Neutrino mass generation with large $SU(2)_L$ multiplets under local $U(1)_{L_\mu - L_\tau}$ symmetry

We propose a model of neutrino mass matrix with large $SU(2)$ multiplets and gauged $U(1)_{L_\mu - L_\tau}$ symmetry, in which we introduce $SU(2)$ quartet scalar and quintet fermions with nonzero $L_\mu - L_\tau$ charge. Then we investigate the neutrino mass structure and explore phenomenologies of large multiplet fields at the collider, particularly, focussing on doubly- and singly- charged exotic leptons from the quintet.


I. INTRODUCTION
The mass spectrum and flavor structure of fermions are mysterious issues in the standard model (SM). In particular, at least two non-zero neutrino masses require the existence of physics beyond the SM for its generating mechanism. Actually there are many mechanisms to generate neutrino mass such as Type-I, II and III seesaw models as well as radiatively induced neutrino mass models. One interesting scenario is to generate neutrino mass via interaction among the SM lepton and the exotic fields which are large SU(2) multiplets like quartet, quintet or septet [1][2][3][4][5][6]. Introducing a quartet scalar and quintet fermions, we can realize neutrino mass generation by type-III like seesaw way using the quintet fermions and the quartet scalar with vacuum expectation value (VEV). Remarkably, tiny neutrino mass is realized by two suppression effects; Majorana mass of the quintet fermion and small VEV of the quartet scalar required by the constraint from ρ-parameter, which is similar to the type-II seesaw model.
In this letter, we construct a model of neutrino mass generation with large SU(2) multiplets and gauged U(1) Lµ−Lτ symmetry in which we introduce SU(2) quartet scalar and fermion quintets with L µ − L τ charge. Then we investigate neutrino mass structure of the model and explore the phenomenology of large multiplet fields at the collider experiments.
In particular we focus on doubly-and singly-charged exotic leptons from the quintet. This paper is organized as follows. In Sec. II, we introduce our model, derive some formulas of active neutrino mass matrix, and show the typical order of Yukawa couplings and related masses. In Sec. III, we discuss implications to physics at the Large Hadron Col- lider(LHC) focusing on pair production of charged particles in the multiplets. We conclude and discuss in Sec. IV.

II. MODEL SETUP
In this section, we introduce our model based on U(1) Lµ−Lτ gauge symmetry, and derive a formula of active neutrino mass matrix. The particle contents with charge assignments are shown in Tab. I. In fermion sector, we introduce three right-handed exotic fermions Σ R which are SU(2) quintet with hypercharge Y = 0. 1 In scalar sector, we introduce SU (2) quartet Φ 4 with hypercharge Y = 1/2 and the SM singlet ϕ with U(1) Lµ−Lτ charge 1. Under the gauge symmetries, we can write following Yukawa interactions associated with the SM leptons and exotic fermions, and scalar potential 2 : where V trivial term includes trivial quartic terms, and λ 0 plays a role in evading dangerous Goldstone Boson(GB) from Φ 4 as well as in inducing the VEV of Φ 4 when M 2 4 > 0.
Scalar sector: The scalar fields can be written as where w + , z and z ′ are Nambu-GB (NGB) absorbed by W + , Z and Z ′ bosons, and v, v 4 and v ϕ are VEVs of each field. The VEVs are obtained by applying the conditions where we assumed v 4 << {v, v ϕ }, λ Hϕ ≪ 1 and couplings in the V trivial term are small. The VEV of Φ 4 is restricted by the ρ-parameter which is given by where the experimental value is given by ρ = 1.0004 +0.0003 −0.0004 at 2σ confidence level. Therefore we should require v 4 2.65 GeV, and this bound is naturally satisfied; v 4 ∼ 1 GeV with M 4 ∼ 1 TeV and λ 0 ∼ 0.1. Assuming small contribution from terms in V trivial term , the masses for components in Φ 4 is given by M 4 . The squared mass terms for CP-even scalar bosons {h,φ R } are given by where the mixing with neutral component of Φ 4 is negligibly small due to small v 4 . The above squared mass matrix can be diagonalized by an orthogonal matrix and the mass eigenvalues are given by The corresponding mass eigenstates h and ϕ R are obtained as where α is the mixing angle, and h is identified as the SM-like Higgs boson when α ≪ 1.
For later convenience, we write gauge interactions giving decay process of Φ 4 components such that where c W (s W ) = cos θ W (sin θ W ) with the Weinberg angle θ W and g 2 is the SU(2) L gauge coupling constant.
Z' boson and muon g −2: After ϕ developing VEV, U(1) Lµ−Lτ symmetry is spontaneously broken resulting in massive Z ′ boson. We obtain Z ′ boson such as where g X is the gauge coupling constant associated with U(1) Lµ−Lτ and we have ignored U(1) kinetic mixing assuming it is negligibly small. Gauge interactions among Z ′ and the SM fermions are given by The Z ′ contribution to muon g − 2 is estimated as (II.12) We can explain muon g − 2 with m Z ′ ∼ O(0.1) GeV and 10 −4 g X 10 −3 without conflict to other constraints. In this region Z ′ dominantly decays into neutrino pair.
Fermion quintet: After ϕ developing a VEV, the Majorana mass matrix for Σ R i is obtained as Here we write the quintet Majorana fermions by components such that where the upper indices represent the electric charges and lower indices distinguish components with the same electric charge for each component. The quintet is also written as (Σ Ra ) ijkl where the indices {i, j, k, l} take 1 or 2 corresponding to SU(2) L doublet index 3 .
The mass term can be expanded as

A. Neutrino mass matrix
Here we derive formula for the active neutrino mass matrix. Firstly the relevant interaction in Yukawa coupling is given by 16) where the terms in the last line contribute to generate the active neutrino mass matrix.
Then the active neutrino mass matrix m ν is generated as Fig. 1 whose formula is given by If we take the scale of neutrino mass is O(0.1) eV the magnitude of the Yukawa coupling y ν is around 10 −3 for v 4 ∼ 1 GeV and M Σ ∼ 1 TeV. Thus we can obtain tiny neutrino mass 3 Here (Σ R ) ijkl is the symmetric tensor notation which is explicitly given by ( Then one finds that the energy evolution of the gauge coupling g 2 as [6, 42] where N f is the number of Σ R , µ is a reference energy, b SM g 2 = −19/6, and we assume m in. (= m Z ) < m th. =500 GeV, being respectively threshold masses of exotic fermions and bosons for m th. . The resulting flow of g 2 (µ) is then given by the Fig. 2 for N f = 2 and 4 In general, one-loop induced neutrino mass matrix is also induced via λ HΦ4 . But here we simply neglect this term, assuming λ HΦ4 << 1. Even when this contribution is comparable to the tree level, our prediction retains if M Σ is degenerate to the neutral component of Φ 4 [43]. 5 We have confirmed that the gauge coupling of U (1) Y is relevant up to Planck scale.  Thus our theory does not spoil, as far as we work on at around the scale of TeV.

III. PHENOMENOLOGY
In this section, we discuss phenomenology of the model focusing on interactions among quintuplet fermions and Z ′ boson 6 . The relevant interaction is written as where Σ Q i denotes a mass eigenstate with electric charge Q. For illustration, here we show X ij in the case of (M Σ ) 11 as a function of their mass.
GeV. In this case, we obtain The components of the quintuplets can be produced via electroweak gauge interaction which is given by where these terms are diagonal for generations of the quintuplets.
Here we estimate the cross section of production process pp → Σ ++ i Σ −− i (Σ + i Σ − i ) via electroweak interaction using CalcHEP [47] implementing relevant interactions with CTEQ6L PDF [48]. In Fig. 3, we show the cross sections for pair production as a function of exotic fermion mass. We find that production cross section for doubly-charged fermion can be O(1) fb for m Σ i = 1 TeV while that for singly-charged fermion is smaller.
The components Σ Q i decay through gauge interactions and Yukawa interactions in Eq. (II.2). Firstly decay mode Σ Q i → Σ Q∓1 i W ± mode with on-shell W ± is not kinemat-ically allowed since initial and final state exotic fermion masses are degenerated at tree level 7 . At one-loop level, mass difference is induced which can be few 100s MeV scale.
Then decay mode of Σ Q i → Σ Q∓1 i π ± is kinematically allowed which is induced via off-shell The decay width is given by where m π ≃ 140 MeV is charged pion mass, f π = 131 MeV is pion decay constant and G F is the Fermi constant. Adopting ∆M = 166 MeV in quintet case, we obtain Γ Σ Q i →Σ Q∓1 i π ± ∼ 10 −14 GeV. In addition Σ Q i can decay into "Z ′ + lighter generation" and/or "SM lepton + component of Φ 4 " depending on the generation of the Σ Q i . The partial decay widths are then given by where charge of φ Q ′ is determined by final state lepton in second decay mode, and C φΣ is numerical factor which appear each component of Eq. (II.16). The value of Γ Σ Q i →ℓ ± (ν ℓ )φ Q ′ is ∼ 10 −7 GeV taking m Σ i = 1000 GeV, m φ = 500 GeV and V iℓ y ℓℓ = 10 −4 , and it is much larger We can thus neglect decay mode with pion in our analysis. Furthermore we find that the first mode has enhancement factor of m 2 Σ i /m 2 Z ′ which is large for light Z ′ case motivated by explaining muon g − 2. Thus heavier generation of the quintuplet dominantly decay into lighter generation with Z ′ boson. We also find that the BRs of first generation of quintuplets are given by where ℓ ± and ν include all lepton flavor, and φ ± includes φ ± 1,2 and φ 0 = (φ R + iφ I )/ √ 2.
7 Mass difference can appear through mixing among SM leptons but it is negligibly small where Z ′ dominantly decays into the SM neutrinos. These processes give signals of multileptons with/without jets and missing transverse momentum. In Table.  ) ≃ 1 as discussed above. BRs for multilepton final states are relatively small and we need large integrated luminosity to explore the signal.
Thus these signal could be tested at the High-Luminosity LHC. Notice that the masses of doubly-and singly-charged exotic fermion are same in each generation and reconstruction of the mass spectrum is important to confirm our scenario. However, since each signal has many particles in final state, detailed analysis is beyond the scope of this letter.

IV. CONCLUSIONS AND DISCUSSIONS
We have constructed a model of neutrino mass generation via interactions among large multiplet of SU(2) L under U(1) Lµ−Lτ gauge symmetry. The neutrino mass is given by the Yukawa interaction with the quartet scalar and the quintet fermion. Then we realize tiny neutrino mass by suppression factors regarding the small quartet VEV and Majorana mass of the quintet. We find neutrino mass matrix has predictive structure due to the U(1) Lµ−Lτ symmetry.
Then collider physics of the model is investigated, where we have focused on doubly-and singly-charged exotic leptons. We find that heavier generations of them dominantly decay into lighter one and Z ′ boson, while the lightest one decays into SM lepton and components in quartet scalar. The components of quartet scalar decay into SM gauge bosons when we assume they are lighter than the quintet fermions. In such a case, the signal of exotic lepton productions is multi-leptons with/without jets and missing transverse momentum.