Four-loop Neutrino Model Inspired by Diphoton Excess at 750 GeV

We propose a four-loop induced radiative neutrino mass model inspired by the diphoton excess at 750 GeV recently reported by ATLAS and CMS, in which a sizable diphoton excess is obtained via photon fusion introducing multi doubly-charged scalar bosons. Also we discuss the muon anomalous magnetic moment, and a dark matter candidate. The main process to explain the observed relic density relies on the final state of the new particle at 750 GeV. Finally we show the numerical results and obtain allowed region of several physical values in our model.


I. INTRODUCTION
According to the recent announcements by ATLAS and CMS experiments, a new particle could exist at around 750 GeV by the observation of the diphoton invariant mass spectrum from the run-II data in 13 TeV [1,2]. Subsequently a vast of paper along this line of issue has been arisen in Ref. . One of these interpretations is to identify a scalar (or pseudoscalar) as the new particle (S), and the resonance occurs in the process; pp → S + X → 2γ + X, where X is the missing particle. This can be interpreted as the following 13 TeV data in terms of the production cross section of S and its branching ratio of two photons, which is extremely large compared to the previous observations from the run-I data at 8 TeV [138,139]. Also the ATLAS experiment group [1] reported Γ S = 45 GeV that is the best fit value of the decay width of S to the two photons, and Γ S = 5.3 GeV is given as the experimental resolution obtained by the analysis [14]. To achieve such a large signal strength, we have to enlarge the production cross section and (or) its branching ratio. One of the simplest ways to enhance the production cross section is to introduce a vector like exotic quark that couples to S, where such a quark induces the gluon fusion production of S that can be always dominant process [11]. On the other hand, one of the simplest ways to increase the branching ratio to photons is that S should couples to the isospin singlet bosons or fermions with nonzero electric charges, because main modes such as a pair of W ± bosons can be forbidden. However once one can reach the enough branching ratio to the two photons, (which is around ≈ 60 %), the dominant production cross section can also be arisen from the photon fusion process, which is proposed by, i.e., Ref. [37]. This scenario is in favor of leptonic models, especially, radiative seesaw models, when such charged particles also interact with lepton sector. Especially there are some representative radiative seesaw models at the three-loop level [135][136][137]. In this framework, the recent paper [126] has concluded that the O(10 3 − 10 4 ) number of electrically charged bosons that propagate between S and two photons have to be introduced as can be seen in Fig. 1, 1 in order to satisfy the condition of unitarity bound via processes such as k ±± S → k ±± → k ±± S and 2k ±± → S → 2k ±± . Therefore, the trilinear term µ S proportional to Sk ±± k ∓∓ should be nearly equal or less than m S ≈ 750 GeV. The relevant potential per k ±± to generate the diphoton anomaly is simply given by Then the total cross section with m S =750 GeV at 13 TeV is given by [37] In our case the cross section simplifies the following values due to BR(S → 2γ) ≈ 60%, This result could drastically changes the situation of any radiative seesaw models that include electrically charged bosons such as Zee-Babu model [140], which is the first proposal including the doubly charged boson, because the scale of neutrino masses must be enhanced by N CB . To show this issue more clearly, let us consider the Zee-Babu model. The model has the following relevant terms per k ±± : Then the resulting neutrino mass has to be multiplied by N CB , and can be estimated as The diphoton excess is analyzed by rather general way, introducing arbitral number of doubly charged bosons with isospin singlet in this paper, although they fix a specific model in the neutrino sector. Hence one can apply some results to any kind of leptonic models that include charged bosons with isospin singlet even when singly charged bosons. where , and loop factor is order 1. It suggests that the neutrino mass scale is determined by the trilinear coupling µ and the Yukawa couplings, and N CB that almost compensates the two loop suppression effect. Therefore the two loop neutrino mass scale is equivalent to the tree level scale.
Applying this fact, we will discuss our radiative neutrino model at the four loop level in the next section, which could be equivalent to a typical two loop radiative model. Then we will conclude and discuss in Sec. III.

Lepton Fields
Scalar Fields

II. MODEL SETUP AND ANALYSIS
In this section, we explain our model with global U(1) symmetry. The particle contents and their charges are shown in Tab. I. We add a vector-like exotic doubly charged fermion E, a Majorana fermion N R , a singly charged scalar h ± , the N CB number of doubly charged scalars k ±± , and a neutral scalar S to the SM, where all these new fields are iso-spin singlet, and S is identified as a new scalar with 750 GeV mass. We assume that only the SM Higgs Φ and S have vacuum expectation values (VEVs), which are respectively symbolized by v/ √ 2 and v S / √ 2. The quantum number ℓ = 0 of U(1) symmetry is arbitrary, but its assignment for each field is unique to realize our four loop neutrino model.
The relevant Lagrangian and Higgs potential under these symmetries per k ±± are given where τ 2 is a second component of the Pauli matrix. After the global U(1) spontaneous breaking of S, we obtain trilinear terms as well as the Majorana masses as follows:  isospin doublet scalar field can be parameterized as Φ = [w + , v+φ+iz √ 2 ] T where v ≃ 246 GeV is VEV of the Higgs doublet, and w ± and z are respectively absorbed by the longitudinal component of W and Z boson. The isospin singlet scalar field can be parameterized as Here we assume φ is the SM Higgs, therefore we neglect the mixing between φ and s for simplicity. We also assume that the lightest Majorana fermion N R | lightest = X does not couple to E R and k ±± in the fourth term of L Y and does not mix with other N R so that it can be stable and a DM candidate. Such a situation for DM can easily be realized by imposing additional Z 2 odd assignment.

Neutrino mass matrix:
Then the leading contribution to the active neutrino masses m ν is given at four-loop level as shown in Figure 2, and we can respectively estimate the order of masses as follows: Hence we can approximate the neutrino masses as where we take G I = O(1), and m ν should be 0.001 eV m ν 0.1 eV from the neutrino oscillation data [141].
Our formula of muon g − 2 is given by (II.10) Dark matter: Assuming the lightest Majorana particle of N R as our DM candidate, which is denoted by X, we find the dominant mode to explain the observed relic density Ωh 2 ≈ 0.12 [144].
Our dominant non-relativistic cross section comes from 2X → 2s with t-and u-channels 2 , and its formula is given by Then the relic density is formulated by where M P ≈ 1.22 × 10 19 GeV is the Planck mass, g * ≈ 100 is the total number of effective relativistic degrees of freedom at the time of freeze-out, and x F ≈ 25. In our numerical analysis below, we set the allowed region to be 0.11 Ωh 2 0.13, (II. 13) where mass relation Numerical results: Now we randomly select values of the twelve parameters within the corresponding ranges % confidential level, and its formula is given by (II.17) where these sample points satisfy the allowed regions in Eqs. (II.15) and (II.16) respectively.
We also estimate the cross section of doubly charged scalar production, i.e. pp → γ * /Z * → k ++ k −− . Although each pair production cross section is small the sum of the cross section for N CB pair can be sizable. The production cross section is numerically estimated by CalcHEP [146] implementing relevant interactions and using CTEQ6L PDF [147].
The left(right) plots in Fig. 3 show the sum of the k ++ k −− production cross section at the LHC 13 (14) TeV applying N CB = 6000. Note that the total cross section is simply N CB ×(each k ++ k −− production cross section). We thus find that the doubly charged scalar could be produced at the LHC run-II with O(100) fb cross section when m k ±± ∼ 1 TeV.
The doubly charged scalar then decays as k ±± → h ± h ± → ℓ ± ℓ ± νν where ℓ = e, µ and τ . Therefore the signal of the k ++ k −− pair is four charged lepton plus missing transverse energy.
We have proposed a four-loop induced radiative neutrino mass model inspired by the diphoton excess at 750 GeV recently reported by ATLAS and CMS, in which a sizable diphoton excess is obtained via photon fusion introducing multi doubly-charged scalar bosons. The sizable neutrino mass scale has been obtained due to the enhancement of the number of doubly charged bosons N CB . Also we have discussed the muon anomalous magnetic moment, and a dark matter candidate of the lightest fermion X, and we have found that the main process to explain the correct relic density relies on the final state of the new particle at 750 GeV through the t-and u-channels. Finally we have shown the numerical results and have obtained allowed region of several physical values in our model, as can be seen in Eqs (II. 15) for N CB = 6000 and Eqs (II.16) for N CB = 9000 respectively. The doubly charged scalar production cross section has been numerically estimated. Then we have found that sum of the pair production cross section can be as large as O(100) fb for m k ±± ∼ 1 TeV. Therefore our model could be tested at the LHC run-II by searching for the signal of four charged lepton plus missing transverse energy which is obtained as Further analysis of the signal is left as future work.