One-loop neutrino mass model with $SU(2)_L$ multiplet fields

We propose a one-loop neutrino mass model with several $SU(2)_L$ multiplet fermions and scalar fields in which the inert feature of a scalar to realize the one-loop neutrino mass can be achieved by the cancellation among Higgs couplings thanks to nontrivial terms in the Higgs potential. Then we discuss our typical cut-off scale by computing renormalization group equation for $SU(2)_L$ gauge coupling, lepton flavor violations, muon anomalous magnetic moment, possibility of dark matter candidate, neutrino mass matrix satisfying the neutrino oscillation data. Finally, we search for our allowed parameter region to satisfy all the constraints, and discuss a possibility of detecting new charged particles at large hadron collider.


I. INTRODUCTIONS
Radiatively induced neutrino mass models are one of the promising candidates to realize tiny neutrino masses with natural parameter spaces at TeV scale and to provide a dark matter (DM) candidate, both of which cannot be explained within the standard model (SM).
In order to build such a radiative model, an inert scalar boson plays an important role and its inert feature can frequently be realized by imposing additional symmetry such as Z 2 symmetry [1][2][3][4] and/or U(1) symmetry [5][6][7], which also play an role in stabilizing the DM.
On the other hand, once we introduce multiplet fields such as quartet [8,9], quintet [10,11], septet fields [12][13][14] under SU(2) L gauge group, we sometimes can evade imposing additional symmetries [15,16]. Then, the stability originates from a remnant symmetry after the spontaneous electroweak symmetry breaking due to the largeness of these multiplets. In addition, the cut-off scale of a model is determined by the renormalization group equations (RGEs) of SU(2) L gauge coupling, and it implies that a theory can be within TeV scale, depending on the number of multiplet fields. Thus a good testability could be provided in such a scenario.
In this letter, we introduce several multiplet fermions and scalar fields under the SU(2) L gauge symmetry. As a direct consequence of multiplet fields, our cut-off scale is of the order 10 PeV that could be tested by current or future experiments. In our model we do not impose additional symmetry and search for possible solution to obtain inert condition for generating neutrino mass at loop level. Then required inert feature can be realized not via a remnant symmetry but via cancellations among couplings in our scalar potential thanks to several non-trivial couplings [17]. In such a case, generally DM could decay into the SM particles, but we can control some parameters so that we can evade its too short lifetime without too small couplings. Therefore our DM is long-lived particle which represents clear difference from the scenario where the stability of DM is due to an additional or remnant symmetry.
We also discuss lepton flavor violations (LFVs), and anomalous magnetic moment (muon g − 2), and search for allowed parameter region to satisfy all the constraints such as neutrino oscillation data, LFVs, DM relic density, and demonstrate the possibility of detecting new charged particles at the large hadron collider (LHC).
This letter is organized as follows. In Sec. II, we review our model and formulate the Higgs sector, neutral fermion sector including active neutrinos. Then we discuss the RGE of the SU(2) L gauge coupling, LFVs, muon g − 2, and our DM candidate. In Sec. III, we explore the allowed region to satisfy all the constraints, and discuss production of our new fields (especially charged bosons) at he LHC. In Sec. IV, we devote the summary of our results and the conclusion.

II. MODEL SETUP AND CONSTRAINTS
In this section we formulate our model. As for the fermion sector, we introduce three families of vector fermions ψ with (4, −1/2) charge under the SU(2) L ×U(1) Y gauge symmetry. As for the scalar sector, we respectively add an SU ( Table I, where the quark sector is exactly the same as the SM. The renormalizable Yukawa Lagrangian under these symmetries is given by where SU(2) L index is omitted assuming it is contracted to be gauge invariant inside bracket [· · · ], upper indices (a, b) = 1-3 are the number of families, and y ℓ and either of g L/R or M D are assumed to be diagonal matrix with real parameters without loss of generality. Here, we assume g L/R and M D to be diagonal for simplicity. The mass matrix of charged-lepton is Here we assign lepton number 1 to ψ so that the source of lepton number violation is only the terms with coupling g ab and g ′ ab in the Lagrangian requiring the lepton number is conserved at high scale.
Scalar potential and VEVs: The scalar potential in our model is given by where V tri is the trivial quartic terms containing H 4,5,7 . From the conditions of ∂V/∂v 5 = 0 and H 5 = 0, we find the following relation: Then, the stable conditions to the H 4 and H 7 lead to the following equations: where we have ignored contributions from terms in V tri assuming corresponding couplings are negligibly small; we can always find a solution satisfying the inert condition including such terms. Solving Eqs. (3) and (4), one rewrites VEVs and one parameter in terms of the other parameters. In addition to the above conditions, we also need to consider the constraint from ρ parameter, which is given by the following relation at tree level: where the experimental values is given by ρ = 1.0004 +0.0003 −0.0004 at 2σ confidential level [18]. Then, we have, e.g., the solutions of (

A. Neutral fermion masses
Heavier neutral sector: After the spontaneously electroweak symmetry breaking, extra neutral fermion mass matrix in basis of Ψ 0 R ≡ (ψ 0 R , ψ 0c L ) T is given by where µ R ≡ 3 10 g R v 7 and µ L ≡ 3 10 g * L v 7 . Since we can suppose to be µ L/R << M D , the mixing is expected to be maximal. Thus, we formulate the eigenstates in terms of the flavor eigenstate as follows: where ψ 1 R and ψ c 2 L represent the mass eigenstates, and their masses are respectively given by Active neutrino sector : In our scenario, active neutrino mass is induced at one-loop level, where ψ 1,2 and H 5 propagate inside a loop diagram as in Fig. 1, and the masses of real part and imaginary part of electrically neutral component of H 5 are respectively denoted by m R and m I . As a result the active neutrino mass matrix is obtained such that where r α R/I ≡  [19,20] as follows: where O is a three by six arbitrary matrix, satisfying OO T = 1, and |f | √ 4π is imposed not to exceed the perturbative limit.
Beta function of SU(2) L gauge coupling g 2 : Here we estimate the running of gauge coupling of g 2 in the presence of several new multiplet fields of SU(2) L . The new contribution to g 2 from fermions (with three families) and bosons are respectively given by [13,21] ∆b f g 2 = Then one finds that the resulting flow of g 2 (µ) is then given by the Fig. 2 Lepton flavor violations(LFVs): LFV decays ℓ i → ℓ j γ arise from the term f at one-loop level, and its form can be given by [22,23]  where and where a L = a R (m ℓ i → m ℓ j ).
New contributions to the muon anomalous magnetic moment (muon g − 2: ∆a µ ) : We obtain ∆a µ from the same diagrams for LFVs and it can be formulated by the following where a Lµµ = a Rµµ has been applied and we use the allowed range of ∆a µ = (26.1 ± 8.0) × 10 −10 in our numerical analysis below.
where we assume the final states to be massless, m R ≈ m I , M DM is the mass of DM, and h is the SM Higgs. In the numerical analysis, we will estimate the lifetime and show the allowed region, where we take the maximum value of |f 1a |. 1

III. NUMERICAL ANALYSIS AND PHENOMENOLOGY
Here we carry out numerical analysis to discuss consistency of our model under the constraints discussed in previous section. Then we discuss collider physics focusing on charged scalar bosons in the model.

Numerical analyses:
In our numerical analysis, we assume all the mass of ψ 1,2 to be the mass of DM; 2.4 TeV, and all the component of H 5 except m I to be degenerate, where m I = 1.1m R . These assumptions are reasonable in the aspect of oblique parameters in the multiplet fields [18]. Also we fix to be the following values so as to maximize the muon g − 2: where O 12,23,13 are arbitral mixing matrix with complex values that are introduced in the neutrino sector [10,20]. Notice here that we also impose |f | √ 4π not to exceed the perturbative limit. BR(τ → eγ), BR(τ → µγ), and ∆a µ are respectively colored by red, magenta, blue, and black. The black horizontal line shows the current upper limit of the experiment [28,29], while the green one does the future upper limit of the experiment [28,30]. Considering these bounds of µ → eγ, one finds that the current allowed mass range of m R ∼ 4-20 TeV can be tested in the near future. Here the upper bounds of BR(τ → eγ) and BR(τ → µγ) are of the order 10 −8 , which is safe for all the range. The maximum value of ∆a µ is about 10 −12 , which is smaller than the experimental value by three order of magnitude.
1 In case where the neutral component of H 5 is DM candidate, H 5 decays into SM-like Higgs pairs via λ 0 , and its decay rate is given by 800πMX . Then the required lower bound of λ 0 is of the order 10 −19 so that its lifetime is longer than the age of Universe, where DM is estimated as 5 TeV [16]. horizontal line shows the current upper limit of the experiment [28,29], while the green one does the future upper limit of the experiment [28,30]. decays into W ± W ± via (D µ H 4 ) † (D µ H 4 ) term. We thus obtain multi W boson signal from quadruply charged scalar boson production. Mass reconstruction from multi W boson final state is not trivial and detailed analysis is beyond the scope of this paper.

IV. SUMMARY AND DISCUSSIONS
We have proposed an one-loop neutrino mass model, introducing large multiplet fields under SU(2) L . The inert boson is achieved by nontrivial cancellations among quadratic terms. We have also considered the RGE for g 2 , the LFVs, muon g − 2, and fermionic DM candidate, and shown allowed region to satisfy all the constraints as we have discussed above. RGE of g 2 determines our cut-off energy that does makes our theory stay within the order 10 PeV scale, therefore our model could totally be tested by current or near future experiments. Due to the multiplet fields, we have positive value of muon g − 2, but find its maximum value to be of the order 10 −12 that is smaller than the sizable value by three order of magnitude. For the LFVs, the most promising mode to be tested in the current and future experiments is µ → eγ at the range of 3.2 TeV m R 11 TeV. We have also discussed possible decay mode of our DM candidate and some parameters are constrained requiring DM to be stable on cosmological time scale. Notice that the decay of DM is one feature of our model and we would discriminate our model from models with absolutely stable DM by searching for signal of the DM decay. Finally, we have analyzed the collider physics, focussing on multi-charged scalar bosons H 4 and H 7 . We find that sizable production cross section for quadruply charged scalar pair can be obtained adding the photon fusion process that is enhanced by large electric charge of φ ±±±±