# ALP-portal majorana dark matter

Shivam Gola<EMAIL_ADDRESS>The Institute of Mathematical Sciences, C.I.T
Campus, Taramani, Chennai 600 113, India  Homi Bhabha National Institute, BARC
Training School Complex, Anushakti Nagar, Mumbai 400094, India  Sanjoy Mandal
<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular,
CSIC/Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia), Spain  Nita Sinha
<EMAIL_ADDRESS>The Institute of Mathematical Sciences, C.I.T Campus,
Taramani, Chennai 600 113, India  Homi Bhabha National Institute, BARC
Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

Axion like particles(ALPs) and right handed neutrinos (RHNs) are two well-
motivated dark matter(DM) candidates. However, these two particles have a
completely different origin. Axion was proposed to solve the Strong CP
problem, whereas RHNs were introduced to explain light neutrino masses through
seesaw mechanisms. We study the case of ALP portal RHN DM (Majorana DM) taking
into account existing constraints on ALPs. We consider the leading effective
operators mediating interactions between the ALP and SM particles and three
RHNs to generate light neutrino masses through type-I seesaw. Further, ALP-RHN
neutrino coupling is introduced to generalize the model which is restricted by
the relic density and indirect detection constraint.
Keywords: Axion like particle, Heavy Neutrinos, Dark matter

##  1 Introduction

Dark matter is one of the most important issues of modern particle physics and
cosmology. Although a wide variety of experiments ranging from sub-galactic
scale to a large cluster of galaxies have accumulated data in support of DM’s
existence [1, 2, 3, 4, 5], its microscopic properties still remain unknown.
Several DM candidates have been proposed and searched for, however, no
completely satisfactory DM candidate has been found so far. Among a lot of
possibilities, the axion and sterile neutrinos can be regarded as a leading
candidates for DM. These particles arise in well-motivated extensions of the
Standard Model and have very rich phenomenology.

It is known from the cosmological observations that the sum of the active
neutrino masses $\sum m_{\nu}\leq 0.12$ eV [6, 7] and contribution to the
relic density $\leq 4.5\times 10^{-3}$, which is too low to explain the DM
abundances today, hence there is no explanation for DM within the SM. However,
a new heavier neutrino field could explain DM and it is naturally required to
explain the masses of active neutrinos as inferred from the oscillation
experiments [8]. The discovery of neutrino oscillations confirming the
existence of at least two non-vanishing neutrino mass-squared differences
necessitate physics beyond the Standard Model (BSM). In principle neutrino
mass could be simply generated by addition of right-handed neutrinos (RHNs) to
the SM particle content. These RHNs interact with SM fields via mixing with
active neutrinos. Since RHNs are SM singlet, they allow Majorana mass term
along with usual Dirac mass term. This is known as type-I seesaw mechanism [9,
10, 11, 12]. Mass of these RHNs could range from eV to GUT scale depending on
the models [13, 14, 15, 16]. RHNs can also play the role of warm dark
matter(WDM), which is singlet under the SM gauge symmetry and has tiny mixing
with the SM neutrinos leading to a long lifetime [17, 18, 19]. Also, KeV scale
RHNs have been studied as a viable DM candidate [20, 21, 22]. In this work we
have instead focused on the prospects of having GeV scale RHNs as a Weakly
interacting massive particle (WIMP) DM candidate.

Axion [23] was postulated in the Peccei-Quinn (PQ) mechanism to solve the
strong CP problem [24, 25, 26, 27] of quantum chromodynamics (QCD). This axion
can be identified as a (pseudo) Nambu-Goldstone boson associated with the
spontaneous breaking of the $U(1)_{\text{PQ}}$ global symmetry [28, 29, 30].
This QCD axion gets a tiny mass from the explicit breaking of this global
symmetry due to QCD anomaly. Astrophysical and experimental searches have not
favoured the PQ model. To resolve issues with PQ model, other popular
solutions like KSVZ [31, 32], DFSZ [33] etc. invoking axion were also proposed
and studied afterwards. The magnitude of the couplings of axions to ordinary
matter is inversely proportional to the axion decay constant $f_{a}$ which is
associated with the $U(1)_{\text{PQ}}$ symmetry breaking scale. Hence, the
couplings are highly suppressed if $f_{a}$ is sufficiently large and this
features make the axion suitable to be a DM candidate. Many BSM extensions
which features spontaneously broken global $U(1)$ symmetry predict massless
Nambu-Goldstone bosons whose couplings are not constrained unlike the original
QCD axion. These kind of particles are known as axion like particles (ALPs).
Mass of these ALPs are not related to its symmetry breaking scale unlike PQ
axion. In general they are not supposed to solve the strong CP problem, but
with the introduction of planck scale operators they could solve the strong CP
problem [34, 35]. Here we will consider the most general $SU(2)_{L}\otimes
U(1)_{Y}$ invariant formulation of ALP interactions developed in Refs. [36,
37, 38]. This generic effective ALPs Lagrangian allows ALPs coupling with all
the SM gauge bosons as well as with all the SM fermions. In addition to this
effective ALP Lagrangian, we introduce three RHNs which can generate light
neutrino masses through type-I seesaw. We invoke a $\mathbb{Z}_{2}$ symmetry
under which all SM and BSM particles are even except the lightest RHN. Further
we introduced the RHN-ALP coupling and show that the lightest RHN (Singlet
majorana fermion) which is odd under $\mathbb{Z}_{2}$ can play the role of DM
candidate. Note that the phenomenology of this ALP-mediated DM [39] will be
similar to pseudoscalar-portal DM. Also DM interacting via the exchange of a
light ALP can induce observable signals in indirect detection experiments
while evading the strong bounds from direct DM searches. Note that ALPs with
mass $M_{a}\sim\mathcal{O}(\text{GeV})$ may also show up at colliders [40, 41,
42] or in rare meson decays [43, 44, 45].

Rest of the manuscript is organized as follows. In Sec. 2 we introduced our
model, detailing the new interactions present. In Sec. 3 we summaries the
existing constraints on ALP parameter space coming from various observables
and collider searches. In Sec. 4 we have explored and discussed the feasible
parameter space coming from DM analyses such as relic density, direct and
indirect detection. Finally, we give our conclusions in Sec. 5.

##  2 Model

The model that we consider is the minimal combination of type-I seesaw and
effective ALP interaction with additional $\mathbb{Z}_{2}$ symmetry apart from
the SM gauge symmetry [46, 47]. The matter content of the model is shown in
Table. 1. Let’s first briefly discuss the features of a generic ALP
Lagrangian. We extend the SM particle content by adding an additional ALP
which is a singlet under SM charges and is a pseudo Nambu-Goldstone boson of a
spontaneously broken symmetry at some energy which is higher than the
electroweak scale $v$. In effective theory the operators will be weighted by
powers of $a/f_{a}$, where $f_{a}$ is the scale associated to the physics of
the ALP, $a$. Effective linear Lagrangian with one ALP has been already
discussed in great detail in Ref. [36, 37]. For linear EWSB realizations the
most general linear bosonic Lagrangian involving $a$ is given by,

| Standard Model | New Fermions | New Scalar
---|---|---|---
|  $\ell_{L}$ |  $e_{R}$ |  $q_{L}$ |  $u_{R}$ |  $d_{R}$ |  $H$ |  $N_{1}$ |  $N_{2,3}$ |  $a$
$SU(2)_{L}$ | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1
$U(1)_{Y}$ | -1/2 | -1 | 1/6 | 2/3 | -1/3 | 1/2 | 0 | 0 | 0
$\mathbb{Z}_{2}$ | $+$ | $+$ | $+$ | $+$ | $+$ | $+$ | $-$ | $+$ | $+$
Table 1: Matter content and charge assignment of the considered model.

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{ALP}},$
(1)

where the leading order effective Lagrangian $\mathcal{L}_{\text{SM}}$ is same
as the SM one and with

$\displaystyle\mathcal{L}_{\text{ALP}}\ $ $\displaystyle=\
\frac{1}{2}\partial_{\mu}a\partial^{\mu}a\ -\
\frac{1}{2}M_{a}^{2}a^{2}-\frac{C_{\tilde{G}}}{f_{a}}\
aG_{a\mu\nu}\tilde{G}^{\mu\nu a}-\frac{C_{\tilde{B}}}{f_{a}}\
aB_{\mu\nu}\tilde{B}^{\mu\nu}-\frac{C_{\tilde{W}}}{f_{a}}\
aW_{a\mu\nu}\tilde{W}^{a\mu\nu}$
$\displaystyle+iC_{a\Phi}\times\big{[}(\bar{Q}_{L}Y_{U}\tilde{\Phi}u_{R}-\bar{Q}_{L}Y_{D}\Phi
d_{R}-\bar{L}_{L}Y_{E}\Phi e_{R})\frac{a}{f_{a}}+h.c.\big{]}$ (2)

Here $\Phi$ and $a$ are the Higgs and ALP fields respectively. $C_{i}$ where
$i=G,B,W,a\Phi$ are the corresponding Wilson coefficients for ALP-gauge boson
and ALP-matter interactions. Parameters $M_{a}$ and $f_{a}$ are the ALP mass
and energy scale associated to ALP physics. $Y_{D}$, $Y_{U}$ and $Y_{E}$ are
$3\times 3$ matrices in flavour space which stands for down-type quarks, up-
type quarks and charged leptons, respectively. We see that ALP Lagrangian has
a very generic form than that of the QCD-axion, where the mass of ALP is not
restricted by the new physics scale. The remaining fields and parameters are
the same as in SM. References [48, 37, 49] have discussed the phenomenology of
the various pieces of the model. Now let’s introduce the type-I seesaw
Lagrangian with three additional SM singlet RHNs:

$\displaystyle\mathcal{L}_{\text{RHN}}\ =\
i\sum^{3}_{i=1}\bar{N}_{i}\gamma^{\mu}\partial_{\mu}N_{i}\
-\sum^{3}_{j=2}Y_{\alpha j}\bar{L}_{\alpha}\tilde{\Phi}N_{j}\ -\
\sum^{3}_{i,j=2}M_{ij}\bar{N}^{c}_{i}N_{j}\ -\ M_{N_{1}}\bar{N}^{c}_{1}N_{1}\
+\ \text{h.c.}$ (3)

where $N_{i}$ are the SM singlet RHNs, $Y_{\alpha j}$ is the Dirac Yukawa
coupling and $M_{ij}$ is the Majorana mass term. As, $N_{1}$ is odd under
$\mathbb{Z}_{2}$ symmetry, Dirac type of Yukawa interaction is forbidden for
it unlike that for $N_{2,3}$. As a result of this, one of the light neutrinos
will remain massless and we have enough parameters to describe the neutrino
oscillations data. In addition to this, $\mathbb{Z}_{2}$ symmetry stabilizes
the $N_{1}$ and it can play the role of DM candidate if one allows the
following ALP-RHN interaction,

$\displaystyle\mathcal{L}_{\text{ALP-RHN}}\
=-\sum^{3}_{i=1}\frac{C_{aN_{i}}}{f_{a}}(\bar{N_{i}}\gamma^{\mu}\gamma^{5}N_{i})\partial_{\mu}a,$
(4)

where $C_{aN_{i}}$ denotes the ALP-RHNs Wilson coefficient and through this DM
particle $N_{1}$ can communicate with the ALP sector. We call this ALP-portal
RHN DM. We consider ALP mass of the order of few hundreds of MeV to GeV. We
have considered $N_{1}$ to have mass up to few TeV. In the next section we
discuss the allowed range for the several parameters of the model from the
various phenomenological bounds.

##  3 Existing constraints on ALP parameter space

Before diving into the details of DM analyses, lets first recall the existing
experimental bounds on the couplings of ALPs to gluons, photons, fermions and
also from collider searches with $f_{a}\sim\mathcal{O}(1\,\text{TeV})$ [40,
41, 43, 44, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. ALP photon
coupling is the primary parameter through which astrophysical and cosmological
bounds are set on such particles. The ALP and photon coupling can be deduced

$\mathcal{L}_{a\gamma}=-C_{a\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu},\,\,\text{with}\,\,C_{a\gamma}=\frac{(C_{\tilde{B}}\cos^{2}\theta_{w}+C_{\tilde{W}}\sin^{2}\theta_{w})}{f_{a}}$
(5)

Particularly experiments like CAST, using primakoff process, constrained the
parameter space ($M_{a}-C_{a\gamma}$) heavily for $M_{a}$ smaller then few eV
[62]. For $M_{a}\sim 1$ MeV the best present constraint comes from Beam Dump
experiments, $C_{a\gamma}/f_{a}\leq 10^{-5}\,\text{GeV}^{-1}$ [40] 111For very
low masses, tighter constraint exists but we do not discuss them here as we
are interested in mass range $M_{a}\sim\text{MeV-few GeV}$.. A slightly higher
mass range is constrained by the collider experiments such as LEP and LHC
[40]. In LEP, the process like $e^{-}e^{+}\to\gamma a\to 3\gamma$ is being
analysed to constrain the coupling whereas in LHC it is $pp\to\gamma a$ which
searched for mono, di or tri photon signals. All these constraints are
described in the references [63, 37, 64]. A limite on axion gauage boson
coupling $C_{\tilde{W}}/f_{a}<10^{-5}$ GeV-1 is obtained for $0.175\leq
M_{a}\leq 4.78$ GeV by analysing of the process $B^{\pm}\rightarrow
K^{\pm}a,a\rightarrow\gamma\gamma$ at the BABAR experiment [65]. Constraint on
$C_{\tilde{G}}$ is set by mono jet 8 TeV LHC analysis. Unlike $C_{a\gamma}$,
here it is much more complicated to put bound on $C_{\tilde{G}}$ due to large
numbers of diagrams involved in the process. However, the dominant digram was
found to be $gg\to ag$, which is complicated due to hadronisation that leads
to jets in the final state. These constraints have been analysed in the
reference [40, 55]. The limit from this study reads as
$C_{\tilde{G}}/f_{a}\leq 10^{-4}\,\text{GeV}^{-1}$ for $M_{a}\leq
0.1\,\text{GeV}$. Also the bound on
$\text{BR}(K^{+}\to\pi^{+}+\text{nothing})$ [66] can be used to constrain the
process $K^{+}\to\pi^{+}+\pi^{0}(\pi^{0}\to a)$ which can be reinterpreted in
terms of ALP-gluon coupling $C_{\tilde{G}}$, yielding $C_{\tilde{G}}/f_{a}\leq
10^{-5}\,\text{GeV}^{-1}$ for $M_{a}\leq 60\,\text{MeV}$. Constraints on the
ALP matter coupling $C_{a\phi}$ is studied less compared to gauge boson
coupling, however several processes involving flavors have been studied to put
bound on it [43]. The constraints on ALP-fermion coupling $C_{a\Phi}$ depends
on the ALP mass. The higher mass range is tested through the rare meson
decays. Rare meson decay at Beam Dump experiments (CHARM) sets tight
constraints on $C_{a\Phi}$ for mass range $1\,\text{MeV}\leq M_{a}\leq
3\,\text{GeV}$ as $C_{a\Phi}/f_{a}<(3.4\times 10^{-8}-2.9\times 10^{-6})\
\text{GeV}^{-1}$ [67]. For our interested mass range we summaries the existing
tightest constraints in Table. 2.

Bound on Coupling | ALP Mass Range | Observables
---|---|---
$\frac{C_{\tilde{G}}}{f_{a}}\leq 10^{-4}$ GeV-1 |  $M_{a}\leq 0.1$ GeV | mono-jet 8 TeV@LHC
$\frac{C_{a\gamma}}{f_{a}}\leq 10^{-5}$ GeV-1 |  $M_{a}\sim$ 1 MeV | Beam Dump
$\frac{C_{\tilde{W}}}{f_{a}}\sim 10^{-5}$ GeV-1 | 0.175 GeV $\leq M_{a}\leq$ 4.78 GeV | BABAR Exp.
$\frac{C_{a\Phi}}{f_{a}}\sim 10^{-8}-10^{-6}$ GeV-1 | 1 MeV $<M_{a}<$ 3 GeV | Rare meson decay
Table 2: Summary of existing constraint on ALP couplings.

Benchmark: To start analyzing the model, we choose ALP mass $M_{a}=10$ GeV.
ALP-Gauge boson couplings $C_{\tilde{B}}=C_{\tilde{W}}$, since ALP is mostly
constrained through $C_{a\gamma},C_{\tilde{G}}$ and we also choose
$\frac{C_{\tilde{G}}}{f_{a}},\frac{C_{a\gamma}}{f_{a}}$ of the order of
$10^{-4}$ GeV-1 and ALP-fermion coupling $\frac{C_{a\Phi}}{f_{a}}\sim 10^{-6}$
GeV-1, which satisfy all of the above constraints.

ALP decay: ALP can decay into SM final states when kinematically accessible,
such as leptons, gauge bosons ($W^{\pm},Z,\gamma,g$) and hadrons. Analytical
form of these decay widths are listed in Appendix A. We found that for the
range of couplings considered by us, ALP decay width is small enough to use
the narrow width approximation.

Figure 1: The left and right panels show the variations of branching ratios
for various decay channels with respect to ALP mass $M_{a}$. Different colors
stand for different final states. In the right panel we choose smaller value
of $C_{a\phi}/f_{a}=10^{-3}$ TeV-1 to show that ALP decays dominantly to gauge
bosons.

In Fig. 1, we show the various branching ratios of ALP decay to SM final
states. For left and right panel, we choose
$C_{\tilde{G}}/f_{a}=C_{a\gamma}/f_{a}=0.1\,\text{TeV}^{-1},\,C_{a\Phi}/f_{a}=1\,\text{TeV}^{-1}$
and
$C_{\tilde{G}}/f_{a}=C_{a\gamma}/f_{a}=0.1\,\,\text{TeV}^{-1},C_{a\Phi}/f_{a}=0.001\,\text{TeV}^{-1}$,
respectively. We see from Fig. 1 that ALP mostly decays to gluon and photon
pair. Also $W^{+}W^{-}$, $ZZ$ pair contributes significantly as ALP mass
crosses respective threshold values. Comparing the left panel with right panel
one sees that lepton channels only contributes when $C_{a\Phi}$ is relatively
large.

##  4 Dark matter analysis

So far we have discussed the constraints on ALP couplings to SM field. In this
section we collect the results of our analyses of DM phenomenology. In our
case $N_{1}$ plays the role of DM due to $\mathbb{Z}_{2}$ symmetry protection.
In order to calculate all the vertices, the model is implemented in the
FeynRules package [68]. All DM observables such as the thermal component of
the DM relic abundance are determined using micrOMEGAS [69] which relies on
CalcHEP [70] model file obtained from FeynRules. In the scanning of the
relevant parameters of the model, we have imposed phenomenological bounds we
discussed earlier. The mass of $N_{2,3}$ is always chosen to be greater than
$N_{1}$ for further analysis.

Figure 2: Annihilation diagrams contributing to the relic abundance of
$N_{1}$.

###  4.1 Relic density

The relevant processes responsible for the freeze-out of DM in the early
universe are shown in Fig. 2. All together, they determine the relic abundance
of our assumed DM, $N_{1}$. The annihilation cross section$(\sigma)$ and
thermal average annihilation cross section $\braket{\sigma v}$ and thus the
relic abundance, scales straightforwardly with the parameters of the model.
The exact analytical expressions for all the annihilation cross section are
given in Appendix. B and Appendix. C. For our considered benchmark, only gauge
bosons channels are relevant. We see that annihilation cross section for the
process $N_{1}N_{1}\to f\bar{f}$ is negligible due to very small value of
$C_{a\Phi}$. On the other hand with the limit $M_{N_{1}}>M_{a}$,
$N_{1}N_{1}\to aa$ annihilation channel opens up but is $v^{2}$ suppressed.

Figure 3: Annihilation cross sections for different channels as a function of
DM mass $M_{N_{1}}$. In the right panel we choose smaller value of
$C_{a\Phi}/f_{a}=10^{-3}\,\text{TeV}^{-1}$ to illustrate that annihilation to
gauge bosons dominate. In both panel we fix $C_{aN_{1}}=0.1\,\text{TeV}^{-1}$.

In Fig. 3, we show the annihilation cross section for different channels as a
function of DM mass $M_{N_{1}}$. For left and right panel, we choose
$C_{a\Phi}/f_{a}=1\,\text{TeV}^{-1}$ and
$C_{a\Phi}/f_{a}=0.001\,\text{TeV}^{-1}$ respectively. For both panel we fix
$C_{\tilde{G}}/f_{a}=C_{a\gamma}/f_{a}=C_{aN_{1}}=0.1\,\text{TeV}^{-1}$. For
relatively small value of $C_{a\Phi}/f_{a}$ annihilation cross section to
gauge boson dominates over fermionic channels.

Figure 4: The left panel shows the relic density ($\Omega_{N_{1}}h^{2}$)
behavior with DM mass $M_{N_{1}}$ for the choosen ALP parameters, labeled at
the top. The three colored curves are due to three discrete choice of
ALP-$N_{1}$ coupling. The region inside the horizontal black lines stands for
the measured $3\sigma$ relic density range given by Planck satellite data, Eq.
6. The right panel is done using same analysis but with ALP-$N_{1}$ coupling
varying continuously on the horizontal axis whereas mass of $N_{1}$ has been
chosen discretely.

The left and right panel of Fig. 4 shows the relic density behaviour as a
function of DM mass $M_{N_{1}}$ and ALP-RHN coupling $C_{aN_{1}}$,
respectively. In the left and right panel, three curves stand for three
choices of ALP-RHN coupling and DM masses, respectively. The narrow horizontal
band is the $3\sigma$ range for cold DM derived from the Planck satellite data
[6]:

$\displaystyle 0.1126\leq\Omega_{N_{1}}h^{2}\leq 0.1246.$ (6)

Only for solutions falling exactly within this band the totality of the DM can
be explained by $N_{1}$. We see from Fig. 4 that for smaller value of DM mass
$M_{N_{1}}$, the required values of $C_{aN_{1}}/f_{a}$ is relatively large to
explain the correct relic density.

Figure 5: The left panel describes the colored regions where the relic density
bound ($\Omega_{N_{1}}h^{2}\leq 0.12$) holds on the $m_{N_{1}}$-$C_{aN_{1}}$
plane. We have choosen the three discrete values for the ALP-gluon coupling
($C_{\tilde{G}}$) labeled by the corresponding colors. In the right panel the
same analysis is done but now with three discrete values for ALP-photon
coupling ($C_{a\gamma}$).

In Fig. 5 we show the region where the relic density bound
$\Omega_{N_{1}}h^{2}\leq 0.12$ holds on the $M_{N_{1}}-C_{aN_{1}}$ plane. In
the left panel we have choosen three discrete values for the ALP-gluon
coupling $C_{\tilde{G}}/f_{a}=10^{-3}\,\text{TeV}^{-1}$ (red),
$10^{-2}\,\text{TeV}^{-1}$ (pink) and $10^{-1}\,\text{TeV}^{-1}$ (gray) by
fixing other couplings as $C_{a\gamma}/f_{a}=10^{-1}\,\text{TeV}^{-1}$ and
$C_{a\Phi}/f_{a}=10^{-3}\,\text{TeV}^{-1}$. In the right panel, the same
analysis is done but now we have fixed
$C_{\tilde{G}}/f_{a}=10^{-1}\,\text{TeV}^{-1}$ and choose three discrete
values of ALP-photon coupling, $C_{a\gamma}/f_{a}=10^{-3}\,\text{TeV}^{-1}$
(red), $10^{-2}\,\text{TeV}^{-1}$ (pink) and $10^{-1}\,\text{TeV}^{-1}$
(gray). Note that in left (right) panel for relatively smaller values of ALP-
gluon (ALP-photon) couplings red and pink region overlap. This happens due to
subdominant contribution to the relic density of annihilation channel
$N_{1}N_{1}\to gg$ ($N_{1}N_{1}\to\gamma\gamma$). We find that when $gg$
channel dominates compared to other annihilation channel, the required value
of $C_{aN_{1}}/f_{a}$ is smaller to satisfy the relic density.

###  4.2 Direct detection

The XENON1T experiment [71] currently has the best sensitivity for spin-
independent and spin-dependent DM-nucleon interactions in our interested mass
range of DM. The interaction between DM $N_{1}$ and a quark $q$ can be
described by the following effective Lagrangian:

$\displaystyle\mathcal{L}=\frac{C_{a\Phi}C_{aN_{1}}}{f_{a}^{2}M_{a}^{2}}m_{q}M_{N_{1}}\bar{q}\gamma_{5}q\,\bar{N_{1}}\gamma_{5}N_{1}.$
(7)

Note that this is only valid when mediator ALP mass $M_{a}$ is relatively
large compared to momentum transferred involved in the scattering process.
Following Ref. [72, 73, 74, 43, 75] we found that in non-relativistic limit,
differential scattering cross section to scatter of a nucleus is
$d\sigma/dE_{R}\propto q^{4}$, where $q^{2}=2m_{N}E_{R}$ is the momentum
transfer, $m_{N}$ is the mass of nucleus and $E_{R}$ is the nuclear recoil
energy. In direct detection experiments typical recoil energy is
$\mathcal{O}(10\,\text{KeV})$, hence direct detection cross section is heavily
suppressed.

###  4.3 Indirect detection

If DM $N_{1}$ annihilates to SM final states with annihilation cross section
near the thermal relic benchmark value $\braket{\sigma v}\sim 3\times
10^{-26}\,\text{cm}^{3}/s$, it may be detected indirectly. Perhaps, $\gamma$
rays are the best messengers since they proceed almost unaffected during their
propagation, thus carrying both spectral and spatial information. These
$\gamma$-rays can be produced from DM annihilation, either mono-energetically
from direct annihilation $N_{1}N_{1}\to\gamma\gamma$, $\gamma X$ or with
continuum spectra from decays of the annihilation products $N_{1}N_{1}\to
X\bar{X}(X=\text{SM state})$.

Figure 6: Above plot shows the thermally averaged annihilation cross section
for di photon emmisions. Experimental limite obtained from AMS-02(Grey) [77],
MAGIC(Dark-Pink) [76], FermiLAT(Blue) [78], HESS(Purple) [76] data are shown.
The red and orange lines are the the model predictions for different choices
of ALP-photon couplings $C_{a\gamma}/f_{a}$.

These $\gamma$ rays would be produced preferentially in regions of high DM
density and can be best detectable by Fermi-LAT [78], HESS [76]. The
integrated $\gamma$-ray flux from the DM annihilation in a density
distribution $\rho(\bf r)$ is given by

$\displaystyle\Phi_{\gamma}(\Delta\Omega)=\frac{1}{4\pi}\frac{\braket{\sigma
v}}{2M_{N_{1}}^{2}}\int_{E_{\text{min}}}^{E_{\text{max}}}\frac{dN_{\gamma}}{dE_{\gamma}}dE_{\gamma}.J,$
(8)

where $J=\int_{\Delta\Omega}{\int_{\text{l.o.s}}\rho^{2}(\bf
r)d\ell}d\Omega^{\prime}$ is the line-of-sight (l.o.s) integral through the DM
distribution integrated over a solid angle $\Delta\Omega$. The integral
$\int^{\text{ROI}}\frac{dJ}{d\Omega}d\Omega$ represents the astrophysical
component of the DM flux calculation in particular Region of Interest (ROI).
AMS-02 look for excess positron flux in the positron energy range $\sim$1 GeV
to $\sim$500 GeV over the cosmic positron background. These positron can give
rise two photon in final state after subsequent process. A model independent
study of such bounds is studied[77]. FermiLAT collaboration look for direct DM
annihilation to two photons from dwarf spheroidal galaxies of Milky Way in
photon energy from few GeV to few hundreads of GeV. Higher energy range is
being explored by MAGIC and HESS collaboration. We have considered the current
upper limit from AMS-02, MAGIC, HESS, and annihilation data from FermiLAT
respectively in Fig. 6 on thermally averaged annihilation cross section of
di-$\gamma$ final state along with our model predictions for our chosen
benchmark in Fig. 6. The model prediction for ALP-photon coupling
$C_{a\gamma}/f_{a}=0.1\,\text{TeV}^{-1}$ lie very close to the Fermi-LAT and
HESS upper limit. This suggests that future sensitivities of Fermi-LAT or HESS
can either probe or exclude large parameter space of the model considered by
us.

##  5 Conclusion

We have analyzed heavy neutrino DM candidate in a minimal extension of SM,
which features three RHNs and one ALP. This model is well motivated as it not
only accounts for DM, but it also explains the neutrino oscillations. Hence,
ALP mediated RHN DM is interesting from both the model-building and
phenomenological perspectives. We have considered the lightest RHN as DM which
is odd under $\mathbb{Z}_{2}$ symmetry and identified the region of parameters
where DM predictions are in agreement with DM relic abundance. In addition,
this model also quite naturally explain the null results of LUX and XENON1T
due to the pseudoscalar nature of interactions with quarks. We have
highlighted the importance of complementary searches, for instance via
indirect detection with single and di-photon. Although the current limits from
Fermi-LAT lie above the predicted signals for our choice of parameter space,
future sensitivities of Fermi-LAT might offer promising prospects to probe
both the low as well as high DM mass regions.

###### Acknowledgements.

The work of SM is supported by the Spanish grant FPA2017-85216-P (AEI/FEDER,
UE) and PROMETEO/2018/165 (Generalitat Valenciana).

##  Appendix A ALP Decay Widths

$\displaystyle\Gamma_{agg}=\frac{2C_{\tilde{G}}M_{a}^{3}}{\pi
f_{a}^{2}},\,\,\Gamma_{a\gamma\gamma}=\frac{M^{3}_{a}(C_{\tilde{B}}\cos^{2}{\theta_{w}}+C_{\tilde{W}}\sin^{2}{\theta_{w}})^{2}}{4\pi
f_{a}^{2}},\,\,\Gamma_{aW^{+}W^{-}}=\frac{C_{\tilde{W}}^{2}(M_{a}^{2}-4M_{W}^{2})^{\frac{3}{2}}}{2\pi
f_{a}^{2}}$ $\displaystyle\Gamma_{a\gamma
Z}=\frac{\sin^{2}{\theta_{w}}\cos^{2}{\theta_{w}}(C_{\tilde{B}}-C_{\tilde{W}})^{2}(M^{2}_{a}-M^{2}_{z})^{3}}{2\pi
f_{a}^{2}M^{3}_{a}},\,\,\Gamma_{af\bar{f}}=\frac{N_{c}C_{a\phi}^{2}M_{f}^{2}\sqrt{M_{a}^{2}-4M_{q}^{2}}}{8\pi
f_{a}^{2}}$
$\displaystyle\Gamma_{aZZ}=\frac{(C_{\tilde{W}}\cos^{2}{\theta_{w}}+C_{\tilde{B}}\sin^{2}{\theta_{w}})^{2}(M_{a}^{2}-4M_{Z}^{2})^{\frac{3}{2}}}{4\pi
f_{a}^{2}},\,\,\Gamma_{aN_{i}}=\frac{C_{aN_{i}}^{2}m^{2}_{N_{i}}\sqrt{M_{a}^{2}-4m_{N_{i}}^{2}}}{\pi
f_{a}^{2}}$ (9)

where $N_{c}=3(1)$ for quark(lepton).

##  Appendix B Annihilation cross sections

$\displaystyle\sigma_{gg}=\frac{16C^{2}_{aN_{1}}C^{2}_{\tilde{G}}M^{2}_{N_{1}}s^{2}}{\pi
f^{4}_{a}(M^{2}_{a}-s)^{2}\sqrt{1-\frac{4M^{2}_{N_{1}}}{s}}},\,\,\sigma_{\gamma\gamma}=\frac{2C^{2}_{aN_{1}}[C_{\tilde{B}}\cos^{2}\theta_{w}+C_{\tilde{W}}\sin^{2}\theta_{w}]^{2}M^{2}_{N_{1}}s^{2}}{\pi
f^{4}_{a}(M^{2}_{a}-s)^{2}\sqrt{1-\frac{4M^{2}_{N_{1}}}{s}}}$
$\displaystyle\sigma_{ZZ}=\frac{2C^{2}_{aN_{1}}[C_{\tilde{W}}\cos^{2}\theta_{w}+C_{\tilde{B}}\sin^{2}\theta_{w}]^{2}M^{2}_{N_{1}}s^{2}(1-\frac{4M^{2}_{Z}}{s})^{3/2}}{\pi
f^{4}_{a}(M^{2}_{a}-s)^{2}\sqrt{1-\frac{4M^{2}_{N_{1}}}{s}}},$
$\displaystyle\sigma_{W^{+}W^{-}}=\frac{2C^{2}_{aN_{1}}C^{2}_{\tilde{W}}M^{2}_{N_{1}}s^{2}(1-\frac{4M^{2}_{W}}{s})^{3/2}}{\pi
f^{4}_{a}(M^{2}_{a}-s)^{2}\sqrt{1-\frac{4M^{2}_{N_{1}}}{s}}},\,\,\sigma_{f\bar{f}}=\frac{N_{c}C^{2}_{aN_{1}}C^{2}_{a\phi}M^{2}_{N_{1}}M^{2}_{f}s\sqrt{1-\frac{4M^{2}_{f}}{s}}}{2\pi
f^{4}_{a}(M^{2}_{a}-s)^{2}\sqrt{1-\frac{4M^{2}_{N_{1}}}{s}}},$
$\displaystyle\sigma_{aa}=\frac{4C^{4}_{aN_{1}}M^{2}_{N_{1}}}{\pi
f^{4}_{a}s}\sqrt{\frac{s-4M^{2}_{a}}{s-4M^{2}_{N_{1}}}}\bigg{(}2s-\frac{4M^{4}_{a}M^{2}_{N_{1}}}{M^{4}_{a}-4M^{2}_{a}M^{2}_{N_{1}}+M^{2}_{N_{1}}s}$
$\displaystyle+\frac{8M^{2}_{N_{1}}(2M^{4}_{a}-4M^{2}_{a}s+s^{2})\tanh^{-1}\left(\frac{\sqrt{(s-4M^{2}_{a})(s-4M^{2}_{N_{1}})}}{-s+2M^{2}_{a}}\right)}{(s-2M^{2}_{a})\sqrt{(s-4M^{2}_{a})(s-4M^{2}_{N_{1}})}}\bigg{)}$
(10)

here $\theta_{w}$ is the weinberg angle.

##  Appendix C Thermal average annihilation cross sections

$\displaystyle(\sigma
v)_{gg}\approx\frac{256C^{2}_{aN_{1}}C^{2}_{\tilde{G}}M^{6}_{N_{1}}}{\pi
f^{4}_{a}(M^{2}_{a}-4M^{2}_{N_{1}})^{2}},\,\,(\sigma
v)_{\gamma\gamma}\approx\frac{32C^{2}_{aN_{1}}[C_{\tilde{B}}\cos^{2}\theta_{w}+C_{\tilde{W}}\sin^{2}\theta_{w}]^{2}M^{6}_{N_{1}}}{\pi
f^{4}_{a}(M^{2}_{a}-4M^{2}_{N_{1}})^{2}}$ $\displaystyle(\sigma
v)_{ZZ}\approx\frac{32C^{2}_{aN_{1}}[C_{\tilde{B}}\sin^{2}\theta_{w}+C_{\tilde{W}}\cos^{2}\theta_{w}]^{2}M^{3}_{N_{1}}(M^{2}_{N_{1}}-M^{2}_{Z})^{3/2}}{\pi
f^{4}_{a}(M^{2}_{a}-4M^{2}_{N_{1}})^{2}},$ $\displaystyle(\sigma
v)_{W^{+}W^{-}}\approx\frac{32C^{2}_{aN_{1}}C_{\tilde{W}}^{2}M^{3}_{N_{1}}(M^{2}_{N_{1}}-M^{2}_{W})^{3/2}}{\pi
f^{4}_{a}(M^{2}_{a}-4M^{2}_{N_{1}})^{2}},\,(\sigma
v)_{f\bar{f}}\approx\frac{2N_{c}C^{2}_{aN_{1}}C^{2}_{a\phi}M^{3}_{N_{1}}M^{2}_{f}\sqrt{M^{2}_{N_{1}}-M^{2}_{f}}}{\pi
f^{4}_{a}(M^{2}_{a}-4M^{2}_{N_{1}})^{2}},$ $\displaystyle(\sigma
v)_{aa}\approx\frac{8C^{4}_{aN_{1}}M_{N_{1}}v^{2}\sqrt{M^{2}_{N_{1}}-M^{2}_{a}}(32M^{8}_{N_{1}}-64M^{2}_{a}M^{6}_{N_{1}}+48M^{4}_{a}M^{4}_{N_{1}}-16M^{6}_{a}M^{2}_{N_{1}}+3M^{8}_{a})}{3\pi
f^{4}_{a}(2M^{2}_{N_{1}}-M^{2}_{a})^{4}}$ (11)

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