Generalized Zee-Babu model with 750 GeV Diphoton Resonance

We propose a generalized Zee-Babu model with a global $U(1)$ B-L symmetry, in which we classify the model in terms of the number of the hypercharge $N/2$ of the isospin doublet exotic charged fermions. Corresponding to each of the number of $N$, we need to introduce some multiply charged bosons in order to make the exotic fields decay into the standard model fields. We also discuss the muon anomalous magnetic moment and the diphoton excess depending on $N$, and we show what kind of models are in favor of these phenomenologies.


I. INTRODUCTION
The recent measurements reported by ATLAS and CMS experiments implies that a new particle (Φ New ) might exist at around 750 GeV by the diphoton invariant mass spectrum from the run-II data in 13 TeV [1,2]. And a typical interpretation is known as Furthermore the ATLAS experiment group [1] announced Γ Φ New = 45 GeV, which is the best fit value of the total decay width of Φ New , while the CMS experiment group [2] indicated a rather smaller decay width. It might suggest that Φ New be a scalar (or pseudoscalar) and additional new fields with nonzero electric charges are in favor of being introduced, since sizable branching fraction for Φ New → γγ requires Φ New strongly interacts with such charged fields. In order to provide reasonable explanations (or interpretations), a vast of paper along this line of issue has been recently arisen in Ref. .
Zee-Babu type [182] of radiative seesaw models could provide one of the economical scenarios to include such new exotic fields with nonzero electric charges (bosons or fermions) that are naturally introduced in order not only to explain the diphoton excess but also to explain the tiny neutrino masses. Also the model can easily be extended to the multi-charged bosons and fermons.
In our paper, we propose a generalized Zee- In Sec. II, we show our model, including neutrino sector, and muon anomalous magnetic moment. In Sec. III, we discuss the decay processes of exotic fields. In Sec. IV, we discuss the diphoton excess. We conclude and discuss in Sec. V.

Scalar Fields
is the odd number.

II. MODEL SETUP AND ANALYSIS
In this section, we explain our model with a global U(1) B−L symmetry. The particle contents and their charges are shown in Tab. I. As for the fermions, we add some vector-like exotic isospin doublet charged fermions L ′ with −N/2 hypercharge and isospin singlet fermions E ′ with −N/2 hypercharge, where N(= 1, 3, 5, ...) is generally an arbitrary odd number. As for the scalars, we introduce a ±M( = 0) electric charged scalar h ±M , and a ±2M charged scalar k ±2M with different U(1) B−L quantum numbers, and a neutral scalar ϕ in addition to the SM. Notice here that the electric charge to the fields with the −N/2 hypercharge, i.e., L ′ , is given as Thus we define L ′ as We assume that only the Higgs doublet Φ and ϕ have vacuum expectation values (VEVs), which are respectively symbolized by v/ √ 2 and v ′ / √ 2.
The relevant Lagrangian and Higgs potential under these symmetries are given by The isospin doublet scalar field is parameterized as Φ = [w + , v+h 1 +iz Here h and H denote SM Higgs and heavier CP-even Higgs respectively.
Exotic Charged Fermion mass matrix: The exotic charged fermion mass matrix with ±M electric charges is given by where we assume to be The mass eigenstates E 1 and E 2 are defined by the bi-unitary transformation: where s θ E ≡ sin θ E and c θ E ≡ cos θ E . The mass eigenvalues and the mixing angles θ E are respectively given by where we define M E 1 < M E 2 , and the mass of the ±(M +1) electric charged fermion ψ ±(M +1) is given by M L .

Neutrino mass matrix:
The leading contribution to the active neutrino masses m ν is given at two-loop level as shown in Figure 1, and its formula [184] is given as follows: , where m ν should be 0.001 eV m ν 0.1 eV from the neutrino oscillation data [183].
Reminding the original Zee-Babu model, the loop function Π 2 is the order 1. Once we fix to be Π 2 = 1, we obtain the following parameter region to satisfy the neutrino mass scale as which can easily be realized due to a lot of free parameters.
Our formula of muon g − 2 is given by In fig. 2  In order to make the analysis simplify, we just assume to be 2m h ±M < m k ±2M . Therefore all we have to take care of the decay is how to make the h ±M or E ±M field decay into the SM fields, which quite depends on the number of N. Thus we classify the model in terms of the concrete number of N below. Notice here that N starts from three, since we assume to be M = 0.

A. N=3
This is equivalent to M = 1. In this case, we can add to write the term −L new ≈ y eEĒL e R ϕ + c.c., (III.1) which suggests the mixing between the SM electric charged leptons and the exotic charged fermions. The mixing makes the neutrino mass matrix complicate, and h ± cannot decay into the SM fields without any additional fields such as another doubly charged boson with +2 U(1) B−L charge, and there exist any allowed region to satisfy the muon anomalous magnetic moment in the last section. Thus we do not mention this case furthermore.

B. N=5
This is equivalent to M = 2. In this case, we can add to write the term −L new ≈ g ′ēc R e R h ++ + c.c., (III.2) that suggest that h −− can decay into the same di charged-leptons. Thus we do not need to add any additional fields, and decay processes are as follows: Notice here that g ′ can contribute to the negative contribution of the anomalous magnetic moment, but we can neglect this effect hereafter because this coupling can be take as a free free parameter.

C. N=7
This is equivalent to M = 3. In this case, introducing a new field S ±± that is an isospin singlet doubly charged boson with ±2 U(1) B−L charge, we can add to write the term where S ±± plays as a role in generating the decaying processes for the exotic fields only.
Then the decay processes are as follows: This is equivalent to M = 4. In this case, introducing a new field S ±± that is an isospin singlet doubly charged boson with ±2 U(1) B−L charge, we can add to write the term where S ±± plays as a role in generating the decaying processes for the exotic fields only.
Then the decay processes are as follows: This is equivalent to M = 5. In this case, introducing new fields (S ± S ±± S ±±±± ) that are respectively isospin singlet (singly, double, fourply) charged bosons with the common ±2 U(1) B−L charges, we can add to write the term where additional fields play as a role in generating the decaying processes for the exotic fields only. Then the decay processes are as follows: This is equivalent to M = 6. In this case, introducing new fields (S ±± S ±±±± ) that are respectively isospin singlet (double, fourply) charged bosons with the common ±2 U(1) B−L charges, we can add to write the term where additional fields play as a role in generating the decaying processes for the exotic fields only. Then the decay processes are as follows:

IV. DIPHOTON EXCESS
We discuss the diphoton excess in case of N = 5, 7, 9, 11, 13 as discussed in the previous section where the candidate of 750 GeV scalar boson is heavy CP even scalar H. The couplings relevant to diphoton decay are obtained from quartic couplings including Φ, ϕ and charged scalar fields; where we denote charged scalar field with electric charge Q as φ Q i in general. After Φ and ϕ get VEV, relevant interactions for mass eigenstates are Since we want to suppress contribution to h → γγ, we assume Then interactions contributing to H → γγ become The heavy CP-even scalar H can be produced by gluon fusion process via mixing with SM Higgs. The cross section is then obtained as [192,193] σ(gg → H) ≃ sin 2 θ × 0.85 pb, (IV.4) in-elastic scattering in Ref. [145] σ(pp(γγ) → H → γγ + X) 13TeV = 10.8pb where X indicate any other associated final states. Therefore total cross section for pp → H → γγ is obtained as where σ γ−fusion is given by Eq. (IV.5).
Through mixing with SM Higgs, H decays into SM particles where the dominant partial decay widths are: Here partial decay widths for other fermion channels are subdominant. The decay process H → γγ is induced by charged particle loops. The partial decay width is given by /m 2 H and we omit SM particle contribution since they are small compared with charged scalar contributions. Note that we also have H → Zγ mode which is subdominant contribution and we omit the formula for decay width here.
The constraint from 8 TeV data should be taken into account. The most stringent constraint in our scenario is given by diphoton search at 8 TeV since our BR(H → γγ) is large. The constraint is [198,199] Here the ratios of 13 TeV cross section and that of 8 TeV are written as σ(gg → H) 13TeV /σ(gg → H) 8TeV ≃ 5 [11] and σ 13TeV γ−fusion /σ 8TeV γ−fusion ≡ R γγ . The R γγ is estimated to be ∼ 2 but the uncertainty is large so that it can be ∼ 4 [31,145]. Thus we investigate the constraint with R γγ = 2 and 4.
We then estimate the product of H production cross section and branching ratio for diphoton channel σ total γγ for the cases of N = 3, 5, 7, 9 and 11. For simplicity, we assume couplings µ Hφ Q i take same value for all charged scalars. Also we choose mass of charged scalar as m φ Q i = 380 GeV to enhance loop function inside the diphoton decay width. The contours of σ total γγ and Γ H are shown in Fig. 3 by solid and dashed lines respectively for N = {5, 7, 9, 11, 13}, and the purple dashed and the red dotted lines indicate constraint from diphoton search at 8 TeV for R γγ = 2 and 4 where the region above the lines are excluded. We find that small sin θ region is strongly constrained for R γγ = 2 but we can obtain ∼ 3 fb cross section for all sin θ. On the other hand, we can obtain ∼ 5 fb cross section for R γγ = 4. Furthermore the trilinear coupling can be less than 1 TeV for 7 ≤ N.
We also find that the total decay width Γ H is O(1 − 10) GeV for the parameter region which explain the diphoton excess. We have estimated the product of H production cross section and branching ratio for diphoton channel. Then the O(1) TeV trilinear coupling is required for N = 5 while the coupling can be smaller for larger N. Thus larger N is preferred to satisfy tree level unitarity and explain the diphoton excess. We also find the total decay width in our scenario is O(1 − 10) GeV. Furthermore we investigated constraint from diphoton search at 8 TeV and we find the parameter region which explain diphoton excess and can satisfy the constraint.
A Dirac type of dark matter can be involved in our theory without conflict of any phe-nomenological point of views [195]. However this analysis is beyond the scope.