A radiative seesaw model linking to XENON1T anomaly

We propose an attractive model that excess of electron recoil events around 1-5 keV reported by the XENON1T collaboration nicely links to the tiny neutrino masses based on a radiative seesaw scenario. Our dark matter(DM) is an isospin singlet inert boson that plays an role in generating non-vanishing neutrino mass at one-loop level, and this DM inelastically interacts with a pair of electrons at one-loop level that is required to explain the XENON1T anomaly. It is also demanded that the mass difference between an excited DM and DM has to be of the order keV. Interestingly, the small mass difference $\sim$keV is proportional to the neutrino masses. It suggests that we have double suppressions through the tiny mass difference and the one-loop effect. Then, we show some benchmark points to explain the XENON1T anomaly, satisfying all the constraints such as the event ratio of electrons of XENON1T, a long lived particle be longer than the age of Universe, and relic density in addition to the neutrino oscillation data and lepton flavor violations(LFVs).


I. INTRODUCTION
Dark matter (DM) is one of the important pieces to be understood its nature beyond the standard model (SM) and cosmology. Recently, XENON1T collaboration reported an excess of electron recoil events around 1-5 keV energy over the known backgrounds [1]. After this report, a vast literature has arisen along this line of the subject such as explaining the excess by axions, absorption of keV scale DM, a scattering model, inelastic DM, boosted DM, and so on .
In this paper, we propose a model that this excess by XENON1T marvelously links to tiny active neutrino masses based on a radiative seesaw scenario with a gauged hidden U (1) symmetry [53]. Some of radiative seesaw models are renowned as natural models to connect DM and the active neutrinos at low energy scale [54][55][56][57][58][59]. More concretely, our DM is an isospin singlet inert boson that plays an role in generating non-vanishing neutrino mass at one-loop level, and this DM inelastically interacts with a pair of electrons at one-loop level that is required by the XENON1T anomaly. It also demands that the mass difference between an excited DM and DM has to be of the order keV. The small mass difference ∼keV between DMs is proportional to the miniscule active neutrino masses. It suggests that we have double suppressions through this keV mass difference and the one-loop effect. At first, we show this mechanism and some benchmark points to explain the XENON1T anomaly, satisfying all the constraints such as the event ratio of electrons of XENON1T, a long lived particle be longer than the age of Universe, and relic density in addition to the neutrino oscillation data and LFVs. This paper is organized as follows. In Sec. II, we review our model, and construct our valid Lagrangian, Higgs potential, neutrino sector, LFVs, and Z boson mass. In Sec. III, we discuss our DM candidate to derive required scattering event rate with electrons, lifetime for a long lived particle, and cross sections of relic density. Then, we show our results accommodating all the issues discussed here. In Sec. IV, we devote to the summary of our results and the conclusion.
whereH = iσ 2 H * σ 2 being second Pauli matrix, generation index is omitted, and y is assume to be diagonal matrix without loss of generality due to the redefinitions of the fermions. The scalar potential is also given by An important point of this potential is to have µ term that provides mass difference between the mass of real part and imaginary part of χ. After symmetry breaking the squares of mass eigenvalues are explicitly written by where v is VEV of Higgs field H. Mass difference ∆m ≡ m χ R − m χ I is thus given by where we assumed µv ϕ m χ . The explanation of XENON1T anomaly by inelastic DM scattering requires O(keV) of ∆m which will be analyzed in next section. In addition, we show the neutrino mass matrix is proportional to ∆m below.

III. DARK MATTER
The neutrino mass matrix arises from the following Lagrangian where ψ is mass eigenstate of twelve extra neutral fermions, and V is a unitary mixing matrix by twelve by twelve to diagonalize the extra neutral fermion matrix, as can be seen in Appendix A. Then, we find the following neutrino mass matrix at one-loop diagram in where m ν is diagonalized by a unitary matrix V M N S as D ν = V T M N S m ν V M N S . Applying Casas-Ibarra parametrization [60] to our model, f is rewritten in terms of the other parameters as follows: where O is three by three orthogonal matrix with arbitrary parameters, T is an upper-right triangle matrix arisen from Cholesky decomposition. (See Appendix in ref. [61].) i → j γ process The relevant lepton flavor violating(LFV) process arises from where we consider χ as a complex scalar because of tiny mass difference. Then, the branching rations(BRs) are given by where C 21 = 1, C 31 = 0.1784, C 32 = 0.1736, α em (m Z ) = 1/128.9, and G F is the Fermi The current experimental upper bounds are given by [62][63][64] A. Z boson mass After the spontaneous symmetry breaking of U (1) X by the VEV of ϕ, we find the massive extra gauge boson Z ; where g X is the gauge coupling associated with U (1) X . Note also that we assume vanishing kinetic mixing between U (1) X and U (1) em and U (1) Y .

IV. DARK MATTER
In this section, we discuss our dark matter explaining XENON1T anomaly via inelastic scattering. We also discuss consistency of the model considering relic density, lifetime of excited DM and connection to neutrino mass.

A. Couplings with electrons
In order to fit the data of XENON1T excess, our DMs χ I,R have to interact with electrons.
Such interaction is realized by Z extra gauge boson through one-loop diagram in Fig. 2.
Here, we show relevant effective Lagrangian as follows: [65,66] . Then, inelastic scattering cross section χ I e → χ R e is given by where we have assumed to be m χ < m Z . Here number of excess events observed by XENON1t is N ex 50.
where N ν ∼ 3 is the effective neutrino number, and the process of χ R → χ I eē is kinematically forbidden due to ∆m 2 ∼ O(keV). We can also neglect the other processes such as χ R → χ I 3γ 2 that is suppressed enough. From this constraint, we find f O(0.01)-O(0.1) at M L = 100 GeV depending on mass of Z .

C. Relic density of DM
Since the mass difference between the two dark matters is very small, we can consider the DM as complex scalar field when calculating the relic density of DM. The corresponding DM relic density is obtained by 1 This process would be bounded from SuperKamiokande that its lifetime be O(10 24 ) second, but this bound is valid only for the case where the missing energy of neutrino is greater than O(0.1) GeV. Thus, this even stronger bound is not needed to be applied to our model. 2 Notice here that χ R → χ I γ and χ R → χ I 2γ are forbidden or highly suppressed.
where g * is the effective number of relativistic degrees of freedom, and x f = m χ /T f . Here we take √ g * ∼ 3.8 and x f ∼ 10 as inputs. We have several processes to explain the relic density of DM, but we select one process of χχ * → ν i ν j /ee/µµ via Z boson s-channel exchange.
Since these processes are induced at one-loop level, we make use of the resonant effect around the pole at 2m χ ≈ m Z . In expansion in terms of relative velocity of DM, the cross section is approximately given by where f are all the lepton flavors, kinematical condition is implicitly imposed, and we have assumed massless limit for final lepton masses for simplicity. Notice here that this process can be evaded from the constraint of CMB because of p-wave dominant [67].
Hereafter we mention the other processes, which is potentially taken into consideration.
The cross section via Yukawa coupling f is induced by a process of χχ * → ¯ (νν) whose formula is given by |f | 4 Therefore, the typical cross section is about 10 −14 GeV −2 at m χ = 1 GeV. This is very small compared to 10 −9 GeV −2 that gives 0.12 relic density of DM.
In addition we kinematically forbid the process of χχ * → Z Z via t-channel by assuming m χ m Z , since this process is s-wave dominant that gives stringent constraint from CMB.
Note also that the process of χχ * → ϕZ is basically allowed due to the p-wave dominant and might be found in an appropriate cross section, but we here neglect this process for simplicity. We can always turn this cross section off, by assuming m χ m ϕ , m Z .

D. Analysis
We show our result to accommodate all the constraints as we discussed above, where we also consider the constraint of BaBar [68] and NA64 [69]. Instead of global analysis, we express some benchmark points in order to clearly find tendency of our solution. In GeV and g = √ 4π that is perturbative limit; here we assume universal f for illustration.
Each of the red, blue, and green line represents the cross section to satisfy the relic density from Eq. (7) as where E does not mix with the SM charged leptons thanks to a remnant Z 2 symmetry. The heavier neutral mass matrix, which is Majorana mass matrix, is found as where M D = gv/ √ 2,M D =gv/ √ 2 and M N LL(RR) = h L(R) v ϕ / √ 2. We then rewrite fields by n R ≡ X 1 , n c L ≡ X 2 , N R ≡ X 3 and N c L ≡ X 4 . Then the mass matrix can be diagonalized by acting a unitary matrix as where ψ 0 α is the mass eigenstate.