass DBD
experiments. It is crucial for the rare-event search to reduce (BG) background
counts $B$ per t y to the level of $B\leq N_{S}$ . Backgrounds due to radio-
active impurities to be considered are the $\beta-\gamma$ rays from 214 Bi and
208 Tl. They are known to be serious in case of DBD experiments with the
signals around or below 3 MeV, but are not serious in the present case of the
signals above 5 MeV. Alpha particle backgrounds are separated from the
$\gamma$ -ray signals by a pulse-shape discrimination. The background event
rate for the 8B solar-neutrinos $B_{\nu}$ /t y is expressed Ejiri (2005);
Ejiri et al. (2019); Ejiri and Zuber (2017); Ejiri and Elliott (2014, 2017) as

$B_{\nu}=0.15\times E\Delta,$ (17)

where $E$ is the energy in units of MeV, i.e. 5.5 in the present case, and
$\Delta$ is the ratio of the energy window to E. The 5.5 MeV signal rate for
the realistic axion is too small to be detected beyond the background
neutrinos. Thus the search for such solar axion is not realistic at present.

In case of the 14.4 keV axion search, the BG rate in the energy region is of
the order of or larger than 1/(t y) because of all kinds of radio-active
impurities in addition to the solar-neutrino BG. The background level of
current low-background detectors being used for dark-matter searches is of the
order of $10^{5}/(ty)$. The signal to be considered is the 14.4 keV one from
the axion-induced atomic electron. Then on may search for such axions in the
5eV region, which is interesting but not yet well investigated, by improving
the background level by a few orders of magnitude. Thus it is interesting to
discuss the rate as given in sections IX and X.

##  V Cross sections for nuclear process

Using the effective axion nucleon coupling given by Eq, (15) the nuclear
matrix element (ME) is of the form:

$\mbox{ME}=\langle\Psi_{f}|{\bf O}\cdot{\bf q}|\Psi_{i}\rangle,\,{\bf
O}=\sum_{k}\left(g_{aN}^{0}+g_{aN}^{3}\tau_{3}(k)\right)\mbox{\boldmath$\sigma$}(k)$
(18)

with the summation involving all nucleons. In the case of the MODEL F of Eq.
(14) we find $g_{aN}^{0}=0.073$ and $g_{aN}^{3}=0.197$. This can trivially be
written in the proton-neutron if dictated by nuclear physics.

We will consider two processes
i) nuclear excitation:

$a+A(N,Z)\rightarrow A(N,Z)^{*}$

followed by de-excitation emitting $\gamma$ rays and
ii) axion photoproduction:

$a+A(N,Z)\rightarrow A(N,Z)^{*}+\gamma$

Note that the signals of i) and ii) are 5.5 MeV gammas, which are not
separated. Thus one may consider one of them, the one with the larger signal
rate.

###  V.1 Axion induced nuclear excitation

The capture cross section associated with Fig. 1(a) takes the form

$\sigma(E_{a})=\frac{1}{\upsilon}\frac{1}{2E_{a}}2\pi\delta(E_{a}-\Delta){\cal
M}^{2}$ (19)

where $\Delta$ is the energy transferred to the nucleus. If the nuclear recoil
energy is ignored $\Delta$ corresponds to the nuclear excitation energy.
${\cal M}$ is the invariant amplitude, essentially the nuclear matrix element,
and $1/E_{a}$ is the usual normalization of a scalar field. Summing over the
final substates $M_{f}$ and averaging over the initial substates $M_{i}$ we
find:

$\sigma({E_{a}})=\frac{E_{a}}{qc^{2}}\frac{1}{2E_{a}}2\pi\delta(E_{a}-\Delta)\frac{1}{(2f_{a})^{2}}\frac{1}{3}q^{2}\frac{1}{2J_{i}+1}\langle
J_{f}||{\bf O}||J_{i}\rangle^{2}$ (20)

where $\langle J_{f}||{\bf O}||J_{i}\rangle$ is the spin-reduced matrix
element of the operator ${\bf O}$ discussed above. If the spectrum of the
source is continuous given by a distribution $\rho(E_{a})$ the total cross
section is given by:

$\sigma(\Delta)=\int
dE_{a}\rho(E_{a})\sigma(E_{a})=\frac{\pi}{12f^{2}_{a}}\rho(\Delta)\sqrt{\Delta^{2}-m^{2}_{a}}\frac{1}{2J_{i}+1}\langle
J_{f}||{\bf O}||J_{i}\rangle^{2}$ (21)

with $m_{a}$ the axion mass.

In the present case, however, the spectrum resulting from the process given by
Eq. (1) is monochromatic and we have a resonance. So to proceed further we
assume a Breit-Wigner shape :

$\rho(E)=\frac{1}{2\pi}\frac{\Gamma}{(E-E_{a})^{2}+\Gamma^{2}}$ (22)

with a small width $\Gamma$ determined by the production rate centered at the
monochromatic energy, which is tiny. In fact in the case of 14.4 keV solar
axion for $m_{a}=5$ eV one finds F.T. Avignone III et al. (2018) that the
width $\Gamma=10^{-24}$ eV. The situation for 5.5 MeV 3He solar axions, Eq. 1,
is similar. So it is reasonable to assume a flat distribution centered around
$E_{a}$, that is:

$\rho(E_{a})=\left\\{\begin{array}[]{cc}\frac{1}{2\kappa\Delta},&(1-\kappa)\Delta\leq
E_{a}\leq(1+\kappa)\Delta,\,0<\kappa<1\\\ 0&\mbox{ otherwise}\\\
\end{array}\right.$ (23)

The distribution of Eq. (23) is normalized the same way as that of Eq. (22),
that is to unity. We thus get

$\sigma(\Delta)=\sigma_{aN}\frac{1}{2J_{i}+1}\langle J_{f}||{\bf
O}||J_{i}\rangle^{2},\,\sigma_{aN}=\frac{\pi}{12f_{a}^{2}}\frac{\sqrt{\Delta^{2}-m^{2}_{a}}}{2\kappa\Delta}.$
(24)

For axions with a mass much smaller than its energy we find according to (11)

$\sigma_{aN}=\frac{1}{2\kappa}\times 1.00\times
10^{-42}\mbox{cm}^{2}({m_{a}}/{0.6\mbox{eV}})^{2}$ (25)

or

$\sigma_{aN}\approx\frac{1}{2\kappa}\times 6.28\times\times
10^{-41}\mbox{cm}^{2}({m_{a}}/{5\mbox{eV}})^{2}$ (26)

###  V.2 Axion photoproduction

This process is reminiscent of the old pion photoproduction, see e.g. Ericson
and Rho (1972) for a review and Baer et al. (1975) for nuclear applications.
It involves the absorption of negatively charged pion absorbed by a nucleus
from a bound state in an atomic like orbit. Since the pion is charged the
final nucleus has a different charge than the original one. There exist, of
course, differences in relation to the axion having to do with its coupling,
the energy and the fact that the axion is electrically neutral, so that in
this case the final nucleus coincides with the original one. Thus one expects
the prospect an elastic coherent process as well, with a very large cross
section.

The relevant elementary amplitude at the nucleon level is given in Fig. 1(b).

Figure 1: (a) An axion is absorbed by a nucleus leading to an excited nuclear
state. (b ) A Feynman diagram relevant for axion photoproduction. In both
cases the axion vertex we encounter the coupling given by Eq, (15), while the
EM vertex in (b) we have the usual interaction involving the nuclear magnetic
moment. p, q and k are the initial nucleon, axion and photon 4-momenta
respectively. In both cases of (a) and (b), the out-put signal energies are
the same 5.5 MeV.

The non relativistic reduction of this amplitude leads to an effective
operator with the structure

$\frac{\mu_{B}(N)}{f_{a}2E_{a}}\left(\Omega_{A}+\Omega_{B}+\Omega_{C}\right)$
(27)

with $\mu_{B}(N)$ the Bohr magneton for the nucleon and

$\displaystyle\Omega_{A}$ $\displaystyle=$
$\displaystyle\left(G^{0}_{aN}+G^{3}_{aN}\tau_{3}\right)({\bf q}\times i{\bf
k})\cdot\epsilon_{\lambda},$ $\displaystyle\Omega_{B}$ $\displaystyle=$
$\displaystyle\left(G^{0}_{aN}+G^{3}_{aN}\tau_{3}\right)\mbox{\boldmath$\sigma$}\cdot(i{\bf
k}\times\epsilon_{\lambda})E_{a},\,\Omega_{C}=\left(G^{0}_{aN}+G^{3}_{aN}\tau_{3}\right)\mbox{\boldmath$\sigma$}\cdot\left((\epsilon_{\lambda}\times{\bf
k})\times{\bf q}\right)$ (28)

where $\epsilon_{\lambda}$ and ${\bf k}$ the polarization and momentum of the
photon, $E_{a}$ and q the energy and momentum of the axion. Furthermore

$G^{0}_{aN}=\frac{1}{2}(g_{s}(p)C_{p}+g_{s}(n)C_{n}),\,G^{3}_{aN}=\frac{1}{2}(g_{s}(p)C_{p}-g_{s}(n)C_{n}).$
(29)

Using the values of Eq. (14), MODEL F, we obtain:

$G^{0}_{aN}=0.781,\,G^{3}_{aN}=0.734$ (30)

We notice that in this case we could have elastic transitions involving the
spin independent operator $\Omega_{A}$. So in this case, proceeding as above,
we can write

$\displaystyle{\cal M}^{2}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}_{B}(N)}{f^{2}_{a}4E_{a}^{2}}k^{2}E_{a}^{2}\left(\mbox{ME}_{el}^{2}+\mbox{ME}^{2}_{i\rightarrow
f}\right)\mbox{ with }$ $\displaystyle\mbox{ME}_{el}^{2}$ $\displaystyle=$
$\displaystyle\left\\{\left(G^{0}_{aN}A+G^{3}_{aN}(Z-N)\right)^{2}\frac{q^{2}}{E_{a}^{2}}(1-(\hat{k}\cdot\hat{q})^{2})+\frac{1}{3}\frac{1}{2J_{i}+1}\langle
J_{i}||(G^{0}_{aN}+G^{3}_{aN}\tau_{3})\mbox{\boldmath$\sigma$}||J_{i}\rangle^{2}\left(1+\frac{q^{2}}{E_{a}^{2}}\right)\right\\},$

where $k$ is the photon energy, which in this case is equal to the axion
energy. q is the axion momentum and $q^{2}=E^{2}_{a}-m^{2}_{a}\approx
E^{2}_{a}$ since the axion mass is very small. The first term in the last
expression is reminiscent of the coherent contribution of all nucleons in the
more familiar case of the spin independent contribution in the WIMP-nucleus
scattering (WIMP stands for weakly interacting massive particle). In deriving
this expression we have summed over all polarizations and final magnetic
substates and sum over the initial m-sub-states. The last sum includes the
contribution of both $\Omega_{B}$ and $\Omega_{C}$, which contribute only if
$J_{i}\neq 0$. When averaged over the photon momentum directions the
contribution of the first term is reduced by 2/3.

For transitions to excited states we simply write

$\mbox{ME}_{i\rightarrow f}^{2}=\left\\{\frac{1}{3}\frac{1}{2J_{i}+1}\langle
J_{f}||(G^{0}_{aN}+G^{3}_{aN}\tau_{3})\mbox{\boldmath$\sigma$}||J_{i}\rangle^{2}\left(1+\frac{q^{2}}{E_{a}^{2}}\right)\right\\}$
(32)

The differential cross section, neglecting the energy of the recoiling
nucleus, becomes

$d\sigma(E_{a})=\frac{1}{\upsilon}\frac{1}{2k}\frac{1}{2E_{a}}2\pi\delta(E_{a}-\Delta-k)\frac{1}{(2\pi)^{3}}d^{3}k{\cal
M}^{2},\,\Delta\mbox{ being the excitation energy}\Rightarrow$ (33)
$\displaystyle d\sigma(E_{a})$ $\displaystyle=$
$\displaystyle\frac{E_{a}}{qc^{2}}\frac{1}{2k}\frac{1}{2E_{a}}\frac{1}{(2\pi)^{3}}\frac{\mu^{2}_{B}(N)}{f^{2}_{a}4E_{a}^{2}}k^{2}E_{a}^{2}\left(\mbox{ME}_{el}^{2}+\sum_{f}\mbox{ME}_{i\rightarrow
f}^{2}\right)2\pi\delta(E_{a}-\Delta-k)4\pi k^{2}dk$ (34) $\displaystyle=$
$\displaystyle\frac{E_{a}}{qc^{2}}\frac{1}{4\pi}\frac{\mu^{2}_{B}(N)}{f^{2}_{a}4E_{a}^{2}}\left(\mbox{ME}_{el}^{2}+\sum_{f}\mbox{ME}_{i\rightarrow
f}^{2}\right)kE_{a}\delta(E_{a}-\Delta-k)k^{2}dk\Rightarrow$
$\sigma(E_{a})=\frac{E_{a}}{qc^{2}}\frac{1}{4\pi}\frac{\mu^{2}_{B}(N)}{f^{2}_{a}4E_{a}^{2}}\left(\mbox{ME}_{el}^{2}+\sum_{f}\mbox{ME}_{i\rightarrow
f}^{2}\right)E_{a}(E_{a}-\Delta)^{3},$ (35)

$E_{a}-\Delta$ is, of course, equal to the photon energy. The cross section
can also be expressed in terms of the axion mass:

$\sigma(E_{a})=\sigma_{0}(a\gamma)\left(\mbox{ME}_{el}^{2}+\sum_{f}\mbox{ME}_{i\rightarrow
f}^{2}\right)\left(1-\frac{\Delta}{E_{a}}\right)^{3},\quad\sigma_{0}(a\gamma)=\frac{E_{a}}{qc^{2}}\frac{1}{4\pi}\frac{E_{a}^{2}\mu^{2}_{B}(N)}{4}\frac{1}{f_{a}^{2}}\hskip
3.0pt$ (36)

Thus the scale of the cross section for axion photoproduction for
$f_{a}=10^{7}$ GeV, which corresponds to $m_{a}=0.6$ eV, is

$\sigma_{a\gamma}=6\times
10^{-50}\mbox{cm}^{2}\left(E_{a}/5.5\mbox{MeV}\right)^{2}\left(m_{a}/0.6\mbox{eV}\right)^{2}$
(37)

or

$\sigma_{a\gamma}\approx 4\times
10^{-48}\mbox{cm}^{2}\left(E_{a}/5.5\mbox{MeV}\right)^{2}\left(m_{a}/5\mbox{eV}\right)^{2}$
(38)

This seems suppressed compared, e.g., to the axion-nucleus cross section
discussed above. This is not surprising, since the axion energy of 5.5 MeV is
much smaller then the nucleon mass $E_{a}^{2}\mu^{2}_{B}(N)<<1$. This can be
overcome by the fact that many final states below $E_{a}$ can contribute.
Furthermore the elastic term contains a large coherent contribution, which is
proportional to $A^{2}$. We should also keep in mind that in this case the
cross section includes the radiative part as well.

###  V.3 Weak process induced by axion absorption

This can occur in a process given by the Feynman diagram shown in Fig. 2.

Figure 2: A Feynman diagram relevant for axion induced beta decay

Proceeding as in the previous subsection we find the following possibilities;

$\Omega=\frac{G_{F}}{\sqrt{2}}\frac{1}{f_{a}2E_{a}}C_{n}E_{a}\left(((N\tau_{+}N)j_{0}+(N\tau_{+}\mbox{\boldmath$\sigma$}N)\cdot{\bf
j})-(N\tau_{+}N){\bf j}\cdot\frac{{\bf
q}}{E_{a}}-(N\tau_{+}\mbox{\boldmath$\sigma$}N)\cdot\frac{(-i{\bf q}\times{\bf
j})}{E_{a}}\right)$

where $C_{n}$ is the appropriate axion-neutron coupling appropriate for
negaton decay.

$j_{\mu}=\bar{e}\gamma_{\mu}(1-\gamma_{5})\nu_{e}$

is the usual weak leptonic current. The first two terms are exactly the Fermi
and the GT matrix elements encountered in the allowed beta decay. The other
two contribute similarly, since, for a light axion, $q^{2}/E^{2}_{a}\approx
1$. So one encounters the contribution
$2\left(\mbox{ME}^{2}_{F}+\mbox{ME}^{2}_{GT}\right)$.

The differential cross section takes the form

$d\sigma_{a\beta}=\frac{1}{\upsilon}\frac{1}{2E_{a}}\left(\frac{G_{F}}{\sqrt{2}}\frac{1}{f_{a}}C_{n}\right)^{2}2\left(\mbox{ME}^{2}_{F}+\mbox{ME}^{2}_{GT}\right)m^{2}(p_{e},p_{\nu})2\pi\delta(E_{a}-\Delta-
E_{e}-E_{\nu})\frac{d^{3}{\bf p}_{e}}{(2\pi)^{3}}\frac{d^{3}{\bf
p}_{\nu}}{(2\pi)^{3}}$ (39)

where $\Delta$ is the nuclear energy difference involved,
$m^{2}(p_{e},p_{\nu})$ the standard leptonic matrix element. The total cross
section can be cast in the form:

$\sigma_{a\beta}=\sigma_{0a\beta}\left(\mbox{ME}^{2}_{F}+\mbox{ME}^{2}_{GT}\right)\frac{1}{m_{e}^{5}}\int
p_{e}^{2}dp_{e}p^{2}_{\nu}dp_{\nu}m^{2}(p_{e},p_{\nu})2\pi\delta(E_{a}-\Delta-
E_{e}-E_{\nu})$ (40)

with

$\sigma_{0a\beta}=\frac{1}{2}\frac{1}{(2\pi)^{3}}\left(G_{F}m_{e}^{2}\right)^{2}\frac{m_{e}}{E_{a}}\frac{C^{2}_{n}}{f^{2}_{a}}$

Thus for $f_{a}=10^{7}$ GeV we find:

$\sigma_{0a\beta}=1.1\times 10^{-64}\mbox{cm}^{2}(E_{a}/5.5\mbox{MeV})^{-1}$

Since this corresponds to $m_{a}=0.6$ eV we find

$\sigma_{0a\beta}=1.1\times
10^{-64}\mbox{cm}^{2}(E_{a}/5.5\mbox{MeV})^{-1}(m_{a}/0.6\mbox{eV})^{2}$

or

$\sigma_{0a\beta}=7.6\times
10^{-63}\mbox{cm}^{2}(E_{a}/5.5\mbox{MeV})^{-1}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$

This is much smaller than the $\sigma_{0\gamma}$ discussed above, which is not
surprising due to the fact that we now deal with a weak process. The scale of
this cross section has perhaps been underestimated, scaling it with $m_{e}$
rather than the energy available in the decay. This, however, is not lost,
since it will show up in the integral involved in Eq. (40).

In any case it will be of interest to compare this process with the standard
beta decay. Leaving aside the nuclear matrix elements we find:

$\frac{R_{0a,\beta}}{R_{0\beta}}=\frac{N_{A}\sigma_{0a\beta}\Phi(E_{a})}{\frac{1}{2\pi^{3}}(G_{F}m_{e}^{2})^{2}m_{e}}\approx\frac{10^{27}\times
7.6\times 10^{-63}\times 1.3\times 10^{9}\mbox{s}^{-1}}{0.016\times 9.62\times
10^{-24}\times 7.8\times 10^{20}}=8.3\times 10^{-23}$ (41)

Experimentally, the 5.5 MeV state, if excited , decays immediately by gamma
emission with $10^{-15}$ sec, and thus the weak decays are not considered
experimentally as in many other cases.

Let us hope that a target with $N_{A}>>10^{27}$ is possible111In high energy
experiments searching, e.g., for proton decay, about $10^{32}-10^{33}$ nuclei
have been accumulated, but in this case the decay products are characterized
by high energy, so their detection becomes easier and, in addition. the
radioactivity background is absent.. If this optimistic scenario holds, the
axion induced weak offers many advantages:

  * i)

Many stable nuclei might become unstable. Provided that transitions
$A(N,Z)(0^{+})\rightarrow A(N-1,Z+1)(1^{+})$ become energetically available
with an axion energy of 5.5 MeV. One may think that the targets involved in
double beta decay Vergados et al. (2012) may undergo axion induced weak
transitions. The relevant rates, however, are quite slow since the two
neutrino double beta decay is faster and, in fact, it has been observed in
many systems.
Better yet such transitions involving odd nuclei with many final states, some
of which maybe populated even via the Fermi transition, become available.

  * ii)

In the case all ordinary allowed beta decay transitions the end point energy
will be shifted by 5.5 MeV in the presence of axions.

  * iii)

Axion induced electron capture will exhibit a spectacular feature, which is
the population of an excited state in the final nucleus, not seen before, i.e.
a 5.5 MeV above the highest one hitherto observed, if such a state with a
similar structure exists.This can be a state not populated before, which will
lead to a de-excitation $\gamma$ ray of higher energy than hitherto observed
in the usual electron capture. The states already seen can also be populated
in the exotic process. Thus in this case

$\frac{R^{E_{x}}_{0a,\beta}}{R^{E_{x}}_{0\beta}}=\frac{R_{0a,\beta}}{R_{0\beta}}\left(\frac{E_{a}+\Delta+m_{e}-b-E_{x}}{\Delta+m_{e}-b-E_{x}}\right)^{2}$
(42)

Where $\Delta$ is the available nuclear energy (Q-value), $b$ the binding
energy of the electron and $E_{x}$ is the excitation energy of the final
nuclear state. If the nucleus is chosen so that $\Delta+m_{e}-b-E_{x}$ is very
small for some state, preferably that with the highest excitation energy, one
may get a large enhancement in the rate. This is similar to the well know
nuclear excitation by neutrinos, photons, mesons and neutrons. The excited
states decay mostly by gamma transitions and the weak decay branch is well
know to be negligible ( of the order of $10^{-15}$). The gamma decays are non-
negligible in the double beta decay and axion experiments. The weak decays are
known to be negligible in double beta decays and axion search experiments, and
thus are not considered there.

Before concluding this section we should mention that for axion induced weak
processes the obtained rates are undetectable, especially if one uses the
axion 5.5 MeV Borexino flux. We are not going to elaborate here on this point,
but it will become apparent after the discussion of section VIII.2.

##  VI  The nuclear model

For the evaluation of the cross sections of the processes discussed above, an
essential input is the nuclear matrix elements for some targets of
experimental interest, preferably those that have been used in connection with
dark matter searches. Those are going to be obtained in the context of the
large basis shell model.

###  VI.1 The target 127I

In this work we study the experimentally relevant 127I for axion detection,
The coherent contribution is expected to dominate in radiative axion capture,
but this is pretty much independent of the details of the wave function. The
spin-dependent (SD), however, depends on the details of the nuclear structure
involved.

The construction of the nuclear wave functions needed for the evaluation of
matrix elements entering in the corresponding scattering cross section is
accomplished in the framework of shell model. Specifically, the calculation
includes the valence space $0g_{7/2},1d_{5/2},2s_{1/2},1d_{3/2},0h_{11/2}$ for
protons and neutrons on top of a 100Sn core. The diagonalization of the
Hamiltonian was performed using the shell model code ANTOINE Caurier et al.
(2005). In order to make the calculations feasible, the neutron configurations
were restricted to allow at most two-particle-two-hole excitations from the
full $0g_{7/2},1d_{5/2}$ shells. The number of excitations into the orbitals
$2s_{1/2},1d_{3/2},0h_{11/2}$ leads to a matrix dimension of $7.3\times
10^{7}$. A more extended configuration space has been obtained in Ref. Klos
and et al (2013). Proton many-body configurations were not restricted due to
small number of valence protons. The mass depended Bonn-C interaction Caurier
et al. (2008) was used as the two-body interaction, while the single particle
energies (SPEs): $\epsilon_{0g_{9/2}}=-0.3$ MeV, $\epsilon_{1d_{5/2}}=0.0$
MeV,$\epsilon_{2s_{1/2}}=1.3$ MeV, $\epsilon_{1d_{3/2}}=1.5$ MeV, and
$\epsilon_{0h_{11/2}}=1.9$ MeV are taken from Ref. Ressell and Dean (1997).

To test the quality of the structure calculations we compare the energy
spectrum and three known magnetic moments of relevant 127I isotope. Three
magnetic moments are known for 127I. Experimental and calculated magnetic
moments are given in Table 1 while in Fig.3 the corresponding energy spectrum
is presented. Most of our theoretical levels agree with the experimental ones
to about 100 keV. An extension of the valence space to include configurations
that arise from excitations of one or more particles probably might improve
the agreement between theory and experiment although there are experimental
states that are still uncertain in spin and parity assignment. Note that the
spin and parity of the low lying experimental states at 0.295 MeV and 0.473
MeV are not known. The identification of these states will be a very useful
test for our calculations.

Figure 3: (Color on line) The low lying experimental (left) Hashizume (2011) and theoretical (right) energy spectrum of 127 I. The levels in the figure are labeled by $2J$. Note that the spin and parity of the experimental states at 0.295 MeV and 0.473 MeV are not known.  Table 1: Experimental Hashizume (2011) and calculated magnetic moments in units of nuclear magneton $\mu_{B}(N)$. The magnetic moments are calculated using the standard g factors $g_{s,n}=-3.826$, $g_{s,p}=5.586$, $g_{l,n}=0$ and $g_{l,p}=1$ in nuclear magneton units. state | Experiment | Theory
---|---|---
$5/2^{+}_{gs}$ | 2.81 | 2.92
$3/2^{+}_{1}$ | 0.97 | 0.99
$7/2^{+}_{1}$ | 2.54 | 1.72
Table 2:  Isoscalar and isovector spin depended matrix elements of 127I from ground ($5/2^{+}_{gs}$) to excited states up to to the axion energy of 5.5 MeV, which are relevant for axion photoproduction. We also include the total nuclear matrix elements ME${}^{2}(E_{x})$, including that of the spin independent entering in the coherent mode, which contain the factor $(1-\frac{E_{x}}{E_{a}})^{3}$ related to the photon energy, and the corresponding cross sections. $E_{x}$(MeV) | $\Omega_{0}^{2}$ | $\Omega_{1}^{2}$ | $\Omega_{0}\Omega_{1}$ | ME${}^{2}(E_{x})$ |  $\sigma$$\times 10^{-48}$cm2
---|---|---|---|---|---
0 (coherent) |  |  |  | 4000 | 18714
0.0000 | 1.0056 | 0.4140 | 0.6452 | 0.8042 | 3.217
0.1800 | 0.1076 | 0.1243 | 0.1157 | 0.1326 | 0.480
0.3120 | 0.0194 | 0.0583 | 0.0336 | 0.0417 | 0.140
0.3150 | 0.0007 | 0.0212 | -0.0038 | 0.0064 | 0.022
0.3750 | 0.0472 | 0.0314 | 0.0385 | 0.0452 | 0.146
0.3830 | 0.0023 | 0.0194 | 0.0067 | 0.0105 | 0.034
0.6490 | 0.0029 | 0.0198 | -0.0076 | 0.0054 | 0.015
0.6600 | 0.0567 | 0.0481 | 0.0523 | 0.0603 | 0.164
0.8160 | 0.0241 | 0.0996 | 0.0490 | 0.0643 | 0.159
0.8340 | 0.0793 | 0.1188 | 0.0970 | 0.1120 | 0.274
0.8610 | 0.0753 | 0.0135 | 0.0319 | 0.0477 | 0.114
0.9290 | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.000
0.9780 | 0.3952 | 0.3163 | 0.3536 | 0.4094 | 0.910
1.0440 | 0.0630 | 0.0523 | 0.0574 | 0.0663 | 0.141
1.1570 | 0.0364 | 0.0167 | 0.0247 | 0.0302 | 0.060
1.1700 | 0.0006 | 0.0006 | 0.0006 | 0.0007 | 0.001
1.1930 | 0.0026 | 0.0562 | -0.0121 | 0.0166 | 0.032
1.2790 | 0.0121 | 0.0112 | 0.0117 | 0.0134 | 0.024
1.3660 | 0.0248 | 0.0093 | 0.0152 | 0.0192 | 0.033
1.3910 | 0.0029 | 0.0007 | 0.0014 | 0.0020 | 0.003
1.4780 | 0.0019 | 0.0859 | 0.0127 | 0.0365 | 0.057
1.6380 | 0.0004 | 0.0035 | -0.0012 | 0.0010 | 0.001
1.6780 | 0.0005 | 0.0100 | 0.0022 | 0.0046 | 0.006
1.7610 | 0.0008 | 0.0055 | 0.0021 | 0.0031 | 0.004
1.9370 | 0.0003 | 0.0019 | -0.0008 | 0.0005 | 0.001
1.9650 | 0.0006 | 0.0047 | -0.0017 | 0.0013 | 0.001
2.1040 | 0.0164 | 0.0202 | 0.0182 | 0.0209 | 0.020
2.2370 | 0.0013 | 0.0097 | -0.0035 | 0.0027 | 0.002
2.3140 | 0.0001 | 0.0002 | 0.0002 | 0.0002 | 0.000
2.4280 | 0.0054 | 0.0036 | 0.0044 | 0.0052 | 0.004
2.6030 | 0.0004 | 0.0002 | 0.0003 | 0.0004 | 0.000
2.6650 | 0.0030 | 0.0001 | -0.0005 | 0.0011 | 0.001
2.7420 | 0.0007 | 0.0002 | 0.0004 | 0.0005 | 0.000
3.0070 | 0.0003 | 0.0005 | 0.0004 | 0.0004 | 0.000
3.0620 | 0.0003 | 0.0000 | -0.0001 | 0.0001 | 0.000
3.1740 | 0.0019 | 0.0009 | 0.0013 | 0.0016 | 0.000
3.4440 | 0.0002 | 0.0001 | 0.0001 | 0.0002 | 0.000
3.4890 | 0.0002 | 0.0000 | 0.0001 | 0.0001 | 0.000
3.6020 | 0.0040 | 0.0009 | 0.0019 | 0.0027 | 0.000
3.9230 | 0.0001 | 0.0004 | -0.0002 | 0.0001 | 0.000
3.9680 | 0.0000 | 0.0002 | -0.0000 | 0.0001 | 0.000
4.0600 | 0.0001 | 0.0002 | -0.0001 | 0.0001 | 0.000
4.4340 | 0.0004 | 0.0000 | 0.0001 | 0.0002 | 0.000
4.4560 | 0.0003 | 0.0000 | 0.0001 | 0.0001 | 0.000
4.5550 | 0.0008 | 0.0002 | 0.0004 | 0.0005 | 0.000
4.9430 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.000
4.9570 | 0.0000 | 0.0000 | -0.0000 | 0.0000 | 0.000
5.0370 | 0.0003 | 0.0000 | 0.0001 | 0.0002 | 0.000
5.4340 | 0.0003 | 0.0000 | 0.0001 | 0.0002 | 0.000
5.4730 | 0.0000 | 0.0000 | -0.0000 | 0.0000 | 0.000

The isoscalar spin-reduced matrix elements

$\Omega_{0}=\frac{\langle
J_{f}||\mbox{\boldmath$\sigma$}||J_{i}\rangle_{I=0}}{\sqrt{{2J_{i}+1}}}\mbox{
(isoscalar) },\Omega_{1}=\frac{\langle
J_{f}||\mbox{\boldmath$\sigma$}||J_{i}\rangle_{I=1}}{\sqrt{{2J_{i}+1}}}\mbox{
(isovector) }$ (43)

from the ground state $J_{i}=({5/2}^{+})_{gs}$ to the ground state
$J_{i}=({5/2}^{+})_{gs}$ and from the ground and to the final
$J_{f}={3/2}^{+},{5/2}^{+}$ and ${7/2}^{+}$ states up to almost 6 MeV have
been computed. They are presented in Fig.4. We also summarize the obtained
results in table 2. In this table we also present the small nuclear matrix
elements around the axion energy $E_{a}=5.5$MeV, since the scale of the cross
section involved, see Eq. (25), is orders of magnitude larger than the spin
induced cross section involved in radiative axion capture, Eq. (37). We also
include the huge matrix element involved in the coherent axion
photoproduction.

Figure 4: (Color on line) Isoscalar and isovector distribution of 127I spin-
reduced matrix elements from $J_{i}={5/2}^{+}_{gs}$ to $J_{f}$ relevant for
axion photoproduction. We should mention that the energy of the emitted photon
is $E_{\gamma}=E_{a}-E_{x}$, i.e. the maximum photon energy corresponds to the
gs to gs transition.

Figure 5: (Color on line) Isoscalar and isovector spin matrix elements up to 6 MeV relevant for axion absorption are presented. Note that in the range of 4-6 MeV, in order to make the matrix elements visible, we exhibit them in a separate small window, where the small scale is indicated. Table 3:  Isoscalar and isovector spin depended matrix elements of 23Na from ground ($3/2^{+}_{gs}$) to the various excited states entering the axion induced photoproduction. We also include the total nuclear matrix elements ME${}^{2}(E_{x})$, including the spin independent one appearing in the coherent mode, which contain the factor $(1-\frac{E_{x}}{E_{a}})^{3}$ related to the photon energy, and the relevant cross sections as well. Note that states up to the axion energy of 5.5 MeV are relevant. $E_{x}$(MeV) | $\Omega_{0}^{2}$ | $\Omega_{1}^{2}$ | $\Omega_{0}\Omega_{1}$ | ME${}^{2}(E_{x})$ |  $\sigma$$\times 10^{-48}$cm2
---|---|---|---|---|---
0 (coherent) |  |  |  | 200 | 791
0.0000 | 0.4756 | 0.3450 | 0.4051 | 0.4721 | 1.889
0.4120 | 0.2024 | 0.2589 | 0.2289 | 0.2628 | 0.832
2.3240 | 0.1868 | 0.1286 | 0.1550 | 0.1814 | 0.140
2.7470 | 0.4689 | 0.3665 | 0.4145 | 0.4807 | 0.241
3.8930 | 0.0934 | 0.0928 | 0.0931 | 0.1069 | 0.011
4.3110 | 0.5066 | 0.4589 | 0.4821 | 0.5551 | 0.022
5.2460 | 0.0931 | 0.1184 | 0.1050 | 0.1205 | 0.000
5.7460 | 0.0474 | 0.0708 | 0.0579 | 0.0668 | -
5.9760 | 0.0105 | 0.0046 | 0.0070 | 0.0086 | -

###  VI.2 The target 23Na

The valence space of 23Na is the $sd$ shell, which comprises the $0d_{5/2}$,
$1s_{1/2}$, and $0d_{3/2}$ orbitals, with a 16O core. Full calculations for
this isotope have previously been performed in the work Divari et al. (2000).

Figure 6:  Isoscalar and isovector distribution of 23Na spin-reduced matrix
elements from $J_{i}={3/2}^{+}_{gs}$ to $J_{f}$ relevant for axion
photoproduction. We should mention that the energy of the emitted photon is
$E_{\gamma}=E_{a}-E_{x}$, i.e. the maximum photon energy corresponds to the gs
to gs transition.

As expected from weak interactions the spin induced rates at low excitation
energies are small compared to some canonical value (sum rule). Indeed

$\displaystyle\frac{1}{2J_{i}+1}\sum_{f}\langle J_{f}||{\bf
O}||J_{i}\rangle^{2}$ $\displaystyle=$
$\displaystyle\sum_{M_{i},M_{,}q,f}(-1)^{q}\langle J_{f}M_{f}||{\bf
O}_{q}|J_{i}M_{i}\rangle^{2}=\sum_{M_{i},M_{,}q,f}(-1)^{q}\langle
J_{i}M_{i}|{\bf O}_{-q}|J_{f}M_{f}\rangle|\langle J_{f}M_{f}|{\bf
O}_{q}|J_{i}M_{i}\rangle$ (44) $\displaystyle=$
$\displaystyle\sum_{q}(-1)^{q}\langle J_{i}M_{i}|{\bf O}_{-q}{\bf
O}_{q}|J_{i}M_{i}\rangle=\langle J_{i}M_{i}|{\bf O}\cdot{\bf
O}|J_{i}M_{i}\rangle$

with

${\bf
O}=\sum_{k}\left(g^{0}_{aN}+g^{3}_{aN}\tau_{3}(k)\right)\mbox{\boldmath$\sigma$}(k)$
(45)

As a result Eq. (44) produces an one body part which is

$\left(\left(g^{0}_{aN}\right)^{2}+\left(g^{3}_{aN}\right)^{2}\right)A+2g^{0}_{aN}g^{3}_{aN}(2Z-A),$
(46)

and a two body part which is $2V$ with

$V=\langle
J_{i}M_{i}|\sum_{k\ell}\left((g^{0}_{aN}+g^{3}_{aN}\tau_{3}(k))\mbox{\boldmath$\sigma$}{(k)}\cdot(g^{0}_{aN}+g^{3}_{aN}\tau_{3}(\ell))\right)\mbox{\boldmath$\sigma$}{(\ell)}|J_{i}M_{i}\rangle.$
(47)

Since $V$ is expected to be much smaller than that given by Eq, (46), we find
to a ggod approximation

$S\approx\left(\left(g^{0}_{aN}\right)^{2}+\left(g^{3}_{aN}\right)^{2}\right)A+2g^{0}_{aN}g^{3}_{aN}(2Z-A)$
(48)

We do not know the energy where this strength is located, but is expected to
be in the higher part of the spectrum.

In the special case of interest to us using the couplings given by Eq. (15)
and model F of (14) we find

$\displaystyle S(I)$ $\displaystyle=$
$\displaystyle(0.083^{2}+0.192^{2})\times 127+2\times 0.083\times
0.192\times(-11)=5.1,$ $\displaystyle S(Na)$ $\displaystyle=$
$\displaystyle(0.083^{2}+0.192^{2})\times 23+2\times 0.083\times
0.192\times(-1)=1.0$ (49)

##  VII Event rates

Noting that the oncoming beam of axions is monochromatic, the event rate is
given by:

$R=\sigma\Phi(E_{a})N_{A}$ (50)

where $\sigma$ is cross section obtained above, $N_{A}$ the number of nuclei
in the target and $\Phi(E_{a})$ the oncoming axion flux. We will consider two
cases:
i) the 14.4 keV Fe*57 flux

$\Phi(E_{a})=0.7\times
10^{9}\left(\frac{10^{7}\mbox{GeV}}{f_{a}}\right)^{2}\mbox{cm}^{-2}\mbox{s}^{-1}\mbox{
(14.4 keV solar axion)}$ (51)

The flux has been obtained using the calculated matrix elements F.T. Avignone
III et al. (2018) for the 57Fe* de-excitation and realistic axion nucleon
coupling constants obtained in a variety of particle models.

ii) the 5.5 MeV He-3 flux
The flux of the 5.5 MeV solar axions is obtained under the assumption that the
axion production is dominated by the isovector transition and an axion mass
much smaller than its energy. The standard formula for axion to photon
production rate Peccei (1996); F.T. Avignone III et al. (2018) can be cast in
the form:

$\frac{\Gamma_{a}}{\Gamma_{\gamma}}\approx 8\times
10^{-13}\left({g^{3}_{aN}}\right)^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$
(52)

From this one can obtain the axion flux, if the photonic rate for the process
of Eq. (1) is known. In the Standard Solar Model (SSM) this constitutes the
second stage of the pp solar fusion chain, with the first stage provided by
the two reactions $p+p\rightarrow d+e^{+}+\nu_{e}$ and $p+p+e^{-}\rightarrow
d+\nu_{e}$. As the deuterons produced via this first stage capture protons
within $\tau=6$s, the axion flux resulting from (1) can be expressed in terms
of the known pp neutrino flux Bellini et al. (2012b); Bhusal et al. (2020):

$\Phi_{\nu pp}=6.0\times 10^{10}\mbox{ cm }^{-2}\mbox{s}^{-1}\mbox{ (solar pp
neutrino flux)}$

The extracted flux from the Borexino experiment Bellini et al. (2012b) for the
5.5 MeV axions is given by

$\Phi=2.45\times 10^{-5}\mbox{cm}^{-1}\mbox{s}^{-1}(m_{a}/\mbox{1
eV})^{2}=6.1\times 10^{-4}\mbox{cm}^{-2}\mbox{s}^{-1}(m_{a}/\mbox{5
eV})^{2}=1.86\times 10^{4}\mbox{cm}^{-2}\mbox{y}^{-1}(m_{a}/\mbox{5\ eV})^{2}$
(53)

with the last re-scaled value obtained for $m_{a}=$5eV. This is many orders of
magnitude smaller than that of Eq. (51).

We will consider $N_{A}=4\times 10^{27}$, which is close to 1ton of NaI in the
target.

##  VIII Some results for nuclear transitions

NaI seems to be a ideal material for the detection of $\gamma$ rays. Such a
target has already been employed by the DAMA experiment in dark matter
searches, see e.g.Bernabei et al. (1999), and later in the DAMA/LIBRA
collaboration Bernabei et al. (2008).

###  VIII.1 Axion photoproduction

Let us begin with the axion photoproduction.

A) We will first examine the component 127I of the target. We will consider
two cases
a) Elastic scattering contributions.
The first term is the coherent contribution which amounts to

$\mbox{ME}^{2}_{\mbox{coh}}=\frac{2}{3}(0.781\times
127+0.734\times(-21))^{2}\approx 4\times 10^{3}$ (54)

The second term involves the gs$\rightarrow$gs spin induced transition
obtained from Fig 5 we find:

$\displaystyle\mbox{ME}^{2}_{\mbox{gs}}=\frac{2}{3}\left((G^{0}_{aN}\Omega_{0})^{2}+(G^{3}_{aN}\Omega_{1})^{2}+G^{0}_{aN}G^{3}_{aN}\Omega_{0}\Omega_{1}\right)=$
(55)
$\displaystyle\frac{2}{3}\left((G^{0}_{aN})^{2}1.005+(G^{3}_{aN})^{2}0.414+G^{0}_{aN}G^{3}_{aN}0.645\right)=0.804$

b) Similarly for the transition to the excited state at 0.978 MeV we obtain:

$\displaystyle\mbox{ME}^{2}_{1}=\frac{2}{3}\left((G^{0}_{aN}\Omega_{0})^{2}+(G^{3}_{aN}\Omega_{1})^{2}+G^{0}_{aN}G^{3}_{aN}\Omega_{0}\Omega_{1}\right)=$
(56)
$\displaystyle\frac{2}{3}\left((G^{0}_{aN})^{2}0.395+(G^{3}_{aN})^{2}0.316+G^{0}_{aN}G^{3}_{aN}0.354\right)=0.409$

Using now Eq. (50) we obtain and considering only the above the dominant
contributions we obtain the event rate

$\displaystyle E_{\gamma}=5.5\mbox{MeV}\leftrightarrow R_{\mbox{\small coh}}$
$\displaystyle=$ $\displaystyle 2.9\times 10^{-16}\times 4\times 10^{3}\approx
1.2\times 10^{-12}\mbox{(t-y)}^{-1}$ $\displaystyle
E_{\gamma}=5.5\mbox{MeV}\leftrightarrow R\Big{(}{\frac{5}{2}}^{+}\Big{)}_{gs}$
$\displaystyle=$ $\displaystyle 2.9\times 10^{-16}\times 0.804\approx
2.4\times 10^{-16}\mbox{(t-y)}^{-1},$ $\displaystyle
E_{\gamma}=4.5\mbox{MeV}\leftrightarrow R\Big{(}{\frac{3}{2}}^{+}\Big{)}_{4}$
$\displaystyle=$ $\displaystyle 2.9\times 10^{-16}\times
0.409\times\left(1-\frac{E_{x}}{E_{a}}\right)^{3}=2.9\times 10^{-16}\times
0.409\times 0.55\approx 0.6\times 10^{-16}\mbox{(t-y)}^{-1}$

A state by state evaluation of the rates can be similarly obtained from the
cross sections is given in Table 2.

B) The component 23Na of the target.
The obtained rates can also be obtained from the cross sections summarized in
Table 3. Using these data we find for the dominant contribution

$R_{\mbox{\small coh}}=6\times 10^{-14}\mbox{(t-y)}^{-1},$

expected due to the smaller A involved.

It is clear that the coherent ME is very large and dominates. We do not know
of any other process that exhibits such a coherence. In pion photoproduction
see e.g. Ericson and Rho (1972) and Baer et al. (1975) one encounters a spin
independent contribution, but this does not lead to coherence since the pion
is negatively charged and stop in an atomic orbit before it is absorbed by the
nucleus. Anyway a change of the nucleus is involved. The process
$(\gamma\pi^{0})$ is somewhat different, but it leads to coherence. The
coherent scattering of the axion is the same as the well-known coherent
scattering of low-energy (large wave-lengths) neutrons, photons, electrons,
neutrinos and others. They are traditionally called as elastic scattering.

In any case we believe that the coherence is there, but, in spite of the fact
that the axion photoproduction is favored by the nuclear physics, the obtained
rate is not very large. This can be traced to the small axion mass which makes
the relevant coupling small. The above rates were obtained with $4\times
10^{27}$ particles in the target, which amounts to 1ton of NaI. Close to such
amounts of material have in fact been used in WIMP (weakly interacting massive
particles) dark matter searches Bernabei et al. (2008, 2018); Kim et al.
(2012).

###  VIII.2 Axion induced nuclear excitation

A) We will begin with the target 127I
In this case we consider the excitation of the nucleus which can subsequently
de-excite by $\gamma$ emission. In this case, since the axion source is
monochromatic, only states around $E_{a}$ can be excited. The nuclear matrix
elements can be obtained form those of Fig. 5 Thus we obtain

$\left(g^{0}_{aN}\right)^{2}(0.8+0.3)\times
10^{-4}+\left(g^{3}_{aN}\right)^{2}4\times
10^{-4}+\left(g^{0}_{aN}g^{3}_{aN}\right)4\times 10^{-4}=2.2\times 10^{-5}.$

Thus for $\kappa=0.1$ in Eq. (26) and using Eqs (50) and (53) we obtain

$R_{5}=1.86\times 10^{4}\times\frac{1}{0.2}\times 6.28\times 10^{-41}\times
2.2\times 10^{-5}\times 4\times 10^{27}\mbox{y}^{-1}=5\times
10^{-13}\mbox{(t-y)}^{-1}$ (58)

This time the scale of the cross section is favorable, but the nuclear matrix
elements are suppressed.

B) We will continue with the target 23Na

In this case the last states of Table 3 are relevant. Proceeding as in the
previous case we obtain:

$(6.4,\,1.5,\,0.9,\,0.07)\times 10^{-10}\mbox{(t-y)}^{-1}\mbox{in that order}$
(59)

Unlike the case of 127I, the above rates for 23Na appear larger, but still
small.

In view of the very small 5.5 MeV axion flux for $m_{a}=$5 eV axions, as given
by eq. (53), axion motivated nuclear physics experiments look doomed.

As we have seen, however, the axion mass is not known. The obtained rates
scale with $m^{2}_{a}\times m^{2}_{a}$, one factor coming from the axion flux
and the other of the axion-nucleus cross section. The obtained rates as a
function of the axion mass for two typical cases are exhibited in Fig. 7.

Figure 7: (Color on line) (a)Event rates for the last four states 4.31MeV,
5.24MeV, 5.74MeV, and 5.97MeV of 23Na (see Table 3) versus different axion
masses.(b) The rate for axion photoproduction on 127I. As expected this is
quite a bit smaller than in (a), due to the fact that the axion energy of 5.5
MeV is much smaller than the nucleon mass. Note, however, that in this case we
have the extra signature of the produced photon. Since the rate is dominated
by the coherent mode, the photon energy is approximately equal to the axion
energy.

For a larger mass, e.g. 5 keV, the expected rates become:
(a) For the last four states 4.31MeV, 5.24MeV, 5.74MeV, and 5.97MeV of 23Na
(see Table 3) as follows

$R=(6.4,\,1.5,\,0.9,\,0.07)\times 10^{2}\mbox{(t-y)}^{-1}$ (60)

In the case of the expressions (LABEL:Eq:ResPhotoI), (58) and (59)
respectively.
The rates for 23Na excitation are higher than those for 127I, since it so
happens that in the former case states with larger nuclear matrix element
happened to be around 5.5 MeV.

We remind the reader that only excited states with energies in a narrow window
around 5.5 MeV can contribute to the process. The structure of all the other
states, especially of the low lying ones, whose energies are experimentally
known and can be compared to the calculated ones, is immaterial.

(b) For the axion photoproduction rate, dominated by the coherent mode of
127I, as follows:

$R\approx R_{coh}=1.2\mbox{(t-y)}^{-1}$ (61)

This rate is expected to be the similar to that of nuclear targets with (A,Z)
close to that of 127I.
As expected this rate is quite a bit smaller than in (a), due to the fact that
the axion energy of 5.5 MeV is much smaller than the nucleon mass. Note,
however, that in this case we have the extra signature of the produced photon.
Since the rate is dominated by the coherent mode, the photon energy is
approximately equal to the axion energy.
Note that if only the total $\gamma$-ray energy is measured, one cannot
distinguish between (a) and (b).

It seems that axion masses in the keV region are not allowed by astrophysical
and cosmological axion limits, see e.g. Fig. 3 of the review by Raffelt G. G.
Raffelt (2008). So, even though such limits are somewhat model dependent, we
have to concede that axion masses in the keV region are not realistic and high
detection rates as those mentioned above in Eqs (60) and (61) are not
expected.

In addition to the 5.5 MeV solar axions employed here, other solar axion
sources of quite a bit lower energies can be employed in nuclear physics
experiments, like the 477.6 keV 7Li solar axion. A search for such axions has
been carried out M. Krcmar et al. (2001); A. V. Derbin et al. (2005); P. Belli
et al. (2008) by using the resonant axion absorption process. No such axions
have been detected, but the data collected allowed the authors to set a limit
on the 7Li solar axion mass of 13.9 keV at 90% C.L. P. Belli et al. (2008). It
is worth mentioning that a LiF powder of 243 g measured during 722 h was
employed, i.e. $2.0\times 10^{-5}$ (t-y). This corresponds to a detector
sensitivity of about $1.5\times 10^{6}$ (t-y)-1. It is obvious that this
sensitivity needs to be improved by many orders of magnitude to reach the
level of, e.g., $10^{2}$(t-y)-1 expected for 5 eV axions and relevant coupling
to the electron of $g_{ae}=024$, see section X.5 below. This proposal is
allowed and interesting, expected to be considered in future experiments. We
are not going to employ the 13.9 keV axions in this work, since the authors
themselves classify their work as " Preliminary results and feasibility
studies".

If, instead, we use lower masses, e.g. the value of $m_{a}=0.25$keV, a model
dependent limit extracted by the CUORE experiment Cuore Collaboration (2013),
see the discussion following Eq. (69) below, we find

$R=(6.4,\,1.5,\,0.9,\,0.07)\times 6.25\times
10^{-4}\mbox{(t-y)}^{-1},\,R_{coh}=7.5\times 10^{-6}\mbox{(t-y)}^{-1}$ (62)

quite a bit smaller than the values given by Eqs (60) and (61) respectively.

For such axion masses the detection rates in experiments involving nuclear
targets may be exceedingly difficult. This, of course, can be traced back
mainly to the small Borexino 5.5. MeV axion flux.

##  IX The axion induced processes in atomic targets

The presence of axion electron interaction of the form of Eq, 4 gives rise to
some interesting processes involving atomic targets. The most interesting are:
i) The process:

$a+A(Z)\rightarrow A^{\prime}(Z)+\gamma$ (63)

with $A^{\prime}(Z)$ the ground state or excited state of the neutral atom.
This process is historically known as inverse Compton scattering or axion
induced Compton scattering.
ii) The process

$a+A(Z_{e})\rightarrow A^{*}(Z_{e}-1)+e^{-}$ (64)

where $Z_{e}$ is the number of electrons in the atom. In other words the final
atom contains an electron hole. This process is historically known as axio-
electric effect or axion induced atomic excitation.

###  IX.1 The axion electron coupling

Before proceeding with the evaluation of the cross section for axion induced
atomic process, we will consider the axion electron coupling, see section II
which can be written as

${\cal L}=G_{e}\mbox{\boldmath$\sigma$}\cdot{\bf
q},\,G_{e}=\frac{g_{ae}}{2f_{a}},\,$ (65)

where $g_{e}$ is a dimensionless coupling constant, which can be computed in
axion models, e.g. Dine and Fischler (1983). It argued in the context of the
DFSZ model Srednicki (1985); di Cortona et al. (2016); Ringwald and Saikawa
(2016) that It can be given by a relation of the form

$g_{ae}=\frac{1}{3}\left(1-\frac{\tan^{2}{\beta}}{1+\tan^{2}{\beta}}\right),\,\tan{\beta}=\frac{\upsilon_{2}}{\upsilon_{1}},$
(66)

where $\upsilon_{2}$ and $\upsilon_{1}$ are the vacuum expectation values of
the two doublets of the DFSZ model. These are not known. In such scenario the
maximum value of $g_{ae}$ can be 1/3, but small values corresponding to the
large values $\tan^{2}{\beta}$ are perhaps favored, as had been the case with
supersymmetry. Be that as it may, we have decided to be conservative and adopt
a relatively large value of $\tan{\beta}$, namely $\tan^{2}{\beta}=12.9$. This
yields the value

$g_{ae}=0.024$ (67)

In any case, it will be treated as a parameter to be fixed by experiment. To
make this clear, in all expressions it is going to be used, there will be a
relevant accompanying factor, some power of
$\left(\frac{g_{ae}}{0.024}\right)$

On the other hand $f_{a}$ is the axion decay constant, which is related to the
axion mass via Eq. (11). This relation is close to that between $m_{a}$ and
$f_{a}$ used in the analysis of the CUORE Cuore Collaboration (2013) and
Borexino Bellini et al. (2012b) experiments. Thus for $f_{a}=10^{7}$ GeV we
get $m_{a}=0.6$ eV and, selecting the value $g_{ae}=0.024$, we find

$G_{e}=1.2\times
10^{-12}\mbox{MeV}^{-1}\left(\frac{g_{ae}}{0.024}\right)\frac{m_{a}}{0.6\mbox{eV}}=1.0\times
10^{-11}\mbox{MeV}^{-1}\left(\frac{g_{ae}}{0.024}\right)\frac{m_{a}}{5\mbox{eV}}$

Thus the scale of the cross section is

$(\sigma_{0})_{e}=G_{e}^{2}=1.0\times
10^{-22}\mbox{MeV}^{-2}\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{2}=3.8\times 10^{-44}\mbox{cm}^{2}\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{2}\left(\frac{g_{ae}}{0.024}\right)^{2}$ (68)

We should mention at this point that all cross sections calculated below will
remain the same, when $g_{ae}$ and $m_{a}$ are suitably scaled. For example if
$g_{ae}$ happens to be a thousand times smaller, $g_{ae}=2.4\times 10^{-5}$,
provided, of course, that the axion mass is three orders of magnitude larger,
i.e. $m_{a}=5$ keV.

The 14.4 keV axion source appears to be ideal for atomic physics experiments.
The 14.4 keV source is monochromatic, produced in the sun via the 57Fe* F.T.
Avignone III et al. (2018), with a flux given by Eq. (51).

###  IX.2 A brief summary of limits on the parameters $m_{a}$ and $f_{a}$
extracted from experiments

In all the calculations of the type we have performed a crucial parameter is
the axion mass $m_{a}$, which is not known. We have been using a value of
$m_{a}=5$ eV, but both lower and higher values are not excluded. Limits much
smaller than 5 eV, of the order of meV, typically come form astrophysics
Borisov and Grishina (1996) (cooling of neutron stars etc via axion emission)
or supernova (SN) explosions. Recently bounds come from (SN) data Carenza et
al. (2019) which vary from 10 to 50 meV in the context of KVSZ model. Such
light axions are, of course, not relevant for nuclear and atomic processes
considered in this work.

Higher bounds have also been given:
a) The CUORE Cuore Collaboration (2013) collaboration using the 14.4 keV solar
axions and measuring the axio-electric effect in the TeO2 bolometers, provides
two limits on $m_{a}$:

$m_{a}\leq 19.2\mbox{ eV and }m_{a}\leq 250\mbox{ eV both at }95\%\mbox{
C.L.}$ (69)

Strictly speaking what the experiment yielded are limits on $f_{a}$ by using
the DFSZ and KSVZ model for $g_{Ae}$, i.e. values of $f_{a}\geq 3.125\times
10^{5}$ GeV and $\geq 2.4\times 10^{4}$ GeV respectively. Then they extracted
the above values of $m_{a}$ as implied by Eq. (11).
b) Axions of energy 14.4 keV were employed using the axion induced excitation
of ${}^{57}\mbox{Fe}$, the inverse of the axion production in the sun. That is
the process
$a+^{57}\mbox{Fe}\rightarrow^{57}\mbox{Fe}^{*}\rightarrow^{57}\mbox{Fe}+\gamma\mbox{(14.4
keV)}$ with a sectioned Si(Li) detector arranged in a low-background facility
Derbin et al. (2011b). This approach succeeded in setting a new limit on the
axion couplings to nucleons, $|-1.19g^{0}_{aN}+g^{3}_{aN}|\leq 3.0\times
10^{-6}$. Within the hadronic-axion model, the respective constraint on the
axion mass is $m_{a}\leq 145$ eV (at a $95\%$ C.L.). An analysis of the CUORE
sensitivity to 14.4 keV solar axion detected by inverse coherent Bragg-
Primakoff conversion in single-crystal TeO2 has been performed. The axion
coupling of $g_{a\gamma\gamma}g^{eff}_{aN}$ to be studied by CUORE was
calculated (evaluated ) to be $<1.105\times 10^{-16}$/GeV. In this analysis an
axion mass of less than 500 eV has been used Li et al. (2016).

In another search for solar axions Derbin et al. (2011a) produced by Compton
($\gamma+e^{-}\rightarrow e^{-}+a$ ) and bremsstrahlung- like (
$e^{-}+Z\rightarrow e^{-}+Z+a$) processes has been performed. In this case an
upper limit on the axion-electron coupling and on the axion mass, i.e. on
$g_{Ae}\times m_{a}\leq 3.1\times 10^{-7}$ eV at (90$\%$C.L.), has been
obtained. The limits on axion mass are $m_{a}\leq 105$ eV and $m_{a}\leq 1.3$
keV at the (90$\%$ C.L.) for DFSZ and KSVZ axion models, respectively.

The existence of high upper bounds on the axion mass are extremely encouraging
to us in connection with this work, since they allow the possibility
detectable rates involving axions in the 5 eV range, i.e. $f_{a}$ around
$10^{6}$ GeV.

We stress that there exist two components in our calculation. First the size
of the oncoming axion flux, coming from the production mechanism, which can
now be written as

$\Phi(E_{a})=0.7\times 10^{9}\left(\frac{m_{a}}{0.6\mbox{
eV}}\right)^{2}\mbox{cm}^{-2}\mbox{s}^{-1}=2.2\times
10^{16}\left(\frac{m_{a}}{0.6\mbox{
eV}}\right)^{2}\mbox{cm}^{-2}\mbox{y}^{-1}\mbox{ (14.4 keV solar axion)}$ (70)

This, for convenience, can be re-scaled

$\Phi(E_{a})=1.5\times 10^{18}\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{2}\mbox{cm}^{-2}\mbox{y}^{-1}\mbox{ (solar for 14.4 keV axion,
rescaled)}.$ (71)

The value for $m_{a}=5$ eV corresponds to $f_{a}=1.2\times 10^{6}$ GeV.

Second, the calculated cross section, which is also proportional to
$m^{2}_{a}$. These combined for $N_{A}=4\times 10^{27}$ particles in the
target (1ton of NaI) yield a rate scaled by

$R_{0}=1.5\times 10^{18}\times 4\times 10^{27}\times 3.8\times
10^{-44}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}=228\mbox{ events per
t-y}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}$ (72)

It seems that in the atomic experiments one should use the value of
$\sigma_{0}$ given by Eq. (68) and the flux given by (71).

##  X Some results on Axion electron induced transitions in the case N$a$I
target

In our calculations we will consider the following processes:

###  X.1 Axion photoproduction on electrons

This process involves a special mechanism leading to the reaction given by Eq.
(63). In other words in this case the process can proceed in a manner
analogous to Fig. 1(b), with the obvious replacements $N\rightarrow e$,
$g^{3}_{aN}\rightarrow g_{ae}$. It involves the spin dependent axion electron
interaction as given by Eq. (4) and the magnetic moment of the electron in the
photon vertex. The name came from nuclear physics, the old analogous pion
photoproduction process. In the atomic case it is a special case of the axion
induced Compton scattering. It involves the gs or low lying atomic
excitations. Other possible mechanisms leading to reaction given by Eq. (63),
involving, e.g., the inverse Primakoff axion scattering Li et al. (2016); Abe
et al. (2021), will not be considered.

Let us estimate the expected cross sections, by comparing it to the hadronic
process. The new feature is the Bohr magneton for the electron (as opposed to
the nucleon one used before) and the axion energy dependence. Thus for the
14.4 keV axion we get

$\sigma_{0}=\frac{m^{2}_{n}}{m^{2}_{e}}\left(\frac{14.4\mbox{keV}}{5.5\mbox{MeV}}\right)^{2}\times
6.0\times 10^{-50}=1.5\times 10^{-48}\mbox{cm}^{2}$ (73)

see Eq. ( 37). This, however, has been estimated with $f_{a}=10^{7}$ GeV.
Thus, in view of Eq. (11), it can be written as

$\sigma_{0}=1.5\times
10^{-48}\mbox{cm}^{2}\left(\frac{m_{a}}{0.6\mbox{eV}}\right)^{2}=1.1\times
10^{-46}\mbox{cm}^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$ (74)

The dimensionless couplings will be included separately.
For the 5.5 MeV axion we have

$\sigma_{0}=\frac{m^{2}_{n}}{m^{2}_{e}}\times 6.0\times 10^{-50}=2.1\times
10^{-43}\mbox{cm}^{2}$

The second process, of course, is not relevant, since, as we have seen, with
the Borexino flux it is undetectable. There is, of course, a coherence factor,
that goes with the electron axion - photoproduction, which is now proportional
to $Z^{2}$. Thus for 14.4 keV axion proceeding as in the nuclear axion
photoproduction we find:

$\sigma(E_{a})=\sigma_{0}G^{2}_{ae\gamma}\left(\mbox{ME}_{el}^{2}+\frac{1}{3}\frac{1}{2J_{i}+1}\sum_{f}\langle
J_{f}||\mbox{\boldmath$\sigma$}||J_{i}\rangle^{2}\left(1-\frac{Ex}{E_{a}}\right)^{3}\right),$
(75)

with $\sigma_{0}$ given by Eq. (74) and the effective axion-electron-photon
coupling, analogous to those of Eq. (18), is

$G_{ae\gamma}=g_{s}g_{ae}=2\times 0.024=0.048$ (76)
$\mbox{ME}_{el}^{2}=\left\\{\frac{2}{3}Z^{2}+\frac{1}{3}\frac{1}{2J_{i}+1}\langle
J_{i}||\mbox{\boldmath$\sigma$}||J_{i}\rangle^{2}\right\\}.$ (77)

Where in the atomic system $E_{x}<<E_{a}$.
Compare this formula with the analogous expression in nuclear transitions, Eq,
(LABEL:Eq:elasticNucleus), and notice that in the present case the spin
independent operator, see the analogous operator $\Omega_{A}$ in Eq. (28), is
just a multiple of the identity, yielding the $Z^{2}$ dependence of the square
of the relevant matrix element.

The spin reduced ME for single particle transitions are given by:

$\left(\begin{array}[]{ccc|c}\ell&j&j^{\prime}&\langle\ell,j^{\prime}||\sigma||\ell,j\rangle^{2}\\\
\hline\cr 0&\frac{1}{2}&\frac{1}{2}&6\\\
1&\frac{1}{2}&\frac{3}{2}&\frac{16}{3}\\\
1&\frac{1}{2}&\frac{1}{2}&\frac{2}{3}\\\
1&\frac{3}{2}&\frac{1}{2}&\frac{16}{3}\\\
1&\frac{3}{2}&\frac{3}{2}&\frac{20}{3}\\\
2&\frac{3}{2}&\frac{5}{2}&\frac{48}{5}\\\
2&\frac{3}{2}&\frac{3}{2}&\frac{12}{5}\\\
2&\frac{5}{2}&\frac{3}{2}&\frac{48}{5}\\\
2&\frac{5}{2}&\frac{5}{2}&\frac{42}{5}\\\
3&\frac{5}{2}&\frac{7}{2}&\frac{96}{7}\\\
3&\frac{5}{2}&\frac{5}{2}&\frac{30}{7}\\\
3&\frac{7}{2}&\frac{5}{2}&\frac{96}{7}\\\
3&\frac{7}{2}&\frac{7}{2}&\frac{72}{7}\\\ \end{array}\right)$ (78)

Even for Na, the lighter component of the target, we see that the coherent
contribution becomes dominant.
Thus the obtained coherent rates, obtained with the flux given by (71), are

$\displaystyle R_{I}$ $\displaystyle\approx
2.78\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{4}\mbox{ events per (t-y)},\,R_{Na}\approx
0.097\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{4}\mbox{ events per (t-y)}\Rightarrow$ $\displaystyle R$
$\displaystyle\approx 2.92\left(\frac{m_{a}}{5\mbox{
eV}}\right)^{4}\left(\frac{g_{ae}}{0.024}\right)^{2}\mbox{ events per (t-y)}$
(79)

In this mechanism the photons are radiated off electrons directly. As far as
we know, nobody has previously considered this axion photoproduction
mechanism.

The above expression does not depend on the details of the atomic structure of
the target, but due to coherence only to the atomic number. Also this process
leads directly to photons with an energy equal to the axion energy $E_{a}$.
Recall, however that the scale of the cross section is proportional to
$E^{2}_{a}$. Recalling Eq. (72) we get a sort of global expression:

$R(Z)\approx
2.50\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{Z}{50}\right)^{2}\left(\frac{Ea}{14.4\mbox{keV}}\right)^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}\mbox{
events per y}$ (80)

for $N_{A}=4\times 10^{27}$

###  X.2  Ejection of an electron from an interior orbit

This process is described by Eq, (64), but with 14.4 keV axion source the
electron hole can be in a deep bound orbit and as result the outgoing electron
energy is $E_{a}+\epsilon_{b}$. Such an electron can be observed directly. In
addition one may observe the X-rays and Auger electrons following the de-
excitation of the final atom.

We will now proceed with estimates of the total cross section. The essential
mathematical steps are given in the Appendix, section XII.
For bound electrons the cross section depends on two parameters: a) the form
factor like dependence involving the bound state wave function in momentum
space $F_{n\ell}=a^{3}\Phi^{2}_{n\ell}(a,k)$, with $\Phi_{n\ell}(a,k)$ the
electron wave function in momentum space. Here ${\bf k}=\bf{q-p}_{e}$ is the
momentum transfer with ${\bf q}$ the axion momentum and ${\bf p}_{e}$ the
momentum of the outgoing electron and b) The phase space factor $K_{n}$, which
has been evaluated with the same binding for all states in a given $n$.
Hydrogenic bound state wave functions we employed with $a=\frac{\alpha
Zm_{e}}{n}$. The function $F_{n\ell}$ for convenience has been chosen to be
dimensionless with a compensating factor $a^{-3}$ in the phase space part to
make it also dimensionless.

###  X.3 The cross section for ionization of the Iodine Component

Bound states up to $n=4$ were considered, with the last one being partially
filled. The form factor in the case of the 14.4 keV axions is dominated by the
outgoing electron momentum $p_{e}=\sqrt{2m_{e}(E_{a}+\epsilon_{b})}$. The
corresponding electron energies are:

$T=\\{4.84,10.15,12.01\\}\mbox{ keV for }n=2,3,4\mbox{ resectvely}$

In the case of the 5.5 MeV axions the situation is a bit complicated and we
made the simplifying assumption $k=\sqrt{q^{2}+p_{e}^{2}}$. In this case the
binding energy becomes irrelevant and the electron momentum
$p_{e}=\sqrt{E_{a}(E_{a}+2m_{e})}$.

We should mention that we summed over all states in a given full shell by
introducing the $(4\ell+2)$ factor. The form factor $F_{n\ell}$ and the
kinematical factor $K_{n}$ are given as follows:

$F_{n\ell}=\left(\begin{array}[]{cccccc}n&K(n)&F_{S}(n)&F_{P}(n)&F_{D}(n)&F_{F}(n)\\\
\hline\cr 2&0.536&1.69994&14.1273&-&-\\\
3&2.619&0.0166029&0.517959&0.854939&-\\\
4&6.7547&0.00127243&0.0770985&0.135348&0.0335818\\\ \end{array}\right)$ (81)

The $S_{n\ell}$ factors for the cross section are given by

$S_{n\ell}=\left(\begin{array}[]{ccccc}n&S(S(n))&S(P(n))&S(D(n))&S(F(n))\\\
\hline\cr 2&0.911049&7.57128&-&-\\\ 3&0.0434898&1.35675&2.23944&-\\\
4&0.00859345&0.52069&0.914085&0.226797\\\ \end{array}\right)$ (82)

The corresponding form factor in the case of 5.5 MeV energy is given by:

$F_{n\ell}=\left(\begin{array}[]{ccccc}n&F_{S}(n)&F_{P}(n)&F_{D}(n)&F_{F}(n)\\\
\hline\cr 1&3.5376\times 10^{-12}&0&0&0\\\ 2&5.5345\times
10^{-14}&2.3850\times 10^{-17}&0&0\\\ 3&4.8600\times 10^{-15}&2.4813\times
10^{-18}&2.1373\times 10^{-22}&0\\\ 4&4.6504\times 10^{-16}&4.6580\times
10^{-19}&5.4167\times 10^{-23}&1.6663\times 10^{-27}\\\ \end{array}\right)$
(83)

The total $S$ factor for the cross section is given by

$S_{n\ell}=\left(\begin{array}[]{ccccc}n&S(S(n))&S(P(n))&S(D(n))&S(F(n))\\\
\hline\cr 1&7.6183\times 10^{-8}&0&0&0\\\ 2&9.53514\times 10^{-9}&4.1075\times
10^{-12}&0&0\\\ 3&2.8259\times 10^{-9}&1.4428\times 10^{-12}&1.2428\times
10^{-16}&0\\\ 4&1.1923\times 10^{-9}&6.4200\times 10^{-13}&7.4657\times
10^{-17}&2.2966\times 10^{-21}\\\ \end{array}\right)$ (84)

We see that the largest $S_{n\ell}$-factor is $7.6\times 10^{-8}$ coming for
the ejection of an $0s$ electron. The form factor for this large momentum
transfer leads to a great suppression of the cross section, in spite of the
fact that the kinematical factor is favored,
$K_{n}=0.022,\,0.17,\,0.58,\,1,37\times 10^{6}$, for $1,2,3,4$ respectively.
In addition the 5.5 MeV axions are not favored in atomic physics experiments
due to the very small flux involved.

In the case of the 14.4 keV axions the cross section is

$\sigma_{n,\ell}=\sigma_{0}\frac{1}{2\pi}S_{n,\ell}$ (85)

(see Eq. ( 93), with $\sigma_{0}$ given by Eq. (68). Thus we get

$\sigma_{n,\ell}=\\{0.549,4.580,0.0279,0.8200,1.353,0.005,0.314,0.553,0.135\\}\times
10^{-44}\mbox{cm}^{2}\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$
(86)

in the order of the orbitals appearing in Eq. (82).

###  X.4 The cross section for ionization of the Na Component

We do not expect a large cross section in this case because the energy of the
outgoing electron is larger due to the smaller binding energy involved. Thus
one expects a lager damping on the form factors of the available $n=1$ end
$n=2$ orbitals. In this case we have $\epsilon_{n}=(-1.646,-0.420)$keV
respectively for $n=1,2$ . On the other hand the outgoing electron energies
are:

$T=\\{12.75,13.99\\}\mbox{ keV for }n=1,2\mbox{ resectvely}$

The suppression of the form factors at high momentum transfer, here
essentially the momentum of the outgoing electron, can be seen in Fig. 8 as a
function of $X$, the momentum transfer in units of the electron mass. The
actual electron momenta involved here are $X=(0.2234,0.2340)$ for electrons
ejected from the $n=1$ and $n=2$ respectively.

$X=\frac{P_{e}}{m_{e}}\longrightarrow$

Figure 8: (a) We exhibit the dependence of the form factors encountered in the
ionization of Na as a function of the momentum transfer, here momentum of the
emitted electron $p_{e}$ in units of the electron mass. (b) The same figure
restricted to higher momentum transfers. The solid line corresponds to $n=1$,
the thick solid line to $n=2,\ell=0$ and the dashed line to $n=2,\ell=1$.

In view of this, it is not surprising that, in spite of the fact the
kinematical function $K_{n}$ is quite large, the $S_{n\ell}$ factors are
small:

$S_{n\ell}=\left(\begin{array}[]{cccc}n&K(n)&S(S(n))&S(P(n))\\\ \hline\cr
1&12.2&0.0424184&-\\\ 2&101.9&0.00493614&0.000617041\\\ \end{array}\right)$
(87)

Thus using Eq. (85) we find

$\sigma_{n\ell}=\\{2.567,0.298,0.036\\}\times
10^{-46}\mbox{cm}^{2}\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$

The total cross section is

$\sigma=\sum_{n\ell}\sigma_{n\ell}=8.36\times
10^{-44}\mbox{cm}^{2}\left(\frac{g_{ae}}{0.024}\right)^{2}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{2}$
(88)

This cross section scaled to $m_{a}=5$keV becomes $8.36\times
10^{-38}\mbox{cm}^{2}$ which between the values $10^{-36}\mbox{cm}^{2}$ (DFSZ)
and $10^{-41}\mbox{cm}^{2}$ (KSVZ) reported by CUORE Cuore Collaboration
(2013).

###  X.5 Some results regarding the rates for atomic ionization

Using now the 14.4 keV flux given by Eq. (71)) and $N_{A}=4\times 10^{27}$ as
above we find:

i) For the iodine component:

$R_{n,\ell}=\\{0.337,2.810,0.016,0.503,0.830,0.003,0.194,0.340,0.082\\}\times
10^{2}\mbox{per
t-y}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}\left(\frac{g_{ae}}{0.024}\right)^{2}$
(89)

in the same order as in Eq, (86).

ii) For the Na component

$R_{n,\ell}=\\{1.575,0.182,0.022\\}\mbox{per
t-y}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}\left(\frac{g_{ae}}{0.024}\right)^{2}$
(90)

The obtained rates depend on the atomic wave function and in particular the
form factor of the orbit involving the ejected electron. We have seen that a
great suppression of the form occur, if the axion energy is high, e,g, 5 MeV.
The same thing happens in the case of a 14,4 keV axion for a small $Z$ atom,
since again, due to the small binding, the form factor is suppressed due large
momentum transfer, see Fig. 8. Thus, for the Iodine atom, as well as others
with nearby $Z$ and the 14.4 keV energy, we have a favorable situation. One
can write for the dominant contributions:

$R_{n,\ell}=\\{0.337,2.810,0.503,0.829,0.340\\}\times 10^{2}\mbox{per
t-y}\left(\frac{m_{a}}{5\mbox{eV}}\right)^{4}\left(\frac{g_{ae}}{0.024}\right)^{2}$
(91)

in the order $2S,\,2P,\,3P,\,3D,\mbox{ and }4D$ for the orbits.
We notice the dramatic dependence of the event rate on the axion mass.
Deceasing the axion mass by a factor of 5 will lead to a reduction of the rate
by 3 orders of magnitude and make it un-observable, an increase by the same
factor makes one wonder how come such an axion has not already been seen.

##  XI Concluding remarks

In the present work we studied various axion induced nuclear and atomic
processes, assuming an axion mass of 5 eV.

In the case of the nuclear processes we considered the mono-energetic axion
source of 5.5 MeV. The obtained nuclear cross sections were found to be
reasonable, but the obtained rates were very small mainly due to the small
flux on the earth of such axions, obtained with the Borexino detector Bellini
et al. (2012b). So, such processes are doomed to be non detectable for axion
masses in the few eV range. Had the mass of the axion been in the keV region,
such processes would become detectable with rates as given by Fig. 7 and, in
particular, by Eqs (60) and (61). We have seen, however, that such masses are
excluded by astrophysical data. For lower axion masses in the subkeV region,
e.g. the value of $m_{a}=0.25$keV, a model dependent limit extracted by the
CUORE experiment Cuore Collaboration (2013), the obtained total excitation and
photoproduction rates in the case of NaI target become $R\approx 6\times
10^{-3}$ and $R_{coh}\approx 10^{-5}\mbox{(t-y)}^{-1}$ respectively,
exceedingly difficult to detect.

In the case of atomic experiments, with inner electron binding energies in the
keV region, we found appropriate and we used the 14.4 keV 57Fe axion source.
In this case we find sizable rates, e.g.:
i) Axion induced photoproduction. We get the rate given by Eq. (80) for
directly producing X-rays with energy equal to the axion energy $E_{a}$.
Smaller rates are expected from the excited states with about the same energy.
This is independent of the details of the structure of the atom. For $m_{a}=5$
eV axions ($f_{a}=1.2\times 10^{6}$ GeV) this leads to a rate $R\approx
2.50\mbox{ events per (t-y)}$ in the case of NaI target, assuming a coupling
$g_{ae}=0.024$.
ii) Electron ejection from the atom. In this case we find sizable rates for
ejecting an electron from the relevant orbits of the Iodine, given by Eq.
(91), the maximum being 280 events per t-y for $m_{a}=5$ eV ( $f_{a}=1.2\times
10^{6}$ GeV), again for $g_{ae}=0.024$, coming from the $n=2,\ell=1$ orbit,
and slightly smaller rates from some of the other orbits. Much smaller rates
are expected for the Na component.

Experiments like those discussed above have already been done by CUORE-Te
Cuore Collaboration (2013) and others Li et al. (2016), Derbin et al. (2011b).
As given in Eq.(69), the 14.4 keV axion with the rate larger than $R=2\times
10^{5}(t-y)^{-1}(m_{a}<19$ eV DFSZ) has been excluded Cuore Collaboration
(2013). Then, axions in the region of $R=(10^{2}-10^{4})(t-y)^{-1}$, i.e.
$m_{a}=5-10$ eV, $g_{ae}=0.024$, are very interesting and their detection in
the case of atomic excitations is quite realistic, by improving detector
sensitivities by a few orders of magnitude.

If that is the case, we expect a very interesting axion signature, i.e. a
signal consisting of the detection of the primarily produced electrons from
each orbit as well as the detection of the X-rays or Auger electrons produced,
when the created hole is filled by the de-excitation of the atom. The relevant
rates depend, of course, on the assumed value of the elementary coupling
$g_{ae}$, which is not really known.

##  XII Appendix: The formalism of the axion electron cross section from bound
orbits

The axion electron scattering cross section for relativistic axions is given
by

$d\sigma=\sigma_{0}\int{\cal M}(k^{2})^{2}\frac{1}{2E_{a}}\frac{d{\bf
p}_{e}^{3}}{(2\pi)^{3}}\frac{d{\bf
p}_{A}^{3}}{(2\pi)^{3}}(2\pi)\delta(E_{a}+\epsilon_{b}-T)(2\pi)^{3}\delta({\bf
p}_{a}-{\bf p}_{A}-{\bf p}_{e})$ (92)

where ${\cal M}(k^{2})$ is the invariant amplitude, which depends on the
momentum transfer, ${\bf p}_{e}$ and $T$ are the momentum and the energy of
the outgoing electron and ${\bf p}_{A}$ of the atom, $\epsilon_{b}$ is the
binding energy of the electron in the target. $E_{a}$ is the energy of the
axion, normally being introduced as a normalization of the scalar field. We
have neglected the energy of the outgoing atom. Thus integrating over ${\bf
p}_{A}$ with the use of the momentum conserving $\delta$ function we obtain

$\sigma=\sigma_{0}\frac{1}{2}\frac{1}{(2\pi)^{2}}4\pi
S=\frac{1}{2\pi}S,\,S=\frac{1}{4\pi}\int{\cal
M}(k^{2})^{2}\frac{1}{E_{a}}d{\bf p}_{e}^{3}\delta(E_{a}+\epsilon_{b}-T)$ (93)

Let us begin with the discussion of ${\cal M}(k^{2})^{2}$. The relevant matrix
element (ME) involving the initial bound electron and the outgoing plane wave
electron is given by:

$\mbox{ME}=\int\psi_{n\ell m}({\bf r})e^{i{\bf q\cdot{\bf
r}}}\langle\frac{1}{2}m_{s}|\mbox{\boldmath$\sigma$}\cdot{\bf
q}|\frac{1}{2}m_{s}^{\prime}\rangle\frac{1}{(2\pi)^{3/2}}e^{-i{\bf
p_{e}\cdot{\bf r}}}=\phi_{n\ell m}({\bf
k})\langle\frac{1}{2}m_{s}|\mbox{\boldmath$\sigma$}\cdot{\bf
q}|\frac{1}{2}m_{s}^{\prime}\rangle,\,{\bf k}={\bf q}-{\bf p}_{e}$ (94)

Where $\phi_{n\ell m}({\bf k})$ is the initial electron bound wave function in
momentum space. The last term involves the axion electron interaction.We thus
find that transition probability averaging over the initial m-substates and
summing over the final ones becomes:

$\frac{1}{(2j+1)}\sum_{m,m_{\ell},m_{s}.m_{s}^{\prime}}\langle\ell
m_{\ell},\frac{1}{2}m_{s}|jm\rangle^{2}\mbox{ME}^{2}=|\phi_{n\ell m}({\bf
k})|^{2}\langle\frac{1}{2}\sum_{m_{s},m_{s}^{\prime}}\frac{1}{2}m_{s}|\mbox{\boldmath$\sigma$}\cdot{\bf
q}|\frac{1}{2}m_{s}^{\prime}\rangle^{2}$ (95)

The last result was obtained using the properties of the Clebcsh-Gordan
coefficients.The last matrix element can easily be evaluated yielding
$q^{2}\approx E_{a}^{2}$ We thus find

$S=\frac{1}{4\pi}\int|\phi_{n\ell m}({\bf k})|^{2}E_{a}d{\bf
p}_{e}^{3}\delta(E_{a}+\epsilon_{b}-T)$ (96)

We will specialize it in the case of 14.4 keV axion. In this case the outgoing
electron is non relativistic $T=p_{e}^{2}/(2m_{e})$, with the use of $\delta$
function we get $p_{e}=\sqrt{2m_{e}(E_{a}+\epsilon_{b})}$, which is much
larger than the axion momentum. So $k\approx p_{e}$. So after the integration
over the angles we get:

$S=\left(\phi_{n\ell
m}(\sqrt{2m_{e}(E_{a}+\epsilon_{b})})\right)^{2}E_{a}m_{e}\sqrt{2m_{e}(E_{a}+\epsilon_{b})}$
(97)

At this point we recall that the bound state wave function can be written as
$\phi_{n\ell}(a,k)$, where $k$ is the momentum and $a=\frac{\alpha
Zm_{e}}{n}$. We prefer to write it in dimensionless form and write is as

$F_{n\ell}(a,k)=a^{3}\phi^{2}_{n\ell}(a,k)\Rightarrow
S=F_{n\ell}K_{n},\,K_{n}=n^{3}\frac{1}{(\alpha
Z)^{3}}\frac{E_{a}}{m_{e}}\sqrt{2\left(\frac{E_{a}+\epsilon_{b}(n)}{m_{e}}\right)}$
(98)

A compensating $1/a^{3}$ factor has been introduced in the kinematical portion
of Eq, (97). This way both $F_{n\ell}$ and $K_{n}$ are dimensionless.

In the case of the 5.5 MeV axion we proceed in an analogous way. Now the
binding becomes irrelevant and the outgoing electron caries all the energy,
Its momentum however is $\sqrt{E_{a}(E_{a}+2m_{e})}$, approximately the same
with the axion momentum. So the angular integral is quite complicated and we
made the simplifying assumption $k=\sqrt{q^{2}+p_{e}^{2}}$ Now

$K_{n}=n^{3}\frac{1}{(\alpha Z)^{3}}\left(\frac{E_{a}}{m_{e}}\right)^{3}$ (99)

As expected $K_{n}$ becomes large, but the form factor $F_{n\ell}(a,k)$ at
this high momentum transfer is tiny, so that the cross section becomes
negligibly small.

The relevant for factors employed are given below:

$F_{1s}=\frac{128a^{8}(a-k)^{2}(a+k)^{2}}{\pi\left(a^{2}+k^{2}\right)^{6}},\,F_{0p}=\frac{512a^{10}k^{2}}{3\pi\left(a^{2}+k^{2}\right)^{6}},\,F_{2s}=\frac{32a^{8}\left(3a^{4}-10a^{2}k^{2}+3k^{4}\right)^{2}}{\pi\left(a^{2}+k^{2}\right)^{8}}$
$F_{1p}=\frac{1024a^{10}k^{2}(a-k)^{2}(a+k)^{2}}{\pi\left(a^{2}+k^{2}\right)^{8}},\,F_{0d}=\frac{4096a^{12}k^{4}}{5\pi\left(a^{2}+k^{2}\right)^{8}},\,F_{3s}=\frac{512a^{8}\left(a^{6}-7a^{4}k^{2}+7a^{2}k^{4}-k^{6}\right)^{2}}{\pi\left(a^{2}+k^{2}\right)^{10}}$
$F_{2p}=\frac{2048a^{10}k^{2}\left(5a^{4}-14a^{2}k^{2}+5k^{4}\right)^{2}}{15\pi\left(a^{2}+k^{2}\right)^{10}},\,F_{1d}=\frac{32768a^{12}k^{4}\left(a^{2}-k^{2}\right)^{2}}{5\pi\left(a^{2}+k^{2}\right)^{10}},\,F_{0g}=\frac{131072a^{14}k^{6}}{35\pi\left(a^{2}+k^{2}\right)^{10}}$

Acknowledgments: J. D. V. would like to thank Georg Raffelt for his help in
getting the axion mass dependence of the 14.4 keV axion solar flux.

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