Longevity Problem of Sterile Neutrino Dark Matter

# Longevity Problem of Sterile Neutrino Dark Matter

Hiroyuki Ishida 111email: h_ishida@tuhep.phys.tohoku.ac.jp, Kwang Sik Jeong 222email: ksjeong@tuhep.phys.tohoku.ac.jp, and Fuminobu Takahashi 333email: fumi@tuhep.phys.tohoku.ac.jp Department of Physics, Tohoku University, Sendai 980-8578, Japan
Kavli IPMU, TODIAS, University of Tokyo, Kashiwa 277-8583, Japan
###### Abstract

Sterile neutrino dark matter of mass  keV decays into an active neutrino and an X-ray photon, and the non-observation of the corresponding X-ray line requires the sterile neutrino to be more long-lived than estimated based on the seesaw formula: the longevity problem. We show that, if one or more of the BL Higgs fields are charged under a flavor symmetry (or discrete R symmetry), the split mass spectrum for the right-handed neutrinos as well as the required longevity is naturally realized. We provide several examples in which the predicted the X-ray flux is just below the current bound.

###### pacs:
preprint: TU-945, IPMU13-0176

## I Introduction

One of the central issues in modern cosmology and particle physics is the identity of dark matter. If dark matter is made of as-yet-unknown species of particles, they must be stable on a cosmological time scale. The required longevity can be attributed to their light mass and/or extremely weak interactions, and the elusiveness of dark matter is probably related to its longevity to some extent. This however does not necessarily imply that dark matter is completely stable; it may have a long but finite lifetime, decaying into lighter particles. If so, it will enable us to identify dark matter by detecting the signal of the decay products.

Sterile neutrino is one of the plausible candidates for dark matter, and it has been extensively studied from various aspects such as the structure formation and baryogenesis. See Refs. Boyarsky:2009ix (); Kusenko:2009up (); Abazajian:2012ys (); Drewes:2013gca (); Merle:2013gea () for a review. Interestingly, sterile neutrino dark matter decays into an active neutrino and an X-ray photon through mixing with active neutrinos Lee:1977tib (); Marciano:1977wx (); Petcov:1976ff (); Pal:1981rm (). So far, the corresponding X-ray line has not been observed, which places severe constraints on the mixing angle, or equivalently, its neutrino Yukawa couplings.

The smallness of the neutrino Yukawa couplings can be partially understood by a simple Froggatt-Nielsen (FN) type flavor model Froggatt:1978nt () or the split seesaw mechanism Kusenko:2010ik (), in which the right-handed neutrinos are charged under a flavor symmetry or propagate in an extra dimension, while the other standard model (SM) particles are neutral or reside on the four dimensional brane. One of the interesting features of these models is that the beauty of the seesaw formula seesaw (), which relates the light neutrino masses to the ratio of the electroweak scale to the GUT (or BL) scale, is preserved even for a split mass spectrum of the right-handed neutrinos, e.g. , where and represent the generation index. This is because both the light sterile (or right-handed) neutrino mass and the corresponding neutrino Yukawa couplings are suppressed simultaneously in such a way that the seesaw formula remains intact. However, the suppression is not sufficient to avoid the X-ray constraint; the observation requires the sterile neutrino dark matter to be more long-lived than naively expected. The gap becomes acute for a heavier mass. As we shall see shortly, for the sterile neutrino mass of  keV, the corresponding neutrino Yukawa couplings must be more than two orders of magnitude smaller than estimated based on the seesaw formula. If there is no correlation among different elements of the neutrino Yukawa matrix as in the neutrino mass anarchy hypothesis Hall:1999sn (); Haba:2000be (), it would amount to fine-tuning of order . We call this fine-tuning associated with the neutrino Yukawa couplings of the sterile neutrino dark matter as “the longevity problem.”

Taken at a face value, the longevity problem of the sterile neutrino dark matter suggests an extended structure of the theory, such as an additional symmetry forbidding the neutrino Yukawa couplings. In particular, it requires a slight deviation from the seesaw formula for the sterile neutrino dark matter.

In fact, it is well known that, if the sterile neutrino comprises all the dark matter, its contribution to the light neutrino mass must be negligible in order to satisfy the X-ray bounds Asaka:2005an (); Boyarsky:2006jm (). The point of this paper is to take the observational constraint seriously and construct theoretical models that could realize both the required split mass hierarchy and the longevity simultaneously. In Ref. Araki:2011zg (), it was shown that the mass spectrum and the mixing angles in the so called MSM Asaka:2005an (), where the lightest sterile neutrino has a mass of order keV and the other two heavy sterile neutrinos have quasi-degenerate masses of  GeV, can be realized by introducing , , and flavor symmetries as well as four SM singlet scalars. Importantly, the longevity problem was solved in their flavor model. On the other hand, our purpose is to solve the longevity problem and not to realize the quasi-degenerate mass for the two heavy sterile neutrinos, and so, we will consider a relatively simple model in which the SM is extended by introducing three right-handed neutrinos, a gauged U(1) symmetry, and an extra flavor symmetry. Actually one can easily make the lightest sterile neutrino completely stable by assigning a discrete symmetry such as  Allison:2012qn (), which however implies that one cannot observe the sterile neutrino dark matter through its decay. Also an additional mechanism is required to realize the split mass spectrum for the sterile neutrinos. Instead, we will construct models in which a single flavor symmetry realizes both the split mass spectrum and the longevity of the lightest sterile neutrino. In particular, the predicted X-ray flux can marginally satisfy the observational bounds, so that the X-ray observation still remains a viable probe of the sterile neutrino dark matter scenario.

In this paper we show that the longevity problem can be solved naturally if one or more of the BL Higgs fields is charged under a flavor symmetry which also realizes the split mass spectrum, . The main difference from the simple FN model is that the scalar charged under the flavor symmetry has a non-zero BL charge, and we call such mechanism achieving the split mass spectrum for the right-handed neutrinos with a sufficiently long lifetime as “split flavor mechanism” in order to distinguish it from the simple FN model. As we shall see shortly, the split flavor mechanism works well for both continuous and discrete flavor symmetries, and we provide several examples which solve the longevity problem and predict the X-ray flux just below the current bound.

## Ii Longevity problem

We consider an extension of the SM with three right-handed neutrinos, and assume the seesaw mechanism seesaw () throughout this paper. The relevant interactions for the seesaw mechanism are given by

 L = i¯NIγμ∂μNI−(λIα¯NILαH+12MI¯NcINI+h.c.), (1)

where , and are the right-handed neutrino, lepton doublet and Higgs scalar, respectively, denotes the generation of the right-handed neutrinos, and runs over the lepton flavor, , and . The sum over repeated indices is understood. Here we adopt a basis in which the right-handed neutrinos are mass eigenstates, and is set to be real and positive. If there is a U(1) gauge symmetry, the breaking scale is tied to the right-handed neutrino mass, as long as the coupling of the BL Higgs to the right-handed neutrinos is not suppressed.

Integrating out the massive right-handed neutrinos yields the seesaw formula for the light neutrino mass:

 (mν)αβ=λαIλIβv2MI, (2)

where  GeV is the vacuum expectation value (VEV) of the Higgs field. The solar and atmospheric neutrino oscillation experiments clearly showed that at least two neutrinos have small but non-zero masses, and the mass splittings are given by and . The seesaw mechanism then suggests that a typical mass scale of the right-handed neutrinos or the BL breaking scale is around  GeV, close to the GUT scale, for . An attractive feature of the seesaw formula is that it explains the smallness of the neutrino masses by relating them to the ratio of the electroweak scale to the GUT (or BL) scale. Furthermore, the baryon asymmetry of the Universe can be generated via leptogenesis by out-of-equilibrium decays of such heavy right-handed neutrinos Fukugita:1986hr ().

The above argument does not necessarily mean that all the right-handed neutrinos have a mass of order  GeV. In fact, it is known that the above mentioned feature of the seesaw formula can be preserved even for a split mass spectrum of the right-handed neutrinos in the simple FN model Froggatt:1978nt () or the split seesaw mechanism Kusenko:2010ik (). Most importantly, the lightest right-handed neutrino can be dark matter, as it becomes stable in a cosmological time scale for a sufficiently light mass. Thus an interesting scenario is that sterile neutrinos have a split mass spectrum so that the lightest one contributes to the dark matter while the other two implement leptogenesis. Intriguingly, this may explain why there are three generations Kusenko:2010ik ().

In the simple FN model or the split seesaw mechanism, transforms differently from () under some symmetry or has an exponentially different localization property due to slightly different bulk masses, respectively. The mass and Yukawa couplings of the lightest right-handed neutrino are then suppressed as

 M1 = x2M, (3) |λ1α| = xα, (4)

where represents the suppression factor, and is the U breaking scale. The relation arises from the crucial assumption that the suppression mechanism is independent of the U symmetry and its breaking. The light neutrino masses are still related to the ratio of the electroweak scale to the GUT (or BL) scale, since the dependence on and is cancelled in the seesaw formula (2) as long as .

On the other hand, the mixing angle between and active neutrinos is given by

 θ21 ≡ ∑α|λ1α|2v2M21 (5) = 10−5ϵ2(mseesaw0.1eV)(M110keV)−1,

where we have defined , and denotes the typical neutrino mass induced by the seesaw mechanism,

 mseesaw ≡ v2M≃0.03eV(M1015GeV)−1. (6)

Through the mixing , the sterile neutrino decays into three active neutrinos, and also radiatively into active neutrino plus photon Lee:1977tib (); Marciano:1977wx (); Petcov:1976ff (); Pal:1981rm (). The latter process is strongly constrained by the non-observation of the corresponding X-ray line Abazajian:2012ys () (see also Refs. Loewenstein:2008yi (); Loewenstein:2009cm (); Loewenstein:2012px ()), leading to a tight upper bound on the mixing angle as shown Fig. 1. The bound can be conveniently parameterized by Boyarsky:2009ix ()

 θ21 ≲ 1.8×10−10(M110keV)−5. (7)

Therefore, should be much smaller than unity to satisfy the X-ray bound for a few keV:

 ϵ ≲ 4×10−3(mseesaw0.1eV)−12(M110keV)−2. (8)

This requires a deviation from the seesaw formula (2) for the sterile neutrino dark matter , and the gap becomes acute for a heavier . Note that the Lyman alpha bounds on reads  keV (C.L.), assuming the non-resonant production for the sterile neutrino dark matter Boyarsky:2008xj ().444 The bound is relaxed for the production from the singlet Higgs decay Kusenko:2006rh (); Petraki:2007gq () or the resonant production which works in the presence of large lepton asymmetry Boyarsky:2008xj (). Therefore must be much smaller than unity, which implies the neutrino Yukawa couplings should be suppressed by about with respect to that estimated from the seesaw formula. For instance, for  keV, we need smaller than . If takes a value of order unity randomly as in the neutrino mass anarchy, it would require a fine-tuning of order . We call this fine-tuning problem as the longevity problem. Importantly, the problem cannot be resolved in the split seesaw mechanism or the simple FN model. As we shall see in the next section, the split mass spectrum as well as the required longevity can be naturally explained if one or more of the BL Higgs is charged under a flavor symmetry; the key is to combine the flavor symmetry with the BL symmetry.

## Iii Split flavor mechanism

In this section, we present a modified seesaw model which realizes the split mass spectrum for while solving the longevity problem. We consider an extension of the SM with three right-handed neutrinos for , the U(1) gauge symmetry, and two BL Higgs fields and whose VEVs provide masses to the sterile neutrinos. The reason why two BL Higgs fields are needed will be clarified soon. In a supersymmetric theory, two Higgs fields are anyway required for the anomaly cancellation. Here we adopt a flavor basis for , but the mixing between and is suppressed in the models considered below. In the split flavor mechanism, we will introduce a flavor symmetry, under which only the fields in the seesaw sector are charged, and the SM fields are assumed to be neutral. The role of the flavor symmetry is to suppress both the mass and mixings of to satisfy the X-ray bound (7), and the key is to assign a flavor charge on one or more of the BL Higgs fields. As reference values we take  keV and , but it is straightforward to further impose a usual FN flavor symmetry, e.g., in order to make much lighter than .

### iii.1 Non-supersymmetric case

We adopt a flavor symmetry under which only and are charged while the others are singlet:

   U(1)B−L Z4 Φ Φ′ N1 Ni Lα H 2 −2n −1 −1 −1 0 0 −1 1 0 0 0

with being a positive integer, and . Then the seesaw sector is described by

 −ΔL=12κiΦ¯NciNi+λi¯NiLH+12κ1(Φ2n−1Φ′2)∗Λ2n¯Nc1N1+~λ(ΦnΦ′)∗Λn+1¯N1LH+h.c., (9)

for a cut-off scale . Here , , and are numerical coefficients of order unity, and we have dropped the lepton flavor indices. Note that the term has been omitted as it can be removed by redefining without any significant effects on the above interactions.

The U(1) gauge symmetry is spontaneously broken when and develop a non-zero VEV. Here we assume . As a result, the mass of the two heavy right-handed neutrinos is set by , and the light neutrino masses are nicely explained by the seesaw mechanism. The above neutrino interactions lead to the mass and mixing of the as

 M1 ≈ (MΛ)2(n−1)(⟨Φ′⟩Λ)2M, (10) λ1α ≈ (MΛ)n⟨Φ′⟩Λ, (11)

implying

 ϵ ≈ MΛ. (12)

Therefore the suppression of is achieved for , and consequently the active-sterile neutrino mixing is estimated to be

 θ21 ≈ 10−5(MΛ)2(mseesaw0.1eV)(M110keV)−1, (13) ≃ 2×10−12(mseesaw0.1eV)(M110keV)−1(M1015GeV)2,

where we have set to be the Planck scale, , in the second equality. Note that the mixing angle depends on only through . For instance, in the case of , is around 10 keV when both and have a VEV around  GeV. Fig. 2 shows the property of for the case with , assuming that and have VEVs of a similar size. Also,  keV can be realized for or 2 if is at an intermediate scale, which is possible because there is no dynamical reason to relate to in contrast to supersymmetric cases.

It is possible to consider a general discrete symmetry under which only and are charged. A proper charge assignment makes have a small Yukawa coupling induced from the term after BL breaking. Here carries a BL charge equal to for coprime positive integers and . Then it is obvious that always receives contribution from . If it is the dominant contribution, one obtains as in the simple FN model, and thus the longevity problem is not solved. This holds also when one uses a global U instead of . We note that a suppression of can be achieved by taking a charge assignment such that gets a mass dominantly either from or from .

### iii.2 Supersymmetric case

The seesaw mechanism can be embedded into a supersymmetric framework. For the anomaly cancellation, and must be vector-like under U. Interestingly enough, it is then possible to suppress as well as the active-sterile neutrino mixing by both supersymmetry (SUSY) breaking effects and a flavor symmetry. We will also show that a discrete R-symmetry can do the job.

#### iii.2.1 Discrete flavor symmetry

Let us first consider a flavor symmetry with , under which only and transform non-trivially and the others are neutral:

U 0
0 1 1 0 0 0

with being the up-type Higgs doublet superfield. Such discrete symmetry acting on one of the BL Higgs fields was considered in the BL Higgs inflation models Nakayama:2012dw (). Note that , and are left-chiral superfields, and in particular, the fermionic component of is the left-handed anti-neutrino. That is why the BL charge assignment on these fields is different from the non-supersymmetric case.

With the above charge assignment, the relevant terms in the Kähler and super-potentials of the seesaw sector are given by

 ΔK = Φ′∗ΛN1Ni+12(ΦΦ′2)∗Λ3N1N1+h.c., ΔW = 12ΦNiNi+NiLHu+(ΦΦ′)k−1Λ2k−2N1LHu+12Φ(ΦΦ′)k−2Λ2k−4N1N1, (14)

where we have omitted coupling constants of order unity.555 Instead of the discrete symmetry, one can take a global U symmetry under which and have the same charge and the other fields are neutral. Then the terms in are still allowed while the last two terms in are forbidden. The Nambu-Goldstone boson associated with U may contribute to dark radiation Nakayama:2010vs (); Weinberg:2013kea (). Though we have not considered here, one may impose a U symmetry under the assumption that it is broken by a small constant term in the superpotential, i.e. by the gravitino mass . As we shall see shortly, in such case, both of the terms in can be further suppressed by if the superpotential is to possess the term . Note here that the gravitino mass represents the explicit U(1) breaking by two units.

To examine the property of sterile neutrino dark matter, it is convenient to integrate out the U sector. The U is broken along the -flat direction , which is stabilized by higher dimension operators, or by a radiative potential induced by the interaction. For much larger than the gravitino mass , the effective theory of neutrinos is written as

 ΔWeff = 12κiMNiNi+λiNiLHu+12M1N1N1+λ1αN1LHu, (15)

at energy scales around and below , where the sterile neutrino obtains

 M1 = m3/2M3Λ3+M2k−3Λ2k−4, (16) λ1α = m3/2Λ+M2k−2Λ2k−2 (17)

omitting numerical coefficients of order unity. Here the terms proportional to arise from after redefining to remove mixing terms in the effective superpotential. In contrast to the non-supersymmetric case, there are two important effects here. One is the holomorphic nature of the superpotential, and the other is the SUSY breaking effects represented by the gravitino mass.

Depending on the values of , , and , there are various possibilities. To simplify our analysis, let us focus on the case of the reference values, and . Then  keV is realized for  TeV and ,666 This may provide a motivation to consider SUSY around  TeV, which is consistent with the recent discovery of the SM-like Higgs boson of mass  GeV. If the SUSY breaking was much higher, the sterile neutrino could not be dark matter because of its too short lifetime. Note that the decay rate is proportional to . for which the neutrino Yukawa coupling receives the dominant contribution from the SUSY breaking effect, i.e., from the first term in Eq. (17). Note also that is determined entirely by the SUSY breaking effect for . In the following we consider  TeV and . The parameter and active-sterile neutrino mixing angle then read

 ϵ ≈ 4×10−4(m3/2100TeV)56(M110keV)−13, (18) θ21 = ϵ2mseesawM1≈10−12(mseesaw0.1eV)(m3/2100TeV)53(M110keV)−53. (19)

Thus, the observational constraint (7) is naturally satisfied if the gravitino mass is smaller than or comparable to  TeV. In particular, the predicted X-ray flux is just below the observational bound for  TeV. See Fig. 3, where the contours of and are shown in the plane. On the other hand, the squarks and sleptons acquire soft SUSY breaking masses in the range between about and , depending on mediation mechanism. It is interesting to note that the gravitino mass around  TeV leads to TeV to sub-PeV scale SUSY, which can accommodate a SM-like Higgs boson at 126 GeV within the minimal supersymmetric SM (MSSM).

Lastly we comment on the case with an approximate global U(1) broken by a constant superpotential term. The neutrino interactions are then further constrained. For instance, let us consider the case where and have the same R charge equal to one while , and are neutral. Then both the terms in are further suppressed by the gravitino mass. As a result, the sterile neutrino mass as well as the neutrino Yukawa couplings are determined by the ratio of the BL breaking scale to the cut-off scale, and the effect of SUSY breaking is negligibly small. That is to say, and receive the dominant contributions from the second terms in (16) and (17), respectively. For the reference values and , must be equal to to realize  keV unless is extremely heavy (say, or heavier). Then the neutrino Yukawa couplings will become extremely small so that sterile neutrino dark matter becomes practically stable and the predicted X-ray flux is negligibly small. Although not pursued here, it may be interesting to consider the case of where a sterile neutrino dark matter is much heavier than keV and has a sufficiently small mixing angle.777See Ref. Essig:2013goa () for the latest X-ray and gamma-ray constrains on such heavy sterile neutrino dark matter.

#### iii.2.2 Discrete R symmetry

Next let us consider a case of discrete R symmetry. The discrete R symmetry has been extensively studied from various cosmological and phenomenological aspects. See e.g. Refs. Kurosawa:2001iq (); Izawa:1996dv (); Dine:2009swa (); Dine:2010eb (); Harigaya:2013vja (); Takahashi:2013cxa (). Now we show that the split flavor mechanism can be implemented by the discrete R symmetry with the following charge assignment,

   U(1)B−L ZkR Φ Φ′ N1 Ni Lα Hu −2 2 1 1 −1 0 0 p q 1 1 0

where and are integers mod . To simplify our analysis, we assume that the cut-off scale for higher dimensional operators is given by the Planck scale, , and the BL breaking scale is about . The gravitino mass is assumed to be below PeV scale.

Note that the discrete symmetry ( is explicitly broken by the constant term in the superpotential, . Therefore, the mass and neutrino Yukawa couplings generically receive two contributions; one is invariant under , and the other is not invariant and is proportional to the gravitino mass.

The sterile neutrino mass  keV is numerically close to or , and the mass of this order can be generated if one or more of the following operators are allowed:

 ΔK = (ΦΦ′2)∗M3pN1N1+h.c., ΔW = Φ(ΦΦ′)3M6pN1N1orm3/2Φ2Φ′M3pN1N1. (20)

Similarly, the neutrino Yukawa coupling of the desired magnitude can be induced from the following operators,

 ΔK = Φ′∗MpN1Ni+h.c., ΔW = (ΦΦ′)2M4pN1LHuorm3/2MpN1LHu. (21)

In order for one or more of the above operators to give the dominant contribution to and , the following operators must be forbidden by the discrete R-symmetry:

 ΔKforbidden = Φ′∗MpN1N1+h.c., ΔWforbidden = ΦN1NI+Φ2Φ′M2pN1NI+Φ(ΦΦ′)2M4pN1N1+N1LHu+ΦΦ′M2pN1LHu, (22)

which puts constraints on and .

To summarize, we need to find a set of satisfying

 2p−2q≡0   or   3p+2q≡2   or   p+2q≡0, (23) p−q−1≡0   or   2p+q+1≡2   or   q+1≡0, (24) p−2q≢0,  2q≢2,  q+1≢2,  p+2q≢2,  p+q+1≢2,  2p+2q≢2, (25)

where all the equations are mod . Some of the solutions of the above conditions are888 If we forbid a SUSY mass in the superpotential, the solutions with should be excluded.

 (k,p,q)=(5,2,2),(5,4,3),(7,3,2),(7,5,4),(7,5,5),(7,6,6),⋯. (26)

In fact there is no solution for which both and are generated by the invariant operators. That is to say, either or both of them should be generated by the SUSY breaking effect proportional to the gravitino mass.

Let us focus on the case of . Then the relevant terms in the superpotential are given by

 ΔW = 12ΦNiNi+NiLHu+12m3/2Φ2Φ′M3pN1N1+(ΦΦ′)2M4pN1LHu, (27)

where we have dropped numerical coefficients of order unity. The other interactions in the Kähler and super-potentials are either forbidden or irrelevant for the following discussion. The mass and neutrino Yukawa couplings for are given by

 M1 ≈ 10keV(m3/2100TeV)(M1015GeV)3, (28) λ1α ≈ 10−14(M1015GeV)4, (29)

from which one finds

 ϵ ≃ 3×10−4(m3/2100TeV)−12(M1015GeV)3, (30)

using the D-flat condition, . Therefore the mass is close to  keV and for the reference values and . Finally, the mixing angle reads

 θ21 ≈ (31)

We show the contours of and the mixing angle are shown in the - plane in Fig. 4. It is interesting to note that  TeV and lead to the sterile neutrino mass  keV with the predicted X-ray line flux just below the current bound.

## Iv Cosmological aspects

We have so far focused on the mass and mixing angles of the sterile neutrinos. In order for the lightest sterile neutrino to account for the observed dark matter, a right amount of must be produced in the early Universe. The density parameter of dark matter is related to the number to entropy ratio as

 ΩDMh2 ≃ 0.14(M110keV)(nN1/s5×10−5), (32)

where is the dimensionless Hubble parameter in the units of , and and are the number density of and the entropy density, respectively. The latest observations give  Ade:2013zuv ().

The thermal production known as the Dodelson-Widrow mechanism Dodelson:1993je () is in tension with the X-ray bound for  keV, as can be seen from Fig. 1. Therefore we need another production mechanism. One possibility is that the is produced via the s-channel exchange of the BL gauge boson Kusenko:2010ik (). The number to entropy ratio of the sterile neutrino produced by this mechanism is roughly estimated as

 nN1s ∼ 10−4(g∗100)32(M1015GeV)−4(TR5×1013GeV)3, (33)

where counts the relativistic degrees of freedom at the reheating, and denotes the reheating temperature. The numerical solution of the Boltzmann equation gives a consistent result Khalil:2008kp (). The assumption here is that the BL symmetry is spontaneously broken during and after inflation. This production mechanism works both for supersymmetric and non-supersymmetric cases. Also, a right amount of the baryon asymmetry can be created via thermal leptogenesis due to the two heavy right-handed neutrinos and for such high reheating temperature Endoh:2002wm (); Raidal:2002xf ().999 Thermal leptogenesis in the neutrino mass anarchy hypothesis was studied in Ref. Jeong:2012zj ().

On the other hand, if the BL symmetry is restored during or after inflation, the sterile neutrinos will be in thermal equilibrium through the U gauge interactions. The thermal abundance is given by

 n(eq)N1s≃2×10−3(g∗100)−1. (34)

So, if there is an entropy dilution of the order of a few tens, the right amount of can be generated. In the non-supersymmetric case, such entropy dilution can be easily realized by the BL Higgs dynamics. Suppose that the mass of the BL Higgs is slightly smaller than the BL breaking scale. Then it remains trapped at the origin due to the thermal mass induced by the BL gauge boson loop, dominating the Universe for a while. This is a mini-thermal inflation.101010 See Ref. Yamamoto:1985rd () for the usual thermal inflation. The entropy production due to the bubble formation was discussed in Ref. Kusenko:2010ik (). When the plasma temperature becomes lower than the mass, the BL Higgs develops a large VEV, and the subsequent decays of the BL Higgs produce the entropy. Also, thermal and/or non-thermal leptogenesis works successfully in this case. Since we have imposed a discrete symmetry on the BL Higgs, domain walls are generally produced. The domain walls will annihilate if we add a small breaking of the discrete symmetry. Interestingly, gravitational waves GW () are likely produced during the violent annihilation processes of the domain walls Gleiser:1998na (); Takahashi:2008mu (); Dine:2010eb (); Hiramatsu:2010yz (); Kawasaki:2011vv (); Hiramatsu:2013qaa (), which may be within the reach of the future and planned gravitational wave experiments. After the domain wall annihilation, we are left with the cosmic strings whose tension is consistent with the CMB observation Ade:2013xla () for  GeV.

In a supersymmetric case, on the other hand, the stabilization of the BL Higgs is slightly more involved. To be concrete, let us consider the model based on the discrete R symmetry and adopt in the following. The -flat direction composed of and can be stabilized by the balance between non-renormalizable superpotential term and SUSY breaking effect (negative soft mass squared at the origin, or the -term associated with the superpotential term):

 V=−m2ϕ|ϕ|2−(m3/2ϕ6M3p+h.c.)+|ϕ|10M6p, (35)

where parameterizes the -flat direction, represents the soft mass for the -flat direction, and we have dropped numerical coefficients of order unity. The BL Higgs is then stabilized at

 M=⟨ϕ⟩∼1015GeV(m3/2100TeV)14. (36)

If the U(1) symmetry is restored during or after inflation, thermal inflation generically takes place because has a relatively flat potential. Then the entropy dilution factor tends to be large, and any pre-existing will be diluted away. The subsequent domain walls can be erased if we introduce a breaking of the discrete symmetry.111111 In the case of the discrete R symmetry, the constant term in the superpotential provides such breaking terms. Unfortunately, however, its size is too small to make domain walls to annihilate before dominating the Universe.

In the supersymmetric case, the lightest supersymmetric particle (LSP) in the MSSM contributes to the dark matter abundance. Even though the R-parity is broken in the case of the discrete symmetry, the MSSM-LSP is stable due to the residual since U(1) is spontaneously broken only by and with the BL charge two. In order for the lightest sterile neutrino to be the dominant component of dark matter, the MSSM-LSP abundance must be suppressed. If the reheating temperature is as high as  GeV, the Universe becomes gravitino-rich, and the MSSM-LSPs tend to be overproduced by the gravitino decay Jeong:2012en (). The MSSM-LSP abundance can be suppressed if it is a Wino-like or Higgsino-like neutralino of mass  GeV and the gravitino mass is of order PeV. Since they comprise only a fraction of the total dark matter, the constraints from indirect dark matter searches are relaxed. It would be interesting if we could see the indirect dark matter signatures for both the sterile neutrino and the Wino-like or Higgsino-like neutralino. On the other hand, if the gravitino mass is of  TeV, the MSSM-LSPs are overproduced by the gravitino decay. It is actually possible to make the MSSM-LSP unstable. Let us consider the case of the discrete R symmetry with . Then, this can be achieved by introducing another vector-like pair of the BL Higgs and where the BL and R-charges are shown in the parenthesis, respectively. If and have a nonzero VEV, say, of  GeV, the trilinear R-parity violating operators are allowed, and the MSSM-LSP decays before the big bang nucleosynthesis. The constraints from the proton decay can be safely satisfied Hinchliffe:1992ad (). Alternatively, if there is another production mechanism of the sterile neutrino dark matter which works at a temperature below , the Universe is not gravitino-rich, and we can avoid the overproduction of the MSSM-LSPs from the gravitino decay.

## V Conclusions

The sterile neutrino dark matter of mass  keV generically decays into an active neutrino and an X-ray photon, but the non-observation of the X-ray line requires the sterile neutrino to be more long-lived than estimated based on the seesaw formula. Specifically, the neutrino Yukawa couplings must be suppressed by more than two orders of magnitude than naively estimated for  keV. We call this tension as the longevity problem for the sterile neutrino dark matter. It is worth noting that the longevity problem is not solved by the simple FN model and the split seesaw mechanism, both of which preserve the seesaw formula. In this paper we have quantified the longevity problem and proposed the split flavor mechanism as a possible solution. In this mechanism, we have introduced a single flavor symmetry (or discrete R symmetry) under which one or more of the BL Higgs is charged. As a result, the split mass spectrum for the sterile neutrinos as well as the longevity required for the lightest sterile neutrino dark matter are realized. The key is to combine the BL symmetry with the flavor symmetry. We have provided several examples in which the lightest sterile neutrino of mass is keV and the predicted X-ray flux is just below the current bound. Therefore it may possible to test our models in the future X-ray observations.

## Acknowledgment

This work was supported by Grant-in-Aid for Scientific Research (C) (No. 23540283) [KSJ], Scientific Research on Innovative Areas (No.24111702 [FT], No. 21111006 [FT] , and No.23104008 [KSJ and FT]), Scientific Research (A) (No. 22244030 and No.21244033) [FT], and JSPS Grant-in-Aid for Young Scientists (B) (No. 24740135) [FT], and Inoue Foundation for Science [HI and FT]. This work was also supported by World Premier International Center Initiative (WPI Program), MEXT, Japan [FT].

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