Linking axionlike dark matter to neutrino masses

TL;DR

This paper proposes a framework connecting axionlike particles to neutrino masses via the inverse seesaw mechanism, offering models that explain dark matter relic density and are testable by upcoming experiments.

Contribution

It introduces minimal inverse seesaw models where gravity-induced operators generate neutrino and dark matter masses involving ALPs, with some models allowing ALPs and sterile neutrinos as dark matter.

Findings

Gravity-stable models match observed dark matter density.

Models are consistent with neutrino and ALP phenomenology.

Some ALP parameters are within reach of future experiments.

Abstract

We present a framework linking axionlike particles (ALPs) to neutrino masses through the minimal inverse seesaw (ISS) mechanism in order to explain the dark matter (DM) puzzle. Specifically, we explore three minimal ISS cases where mass scales are generated through gravity-induced operators involving a scalar field hosting ALPs. In all of these cases, we find gravity-stable models providing the observed DM relic density and, simultaneously, consistent with the phenomenology of neutrinos and ALPs. Remarkably, in one of the ISS cases, the DM can be made of ALPs and sterile neutrinos. Furthermore, other considered ISS cases have ALPs with parameters inside regions to be explored by proposed ALPs experiments.

Figures

Tables

Table 1. Table 1: Two of the continuous symmetries of the Lagrangian in Eq. ( 1 ). B and 𝕃 𝕃 \mathbb{L} are the baryon number and the generalized lepton number, respectively. The charges a 𝑎 a , b 𝑏 b , c 𝑐 c and d 𝑑 d are given by a = q ​ d / 2 , 𝑎 𝑞 𝑑 2 a=qd/2, b = s ​ d + c 𝑏 𝑠 𝑑 𝑐 b=sd+c , c = 1 − r ​ d 𝑐 1 𝑟 𝑑 c=1-rd and d = ( p − q / 2 ) − 1 𝑑 superscript 𝑝 𝑞 2 1 d=\left(p-q/2\right)^{-1} .

	$Q_{\text{L}i}$	$d_{\text{R}i}$	$u_{\text{R}i}$	$H_i$	$L_i$	$l_{\text{R}i}$	$N_{\text{R}\beta }$	$S_{\text{R}\alpha }$	$E_{\text{L}}$	$E_{\text{R}}$	$\sigma$
B	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$𝕃$	$0$	$0$	$0$	$0$	$1$	$1$	$1$	$a$	$b$	$c$	$d$

Table 2. Table 2: The appropriated ( D , v σ ) 𝐷 subscript 𝑣 𝜎 \left(D,\,v_{\sigma}\right) values in order to provide Ω a , DM ​ h 2 = Ω DM Planck ​ h 2 subscript Ω 𝑎 DM superscript ℎ 2 superscript subscript Ω DM Planck superscript ℎ 2 \Omega_{a,\text{DM}}h^{2}=\Omega_{\text{DM}}^{\text{Planck}}h^{2} (first and second columns) and Ω a , DM ​ h 2 = 0.57 × Ω DM Planck ​ h 2 subscript Ω 𝑎 DM superscript ℎ 2 0.57 superscript subscript Ω DM Planck superscript ℎ 2 \Omega_{a,\text{DM}}h^{2}=0.57\times\Omega_{\text{DM}}^{\text{Planck}}h^{2} (first and third columns). It has been considered 10 − 3 ≤ g ≤ 2 superscript 10 3 𝑔 2 10^{-3}\leq g\leq 2 and Ω DM Planck ​ h 2 superscript subscript Ω DM Planck superscript ℎ 2 \Omega_{\textrm{DM}}^{\textrm{Planck}}h^{2} at 3 σ 𝜎 \sigma level.

$D$	$v_{\sigma }$ $\left(\text{GeV}\right)$	$v_{\sigma }$ $\left(\text{GeV}\right)$
8	$\left(1.9-3.3\right)\times {10}^{10}$	$\left(1.5-2.8\right)\times {10}^{10}$
9	$\left(0.6-1.1\right)\times {10}^{11}$	$\left(5.5-9.5\right)\times {10}^{10}$
10	$\left(1.9-3.2\right)\times {10}^{11}$	$\left(1.6-2.8\right)\times {10}^{11}$
11	$\left(5.0-8.2\right)\times {10}^{11}$	$\left(4.4-7.2\right)\times {10}^{11}$
12	$\left(1.2-1.9\right)\times {10}^{12}$	$\left(1.0-1.7\right)\times {10}^{12}$
13	$\left(2.6-4.0\right)\times {10}^{12}$	$\left(2.3-3.6\right)\times {10}^{12}$
14	$\left(5.2-7.9\right)\times {10}^{12}$	$\left(4.6-7.0\right)\times {10}^{12}$
15	$\left(1.0-1.4\right)\times {10}^{13}$	$\left(0.9-1.3\right)\times {10}^{13}$
16	$\left(1.7-2.5\right)\times {10}^{13}$	$\left(1.6-2.3\right)\times {10}^{13}$
17	$\left(2.9-4.2\right)\times {10}^{13}$	$\left(2.7-3.8\right)\times {10}^{13}$
18	$\left(4.8-6.7\right)\times {10}^{13}$	$\left(4.3-6.1\right)\times {10}^{13}$
19	$\left(0.7-1.0\right)\times {10}^{14}$	$\left(6.8-9.4\right)\times {10}^{13}$

Table 3. Table 3: Discrete and continuous charge assignments of the fields in the different models.

Model	Symmetry	$Q_{\text{L}i}$	$d_{\text{R}i}$	$u_{\text{R}i}$	$H$	$L_i$	$l_{\text{R}i}$	$N_{\text{R}\beta }$	$S_{\text{R}\alpha }$	$E_{\text{L}}$	$E_{\text{R}}$	$\sigma$
(2,2)	$Z_{17}$	$1$	$2$	$0$	$16$	$14$	$15$	$13$	$8$	$15$	$12$	$7$
	U${\left(1\right)}_A$	$0$	$0$	$0$	$0$	$11/2$	$11/2$	$11/2$	$-5/2$	$11/2$	$15/2$	$1$
	$Z_{19}$	$1$	$14$	$7$	$6$	$16$	$10$	$3$	$9$	$17$	$2$	$4$
	U${\left(1\right)}_A$	$0$	$0$	$0$	$0$	$11/2$	$11/2$	$11/2$	$-5/2$	$5/2$	$7/2$	$1$
(3,3)	$Z_{17}$	$1$	$10$	$9$	$8$	$14$	$6$	$5$	$7$	$2$	$15$	$4$
	U${\left(1\right)}_A$	$0$	$0$	$0$	$0$	$11/2$	$11/2$	$11/2$	$-5/2$	$9/2$	$7/2$	$1$
	$Z_{19}$	$1$	$13$	$8$	$7$	$16$	$9$	$4$	$12$	$9$	$11$	$18$
	U${\left(1\right)}_A$	$0$	$0$	$0$	$0$	$11/2$	$11/2$	$11/2$	$-5/2$	$11/2$	$7/2$	$1$
(2,3)	$Z_{10}$	$1$	$2$	$0$	$9$	$7$	$8$	$6$	$6$	$8$	$6$	$6$
	$Z_4$	$0$	$1$	$3$	$3$	$0$	$1$	$3$	$1$	$1$	$1$	$2$
	U${\left(1\right)}_A$	$0$	$0$	$0$	$0$	$7/2$	$7/2$	$7/2$	$-3/2$	$7/2$	$3/2$	$1$

Table 4. Table 4: Main features of the models discussed in the text. We have considered the constant g ⊂ [ 10 − 3 , 2 ] 𝑔 superscript 10 3 2 g\subset[10^{-3},2] in the mass term of the ALPs, the η 𝜂 \eta Yukawa at order 10 − 2 superscript 10 2 10^{-2} in the M23a(b) models, and Ω DM Planck ​ h 2 superscript subscript Ω DM Planck superscript ℎ 2 \Omega_{\textrm{DM}}^{\textrm{Planck}}h^{2} at 3 ​ σ 3 𝜎 3\sigma .

ISS	$m_a$	$g_{a\gamma }$	$M$ (TeV), ${\mu }_S$ (keV)
Model	$\times {10}^{-11}$ eV	$\times {10}^{-17}$ GeV-1	$m_{\nu \text{light}}$ (eV)
M22a	$\left(19.0-56.0\right)$	$\left(5.5-7.8\right)$	$\left(1.5-4.5\right),\left(0.1-0.7\right),$
			$\left(1.0-1.4\right)$
M22b	$\left(0.52-1.4\right)$	$\left(1.1-1.6\right)$	$\left(24.3-65.6\right),\left(11.3-58.9\right),$
			$\left(0.4-0.6\right)$
M33a	$\left(19.0-56.0\right)$	$\left(2.7-3.9\right)$	$\left(1.5-4.5\right),\left(0.1-0.7\right),$
			$\left(1.0-1.4\right)$
M33b	$\left(0.52-1.4\right)$	$\left(2.2-3.1\right)$	$\left(24.3-65.6\right),\left(11.3-58.9\right),$
			$\left(0.4-0.6\right)$
M23a	$\left(0.06-0.30\right)\times {10}^{11}$	$\left(720-1200\right)$	$\left(7.5-21.0\right),\left(4.12-19.41\right),$
			$\left(1.33-2.24\right)$
M23b	$\left(0.03-0.2\right)\times {10}^{11}$	$\left(830-1400\right)$	$\left(5.6-15.8\right),\left(2.7-12.7\right),$
			$\left(1.4-2.5\right)$

Equations

$$\mathcal{L}\supset {\mathcal{L}}_{\text{SM}}^{\text{Yuk}}+{\mathcal{L}}_{\sigma }+{\mathcal{L}}_{\text{ISS}}+{\mathcal{L}}_E,$$

$${\mathcal{L}}_{\text{SM}}^{\text{Yuk}}=Y_{ij}^{(u)}\overline{Q_{\text{L}i}}\tilde{H}{u}_{\text{R}j}+Y_{ij}^{(d)}\overline{Q_{\text{L}i}}H{d}_{\text{R}j}+Y_{ij}^{(l)}\overline{L_i}H{l}_{\text{R}j}+\text{H.c.},$$

$${\mathcal{L}}_{\sigma }\supset g\frac{{\sigma }^D}{M_{\text{Pl}}^{D-4}}+\text{H.c.},$$

$$10^9\lesssim \sqrt{2}⟨\sigma ⟩\equiv v_{\sigma }\lesssim 10^{14}\text{ GeV}.$$

$$m_a=|g{|}^{\frac{1}{2}}D{M}_{\text{Pl}}{\lambda }^{\frac{D}{2}-1},$$

$$g_{a\gamma }=\frac{\alpha }{2\pi }\frac{C_{a\gamma }}{v_{\sigma }},$$

$$C_{a\gamma}=2\sum_{\psi}\Big{(}X_{\psi_{\textrm{L}}}-X_{\psi_{\textrm{R}}}\Big{)}\Big{(}C_{\textrm{em}}^{(\psi)}\Big{)}^{2},$$

$${\mathcal{L}}_{\text{ISS}}=y_{i\beta }\overline{L_i}\tilde{H}{N}_{\text{R}\beta }+{\zeta }_{\alpha \beta }\frac{{\sigma }^p}{M_{\text{Pl}}^{p-1}}\overline{S_{\text{R}\alpha }}(N_{\text{R}\beta }{)}^{\text{C}}+\eta {}_{\alpha {\alpha }^{\prime }}\frac{{\sigma }^q}{2M_{\text{Pl}}^{q-1}}\overline{S_{\text{R}\alpha }}(S_{\text{R}{\alpha }^{\prime }}{)}^{\text{C}}+{\theta }_{\beta {\beta }^{\prime }}\frac{{\sigma }^r}{2M_{\text{Pl}}^{r-1}}\overline{N_{\text{R}\beta }}(N_{\text{R}{\beta }^{\prime }}{)}^{\text{C}}+\text{ H.c.},$$

$$M_{\nu }$$

$$\displaystyle M_{D}\equiv\left[\begin{array}[]{c}m_{D}\\ 0\end{array}\right]$$

$$m_{Di\beta }=y_{i\beta }\frac{v_{\text{SM}}}{\sqrt{2}},$$

$${\mu }_{S\alpha {\alpha }^{\prime }}=\eta {}_{\alpha {\alpha }^{\prime }}{M}_{\text{Pl}}{\lambda }^{|q|},$$

$$m_{\nu \text{light}}$$

$$M\gtrsim \sqrt{10\frac{{\mu }_S}{\text{keV}}}\text{TeV},$$

$$1\le M\le 25\text{TeV},0.1\le {\mu }_S\le 50\text{keV.}$$

$$(p,|q|)$$

$$m_{\nu\textrm{light}}=\big{[}y^{\intercal}\zeta^{-1}\eta\left(\zeta^{\intercal}\right)^{-1}y\big{]}\frac{v_{\text{SM}}^{2}}{\sqrt{2}v_{\sigma}}.$$

$${\mathcal{L}}_E$$

$$A_2(Z_N)=A_3(Z_N)=A_{\text{grav}}(Z_N)=0\text{Mod}\frac{N}{2},$$

$$A_2(Z_N)$$

$$A_{\text{grav}}(Z_N)$$

$${\Omega }_{a,\text{DM}}{h}^2\approx 0.16{\left[\frac{{\Theta }_i}{\pi }\right]}^2\times {\left[\frac{m_a}{\text{eV}}\right]}^{1/2}{\left[\frac{v_{\sigma }}{10^{11}\text{GeV}}\right]}^2,$$

$${\Omega }_{{\nu }_S,\text{DM}}{h}^2=$$

$${\Omega }_{{\nu }_S,\text{DM}}{h}^2=0.3\left[\frac{{\sin }^{2}2\theta }{10^{-10}}\right]{\left[\frac{m_{{\nu }_S}}{100\text{ keV}}\right]}^2.$$

$${\Omega }_{\text{DM}}^{\text{Planck}}{h}^2$$

$$\text{mbox}Br({\ell }_{\beta }\to {\ell }_{\alpha }\gamma )=$$

$${\Omega }_{a,\text{DM}}{h}^2\simeq$$

$$g_{a\gamma }\cong 7.54\times 10^{-17}\left[\frac{3.08\times 10^{13}\text{ GeV}}{v_{\sigma }}\right]{\text{ GeV}}^{-1},$$

$$m_a\cong 5.59\times 10^{-10}|g{|}^{\frac{1}{2}}{\left[\frac{v_{\sigma }}{3.08\times 10^{13}\text{ GeV}}\right]}^{15/2}\text{eV}.$$

$$M\cong \zeta \times 1.73\text{TeV},{\mu }_S\cong \eta \times 0.13\text{keV},$$

$$\displaystyle\ m_{\nu\textrm{light}}\simeq\big{[}y^{\intercal}\zeta^{-1}\eta\left(\zeta^{\intercal}\right)^{-1}y\big{]}\times 1.38\ \textrm{eV},$$

$$\text{mbox}Br(\mu \to e\gamma )\sim 1.1\times 10^{-13}{\left(\frac{m_D}{10\text{mbox}GeV}\right)}^4{\left(\frac{3\text{mbox}TeV}{M}\right)}^4,$$

Full text

††preprint: APS/123-QED

I Introduction

The discovery of neutrino oscillations Fukuda et al. (1998); Ahmad et al. (2002) and the fact that baryonic matter only yields a few percent contribution to the energy density of the Universe Ade et al. (2016) are two experimental evidences calling for physics beyond the standard model (SM). On the theoretical side, the apparent absence of CP violation in the QCD sector is also a strong motivation for going beyond the SM since it can be dynamically explained by the Peccei-Quinn mechanism Peccei and Quinn (1977), which requires to extend the SM gauge group with a global symmetry and the existence of a pseudo-Nambu-Goldstone boson, the axion Weinberg (1978); Wilczek (1978). Besides elegantly solving the strong CP problem Kim (1979); Shifman et al. (1980); Dine et al. (1981); Zhitnitsky (1980), the Peccei-Quinn mechanism may be also related to the solution of DM and neutrino puzzles by offering a candidate for cold DM, the axion itself, Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983) and a connection to the neutrino mass generation Mohapatra and Senjanovic (1983); Shafi and Stecker (1984); Langacker et al. (1986); Shin (1987); He and Volkas (1988); Berezhiani and Khlopov (1991); Bertolini and Santamaria (1991).

In the same vein, ALPs, arising from spontaneous breaking of approximate global symmetries, are also theoretically well motivated since these appear in a variety of ultraviolet extensions of the SM Jaeckel and Ringwald (2010); Ringwald (2012); Dias et al. (2014); Montero and Sánchez-Vega (2011) and, as in QCD axion models, these can make up all of Universe DM Arias et al. (2012), or be a portal connecting the DM particle to the SM sector Nomura and Thaler (2009). Moreover, there are some astrophysical phenomena such as the cosmic $\gamma$-ray transparency De Angelis et al. (2007); Simet et al. (2008); Sanchez-Conde et al. (2009); Meyer et al. (2013), the x-ray excess from the Coma cluster Conlon and Marsh (2013); Angus et al. (2014) and the x-ray line at 3.55 keV Bulbul et al. (2014); Boyarsky et al. (2014) that suggest the presence of ALPs. These hints have led to a plethora of search strategies involving astrophysical observation production and detection in laboratory experiments Jaeckel and Ringwald (2010); Ringwald (2012); Essig et al. (2013) with the aim of establishing the ALPs properties.

In the context of ALPs models, the approximate continuous symmetry is typically assumed to be remnant of an exact discrete gauge symmetry as gravity presumably breaks the global symmetries through Planck-scale suppressed operators. In other words, since the global symmetry is highly unstable it is usually stabilized by imposing a discrete gauge symmetry Ibáñez and Ross (1991); Banks and Dine (1992); Ibáñez (1993); Luhn and Ramond (2008) such as a $Z_{N}$ symmetry Dias et al. (2014); Carvajal et al. (2015); Kim and Marsh (2016) (see Refs. Dine (1992); Dias et al. (2004, 2003); Babu et al. (2003); Dias and Pleitez (2004); Carpenter et al. (2009); Harigaya et al. (2013); Ringwald and Saikawa (2016) for $Z_{N}$ realizations in QCD axion models).

This discrete gauge symmetry protects the ALPs mass against large gravity-induced corrections and it can also be used to stabilize other mass scales present in the theory. In particular, with the aim of generating neutrino masses the authors in Refs. Dias et al. (2014); Carvajal et al. (2015) used these types of discrete gauge symmetries in order to protect the associated lepton-number-breaking scale. In this work, we go further by building a self-consistent framework of ALPs DM111Note that in Dias et al. (2014); Carvajal et al. (2015) the ALPs is used to explain some astrophysical anomalies and not to give account for the entire DM abundance. and neutrino masses via the ISS mechanism Mohapatra (1986); Mohapatra and Valle (1986). For this purpose we make use of appropriate $Z_{N}$ discrete gauge symmetries to protect the suitable ALPs mass reproducing the correct DM relic abundance as well as to stabilize the mass scales present in the ISS mechanism. It turns out that the ISS mass terms are determined -up to some factor- by $v^{n}_{\sigma}/M^{n-1}_{\textrm{Pl}}$, where $v_{\sigma}$ is the vacuum expectation value (VEV) of the scalar field $\sigma$ that spontaneously breaks the global symmetry U$(1)_{A}$ and hosts the ALPs, $a$. $n$ is an integer that is determined by the invariance of such terms under the symmetries of the model and some phenomenological constraints.

In order to implement the ISS mechanism we extend the SM matter content by introducing $n_{N_{R}}(n_{S_{R}})$ generations of SM-singlet fermions $N_{\textrm{R}}(S_{\textrm{R}})$ as it is usual. In this work, we consider the minimal number of singlet fermionic fields that allows to fit all the experimental neutrino physics: the $(n_{N_{R}},n_{S_{R}})=(2,2),$ $(2,3)$ and $(3,3)$ cases Abada and Lucente (2014). In each case, the ALPs plays the role of the DM candidate. Moreover, for the (2,3) ISS case there is a possibility of having a second DM candidate: the sterile neutrino (the unpaired singlet fermion) Kusenko (2009); Abazajian et al. (2012); Drewes et al. (2017). Motivated by that, we also build a multicomponent DM framework where the DM of the Universe is composed by ALPs and sterile neutrinos, with the latter being generated through the active-sterile neutrino mixing Dodelson and Widrow (1994) and accounting for a fraction of the the DM relic density.

As far as phenomenological issues are concerned, since in each framework the approximate continuous symmetry is anomalous respect to the electromagnetic gauge group (through an exotic vectorlike fermion) instead of being anomalous respect to QCD, it is possible to build an effective interaction term involving the axion field and the electromagnetic field strength and its dual Srednicki (1985); Kaplan (1985). This in turn implies that the ALPs may be detected in current- and/or proposed- experiments that use the ALPs-photon coupling as their main interaction channel to search for ALPs Jaeckel and Ringwald (2010); Ringwald (2012); Essig et al. (2013); Graham et al. (2015). Moreover, considering this particle as the dark matter candidate it can be part of the Milky Way DM halo and could resonantly convert into a monochromatic microwave signal in a microwave cavity permeated by a strong magnetic field Sikivie (1983, 1985, 2008). On the other hand, since a large portion of the parameter space of ALPs (e.g. low masses and couplings) is relatively unconstrained by experiment since the conventional experiments, Helioscopes, Haloscopes and others Graham et al. (2015), are only sensitive to axion particles whose Compton wavelength is comparable to the size of the resonant cavity, it is important looking for new search strategies in order to cover other regions of the parameter space. To reach smaller values for the ALPs mass and ALPs-photon coupling is necessary a different experimental approach like the ones associated to the ABRACADABRA proposal Kahn et al. (2016); Sikivie et al. (2014), where it is suggested a new set of experiments based on either broadband or resonant detection of an oscillating magnetic flux, designed for the axion detection in the range $m_{a}\in[10^{-14},10^{-6}]$ eV. And it is precisely these kinds of searches that can be used to probed the benchmark regions that we study within the (2,2) and (3,3) ISS cases.

The rest of the paper is organized as follows: in Sec. II we discuss phenomenological and theoretical conditions that lead to a successful protection of the ALPs mass and the ISS texture against gravity effects. In Sec. III we search for viable models simultaneously compatible with DM phenomenology, neutrino oscillation observables and lepton-flavor-violating processes. Finally, we present our discussion and conclusions in Sec IV.

II Framework

The goal of this section is to present the main ingredients of a SM extension in order to link ALPs to neutrino mass generation, and at the same time, to offer an explanation for the current DM relic density reported by Planck Collaboration Ade et al. (2016). In order to achieve that, the SM matter content must to be extended with some extra fields. Besides the scalar $\sigma$ and fermionic $S_{\textrm{R}\alpha}$ and $N_{\textrm{R}\beta}$ fields, an extra electrically charged fermion $E$ is also added to the SM to make possible the coupling of ALPs to photons, $g_{a\gamma}$. That is necessary because $g_{a\gamma}$ is anomaly induced and there is no any U$\left(1\right)_{A}$ symmetry anomalous in the electromagnetic group just with the SM charged fermions. The main role of the anomalous U$(1)_{A}$ symmetry is to induce an ALPs coupling to two photons. This brings as a consequence that the ALPs can be found, in principle, in current and/or proposed experiments that make use of the ALPs-photon coupling. We will show target regions of some experiments searching for ALPs in Fig 1. Also, it will be found that ALPs in some ISS cases discussed in this paper are inside the regions of planned experiments Kahn et al. (2016); Sikivie et al. (2014).

Another key point of the framework is the existence of a $Z_{N}$ discrete gauge symmetry. In order to understand its role, firstly, note that to impose an anomalous U$\left(1\right)_{A}$ symmetry to the Lagrangian does not seem sensible in the sense that in the absence of further constraints on very high energy physics we should expect all relevant and marginally relevant operators that are forbidden only by this symmetry to appear in the effective Lagrangian with coefficient of order one. However, if this symmetry follows from some other free anomaly symmetry, in our case from the a $Z_{N}$ discrete gauge symmetry, all terms which violate it are then irrelevant in the renormalization group sense. Secondly, the $Z_{N}$ symmetry also protects both the ALPs mass and the ISS texture against gravity effects as we will explain in more detail later on. For these reasons, the effective Lagrangian will be invariant under a $Z_{N}$ discrete gauge symmetry. Due to the ALPs mass is very low and only protected by the U$(1)_{A}$ symmetry which is explicitly broken by gravity effects, the ZN symmetry will have a high order. This fact also happens in models with QCD axions and it is shared by all models with this type of stabilization mechanism Di Luzio et al. (2017); Dias et al. (2014); Ringwald and Saikawa (2016); Dias et al. (2004, 2003); Carvajal et al. (2015).

II.1 Lagrangian

The effective Lagrangian that we consider to relate the ISS mechanism to ALPs DM reads

[TABLE]

where $\mathcal{L}_{\text{SM}}^{\text{Yuk}}$ is nothing more than the Yukawa Lagrangian of the SM

[TABLE]

with the usual $Q_{\textrm{L}i},u_{\textrm{R}i},d_{\textrm{R}i}$ and $L_{i},l_{\textrm{R}i}$ fields denoting the quarks and leptons of the SM, respectively. $H$ is the Higgs SU$(2)_{L}$ doublet with $\widetilde{H}=i\tau_{2}H^{*}$ ($\tau_{2}$ is the second Pauli matrix).

The term in $\mathcal{L}_{\sigma}$ (Lagrangian involving the $\sigma$ field) which is relevant in our discussion is the following non-renormalizable operators

[TABLE]

with $g=e^{i\delta}\left|g\right|$ and $D$ being an integer. The $\sigma$ field is parametrized as $\sigma(x)=\frac{1}{\sqrt{2}}[v_{\sigma}+\rho(x)]e^{i\frac{a(x)}{v_{\sigma}}},$ with $a(x)$ being the ALPs field and $\rho(x)$ the radial part that will gain a mass of order of the vacuum expectation value Ringwald (2014); Dias et al. (2014); Marsh (2016)

[TABLE]

With the operators in Eq. (3) and the $\sigma(x)$ parametrization, the ALPs mass term is written as follows Carvajal et al. (2015)

[TABLE]

where $10^{-10}\lesssim\lambda\equiv\frac{v_{\sigma}}{\sqrt{2}M_{\textrm{Pl}}}\lesssim 10^{-5}$ and $M_{\textrm{Pl}}=2.44\times 10^{18}$ GeV is the reduced Planck scale.

Now, we turn our attention to the coupling of ALPs to photons which is determined by the interaction term $\frac{g_{a\gamma}}{4}a\,F_{\mu\nu}\widetilde{F}^{\mu\nu}$, where $F_{\mu\nu}$ and $\widetilde{F}^{\mu\nu}$ are the electromagnetic field strength and its dual, respectively. This term is anomaly induced and given by 222Higher corrections to the $g_{a\gamma}$ coupling are possible. For an extensive study of them to see Bauer et al. (2017). However, for the suitable ALPs masses in order to explain the observed DM relic density, all of them can be safely neglected.

[TABLE]

with $\alpha\approx 1/137$. Here, the electromagnetic anomaly coefficient $C_{a\gamma}$ reads as Srednicki (1985); Marsh (2016):

[TABLE]

where $C_{\textrm{em}}^{(\psi)}$ is the electric charge of the fermion $\psi$, and $X_{\psi_{L,R}}$ is its charge under the U$(1)_{A}$ symmetry. This anomaly coefficient is of order of one (1 or 2 more specifically) in our models and it directly determines the width of the red band in Figure 1 where ALPs are DM candidates. Also, it is important to note that the existence of a non-null anomaly coefficient guarantees that $g_{a\gamma}\neq 0$. This is the reason for the total Lagrangian in Eq. (1) is invariant under an anomalous U$(1)_{A}$ global symmetry. Nevertheless, only with SM model fermions and the neutral $S_{\textrm{R}\alpha}$ and $N_{\textrm{R}\beta}$ fermions is not possible to have an anomalous U$(1)_{A}$ symmetry in the electromagnetic group. Therefore, we need include the SU$(2)_{L}$ singlet fermion, $E$, with an unit of electric charge.

On the other hand, the dimension $D$ of the gravity-induced mass operator in Eq. (3) must be, in general, larger than $4$ because the astrophysical and cosmological constraints on the properties of ALPs. To be more specific, we show, in Figure 1, some regions of the ALPs space of parameters $-g_{a\gamma}\text{ vs }m_{a}-$ where ALPs give an explanation for some astrophysical anomalies and others forbidden regions Graham et al. (2015); Budker et al. (2014); Graham and Rajendran (2013); Ringwald (2012).

Regarding the neutrino mass generation, we have that, once introduced the $N_{\textrm{R}\beta}$ and $S_{\textrm{R}\alpha}$ fields, the $\mathcal{L}_{\textrm{ISS}}$ Lagrangian reads as:

[TABLE]

where the $y_{i\beta}$, $\zeta_{\alpha\beta}$, $\eta{}_{\,\alpha\alpha^{\prime}}$, $\theta_{\,\beta\beta^{\prime}}$, coupling constants, with $i,j=1,2,3$, $\alpha,\alpha^{\prime}=1,2,(\text{or }3)$ and $\beta,\beta^{\prime}=1,2,(\text{or }3)$, are generically assumed of order one. The exponents $p,q,r$ are integer numbers chosen for satisfying some phenomenological constraints discussed below. Negative values for these exponents will mean that the term is $\sim\sigma^{*\,n}$ instead of $\sim\sigma^{n}$. Note that, without loss of generality, the exponent $p$ can be assumed to be positive. We will only consider the minimal number of neutral fermionic fields, $S_{\textrm{R}\alpha}$ and $N_{\textrm{R}\beta}$, that allow to fit all the experimental neutrino physics Abada and Lucente (2014). Specifically, we study the $(2,2),$ $(2,3)$ and $(3,3)$ cases.

As the $\sigma$ field gets a VEV the gravity-induced terms in Eq. (8) give the mass matrix for light (active) and heavy neutrinos Carvajal et al. (2015). Specifically, we can write the mass matrix in the $(\nu_{\textrm{L}},N_{\textrm{R}}^{\textrm{C}},S_{\textrm{R}}^{\textrm{C}})$ basis as

[TABLE]

where $m_{D}$, $M$, $\mu_{N}$ and $\mu_{S}$ are matrices with dimension equal to $n_{N_{R}}\times 3$, $n_{N_{R}}\times n_{S_{R}}$, $n_{N_{R}}\times n_{N_{R}},$ $n_{S_{R}}\times n_{S_{R}},$ respectively. The energy scales of the entries in these matrices are determined essentially by $\sqrt{2}\left\langle H\right\rangle\equiv v_{\text{SM}}\simeq 246$ GeV, $\lambda$ (or $v_{\sigma}$) and $M_{\textrm{Pl}}$ GeV as follows

[TABLE]

The mass matrix in Eq. (11) allows light active neutrino masses at order of sub-eV without resorting very large energy scales in contrast to the type I seesaw mechanism Minkowski (1977); Gell-Mann et al. (1979); Yanagida (1979); Mohapatra and Senjanovic (1980); Schechter and Valle (1980, 1982). In more detail, assuming the hierarchy $\mu_{N}\lesssim\mu_{S}\ll m_{D}<M$ (note that making $\mu_{S}$ and $\mu_{N}$ small is technically natural) and taking a matrix expansion in powers of $M^{-1}$, the light active neutrino masses, at leading order, are approximately given by the eigenvalues of the matrix Grimus and Lavoura (2000); Boucenna et al. (2014)

[TABLE]

On the other hand, the heavy neutrino masses are given by the eigenvalues of $m_{\nu\textrm{heavy}}\simeq M_{R}$. Note from Eq. (19) that $\mu_{N}$ does not contribute to the light active neutrino masses at the leading order Grimus and Lavoura (2000); Boucenna et al. (2014). Actually, the presence of $\mu_{N}$ term gives a subleading contribution to $m_{\nu\text{light}}$ of the order of $m_{D}^{\intercal}M^{-1}\mu_{S}(M^{\intercal})^{-1}\mu_{N}M^{-1}\mu_{S}(M^{\intercal})^{-1}m_{D}$, which is a factor $\mu_{S}\mu_{N}/M^{2}$ smaller than the leading contribution Carvajal et al. (2015).

Very motivated scales for $M$ and $\mu_{S},\,\mu_{N}$ are TeV and keV scales, respectively. These scales allow getting active neutrino masses in the sub-eV scale without considering smaller Yukawas and, in some scenarios, such as the $(2,3)$ ISS case, the existence of a keV sterile neutrino as a warm dark matter (WDM) candidate Abada et al. (2014). In addition, $M$ has to satisfy

[TABLE]

because light active neutrino masses are in sub-eV scale and $m_{D}$ is of order of ${\cal O}(v_{\text{SM}})$.

Another constraint on the $M$ scale comes from the fact that the mixing matrix that relates the three left-handed neutrinos with the three lightest mass-eigenstate neutrinos is not longer unitary. This implies that deviations of some SM observables may be expected, such as additional contributions to the $\ell\nu W$ vertex and to lepton-flavor and CP-violating processes, and non-standard effects in neutrino propagation Hettmansperger et al. (2011); de Salas et al. (2017). For example, in the inverse seesaw model, the violation of unitary is of order of $\epsilon^{2}$, with $\epsilon\equiv m_{D}M^{-1}$ being approximately the mixing between light active and heavy neutrinos Boucenna et al. (2014). Roughly speaking, $\epsilon^{2}$ at the percent level is not excluded experimentally Hettmansperger et al. (2011); Ibarra et al. (2010); Das and Okada (2013); Das et al. (2014).

Taking into account the previous considerations, the ranges chosen for $M$ and $\mu_{S}$ are

[TABLE]

Once established that scales of the mass matrices and using Eqs. (17) and (18) (and following a similar procedure as in Ref. Carvajal et al. (2015)), the integers $p$ and $q$ in Eq. (8) can only take the values

[TABLE]

That happens because the same VEV simultaneously provides $M$ and $\mu_{S}$ scales. Note that for both possibilities in Eqs. (LABEL:eq9) and (LABEL:eq10) the light active neutrino mass matrix in Eq. (19) is simplified to

[TABLE]

Moreover, the exponent $r$ of the term that generates $\mu_{N}$ in Eq. (8) is also constrained to be $r\geq|q|$, because $\mu_{N}$ must be $\lesssim\mu_{S}$.

Finally, we have that $\mathcal{L}_{E}$, the Lagrangian involving the $E$ charged fermion, is written as

[TABLE]

where $\vartheta_{i}$ and $\kappa$ are Yukawas, in principle, assumed of order one. These two terms are also subjected to phenomenological and theoretical constraints as follow. Because the term $\sim\sigma^{t}\overline{E_{\textrm{L}}}E{}_{\textrm{R}}$ must give a mass large enough for the $E$ fermion to satisfy its experimental constrains. For stable charged heavy lepton, $m_{E}>102.6$ GeV at 95$\%$ C.L. Patrignani et al. (2016), or for charged long-lived heavy lepton, $m_{E}>574$ GeV at 95$\%$ C.L. assuming mean life above $7\times 10^{-10}-3\times 10^{-8}$ s Chatrchyan et al. (2013); Perl et al. (2001), $t$ must be less or equal than $3$. It must be different from zero because the electromagnetic anomaly must be present. On the other hand, $s$ can take the values $1$ or $2$ because $\sim\sigma^{s}\overline{L}HE_{\textrm{R}}$ determines the interaction of the $E$ fermion with the SM leptons, and whether $s$ is larger than $2$, the charged $E$ fermion becomes stable enough to bring cosmological problems, unless its mass is $\lesssim\,$TeV. Another constraint comes from searches for long-lived particles in pp collisions Chatrchyan et al. (2013); Perl et al. (2001).

Now an important discussion about the stability of both the ISS mechanism and the ALPs mass is in order. In general, the gravitational effects must be controlled to give a suitable ALPs mass. With this aim, we introduced a gauge discrete $Z_{N}$ symmetry assumed as a remnant of a gauge symmetry valid at very high energies Krauss and Wilczek (1989). Thus, to truly protect the ALPs mass against those effects, $Z_{N}$ must at least be free anomaly Ibáñez and Ross (1991); Banks and Dine (1992); Ibáñez (1993); Luhn and Ramond (2008), i.e.,

[TABLE]

where $A_{2},$ $A_{3}$ and $A_{\textrm{grav}}$ are the $[\text{SU}(2)_{\textrm{L}}]^{2}\times Z_{N}$, $[\text{SU}(3)_{\textrm{C}}]^{2}\times Z_{N}$ and $[\textrm{gravitational}]^{2}\times Z_{N}$ anomalies, respectively. Other anomalies, such as $Z_{N}^{3}$, do not give useful low-energy constraints because these depend on some arbitrary choices concerning to the full theory.

Gravitational effects can also generate terms such as $\frac{\sigma^{n}}{M_{\textrm{Pl}}^{n-1}}\overline{S_{\textrm{R}}}S_{\textrm{R}}^{\textrm{C}}$, $\frac{\sigma^{n}}{M_{\textrm{Pl}}^{n-1}}\overline{S_{\textrm{R}}}N_{\textrm{R}}^{\textrm{C}}$, $\frac{\sigma^{n}}{M_{\textrm{Pl}}^{n-1}}\overline{N_{\textrm{R}}^{\textrm{C}}}N_{\textrm{R}}$ or $\frac{\sigma^{n}}{M_{\textrm{Pl}}^{n}}\overline{L}\widetilde{H}S_{\textrm{R}}$ (with $n$ smaller than those in the Lagrangian (8)) that jeopardize both the matrix structure - Eqs. (11) and (16)

• and the scales of the ISS mechanism. Thus, $Z_{N}$ will be chosen such that it also prevents these undesirable terms from appear.

In general, the $Z_{N}$ symmetry can be written as a linear combination of the continuous symmetries in the model: the hypercharge $Y$, the baryon number B and the generalized lepton number $\mathbb{L}$. The charge assignments for B and $\mathbb{L}$ symmetries are shown in Table 1, whereas the assignment for Y symmetry is the canonical one. Nevertheless, since the hypercharge is free anomaly by construction, the $Z_{N}$ charges $(Z)$ of the fields can be written as $Z=c_{1}\textrm{B}+c_{2}\mathbb{L}$, where $c_{1,2}$ are rational numbers in order to make the $Z_{N}$ charges integers Carvajal et al. (2015). Now, substituting the charges in Table 1 into the general form of the $Z_{N}$ symmetry (see Refs. Ibáñez and Ross (1991); Banks and Dine (1992); Ibáñez (1993); Luhn and Ramond (2008)) we can obtain the anomaly coefficients. Doing so, we find that $A_{3}(Z_{N})=0$ and

[TABLE]

Note that $A_{2}(Z_{N})$ and $A_{\textrm{grav}}(Z_{N})$ are not, in general, [math] Mod $N/2$ which implies strong constraints on the choice of the $Z_{N}$ discrete symmetry.

II.2 ALPs and sterile neutrino dark matter

Since the ALPs are very weakly interacting slim particles and cosmologically stable, they can be considered as DM candidates Arias et al. (2012). In fact, ALPs may be nonthermally produced via the misalignment mechanism in the early Universe and survive as a cold dark matter population until today. Specifically, its relic density is determined from the following equation Sikivie and Yang (2009); Sikivie (2008); Hoeneisen (2015); Bernabei et al. (2003); Arias et al. (2012); Masso (2008); Wantz and Shellard (2010); Marsh (2016)

[TABLE]

where $\Theta_{i}$ is the initial misalignment angle, which is taken as $\frac{\pi}{\sqrt{3}}$, because we are assuming a post-inflationary symmetry-breaking scenario, favorable for models with $v_{\sigma}\lesssim 10^{14}$ GeV Arias et al. (2012); Sikivie (2008).

On the other hand, the fraction of DM abundance in form of sterile neutrino depends on its mass, $m_{\nu_{S}}$, and its mixing angle with the light active neutrino, $\theta$. Specifically, $\nu_{S}$ as a WDM candidate can be generated through the well-known Dodelson-Widrow (DW) mechanism Dodelson and Widrow (1994), which is present as long as active-sterile mixing is not zero Kusenko (2009); Abazajian et al. (2012); Drewes et al. (2017). In the $(2,3)$ ISS case, the sterile neutrino through the DW mechanism can account at maximum for $\approx 43\%$ of the observed relic density without conflicting with observational constraints Abada et al. (2014). This DM amount can be slightly increased to $\approx 48\%$ when including effect of the entropy injection of the pseudo-Dirac neutrinos provided the lightest pseudo-Dirac neutrino has mass $1-10$ GeV Abada et al. (2014). We are not going to consider these effects here. For $m_{\nu_{S}}>0.1$ keV, the relic density produced in the usual DW mechanism is given by Asaka et al. (2007); Abada et al. (2014)

[TABLE]

where $C_{\alpha}(m_{S})$ are active flavor-dependent coefficients which are calculated solving numerically the Boltzmann equations (an appropriated value in this case is $C_{\alpha}(m_{S})\simeq 0.8$ Asaka et al. (2007)). We also have that the sum of $U_{\alpha S}$, the elements of the leptonic mixing matrix, is the active-sterile mixing, i.e., $\sum_{\alpha}|U_{\alpha S}|^{2}\sim\sin^{2}(2\theta)$. For the case $m_{\nu_{S}}<0.1$ keV there is a simpler expression written as follows Abazajian et al. (2001); Abada et al. (2014)

[TABLE]

After imposing bounds coming from stability, structure formation and indirect detection, in addition to the constraints arising from the neutrinos oscillation experiments, it was found that the sterile neutrino as WDM in the $(2,3)$ ISS provides a sizable contribution to the DM relic density for $2\lesssim m_{\nu_{S}}\lesssim 50$ keV and active-sterile mixing angles $10^{-8}\lesssim\sin^{2}(2\theta)\lesssim 10^{-11}$ Abada et al. (2014), where the maximal fraction of DM made of $\nu_{S}$ is achieved when $m_{\nu_{S}}\simeq 7$ keV Boyarsky et al. (2009a, b); Horiuchi et al. (2014); Boyarsky et al. (2012).

Once established the DM candidates and the parameters that determine the relic density in each case, we are going to search for models satisfying all mentioned conditions in Section II as well as

[TABLE]

where $\Omega_{\text{DM}}^{\text{Planck}}h^{2}=0.1197\pm 0.0066$ (at $3\sigma$) is the current relic density as reported by Planck Collaboration Ade et al. (2016).

III Models

In the previous section, we have introduced the general and minimal constraints that models have to satisfy. Now, we proceed to find specific models that give an explanation to the dark matter observed in the Universe. In particular, the $(2,2)$, $(3,3)$ and $(2,3)$ cases of the ISS mechanism are studied in detail. For each model we check the compatibility (at 3$\sigma$) with the experimental neutrino physics de Salas et al. (2017) for the normal mass ordering and vanishing CP phases by varing the free Yukawa couplings $y_{i\beta},\zeta_{\alpha\beta},\eta_{\alpha\alpha},\theta_{\beta\beta}$ in the range $\sim(0.1,3.5)$. Additionally, we also analize the lepton flavor violating processes such as $\ell_{\beta}\to\ell_{\alpha}+\gamma$, which are induced at one loop by the $W$ boson and the heavy neutrinos.

The correspondig decay rates read Gonzalez-Garcia and Valle (1992); Deppisch and Valle (2005)

[TABLE]

where $G(x)=x(1-6x+3x^{2}+2x^{3}-6x^{2}\log(x))/[4(1-x)^{4}]$, $\Gamma_{\ell_{\beta}}$ is the total decay width of $\ell_{\beta}$ and $U$ represents the lepton mixing matrix. We verify that each ISS model is compatible with the current experimental limits Br$(\mu\to e\gamma)<5.7\times 10^{-13}$ Adam et al. (2013), Br$(\tau\to e\gamma)<3.3\times 10^{-8}$ and Br$(\tau\to\mu\gamma)<4.4\times 10^{-8}$ Aubert et al. (2010).

III.1 $(2,2)$ ISS case

Among the minimal configuration of the ISS mechanism consistent with the experimental neutrino physics and lepton-flavor-violating (LFV) processes Abada and Lucente (2014); Abada et al. (2014); Das et al. (2017) (for a recent review see Ref. Lindner et al. (2016)), we, firstly, study the $(2,2)$ ISS case because this is the minimal configuration that satisfy all the constraints coming from experimental neutrino physics. For this case, in the neutrino mass spectrum there are two heavy pseudo-Dirac neutrinos with masses $\sim M$ and three light active neutrinos with masses of order of sub-eV coming from the mass matrix in Eq. (19) Abada and Lucente (2014). Because in this case $n_{N_{R}}=n_{S_{R}}=2$ (similarly for the $(3,3)$ ISS case) there is not a light sterile neutrino $\nu_{S}$ in the mass spectrum. Therefore, all the current DM abundance must be constituted by ALPs, i.e. $\Omega_{\text{DM}}^{\text{Planck}}h^{2}=\Omega_{a,\text{DM}}h^{2}$.

In order to find the main features of the model, we find useful to rewrite $\Omega_{a,\text{DM}}h^{2}$ in terms of $D$ - the exponent of the mass operator for $\sigma$, Eq. (3) - and $m_{a}$. Thus, substituting Eqs.(3) and (5) in Eq. (29), we find that

[TABLE]

where $g$ is assumed to be $10^{-3}\leq\left|g\right|\leq 2$. Thus, we can see that $\Omega_{a,\text{DM}}h^{2}$ only depends on $\left(\left|g\right|,\,D,\,v_{\sigma}\right)$. In Table 2 we show $\left(D,\,v_{\sigma}\right)$ values for the cases where $\Omega_{a,\text{DM}}h^{2}=\Omega_{\text{DM}}^{\text{Planck}}h^{2}$ and $\Omega_{a,\text{DM}}h^{2}=0.57\times\Omega_{\text{DM}}^{\text{Planck}}h^{2}$. The last case applies only for the $(2,3)$ ISS case and will be discussed in Section III.3.

In order to obtain the Lagrangian in this scenario, we search for discrete symmetries for the two possibilities showed in Eqs. (LABEL:eq9-LABEL:eq10) and different values of $r,s,t$ according its respective constraints as follow: considering Eqs. (LABEL:eq9-LABEL:eq10) and the Table 2 we can see that, for the range of values of $v_{\sigma}$ established in the Section II.1, only the values $D=9,10,16-19$ are allowed to reproduce the correct relic density to ALPs. Thus we searched for discrete symmetries $Z_{N}$ that allows the mass operators with those dimensions $D$, with the following results: The $Z_{9,10,}$ symmetries allow terms such as $\frac{\sigma^{*}}{M_{\textrm{Pl}}}\bar{L}\widetilde{H}S_{R}$, $\frac{\sigma^{*2}}{M_{\textrm{Pl}}}\bar{N}_{R}N_{R}^{C}$, $\bar{L}\widetilde{H}S_{R}$, and since $H$ and $\sigma$ get VEVs, these terms do not give the appropriate zero texture of the ISS mechanism shown in Eqs. (11) and (16). We have searched for all the possible combinations of $r,s,t$ values in the Lagrangian of Eq. (8) without any success. On the other hand, the $Z_{16,18}$ symmetries are not free of the gravitational anomaly. In fact, the $Z_{N\leq 20}$ discrete symmetries that satisfy all the anomaly constraints and stabilize the ISS mechanism are $Z_{17,19}$.

In the case of $Z_{17}$ symmetry, the Lagrangian, $\mathcal{L}{}_{Z_{17}}$, is given by Eq. (1) with the parameters $D=17$ and $(p,q,r,s,t)=(3,-5,-6,2,2)$ in Eqs. (3), (8) and (25), respectively. An assignment of the $Z_{17}$ (with $Z_{17}=6\text{B}+11\text{U}(1)_{\mathbb{L}}$) charges and the anomalous U$(1)_{A}$ symmetry for this case is shown in Table 3. Note that, for this model the term $\sim\sigma^{*6}\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$ in Eq. (8) gives a negligible contribution for the light active neutrino masses.

The corresponding $g_{a\gamma}$ and $m_{a}$ for this model is given by

[TABLE]

The benchmark region for this case is denoted as M$22$a in the Figure 1 where we have considered $10^{-3}\leq\left|g\right|\leq 2$ and $2.9\times 10^{13}\lesssim v_{\sigma}\lesssim 4.2\times 10^{13}$ GeV. These values for $g_{a\gamma}$ and $m_{a}$ allow that the ALPs explain $100\%$ of the DM relic density.

Sharp predictions for neutrinos masses are not possible with just the knowledge of the $p,q,r,s,t$ values and $v_{\sigma}$. However, the order of magnitude of the mass matrices can be estimated from Eqs. (17) and (18) to be (using $v_{\sigma}\cong 3.08\times 10^{13}\textrm{ GeV}$)

[TABLE]

which is appropriate to satisfy the constraints coming from experimental neutrino physics and unitarity without resorting a fine tuning in couplings. Nevertheless, we have to admit that some care must be taken in order to generate the benchmark region M$22$a in agreement with bounds coming from LFV processes such as $\mu\rightarrow e+\gamma$. Specifically, due to $m_{N_{i}}\sim M\gg m_{W}$ the loop function tends to $G(x)\to 1/2$ and the mixing terms are generically given by $U\sim m_{D}/M$. This leads to the decay rate for $\mu\rightarrow e+\gamma$ of the order of

[TABLE]

which implies that small $y\sim 0.1$ couplings must be required.

For the case with $Z_{19}$, the effective Lagrangian is characterized by $(p,q,r,s,t)=(3,-5,-8,-2,1)$, and the results, roughly speaking, are quite similar to the model with $Z_{17}$, in the sense that as the $p,q,|s|$ values are equals for both models, the neutrino spectrum is similar in both cases. Nevertheless, since $D,t$ values are not equals, we have as a consequence that: the ALPs mass, the mass term for the exotic fermion $E$ and the ALPs-photon coupling, $g_{a\gamma}$, are different. Specifically, from the Table 3 and Eq. (7), in the $Z_{19}$ model the ALPs parameters are

[TABLE]

The benchmark region in this model corresponding to this case in Figure 1 is denoted as M$22$b. We also show the values for the neutrino mass spectrum in Table 4. Concerning to the upper bound on $\mu\rightarrow e+\gamma$, it is easily fulfilled due to the larger supression coming from $M\sim 50$ TeV.

III.2 $(3,3)$ ISS case

Regarding the neutrino mass spectrum the $(3,3)$ ISS case is quite similar to the previous one in the sense that there is not a light sterile neutrino in the mass spectrum because $n_{N_{R}}=n_{S_{R}}=3$. Therefore, all the DM abundance in this model has to be made of ALPs.

Proceeding in a similar manner to the (2,2) ISS case and taking into account that $A_{\textrm{grav}}(Z_{N})$ is now different (see Eq. (28)), we have searched for all anomaly-free $Z_{N}$ discrete symmetries, with $N\leq 20$ and with $(p,\,q,\,r,\,s,\,t)$ values established according to the constraints in Section II.1. Doing that, we found the following results: the $Z_{9}$ symmetry is not free of gravitational anomalies, while the $Z_{10}$ symmetry allows dangerous terms such as $\bar{L}\widetilde{H}S_{R}$, $\frac{\sigma^{*}}{M_{\textrm{Pl}}}\bar{L}\widetilde{H}S_{R}$, $\sigma\bar{N}_{R}N_{R}^{C}$, and others that jeopardize the matrix structure in Eqs. (11) and (16), therefore the possibility of building a model for the solution in Eq. (LABEL:eq9) is not realized. On the other hand, the $Z_{16,18}$ symmetries corresponding to the solution in Eq. (LABEL:eq10), are not free of gravitational anomalies, therefore these are not suitable symmetries. However, the $Z_{17,19}$ symmetries forbid the dangerous terms and allow an effective Lagrangian.

In the case of $Z_{17}$ symmetry, the Lagrangian in Eq. (1) is characterized by the parameters $(p,\,q,\,r,\,s,\,t)=(3,\,-5,\,7,\,2,\,1)$ and $D=17$. Note then that this model has a Lagrangian very similar to the $(2,2)$ ISS Lagrangian. However, in this case, the mass term for the exotic fermion $E$ has the exponent equal to one and the term associated with $\mu_{N}$ is not allowed with dimension less than seven. Because the parameters $(p,\,|q|)$ are equals in both cases, the neutrino spectrum is the same as in the M22a model (see Eqs. (36)). Moreover, note that in this case, the term $\sim\sigma^{7}\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$ gives a negligible contribution for the light active neutrino masses. On the other hand, the fact that the mass term for the exotic fermion differs from (2,2) ISS model imply that the anomaly coefficient $C_{a\gamma}$ be different (see charges in the Table 3 and Eq. (7)), such that the ALPs-photon coupling has also a different value. Possible assignments for the $Z_{17}$ and U$(1)_{A}$ symmetries are shown in Table 3, with $Z_{17}=9\text{B}+11\text{U}(1)_{\mathbb{L}}$.

The corresponding $v_{\sigma}$ value is the same that in the $(2,2)$ ISS case showed in the Table 2 for $D=17$, implying also that the $m_{a}$ is equal to it given in Eq. (36). Nevertheless, the $g_{a\gamma}$ turn to be

[TABLE]

because the anomaly coefficient now has a different value. A benchmark region for this case is denoted as M$33$a in Figure 1, where these values for $g_{a\gamma}$ and $m_{a}$ allow that the ALP explains $100\%$ of the DM relic density.

For the $Z_{19}$ case, we find that the model is determined by the parameters $(p,\,q,\,r,\,s,\,t)=(3,\,-5,\,-8,\,2,\,2)$, which brings similar conclusions that the M22b model, with some differences coming from the anomaly coefficient $C_{a\gamma}$. Specifically, the coupling

[TABLE]

The other parameters associated to neutrino spectrum and $m_{a}$ are similar than in the M22b case, and are shown in Table 4. The benchmark region for this case is denoted as M$33$b in Figure 1.

On the other hand, the constraints and prospects regarding lepton flavor violating processes are similiar to the ones in case (2,2) since the mass scale $M$ of the benchmark regions M33a and M33b are the same of the benchmark regions M22a and M22b, respectively.

We remark that a similar effective Lagrangian for the $(3,3)$ ISS case was worked in the Ref. Carvajal et al. (2015) with the aim of explaining some astrophysical phenomena. However, in that case, the DM abundance via ALP was not considered.

III.3 $(2,3)$ ISS case

For this case, because there are $n_{N_{R}}=2$ and $n_{S_{R}}=3$ neutral fermions, the neutrino mass spectrum contains two heavy pseudo-Dirac neutrinos with masses $\sim M$ and three light active neutrinos with masses of order of sub-eV. In addition, there is a sterile neutrino, $\nu_{S}$, with mass of order $\sim\mu_{S}$. Then, for this model, the presence of both the $\nu_{S}$ and the ALPs, $a$, brings the possibility of having two DM candidates in the $(2,3)$ scenario Abada et al. (2014); Bhupal Dev and Pilaftsis (2013).

First, let’s consider the case $\Omega_{\text{DM}}^{\text{Planck}}h^{2}=\Omega_{a,\text{DM}}h^{2}$, i.e., when the DM abundance is totally constituted by ALPs. Now, from Eqs. (LABEL:eq9) and (LABEL:eq10) and Table 2, we can see that $\left(D,\,v_{\sigma}\right)=\left(9,\,(0.6-1.1)\times 10^{11}\text{ GeV}\right)$ and $\left(D,\,v_{\sigma}\right)=\left(10,\,(1.9-3.2)\times 10^{11}\text{ GeV}\right)$ corresponds to the $(p,|q|)=(2,3)$ solution in Eq. (LABEL:eq9) (note that $v_{\sigma}$ corresponding to $D=10$ is slightly out of allowed range in Eq. (LABEL:eq9)). Moreover, the values $D=9,10$ restrict the symmetry to be $Z_{9,10}$. For these discrete symmetries we find solutions for anomaly free $Z_{9\text{ and }10}$ charges, i.e, solutions to Eqs. (27) and (28) with $(p,|q|)=(2,3)$. Nevertheless, all the solutions for the $Z_{9\text{ and }10}$ charges allow terms such as $\sim\sigma\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$, $\sim\frac{\sigma^{*2}}{M_{\textrm{Pl}}}\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$, $\sim\frac{\sigma^{*}}{M_{\textrm{Pl}}}\overline{L_{i}}\widetilde{H}S_{\textrm{R}\alpha}$ and other terms in the Lagrangian that do not give the correct texture to the mass matrix in the ISS mechanism. We also have searched for all the possible combinations of $r,s,t$ values in the Lagrangian (1) with $(p,|q|)=(2,3)$ without any success. Therefore, the $(p,|q|)=(2,3)$ case cannot offer a realization for an effective model providing all the observed DM abundance via ALPs when all the constraints in Section II are considered. However, from Table 2 we see that for $D=15,\dots,19$ with a larger value of $v_{\sigma}$ the second solution $(p,|q|)=(3,5)$, cf. Eq. LABEL:eq10, can, in principle, offer a model (note that, strictly speaking, the $v_{\sigma}$ value corresponding to $D=15$ is slightly out of allowed range in Eq. (LABEL:eq10)). Moreover, the cases of $Z_{17}$ and $Z_{19}$ are excluded because the condition for the gravitational anomaly is never satisfied, while in the $Z_{16,18}$ cases terms as $\sim\overline{L_{i}}\widetilde{H}S_{\textrm{R}\alpha}$ and $\sim\sigma\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$ give an incorrect texture for the ISS mass matrix. In fact, after imposing all the constraints, we find that the only symmetry that provides a solution is $Z_{15}$. In more detail, we find that the discrete symmetry can be written as $Z_{15}=9\text{B}+11\text{U}(1)_{\mathbb{L}}$ (other combinations for $Z_{15}$ are possible). This model has the effective Lagrangian, $\mathcal{L}{}_{Z_{15}}$, given by Eqs. (1), (8) and (25) with $(p,q,r,s,t)=(3,-5,-4,\,2,\,2)$. Note that the term $\sim\sigma^{*4}\overline{N_{\textrm{R}\beta}}(N_{\textrm{R}\beta^{\prime}})^{\textrm{C}}$ gives a negligible contribution for the light active neutrino masses.

The $\mathcal{L}{}_{Z_{15}}$ is also invariant under a U$(1)_{A}$ symmetry which is anomalous in the electromagnetic group as must be to generate a non-null coupling between photons and ALPs, $g_{a\gamma}$ (see Sec. II.1). Specifically, for this case we have that the ALPs parameters are given by

[TABLE]

We also check that the neutrino mass spectrum for this model is

[TABLE]

where we have used the particular value $v_{\sigma}\simeq 1.03\times 10^{13}$ GeV, which is one of the suitable values given in Table 2 for $D=15$ giving the $100\%$ of the current DM abundance. For this case, the sterile neutrino as DM candidate has a negligible contribution because the small scales in Eq. (42) imply that the mixing angle between the active and sterile neutrinos has a great suppression. Moreover the mass scale of the sterile neutrino, $\mu_{S}$, is very small to bring a considerable contribution to DM.

Now, from values of $M,\,\mu_{S},\,m_{\nu\textrm{light}}$ in Eq. (42) we note that in this scenario there is a some tension to satisfy the unitarity constraint. In more detail, $\left|\frac{y}{\zeta}\right|<\frac{M}{v_{\text{SM}}}\times 10^{-1}=2.6\times 10^{-2}$ where we have been conservative choosing a $\epsilon^{2}$ value of $1\%$ (recall $\epsilon\equiv m_{D}M^{-1}$). However, this upper bound on $\left|\frac{y}{\zeta}\right|$ implies a lower bound on $\eta>\left|\frac{y}{\zeta}\right|^{-2}\frac{m_{\nu\textrm{light}}}{4.15}\approx\left|\frac{y}{\zeta}\right|^{-2}\frac{\sqrt{\Delta m_{\text{atm}}^{2}}}{4.15}\approx 17.17$ (with $\Delta m_{\text{atm}}^{2}=2.32\times 10^{-3}$ eV2 being the atmospheric squared-mass difference) which is not a perturbative value for $\eta$. This happens because the values for $v_{\sigma}$ corresponding for $D=15$ is smaller than the values allowed in the range in Eq. (LABEL:eq10). Similar conclusions are found if we consider the case when $\Omega_{a,\text{DM}}h^{2}<\Omega_{\text{DM}}^{\text{Planck}}h^{2}$. Therefore, the effective Lagrangian $\mathcal{L}{}_{Z_{15}}$ can not provide a natural framework for DM and the neutrino masses in $(2,3)$ ISS case. For this reason we do not show the benchmark region for this model in Figure 1.

However, models explaining the DM relic density via ALPs and/or sterile neutrinos for the $\left(2,3\right)$ ISS case can be found provided we slightly relax some constraints mentioned in Section II. Actually, if an extra $Z_{N}$ symmetry is allowed, we found that, for example, the solution $\left(p,\left|q\right|\right)=\left(2,3\right)$ in Eq. (LABEL:eq9) makes possible a model with $D=10$ and $(p,q,r,s,t)=(2,-3,-3,2,2)$ in Eqs. (1), (8) and (25), where the discrete gauge symmetry $Z_{10}\times Z_{4}$, with the corresponding charges given in Table 3, must be considered with the aim of get the correct DM relic density using $D=10$ to calculate the ALPs mass. It is straightforward to check that for this model, ALPs provide $100\%$ of the DM abundance provided $v_{\sigma}\cong 2.03\times 10^{11}$ GeV with $g$ of order one. In more details, for this benchmark point, we have that

[TABLE]

with the neutrino mass spectrum given by

[TABLE]

We note that for $\eta\leq 10^{-2}$ and the other coupling constants of order one, a suitable neutrino mass spectrum is achieved.

In this case, we have also check that the sterile neutrino has a negligible contribution to DM relic density because the mixing angle between the active and sterile neutrinos is smaller than the limits established to consider $\nu_{S}$ as a DM candidate ($10^{-8}\lesssim\sin^{2}(2\theta)\lesssim 10^{-11}$, see ref. Abada et al. (2014) for more details). For this model, we have shown in Figure 1 a benchmark region denoted as M$23$a where ALPs provide $100\%$ of DM abundance.

For the case that the DM abundance is made of ALPs and sterile neutrinos, the scenario slightly changes. We have chosen the case when the DM is made of $\approx 43\%$ of sterile neutrinos and $\approx 57\%$ of ALPs as an illustrating example. However, these can take other values provided the DM abundance made of sterile neutrinos is $\lessapprox 50\%$, consistently with the constraints over its parameter space Abada et al. (2014); Merle et al. (2016); Boyarsky et al. (2009b). Doing a similar procedure as in the previous cases, we can obtain

[TABLE]

and

[TABLE]

In this case for $\eta\approx 10^{-2}$ the sterile neutrino has $m_{\nu_{S}}\approx 7.1\,$KeV. In particular, this mass for the sterile neutrino may explain the recently indicated emission lines at 3.5 keV from galaxy clusters and the Andromeda galaxy Bulbul et al. (2014); Boyarsky et al. (2014). The benchmark region for this model is denoted as M$23$b in Figure 1.

It is worth to mention that for both benchmark regions in those models, the constraints and prospects regarding lepton-flavor-violating processes are also similiar to the ones in case (2,2). This happens because the contribution of the sterile neutrino to Br$(\ell_{\beta}\to\ell_{\beta}\gamma)$ is negligible since $G(m_{\nu_{S}}^{2}/m_{W}^{2})\to 0$ for $m_{\nu_{S}}\ll m_{W}$.

Finally, for clearness, we show in Table 4 an overview of the main results of all considered models. Specifically, we show energy scales for the neutrino masses and the ALPs parameter space for each ISS case.

IV Discussion and summary

We have connected two interesting motivations for going beyond the standard model: neutrino masses and ALPs as dark matter. A natural scenario for achieving that is the ISS mechanism. In particular, we have considered the minimal versions of the ISS mechanism in agreement with all the neutrino constraints. Nevertheless, in the considered framework, the mass scales for the ISS mechanism are generated by gravity-induced non-renormalizable operators when the scalar field containing the ALPs gets a vacuum expectation value, $v_{\sigma}$. Naturalness of these scales imposes strong constraints on these operators and, when combining these with the ALPs acceptable range for $v_{\sigma}$, only two solutions are possible: $(p,|q|)=(2,3)\ \ \text{for}\ \ 6\times 10^{10}\lesssim v_{\sigma}\lesssim 1\times 10^{11}\ \textrm{GeV}$ and $(p,|q|)=(3,5)\ \ \text{for}\ \ 2\times 10^{13}\lesssim v_{\sigma}\lesssim 8\times 10^{13}\ \textrm{GeV}$. This implies that operators given $M$ and $\mu_{S}$ scales can only belong to these two categories. Then, a simultaneous application of constraints coming from the texture of ISS mass matrix, the violation of the unitarity, the mass of exotic charged leptons, the stability of the effective Lagrangian against gravitational effects and the suitable ALPs parameter space ($m_{a}$ and $g_{a\gamma}$) to provide the total DM density almost set the rest of terms in the Lagrangian, only leaving a few of possibilities for all of ISS cases. These constraints ultimatelly lead to a concrete prediction for the viable ALPs masses and ALPs-photon couplings and also for the mass scale of the heavy neutrinos necessary to explain the neutrino oscillation data. In other words, both sectors are deeply connected and the observation of a hypothetical signal of the ALP existence within the proper regions will automatically lead to the existence of heavy neutrino states in the TeV and multi TeV scales. In the same way, the nonobservation of an ALP within such regions or the observation of heavy neutrinos below the TeV scale would disfavour the possible linkage between ALP DM and neutrino masses suggested in this work.

Among the minimal ISS mechanisms, the $(2,2)$ and $(3,3)$ ISS cases are quite similar. It is due to the fact that in both of them $n_{N_{R}}=n_{S_{R}}$ implying that neutrino mass spectrum is characterized by only two mass scales, $M$ and $m_{\nu\text{light}}$. Thus, the results obtained are almost identical. Although, there is a slightly difference in the value of $g_{a\gamma}$ due to the presence of more fermions in the $(3,3)$ ISS case. In both cases, we find two effective models denoted as M22a,b and M33a,b in Table 4. Since there is not sterile neutrino in these cases, the total DM density is made of ALPs. We also remark that, although, the ALPs in these models can decay to two photons and, in the $(2,2)$ ISS mechanism, to two massless active neutrinos, these are cosmologically stable because those decays are strongly suppressed by factors of $1/M_{\textrm{Pl}}^{2}$ and/or $1/v_{\sigma}^{2}$.

On the other hand, the $(2,3)$ ISS case is phenomenologically more interesting due to the presence of a sterile neutrino in the mass spectrum. It implies that the DM density can be made of ALPs and $\nu_{S}$. We have found a model satisfying all of previously mentioned constraints and, at the same time, offering the total DM. Because sterile neutrinos in the $(2,3)$ ISS mechanism can give, roughly speaking, at most $\approx 43\%$ of the DM density, it is necessary that the remaining $\approx 57\%$ of DM be made of ALPs. It is also possible that ALPs give the total DM density. It occurs when the mixing angle between active and sterile neutrinos is very suppressed in order to make the the Dodelson-Widrow mechanism inefficient. Both cases were studied in detail and denoted as M23a and M23b, respectively.

Regarding the search for ALPs, the benchmark regions in Figure 1 are out of reach of the current and future experimental searches for axion/ALPs such as ALPS II, IAXO, CAST Graham et al. (2015), since these currently have not enough sensitivity to probe the ALPs/axion-photon couplings and masses that are motivated in models with scales $v_{\sigma}\gtrsim 10^{13}$ GeV. Nevertheless, for the (2,2) and (3,3) ISS cases the benchmark regions are remarkably within the target regions in proposed experiments based on LC circuits Kahn et al. (2016); Sikivie et al. (2014), which are designed to search for QCD axions and ALPs and cover many orders of magnitude in the parameter space of these particles, beyond the current astrophysical and laboratory limits Budker et al. (2014); Graham et al. (2015); Graham and Rajendran (2013). Specifically, the ABRACADABRA experiment Kahn et al. (2016) may explore ALPs masses as low as $\sim 10^{-10}$ eV for a coupling to photons of the order of $\sim 10^{-18}$ GeV*-1*, which are well below our benchmark regions (Figure 1).

Finally, despite the fact that neutrino mass spectrum is not completely predicted in the models found, the matrix scales in the ISS mechanism are estimated to be in agreement with the neutrino constraints 333It is worth mentioning that for all the models the normal spectrum is the preferred neutrino mass spectrum Abada and Lucente (2014) which in turn implies that our scan results are also compatible with the cosmological upper bound on the neutrino mass sum Hannestad and Schwetz (2016); Vagnozzi et al. (2017). Moreover, we have numerically checked, in all models, that there are solutions with coupling constants of order one that also satisfy LFV processes and the unitary condition. These processes can easily avoid without fine-tuning in the models discussed in this paper. Specifically, we have found that the BR($\mu\rightarrow e\gamma)$ in all cases are as small as $\sim 10^{-20}-10^{-15}$ which are consistent with the current experimental value BR($\mu\rightarrow e\gamma)<5.7\times 10^{-13}$ Adam et al. (2013) and with future sensitivities around $6\times 10^{-14}$ Baldini et al. (2013).

Acknowledgements.

B. L. S. V. would like to thank Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil, for financial support. C. D. R. C. acknowledges the financial support given for the Departamento Administrativo de Ciencia, Tecnología e Innovación - COLCIENCIAS (doctoral scholarship 727-2015), Colombia, and the hospitality of Laboratori Nazionali di Frascati, Italy, in the final stage of this work. O. Z. has been partly supported by UdeA/CODI grant IN650CE and by COLCIENCIAS through the Grant No. 111-565-84269.