# Probing the axion-nucleon coupling with the next generation of axion
helioscopes

Luca Di Luzio<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia ‘G.
Galilei’, Università di Padova, Italy  Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, Italy  Javier Galan<EMAIL_ADDRESS>Center for
Astroparticles and High Energy Physics (CAPA), Universidad de Zaragoza, 50009
Zaragoza, Spain  Maurizio Giannotti<EMAIL_ADDRESS>Physical Sciences,
Barry University, 11300 NE 2nd Ave., Miami Shores, FL 33161, USA  Igor G.
Irastorza<EMAIL_ADDRESS>Center for Astroparticles and High Energy
Physics (CAPA), Universidad de Zaragoza, 50009 Zaragoza, Spain  Joerg Jaeckel
<EMAIL_ADDRESS>Institut für theoretische Physik,
Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany  Axel
Lindner<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany  Jaime Ruz<EMAIL_ADDRESS>Lawrence
Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94551  Uwe
Schneekloth<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Lukas Sohl<EMAIL_ADDRESS>Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Lennert J.
Thormaehlen<EMAIL_ADDRESS>Institut für theoretische
Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
Julia K. Vogel<EMAIL_ADDRESS>Lawrence Livermore National Laboratory, 7000
East Avenue, Livermore, CA 94551

###### Abstract

A finite axion-nucleon coupling, nearly unavoidable for QCD axions, leads to
the production of axions via the thermal excitation and subsequent de-
excitation of 57Fe isotopes in the sun. We revise the solar bound on this flux
adopting the up to date emission rate, and investigate the sensitivity of the
proposed International Axion Observatory IAXO and its intermediate stage
BabyIAXO to detect these axions. We compare different realistic experimental
options and discuss the model dependence of the signal. Already BabyIAXO has
sensitivity far beyond previous solar axion searches via the nucleon coupling
and IAXO can improve on this by more than an order of magnitude.

##  I Introduction

Axions [1; 2] are the direct and testable prediction following from the
Peccei-Quinn (PQ) mechanism [3; 4] proposed as a possible solution to the
strong CP problem of the Standard Model (SM) of particle physics. This
problem, still one of the most puzzling in modern particle physics, concerns
the apparent absence of CP-violating effects in Quantum Chromodynamics (QCD).
To emphasize their role in explaining the observed QCD behavior, such
particles are also dubbed _QCD axions_. Other kinds of pseudoscalar particles,
with properties very similar to the QCD axion but no relation to the strong CP
problem, emerge naturally in several extensions of the SM, in particular in
compactified string theories [5; 6; 7; 8; 9; 10; 11; 12]. To distinguish them
from the QCD axion, these are often called axion-like particles (ALPs). The
considerations discussed in this paper apply, in large part, to the QCD axion
as well as to ALPs, except when specific QCD axion models are discussed. In
the interest of brevity, we will refer to both QCD axions and ALPs as “axions”
throughout this paper.

The axion phenomenology and experimental landscape are discussed in several
recent reviews [13; 14; 15; 16; 17]. In summary, axions are expected to couple
to the SM fields with model dependent couplings,

$\mathcal{L}_{\rm
int}=-\frac{1}{4}g_{a\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}-\sum_{f}i\,g_{af}\,a\,\bar{f}\gamma_{5}f\,,$
(I.1)

where $F$ is the electromagnetic field tensor and $f$ are the SM fermions (for
the present work, it is sufficient to consider couplings to protons, neutrons
and electrons, so we can assume $f=p,n,e$).

Particularly appealing is the axion coupling to photons, since it allows very
promising experimental strategies for the axion search. However, the couplings
to electrons and nucleons are also employed in experimental axion searches.

In this work, we examine the axion coupling to nucleons (for a recent
calculation see [18]) and explore the potential to probe it with axion
helioscopes [19] which, as suggested by the name, search for axions produced
in the sun.

From a theoretical point of view the nucleon coupling is of particular
interest because it receives, similar to the photon coupling, an unavoidable
contribution from the defining coupling of QCD axions to gluons. While it is
possible to conceive models where this is (partially) cancelled, thereby
having QCD axions with a suppressed coupling to nucleons [20], a large nuclear
coupling is expected in most models which solve the strong CP problem and in
certain cases can be considerably enhanced (see, e.g., Ref. [21]). A sizable
nucleon coupling is therefore an expected feature of a QCD axion. In this
sense, measuring the axion-nucleon coupling would be a good indication of the
QCD axion nature. Furthermore, such a detection by a helioscope would likely
be accompanied by a signal from Primakoff (and possibly
Compton/Bremsstrahlung) axions. All these spectra have well defined shapes and
so, given enough statistics and energy resolution, the data analysis might
help understand specific properties of the detected particle (see discussion
in Ref. [22] for a similar analysis in the case of the axion-photon and axion-
electron couplings).

The axion coupling to nucleons can be probed indirectly in astrophysics,
through the effects on the cooling of neutron stars (NS) [23; 24; 25; 26; 27;
28] and from the analysis of the observed neutrino signal from SN 1987A [29;
30; 31; 32; 33; 34; 35]. A direct detection is also possible, in principle,
through experiments such as CASPEr-gradient [36] or ARIADNE [37].111Note,
however, that both experiments rely on important assumptions which strongly
affect the predicted sensitivities. The CASPEr detection potential depends on
the axion primordial abundance, and it is strongly reduced if axions
constitute only a small fraction of the total dark matter in the galaxy. The
ARIADNE potential depends on the CP-violating axion-scalar coupling, which is
presently very poorly known (see. e.g., Ref. [38]).

Axions coupled to nucleons could also be produced in the sun, for example
through the decay of excited nuclear states. As pointed out already in
Weinberg’s seminal paper [1], being pseudoscalars, axions can be emitted in
magnetic nuclear transitions. The best known example, and the one adopted
here, is the decay of the first excited state of 57Fe with the emission of a
14.4 keV axion. Other nuclear transitions turn out to generate a substantially
smaller axion flux (see also Appendix A).

The search for these 57Fe axions has a long history. The currently most
powerful helioscope, the CERN Axion Solar Telescope (CAST) [39], as well as
CUORE [40] and, more recently, XENON1T [41], have searched for axions produced
in this transition and provided constraints on the axion-nucleon coupling.

In this paper, we assess the potential of the next generation of axion
helioscopes, BabyIAXO [42; 43] and IAXO [44; 45; 46], to detect such axions.
An important motivating factor is that with BabyIAXO the transition from
concept to real experiment is imminent, as its construction is expected to
start in 2022\. We therefore believe it is of great importance to assess its
potential for an axion detection in all possible channels. Furthermore, the
BabyIAXO and IAXO potential to probe this coupling are expected to be
considerably superior to that of CAST [39], allowing to probe large regions of
additional axion parameter space. We provide a guide for the best setups
required to maximize the efficiency to detect axions from 57Fe. We also use
this opportunity to include new theoretical developments. In particular, the
matrix elements for the relevant transition have recently been revised [47],
and show a $\sim$ 30 % increase in the axion emission rate. In addition, we
use a more recent solar model than previous publications. Finally, we discuss
the model dependence of the 57Fe signal and identify a class of QCD axion
models which yield an enhanced signal compared to the standard KSVZ [48; 49]
and DFSZ [50; 51] axion models.

The paper is organized as follows. In section II, we revisit the problem of
the solar axion production in nuclear transitions, provide an updated
expression for the flux from 57Fe and discuss its dependence on the specific
axion model; in section III, we present a brief discussion of the current
stellar bounds; section IV is dedicated to the sensitivity estimates for
BabyIAXO and IAXO, assuming different experimental setups; finally, in section
V we provide our conclusions. A discussion of other nuclei that could
contribute spectral lines to the solar axion flux can be found in Appendix A.

##  II Axions from nuclear transitions in the sun

Axion helioscopes [19] such as the planned International Axion Observatory
(IAXO) [44; 42; 45; 46; 43] are searching for very light (sub-eV) axions
produced in the sun. Axions can be produced in the solar plasma by various
processes. Helioscopes search primarily for axions originating from the
Primakoff effect (cf, e.g. [52; 53; 54; 55; 56; 57; 58]) or from axion-
electron interactions [59; 60; 61; 22]. This makes them sensitive to both the
axion-photon coupling $g_{a\gamma}$ and to the product of photon and electron
couplings $g_{a\gamma}g_{ae}$ [62; 22].

Axions can also be emitted in solar nuclear processes. There are two
noteworthy mechanisms for this: nuclear fusion and decays as well as thermal
excitation and subsequent de-excitation of the nuclei of stable isotopes.

Axions produced in nuclear fusion and nuclear decay processes typically carry
an energy of the order of $\sim$ MeV. The axion flux from the $p+d\to$
${}^{3}{\rm He}+a$ (5.5 MeV) reaction, which provides one of the most intense
axion fluxes from nuclear reactions, has been experimentally searched for by
Borexino [63]. More recently, the SNO data [64] has also been scrutinized for
traces of these 5.5 MeV axions. Helioscopes can, in principle, also be
equipped with a $\gamma$-ray detector to look for such axions. For example,
the CAST helioscope installed a $\gamma$-ray calorimeter for some time to gain
sensitivity to these high-energy axions [65]. However, in general all these
analyses provided somewhat weak bounds on the axion-nucleon coupling, since
the solar flux from nuclear reactions is not expected to be very large.
Indeed, Ref. [65] estimated an axion flux of the order of
$10^{10}(g^{3}_{aN})^{2}/({\rm cm^{2}s})$ for the reaction mentioned above.
This is more than 10 orders of magnitude smaller than the flux we will find
below, Eq. (II.3).

A perhaps more promising direction is to look for low lying nuclear
excitations of stable isotopes with a significant abundance inside the sun
that can be thermally excited. Two candidates have been proposed for this in
the past, 57Fe [66; 67] and 83Kr [68]. The former has a first nuclear
excitation energy $E^{*}$ of 14.4 keV and the latter of 9.4 keV. In the solar
core, at temperatures $T\sim 1.3$ keV, these excited states have a small but
non-vanishing occupation number that can be calculated from the Boltzmann
distribution. The amount of axions produced is proportional to the occupation
number, the isotope abundance and the inverse lifetime of the excited state.
By combining a list of possible elements and their nuclear transitions [69]
with the solar abundances in [70], it becomes clear that for IAXO it is the
57Fe transition that would produce the strongest signal (see Appendix A for
more details).

###  II.1 Effective axion coupling in the 57Fe nuclear transition

To describe the axion interactions with nuclei it is convenient to rewrite the
relevant terms in the Lagrangian (I.1) as

$\displaystyle\mathcal{L}_{aN}=-ia\bar{N}\gamma_{5}\left(g^{0}_{aN}+g^{3}_{aN}\tau^{3}\right)N\,.$
(II.1)

Here, $N=(p,n)^{T}$ is the nucleon doublet, $g^{0}_{aN}$, $g^{3}_{aN}$ are the
iso-scalar and iso-vector couplings, respectively, and $\tau^{3}$ is the Pauli
matrix.

The axion-to-photon branching ratio for the decay rates of the first excited
state of 57Fe can then be expressed as [71; 72]

$\displaystyle\frac{\Gamma_{a}}{\Gamma_{\gamma}}=\left(\frac{k_{a}}{k_{\gamma}}\right)^{3}\frac{1}{2\pi\alpha}\frac{1}{1+\delta^{2}}\left[\frac{\beta\,g_{aN}^{0}+g_{aN}^{3}}{\left(\mu_{0}-\frac{1}{2}\right)\beta+\mu_{3}-\eta}\right]^{2},$
(II.2)

where $k_{a},~{}k_{\gamma}$ are the axion and photon momenta, $\mu_{0}$ and
$\mu_{3}$ are the isoscalar and isovector nuclear magnetic moments (expressed
in nuclear magnetons), $\delta$ is the E2/M1 mixing ratio for the 57Fe nuclear
transition, while $\beta$ and $\eta$ are constants dependent on the nuclear
structure [72]. The constant $\delta\simeq 0.002$ can be safely neglected to
the level of precision required in this work. The other constants were
recently reevaluated in Ref. [47]. Assuming $(k_{a}/k_{\gamma})\simeq 1$,
which applies to ultrarelativistic axions, and adopting the most recent values
in Ref. [47] for the relevant nuclear constants, we find

$\displaystyle\frac{\Gamma_{a}}{\Gamma_{\gamma}}$
$\displaystyle=2.32\left(-1.31\,g_{aN}^{0}+g_{aN}^{3}\right)^{2}$ (II.3)
$\displaystyle=2.32\left(0.16\,g_{ap}+1.16\,g_{an}\right)^{2}\,,$ (II.4)

where the second line is expressed in terms of the more common axion couplings
to neutrons,

$g_{an}=g_{aN}^{0}-g_{aN}^{3}\;,$ (II.5)

and to protons,

$g_{ap}=g_{aN}^{0}+g_{aN}^{3}\;.$ (II.6)

From the form of Eq. (II.4), it is convenient to define the effective nucleon
coupling as222We stress that this combination is specific to the 14.4 keV
transition in iron. However, as this is by far the most promising transition
we use it in the following without a specific label.

$\displaystyle g_{aN}^{{\rm eff}}=0.16\,g_{ap}+1.16\,g_{an}\,,$ (II.7)

which is the coupling combination that controls the axion emission rate in
this transition. Notice that the updated branching ratio is 27 % larger than
the one found in Ref. [71] and used in the previous experimental analyses,
including CAST [39], CUORE [40] and more recently XENON1T [41]. Furthermore,
the relative importance of the coupling to protons has strengthened in this
new analysis even though, as evident from Eq. (II.4), it is still considerably
less relevant than the coupling to neutrons.

###  II.2 Axion model dependence of the 57Fe transition rate

Before proceeding with the experimental sensitivity, it is interesting to look
at the axion emission rate for some benchmark axion models.

Expressing the axion-nucleon couplings in terms of the dimensionless axion-
quark coefficients $c^{0}_{q}$ [73] (defined via the Lagrangian term
$\frac{\partial_{\mu}a}{2f_{a}}c^{0}_{q}\bar{q}\gamma^{\mu}\gamma_{5}q$, with
$f_{a}$ the axion decay constant), Eq. (II.4) can be cast as333In deriving
this equation, we use the standard relation (cf., e.g., [18])  $\displaystyle
m_{a}\simeq 5.7\,\mu{\rm eV}\times\left(\frac{10^{12}\,{\rm
GeV}}{f_{a}}\right)$ (II.8) between the axion mass and decay constant.

$\displaystyle\frac{\Gamma_{a}}{\Gamma_{\gamma}}=5.81\times
10^{-16}(1+3.28\,c^{0}_{u}-9.97\,c^{0}_{d}+0.52\,c^{0}_{s}$
$\displaystyle+0.16\,c^{0}_{c}+0.12\,c^{0}_{b}+0.048\,c^{0}_{t})^{2}\left(\frac{m_{a}}{1\,\text{eV}}\right)^{2}\,,$
(II.9)

where $m_{a}$ is the axion mass. In the KSVZ model [48; 49] one has
$c^{0}_{q}=0$, yielding

$\left.\frac{\Gamma_{a}}{\Gamma_{\gamma}}\right|_{\rm KSVZ}=5.81\times
10^{-16}\left(\frac{m_{a}}{1\,\text{eV}}\right)^{2}\,,$ (II.10)

while in the DFSZ model [50; 51], $c^{0}_{u,c,t}=\frac{1}{3}\cos^{2}\beta$ and
$c^{0}_{d,s,b}=\frac{1}{3}\sin^{2}\beta$ corresponding to

$\displaystyle\left.\frac{\Gamma_{a}}{\Gamma_{\gamma}}\right|_{\rm
DFSZ}=5.81\times 10^{-16}$ (II.11)
$\displaystyle\times(1+1.16\cos^{2}\beta-3.11\sin^{2}\beta)^{2}\left(\frac{m_{a}}{1\,\text{eV}}\right)^{2}\,,$

with $\tan\beta$ defined in the perturbative domain $\in[0.25,170]$ [15].

Note that in KSVZ models the axion emission rate is accidentally suppressed as
in these models the neutron coupling is very small. In DSFZ models it can get
enhanced by up to a factor of $\sim$ 4 with respect to the KSVZ.

Eq. (II.2) suggests that a strong enhancement of the axion emission rate can
be achieved for “down-philic” axions, with $c^{0}_{d}\gg c^{0}_{u}$. More
generally this holds if a sizable cancellation with the model-independent
factor normalized to 1 in Eq. (II.2) is avoided. This last possibility
naturally happens in a class of non-universal DFSZ models with
$c^{0}_{u}+c^{0}_{d}=1$ that have been analyzed in Refs. [20; 74; 75] (for a
summary of axion couplings in those models see Table 3 in [76]). For instance,
in the non-universal M1 model of Ref. [20] one has
$c^{0}_{u}=c^{0}_{c}=\sin^{2}\beta$, $c^{0}_{t}=-\cos^{2}\beta$,
$c^{0}_{d}=c^{0}_{s}=\cos^{2}\beta$, $c^{0}_{b}=-\sin^{2}\beta$, with
$\tan\beta\in[0.25,170]$ (also, $c^{0}_{e}=-\sin^{2}\beta$ and $E/N=2/3$),
leading to

$\displaystyle\left.\frac{\Gamma_{a}}{\Gamma_{\gamma}}\right|_{\rm
M1}=5.81\times 10^{-16}$ (II.12)
$\displaystyle\times(1+3.32\sin^{2}\beta-9.50\cos^{2}\beta)^{2}\left(\frac{m_{a}}{1\,\text{eV}}\right)^{2}\,,$

which at small $\beta$ yields an $\mathcal{O}(60)$ enhancement of the axion
emission rate with respect to the KSVZ model.444Again keeping $\beta$ in the
perturbative range. Other non-universal DFSZ models, among those mentioned
above, feature a similar enhancement of the $g^{\rm eff}_{aN}$ coupling, but
they have different values for the axion-photon coupling (that is important in
detection for the IAXO setup). In particular, the non-universal model
$\mathcal{T}_{2}^{(u)}$ of Ref. [74] features the largest axion coupling to
photons among the general class of non-universal DFSZ models with two Higgs
doublets (see Table 5 in [76]).

###  II.3 Solar 57Fe axion flux and limits

To calculate the resulting flux from thermal excitation and subsequent de-
excitation in detail, we follow the derivation by Moriyama [66], also used by
the CAST collaboration in their search for this source [39], but adopt updated
nuclear matrix elements derived in Ref. [47]. The axion emission rate per unit
mass of solar matter is given by

$\mathcal{N}_{a}=\mathcal{N}\,\omega_{1}(T)\,\frac{1}{\tau_{0}}\frac{1}{1+\alpha}\frac{\Gamma_{a}}{\Gamma_{\gamma}},$
(II.13)

where $\mathcal{N}$ is the 57Fe number density per solar mass, $\omega_{1}$
the occupation number of the first excited state, $\tau_{0}$ the lifetime of
the excited state, $\alpha$ the internal conversion coefficient and
$\frac{\Gamma_{a}}{\Gamma_{\gamma}}$ the branching ratio of axion to photon
emission, given in Eq. (II.4). A detailed description of these parameters is
given in Appendix A.

The axion signal from this nuclear transition is expected to be very narrow.
The natural line width is negligible compared to the Doppler broadening
$\sigma$ of

$\sigma(T)=E^{*}\sqrt{\frac{T}{m_{\text{Fe57}}}}\sim 2\ \text{eV}\,,$ (II.14)

corresponding to a full width at half maximum (FWHM) of $2.35\,\sigma\sim 5$
eV. Consequently, the spectral axion flux at a distance of one astronomical
unit $d_{\odot}$ from the sun is an integral of a Gaussian peak over the solar
radius $R_{\odot}$,555We use the solar axion flux library [60] which is
publicly available at https://github.com/sebhoof/SolarAxionFlux to perform all
axion flux calculations.

$\displaystyle\frac{\operatorname{d\\!}\Phi_{a}(E_{a})}{\operatorname{d\\!}E_{a}}=$
$\displaystyle\frac{1}{4\pi
d_{\odot}^{2}}\int_{0}^{R_{\odot}}\mathcal{N}_{a}(T(r))\frac{1}{\sqrt{2\pi}\sigma(T(r))}$
$\displaystyle\times\exp\left(-\frac{(E_{a}-E^{*})^{2}}{2\sigma(T(r))^{2}}\right)\rho(r)4\pi
r^{2}\text{d}r.$ (II.15)

Although, depending on the optics adopted, IAXO’s field of view may cover only
part of the sun, we integrate over the entire solar radius since – as
illustrated in Fig 1 – more than 99 % of the total flux originates from a
circle around the solar centre with radius 0.15 $R_{\odot}$.

Figure 1: Radial dependence of the solar axion flux from 57Fe transitions and
the Primakoff effect. We show the fraction of the flux inside the field of
view which is a circle around the solar center with radius $r$.

The solar model (in our case B16-AGSS09 [77]) fixes temperature $T$, density
$\rho$ and iron abundance at every radius.

It is noteworthy that this axion source is particularly sensitive to the
temperature. This is because of the thermal occupation number
$\omega_{1}\propto e^{-E^{*}/T}$ of the excited nuclear state. For instance,
the latest high and low metallicity solar models (B16-GS98 and B16-AGSS09
[77]) only differ by about 1 % in their respective core temperature but this
alone results in a difference of 12 % in the total flux.

We do not expect the narrow, Doppler-broadened peak to be resolved.666Metallic
magnetic calorimeters [78] may reach an energy resolution of few eV which
would enable them to coarsely resolve the Doppler peak. Hence, only the total
flux is of interest and the integral over $E_{a}$ can be performed to get the
total solar axion flux from the 57Fe nuclear transition [39],

$\displaystyle\Phi_{a}=$ $\displaystyle\frac{1}{4\pi
d_{\odot}^{2}}\mathcal{N}\frac{1}{\tau_{0}}\frac{1}{1+\alpha}\frac{\Gamma_{a}}{\Gamma_{\gamma}}$
$\displaystyle\times\int_{0}^{R_{\odot}}\omega_{1}(T(r))\rho(r)4\pi
r^{2}\text{d}r$ (II.16) $\displaystyle=$ $\displaystyle 5.06\times 10^{23}\
(g_{aN}^{{\rm eff}})^{2}\ \rm{cm}^{-2}\rm{s}^{-1}\ .$ (II.17)

This numerical result is larger than the one previously derived by the CAST
collaboration [39] because we worked with the updated nuclear matrix elements.
In addition, we integrated over a more recent solar model, namely B16-AGSS09
[77], whose core temperature and iron abundance is smaller compared to the
values adopted previously. This explains why the overall flux is not equally
enhanced as the updated nuclear matrix elements. The following calculations
are all done with this updated axion flux and the cited bounds derived by the
CAST collaboration in [39] were rescaled accordingly.

##  III Confronting stellar bounds

Let us start by quickly updating the energy loss constraint. Using the revised
axion rate, Eq (II.4), and the solar model B16-AGSS09 we find a total axion
luminosity (energy loss rate),

$L_{a}=8.38\times 10^{9}(g_{aN}^{{\rm eff}})^{2}\,L_{\odot}$ (III.1)

via the 57Fe transition. The recent study in Ref. [79] constrains an exotic
energy loss to a maximum of 3 % of the standard solar luminosity $L_{\odot}$.
This leads to an improved bound on the axion effective coupling with nucleons,

$g_{aN}^{{\rm eff}}\leq 1.89\times 10^{-6}\,.$ (III.2)

Notice that the result in Eq. (III.2), known as the _solar bound_ on
$g_{aN}^{{\rm eff}}$, is about a factor of two more stringent than the
previous constraint, $g_{aN}^{{\rm eff}}\leq 3.6\times 10^{-6}$ [39]. Besides
the enhanced emission rate [47] and the updated solar model [77], this is also
due to Ref. [39] having excluded only $L_{a}>0.1\ L_{\odot}$.

The axion-nuclear coupling can also be constrained from other stellar
observations. In particular, strong bounds were derived from X-ray
observations of various NS [23; 24; 25; 26; 27; 28]. These bounds are,
however, subject to several uncertainties and do not always agree with each
other, not even when referring to the same star [76]. In any case, all these
analyses suggest a limit of $\sim 10^{-9}$ on some combination of axion-
nucleon couplings.777In most of these analyses, the limit applies only to the
axion-neutron coupling.

A similar bound can be deduced from the analysis of the neutrino signal
observed in coincidence with the SN 1987A event [31; 29; 32; 80; 81; 82; 83;
84; 33; 35]. Here, we refer specifically to the most recent analysis [33],
which derived the bound

$g_{an}^{2}+0.6\,g_{ap}^{2}+0.5\,g_{an}\,g_{ap}\lesssim 8.3\times 10^{-19}\,.$
(III.3)

This is a very strong constraint and, as we will see, only advanced setups may
allow the exploration of smaller couplings. It is worth noticing, however,
that the coupling expected in the 57Fe transition differs from the coupling in
Eq. (III.3) and that, for some specific models, the coupling relevant for the
solar axion searches may be enhanced or suppressed with respect to what is
constrained by the supernova (SN) argument (see Fig. 2). For now, we will not
investigate this argument further but we stress that, due to the peculiar
effective coupling appearing in the 57Fe transition, the comparison with the
astrophysical bounds is model dependent.

Figure 2:  Effective axion coupling entering in the 57Fe transition as a
function of the angle $\beta$, which defines the model dependent couplings
(Cf. Sec. II.2). In the figure, the colored lines indicate the value of the
coupling $g_{aN}^{\rm eff}$ calculated assuming that the specific axion model
saturates the SN bound, given in Eq. (III.3).

That said, Eq. (III.3) can be translated into a limit on $g^{\rm eff}_{aN}$ by
choosing the ratio between the proton and neutron couplings such that the left
hand side of this equation is minimal, $g_{ap}/g_{an}\approx-0.2$ while
keeping $g^{\rm eff}_{aN}$ constant. This yields,

$\left|g^{\rm eff}_{aN}\right|\lesssim 1.1\times 10^{-9}.$ (III.4)

The only model in Fig. 2 which saturates this bound is the M1 model.

##  IV Sensitivity estimates

In this section, we present a detailed discussion of the helioscopes potential
to detect the 57Fe axion flux. A particular emphasis is given to BabyIAXO,
which is expected to start construction soon at DESY in Hamburg. Estimates for
the more advanced IAXO and IAXO+ configurations [45] will also be presented.

Before going into specific experimental aspects let us make a few general
considerations that can provide some help with the experimental design. To
detect the 57Fe line we want to maximize the signal to noise ratio in the
single relevant energy bin which contains all of the signal events. There are
two main contributions to the background for the measurement of the nucleon
line. One is the usual background rate of the detector itself (e.g., cosmic
rays, environmental gammas, intrinsic detector radioactivity) that usually
grows linearly with the area of the detector and it is typically proportional
to the spectral size of the signal bin. This means that, in the case of the
expected narrow signals, it can be reduced by making use of good energy
resolution detectors. Second, there is the physics background due to Primakoff
production. This grows with the photon coupling. Importantly, as the Primakoff
spectrum is continuous, it also grows linearly with worsening energy
resolution. Combining these two effects leads us to the following figure of
merit,

$f\propto\frac{S}{\sqrt{B}}\propto\frac{\epsilon_{o}\epsilon_{d}\
g^{2}_{a\gamma}}{\sqrt{\Delta
E_{d}}\sqrt{ba+g^{4}_{a\gamma}\kappa\epsilon_{o}\epsilon_{d}}}.$ (IV.1)

Here, $\epsilon_{o,d}$ are the optics and detector efficiencies, $\Delta
E_{d}$ is the energy resolution of the detector888In the numerical
calculations in Sects. IV.4 and IV.5 the detector resolution is implemented by
a sharp signal bin of width $\Delta E_{d}$ centered around the 57Fe peak., $b$
is the (spectral) background rate per area, and $a$ is the signal spot area on
the detector. $\kappa$ quantifies the Primakoff flux in the 57Fe signal bin
and is implicitly defined in Eq. (IV.6). From this we can identify the
parameters to be optimized. While good energy resolution is critical this
should not be offset by too large a background. Similarly, with focusing
optics we can reduce the detector area and therefore the background
contribution. However, again there is a balance because such X-ray optics may
have an efficiency significantly smaller than $1$. We will see all this more
concretely when considering explicit setups below.

###  IV.1 BabyIAXO

Conceived as an intermediate stage towards the full IAXO experiment, BabyIAXO
[42; 43] is nevertheless expected to substantially advance the exploration of
the axion parameter space. With its sensitivity, the experiment will be able
to study QCD axion models, and to investigate stellar cooling hints and other
well motivated sections of the parameter space [45; 15].

The BabyIAXO experiment is mainly designed to measure Primakoff axions in the
energy range from 1 keV to 10 keV with the peak of the solar axion flux
spectrum at $\sim$3 keV. BabyIAXO will consist of two magnetic bores of 10 m
length and 70 cm diameter, each with an average magnetic field strength of
about 2 T. Together with newly developed X-ray optics and detector systems
providing higher energy resolution and lower background, BabyIAXO will be the
first helioscope to exceed the sensitivity of the CAST experiment. The magnet
bores of BabyIAXO are of similar diameter as those of IAXO (60 cm) and IAXO+
(80 cm), thus the experience from the optics and detector development for
BabyIAXO can later be applied to IAXO [42]. The first detection system for
BabyIAXO is chosen to be microbulk micromegas technology. This detector
technology has proven background levels as low as
$10^{-7}/(\textrm{keV}\,\textrm{cm}^{2}\,\textrm{s})$ [85] and a high
detection efficiency for Primakoff photons $<$10 keV. A variety of other
detector types, like silicon drift detectors (SDD), metallic magnetic
calorimeters (MMC) and transition edge sensors (TES), are also studied for
BabyIAXO aiming to optimize the energy resolution for precision measurements
of the axion spectrum [42].

###  IV.2 BabyIAXO configurations for 57Fe detection

For the measurement of 57Fe axions at 14.4 keV, the detection efficiency in
the baseline BabyIAXO configuration is not optimal and an enhancement of the
detection system is necessary. Indeed, GEANT4 simulations of the Micromegas
detector to be used in the baseline BabyIAXO configuration show that the
ionization probability of 14.4 keV photons at a conversion length of 3 cm at
the given Argon gas mixture and pressure is only around 15 %. Additionally,
the current designs of the BabyIAXO telescopes are not optimized for energies
above $10$ keV [86]. The expected optics efficiency at 14.4 keV is a mere
$1.3\,\%$ (cf. “baseline”/BabyIAXO0 configuration in Tab. 1).

In the following we discuss different adaptations to the detector system of
BabyIAXO for the measurement of 14.4 keV photons with enhanced efficiency,
energy resolution and lower background.

Let us start with a relatively minimal modification. The sensitivity of the
Micromegas detector for 14.4 keV photons can be enhanced by changing the gas
mixture and adjusting the pressure. Gas mixtures with an inert gas of higher
Z-value, like Xenon, show a higher conversion efficiency for 14.4 keV photons,
compared to Argon. A suitable high ionization probability for 14.4 keV photons
of $>$90 % should be reachable with a 10 cm photon conversion length and a
Xenon based gas mixture at atmospheric pressure. Currently, microbulk
Micromegas [87] are developed to have a square active area of 25 cm x 25 cm
[88] and the whole bore opening can be covered with a few detector tiles. As
BabyIAXO features two bores, one could be used for operation without X-ray
optics and this detector could be operated in parallel on the second magnetic
bore. Removing the optics and covering of the whole area improves the
detection efficiency, as this avoids the losses from inefficient X-ray optics
at this energy. However, the detector will have a higher background due to the
larger conversion volume than the smaller micromegas. This “no optics”
configuration is denoted as “no optics”/BabyIAXO1 in Tab. 1 and in the
figures.

The second detection concept for 14.4 keV photons at BabyIAXO is based on
silicon drift detectors (SDD). This detector consists of a thick negative
doped layer, that is fully depleted by a negative bias voltage, and positive
doped contacts and strips on both sides of the layer. The incoming X-ray
radiation generates electrons in the depleted zone that are then drifted
towards the anode at the end of the layer. An SDD detector of 300 µm thickness
may reach 50 % sensitivity for 15 keV photons [89]. This detector type is also
considered as an additional detector for the baseline measurements at BabyIAXO
[42]. Efficient X-ray optics focusing the photons on the small detector
structure are mandatory, as SDDs are only produced in small pixels with a
detector size in the range of millimeters. Preliminary studies showed that the
optimization of the X-ray optical system for high energy X-rays is possible,
using multilayer coating techniques [90]. The Nuclear Spectroscopic Telescope
Array (NuSTAR) [91] has already used X-ray optics sensitive to photons in the
range of 5-80 keV. A dedicated instrument with non-imaging optics can further
boost the throughput at 14.4 keV. A realistic figure for the optics efficiency
is $\epsilon_{0}=0.3$, which is the value adopted in our analysis. In our
study here, we assume that the optimized optics will be used in just one of
the two BabyIAXO bores. This assumption will be lifted in our analysis of IAXO
and IAXO+. With this included we obtain the “optimized optics”/BabyIAXO2
configuration (cf. Tab. 1).

Finally, a third detection concept suitable for the detection of 14.4 keV
photons at BabyIAXO is based on Cadmium-Zinc-Telluride (CZT) semiconductors.
The detection principle is similar to silicon-based ionization detectors. With
a higher Z-value compared to silicon, CZT provides a higher ionization
probability for the photons and CZT of only 300 µm thickness have an
ionization probability for 14.4 keV of $>$99 %. Indeed CZT detectors are
already used in experiments focusing on the detection of hard X-rays like
NuSTAR [91]. These detectors can reach an energy resolution down to 2 % in the
relevant energy range [92], which makes them optimal for discrimination of the
Primakoff background. However, the good energy resolution comes with the price
of a somewhat increased background. Furthermore, currently it is only possible
to produce detectors with an active area of up to 2 cm x 2 cm [93], which
again makes an efficient X-ray optic necessary for the use of CZT detectors at
BabyIAXO. Coupled with the same NuSTAR-like optics adopted in BabyIAXO2, this
is our “energy resolution”/BabyIAXO3 configuration. Notice that, just like in
the case of BabyIAXO2, the detector and optics discussed above will be
implemented in only one of the two BabyIAXO bores (see Tab. 1).

###  IV.3 IAXO and IAXO+

The full scale helioscope IAXO is expected to begin construction during the
BabyIAXO data-taking period. It adopts realistic components which will allow a
considerably better performance than BabyIAXO in the entire mass range. IAXO+
is a more aggressive setup which will allow to increase the sensitivity and a
deep exploration of physically motivated axion parameters (see. e.g., Refs.
[45; 76]).

Here, we consider two possible setups for IAXO/IAXO+, summarized in table 1.
The first, indicated as IAXO${}_{\textrm{b}}$, uses the benchmark
configuration parameters discussed in the most recent IAXO publication [45].
In this case, we assume an energy resolution of 2 % in the relevant energy
range, which is a very realistic figure for CZT detectors, as discussed in
Sec. IV.1. In addition we assume, somewhat optimistically, that significant
improvements in the backgrounds of these detectors can be achieved.

Alternatively, we consider a further configuration, indicated as
IAXO${}_{\textrm{r}}$, with optimized energy resolution. Though generally
realistic, we nevertheless note that the energy resolution in this
configuration is non-trivial and may require new technologies such as
microcalorimeters operated at mK temperatures [78; 94]. Magnetic
microcalorimeters (MMCs) are also studied in the IAXO collaboration for
precision measurements of the axion spectrum in the energy range $<$10 keV.
First measurements have shown an energy resolution of 6.1 eV (FWHM) at the 5.9
keV 55Fe-peak [78]. To achieve the sufficient energy resolution, advanced
cryogenic setups operating in the milli-Kelvin regime will need to be
implemented and a sufficiently low background still needs to be established.
We therefore have allowed for a somewhat larger background rate.

For IAXO+ we anticipate also additional improvements of the detection system
combined with the enhanced magnetic field, area etc. outlined in [45].

###  IV.4 Results (massless axions)

For our sensitivity study in the coupling space spanned by axion-photon
$g_{a\gamma}$ and effective axion-nucleon coupling $g^{\text{eff}}_{aN}$, we
are going to assume that the axion is very light ($\lesssim$20 meV) or
massless. We will briefly comment on the effects of the mass later, in Sec.
IV.5. For the moment, let us nevertheless note that in case of a discovery of
an axion with a non-vanishing mass one can employ a suitable amount of buffer
gas so that further measurements such as the one of the nucleon coupling can
essentially be done as in the case of vanishing mass.

In the massless case, the conversion probability of an axion to a photon
$P_{a\rightarrow\gamma}$ is energy independent,

$P_{a\rightarrow\gamma}=\frac{g_{a\gamma}^{2}B^{2}L^{2}}{4},$ (IV.2)

where $B$ is the magnetic field and $L$ the length of the conversion volume.
Other experimental parameters that we used in the calculation include the
cross-section of the magnet bores $A$, the total observation time $t$, optics
and detector efficiencies, $\epsilon_{o}$ and $\epsilon_{d}$, the size of the
focal spot $a$, the background level $b$ and the relative energy resolution at
14.4 keV $r_{\omega}$.

We have considered several different combinations of parameters which are
listed in Tab. 1. The benchmark values for the BabyIAXO, IAXO and IAXO+
magnets were taken from the most recent IAXO publication [45].

Because the FWHM of the Doppler-broadened iron peak is only $\sim 5$ eV, it
can be assumed that the whole signal is always in one energy bin.999This is
only approximately true for the configurations IAXOr and IAXO${}^{+}_{\rm r}$
in Tab. 1 for which we assumed an energy resolution equal to the FWHM of the
emission peak. We can therefore calculate the expected number of signal events
$\mu_{\text{signal}}$ directly from the total flux given in Eq. (II.3)

$\displaystyle\mu_{\text{signal}}$ $\displaystyle=\Phi_{a}\
P_{a\rightarrow\gamma}\ A\ t\ \epsilon_{o}\ \epsilon_{d}$ (IV.3)
$\displaystyle\propto(g_{a\gamma}g_{aN}^{{\rm eff}})^{2}.$ (IV.4)

As already discussed at the beginning of the section, to find the 57Fe
detection sensitivity, we also have to take all possible background sources
into account. First, the detector background, which is quantified by the
background level $b$ and which can be measured accurately at times when the
magnet bores are not pointed at the sun. Second, the tail of the Primakoff
spectrum, which may act as an additional background. The expected background
events $\mu_{\rm back}$ are therefore given by

$\displaystyle\mu_{\rm back}=$ $\displaystyle\int_{E^{*}-\frac{\Delta
E_{d}}{2}}^{E^{*}+\frac{\Delta E_{d}}{2}}\left(\frac{\textrm{d}\Phi_{a}^{\rm
P}}{\textrm{d}\omega}\ \epsilon_{o}\epsilon_{d}\right)\textrm{d}\omega\
P_{a\rightarrow\gamma}At$ $\displaystyle+bat\Delta E_{d}$ (IV.5)
$\displaystyle\simeq$
$\displaystyle\left(g_{a\gamma}^{4}\kappa\epsilon_{o}\epsilon_{d}+ba\right)\Delta
E_{d}t\;.$ (IV.6)

The Primakoff flux $\Phi_{a}^{\rm P}$ as well as the efficiencies
$\epsilon_{o}$ and $\epsilon_{d}$ are in general functions of $\omega$. In
case of a sufficiently small energy resolution $\Delta E_{d}$, we can average
over the energy and describe the Primakoff background using the constant
$\kappa$, as done in Eq.(IV.6). If the Primakoff background to the 57Fe-peak
is detectable at 14.4 keV, it is clear that there will be a much stronger
Primakoff signal at smaller energies and we will have measured $g_{a\gamma}$
very precisely. Therefore, we either know the expected contribution from
Primakoff axions to the number of background events $\mu_{\text{back}}$ or it
is negligible compared to the intrinsic detector background.

The $p$-value of the Poisson-distributed observed number of counts $k$ in the
signal bin is given by

$\displaystyle p=$
$\displaystyle\sum_{n=k}^{\infty}\frac{\mu_{\textrm{back}}^{n}e^{-\mu_{\textrm{back}}}}{n!}\
,$ (IV.7)

where $k\in\mathbb{N}$. In order to find the sensitivity in parameter space,
we have to calculate the expectation value of $p$ assuming a Poisson
distribution for $k$ with mean $\mu=\mu_{\textrm{back}}+\mu_{\textrm{signal}}$
for each possible value of the two couplings. Note that $\mu_{\textrm{back}}$
is the sum of the usual detector background and Primakoff background. We
regard the experiment as sensitive to a set of axion couplings if the expected
$p$-value, $\langle p\rangle_{k}$, is smaller than 0.05.101010A measurement
with $p=0.05$ would strictly speaking only amount to a 2$\sigma$ anomaly.
Nevertheless, it is common to define the sensitivity in this way because it
coincides with the expected exclusion limits in the case of a null result. The
resulting sensitivity curves are plotted in Fig. 3.

| BabyIAXO | IAXO | IAXO+
---|---|---|---
| baseline |  no optics |  optimized optics |  high energy resolution |  low background |  high energy resolution |  low background |  high energy resolution
Label | BabyIAXO0 | BabyIAXO1 | BabyIAXO2 | BabyIAXO3 | IAXOb | IAXOr | IAXO${}^{+}_{\rm b}$ | IAXO${}^{+}_{\rm r}$
$B$ [T] | 2 | 2 | 2 | 2 | 2.5 | 2.5 | 3.5 | 3.5
$L$ [m] | 10 | 10 | 10 | 10 | 20 | 20 | 22 | 22
$A$ [m2] | 0.77 | 0.38 | 0.38 | 0.38 | 2.3 | 2.3 | 3.9 | 3.9
$t$ [year] | 0.75 | 0.75 | 0.75 | 0.75 | 1.5 | 1.5 | 2.5 | 2.5
$b$ [$\frac{1}{\textrm{keV}\textrm{cm}^{2}\textrm{s}}$] | $10^{-7}$ | $10^{-6}$ | $10^{-7}$ | $10^{-5}$ | $10^{-8}$ | $10^{-6}$ | $10^{-9}$ | $10^{-6}$
$\epsilon_{d}$ | 0.15 | 0.9 | 0.5 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99
$\epsilon_{0}$ | 0.013 | 1 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3
$a$ [cm2] | 0.6 | 3800 | 0.3 | 0.3 | 1.2 | 1.2 | 1.2 | 1.2
$r_{\omega}=\frac{\Delta E_{d}}{14.4\,{\rm keV}}$ | 0.12 | 0.12 | 0.12 | 0.02 | 0.02 | $\frac{5}{14400}$ | 0.02 | $\frac{5}{14400}$
Table 1:  List of experimental parameters adopted for all helioscope
configurations which are considered in Figs. 3 and 4. As usual $B$ is the
magnetic field of the helioscope, $L$ its length and $A$ the area. $t$ is the
time that the helioscope is pointed at the sun. For these parameters we use
values based on [45]. As already mentioned below Eq. (IV.1) $\epsilon_{o,d}$
are the efficiencies of the optics and detector, $b$ is the spectral
background rate per detector area, and $r_{\omega}$ is the relative spectral
resolution of the detector. Setup BabyIAXO0 is the baseline BabyIAXO,
BabyIAXO1 is a version without optics, BabyIAXO2,3 assume optics optimized for
the 14.4 keV line, with BabyIAXO3 including also a good energy resolution. In
addition we show parameters from more advanced setups of IAXO and IAXO+.
Figure 3: Model independent prediction for the sensitivity to the axion
couplings for light axions ($\lesssim 20$ meV). The different regions refer to
the setups presented in Tab. 1. The dark red region is the solar bound
discussed in the text (cf. Eq. (III.2)). The dark blue region represents the
latest CAST exclusion regions from searches for the Primakoff flux [95] and
the 57Fe peak [39], which we have rescaled to match the updated axion flux
from 57Fe transitions. The dashed horizontal green line indicates the expected
sensitivity to the pure Primakoff flux. The supernova limit, Eq. (III.4),
$g^{\rm eff}_{aN}\lesssim 1.1\times 10^{-9}$ is not shown.

Since we have assumed massless axions for these sensitivity estimates, we do
not show the parameter space of DFSZ and M1 models (cf. Sec. II.1). These
models require large masses – $m_{a}\gtrsim 100\,$meV – at those couplings,
and helioscopes quickly lose sensitivity above $\sim$ 20 meV (cf. Fig. 4).
This problem could be eased by filling the helioscope bore with a buffer gas
[96].

Our analysis shows the potential of BabyIAXO to study areas of the parameter
space, well beyond the solar bound and the region probed by CAST. The
different green shaded areas show the experimental potential for different
configurations, summarized in Tab. 1. The most efficient setups for BabyIAXO
are the ones with optimized optics (labeled BabyIAXO2,3 in Tab. 1). As evident
from the table, these setups allows for an enormous reduction of the total
background by limiting the focal spot area $a$ by $\sim$ 4 orders of magnitude
with respect to the no-optics solution. Adopting this setting, BabyIAXO would
be able to extend its detection potential also to regions of the parameter
space below the Primakoff sensitivity, for $g_{aN}^{\rm eff}\gtrsim 10^{-7}$.
It is, therefore, possible, at least in principle, that BabyIAXO could
discover axions through the 57Fe channel, before the Primakoff flux can be
detected. If, on the other hand, axions have couplings in the green shaded
area above the Primakoff sensitivity line (dashed green), one might have the
opportunity to extract both couplings and derive information about the
underlying axion model.

The more optimistic IAXO and IAXO+ configurations can explore an even larger
area of parameter space. A noteworthy feature of these exclusion curves is
their behaviour at different values of $g_{a\gamma}$. At values of
$g_{a\gamma}\lesssim 10^{-11}\ \textrm{GeV}^{-1}$ the detector background
dominates over the Primakoff background. Therefore the figure of merit in Eq.
(IV.1) becomes $f\propto\frac{\epsilon_{o}\epsilon_{d}\
g^{2}_{a\gamma}}{\sqrt{\Delta E_{d}ba}}$. With the parameters given in Table
1, the configurations with minimized background slightly outperform the ones
with optimized energy resolution in this regime (cf. Fig. 3). However, at
$g_{a\gamma}\gtrsim 10^{-11}\ \textrm{GeV}^{-1}$ the Primakoff background
starts to play a role and eventually dominates. In this regime the figure of
merit is given by $f\propto\sqrt{\frac{\epsilon_{o}\epsilon_{d}}{\Delta
E_{d}\kappa}}$. The detector background becomes negligible and the
configurations with optimized energy resolution are significantly more
sensitive than the ones with minimized background. Therefore, the ideal
detector for the 57Fe line crucially depends on the value of $g_{a\gamma}$. If
BabyIAXO detects Primakoff axions, a detector with good energy resolution may
be required to supress the Primakoff background to the 57Fe line. If, on the
other hand, BabyIAXO only puts a stronger bound on $g_{a\gamma}$, the energy
resolution becomes less important and the low background detectors may be
advantageous.

###  IV.5 Effects of a finite axion mass

On the production side, the axion mass only becomes relevant at scales of the
solar temperature $\sim$ keV. Therefore, we can safely regard the solar axion
flux as independent of the mass. However, on the detection side, a finite
axion mass can cause decoherence between the photon and axion wave functions
inside the magnet bores and lead to a signal suppression. The full expression
for the conversion probability $P_{a\rightarrow\gamma}$ of axions into photons
in the helioscope reads [97; 96]

$P_{a\rightarrow\gamma}=\frac{g_{a\gamma}^{2}B^{2}L^{2}}{4}\times\frac{2(1-\cos(qL))}{(qL)^{2}}\
,$ (IV.8)

where $q$ is the transferred momentum given by $m_{a}^{2}/(2\omega)$ and
$\omega$ is the energy of the axion. The suppression becomes relevant for
masses above $\sim$ 20 meV, the exact value depending on the length of the
magnet.

The decoherence effect can be compensated at the cost of some absorption by
feeding a buffer gas into the bores [96; 98]. This is why the sensitivity
study to effectively massless (i.e. $m_{a}\lesssim 20\ \mathrm{meV}$) axions
in the previous section serves as a good benchmark. Nevertheless, we also want
to explicitly investigate the sensitivity to massive axions without a buffer
gas. To do this, we assume that the background from Primakoff axions is
negligible and that instead the detector background dominates. In this case,
the background is independent of any axion properties and the signal depends
on the product of the two couplings, $g_{a\gamma}g^{\mathrm{eff}}_{aN}$, as
well as the mass. The statistical analysis is equivalent to the one in the
massless case. From the expected signal and background we calculate the
expected $p$-value and draw the sensitivity curves where $p$ is expected to be
smaller than 0.05. The results for a selection of viable setups are plotted in
Fig. 4. The regions shaded in yellow indicate the coupling relations for the
DFSZ, M1 and $\mathcal{T}_{2}^{(u)}$ models (cf. Sec. II.1).

The sensitivity curves are very similar to typical helioscope exclusion plots
in the coupling vs. mass plane with two noteworthy exceptions. Because the
57Fe line is highly energetic at 14.4 keV, the transferred momentum $q$ is
smaller than for axions of the same mass from other solar processes. As a
result the decoherence effect only becomes relevant at slightly higher masses
in comparison to – for instance – Primakoff axions. To illustrate this effect,
we have plotted the expected sensitivity of the IAXO${}^{+}_{\rm b}$ setup
with a decoherence factor from a Primakoff spectrum as a dashed black line in
Fig. 4. Furthermore, the oscillations of the form factor for large $qL$ are
clearly visible in the 57Fe exclusion lines while they are washed out in the
case of the broadband Primakoff spectrum.

Figure 4: Model independent prediction of the IAXO sensitivity to the 57Fe
peak, assuming that axions are produced only through the axion coupling to
nucleons. In contrast to Fig. 3, we directly show the sensitivity of the
various setups to the coupling combination $g_{a\gamma}g_{aN}^{\textrm{eff}}$
under the assumption that the Primakoff background is negligible. The
oscillations at higher masses are due to the form factor in the conversion
probability. For comparison we show the effect of decoherence for a Primakoff
spectrum as a dashed black line. The dark blue region represents the rescaled
CAST result [39]. The dark yellow region indicates the parameter space
expected for the DFSZ model. In brighter shades of yellow, we show the flavor
non-universal DFSZ models M1 and $\mathcal{T}_{2}^{(u)}$. The dashed yellow
line, on the other hand, shows the expected coupling for a nucleophilic QCD
axion model of the kind presented in Ref. [21], with $n=3$. All models with
$n>3$ would be already accessible to BabyIAXO. Note that the experimental
sensitivity estimates here do not assume the use of a buffer gas, which would
extend the sensitivity to higher masses.

##  V Discussion and conclusion

In this work, we have presented the first dedicated investigation of the
BabyIAXO and IAXO potential to detect 14.4 keV axions from 57Fe transitions in
the sun. The analysis is based on a recent reevaluation of the matrix elements
for the nuclear transition [47] and an updated solar model [77]. We have
carefully considered different realistic setups for the detectors and optics
to evaluate which combination gives the best performance for 14.4 keV axions.
The configuration parameters are listed in Tab. 1.

Our results, summarized in Fig. 3 and 4, show that already BabyIAXO will be
able to study a large section of interesting axion parameter space, well
beyond the region accessed by CAST, particularly in the configurations with an
optimized X-ray optics. The potential will be greatly improved with IAXO and
IAXO+.

The figures also show representative QCD axion models. The yellow band in Fig.
4 represents the parameter area spanned by generalized DFSZ models [20; 76]
(QCD axion models with two Higgs doublets plus a singlet scalar field,
allowing for flavor non-universality). There, we present the specific examples
of the classical DFSZ axion model, and of the M1 and $\mathcal{T}_{2}^{(u)}$
models discussed in Sec. II.1. We are not showing the well known KSVZ axion
model since an accidental cancellation reduces its effective coupling to
nucleons relevant in the 57Fe transition (see Sec. II.1).

Although BabyIAXO is expected to have enough sensitivity to explore large
sections of the parameter space for these models [76], a sizable axion flux
from 57Fe transitions requires large axion masses, where BabyIAXO loses
sensitivity. This problem can be eased with the use of a buffer gas [96], a
technique already tested in CAST [95]. The sensitivities shown in the figures
do not account for this option in BabyIAXO nor in its scaled up versions. A
dedicated study may show if a buffer gas may allow to probe the M1 or other
DFSZ-like models through the 57Fe line in the near future.

Less minimal models for QCD axions may present larger couplings to nucleons
and be better accessible through the 57Fe channel. For example, the
nucleophilic QCD axion models presented in Ref. [21] (and shown in Fig. 4),
have exponentially large couplings to nucleons111111In model A of Ref. [21],
one has  $\displaystyle g_{a\gamma}g_{aN}^{\rm eff}\sim\frac{2^{2n}\alpha_{\rm
em}}{2\pi\,f_{a}}\frac{m_{n}}{f_{a}}\sim 2^{2n+2}\cdot
10^{-17}\,\left(\frac{m_{a}}{\rm eV}\right)^{2}\,{\rm GeV^{-1}}$ (V.1) where
$n+1$ is the number of Higgs doublets in the model. The parameter $n$ is
constrained to $n\lesssim 50$ in order to avoid sub-Planckian Landau poles
[99].  and are efficiently produced in 57Fe transitions even at lower axion
mass. As shown in Fig. 4, practically, the entire class of these models will
be accessible already to BabyIAXO, even without the need for a buffer gas.

One should nevertheless keep in mind that most of the region shown in the
figures is in tension with astrophysical considerations, in particular,
SN1987A (cf. Eq. (III.4)). As these are affected by their own uncertainties
(e.g. relying on a single supernova event) it would nevertheless be comforting
to have independent confirmation in more controlled setups. In addition, the
IAXO+ setup shown in Fig. 3 approaches a level of sensitivity comparable to
Eq. (III.4). This shows a pathway to pushing beyond the astrophysical limits.

In conclusion, helioscopes of the next generation may offer a unique chance to
probe an interesting range of the $g_{aN}^{\rm eff}$-$g_{a\gamma}$ parameter
space. While this potential is expected to be greatly improved with its scaled
up versions, IAXO and IAXO+, already BabyIAXO will have enough sensitivity to
detect axions with couplings to nucleons over an order of magnitude below the
solar bound (see, Fig. 3). Furthermore, our analysis shows that already
BabyIAXO, especially if equipped with optimized optics, has the potential to
detect through the 57Fe channel axions too weakly coupled to photons to give a
sizable Primakoff flux (region below the dashed green line in Fig. 3). In a
more likely scenario, a detection of axions through 57Fe will be accompanied
by a (larger) signal from Primakoff axions, allowing to extract important
information about its couplings to both photons and nucleons.

Ending on an optimistic outlook, we note that discovery of an axion and its
nucleon-coupling induced lines could perhaps also shed light on properties of
the sun.121212See [61] for an opportunity to use axions with electron
couplings to measure the solar metallicity. For example, the strong
temperature dependence may make this a good way to measure the sun’s core
temperature.

## Acknowledgements

We would like to thank José Crespo López-Urrutia for helpful comments and
discussions. LT is funded by the Graduiertenkolleg Particle physics beyond the
Standard Model (GRK 1940). The work of M.G. is supported by funding from a
grant provided by the Fulbright U.S. Scholar Program and by a grant from the
Fundación Bancaria Ibercaja y Fundación CAI. M.G. thanks the Departamento de
Física Teórica and the Centro de Astropartículas y Física de Altas Energías
(CAPA) of the Universidad de Zaragoza for hospitality during the completion of
this work. I.G.I and J. G. acknowledge support from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme, grant agreement ERC-2017-AdG788781 (IAXO+), as well as the Spanish
Agencia Estatal de Investigación under grant PID2019-108122GB. J.J. is
grateful to be part of the ITN HIDDeN which is supported within the European
Union’s Horizon 2020 research and innovation programme grant agreement No
860881. Part of this work was performed under the auspices of the US
Department of Energy by Lawrence Livermore National Laboratory under Contract
No. DE-AC52-07NA27344.

|  ${}^{\textbf{57}}$Fe |  ${}^{\textbf{83}}$Kr |  ${}^{\textbf{169}}$Tm |  ${}^{\textbf{187}}$Os |  ${}^{\textbf{201}}$Hg
---|---|---|---|---|---
$E^{*}$ [keV] | 14.4 | 9.4 | 8.4 | 9.7 | 1.6
$J_{0}$ | 1/2 | 9/2 | 1/2 | 1/2 | 3/2
$J_{1}$ | 3/2 | 7/2 | 3/2 | 3/2 | 1/2
$\tau_{0}$ [ns] | 141 | 212 | 5.9 | 3.4 | 144
$\alpha$ | 8.56 | 17.09 | 285 | 264 | 47000
$\epsilon$ | $10^{-4.5}$ | $10^{-8.75}$ | $10^{-11.9}$ | $10^{-10.6}$ | $10^{-10.83}$
$a$ [%] | 2.14 | 11.55 | 100 | 1.6 | 13.2
$\mathcal{N}_{a}(r=0)$ [relative to 57Fe] | 1 | $1.8\text{\times}{10}^{-3}$ | $1.3\text{\times}{10}^{-4}$ | $3.0\text{\times}{10}^{-5}$ | $1.9\text{\times}{10}^{-6}$
Table 2: Isotopes with a nuclear M1 transition and $E^{*}<20$ keV. The element
abundances $\epsilon$ are taken from Ref. [70]. All other values are tabled in
the appendix of Ref. [69]. The values in the last row were calculated by
evaluating Eqs. (A.1) and (A.2) with the solar core temperature $T(r=0)=1.33$
keV.

##  Appendix A Axions from other nuclei

The phenomenological discussion in this work centers around the 14.4 keV line
of 57Fe because it is expected to give the strongest signal. In order to
ensure that this is true, especially considering that the 57Fe-line suffers
from strong thermal suppression, we systematically searched for alternative
nuclear M1 transitions which may also generate a line in the solar axion flux.

A list of potential candidates is provided in the appendix of Ref. [69] in
form of a list of isotopes featuring low-energy nuclear transitions. All of
the calculations in Sect. II equally apply to M1 transitions of nuclei other
than 57Fe. Hence, we only need to compare their respective axion flux per
mass. This is expressed in Eq. (II.13) above as

$\mathcal{N}_{a}=\mathcal{N}\omega_{1}\frac{1}{\tau_{0}}\frac{1}{1+\alpha}\frac{\Gamma_{a}}{\Gamma_{\gamma}}\
.$ (A.1)

The nuclear matrix elements entering in the ratio of axion to photon emissions
have not been computed with equal precision for the various nuclear
transitions. It is however reasonable to assume values of order 1 for the
dimensionless constants $\beta$ and $\eta$. Furthermore, the E2/M1 mixing
ratio is already close to its ideal value of zero for the 57Fe line. The
factor $\frac{\Gamma_{a}}{\Gamma_{\gamma}}$ is therefore expected to be
comparable (or smaller than the one of 57Fe in case of a large $\delta$) for
all nuclear transitions and we can focus on the combination of abundance,
occupation number and inverse lifetime.

First sizable differences appear in the isotope’s number density
$\mathcal{N}$. To estimate its value for all radii in the sun, we need to make
two assumptions. First that the contribution of one isotope to the total
element abundance is constant throughout the sun and identical to the one
found on earth, which we denote as $a$. And second that the radial density
profiles of heavy elements only differ from the one of iron by a constant
factor. These two reasonable assumptions allow us to compare the respective
values of $\mathcal{N}$ by just multiplying the photospheric abundance with
the isotope abundance on earth. The former is tabulated in Ref. [70] on a
logarithmic scale. We give the ratio of the number density of the element in
question normalized to the one of hydrogen, i.e. $\epsilon\equiv N_{X}/N_{H}$,
in table 2.

The thermal occupation number $\omega_{1}$ crucially depends on the transition
energy $E^{*}$ and can be written as [66; 39]

$\omega_{1}=\frac{(2J_{1}+1)e^{-E^{*}/T}}{(2J_{0}+1)+(2J_{1}+1)e^{-E^{*}/T}},$
(A.2)

where $J_{0}$ and $J_{1}$ are the total angular momentum quantum numbers of
the ground and excited state, respectively.

We collected all isotopes from the appendix of Ref. [69] with a nuclear M1
transition below 20 keV in Tab. 2. The last row indicates the size of the
axion flux from these nuclear transitions relative to the one from 57Fe. It
becomes clear that the strong Boltzmann suppression in the case of 57Fe is
compensated by a relatively large abundance and a small internal conversion
coefficient. Even though the values in Tab. 2 should be understood as very
rough estimates, this clearly indicates that the 57Fe line generates the
strongest axion signal from nuclear de-excitations.

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