A generalized one-loop neutrino mass model with charged particles

We propose a radiative neutrino-mass model by introducing 3 generations of fermion pairs $E^{-(N+1)/2} E^{+(N+1)/2}$ and a couple of multi-charged bosonic doublet fields $\Phi_{N/2}, \Phi_{N/2+1}$, where $N=1,3,5,7,9$. We show that the models can satisfy the neutrino masses and oscillation data, and are consistent with lepton-flavor violations, the muon anomalous magnetic moment, the oblique parameters, and the beta function of the $U(1)_Y$ hypercharge gauge coupling. We also discuss the collider signals for various $N$, namely, multi-charged leptons in the final state from the Drell-Yan production of $E^{-(N+1)/2} E^{+(N+1)/2}$. In general, the larger the $N$ the more charged leptons will appear in the final state.

In a previous work, we proposed a simple extension of the SM by introducing 3 generations of doubly-charged fermion pairs E −− E ++ and three multi-charged bosonic fields k ++ , Φ 3/2 , Φ 5/2 , in which Φ are the SU (2) doublet fields and k ++ is a singlet field [6]. The model can explain the small neutrino masses and oscillations, muon anomalous magnetic moment, and is consistent with the lepton-flavor violations and the oblique parameters.
Here in this work we generalize the model to E − N+1 2 E + N+1 2 and Φ N/2 , Φ N/2+1 , with N = 1, 3, 5, 7, 9. The previous work [6] corresponds to the case of N = 3. The generalization is indeed nontrivial. Especially, for the case of N = 1 in which a Z 2 parity is required to distinguish between the Higgs doublet H and the doublet Φ 1/2 . The Z 2 assignment thus gives rise to a lightest Z 2 -odd particle, which is stable and can be a dark matter candidate. We will explore the dark matter phenomenology of N = 1 case. The other cases share some similar features as N = 3, but they do have different features that deserve separate discussion. In general, a larger N would rise to a final state with more charged leptons for Drell-Yan production of E +(N +1)/2 E −(N +1)/2 . This paper is organized as follows. In Sec. II, we review the model and describe the constraints.
In Sec. III, we describe the physics for each N . In Sec. IV, we present the numerical analyses and valid parameter space for each N . We discuss the collider signals in Sec. V, and conclude in Sec. VI.

II. MODEL SETUP AND CONSTRAINTS WITH COMMON PART
In our set up of the model, we introduce three families of doubly-charged fermions E 1 , and two types of new bosons Φ N/2 and Φ N ′ /2 with N ′ ≡ N + 2, as shown in Table I. Notice here that one has to impose an additional symmetry such as Z 2 to discriminate between Φ N/2 and H only in case of N = 1, as we will see later. The renormalizable Lagrangian in the lepton sector and the Higgs potential are given by −L Y = (y ℓ ) iiLi HP R e i + f iaLi Φ N/2 P R E a + g iaLi · Φ ′ * N ′ /2 P R E c a + M EaĒa E a + h.c., where (i, a) = 1 − 3 are generation indices and the multiplication symbol · represents iσ 2 with σ 2 being the second Pauli matrix. The first term in the Yukawa Lagrangian, which is assumed to be diagonal for convenience, provides the masses for the charged leptons (m ℓ i ≡ y ℓ ii v/ √ 2) by developing a nonzero vacuum expectation value (VEV) of H, which is symbolized by Also, we work in the basis where all the coefficients are real and positive for simplicity hereafter.
We can parameterize the scalar fields as where the superscript for each component represents the electric charges. Due to the λ ( ′ ) 0 term in Eq. (II.1), the two N +1 2 -charged bosons in basis of (φ N+1 2 , φ ′ N+1 2 ) mix with each other. Their mixing matrix and mass eigenstates are defined as Therefore one can redefine these bosons as the mass eigenstates as follows: where we have used the short-hand notation: 2), and their masses to be

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 [6] − . (II.6) The mass matrix M ν is diagonalized by the neutrino mixing matrix . Then one can parameterize Yukawa coupling in terms of arbitrary antisymmetric matrix with complex values A; (A + A T = 0), as follows [7] In the numerical analysis, we shall use the latter relation for convenience, and use the data in the global analysis [8].
It is worthwhile to describe the differences between the Ma model and our model. Once we assign the lepton number +2 for Φ ′ N ′ /2 and zero for other bosons, the lepton number is explicitly broken at the λ 0 and λ ′ 0 terms by 2 units. On the other hand, if all the bosons were assigned zero charges under the lepton number, then the breaking of lepton number would occur at the Yukawa coupling term g. Here, we suppose that the extra vector fermions E has −1 charge under the lepton number, which is the same as the SM charged leptons. The major difference between our model and the Ma model is that the lepton number in the Ma model is violated via the mass term of the right-handed Majorana neutrinos while in our model it is violated via the boson mass or the Yukawa term.
where we have simplified the notation φ ≡ φ N−1 2 , and φ ′ ≡ φ ′ N+3 2 , and assumed m ℓτ << M Ea , m φ , m φ ′ in the equations above. 2 The current experimental upper bounds are given by [10,11] The muon anomalous magnetic moment (∆a µ ): We can also estimate the muon anomalous magnetic moment through M, which is given by The 3.3σ deviation from the SM prediction is ∆a µ = (26.1 ± 8) × 10 −10 [12] with a positive value.
Obviously, f contributes to the ∆a µ positively, while g does negatively. To achieve the agreement with experimental result, one has to enhance the contributions from f compared to those from g.

C. Flavor-Changing Leptonic Z Boson Decays
In this subsection, we consider the Z boson decay into two charged leptons with different flavors through the Yukawa terms f and g at one-loop level [13]. Since some components of f and g are expected to be large so as to obtain a sizable ∆a µ , the experimental bounds on lepton-flavor violating Z boson decays could be of concern at one loop level. First of all, the relevant Lagrangian is given by (II.14) where s(c) W ≡ sin(cos)θ W ∼ 0.23 stands for the sine (cosine) of the Weinberg angle. Combining all the diagrams in Fig. 2, the ultraviolet divergence cancels out and only the finite part remains [13].
The resulting form is given by and the total Z decay width Γ tot Z = 2.4952 ± 0.0023 GeV [14]. From Eqs. (II.9), (II.10) and (II.15), one finds that the same combinations of Yukawa couplings f ia f * ja and g ia g * ja appear in ℓ i → ℓ j γ and Z → ℓ i ℓ j . Note that the predictions for BR(Z → µτ ) can be rather large and tested in future experiments such as Giga-Z [15], due to different properties of loop functions [13]. The current bounds on the lepton-flavor-changing Z boson decay branching ratios at 95 % CL are given by [14]: (II. 16) We include these constraints in the global analysis.

D. 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 component Φ N/2 and Φ N ′ /2 .
Here we focus on the new physics contributions to the S and T parameters in the case of ∆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 ± bosons, which are respectively given by [16,17] is electric charge, and α(= 1, 2) should be summed up. Fixing ∆U = 0, the experimental bounds on ∆S and ∆T are given by [18] ∆S = (0.05 ± 0.09), ∆T = (0.08 ± 0.07), (II. 22) with a correlation coefficient of +0.91. The ∆χ 2 can be calculated as [19] and we impose the 99% confidence level limit that corresponds to ∆χ 2 = 9.210 in our numerical analysis.

E. Beta function of g Y
Here we estimate the effective energy scale by evaluating the Landau pole for g Y in the presence of new exotic fields with nonzero multiple hypercharges. Each contribution of the new beta function of g Y from one SU (2) L doublet fermion or boson with ±N/2 hypercharge is given by [20] ∆ where the superscript of ∆b represents the fermion (f) or the boson (b), respectively. Similarly, the contribution to the beta function from one SU (2) L singlet boson with ±N/2 hypercharge(=electric charge) is given by where 3/5 is the rescaled coefficient. Here let us include a doubly-charged gauge-singlet boson k ±± in order to make appropriate decays into the SM fields as we will discuss later. 4 Then one finds the energy evolution of the gauge coupling g Y as [21] 1 perturbativity allows the cutoff scale up to Planck mass. However, the case of N = 5, 7, 9, 11 is valid up to around 3 × 10 11 GeV, 3 × 10 8 GeV, 3 × 10 5 GeV, and 10 5 GeV, respectively. Notice here that g 2 running is almost the same the one of SM.

III. PHYSICS OF EACH N
Here we investigate each of N (1, 3, 5, 7, 9) below, where we introduce a gauge-singlet doublycharged boson k ++ for 3 ≤ N . We assume that the mixing between k ++ and a component of doubly-charged boson with isospin doublet Φ N/2 and/or Φ ′ N ′ /2 are small enough to be neglected for the neutrino oscillations, LFVs, and ∆a µ , although the mixing could play an important role in appropriately making exotic fields decay into the SM fermions. 5 When one considers the case of 11 ≤ N , one has to introduce more particles in order to make appropriate decays of exotic particles into the SM particles. For example, the minimum extension in the case of N = 11 is to add a gauge-singlet quartic-charged boson h ±±±± . In this case, the valid scale (=at the Landau pole) decreases down to 10 4 GeV.
The case of N = 1 is different from other N 's, because one cannot distinguish between the SM Higgs field H and Φ N/2 unless they are distinguished by some additional symmetries. This category of models has been analyzed in Ref. [22], which we follow and impose a Z 2 symmetry on the new exotic fields E, Φ 1/2 , and Φ ′ 3/2 . The corresponding Lagrangian and potential retain the general forms in Eqs. (II.1). Here we focus on the analysis of the DM candidate and explore the allowed region combining all the predictions and constraints as discussed before. Since the relic density from the kinetic term had been analyzed extensively in Ref. [23], we neglect the contributions from the potential terms with small couplings in order to satisfy the constraints of direct detection experiments such as LUX [24], XENON1T [25], and PandaX-II [26]. Therefore, we concentrate on the Yukawa coupling term f with the mass range m Z /2 M X m W . 6 Moreover, we have to give a larger mass difference between CP-even and CP-odd bosons of φ 0 ∼ φ R + iφ I , (either of which can be the DM), in order to evade large cross sections via interaction of Z − φ R − φ I , otherwise the coannihilation that gives a very small relic density. 7 The relevant interaction of the DM φ R = X is then given by Then the dominant contribution is found to be the effective d-wave coefficient after expanding the cross section in terms of the relative velocity. Its form is given by [28] where (f i , f j ) = (e, µ, τ ) run over all the charged-lepton mass eigenstates, and we have taken the charged-lepton masses in the final state to be zero. The resulting relic density is given by where we have used several physical parameters given in Ref. [28]. The measured relic density is about 0.12 [29], and we apply the constraint as 0.11 Ωh 2 0.13 in our numerical analysis.
Since this mode is d-wave suppressed, so a rather large f is required, which is in favor of generating a sizable ∆a µ . Notice here that we have to replace φ by φ R / √ 2 and φ I / √ 2, due to the mass difference between φ R and φ I . Hence all the general forms, except for the neutrino mass matrix, should be modified according to the replacement. For example, /2 in Eq. (II.9), and (II.10), in Eq. (II.18).
The N = 3 case is the minimal scenario without the need of additional symmetries. Several new terms are added as follows: where the terms κ, µ, µ ′ mainly contribute to the appropriate decays into the SM particles as mentioned above. The whole analyses have already been studied [6], and so we abbreviate this case for the moment.
C. N = 5 In N = 5 case, additional terms are given by where the terms h, κ, µ mainly contribute to the appropriate decays into the SM particles. Especially, h can also positively contribute to the muon anomalous magnetic moment as well as LFVs.
The formula for M h can be expressed in a similar way as the generic form of M f in Eq. (II.9): where F lf v is defined in Eq. (II.11).
Flavor-Changing Leptonic Z Boson Decays are modified due to the contribution of h and the resulting form is given by Decay modes: Possible decay modes of the exotic particles are where upper component of the Φ 5/2 doublet and the lower one of Φ ′ 7/2 mix with each other. Both always decay into φ −− and/or φ ′−4 via the kinetic terms, or they can directly decay into the SM leptons and E ±3 through f and g respectively, which are the same modes as the case N = 3.

D. N = 7
In N = 7 case, additional terms are given by where the terms κ, λ 1 , λ 2 mainly contribute to the appropriate decays into the SM particles.
Decay modes: Possible decay modes of the exotic particles are where the other modes are the same as the general N .

E. N = 9
In N = 9 case, additional terms are given by where the terms κ, λ 1 , λ 2 mainly contribute to the appropriate decays into the SM particles.
Decay modes: Possible decay modes of the exotic particles are where the other modes are the same as the general N .
IV. NUMERICAL ANALYSIS OF N = 1 ∼ 9 Here we discuss the features of the global analysis for N = 1 and N = 3, 5, 7, 9 separately, because N = 1 involves the constraint of the dark matter relic density.

A. N=1
The N = 1 case is different from the other N 's, because of the additional Z 2 symmetry imposed in order to distinguish between the Higgs field and the Φ 1/2 . Due to the Z 2 symmetry, the model has a dark matter candidate.
The dimensionless parameters are chosen to be scanned over the following moderate ranges without violating naturalness or perturbativity, as well as the mass parameters: where we have assumed m H 1 ≤ m H 2 ≤ m φ ′ , and i = 1 ∼ 3. We show a few plots of correlations among the observables or parameters in Fig. 4, where all the constraints as discussed above (neu-trino oscillation data, LFVs, relic density with 0.11 ≤ Ωh 2 ≤ 0.13, the oblique parameters, and ∆a µ ≤ (26.1 + 8) × 10 −10 ) 8 are satisfied.
One remarkable feature could be found in the first panel that a sizable ∆a µ (∆a µ > 10 × 10 −10 ) can be achieved in good agreement together with the current relic density of DM due to the d-wave Next we investigate the case of N = 3, 5, 7, 9. The dimensionless parameters are scanned similarly to the N = 1 case, and the mass parameters as well:

V. COLLIDER SIGNALS
We expect that the largest production rate of the exotic particles comes from pair production of the lightest fermions E 1 . Drell-Yan (DY) production of E 1 E 1 occurs via γ, Z exchanges. The interactions can be obtained from the kinetic term of the fermion E 1 . Since E 1 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 1 . The following applies for N = 1, 3, 5, 7, 9 and E = E 1 for simplicity.
The square of the scattering amplitude, summed over spins, for q(p 1 )q(p 2 ) → E(k 1 )E(k 2 ) can be written as whereŝ,t,û are the usual Mandelstam variables for the subprocess, and g q L,R are the chiral couplings of quarks to the Z boson. The subprocess differential cross section is given by 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 K ≃ 1.3 at the LHC energies. The production cross sections for pp → E 1Ē1 at √ s = 13 TeV scale as Q 2 E , where Q E = (N + 1)/2. In Ref. [6], the production cross sections for E −− E ++ are shown, and so therefore for other N = 1, 5, 7, 9 can be easily estimated by simple scaling.
We can also derive the decay width of E 1 for a general N . The major decay modes are those via the terms f i1 and g i1 in the Lagrangian in Eq. (II.1). The decay modes are The decay width is given by Nevertheless, the contributions from f i1 dominate because of the requirement of the ∆a µ .
Here we start with the detection of the final states for each case of N = 1, 3, 5, 7, 9. We focus on the lightest state E 1 .
The exotic scalar doublet fields take the form: The neutral component φ 0 1/2 is stable because of the Z 2 symmetry. The exotic fermion E − can decay via the terms f i1 and g i1 : Assuming |f | ≫ |g| due to the requirement of ∆a µ , the exotic fermion E − 1 decays via the term f i1 as where φ − 1/2 → φ 0 1/2 W * and φ 0 1/2 is the lightest exotic particle being stable. Therefore, Drell-Yan production of E − 1 E + 1 gives rise to a pair of charged leptons (not necessary of the same flavor) plus missing energies.

N = 3
This particular case has been given in detail in Ref. [6]. In summary, the exotic fermion pair is produced via the Drell-Yan process. The fermion E −− 1 decays via the terms f i1 and g i1 : The decays via the dominant f i1 term are followed by (via the mixing with scalar k −− ) Each E −− 1 decays into two or four charged leptons. Therefore, the final state can consist of 4, 6, or 8 charged leptons plus missing energies.

N = 5
This case is similar to N = 3. The relevant terms in the Lagrangian responsible for the decay of the exotic fermion E 3− are in which φ 2+ 5/2 and k ++ mix. The exotic fermion E 3− 1 so produced will decay via Each E 3− 1 decays into 3 charged leptons, without or with small missing energy. Therefore, the final state of E 3− E 3+ pair production consists of 6 charged leptons, mainly without missing energies.

N = 7
In this case, there are additional terms in the Lagrangian that allow the quartic-charged φ or φ ′ to decay into a pair of k ++ k ++ , which further decay into 4 charged leptons.

N = 9
In this case, there is one additional term in the Lagrangian that allows the quartic-charged φ to decay into a pair of k ++ k ++ , which further decay into 4 charged leptons.
Each exotic fermion E 5− 1 decays via Thus, each E 5− decays into 5 charged leptons. Therefore, the final state of E 5− 1 E 5+ 1 production consists of 10 charged leptons.

VI. CONCLUSIONS
We have generalized a radiative neutrino-mass model by introducing 3 generations of fermion pairs E −(N +1)/2 E +(N +1)/2 and a couple of multi-charged bosonic doublet fields Φ N/2 , Φ N/2+1 , where N = 1, 3, 5, 7, 9. We have shown that the model can satisfy the neutrino masses and oscillations, lepton-flavor violations, the oblique parameters, and the ∆a µ . We also made predictions for the collider signatures of the model. In general, searches for multi charged leptons in the final state, not necessarily of the same flavor, are interesting probes of the model.
We offer a few more comments as follows.
1. For N = 1, where there is an additional Z 2 symmetry to distinguish the Higgs field from Φ 1/2 , there is a possible dark matter candidate. On the other hand, for N = 3, 5, 7, 9 cases we require extra terms in the Lagrangian in order to make sure no stable charged particles remain. The real part of the neutral component of φ 1 serves as the dark matter, and its mass is around 60 − 80 GeV, in order to satisfy the relic density of the Universe.