An inverse seesaw model with $U(1)_R$ gauge symmetry

We propose a natural realization of inverse seesaw model with right-handed and flavor dependent $U(1)$ gauge symmetries, in which we formulate the neutrino mass matrix to reproduce current neutrino oscillation data in a general way. Also we study a possibility to provide predictions to the neutrino sector by imposing an additional flavor dependent $U(1)_{L_\mu-L_\tau}$ gauge symmetry that also satisfies the gauge anomaly cancellation conditions associated with $U(1)_R$. Then we analyze collider physics on a extra gauge boson and show a possibility of detection.

On the other hand, nature of these two gauge sectors are so different each other, and one might be able to test their differences via current or future experiments so as to make use of the polarized electron/positron beam at e.g., ILC [15]. Indeed, U (1) B−L is chirality-universal in a kinetic term, while U (1) R has right-handed chirality only. In this sense, it would be worthwhile for us to construct models with gauged U (1) B−L and/or U (1) R symmetry as many as possible, so that we can distinguish these two extra symmetries in a various phenomenological points of view.
In this paper, we construct an inverse seesaw model with U (1) R symmetry, in which we formulate the neutrino mass matrix to reproduce current neutrino oscillation data [16] in a general way. Inverse seesaw requires a left-handed neutral fermions S L in addition to the right-handed ones N R , and provides us more complicated neutrino mass matrix. Therefore, each of mass hierarchies are softer than the other models such as canonical seesaw and it could provide abundant phenomenologies such as unitarity constraints. Note that we expect S L has nonzero U (1) B−L charge, because it is a kind of partner of N R . In that case, however, U (1) B−L can not be gauged since anomaly cancellation condition can not * Electronic address: nomura@kias.re.kr † Electronic address: hiroshi.okada@apctp.org where the upper index a is the number of family that runs over 1-3. Singlet scalar ϕ2 is required when we add U (1)L µ −Lτ gauge symmetry.
be satisfied. Therefore introduction of left-handed singlet fermion is more natural in gauged U (1) R symmetry case compared with gauged U (1) B−L symmetry case since the left-handed singlet fermion cannot have lepton number in the latter case. Also we study a possibility to provide predictions to the neutrino sector by imposing an additional flavor dependent U (1) Lµ−Lτ gauge symmetry that also satisfies the gauge anomaly cancellations among U (1) R .
[39] Then we analyze collider physics on an extra gauge boson and show a possibility of detection. This letter is organized as follows. In Sec. II, we review our model and formulate the lepton sector. Then we discuss phenomenologies of neutrinos and an extra neutral gauge boson at colliders. Finally we devote the summary of our results and the conclusion.

II. MODEL SETUP AND CONSTRAINTS
In this section we formulate our model. At first, we add three families of right(left)-handed fermions N R (S L ) with 1(0) charge under the U (1) R gauge symmetry, and an isospin singlet boson ϕ 1 with 1 charge under the same symmetry. Here we denote each of vacuum expectation value to be H ≡ v H / √ 2, and ϕ 1 ≡ v ϕ1 / √ 2. Furthermore, the SM Higgs boson H also has 1 charge to induce the masses of SM fermions from the Yukawa Lagrangian after the spontaneously symmetry breaking. [40] All the field contents and their assignments are summarized in Table I. The relevant Yukawa Lagrangian under these symmetries is given by [41] − L ℓ = y ℓaaL a L He a R + y D abL whereH ≡ iσ 2 H, and upper indices (a, b) = 1-3 are the number of families, and y ℓ and y SN can be diagonal matrix without loss of generality due to the redefinitions of the fermions. Each of the mass matrix is defined by Notice that S L is singlet under all gauge symmetry and it does not interact with any gauge interactions without mixing among neutral fermions. Here we denote S L as left-handed in a sense it dose not have U (1) R charge. As we discuss below heavy extra neutral fermion mass is approximately given by M N S which is taken to be TeV scale. In addition, Majorana mass term of S L breaks lepton number as we assign lepton number to S L .
After the spontaneously symmetry breaking, neutral fermion mass matrix with 9×9 is given by Then the active neutrino mass matrix can approximately be found as where µ << m D M N S is expected [42]. The neutrino mass matrix is diagonalized by unitary matrix U MN S ; . One of the elegant ways to reproduce the current neutrino oscillation data [16] is to apply the Casas-Ibarra parametrization [18] without loss of generality, and find the following relation Here O mix is an arbitrary 3 by 3 orthogonal matrix with complex values, I N is a diagonal matrix, and L N is a lower unit triangular [32], which can uniquely be decom- Note here that all the components of m D should not exceed 246 GeV, once perturbative limit of y D is taken to be 1.
Mass scale of heavy neutral fermions is approximately given by M N S which is taken to be O(1) TeV in our scenario. Then neutrino mass scale is Thus we can realize neutrino mass scale ∼ 0.1 eV with µ ∼ 1(0.0001) GeV for m D ∼ 0.01(1) GeV. In addition, new fermions are not decoupled at TeV scale even if scale of v ′ is as large as 18 TeV.
Here we introduce local U (1) Lµ−Lτ symmetry to restrict neutrino mass structure in inverse seesaw scenario [19,20] where we add SM singlet scalar ϕ 2 with L µ − L τ charge 1 to break the symmetry spontaneously. Then Yukawa interactions and Majorana masses are constrained, and we have new Yukawa interactions; where index i(j) is determined to satisfy gauge invariance. Thus once we impose U (1) Lµ−Lτ gauge symmetry as shown in table II [43], the mass matrices m D , M SN , µ are specified to be where µ 2,3 is induced only after the U (1) Lµ−Lτ spontaneously symmetry breaking. Therefore, the neutrino mass matrix directly reflects the form of µ as Thus we can predict inverted neutrino ordering and specific value of Dirac phase by analyzing the two-zero texture [19,21]. Here the number of parameters in the neutrino mass matrix is nine real parameters (that are equivalent of four complexes and one real). Then one more phase is there in addition to the Dirac phase and two Majorana phases. Note here that this two-zero texture originates from µ in the inverse seesaw model that cannot be reproduced by a canonical seesaw model.

C. Non-unitarity
Here, let us briefly discuss non-unitarity matrix U ′ MN S . This is typically parametrized by the form where F ≡ (m T N S ) −1 m D is a hermitian matrix, and U ′ MN S represents the deviation from the unitarity. The global constraints are found via several experimental results such as the SM W boson mass M W , the effective Weinberg angle θ W , several ratios of Z boson fermionic decays, invisible decay of Z, electroweak universality, measured Cabbibo-Kobayashi-Maskawa, and lepton flavor violations [37]. The result is then given by [38] |F F † | ≤ Once we conservatively take F ≈ 10 −5 , we find µ ≈1-10 GeV to satisfy the typical neutrino mass scale, which can be easy to realize. In addition, for M N S ∼ 1 TeV, we require y D ∼ 10 −4 which is slightly larger than case of Type-I seesaw [23].

D. Collider physics
Here we discuss collider physics of our model mainly focusing on Z ′ R boson from U (1) R which obtain its mass via the vacuum expectation value of ϕ 2 . The gauge interaction associated with Z ′ R is given by where g R is gauge coupling constant for U (1) R , and flavor index is omitted. Z ′ R physics at the LHC : In our model Z ′ R can be produced via qq → Z ′ R process, and it will decay into SM fermions and N R if kinematically allowed. Then stringent constraint is given by di-lepton resonance search at the LHC. We estimate the cross section with CalcHEP [24] by use of the CTEQ6 parton distribution functions (PDFs) [25], implementing relevant interactions. In addition, we find branching ratio (BR) for the decay mode Z ′ R → e + e − /µ + µ − is ∼ 4.8% for both electron and muon when we assumeN R N R mode is not kinematically allowed; even if we includeN R N R mode the BR does not change much as BR(Z ′ R → e + e − /µ + µ − ) 4.2%. In Fig. 1, we show for several values of g R where the BR is sum of electron and muon mode and the red curve indicate the LHC limit obtained from ref. [26]. We find that Z ′ R mass should be heavier than ∼ 3.8 TeV for g R = 0.1 where corresponding production cross section is σ(pp → Z ′ R ) 1 fb.
Here we discuss production of heavy neutrino ν H at the LHC via Z ′ R boson. If masses of ν Hi are sufficiently lighter than m Z ′ R /2, BR(Z ′ R → ν H ν H ) is around 4% for each mass eigenstate. Then ν H decays such that ν H → W ± ℓ ∓ and ν H → Zν L via mixing in neutrino sector. As we discussed above, Z ′ R production cross section is less than ∼ 1 fb for m Z ′ R being several TeV scale, and ν H production cross section will be σ · BR 0.04 fb for  1: The product of Z ′ R production cross section and BR(Z ′ R → ℓ + ℓ − ) where region above red curve is excluded by the latest data [26]. each mass eigenstate. Thus large integrated luminosity is required to obtain sufficient number of events to analyze the signal. It is also important to confirm the ratio of the BR of each decay mode of Z ′ R to distinguish it from other Z ′ boson like that from U (1) B−L gauge symmetry where the approximated BRs are given in Table. III.
Here we also comment on Z ′ µ−τ boson from U (1) Lµ−Lτ gauge symmetry. Gauge interactions among Z ′ µ−τ and fermions are written by where g µ−τ is the gauge coupling constant of U (1) Lµ−Lτ and fermions are flavor eigenstates. It is difficult to detect Z ′ µ−τ when we consider it to be light as O(10)-O(100) MeV so that muon g − 2 can be explained [22]. The Z ′ µ−τ interaction induces flavor violating decay of heavy neutrino such as ν Hi → Z ′ µ−τ ν Hj where m νH i > m νH j . Thus phenomenology of heavy neutrino at the LHC would be affected by the gauge boson. However detailed analysis is beyond the scope of the paper. We note that the Z ′ µ−τ gauge interaction with charged lepton is flavor diagonal and do not induce any flavor violations(LFVs) even at loop levels, once we do not seriously take mixings among neutral gauge bosons into consideration. Contribution via W ± and neutral fermions at one-loop level also gives no LFVs, considering negligible mixing among neutral fermions. Even when we consider the mixing among gauge bosons or neutral fermions, their mixings are so small because they are respectively proportional to (m Z ′ µ−τ /m Z(Z ′ R ) ) 2 10 −6 (10 −8 ) and (m D /M SN ) 2 10 −6 [28].
Mode ℓ − ℓ + qq νHνH BR 0.042 0.13 0.042 Z ′ R physics at lepton collider : Although it would be difficult to produce Z ′ R directly at lepton colliders we can explore the effective interaction induced from Z ′ R exchange; where f indicates all the fermions in the model, and only the right-handed chirality appears due to the nature of U (1) R symmetry. For example, the analysis of data by LEP experiment in ref. [27] provides the constraint m Z ′ R gR 3.7 TeV. Furthermore chirality structure of the effective interaction could be tested by measuring the process e + e − → ff at the International Linear Collider (ILC) using polarized initial state. The partially-polarized differential cross section can be defined as [29] dσ(P e − , P e + ) d cos θ where P e − (e + ) is the degree of polarization for the electron(positron) beam and σ σ e − σ e + indicates the cross section when the helicity of initial electron(positron) is σ e − (e + ) and the helicity of final states is summed up; more detailed form is found in ref [29]. The polarized cross sections σ L,R is given by following two cases as realistic values at the ILC [30]: (15) Then we apply σ R to study the sensitivity to Z ′ R since it is sensitive to right-handed current interactions [5]. To investigate the effect of the new interaction we consider the measurement of a forward-backward asymmetry at the ILC which is given by where a kinematical cut c max = 0.5(0.95) is chosen to maximize the sensitivity for electron(muon) [31], L is an integrated luminosity and ǫ is an efficiency depending on the final states which is assumed to be ǫ = 1 for electron and muon final states. The sensitivity to Z ′ R contribution is estimated by where and A SM F B are forward-backward asymmetry for "SM + Z ′ R " and SM cases respectively. We compare ∆A F B with a statistical error of the asymmetry in only SM case and we focus on muon final state which is the most sensitive one. We find that it is difficult to get ∆A F B > δ SM AF B for √ s = 250 or 500 GeV in the region which satisfy the LHC constraint even if the integrated luminosity is O(10) ab −1 . On the other hand, for √ s = 1 TeV, ∆A F B ∼ 2δ SM AF B can be obtained with the integrated luminosity of 5 ab −1 with m Z ′ R /g R = 40 TeV. Therefore to investigate the chirality structure, we need √ s = 1 TeV with large integrated luminosity which would be achieved if the ILC is upgraded [15].

III. SUMMARY AND CONCLUSIONS
We have constructed an inverse seesaw model with U (1) R symmetry, in which we have formulated the neutrino mass matrix to reproduce current neutrino oscillation data in a general way. Also we have found a predictive two-zero neutrino mass matrix, by imposing an additional flavor dependent U (1) Lµ−Lτ gauge symmetry that also satisfies the gauge anomaly cancellations among U (1) R . Then we have analyzed collider physics on an extra gauge boson and show a possibility of detection, Although the result of collider physics is almost the same as the one of our canonical seesaw model [5]. the neutrino predictions originate from the inverse seesaw model that could be difficult to reproduce any canonical seesaw models.