Neutrino Mass and Dark Matter from Gauged B-L Breaking 1footnote 11footnote 1This talk is based on Ref. Kanemura:2014rpa ().

# Neutrino Mass and Dark Matter from Gauged B−L Breaking 111This talk is based on Ref. Kanemura:2014rpa ().

Toshinori Matsui Department of Physics, University of Toyama, Toyama 930-8555, Japan
###### Abstract

We discuss a new radiative seesaw model with the gauged BL symmetry which is spontaneously broken. We improve the previous model by using the anomaly-free condition without introducing too many fermions. In our model, dark matter, tiny neutrino masses and neutrino oscillation data can be explained simultaneously, assuming the BL symmetry breaking at the TeV scale.

## I Introduction

The neutrino oscillation data Aharmim:2011vm (); Abe:2014ugx (); An:2013zwz () have shown us neutrinos have tiny masses. If are introduced to the standard model of particle physics (SM), there are two possible mass terms for neutrinos (See e.g., Ref. Ref:seesaw ()), the Dirac type and the Majorana type . In radiative seesaw models (See e.g., Refs. Ref:KNT (); Ref:Ma (); Ref:AKS (); Aoki:2011yk (); Kanemura:2012rj (); Ref:KNS (); Kanemura:2013qva ()), an ad hoc unbroken symmetry forbids generating neutrino masses at the tree level and explains the dark matter (DM) stability. A model in Ref. Ref:KNS () was constructed such that the breaking of the gauge symmetry gives a residual symmetry for the DM stability and the Majorana neutrino mass of . However, the anomaly cancelation for the gauge symmetry requires to introduce more additional fermions except for particles for the radiative neutrino mass.

In this talk, we propose a new model which is an improved version of the model in Ref. Ref:KNS () from the view point of the anomaly cancellation. With appropriate charge assignments, there exists an unbroken global symmetry even after the breakdown of the symmetry. The global symmetry stabilizes the DM, so that we hereafter call it . In our work, the DM candidate is a new scalar boson. Furthermore, the Dirac mass term of neutrinos is radiatively generated at the one-loop level due to the quantum effect of the new particles. Tiny neutrino masses are explained by the two-loop diagrams with a Type-I-Seesaw-like mechanism. We find that the model can satisfy current data from the neutrino oscillation, the lepton flavor violation (LFV), the relic abundance and the direct search for the DM, and the LHC experiment.

## Ii Model

We introduce new particles which listed in Table 2. We determine assignment of charges from conditions for cancellation of the and anomalies;

 3−13NνR−23Nψ=0,3−127NνR+(−2x2−43x−827)Nψ=0, (1)

where is the number of (the same as the number of ), and is the number of .

There are four solutions as presented in Table 2. Except for Case III, the charges of some new particles are irrational numbers while the symmetry is spontaneously broken by the vacuum expectation value (VEV) of whose charge is a rational number. Therefore, the irrational charges are conserved, and the lightest particle with an irrational charge becomes stable so that the particle can be regarded as a DM candidate. In this talk, we take Case IV as an example.

In addition to the SM one, the new Yukawa interactions are given by

 LY=−(yR)i¯¯¯¯¯¯¯¯¯¯¯¯(νR)i(νR)ci(σ0)∗−(yψ)i¯¯¯¯¯¯¯¯¯¯¯¯(ψR)i(ψL)i(σ0)∗−hij¯¯¯¯¯¯¯¯¯¯¯¯(ψL)i(νR)js0−fℓi¯¯¯¯¯¯¯¯¯¯¯¯¯(LL)ℓ(ψR)i~η+h.c., (2)

where . The scalar potential in our model is the same as that in the previous model Ref:KNS ():

 V=−μ2ϕΦ†Φ+μ2s|s0|2+μ2ηη†η−μ2σ|σ0|2+μ3(s0η†Φ+h.c.)+λϕ(Φ†Φ)2+λs|s0|4+λη(η†η)2+λσ|σ0|4 +λsσ|s0|2|σ0|2+λsη|s0|2η†η+λsϕ|s0|2Φ†Φ+λση|σ0|2η†η+λσϕ|σ0|2Φ†Φ+λϕϕ(η†η)(Φ†Φ)+ληϕ(η†Φ)(Φ†η). (3)

Neutral scalar fields are given by . Two scalar fields and obtain VEVs  [] and  []. The VEV provides a mass of the gauge boson as , where is the gauge coupling constant. After the gauge symmetry breaking with and , we can confirm in Eqs. (2) and (3) that there is a residual global symmetry, for which irrational -charged particles (, , , and ) have the same -charge while the other particles are neutral.

Two CP-even scalar particles and are obtained by - mixing as . Two neutral complex scalars and are obtained by - mixing as . Scalar masses are given by

 m2h0,H0=λϕv2ϕ+λσv2σ∓√(λϕv2ϕ−λσv2σ)2+λ2σϕv2ϕv2σ,m2H01,H02=12(m2η+m2s∓√(m2η−m2s)2+2μ23v2ϕ), (4)

where , . The mass of the charged scalar  is . Nambu-Goldstone bosons and are absorbed by and bosons, respectively.

## Iii Phenomenology

### iii.1 Neutrino masses

Tiny neutrino masses are generated by two-loop diagrams in Fig. 1 Ref:KNS (). The mass matrix is expressed in the flavor basis as

 (mν)ℓℓ′ = ∑i,j,afℓihia(mR)a(hT)aj(fT)jℓ′[(I1)ija+(I2)ija]/(16π2)2, (5)

where explicit formulas of and are shown in Ref. Kanemura:2014rpa (). The neutrino mass matrix is diagonalized by a unitary matrix , the so-called Maki-Nakagawa-Sakata (MNS) matrix Maki:1962mu (), as . We take  () to be real and positive values. Two differences of three phases are physical Majorana phases. In our analysis, the following values Abe:2014ugx (); An:2013zwz (); Aharmim:2011vm () obtained by neutrino oscillation measurements are used in order to search for a benchmark point of model parameters:

 sin22θ23=1,sin22θ13=0.09,tan2θ12=0.427,δ=0,{α1,α2,α3}={0,0,0}, (6) m1=10−4eV,Δm221=7.46×10−5eV2,Δm232=+2.51×10−3eV2,where Δm2ij≡m2i−m2j. (7)

By using an ansatz Kanemura:2014rpa () for the structure of Yukawa matrix , we found a benchmark point as

 (8) (9) (10)

The values of correspond to , and . The values of and can be produced by , , and .

### iii.2 Lepton flavor violation

We consider the condition of the LFV decays of charged leptons. The charged scalar contributes to the branching ratio (BR) of whose formula have been calculated Hisano:1995cp (). At the benchmark point, we have which satisfies the current constraint  (90% C.L.) Ref:MEG ().

### iii.3 Dark matter

In our model, the scalar turns out to be the DM candidate due to the following reason. If the DM is the fermion , it annihilates into a pair of SM particles via the -channel process mediated by and . Even for a maximal mixing  Ref:BL-2 (), the observed abundance of the DM Ade:2013zuv () requires . The current constraint from direct searches of the DM Ref:LUX () requires larger in order to suppress the contribution.

The scalar DM at the benchmark point is dominantly made from which is a gauge-singlet field under the SM gauge group, because of the tiny mixing . The annihilation of into a pair of the SM particles is dominantly caused by the -channel scalar mediation via  Kanemura:2010sh () because is assumed to be heavy. The coupling constant for the interaction controls the annihilation cross section, the invisible decay in the case of kinematically accessible, and the contribution to the spin-independent scattering cross section on a nucleon. In Ref. Cline:2013gha (), for example, we see that with and can satisfy constraints from the relic abundance of the DM and the invisible decay of . We see also that the contribution to is small enough to satisfy the current constraint for  Ref:LUX (). Although the scattering of on a nucleon is mediated also by the boson in this model, the contribution can be suppressed by taking a large . The benchmark point corresponds to and gives about for the scattering cross section via , which is smaller than the current constraint Ref:LUX () by an order of magnitude. Thus, the constraint from the direct search of the DM is also satisfied at the benchmark point.

### iii.4 Z’ and νR search

The LEP-II bound  LEPZp () is satisfied at the benchmark point because of which we take for a sufficient suppression of for the direct search of the DM. The production cross section of with and is about at the LHC with  Ref:BL-Pheno (). Notice that the current bound at the LHC LHCZp () is for the case where the gauge coupling for is the same as the one for , namely . Decay branching ratios of are shown at the benchmark point in Table 3. Decays of are dominated by with the Yukawa coupling constants because for are small in order to satisfy the constraint. The  () decays into via the trilinear coupling constant . The main decay mode of is through the mixing between and .

The decay into is forbidden because it is heavier than at the benchmark point. Since the BL charge of is rather small, is not produced directly from . However, can be produced through the decays of . As a result, about of produces . For  () followed by the hadronic decay of  (), the would be reconstructed. In this model, an invariant mass of a pair of the reconstructed is not at in contrast with a naive model where only three with are introduced to the SM. This feature of also enables us to distinguish this model from the previous model in Ref. Ref:KNS () where with can be directly produced by the decay.

## Iv Conclusions

We have improved the model in Ref. Ref:KNS () by considering anomaly cancellation of the gauge symmetry. We have shown that there are four anomaly-free cases of BL charge assignment, and three of them have an unbroken global symmetry. The guarantees that the lightest -charged particle is stable such that it can be regarded as a DM candidate. The spontaneous breaking of the symmetry generates the Majorana mass term of and masses of new fermions . In addition, the Dirac mass term of neutrinos is generated at the one-loop level where the DM candidate involved in the loop. Tiny neutrino masses are obtained at the two-loop level.

The case of the fermion DM is excluded, and the lightest -charged scalar should be the DM in this model. We have found a benchmark point of model parameters which satisfies current constraints from neutrino oscillation data, lepton flavor violation searches, the relic abundance of the DM, direct searches for the DM, and the LHC experiments. In such radiative seesaw models, would be produced at the LHC. In our model, cannot be directly produced by the decay, but can be produced by the cascade decay . By the unusual BL charge of , the invariant mass distribution of does not take a peak at , which could be a characteristic signal.

###### Acknowledgements.
This work is based on the collaboration with Shinya Kanemura and Hiroaki Sugiyama. I would like to thank them for their support.

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