Radiative Neutrino Mass in Alternative Left-Right Model

We propose a radiative seesaw model in alternative left-right model without any bidoublet scalar fields, in which all the fermion masses in the standard model are generated through a canonical seesaw mechanism at the tree level. On the other hand the observed neutrino masses are generated at two-loop level. In this paper we focus on the neutrino sector and show how to induce the active neutrino masses. Then we discuss the observed neutrino oscillation, constraints from lepton flavor violations, new sources of muon anomalous magnetic moment, a long-lived dark matter candidate with keV scale mass, and collider physics.


I. INTRODUCTION
Current neutrino oscillation data provide strong evidence of tiny but nonzero neutrino masses [1].
Seesaw mechanism is one of the elegant realization to explain such tiny neutrino masses by introducing right-handed neutrinos, which can naturally be embedded into a left-right symmetry SU (2) L × SU (2) R × U (1) B−L as a theory at TeV scale [2]. 1 On the other hand, radiative seesaw models are also one of the natural realizations to explain the tiny neutrino masses at low energy scale where the neutrino mass matrix is generated at loop level, and a vast paper has recently been arisen in Refs. . Moreover some new particles such as dark matter (DM) and/or electrically charged particles, running inside a loop diagram, are introduced in radiative seesaw models. Thus the radiative seesaw models provide a wide variety of interesting phenomenologies correlated with neutrino sector, and these two scenarios are well compatible [13,14]. Thus it is an attractive interpretation that the active neutrino masses are generated by combination of these mechanisms since neutrino masses are very light compared to the other standard model (SM) fermions. In addition, implementing this scenario into left-right model will be phenomenologically interesting.
In this paper, we combine the left-right model and radiative seesaw model, in which active neutrino masses are generated at two loop level while Dirac neutrino masses are generated at one loop, employing a specific left-right model based on Ref. [120] 2 . And a Majorana mass term of right-handed neutrino is obtained at tree level by introducing SU (2) R triplet scalar ∆ R . But we do not assume the exact left-right symmetry and ∆ L is not introduced. Then we find allowed region of parameter spaces by carrying out numerical analysis where we take into account muon anomalous magnetic moment, various lepton flavor violating processes, and a long lived DM candidate to explain the x-ray line at 7.1 keV [121,122], as well as consistency with the current neutrino oscillation data. This paper is organized as follows. In Sec. II, we show our model building including Higgs masses, neutrino mass matrix. In Sec. III, we discuss lepton flavor violations (LFV), muon anomalous magnetic moment, DM, and collider physics and then carry out numerical analysis to search for the parameter space satisfying all the phenomenological constraints. We conclude in Sec. VI. 1 The left-right symmetry can smoothly be extended into larger groups such as SO(10) symmetry, which is typically realized at higher scale such as grand unified theories. 2 The paper also discusses the quark sector.  In this section, we introduce our model where the gauge symmetry is introduced as In this paper, we focus on the lepton sector and the details of the quark sector is found in Ref. [120].
A. Particle contents and scalar sector The particle contents for leptons and bosons are respectively shown in Tab. I and Tab. II. Here all the new fields are singlet under SU (3) C . We introduce SU (2) R doublet fermions of L R and isospin singlet vector-like fermions of E L(R) both of which have three flavors like SM fermions. As for new bosons, we introduce two SU (2) L(R) doublet scalars Φ L and Φ R , an isospin singlet singly-charged scalar h ± , and an SU (2) R triplet scalar ∆ R . Note here that Φ R and ∆ R respectively develop vacuum expectation values (VEVs) (denoted by v R / √ 2 and v ∆ / √ 2) in order to break the SU (2) R symmetry and generate Majorana mass term for the right-handed neutrinos ν R to realize seesaw mechanism with two-loop induced Dirac mass as shown below.
The relevant Lagrangian for Yukawa sector and scalar potential under these assignments are given by where τ 2 is a second component of the Pauli matrix, the index i(j) runs 1-3, and y ∆ R and M E can be diagonal without loss of the generality. It implies that y ∆ R does not contribute to lepton flavor violations.
Notice here that each of f L(R) and g should be anti-symmetric and symmetric. We work on the basis where all the coefficients are real and positive for our brevity. After the left-right symmetry breaking, each of scalar field has nonzero mass. We parametrize these scalar fields as where h L is the SM-like Higgs, and v L is related to the Fermi constant G F by v 2 The VEVs of Φ L(R) are derived from the conditions ∂V/∂v L = 0, ∂V/∂v R = 0 and ∂V /∂v In this paper we require v R ≫ v L which can be achieved if we adopt m 2 Φ R /λ Φ R ≫ m 2 Φ L /λ Φ L and choose rather small value of λ LR . After the symmetry breaking, we have massive gauge bosons W ± L(R) and Z L(R) associated with left-right symmetry. Note that neutral singlet scalar is required to obtain desired symmetry breaking pattern in the model of Ref. [120] while we can realize the symmetry breaking due to the absence of exact left-right symmetry in the scalar potential.
The CP even Higgs boson mass matrix in the basis of (h L , h R , ∆ 0 R ) is denoted by (M 2 ) CP−even , and it is diagonalized by 3 × 3 orthogonal mixing matrix . Thus h L(R) and ∆ 0 R are respectively given by where h 1 ≡ h SM is the SM Higgs and h 2,3 are additional CP even Higgs mass eigenstates.
The CP odd component a L from Φ L does not mix with the other CP odd components. Thus a L is the massless Nombu-Goldstone (NG) boson which is absorbed by Z L boson. The CP odd Higgs boson mass matrix in the basis of (∆ 0 I , a 0 R ) is denoted by (M 2 ) CP−odd , and it is diagonalized by 2 × 2 orthogonal . Therefore a R and ∆ I are given by where only A 1 should be massive, since A 2 is absorbed by Z R boson.
The singly charged scalar boson h + does not mix with other charged scalar bosons. Thus it is the mass eigenstate with mass m h ± . Also the singly charged component φ ± L from Φ L does not mix and it is the massless NG boson absorbed by W ± L . The singly charged scalar boson mass matrix in the basis of (∆ ± , φ ± R ) is denoted by (M 2 ) singly , and it is diagonalized by 2 × 2 unitary mixing matrix U 1 as ). Therefore ∆ ± and φ ± R are given by should be zero, since φ ± 2 is absorbed by W ± R boson. The doubly charged scalar boson ∆ ±± is mass eigenstate with mass eigenvalue m ∆ ±± ≃ m ∆ .

B. Charged lepton sector
First of all, we define the isospin doublet fermions as The charged lepton mass matrix in the basis of (ℓ, E) can be given as Then it can be diagonalized by bi-unitary mixing matrix V L and V R as Here we have used the assumption m L , m R << M E . The resultant charged lepton mass squared is then given by

C. Neutrino sector
The neutral fermion mass matrix in the basis of (ν L , ν R ) is generated by

and the Dirac fermion mass matrix m D is given by
where all the indices are summed over, and we define ( and assume to be m ℓ << M E . Therefore the active neutrino masses can be obtained at two-loop level through two types of the seesaw mechanisms (canonical seesaw with one-loop induced Dirac mass and Notice here that one of three neutrino masses is zero without loss of the generality, because the matrix rank of (m D ) 3×3 is two. where we neglect Dirac phase δ as well as Majorana phase in the numerical analysis for simplicity. The following neutrino oscillation data at 95% confidence level [123] where we assume normal ordering of the neutrino mass eigenstate in our analysis below, therefore m ν 1 = 0.

D. Neutrinoless double beta decay
Here we discuss the non-standard contribution to the neutrinoless double beta decay. The relevant process arises from the same process of the standard interaction just by flipping the chirality L → R, and its formula is given by where the first term in the left side equation is the contribution to the SM and the second term is the one of the new contribution. Furthermore we have used m W L = g L v L /2, m W R = g R v 2 R + 2v 2 ∆ /2, and the mixing among ν R s is assumed to be diagonal for simplicity. When we adopt the typical bound m ββ 0.29 eV [124], we can estimate the upper bound on the mass of ν R 1 once the v R and v ∆ are fixed. We will see a concrete discussion in the section of numerical analysis.

III. PHENOMENOLOGY OF THE MODEL
In this section, we discuss some phenomenologies in our model such as LFV, muon anomalous magnetic moment and DM. Then numerical analysis is carried out to search for allowed parameter space which is consistent with current experimental data.

A. Muon anomalous magnetic moment and Lepton flavor violations
The muon anomalous magnetic moment (muon g − 2) has been measured at Brookhaven National Laboratory. The current average of muon g − 2 experimental results is found as [125] a exp µ = 11659208.0(6.3) × 10 −10 .
Two discrepancy between the experimental data and the prediction in SM; ∆a µ ≡ a exp µ − a SM µ , have been respectively computed in Ref. [126] as ∆a µ = (29.0 ± 9.0) × 10 −10 at 3.2σ C.L., (III.1) and in Ref. [127] as In our model, we have new contributions to ∆a µ coming from the Yukawa coupling of h L(R) and f L(R) , 8   TABLE III: Summary of ℓ i → ℓ j γ process and the lower bound of experimental data [129]. and its contribution is given as 5 ) 2 and we have assumed m ν L << m ℓ << }. Note here that the contribution of ∆a ∆ µ is negligibly small because of the small mixing.
Lepton flavor violation processes (LFVs) ℓ i → ℓ j γ and ℓ − i → ℓ − j ℓ + k ℓ − ℓ at the one-loop level are measured precisely and severely constrained. Each of flavor dependent process has to satisfy the current upper bound, as can be seen in Table III and IV. The branching ratio (BR) for the ℓ i → ℓ j γ can be written as where C i ≈ (1, 1/5) for i = (µ, τ ) [130], G F is Fermi constant, and α em is the fine-structure constant. On the other hand, the BR for the process ℓ − i → ℓ − j ℓ + k ℓ − ℓ is given by indicating the correlation between LFV and neutrino mass matrix.

B. Dark Matter
We consider a fermionic DM candidate X(≡ ν R 1 ), which is assumed to be the lightest particle of ν R i .
Since DM can decay into neutrinos and photon through the Dirac mass term at the one-loop level, DM has to be long-lived. Hence we focus on the explanation of the X-ray line at 3.55 keV, since X decays into active neutrinos and photon at the one-loop level after the symmetry breaking. Then the mass of DM M X (≡ M ν R 1 ) is fixed to be around 7.1 keV with a small value of the decay rate divided by M X ; i.e., 4.8 × 10 −48 Γ(X→ν k γ) M X 4.6 × 10 −46 [133]. 6 We also note that such a DM candidate will be overabundant if one estimates thermal relic density through the gauge interactions. However this problem can be evaded by the entropy production due to the late decay of ν R 2,3 [131,132]. In our analysis, we assume the right relic density can be obtained by this mechanism and the constraints on the decay rate of DM is taken into account. Then the decay rate is derived as (III.13) where we define a ≡ (M X /m h ± ) 2 , b j ≡ (m ℓ j /m h ± ) 2 , under the assumption M X , m ν L << m ℓ , m h ± . Thus the decay ratio is correlated to neutrino mass matrix, ∆a µ and LF V through the Yukawa coupling f L(R) .

C. Collider Physics
Here we discuss the signature of our model at the LHC 13 TeV. Then we focus on the doubly charged Higgs boson ∆ ±± , which decays into the same sign lepton pair with right-handed chirality. Particularly the process pp → Z R → ∆ ++ ∆ −− is interesting since it provides clear four lepton signal where invariant masses of same sign leptons and of four leptons respectvely give mass of ∆ ±± and m Z R 7 . This is unlikely to neither the type II seesaw scenario nor the Zee-Babu type case with k ++ e c R e R , because the type II decay mode comes from the left-handed chirality, and the Zee-Babu type doubly charged Higgs is produced via gauge interaction with only U (1) Y . Furthermore each of the component y ∆ R can be determined through the neutrino oscillation data, CLFVs processes, and DM phenomenology such as X-ray line search. Thus we expect that collider signature further test the structure of the Yukawa coupling.
The gauge interactions associated with Z R are written as [134,135] g R is SU (2) R gauge coupling. Then we estimate the production cross section of Z R and its branching ratio (BR) with CalcHEP [136] implementing the interaction and applying CTEQ6L PDF [137]. In Fig. 1, several values of r ≡ g R /g L with fixed doubly charged Higgs mass m ∆ = 1 TeV where constraint on from LHC experiment is indicated by red curve [138]. We find that Z R should be heavier than around 3.5 TeV where the lower mass limit depends on r. and structure of the Yukawa coupling y ∆ R would be investigated by measuring the BR of ∆ ±± . The detailed simulation analysis including SM background is beyond the scope of our paper and it will be investigated in future work.

D. Numerical analysis
Now that all the formulae have been provided, we carry out numerical analysis to search for parameter region satisfying all the constraints. First of all, we fix the following parameters in the scalar sector: Before discussing the numerical analysis, we comment on the neutrinoless double beta decay. Once we apply these above values, we can estimate the the neutrinoless double beta decay as which are found as preferred parameter range to satisfy the constraints. Then we have examined 10 6 sampling points to investigate how much parameter space is allowed. We find 311 points that satisfy the current LFV constraints and the neutrino oscillation data. Fig. 2 shows the correlation between ∆a µ and In this section, we discuss the structure of Yukawa couplings of our model. Our neutrino masses are Generally, a matrix M is factorized by the following form, where D is diagonal matrix and L (U) is upper (lower) triangular matrix with unit diagonal components.
The factorization is called LDU decomposition. We can factorize m D using the LDU decomposition as follows: D ) T and the diagonal matrix Z D is written by We assume F L h L and F R h R are lower triangular matrices. The components of F L(R) and h L(R) are written by the following form, 32 h L(R) 12 , 32 h L(R) 12 , 32 h L(R) 13 , h L(R) 33 = h L(R) 23 h L(R) 32 h L(R) 22 , In this case, F L(R) h L(R) becomes a lower triangular matrix: l L(R) ij are determined by neutrino oscillation experiments. Therefore we have 8 free parameters: h L(R) 11 , h L(R) 12 , h L(R) 13 and F L(R) 23 .
Here we summarize the loop factors appearing in the formula of lepton flavor violating decay (B.10) (B.11) Here the factors {a, b, .., h} have been defined as