A modular $A_4$ symmetric model of dark matter and neutrino

We propose a model based on modular $A_4$ symmetry containing a dark matter candidate and realizing radiatively induced neutrino mass at one-loop level. One finds that stability of dark matter candidate can be assured by nonzero value of modular weight and its mass tends to be much smaller than the other two masses in the triplet fields under the $A_4$ group. Therefore we clearly identify single dark matter field in this kind of model. Then we discuss several phenomenological aspects.


I. INTRODUCTION
The standard model (SM) of particle physics has been successfully confirmed by the experimental data including the discovery of the Higgs boson at the Large Hadron Collider (LHC). However physics beyond the SM is also indicated by some issues such as existence of dark matter (DM), non-zero tiny neutrino masses and origin of flavor structure. In describing physics beyond the SM, symmetry would be a key aspect as the SM is based on the gauge symmetry. In fact an additional symmetry such as discrete Z 2 can guarantee stability of DM and forbid neutrino mass generation at tree level, and it is often used in construction of radiative seesaw model [1]. Furthermore non-abelian discrete symmetries have been applied to explain flavor structure in the SM [2][3][4][5][6][7][8][9].
Recently an interesting framework of symmetry has been considered in which modular group is applied and non-abelian discrete symmetries are obtained as their subgroups [10][11][12]. Then some works have been done applying the framework to flavor structure of leptons and/or quarks in S 3 [13,14], S 4 [14][15][16], A 4 [13,[17][18][19][20][21], and A 5 [22,23]. One interesting feature of the framework is that couplings can be written as modular forms which are functions of modulus τ and transform non-trivially under the modular group. Then we can use such a non-trivial structure of couplings to restrict interactions in a model. Furthermore we do not need many scalar filed to break non-abelian discrete symmetries called flavon, breaking the symmetry by vacuum expectation value (VEV) of modulus τ .
In this letter, we apply the framework of modular A 4 symmetry to construct a radiative seesaw model including a DM candidate. The right-handed neutrinos are introduced as triplet of A 4 with modular weight −1. We also introduce some scalar fields such as an isospin doublet field and the SM singlet fields with non-zero modular weight to realize neutrino mass generation at one loop level. Interestingly assignment of modular weight plays a role of Z 2 symmetry in realizing radiative seesaw mechanism and guaranteeing stability of DM.
Moreover we obtain characteristic mass relation among three right-handed neutrinos since it is assigned as triplet under modular A 4 . We discuss phenomenologies of our model such as mass spectrum of right-handed neutrinos, relic density of DM and generation of the active neutrinos.
This letter is organized as follows. In Sec. II, we give our model set up under modular A 4 symmetry. Then, we discuss right-handed neutrino mass spectrum, lepton flavor violation  Lepton couplings Higgs terms  (LFV), relic density of DM and generation of the active neutrino mass at one loop level.
Finally we conclude and discuss in Sec. III.

II. MODEL
Here we explain our model with modular A 4 symmetry in which some fields have non-zero modular weight and couplings with non-zero modular weight are modular forms. First of all, we introduce three right-handed neutrinos as a triplet under A 4 and with modular weight -1, where all the SM leptons have zero modular weight and assigned three kinds of singlet 1, 1 ′ , 1 ′′ for each flavor under the A 4 group. In the scalar sector, we introduce an isospin doublet field and two singlet fields (η, ϕ, S) having non-zero modular weight (-1,-2,-3), where all the scalar fields are true singlet under A 4 and we assume S to be real for simplicity. We expect that S and η are inert boson to forbid the tree level neutrino mass. Each VEV of H and ϕ is denoted where H is identified as SM-like Higgs. ϕ plays a role in inducing the appropriate mass of η, as we will discuss later. We summarize fields assignments in table I and couplings in table II. In general remarks of couplings, we remind that any couplings with A 4 singlets 1, 1 ′ , 1 ′′ have to start from the number of modular wight k = 4, 6, · · · , while any couplings with A 4 triplet start from k = 2, 4, 6, · · · . This fact is understood as follows; modular form with k = 2 is possible only when the associated coupling is triplet under A 4 and a coupling with 1, 1 ′ , 1 ′′ can be constructed by a tensor products of k = 2 and A 4 triplet ones (pure singlet 1 can be also considered as usual coupling constant which is not modular form without non-zero modular weight). In the framework, an interaction is invariant under modular symmetry when sum of modular weights for each associated fields and coupling is zero and it is invariant under A 4 symmetry. Then, interestingly, modular weight can play a role of Z 2 symmetry restricting interactions and stabilizing DM since any coupling should have even weight and odd number of odd weight fields in an interaction is forbidden. Thus even/odd modular weight correspond to Z 2 even/odd for the fields in the model.
Under these symmetries, one writes renormalizable Lagrangian as follows: where ω = e i 2 3 π and the charged-lepton mass eigenstate is directly given by the first term above. Thus, the observed mixing matrix for lepton sector is found in the neutrino sector only. The modular forms of weight 2 (y η 1 , y η 2 , y η 3 ) transforming as a triplet of A 4 is written in terms of Dedekind eta-function η(τ ) and its derivative [11]: The overall coefficient in Eq. (II.2) is one possible choice; it cannot be uniquely determined.
Thus we just impose the purtabative limit y η 1,2,3 √ 4π in the numerical analysis. It implies that the mass hierarchy among right-handed neutrinos could uniquely be fixed, therefore, one might say that DM candidate is determined by the structure of the modular function.
In the similar way as Yukawa couplings, M is also written by y η i (k) such that where M 0 can be taken as a free parameter determining scale of right-handed neutrino mass.
Thus mass relation among three right-handed neutrinos are given once we fix modulus τ .
Higgs potential is given by where µ H , µ 1 µ 2 , λ H have zero modular weight. Here ϕ plays a role in inducing the mass of η through µ ϕη , after it develops a VEV v ϕ . Clearly, we can derive the inert conditions for S and η 0 ≡ (η R + iη I )/ √ 2, and appropriate masses can be found. In the CP even inert bosons in basis of S, η R , which mixes each other due to µ HηS , we define the mixing and its mass eigenvalue as follows: where s a (c a ) is the short-hand symbol of sin a(cos a).
After the electroweak spontaneous symmetry breaking, the charged-lepton mass matrix is given by while the right-handed neutrino mass matrix is given by where M 1 ≡ M X is expected to be the mass of DM candidate, U N is an unitary matrix, and free real parameter are M 0 and complex τ .
Dark matter candidate is expected to be the lightest fermion X R among three-right handed neutrinos and its mass is denoted by M X . The valid Lagrangian is given by Hereafter, we assume the masses of η R , η I , η ± are same and symbolized by m η for simplicity.
Then the valid cross section to explain the relic density of DM is p-wave dominant in the expansion of relative velocity and its form is given by [27] (II.14) Neutrino mass matrix is generated at one-loop level as Fig. 2 and its formula is given by with loop integration factor 1 where we have assumed the mass insertion approximation; m H 1 ≈ m S and m H 2 ≈ m η R (s a << 1). Then the neutrino mass matrix is diagonalized by an unitary matrix U P M N S as In light of the numerical analysis; M X << M 2 < M 3 , we rewrite D N as follows: in the case of M X << M 2 . Since the second term gives small contribution to the neutrino mass matrix, we approximately rewrite the neutrino mass matrix as follows: If we take M 2 = 10 TeV and (m H 1 , m H 2 ) = (4, 1) TeV, we find F (M 2 ) ≈ 10 −15 . While m ν ≈ O(10 −11 ) GeV is obtained, inserting the observed neutrino mass-squared differences and their mixings [29]. Thus, we get the typical value of µ 2 HηS Y Y T to satisfy the neutrino masses as 9.4 × 10 −3 GeV 2 . In this letter, we have not completely discussed whether our neutrino mass matrix can properly reproduce the two neutrino mass-squared differences and their mixings. Actually, this is quite nontrivial for this realization, because our mass matrix is strongly restricted by the A 4 symmetry. The key of this realization is the mixing of M N , since only this mixing matrix is nontrivially obtained by Eq. (II.7). In fact, we would need tuning of values of modulus τ to fit the neutrino data in our construction. Even if our neutrino mass matrix would not reproduce the neutrino oscillation date, we can extend the Majorana mass sector so that nonzero diagonal elements are switched on. The way of this modification is easily carried out by introducing Yukawa couplings with (1, 1 ′ , 1 ′′ ) and 4 under A 4 and modular weight assignments. Then we can add the following Yukawa terms: After ϕ developing VEV, we will find the diagonal elements of M N and reproduce the neutrino oscillation data more easily. We will analyze this issue with more complete way in future work.