A testable radiative neutrino mass model with multi-charged particles

We propose a radiatively-induced neutrino mass model at one-loop level by introducing a pair of doubly-charged fermions and a few multi-charged bosons. We investigate the contributions of the model to neutrino masses, lepton-flavor violations, muon $g-2$, oblique parameters, and collider signals, and find a substantial fraction of the parameter space that can satisfy all the constraints. Furthermore, we discuss the possibility of detecting the doubly-charged fermions at the LHC.

In this work, we propose a simple extension of the SM by introducing 3 generations of doubly-charged fermion pairs and three multi-charged bosonic fields [5]. All of them participate in generation of neutrino mass at one-loop level. We show that the model can explain the anomalous magnetic moment without conflict constraints of the lepton-flavor violating processes and oblique parameters. Also we discuss the possibility of detecting some of the new fields at the LHC.
This paper is organized as follows. In Sec. II, we review the model, describe several constraints, and show numerical results. In Sec. III, we discuss the collider signatures. We conclude in Sec. IV.

II. MODEL SETUP AND CONSTRAINTS
In the model, we introduce three families of doubly charged fermions E, and three types of new bosons k ++ , Φ 3/2 and Φ 5/2 , in addition to the SM fields, as shown in Table I. Under their charge assignments, the relevant Yukawa Lagrangian and the non-trivial terms of Higgs potential are given by where H is the SM Higgs field that develops a nonzero vacuum expectation value (VEV), which is symbolized by H ≡ v/ √ 2, and (i, a) = 1 − 3 are generation indices. The f and g terms contribute to the active neutrino masses, while the κ term does not contribute to the neutrino sector but plays a role of mediating the decays of the new particles into the SM particles. In this work, all the coefficients are chosen to be real and positive for simplicity.
We parameterize the scalar fields as where the lower index in each component represents the hypercharge of the field. Due to the µ ( ′ ) and λ ( ′ ) 0 terms in Eq. (II.2), the three doubly-charged bosons in basis of (k ++ , φ ++ 3/2 , φ ++ 5/2 ) fully mix with one another. The mixing matrix and mass eigenstates are defined as follows: therefore one can rewrite the Lagrangian in terms of the mass eigenstate as follows: (II.5)

A. Neutrino mixing
The active neutrino mass matrix M ν is given at one-loop level via doubly-charged particles in Fig. 1, and its formula is given by where ζ 12 ≡ s 2 12 s 2 13 s 23 c 23 + 2c 12 s 12 s 13 s 2 23 − c 12 s 12 s 13 + s 2 12 s 23 c 23 , ζ 13 ≡ s 2 13 s 23 c 23 , and ζ 23 ≡ s 23 c 23 . M ν is diagonalized by the neutrino mixing matrix . Then one can parameterize the Yukawa coupling in terms of an arbitrary antisymmetric matrix A with complex values (i.e. (A + A T = 0)) with mass scale, as follows [6,7]: In the numerical analysis, we use the latter relation for convenience, and we use the data in the global analysis [8]. Notice here that the mass scale of A should be rather tiny so that A can be the relevant mass parameter to make a significant contribution to the observed neutrino oscillation data. from the term f mediated by φ + 3/2 and E ++ , while the right side arises from the terms g/f /κ that respectively correspond to The branching ratio is given by where G F ≈ 1.16 × 10 −5 GeV −2 is the Fermi constant, α em ≈ 1/137 is the fine structure constant, C 21 = 1, C 31 = 0.1784, and C 32 = 0.1736. (II.14) The current experimental upper bounds are given by [10,11] The muon anomalous magnetic moment (g − 2) µ : It is known that discrepancy of experimental value and the SM prediction is given by [12] ∆a µ = (26.1 ± 8.0) × 10 −10 . (II.16) We have nonvanishing (g − 2) µ , and its formula is found via M in LFVs as Here the f term contribution provides the positive value of (g − 2) µ that corresponds to Fig. 2 with φ + and E ++ mediators inside the loop, while the other terms g and κ give the negative values of (g − 2) µ . 1 In order to achieve the agreement with the experimental value, one has to enhance f term compared to the g and κ term. However the κ term gives another LFV with three body decay ℓ i → ℓ j ℓ klℓ at tree level and it gives more stringent constraints as shown in Table I of Ref. [9]. Thus we can expect this term to be negligible in (g − 2) µ .

C. Oblique parameters
In order to estimate the testability via collider physics, we have to consider the oblique parameters that restrict the mass hierarchy between each of the components in Φ3 Here we focus on the new physics contributions to S and T parameters in the case U = 0.
Then ∆S and ∆T are defined as where s 2 W ≈ 0.23 is the Weinberg angle and m Z is the Z boson mass. The loop factors Π 33,3Q,± (q 2 ) are calculated from the one-loop vacuum-polarization diagrams for Z and W ± 1 The sign of (g − 2) µ , which is induced at one-loop level, generally depends on sign of the electric charge and the direction of momentum of the particle that emits the photon inside the loop. For example, when a fermion (boson) with negative (positive) electric charge propagates in the same direction as the outgoing muon, one finds positive values for (g − 2) µ . In the opposite case, one obtains negative (g − 2) µ . Through this aspect, one can straightforwardly understand the sign of (g − 2) µ without any computations, and our sign shows the direct consequence of this insight.
bosons, which are respectively given by (II.20) ) . (II.21) The experimental bounds are given by [13] ( and new contributions should be within these ranges.

D. Numerical analysis
In the numerical analysis, we prepare 2 × 10 6 random sampling points for the relevant input parameters in the following ranges:    In Fig. 3, we show the scatter plot in the plane of M E 1 and ∆a µ that satisfy all the constraints as discussed above. We observe that the whole mass range of E 1 that we have taken can give ∆a µ to be within (26.1 ± 8.0) × 10 −10 . Also, the smaller the mass M E 1 the larger the value of ∆a µ will be, as expected by the formula in Eq. (II.11).
In Fig. 4, we show two characteristic correlations among the masses of the charged bosons. This is a consequence of the constraint from the oblique parameters as discussed in Sec. II C.

Theses correlations suggest that the masses between m H
The second feature is that the mass range of m φ + 3/2 is restricted to be less than 1 TeV, even though we have scanned it up to 2 TeV as input parameters. This mainly comes from the experimental value of ∆a µ . Moreover, the value of the loop function in (II.11) decreases when the mass of φ + 3/2 increases. Then m H ++ 2 is also restricted to be in the same range as by the consequence of oblique parameters again.

III. COLLIDER SIGNALS
We first consider the Drell-Yan (DY) production of EE via γ, Z exchanges. The interactions can be obtained from the kinetic term of the fermion E. Since E is a singlet, the interactions with γ and Z are given by where s W and c W are respectively the sine and cosine of the Weinberg angle, and Q E is the electric charge of the fermion E with Q E = −2 in our model.
The square of the scattering amplitude, summed over spins, for q( can be written as (III .1) whereŝ,t,û are the usual Mandelstam variables for the subprocess, and g q L = T 3q − s 2 W Q q and g q R = −s 2 W Q q are the chiral couplings of quarks to the Z boson. The subprocess differential cross section is given by where β = 1 − 4M 2 E /ŝ, and where T 3q is the third component of the isospin of q. This subprocess cross section is then folded with parton distribution functions to obtain the scattering cross section at the pp collision level. The K factor for the production cross sections is expected to be similar to the conventional DY process, which is approximately  We proceed to estimate the decay partial widths of the fermion E −− 1 , which is presumed to the lightest among E −− 1,2,3 . The decay channels of E −− 1 can proceed via the following ≈ φ ++ 5/2 . We also take the simplification that the masses of each components in the doublet are similar, i.e., We compute the partial width of E −− → e i φ − 3/2 and obtain Summing over all lepton and neutrino channels with i = 1, 2, 3 as well as the contributions from the f i1 and g i1 terms, we obtain the total decay width of Next, we compute the subsequent decays of H −− i → e − k e − l (where k, l are flavors) and where the function λ(x, y, z) = (x 2 + y 2 + z 2 − 2xy − 2yz − 2zx) and if the mass difference Naively, since g i1 ≪ f i1 due to lepton-number violation, we expect E −− → e i φ − 3/2 , ν i φ −− 3/2 dominantly. Therefore, the branching ratio for E −− → 2e i + E is about 1/2, for E −− → 4e i + E is about 1/6 (including e i = e, µ, τ ). Now we can estimate the event rates at the 13 TeV LHC with a luminosity of 3000 fb −1 (HL-LHC). We have about 0.2×3000×(1/2) 2 = 150 events for 4e i final state, 0.2 × 3000 × 1/2 × 1/6 × 2 = 100 events for 6e i final state, and 0.2 × 3000 × 1/6 × 1/6 ≃ 17 events for 8e i final state.

IV. CONCLUSIONS
In this work, we have proposed a simple extension of the SM by introducing 3 generations of doubly-charged fermion pairs and three multi-charged bosonic fields. We have investigated the contributions of the model to neutrino mass, lepton-flavor violations, muon g −2, oblique parameters, and collider signals, and found a substantial fraction of the parameter space that can satisfy all the constraints.
The design of the κ term in the Lagrangian is to make sure that all new charged particles will decay into SM particles so that no stable charged particles were left in the Universe.
Because of this κ term the new charged particles will decay into charged leptons in collider experiments, thus giving rise to spectacular signatures. Pair production of E ++ 1 E −− 1 can give 4e i , 6e i , or 8e i plus missing energies in the final state. The event rates are 17 − 150 for an integrated luminosity of 3000 fb −1 .