Muon $g-2$ in $U(1)_{\mu-\tau}$ Symmetric Gauged Radiative Neutrino Mass Model

We explore muon anomalous magnetic moment (muon $g-2$) in a scotogenic neutrino model with a gauged lepton number symmetry $U(1)_{\mu-\tau}$. In this model, a dominant muon $g-2$ contribution comes not from an additional gauge sector but from a Yukawa sector. In our numerical $\Delta \chi^2$ analysis, we show that our model is in favor of normal hierarchy with some features. We demonstrate one benchmark point, satisfying muon $g-2$ at the best fit value $25.1\times10^{-10}$.

In this paper, we propose a scotogenic neutrino model with U (1) µ−τ gauge symmetry that explains muon g − 2 by the Yukawa sectors not by the new gauge sector. In order to get the sizable muon g − 2, we need a diagram without chiral suppression and we introduce several exotic fermions and scalars. These several exotic fermions can also play a role in radiative neutrino mass generation at one-loop level [31]. Heavier neutral fermions run inside the loop and the lightest one could be a promising dark matter (DM) candidate. Finally, we show several features of our model by demonstrating the numerical ∆χ 2 analysis.
This paper is organized as follows. In Sec. II, we present the model set up and how to generate neutrino masses in normal and inverted hierarchy. In Sec. III, we discuss constraints from lepton flavor violations (LFVs) and address the muon anomalous magnetic moment. We carry out numerical ∆χ 2 analysis and present the allowed region satisfying the neutrino oscillation data, LFVs and muon g − 2. Conclusions and discussions are given in Sec. IV, briefly mentioning a possibility of DM candidate and how to explain the correct relic density, satisfying direct detection bounds.

Leptons
Fermions In this section, we set up our model Lagrangian and focus on lepton sector and Higgs sector which are crucial for generating neutrino mass and muon g-2 contribution. And we address detail discussion for the neutrino mass matrix. symmetry. In addition, we impose Z 2 odd for new fermions where even is assigned to the SM fermions. The Z 2 symmetry plays a role in generating the active neutrino mass matrix not at treelevel but one-loop level. Furthermore, the lightest field with odd Z 2 can be a dark matter candidate.
Notice here that the U (1) µ−τ anomaly is independently canceled among the SM fermions or N R . In Under these symmetries, our renormalizable Lagrangian is given by where σ 2 is the second Pauli matrix and the charged-lepton mass matrix is diagonal due to the µ−τ after the phase redefinition. Therefore, the neutrino oscillation data is induced via neutrino sector.

B. Higgs sector
Our Higgs potential is also given by and V tri 2 and V tri 4 are respectively trivial quadratic and quartic terms of the Higgs potential; Notice here that h + , z 0 , and z ϕ are respectively absorbed by the longitudinal degrees of freedom in gauge sectors. Consequently, we have massive gauge bosons W ± , Z in the SM and Z in the U (1) µ−τ gauge symmetry. The λ 0 term plays an important role in generating the non-vanishing neutrino mass matrix. In our model, the neutrino mass matrix is proportional to the mass-squared difference between η R and where m R,I is the mass eigenstate of η R,I [31]. Even though there is mixing between χ ± and η ± from λ 0 , we suppose that the mixing is negligibly tiny. 2 After the phase redefinition of the neutral fermions, the mass matrix in basis of is found as follows: |y | are real mass parameters, while ξ, ζ are physical phases. The mass matrix M N is then diagonalized by introducing a unitary matrix V . This matrix satisfies Here, the mass eigenstate ψ R is defined by N R i = 5 k=1 V ik ψ R k , and its mass eigenvalue is defined by M k (k = 1, 2, 3, 4, 5). Then, the valid Lagrangian is rewritten in terms of mass eigenstates as follows 3 : where F is three by five matrix, and G is two by five matrix, therefore a runs over e, µ. The first term of Eq. (II.5) contributes to the neutrino mass matrix, while the other terms induce LFVs, and muon g − 2 as can be seen below. The active neutrino mass matrix is given by [31] ( where we assume to be λ 0 v 2 H = m 2 R − m 2 I m 2 0 ≡ (m 2 R + m 2 I )/2 in the second line, and m ν is diagonalzied by a unitary matrix U PMNS [33]; D ν ≡ U T PMNS m ν U PMNS . Here, we define dimensionless 3 Notice here that hνµLE C L χ − does not contribute to the neutrino mass matrix, since E C L cannot propagate in the loop.
neutrino mass matrix as m ν ≡ (λ 0 v H )m ν ≡ κm ν . Since κ does not depend on the flavor structure, we rewrite this diagonalization in terms of dimensionless form (D ν 1 ,D ν 2 ,D ν 3 ) ≡ U T PMNSm ν U PMNS . Thus, we fix κ by where ∆m 2 atm is the atmospheric neutrino mass-squared difference. Here, NH and IH stand for the normal hierarchy and the inverted hierarchy, respectively. Subsequently, the solar neutrino mass-squared difference is depicted in terms of κ as follows: This should be within the experimental value. The neutrinoless double beta decay is also given by m ee = κ D ν 1 cos 2 θ 12 cos 2 θ 13 +D ν 2 sin 2 θ 12 cos 2 θ 13 e iα 2 +D ν 3 sin 2 θ 13 e i(α 3 −2δ CP ) , (II.10) which may be able to observed by KamLAND-Zen in future [34].

C. Numerical analysis
We show our numerical ∆χ 2 analysis to satisfy the neutrino oscillation data, LFV (µ → eγ only in our case) as well as muon g − 2.
Here we fix ∆a µ as BF value 25.1 × 10 −10 , where we adopt five known observables sin 2 θ 12 , sin 2 θ 23 , sin 2 θ 13 , ∆m 2 atm , ∆m 2 sol in Nufit 5.0 [42] at 3σ confidence level. Notice that we do not include the Dirac CP phase in ∆χ 2 analysis because of big ambiguity at 3σ interval. Furthermore, we employ Gaussian approximations for charged-lepton masses. However since we have found the allowed region within 4.59 ∆χ 2 in case of IH, it does not satisfy the sizable muon g − 2 whose maximum order is 10 −12 in our model. Thus, we focus on NH only. At first, we fix the range of our input parameters as follows: {f In Fig. 1, we show the scatter plots of sum of neutrino masses m i in terms of sin 2 θ 23 (left), neutrinoless double beta decay m ee in terms of sin 2 θ 23 (center), and m ee in terms of m i (right). Here, each of the color plots of green, yellow, and red represents the region at σ of 0-1, 5 In the case of ge = 0, both BR(µ → eγ) and muon g − 2 are proportional to Re[hg * µ ] that does not depend on the flavor structure. Thus, we can take any values for h and gµ in our numerical ranges as far as Re[hg * µ ] does not change. In our benchmark points in Table III,     Before closing our discussion, we briefly mention a DM candidate. Basically, we have two candidates; bosonic and fermionic one.
In case of bosonic DM candidate, the lightest particle of η R and η I is the one. In case of fermionic DM candidate, ψ R 1 is the one. In the case of fermionic DM candidate, we do not need to consider the bound on direct detection, because it does not interact with quark sector directly. The dominant cross section arises from G and H terms that also appear in muon g − 2 and have s-wave dominant.
But this contribution is at most O(5 × 10 −12 ) GeV −2 that is too small to resolve the correct relic density. Thus, our promising DM candidate is bosonic. Here, let us suppose η R to be DM, and we simply neglect any interactions coming from Higgs potential in order to evade any bounds from direct detection experiments. 6 Moreover, we assume that the mass difference between η R and η I is more than 200 keV in order to evade the inelastic direct detection bounds via Z boson portal [43].
Similar to the result of fermionic case, we cannot rely on any contributions from Yukawa couplings because these constributions are too tiny to explain the relic density. Thus, we have to make the use of kinetic interactions from the SM. In this situation, nature of the DM is seriously analyzed by e.g. Ref. [45]. The solutions are uniquely found at the points of half of the Higgs mass; ∼ 63 GeV and 534 GeV where coannihilation processes such as η R η I → Z →f f are taken in consideration.