A ug 2 01 9 KIAS-P 19048 , APCTP Pre 2019-022 A modular A 4 symmetric scotogenic model

A modular A4 symmetric scotogenic model Takaaki Nomura, ∗ Hiroshi Okada, † and Oleg Popov ‡ School of Physics, KIAS, Seoul 02455, Republic of Korea Asia Pacific Center for Theoretical Physics (APCTP) Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784, Korea Institute of Convergence Fundamental Studies, Seoul National University of Science and Technology, Seoul 139-743, Korea (Dated: August 21, 2019)


I. INTRODUCTION
Particle physics experiments and observations have been successfully confirming the standard model (SM) of particle physics. On the other hand, there are some issues indicating an existence of physics beyond the SM such as existence of dark matter (DM), non-zero tiny neutrino masses and origin of flavor structure. In describing these issues, symmetry would play an important role like guaranteeing stability of DM, forbidding neutrino mass at tree level and restricting flavor structure. It is thus interesting to construct a model of physics beyond the SM adopting a new symmetry.
In this paper, we apply modular A 4 symmetry in minimal Scotogenic model [33] in which neutrino mass is generated at one-loop level and DM candidates are contained. We introduce the right-handed neutrinos as triplet of A 4 assigning modular weight −k = −2. Also, nonzero modular weight is assigned to inert Higgs doublet. Interestingly we find that additional Z 2 symmetry is not necessary to realize structure of Scotogenic model due to the nature of modular form. Then numerical analysis for neutrino mass matrix is carried out to show predictions of our model as a result of modular A 4 symmetry.
Manuscript is organized as follows. In Sec. II, we give our model set up under A 4 modular symmetry. We discuss right-handed neutrino mass spectrum, lepton flavor violation (LFV) and generation of the active neutrino mass at one-loop level in Sec. III. Numerial analysis is presented in Sec. IV. Finally, we conclude and discuss in Sec. V.

Bosons
in the lepton and boson sector, where −k is the number of modular weight and the quark sector is the same as the SM.
Couplings Under these symmetries, we write renormalizable Lagrangian as follows: whereη ≡ iσ 2 η * , σ 2 being second Pauli matrix, and charged-lepton matrix is diagonal thanks to the unique representation of A 4 .
The full symmetry of the leptonic sector of the model is SU(2) L × U(1) Y × A 4 , where the Z 2 symmetry of [33] is replaced with modular symmetry Γ 3 ≃ A 4 . A 4 serves three purposes: flavor symmetry, scotogenic symmetry, and dark matter stabilizing symmetry.
The modular forms of weight 2, (y 1 , y 2 , y 3 ), transforming as a triplet of A 4 are written in terms of Dedekind eta-function η(τ ) and its derivative η ′ (τ ) [2]: The overall coefficient in Eq. (II.2) is one possible choice; it cannot be uniquely determined.
Then, any couplings of higher weight are constructed by multiplication rules of A 4 , and one finds the following couplings: Higgs potential of our model is equivalent to the potential of the Scotogenic model [33] without loss of generality, where a quartic coupling that plays the role in generating the nonzero neutrino masses is given by Y term that was forbidden in the inert two higgs doublet model (2HDM) by Z 2 invariance is now forbidden by modular invariance via A 4 . This is due to the fact that A 4 singlet modular form with modular weight 2 can not exist.
The right-handed neutrino mass matrix is given by 3,3 ). Then, the Majorana mass matrix is diagonalized by an unitary matrix as D N ≡ UM N U T , and their mass eigenstates are defined by ψ R , where The Dirac Yukawa matrix is given by where we impose the purtabative limit Max[y η ] √ 4π in the numerical analysis.
In this section we analyze lepton flavor violation and neutrino mass formulating analytic forms of branching ratio (BR) of ℓ i → ℓ j γ process and neutrino mass.
Neutrino mass matrix at one-loop level is given by where m I and m R are, respectively, masses of imaginary and real parts of neutral η.
Then, the neutrino mass matrix is diagonalized by the PMNS unitary matrix, U P M N S , as U P M N S m ν U T P M N S =diag(m ν 1 , m ν 2 , m ν 3 )≡ D ν , since the charged-lepton mass matrix is diagonal. Here Tr[D ν ] 0.12 eV is given by the recent cosmological data [42]. Each of mixing is given in terms of the component of U P M N S as follows: (III.5) Also, the effective mass for the neutrinoless double beta decay is given by m ee = |D ν 1 cos 2 θ 12 cos 2 θ 13 + D ν 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + D ν 3 sin 2 θ 13 e i(α 31 −2δ CP ) |, (III. 6) where its observed value could be measured by KamLAND-Zen in future [43].
To achieve numerical analysis, we derive several relations of the normalized neutrino mass matrix as follows:m where the last line is the first order approximation of the small mass difference between m 2 R and m 2 I ; m 2 R − m 2 I = ∆m 2 . 2 Then, the normalized neutrino mass eigenvalues are given in terms of neutrino mass eigenvalues; diag(m 2 It is found that k 2 3 is given by where normal hierarchy is assumed and ∆m 2 atm is the atmospheric neutrino mass difference square. Comparing Eq.(III.7) and Eq.(III.8), we can rewritten ∆m 2 in terms of other parameters as follows: The solar neutrino mass difference square is also found as In numerical analysis, this value should be within the experimental result, while ∆m 2 atm is expected to be input parameter.

IV. NUMERICAL ANALYSIS
We show numerical analysis to satisfy all of the constraints that we discussed above, where we assume m R ≈ m I ≈ m η ± to avoid the constraint of oblique parameters. Also, we impose  Note that DM candidate in our scenario is inert scalar η R since N 1 is much heavier. The mass value of m R ∼ 530 GeV can accommodate with observed relic density of DM via gauge interaction taking into account coannihilation processes [46]. We also need to assume small Higgs portal coupling to avoid DM direct detection constraints. In principle, our DM phenomenology is the same as of the canonical inert Higgs doublet model and we do not discuss it in this paper.

V. CONCLUSION AND DISCUSSION
We have studied a model based on modular A 4 symmetry in which neutrino masses are generated radiatively at one-loop level via minimal Scotogenic scenario. The modular A 4 symmetry plays a role of restricting interactions, generating neutrino mass and stabilizing DM candidate. We have formulated lepton flavor violation and neutrino mass matrix in the model. Then numerical analysis has been carried out to find prediction of our scenario. In our numerical analyses, we have highlightend several remarks as follows: 1. Three mixings cover all the experimental results by 3σ interval, but the sum of neutrino masses are the range [0.06,0.105] eV that is near to the upper bound of cosmological data of 0.12 eV.
2. Dirac CP and α 31 phases also cover all the region, but when we fix δ ℓ CP = 270[deg] that is favored by T2K, then α 31 is determined to be 180 [deg]. Here, α 21 is zero.
3. We found the following regions; 0.002 m 1 0.016 eV and 0.005 m ee 0.018 eV which can be seen from Fig. 3.
These predictions will be tested in the near future. The DM candidate in our scenario is inert scalar boson and its mass is chosen to be ∼ 530 GeV. DM phenomenology is the same as of the canonical inert Higgs doublet model and our DM mass value can accommodate the observed relic density of DM via gauge interaction taking into account coannihilation processes. We also require small Higgs portal coupling to avoid DM direct detection constraints, which can be easily realized by choosing parameters in the potential.