A radiatively induced neutrino mass model with hidden local $U(1)$ and LFV processes $\ell_i \to \ell_j \gamma$, $\mu \to e Z'$ and $\mu e \to e e$

We investigate a model based on hidden $U(1)_X$ gauge symmetry in which neutrino mass is induced at one-loop level by effects of interactions among particles in hidden sector and the Standard Model leptons. Neutrino mass generation is also associated with $U(1)_X$ breaking scale which is taken to be low to suppress neutrino mass. Then we formulate neutrino mass matrix, lepton flavor violating processes and muon $g-2$ which are induced via interactions among Standard Model leptons and particles in $U(1)_X$ hidden sector that can be sizable in our scenario. Carrying our numerical analysis, we show expected ratios for these processes when generated neutrino mass matrix can fit the neutrino data.


I. INTRODUCTION
A mechanism of generating non-zero neutrino masses is one of the important open questions in particle physics that requires extension of the standard model (SM).In particular the tininess of neutrino masses would provide a hint for structure of physics beyond the SM.
In a neutrino mass model with a hidden U(1) symmetry radiatively generated Majorana neutrino mass is often associated with spontaneous breaking of such U(1) symmetry.For example, a loop diagram includes Majorana mass term of extra neutral fermion generated by a vacuum expectation value (VEV) of a scalar field which spontaneously breaks a hidden U(1) gauge symmetry [11].In such a realization, small VEV has advantage of suppressing neutrino mass in addition to loop factor.Then tiny neutrino mass can be generated naturally and we would have sizable Yukawa interactions between hidden particles and SM leptons which are associated with neutrino mass generation.Remarkably, these interactions with sizable couplings can provide rich phenomenology such as lepton flavor violating (LFV) processes ℓ i → ℓ j γ, ℓ i → ℓ j ℓ k ll and µe → ee.Furthermore we would have light Z ′ boson from hidden U(1) breaking with small VEV and it can also provide LFV process such as µ → eZ ′ at loop level.
In this paper, we construct a neutrino mass model with hidden sector based on local U(1) X symmetry.In our model, Majorana neutrino mass is generated at one-loop level where extra scalar boson and fermions propagate inside a loop.Then neutrino masses are suppressed by loop factor and small mass difference between bosons from real and imaginary part of extra scalar field generated by a VEV breaking the local U(1) X .We formulate neutrino mass matrix, LFV processes and muon g − 2 which are induced via interactions among SM leptons and particles in U(1) X hidden sector.Then we perform numerical analysis searching for allowed parameter region and expected ratios for various LFV processes.This paper is organized as follows.In Sec.II, we show our model and formulate neutrino mass generation mechanism, LFVs and muon g − 2 in addition to scalar sector, U(1) X gauge sector, and extra fermion sector.In Sec.III, we carry out numerical analysis searching for allowed parameter sets and estimate ratios of LFV processes and muon g − 2 with these parameters.In Sec.IV, we provide the summary of our results and the conclusion.

II. MODEL
In this section, we introduce our model in which hidden local U(1) X symmetry is introduced.As for new fermion sector, two kinds of vector fermions L ′ and N ′ with the same T is an isospin doublet and N ′ is an isospin singlet.
We assume these two fermions have three families.As for scalar sector, we introduce SM singlet fields S and ϕ whose U(1) X charges are −Q X and 2Q X respectively, in addition to the SM-like Higgs field H.We summarize the charge assignments of the fields in Table I where quark sector is omitted since it is the same as the SM.Among the scalar fields, we require H and ϕ to develop VEVs while S is an inert scalar field without a non-zero VEV.
Under the symmetries in the model, we write the relevant Yukawa interactions and Dirac mass term associated with extra fermions such that where H = iσ 2 H * σ 2 being second Pauli matrix, generation index is omitted, and y ℓ can be diagonal matrix without loss of generality due to the redefinitions of the fermions.The scalar potential is also given by where we assume all couplings are real.

A. Scalar sector
In this subsection, we discuss mass spectrum in scalar sector of the model.Firstly, we consider scalar bosons associated with H and ϕ.The VEVs of the scalar fields, v and v ϕ , are derived by solving the stationary conditions ∂V /∂v = ∂V /∂v ϕ = 0 such that where these values can be taken to be real positive without loss of generality.We then obtain the squared mass terms for CP-even scalar bosons as which can be diagonalized by an orthogonal matrix providing the mass eigenvalues of the form; The corresponding mass eigenstates h and h D are also given by where α is the mixing angle, and h is identified as the SM-like Higgs boson.In our scenario, the VEV of ϕ is taken to be small as O(100) MeV for suppressing neutrino mass as we discuss below.We also assume λ Hϕ ≪ 1 so that mixing angle α is negligibly small to avoid constraints from the SM Higgs measurements.Thus h is almost SM-like Higgs.The CP-odd components of H and ϕ are identified as Nambu-Goldstone bosons absorbed by Z and Z ′ bosons after symmetry breaking.Note that we have remaining Z 2 symmetry after U(1) X symmetry breaking where {L ′ , N ′ , S} are Z 2 odd, while the other fields including SM fields are Z 2 even due to the charge assignment.
We next consider mass spectrum of inert scalar bosons from S. The mass terms after symmetry breaking are given by Thus masses of S R and S I are where mass difference between real and imaginary part of S is induced by coupling µ.In our numerical analysis below, we parametrize the mass difference as ∆m S ≡ m S R − m S I ∝ µv ϕ /µ S .We will take the mass difference to be as small as O(1) eV to O(100) eV, since µ is expected to be small as the corresponding operator breaks global symmetry in the potential and the scale of v φ is also taken to be low.Such a tiny ∆m S suppresses neutrino mass.

B. Z ′ boson
After U(1) X symmetry breaking by the VEV of ϕ, we obtain massive extra gauge boson where g X is the gauge coupling associated with U(1) X .As we take v ϕ = O(100) MeV, the mass of Z ′ is m Z ′ 100 MeV in our scenario.

C. Extra fermion sector
In this subsection, we discuss mass spectrum in extra fermion sector.The mass term of extra charged lepton is given by Dirac mass term of L ′ as follows In our model E does not mix with SM charged leptons due to remnant Z 2 symmetry.After symmetry breaking, mass terms of extra neutral fermions are obtained such that where We then rewrite fields by , and Majorana mass matrix can be obtained as Then the mass matrix can be diagonalized by acting a unitary matrix as where ψ 0 α is the mass eigenstate.

D. Neutrino mass generation
In our model neutrino masses are generated via one-loop diagram shown in Fig. 1.Here we write the Yukawa interactions for neutrino mass generation in mass basis such that where V is the matrix diagonalizing extra neutral fermion matrix discussed above.We then obtain neutrino mass matrix by calculating the diagram as Then, the Yukawa coupling f is rewritten in terms of the other parameters as follows [21]: where O is three by three orthogonal matrix with an arbitrary parameters.Note that R is suppressed by ∆m S and loop factor.Then Yukawa couplings f ia can have sizable values and significantly affect lepton flavor physics.
E. ℓ i → ℓ j γ and muon g − 2 The relevant interaction to induce ℓ i → ℓ j γ lepton flavor violating(LFV) process is obtained from second term of Eq. ( 2) as Considering one loop diagram, we obtain the BRs such that where where we impose these constraints in our numerical calculation.
In addition, we obtain contribution to muon g − 2, ∆a µ , through the same amplitude where m µ is the muon mass.In our numerical analysis, we also estimate the value.
F. Branching ratio of ℓ i → ℓ j ℓ k ll The LFV three body charged lepton decay processes are induced by box-diagram as shown in Fig. 2. Calculating the one-loop diagram, we obtain BR for ℓ i → ℓ j ℓ k ll process such that where Γ ℓ i is the total decay width of ℓ i , N F = 2 for ℓ i → ℓ j ℓ j lj or ℓ i → ℓ k ℓ k lj and N F = 1 for [25].In our numerical analysis, we impose current experimental constraints [26,27]: FIG. 2: The box diagram inducing ℓ i → ℓ j ℓ k ll decay and effective Lagrangian for µe → ee process.

G. µ → eZ ′
In our scenario, Z ′ is light and ℓ i → ℓ j Z ′ processes can be induced, where we focus on µ → eZ ′ since it will be the clearest signal at experiments.Then its relevant interaction process arises from Eq. ( 20) with U(1) X gauge interaction of E and S. The µ → eZ ′ process is obtained by one-loop diagrams as shown Fig. 3.Here we approximate as m S R ≃ m S I and consider S as complex scalar boson in our calculation.Relevant effective Lagrangian is where coefficients are estimated by calculating the diagrams.We then obtain where m e in our mode and µ → eZ ′ process is dominantly induced by g X /Λ R effect.In terms of the effective couplings, the branching ratio is given as In our numerical analysis below, we impose the constraint This bound is obtained from BR(µ → eX) < 2.6 × 10 −6 with massless particle X [28].

H. µe → ee
In our model µe → ee process in a muonic atom [29] is also induced by Eq. ( 20).We then obtain relevant effective interactions from the same diagram inducing µ → eγ and the box-diagram shown in Fig. 2 such that where the coefficient g 4 in our model is derived as Here A L is suppressed by m e /m µ compared to A R .The ratio of µe → ee width and total decay width of muonic atom can be estimated by A L,R and g 4 where we denote the ratio by where τµ is the lifetime of a muonic atom, which is given in Ref. [30].B 1s ℓ (ℓ = µ, e) is the binding energy of the initial lepton ℓ in a 1s state.For simplicity, we take into account only the 1s electrons, which give the dominant contribution.This formula of R µ − e − →e − e − includes the numerical integration by the energy E 1 of one emitted electron.Once E 1 is fixed, the energy of the other emitted electron is determined by e − E 1 due to the energy conservation.J is the total angular momentum of the lepton system, and κ n (n = 1, 2) indicates the angular momentum of each electron.The explicit formulas of W i s (i = L, R, 4) are given in Refs.[31,32].
The R µ − e − →e − e − gets larger in a muonic atom with a larger proton number.In our calculation, we assume the use of muonic lead ( 208 Pb).

III. NUMERICAL ANALYSIS
In this section numerical analysis is carried out where we search for allowed values of free parameters satisfying neutrino data and show ratios for LFV processes as well as muon g − 2 estimated by the allowed parameter sets.
We scan relevant free parameters in our model in the following region: where we fix v ϕ = 100 MeV and Q X = 1.Note that the scales of mass matrix are chosen taking into account the fact that {M D , MD } ∝ v and M N ′ LL(RR) ∝ v ϕ while M L ′ ,N ′ are bare mass parameters.Then we search for the allowed parameter sets which satisfies neutrino Eq. (19).
In Fig. 4, we provide estimated values of BR(ℓ i → ℓ j γ) and ∆a µ for allowed parameter sets showing correlation on {BR(µ → eγ), ∆a µ } and {BR(τ → eγ), BR(τ → µγ)} plain in left-and right-panel.We find that ∆a µ can be up to ∼ 4 × 10 −11 in our model.In Fig. 5, we also show the correlation between BR(µ → eγ) and (g X /Λ L ) 2 + (g X /Λ R ) 2 on the left panel, and that between BR(µ → eγ) and BR(µ → eZ ′ ) on the right panel.Restricted by µ → eγ constraint, the maximal value of (g X /Λ L ) 2 + (g X /Λ R ) 2 is sufficiently lower than current upper bound of Eq. ( 31), and the maximal value of BR(µ → eZ ′ ) is around 10 −13 that could be further tested in future experiments .In Fig. 6, we show some correlations among R µe→ee , Wilson coefficients A R and g 4 , BR(µ → eγ) and ∆a µ estimated by using allowed parameter sets.In most of the parameter sets, R µe→ee is dominantly determined by the effect of A R indicated by clear correlation in lower-left panel.Thus it is also correlated with BR(µ → eγ) as the lower-right panel since the process is induced by the operator related to A R .The effect of g 4 is found as deviation from the correlation.We also find |g 4 | value tends to be larger when ∆a µ is larger as indicated by the upper-left panel.The largest value of R µe→ee is found to be ∼ 5 × 10 −18 where this value is the maximal value determined in almost model independently since the upper bound of A 4 is given by constraint from µ → eγ process; the upper limit of |g 4 | is also fixed by constraint from µ → eee process.
The expected number of stopped muons is O (10 17 ) to O (10 18 ) in near future experiments for µ − − e − conversion, such as Mu2e [35] and COMET phase-II [36].In these experiments, they are planning to use aluminum targets, which is less suitable for µ − e − → e − e − due to its small proton number.To test the value of R µe→ee , we need next generation experiments providing larger statistics or replacement of target materials to heavier nuclei.

IV. SUMMARY
We have investigated a model based on hidden U(1) X gauge symmetry in which neutrino mass is induced at one-loop level by effects of interactions among particles in hidden sector and the SM leptons.Generated neutrino masses are suppressed by loop factor and small mass difference between bosons from real and imaginary part of extra scalar field generated through U(1) X breaking at low scale.Then we have formulated neutrino mass matrix, LFV processes and muon g −2 which are induced via interactions among SM leptons and particles in U(1) X hidden sector.
Carrying out numerical analysis, we have searched for allowed parameter sets imposing neutrino data and current LFV constraints.In our scenario, we can obtain sizable Yukawa couplings associated with interactions between hidden sector particles and SM leptons when the generated neutrino mass matrix can fit the neutrino data.Then we have discussed LFV processes µ → eZ ′ , ℓ i → ℓ j γ and µe → ee, and muon g − 2 using allowed parameter sets.It is found that these LFV processes could be tested in next generation experiments.

TABLE I :
(2)rge assignments to fields in the model under SU(2) L × U (1) Y × U (1) X where we omitted quark sector since it is the sam as the SM one.