3.55 keV X-ray Line Signal from Excited Dark Matter in Radiative Neutrino Model

We study an exciting dark matter scenario in a radiative neutrino model to explain the X-ray line signal at $3.55$ keV recently reported by XMN-Newton X-ray observatory using data of various galaxy clusters and Andromeda galaxy. We show that the required large cross section for the up-scattering process to explain the X-ray line can be obtained via the resonance of the pseudo-scalar. Moreover this model can be compatible with the thermal production of dark matter and the constraint from the direct detection experiment.


I. INTRODUCTION
In the light of anomalous X-ray line signal at 3.55 keV from the analysis of XMN-Newton X-ray observatory data of various galaxy clusters and Andromeda galaxy [1,2], dark matter (DM) whose mass is in the range from keV to GeV comes into one of the promising candidates. Subsequently, a number of literatures are recently arising around the subject .
As for the keV scale DM, for example, a sterile neutrino can be one of the typical candidates to explain the X-ray anomaly that requires tiny mixing between the DM and the active neutrino; sin 2 2θ ≈ 10 −10 [1]. However these scenarios suggest that neutrino masses cannot be derived consistently with the sterile neutrino DM due to its too small mixing. Moreover, the sterile neutrino DM mass is out of the range in the direct detection searches such as LUX [25], which is currently the most powerful experiment to constrain the kind of Weakly Interacting Massive Particle.
As for the GeV scale DM, on the other hand, the exciting DM scenario which requests a pair of ground state and excited DM is known to explain the X-ray [7]. In this framework, the emission of X-ray is simply realized as follows. After the ground state DM up-annihilates into the excited DM pair, it can decay into photons (X-ray) and the ground state DM. The mass difference among them is assumed to be the energy of the X-ray, 3.55 keV. Since the framework of the exciting DM is simple, this scenario can be applicable to various models such as radiative neutrino models [26][27][28][29]. In this kind of models, small neutrino masses and existence of DM would be accommodated unlike the sterile neutrino DM scenarios above.
Moreover the DM can be testable in direct detection searches because the DM mass is GeV scale.
In this letter, we account for the X-ray anomaly in terms of an excited DM scenario in a simple extended model with radiative neutrino masses [27], in which three right-handed neutrinos, a SU(2) L doublet scalar and a singlet scalar are added to the Standard Model (SM) and the first two lightest right-handed neutrinos are assumed to be a pair of ground state and excited state DM. The particle contents and charge assignments of the model we consider are shown in Tab. I. We introduce three right-handed neutrinos N i (i = 1 − 3) where the first two lightest ones are identified to be a pair of ground state and excited state DM. We also introduce a SU(2) L doublet inert scalar η that is assumed not to have vacuum expectation value (VEV), and a gauge singlet boson Σ with non-zero VEV in addition to the SM like Higgs boson Φ. The Z 2 symmetry is imposed to assure the stability of DM. The Z 3 symmetry plays an important role in forbidding the term (Σ + Σ † )N c i P R N i that leads no pseudo scalar coupling like Σ I N c i γ 5 N i where Σ I is the imaginary part of Σ. As we will see later, the pseudo scalar coupling is important to induce up-scattering process N 1 N 1 → N 2 N 2 . The Z 3 symmetry also allows the cubic term Σ 3 + h.c. that provides the mass of the pseudo scalar component of Σ. The relevant Lagrangian for the discussion is given as follows

A. Model setup
where the generation indices are omitted, and the Yukawa coupling y N can be regarded as diagonal in general. After the electroweak symmetry breaking, the scalar fields can be parametrized as where v ≈ 246 GeV, and G + and G 0 are absorbed in W + boson and Z boson due to the Higgs mechanism. The resulting CP even mass matrix with nonzero VEV is given by where the tadpole conditions ∂V/∂φ 0 | VEV = 0 and ∂V/∂σ| VEV = 0 are inserted. This mass matrix is diagonalized by the rotation matrix, and φ 0 and σ are rewritten by the mass eigenstates h and H as The mass eigenstate h corresponds to the SM-like Higgs and H is an extra Higgs respectively.
The mixing angle sin α is expressed as the function in terms of the other parameters as The pseudo scalar ρ does not mix after the symmetry breaking and the mass is just given The masses of the other Z 2 odd scalars η + , η R and η I are also determined adequately to be The mass splitting between m R and m I is given by The lower bounds of the inert scalar masses are obtained as m η 70 GeV and m R , m I ≥ 45 GeV by the LEP experiment [31][32][33] and the invisible decay of Z boson [33]. In addition, the mass difference between the charged and neutral inert scalars is constrained as roughly less than O(100) GeV by the T parameter [30].

B. Neutrino sector
The right-handed neutrinos obtain the masses after the symmetry breaking due to VEV of Σ, Using the right-handed neutrino masses, the active neutrino masses can be obtained at one-loop level as [27] ( In particular, when the mass splitting between η R and η I is small (λ 5 ≪ 1) and N i are much lighter than η (M i ≪ m R ≈ m I ), the formula can be simplified as follows We will consider the mass hierarchy for the analysis in the next section. The following parameter set is taken for example to be consistent with the sum of the light neutrino masses 0.933 eV [34] M ∼ O(10) GeV, y η ≈ 0.1, λ 5 ≈ 10 −5 , m R ≈ m I ∼ O(1) TeV. (II.12) Note that the Yukawa coupling y η cannot be too small since the lifetime of the decay channel N 2 → N 1 γ becomes too long to explain the X-ray anomaly.
Lepton Flavor Violating processes such as µ → eγ or µ → 3e should be taken into account [35]. One may think that the above parametrization has been already excluded by the strong constraint of µ → eγ whose branting ratio should be Br(µ → eγ) ≤ 5.7 × 10 −13 .
However it can be evaded by considering a specific flavor structure of the Yukawa coupling y η as ref. [36][37][38].

III. DARK MATTER
We identify that N 1 and N 2 are a pair of ground and excited state DM for explaining the X-ray anomaly. Thus their masses are related as M 1 ≈ M 2 < M 3 , and M 2 − M 1 ≡ ∆M = 3.55 keV. Such the situation has been considered for a different motivation in ref. [36][37][38].
The small mass splitting between N 1 and N 2 would be theoretically derived by introducing an extra U(1) symmetry. For example, we can construct the model that the interactions ΣN 1 N 1 and ΣN 2 N 2 are forbidden but ΣN 1 N 2 is allowed, and the small U(1) breaking terms such as N 1 N 1 and N 2 N 2 come from higher dimensional operators. Then after diagonalizing the mass matrix composed by N 1 and N 2 , almost degenerated two mass eigenstates are obtained.
A small momentum of DM is required to lead the up-scattering event where NFW profile is assumed [40]. The above formula suggests that we need a quite large cross section to explain the X-ray line. However our model can obtain such a large cross section via the ρ resonance.
The up-scattering process N 1 N 1 → N 2 N 2 is derived by the massive pseudo-scalar ρ.
Since the required cross section for the process is very large as σv rel ∼ 10 −19 cm 3 /s [7], an enhancement mechanism is required 1 . In our case, we have the resonance in the ρ mediated s-channel which leads s-wave for the cross section. In this sense, the interaction between the pseudo scalar ρ and two DM are crucially important. The up-scattering cross section is given by where s ≈ 4M 2 1 (1 + v 2 rel /4). We define the mass difference between 2m ρ and M 1 as ∆ ≡ 1 − m 2 ρ /4M 2 1 , and focus on the physical pole ∆ > 0. The cross section should be velocity 1 Note here that the pair of η I and η R cannot be used to explain the X-ray line because the decay process η R → η I γ is forbidden by spin statistics. After the up-scattering, N 2 immediately decays into the ground state DM (N 1 ) and photon through η + at one-loop level which is the dominant decay process of the excited DM. The decay width of the process N 2 → N 1 γ is calculated as where µ 12 is the transition magnetic moment between N 1 and N 2 which is calculated as [38] where e is the electromagnetic coupling constant. The lifetime of the excited DM N 2 should be much less than the cosmological timescale τ ∼ 10 17 s so as to decay immediately after the N 2 production. From the requirement, the order of the Yukawa coupling y η is estimated as y η 0.01. This does not conflict with the parameter set of Eq. (II.12). As one can see from Eq. (III.6), a complex phase of the Yukawa coupling y η is necessary to induce the decay Next we consider the thermal relic density of DM. The cross section contributing to the relic density is dominantly given via h and H s-channel. Although there the other contributions through t and u-channel via η exchange [41,42] where only bottom pair is taken into account in fermion pair f f due to the kinematics and strength of Yukawa coupling. As one can see from the equation, we have only p-wave contribution. The co-annihilation with N 2 should be taken into account since the masses among them are degenerated. However the order of the effective cross section including the co-annihilation process is same with Eq. (III.7) as long as y N 1 ≈ y N 2 is assumed. The To obtain the correct relic density Ωh 2 ≈ 0.12, the required cross section is σv rel ≈ 3 × 10 −26 cm 3 /s. In the left panel of Fig. 2, the contours of the required cross section for the thermal relic density are plotted in the plane of DM mass and the mass degeneracy between DM and H. As one can see, stronger degeneracy between DM and H is necessary for smaller mixing angle sin α in order to induce the appropriate cross section for the thermal relic density. The peak at M 1 ≈ 63 GeV is due to the SM Higgs resonance 2M 1 ≈ m h .
The direct detection constraint also should be considered since the scale of our DM is GeV. The elastic cross section with proton is induced by the t-channel Higgs mediation and it is calculated as where C ≈ 0.079. At present, the LUX experiment gives the strongest constraint on the elastic cross section. The constraint of the LUX experiment can be translated to the constraint on the mixing angle sin α in our case as shown in the right panel of Fig. 2. In the figure, the mass of the second Higgs m H is fixed to m H = 2M 1 from the requirement of the thermal relic density. One can see that the mixing angle sin α should be sin α 0.005 in order to evade the direct detection constraint in whole DM mass range.

IV. SUMMARY AND CONCLUSION
We have studied an exciting DM scenario in a radiative neutrino model to explain the X-ray line signal at 3.55 keV recently reported by XMN-Newton X-ray observatory using data of various galaxy clusters and Andromeda galaxy. We have shown that neutrino masses can be radiatively generated by our DM with the mass of O(10) GeV, which is requested by the exciting DM scenario. Also we have shown that the required large cross section to explain the X-ray line can be obtained by the resonance of the massive pseudo scalar ρ that provides s-wave contribution for only the up-scattering process N 1 N 1 → N 2 N 2 . The model can be consistent with the observed relic density as well as the direct detection constraint.
To induce 3.55 keV X-ray line without any inconsistencies, we have found that the mass degeneracies m H ≈ m ρ ≈ 2M 1 are required in the model.