A Two Loop Radiative Neutrino Model

We explore the possibility to explain a bosonic dark matter candidate with a gauge singlet inside the loop to generate the neutrino mass matrix at two-loop level. The mass matrix is suppressed by a small mixing that comes from the bound on {direct detection experiments of the dark matter, and equivalent of the three-loop neutrino model due to the small mixing between neutral inert bosons. Here, our setup is} the Zee-Babu type scenario with $Z_3$ discrete symmetry, in which we consider the neutrino oscillation data, lepton flavor violations, muon $g-2$, $\mu-e$ conversion rate, lepton flavor-changing and conserving $Z$ boson decay and bosonic dark matter candidate.


I. INTRODUCTION
Radiatively induced neutrino masses is one of the promising scenarios which make strong correlations between neutrinos and any fields that are introduced inside loops. If a dark matter (DM) candidate is introduced in the model, its testability is enhanced due to the fact that parameter space is strongly constrained by neutrino data. Especially two-loop induced models that we will focus on in this paper have been widely studied in various aspects .
In this paper, we study the muon anomalous magnetic moment, various lepton flavor violations (LFVs), and DM phenomenology in the framework of Zee-Babu type of neutrino model, emphasizing µ − e conversion rate in T itanium nuclei that will be tested in the near future experiment such as PRISM/PRIME [36]. Despite a two-loop model, we will also show that the scale of neutrino mass in our model is equivalent to a three-loop model. It is due to the small mixing between neutral bosons dictated by the direct detection bound of the DM candidate.
This paper is organized as follows. In Sec. II, we introduce our model, including neutrino sector, LFVs, muon anomalous magnetic moment and lepton flavor-changing and conserving Z boson decay. In Sec. III, we present our numerical analysis and identify regions consistent with the current experiments. We conclude and discuss in Sec. VI.

II. MODEL SETUP
We introduce three families of iso-spin singlet vector-like neutral fermions N i (i = 1, 2, 3), iso-spin doublet scalar η, an isospin singlet neutral scalar χ and charged scalars χ ± in addition to the SM fields. We impose a discrete Z 3 symmetry on all the new particles (N, η, χ, χ ± ) in order to assure the stability of DM (χ in our case). The particle contents and their charge assignments under SU(2) L × U(1) Y × Z 3 are shown in Tab. I 1 . Thus we expect that only the SM-like Higgs Φ has a vacuum expectation value (VEV), which is denoted by Φ 0 = v/ √ 2 [21]. Then the relevant Lagrangian and scalar potential respecting the symmetries are given by where σ 2 is the second Pauli matrix, i, j = 1−3, and the first term of L Y can generate the SM charged-lepton masses m ℓ ≡ y ℓ v/ √ 2 (ℓ = 1 − 3) after the electroweak symmetry breaking.
Both N i and e i can be considered as the mass eigenstates without loss of generality. For simplicity we assume all the parameters in (II.1) are real and positive.
The scalar fields can be parameterized as follows: where η 0 and χ are complex inert neutral bosons, η ± and χ ± are the singly charged bosons, v ≃ 246 GeV is VEV of the Higgs doublet, and φ is the SM Higgs boson with mass m φ ≈ 125.5 GeV. To ensure the stability of DM, the following condition should be at least satisfied: Notice here that we have mixing between inert bosons through µ ηχ and µ, the resulting mass eigenvalues and their rotation matrices are obtained by where (H 1 (2) , H ± 1(2) ) and (α, β) can be written in terms of the parameters in the scalar potential, and we use the short hand notation s α(β) and c α(β) for sin α(β) and cos α(β) below 2 .
DM candidate: The lightest neutral scalar H 1 is our DM candidate. Here we consider constraints on H 1 . As for the direct detection experiment, the dominant elastic scattering cross section comes from Z-boson portal through mixing and found as where m p is a proton mass, s 2 w ≈ 0.23 is the Weinberg angle and G F is the Fermi constant. Note here that s 4 α comes from the kinetic term of η, D µ η † D µ η, where the covariant derivative D µ includes the SM gauge boson Z. Since Z-boson couples only to the isospin doublet scalar, in the effective H 1 − H 1 − Z coupling only η component of H 1 couples to Z-boson. This leads to s 2 α suppression in the effective coupling and s 4 α suppression in the cross section for the DM scattering off the nuclei. In the experiment of LUX [37], the typical upper bound on the cross section is σ SI 10 −45 cm 2 at m H 1 = O(100) GeV. Then the required condition on α is given by 3 (II. 8) It implies that the dominant component of the DM candidate is the gauge singlet boson χ.
Hereafter we will neglect any terms proportional to s 4 α . To explain the relic density, we rely on the resonant effect via s-channel of the SM-Higgs, and we consider only the annihilation 2 See ref. [21] for scalar mass spectra in more details. 3 If we consider the contribution of the Higgs portal [38,40] to the direct detection, the constraint is given Thus one can satisfy the bound of direct detection by tuning the Higgs trilinear and quartic couplings. processes, since our DM is gauge singlet 4 . In this case, to satisfy the relic density Ωh 2 ≈ 0.12, the DM mass should be around the Higgs resonance region m H 1 ≈ m φ /2 ≈ 63 GeV [40].

A. Neutrino masses
The active neutrino mass matrix m ν is generated at two-loop level as shown in fig. 1, and its formula is given by where R N is a parameter with a mass dimension and depends on the parameters we assume y N ≡ y N R ≈ y N L . As we mentioned in Section I, although they are generated at two-loop level, the neutrino masses scale like three-loop model due to s α suppression.
Then the active neutrino mass matrix (m ν ) ij can be diagonalized by the Pontecorvo-Maki- We assume the neutrino masses are normal ordered, neglect the Majorana phases, and fix the Dirac phase δ = −π/2 in the numerical analysis for simplicity. Then we apply the generalized Casas-Ibarra parametrization 5 to our analysis which use the observed neutrino oscillation data with global fit [42]. We impose i=1−3 m ν i < 0.12 eV at 95% C.L. as reported by Planck collaboration [43]. The Yukawa coupling y η can be rewritten in terms of the following parameters: that satisfy OO T = O T O = Diag(1, 1, 1) 6 and R Ch N is Cholesky decomposed matrix. This matrix is a lower triangular matrix and satisfies the following relation (see appendix A), Lepton flavor violations(LFVs) at one-loop level arise from the terms with y η and y χ as shown in the left panel of fig. 2 7 . Between two classes of LFV modes ℓ i → ℓ j γ and ℓ i → ℓ j ℓ k ℓ ℓ , the former tends to give more stringent bounds on the related couplings and masses [44]. Thus we consider only this mode below. The model prediction for the radiative 5 In our case, the central matrix is not diagonal but the symmetric matrix which is proportional to y N . 6 In case the family number of N R is two, O(θ) is an arbitrary 3 × 2 matrix with a complex value θ that satisfies OO T = Diag(0, 1, 1) and O T O = Diag(1, 1). The numerical result for the three families does not change much from this case. 7 Recently sophisticated analysis has been done by refs. [46,47]  decay channel in our case is given by R is obtained approximately to be 8 where Z, Z eff , F (q), and R are given in table II.
Here let us define Y ≡ R BR(µ→eγ) , since their flavor structures are same. Then it is given by Depending on the nuclei, Y ≈ O(0.1), as listed in table II. It suggests that the constraint from the µ − e conversion is always satisfied once we satisfy the constraint of µ → eγ. We will discuss the sensitivity of future experiments, R T i and R Al , in the numerical analysis.
New contribution to the muon anomalous magnetic moment (muon g −2), whose diagram is displayed in the left panel of fig. 2, is given by where only the terms (a ηχ R(L) ) µµ are positive contributions to the muon g − 2. Thus we expect a η R(L)µµ , a χ R(L)µµ ≪ (a ηχ R/L ) µµ to explain the discrepancy between the experimental results and the theoretical predictions which is of order of O(10 −9 ) [50].

III. NUMERICAL ANALYSIS
For the numerical analysis, we generate input parameters randomly in the following ranges: where i, j = 1, 2, 3, y N is a symmetric matrix, θ 1,2,3 are arbitrary complex values in the Casas-Ibarra parametrization. We fixed m H 1 = m φ /2. The lower bound on H ± 1,2 , 80 GeV, comes from the LEP experiment [48,64]. In addition, the LHC gives a mass bound for the charged boson. Especially, the SU(2) L originated charged boson would have a feature similar to the slepton in the supersymmetric model, since it decays into a charged lepton and missing energy. The lower bound of the mass from the CMS collaboration is 450 GeV [65].
Hence we might apply this bound for our case, although the detail analysis is beyond our scope of this paper. Notice here y η should satisfy the perturbative limit; y η √ 4π.
In fig. 4, we show scatter plots BR(τ → eγ)(red) and BR(τ → µγ)(blue) as a function of BR(µ → eγ). As for muon g − 2, the maximum value is at most 5×10 −15 , which is much smaller than the current discrepancy. This is because the positive contribution comes from the mixing term between y η and y χ only.

IV. SUMMARIES AND DISCUSSIONS
We have explored the possibility to explain bosonic dark matter candidate with a gauge singlet inside the loop to generate the neutrino mass matrix at two-loop level. Here, our setup is the Zee-Babu type scenario with Z 3 discrete symmetry, in which we have considered the neutrino oscillation data, DM, and lepton flavor violations.  First of all, we have found the upper bound on s α to be of the order 10 −2 from the direct detection experiment. Thus the only solution to satisfy the observed relic density is to use the SM Higgs resonance with the DM mass around the half of the Higgs mass, m H 1 ≈ m φ /2.
Second, the neutrino mass matrix is reduced by not only the two-loop suppression but also s 2 α ≈ 10 −3 suppression that comes from the direct detection bound of the DM. As a result, the scale of the neutrino masses is equivalent to that of the three-loop neutrino model.
We have found that the positive muon g − 2 thanks to χ ± . But its typical value O (10 −14 ) in our global analysis is too small to explain the ∼ 3σ discrepancy of the muon g −2 between the experiment and the SM.
We briefly mention the possibility to detect our new particles at the LHC or the ILC.
In these kinds of radiative seesaw models, they tend to have large Yukawa couplings in the lepton sector. Therefore, the effect of LFV processes can be large. On the other hand, the bounds from the LHC experiments are typically weaker than the LFV constraints, since new scalar bosons do not couple to quarks. Hence, in this paper we consider only LFV effects. where we assumed all parameters to be real and positive.