Anomaly-free flavor models for Nambu-Goldstone bosons and the 3.5 keV X-ray line signal

We pursue a possibility that a pseudo-Nambu-Goldstone boson is lurking around or below the intermediate scale. To this end we consider an anomaly-free global flavor symmetry, and construct models where the pseudo-Nambu-Goldstone boson is coupled preferentially to leptons. The experimental and astrophysical bounds derived from couplings to photons and nucleons are significantly relaxed. If sufficiently light, the pseudo-Nambu-Goldstone boson contributes to dark matter, and interestingly, it generally decays into photons through couplings arising from threshold corrections. We show that the recent hint for the X-ray line at about $3.5$ keV can be explained by the decay of such pseudo-Nambu-Goldstone boson of mass about $7$ keV with the decay constant of order $10^{10}$ GeV, if the electron is charged under the flavor symmetry.


Introduction
Symmetry plays an important role in physics. Sometimes it is spontaneously broken in the low energy, and as a remnant, there appears a massless Nambu-Goldstone boson. If the symmetry is a local one, it is absorbed by the corresponding gauge boson. On the other hand, if the symmetry is a global and approximate one, there remains a light pseudo-Nambu-Goldstone boson (PNGB), which has been a subject of considerable interest.
PNGBs, if exist, will provide us with invaluable information on the high energy physics.
Various types of global symmetries and the associated PNGBs have been considered so far. One example is the QCD axion, which arises in association with the spontaneous breakdown of the Peccei-Quinn symmetry [1,2]. Importantly, the QCD axion is coupled to gluons and photons through anomalies, as well as to the quarks and the leptons at tree or one-loop level. The interactions are suppressed by the decay constant, which parametrizes the symmetry breaking scale. In other extensions of the standard model (SM), there arise PNGBs with similar properties, the so called axion-like particles, and especially those with couplings to photons have been studied extensively from both theoretical and experimental aspects [3].
The couplings of the QCD axion and the axion-like particles to photons, nucleons, and electrons are tightly constrained by cosmology, astrophysics and the ground-based experiments [3,4,5]. In particular, the astrophysical constraints are extremely tight, pushing the scale of new physics to an intermediate scale or above. Still, there may be other kind of PNGBs with different properties at a scale around or even below the intermediate scale, without any conflict with those constraints.
In this paper we pursue a possibility that a PNGB associated with new physics is lurking around or below the intermediate scale. For this, we need to evade tight astrophysical bounds on the PNGBs. One way is to consider PNGBs, which are not directly coupled to the SM sector, but mainly coupled to a hidden sector [6]. Instead, we want to consider here the case in which some of the SM particles are charged under a global flavor symmetry. The maximal possible flavor symmetry for the SM particles with three right-handed neutrinos is U(3) 6 . We consider an anomaly-free global U(1) F flavor symmetry, which is a subgroup of the maximal flavor symmetry. In particular, a leptophilic PNGB model is simple and phenomenologically interesting, and we will construct concrete models along this lines. Such leptophilic PNGBs without anomalous couplings to photons evade various experimental and astrophysical bounds coming from couplings with nucleons and photons.
We will mainly focus on very light PNGBs with mass lighter than the twice the electron mass. 1 There is an interesting point of the PNGB associated with an anomaly-free global symmetry. Although suppressed, such PNGB is necessarily coupled to photons through threshold corrections. In particular, the decay into two photons can be the main decay mode if the PNGB of mass is less than twice the electron mass. If such light PNGB constitutes dark matter, it mainly decays into two photons, producing a narrow X-ray line.
This can explain the recent hint for the X-ray line at about 3.5 keV [8,9] for the PNGB mass of about 7 keV. As we shall see, the required decay constant is f a = O(10 10 ) GeV if the electron is charged under the symmetry, whereas it is f a = O(10 5 ) GeV if the electron is neutral under the symmetry. This should be contrasted to the fact that the observed X-ray flux can also be explained by the string axion with a decay constant of order 10 14−15 GeV as first pointed out in Ref. [10]. 2 The rest of the paper is organized as follows. In Sec. 2 we discuss the coupling of the PNGB to photons through threshold corrections, and its implications for the 3.5 keV X-ray line. We discuss production of PNGB dark matter in Sec. 3. In Sec. 4, we will build concrete models for leptophilic PNGBs. The last section is devoted for discussion and conclusions.

Couplings of PNGBs to photons
Let us consider a global U(1) F flavor symmetry under which only leptons are charged.
Most important, we assume that the global U(1) F symmetry is anomaly free so that the PNGB coupling to photons is suppressed, evading various observational constraints. The coupling to photons is nevertheless induced by threshold corrections, which we will study 1 Experimental bounds on PNGBs with mass heavier than O(1) MeV including leptophilic ones were studied in Ref. [7]. 2 The X-ray line produced by light modulus decay was studied many years ago by Kawasaki and one of the present authors (TTY) in Ref. [11] (see also Refs. [12,13]). Recently there appeared various possibilities to explain the 3.5 keV X-ray line [14,10,15,16,17,18]. in this section.
Let us study the interactions of the PNGB with leptons in the low energy. Later we will construct concrete flavor models. The relevant low-energy interactions are given by where a is the PNGB associated with the flavor symmetry, f a the decay constant, and q e , q µ , and q τ the coupling constants for electron, muon and tau leptons, respectively. We exclude the case of q e = q µ = q τ = 0 in the following analysis.
We are interested in the case where the PNGB mass is much lighter than twice the electron mass. Integrating out electron, muon, and tau leptons, therefore, we obtain the effective interaction, where the first line corresponds to the anomaly term, and the second line arises from the threshold corrections. We require q e + q µ + q τ = 0 to ensure that the flavor symmetry is anomaly-free. Then the first term in Eq. (2) vanishes, and we are left with the finite threshold corrections. Therefore the PNGB coupling to photons is significantly suppressed for anomaly-free symmetry. As long as we are interested in the decay or production of the on-shell PNGB and photons, we can use their equations of motion. Then the effective interaction for the PNGB to photons becomes for the on-shell PNGB and photons and m 2 a ≪ m 2 e . The PNGB coupling to photons is dominated by the first term if q e = 0; otherwise it is dominated by the second term. Note that both q e and q µ cannot vanish simultaneously to satisfy the anomaly-free condition.
The decay rate of the PNGB into two photons is approximately given by for q e = 0, where we have approximated m 2 e ≪ m 2 µ ≪ m 2 τ and assumed that there is no large hierarchy among the U(1) F charges. Assuming that the PNGB decays mainly into photons via the above interaction, we can estimate the lifetime as for q e = 0 for q e = 0 Thus the PNGB is so long-lived that it can contribute to dark matter. We will show in the next section that, in fact, the right amount of PNGBs can be produced to explain the observed dark matter abundance.
The recent hint for the X-ray line at about 3.5 keV can be explained by dark matter with the following mass and lifetime [8,9]: if it decays into a pair of photons. Therefore, the 3.5 keV X-ray line can be explained by the decay of the PNGB dark matter with m a ≃ 7 keV and f a /q e = 4 × 10 9 GeV − 1 × 10 10 GeV for q e = 0, or f a /q µ = 9 × 10 4 GeV − 3 ×

PNGB dark matter
A light PNGB contributes to dark matter, if it is sufficiently long-lived. In order to explain the observed dark matter density, the right amount of PNGBs need to be produced in the early Universe. There are two important production processes. One is non-thermal production by the initial misalignment mechanism, and the other is thermal production. 3 We will consider these production processes in turn.
The PNGB number density to entropy ratio can be written as where Ω a is the density parameter for the PNGB and h is the reduced Hubble constant.
On the other hand, if the PNGB is in equilibrium, its abundance is given by where g * counts the relativistic degrees of freedom in thermal plasma. Therefore, if the PNGBs constitute the observed dark matter, they should not be in equilibrium, otherwise there must be late-time entropy dilution by a factor 40 for m a ≃ 7keV.
Let us first consider the case of q e = 0. In this case, the decay constant suggested by the observed X-ray line is f a /q e = 4 × 10 9 GeV − 1 × 10 10 GeV. The thermal production process depends on the charge of τ . If the PNGB is directly coupled to τ , the main production process will be through scatterings between leptons and Higgs bosons such as The abundance is roughly estimated as follows where T R is the reheating temperature. Thus, the right amount of PNGBs are thermally produced for T R ∼ 10 6 GeV and f a ∼ 10 10 GeV. Alternatively, if the PNGB is not directly coupled to τ , the abundance is suppressed by ∼ (m µ /m τ ) 2 and given by In this case successful thermal leptogenesis may be possible [20], with a mild degeneracy among the right-handed neutrinos. Note that the thermally produced PNGBs of 7keV mass behave as warm dark matter because of their non-negligible free streaming.
The PNGBs can also be produced by the initial misalignment mechanism. The PNGB starts to oscillate when the Hubble parameter becomes comparable to the mass m a . In the radiation dominated Universe, this happens when T ∼ 2 × 10 6 GeV(m a /7keV) 1/2 .
Therefore, for T R 10 6 GeV, the oscillations starts before the reheating, and the PNGB abundance is given by where θ * ≡ a ini /f a denotes the initial oscillation amplitude. If the U(1) F symmetry is spontaneously broken after inflation, we should replace θ * with its averaged value, For T R 2 × 10 6 GeV, the abundance of PNGBs produced by the initial misalignment mechanism becomes independent of T R . Therefore, the initial misalignment mechanism is subdominant compared to the thermal production for f a = 10 10 GeV. Note that the dependence of the abundance on f a is different between the two production processes, and that for slightly larger values of f a , the initial misalignment mechanism can dominate over the thermal production. This is the case if q e is comparable to ∼ 3 or larger.
Lastly we consider the case of q e = 0. In this case the preferred value of f a is about 10 5 GeV, and the thermal production always dominate over the initial misalignment mechanism unless the anharmonic effect becomes significant [22,23,24,25]. For T R above the weak scale, the PNGBs are thermalized. For m µ < T < m τ , the PNGBs can be produced by scattering processes such as µ + γ → µ + a with a rate given by where T is the temperature. The production through the above process is most efficient at T = m µ , and the production rate exceeds the Hubble parameter at that time if f a 4 × 10 7 GeV.
Therefore, for T R m µ , the PNGBs are thermalized, and we need an additional entropy dilution by a factor of 40. 5 If T R = O(10) MeV, it is possible to produce the right amount of PNGBs to account for the observed dark matter abundance. 4 Recently, the BICEP2 experiment found the primordial B-mode polarization, implying that the inflation scale is about H inf ∼ 10 14 GeV [21]. If this is true, the global U(1) F symmetry must become spontaneously broken after inflation to avoid generating too large isocurvature perturbations. In this case, one needs to introduce extra breaking terms to avoid the cosmological catastrophe induced by domain walls. 5 If the PNGB mass is of O(0.1) eV or lighter, there is no problem even if it is thermalized. It would contribute to hot dark matter [26,27] or the effective neutrino species ∆N eff ≃ 0.39 [6]. Their existence are favored by recent observations [28,29,30,31]. Interestingly, hot dark matter or dark radiation can relax the tension between BICEP2 and Planck.
For any charged assignment satisfying the above conditions, the Yukawa interactions take the diagonal form, Let us normalize the global U(1) F charge so that c = 1. Then the above conditions from (17) to (21) whereH ( with where the B −L Higgs fields are understood to represent their VEVs, and we have dropped O(1) numerical coefficient in each element. If the VEVs are comparable to each other, the large neutrino mixing angles are realized. The light neutrino masses can be explained by the seesaw mechanism [32].
The PNGB resides in the phase of φ(1) and φ(2), and the decay constant f a is approximately given by their VEVs. In fact, the PNGB in this case is similar to the majoron.
The cosmological constraints on the majoron dark matter were studied in e.g. Ref. [33].
One can also introduce the Higgs portal couplings ∼ |φ| 2 |H| 2 . The situation would be similar to the model proposed by Weinberg [6]. For a certain set of parameters, massless PNGBs would contribute the effective neutrino species, ∆N eff .

Case of a = +2
In A couple of comments are in order. In order to give a mass to the PNGB, one needs an explicit U(1) F symmetry breaking. It is interesting to note that the following term breaks the U(1) F symmetry down to the subgroup Z 6 , giving rise to a PNGB mass

Discussion and conclusions
Some comments and discussions are in order. In the case of q e = 0, f a = O(10 9−10 ) GeV is needed to explain the 3.5 keV X-ray line. Since the couplings to photons, gluons and nucleons are suppressed, the PNGBs avoid various astrophysical and ground-based constraints. Still, it may be possible to find them in the future. Interestingly, there is a hint for an extra cooling of white dwarfs, which can be explained by light PNGBs coupled to electrons with the decay constant in this range [34]. 6 If such light PNGBs are coupled with electrons but not with photons, it is possible that they are copiously produced in the Sun, but cannot be detected by experiments using the magnetic field like the CAST experiment [35].
In the case of q e = 0 and q µ = 0, the preferred value of f a is of order 10 5 GeV, much smaller than the previous case. Still, as the effective PNGB coupling to the photon is so weak that the constraint from the cooling of horizontal branch stars can be satisfied [36].
On the other hand, the bound from supernova cooling will be more non-trivial since the PNGB couples to muons directly and the muons might be abundant in the supernova core [37,38]. Although the muon abundance depends sensitively on the temperature, the preferred value of f a may be in tension with the observation. As a rough estimate, we refer to the constraint on the majoron coupling constants to neutrinos from the supernova cooling: it is bounded as g ee 10 −6 where g ee is the yukawa coupling between the majoron and electron neutrinos [39]. In our case, the effective coupling constant between the PNGB and the muon reads m µ /f a ∼ 10 −6 . A more detailed study is needed to test the viability of this model.
So far we have considered the U(1) F flavor symmetry, under which only leptons are charged, and we constructed models in which the lepton mass matrix is (almost) diagonal.
It is possible to extend the models to allow larger off-diagonal terms, or to extend the flavor symmetry to the quark sector, by enlarging the flavor symmetry and adding more Higgs fields. If the actual flavor symmetry group is larger than U(1) F and if it is broken at a scale of O(10 9−10 ) GeV, there may be more PNGBs with different masses with or without couplings to photons and/or gluons. Then it may be possible to provide a unified picture of the QCD axion well as other PNGBs. In this case the light PNGBs can be searched for by flavor-changing processes such as τ → µ + a, µ + → e + + a, K + → π + + a [40].
We have pursued a possibility that a PNGB is lurking below the intermediate scale, evading the astrophysical bounds. Along this lines we have proposed flavor models based on an anomaly-free U(1) F symmetry, where the PNGB is preferentially coupled to the leptons. In particular, its anomalous couplings to gluons and photons are absent, greatly relaxing the astrophysical bounds. We have also pointed out that, although suppressed, the PNGB coupling to photons is induced by threshold corrections. Interestingly, the recent hint for the X-ray line at about 3.5 keV [8,9] can be explained by the PNGB dark matter with m a ≃ 7 keV for the decay constant f a = 10 9−10 GeV (f a = 10 5−6 GeV) if electrons are (not) charged under the flavor symmetry.