Renormalizable Model for Neutrino Mass, Dark Matter, Muon $g-2$ and 750 GeV Diphoton Excess

We discuss a possibility to explain the 750 GeV diphoton excess observed at the LHC in a three-loop neutrino mass model which has a similar structure to the model by Krauss, Nasri and Trodden. Tiny neutrino masses are naturally generated by the loop effect of new particles with their couplings and masses to be of order 0.1-1 and TeV, respectively. The lightest right-handed neutrino, which runs in the three-loop diagram, can be a dark matter candidate. In addition, the deviation in the measured value of the muon anomalous magnetic moment from its prediction in the standard model can be compensated by one-loop diagrams with exotic multi-charged leptons and scalar bosons. For the diphoton event, an additional isospin singlet real scalar field plays the role to explain the excess by taking its mass of 750 GeV, where it is produced from the gluon fusion production via the mixing with the standard model like Higgs boson. We find that the cross section of the diphoton process can be obtained to be a few fb level by taking the masses of new charged particles to be about 375 GeV and related coupling constants to be order 1.

fb −1 . The detailed properties of the diphoton excess was summarized, e.g., in Ref. [3], where the best fit value of the width of the new resonance is about 45 GeV, and the estimated cross section of the diphoton signature is 10 ± 3 fb at ATLAS and 6 ± 3 fb at CMS. If this excess is confirmed by future data, it suggests the existence of a new particle which gives the direct evidence of a new physics beyond the standard model (SM).
The simplest way to explain this excess is to consider an extension of the SM by adding extra isospin scalar multiplets such as a singlet, a doublet and/or a triplet. However, it is difficult to get a sufficient cross section to explain the excess as mentioned in the above in such a simple extension of the SM. For example, if we consider the CP-conserving two Higgs doublet models (THDMs) [4][5][6][7][8], and take the masses of the additional CP-even H and CP-odd A Higgs bosons to be 750 GeV, then the cross section of pp → H/A → γγ is typically three order smaller than the required value [4]. Therefore, we need to further introduce additional sources to get an enhancement of the production cross section and/or the branching fraction to the diphoton mode, e.g., by introducing multi-charged scalar particles [4,6] and vector-like fermions [7]. In Refs. [8], the diphoton excess has been discussed in supersymmetric models.
In this paper, we discuss a scenario to naturally introduce multi-charged particles to get an enhancement of the branching fraction. Namely, we consider a radiative neutrino mass model in which multi-charged particles play a role not only to increase the branching fraction but also to explain the smallness of neutrino masses and the anomaly of the muon anomalous magnetic moment. A dark matter (DM) candidate can also successfully be involved as a part of the model.
There are a few papers discussing the diphoton excess within radiative neutrino mass models [9].
In particular, we discuss a new three loop neutrino mass model 1 whose structure is similar to the model by Krauss, Nasri and Trodden in 2003 [10], because the three loop suppression factor 1/(16π 2 ) 3 is a suitable amount to reproduce the measured neutrino masses, i.e., O(0.1) eV, by order 0.1-1 couplings and TeV scale masses of new particles. In our model, an additional isospin real singlet scalar field can explain the diphoton excess, where it is produced from the gluon fusion process through the mixing with the SM-like Higgs boson.
The plan of the paper is as follows. In Sec. II, we define our model, and give the Lagrangian for the lepton sector and the scalar potential. In Sec. III, we discuss the neutrino masses, the phenomenology of DM including the relic abundance and direct search experiments, and new contributions to the muon g − 2. The diphoton excess is discussed in Sec. IV. Our conclusion is summarized in Sec. V.

Lepton Fields Scalar Fields
The superscripts i and a denote the flavor of the SM fermions and the exotic fermions with i = 1-3 and a = 1-N E , respectively.

II. THE MODEL
Our model is described by the SM gauge symmetry SU (2) L × U (1) Y and an additional discrete symmetry Z 2 which is assumed to be unbroken. This Z 2 symmetry is introduced to avoid tree level contributions to neutrino masses and to enclose the three-loop diagram as shown in Fig. 1.
Because of the Z 2 symmetry, the stability of the lightest neutral Z 2 odd particle is guaranteed, and thus it can be a candidate of DM.
The particle contents are shown in Table I, where L i L and e i R are the SM left-handed lepton doublets and lepton singlets with the flavor of i (i =1-3). In addition, we add the N E flavor of the vector like lepton doublets (singlets) L a with the hypercharge Y = −5/2 (−2) and the right-handed neutrinos N a R (a = 1-N E ). The scalar sector is composed of one isospin doublet field Φ with Y = 1/2 and two complex (one real) isospin singlet scalar fields κ ++ and S ++ with Y = 2 (Σ with Y = 0). The doublet and the real singlet scalar fields are parameterized by where v and v σ are the vacuum expectation values (VEVs) of doublet and singlet scalar fields, respectively, and G + (G 0 ) denotes the Nambu-Goldstone boson which is absorbed into the longitudinal component of the W (Z) boson. The Fermi constant G F is given by the usual relation, i.e., GeV. The singlet VEV v σ does not contribute to the electroweak symmetry breaking. We note that the shift v σ → v ′ σ does not change any physical quantities, because its impact can be absorbed by the redefinition of the parameters in the Lagrangian. We thus take v σ = 0 in the following discussion to make some expressions to be a simple form.
The most general Lagrangian for the lepton fields is given by whereΦ = iσ 2 Φ * . We can take the diagonal form of the invariant masses M a N , M a L and M a E for the vector like leptons L a 5/2 , E a and right-handed neutrinos N a R , respectively, without loss of generality. The SM leptons L L and e R are taken to be the mass eigenstates, so that the Yukawa coupling y i SM is given by the diagonal form. For simplicity, we assume that all the above parameters are real.
The most general Higgs potential is given by where the complex phase of the λ 0 parameter can be absorbed by rephasing the scalar fields. The squared masses of the doubly-charged scalar bosons S ±± and κ ±± are given by In Eq. (3), the V HSM part is given as the same form as in the Higgs singlet model (HSM) involving Φ and Σ as Two CP-even scalar states φ 0 from the doublet and s 0 from the singlet are mixed with each other via the mixing angle α defined as We define h as the SM-like Higgs boson with the mass of about 125 GeV which is identified as the discovered Higgs boson at the LHC. The detailed expressions for the masses of the CP-even Higgs bosons and the mixing angle α in terms of the potential parameters are given, e.g., in Ref. [17].
The masses of the exotic charged leptons are obtained from two sources, i.e., the invariant mass terms M E and M L and the Yukawa interaction terms y 1 and y 2 . The mass of the triply-charged leptons L −−−a is simply given by M a L . For the doubly-charged leptons, there is a mixing between L −−a and E −−a through the y 1 and y 2 terms. The mass matrix is given assuming y ab where M a D = v √ 2 y a E . The mass eigenstates E a 1 and E a 2 are defined by the orthogonal transformation: The mass eigenvalues (M a E 1 ≤ M a E 2 ) and the mixing angles θ a are given by III. NEUTRINO MASS, DARK MATTER, MUON g − 2

A. Neutrino Mass
The leading contribution to the active neutrino mass matrix m ν is given at three-loop level as shown in Fig. 1. One-and two-loop diagrams which have been systematically classified in Refs. [18,19] are absent in our setup. The three-loop diagram is computed as follows and m X is the mass of a particle X, and c αβ = 1 (−1) for α = β (α = β). The three loop function F is given by where The interval of the integrals in Eq. (12) for all the variables is from 0 to 1, i.e., where we assume λ 0 × F = O(1). Therefore, in the range of M max = v-10 v, the magnitude of the mixing factor K ij is required to be O(10 −7 -10 −6 ).

B. Dark Matter
Assuming that the right-handed neutrino N 1 R is the lightest among all the Z 2 odd particles, N 1 R looses its decay modes into any other lighter particles, and then it becomes stable. We thus can regard N 1 R as the DM candidate in our model. The annihilation cross section is then calculated as where m fin is the mass of the final state particle. In the above expression, |M(N 1 R N 1 R → AB)| 2 is the squared amplitude for the following two body to two body processes: The first annihilation process N 1 R N 1 R → κ ++ κ −− happens through the t-and u-channels of the E a α mediation, where the doubly-charged scalar bosons κ ±± decays into the same sign dilepton via the Yukawa coupling h 0 . The squared amplitude of the N 1 R N 1 R → κ ++ κ −− process is given by where s, t and u are the Mandelstam variables, N f c is the color factor, and (p 1 , p 2 ) and (k 1 , k 2 ) are the initial and the final state momenta, respectively. In this expression, we take h ab ≡ h ab 1 = h ab 2 for simplicity. The other cross sections are given through the mixing of α via the s-channel mediation of h and H by where we use the short-hand notations of c α ≡ cos α and s α ≡ sin α. The dimensionful λ ϕ i ϕ j ϕ k at the 2σ level [20].
We also consider the spin independent scattering cross section with a neutron that is induced via the tree level diagram with the Higgs boson h and H exchange. The formula is given by where the neutron mass is m n ≃ 0.939 GeV and the factor C ≃ 0.287 2 is determined by the lattice simulation. The latest upper bound is reported by the LUX experiment that suggests σ n The muon anomalous magnetic moment (muon g − 2) is one of the most promising low energy observables which suggest the existence of new physics beyond the SM. This is because there is the more than 3σ deviation in the SM prediction from the experimental value measured at Brookhaven National Laboratory. The difference ∆a µ ≡ a exp µ − a SM µ has been calculated in Ref. [22] as ∆a µ = (29.0 ± 9.0) × 10 −10 .
This shows the 3.2σ deviation in the SM prediction.
In our model, two diagrams contribute to ∆a µ , where L −−−a -S −−a and ℓ − -κ −−a with ℓ − being the SM lepton are running in the loop. These contributions are calculated by We can see that the contribution from the κ ±± loop gives the negative value which is undesired to explain the muon g − 2 anomaly. We thus neglect the κ ±± loop contribution that can be realized by takingh µi 0 ≪ f µa .

IV. DIPHOTON EXCESS
We discuss how we can reproduce the diphoton excess at around 750 GeV at the LHC. In our model, the additional CP-even Higgs boson H plays the role to explain this excess via the gluon fusion production process by taking its mass of 750 GeV. The cross section σ γγ of the diphoton channel is expressed by using the narrow width approximation as follows Non-zero production cross section σ(gg → H) of the gluon fusion process is given through the mixing α with the SM-like Higgs boson h defined in Eq. (6) as where h SM denotes the SM Higgs boson, and σ(gg → h SM ) does its gluon fusion cross section in which the mass of h SM here is fixed to be 750 GeV in order to derive the cross section for H.
The decay rates of H → PP ′ with H = h or H and PP ′ = ff, W + W − , ZZ or gg are given by where ξ H = sin α (cos α) for H = H (h). For the γγ and Zγ modes, the decay rate is not simply given by the above way due to the additional loop contributions of the new charged particles. In order to simplify the discussion, we take flavor universal valuables for the masses of the exotic charged leptons and the mixing angles, i.e., M a Eα = M Eα and θ a = θ E as we have done it in the previous section. In this case, the decay rates for H → γγ and H → Zγ are given by whereξ H = cos α (− sin α) for H = H (h) and Q X denotes the electric charge, i.e., Q t = 2/3, In the above formulae, The Yukawa couplings y HEαE β and the scalar trilinear couplings λ Hφ ++ φ −− are given by The contribution of the SM particles to H → γγ (F SM ) and H → Zγ (G SM ) are expressed as with I f = 1/2 (−1/2) for f = t (b). The loop functions for the γγ mode are expressed by The value of the Yukawa coupling g S is taken to be 1, 2 and 3 in all the panels. For the right panel, the measured value of µ γγ , i.e., µ exp γγ = 1.14 ± 0.76 [26] at the LHC Run-I experiment is also shown, where the solid and dashed curves denote the central value and the 2σ limit, respectively. We obtain the cross section to be about 0.6, 1.4 and 2.4 fb when | sin α| 0.1, 0.15 and 0.2 in the case of g S = 1, 2 and 3, respectively. Regarding to the width Γ H , its value strongly depends on sin α, while the dependence on g S is quite weak. We find that Γ H ≃ 2.4 (8.5) GeV at | sin α| = 0.1 (0.2) with g S = 1. For σ γγ and Γ H , the sign of sin α does not become important so much, while that for µ γγ does quite important. This can be understood in such a way that the interference effect in the h → γγ process between the W boson loop and the exotic lepton loops becomes constructive (destructive) when sin α is positive (negative). Because of this destructive effect, the value of µ γγ becomes zero at sin α 0, and it rapidly grows when sin α is taken to be a different value from that giving µ γγ = 0. Therefore, the case with sin α taken to be a bit different value from that giving µ γγ = 0 is allowed by the current experimental data µ exp γγ . For the other signal strengths which have been measured at LHC, i.e., µ ZZ , µ W W and µ τ τ , they are calculated by cos 2 α at the tree level. In the range of sin α that we take in Fig. 2, we obtain cos 2 α > 0.91, so that these signal strengths are allowed at the 2σ level from the LHC Run-I data [27,28].
In Fig. 3, we show the contour plots of σ γγ on the sin α-g S plane in the case of λ HS ++ S −− = λ Hκ ++ κ −− = 0. The left, center and right panels respectively show the case of N E = 3, 6 and 9. We The left, center and right panels respectively show the case of N E = 3, 6 and 9. restrict the range of sin α to be 0 to −0.3, because the positive value of sin α is highly disfavored by µ exp γγ as we see in Fig. 2. The shaded region is excluded by µ exp γγ at the 2σ level. We find that the maximally allowed value of the cross section σ γγ is about 1.5 fb, 2.5 fb and 3 fb when N E is taken to be 3, 6 and 9, respectively.

V. CONCLUSIONS
We have constructed the three-loop neutrino mass model whose structure is similar to the model by Krauss, Nasri and Trodden. The neutrino masses of O(0.1) eV are naturally generated by the loop effect of new particles with their couplings and masses to be of order 0.1-1 and TeV, respectively. We have analyzed the Majorana DM candidate, assuming the lightest of N R . The non-relativistic cross section to explain the observed relic density is p-wave dominant, and there are several processes; N 1 R N 1 R → κ ++ κ −− with the t− and u−channels, and N 1 The dominant DM scattering with a nucleus comes from the Higgs boson mediation h and H at the tree level, and we have calculated the spin independent cross section of the process. Furthermore, the anomaly of the muon g − 2 can be solved by the one-loop contribution of the triply-charged exotic leptons and doubly-charged scalar boson. We have found the benchmark parameter set to satisfy the relic abundance of the DM, the constraint from the direct search experiment and to compensate the deviation in the measured value of the muon g − 2 from the SM prediction.
We then have numerically shown the cross section of the diphoton process via the gluon fusion production gg → H → γγ and the width of H under the constraint from the signal strength µ γγ for the SM-like Higgs boson measured at the LHC Run-I experiment. We have obtained the width to be about 3-5 GeV in the typical parameter region, which gives a tension to the measured value, i.e., about 45 GeV. We have found that the cross section of the diphoton process is given to be a few fb level by taking the masses of new charged fermions and scalar bosons to be 375 GeV with an order 1 coupling constant. A bit larger cross section such as about 4 fb is obtained by taking the larger number of flavor N E of the exotic leptons and take a non-zero negative value of the trilinear scalar boson couplings λ HS ++ S −− and λ Hκ ++ κ −− .