An inverse seesaw model with natural hierarchy at TeV scale

We propose a new kind of inverse seesaw model without any additional symmetries. Instead of the symmetries, we introduce several fermions and bosons with higher $SU(2)_L$ representations. After formulating the Higgs sector and neutrino sector, we show that the cut-off energy, which is valid to our model, is at around TeV scale, by examining behavior of $SU(2)_L$ gauge coupling. Then we show the testability of our model at collider physics.

In this letter, we propose an inverse seesaw model without any additional symmetries. Instead, we introduce several fermions (quartet and septet) and bosons (quintet and quartet) with higher SU (2) L representations [30]. Due to such fields, behavior of SU (2) L gauge coupling g 2 is blown up at TeV scale via renormalization group equation (RGE). It suggests that our model is tightly relevant at low energy scale, and the testability of our model is largely expected at various experiments such as the LHC, the ILC, and future colliders. This letter is organized as follows. In Sec. II, we review our model and formulate the lepton sector. Then we discuss phenomenologies of neutrinos. In Sec. III we discuss an extra charged particles at collider experiments. Finally we devote the summary of our results and the conclusion. * Electronic address: nomura@kias.re.kr † Electronic address: okada.hiroshi@apctp.org

II. MODEL SETUP AND CONSTRAINTS
In this section we formulate our model. As for the fermion sector, we introduce three families of vector fermions ψ with (4, −3/2) charge under the SU (2) L × U (1) Y gauge symmetry, and right-handed fermions Σ R with (7, 0) charge under the SU (2) L × U (1) Y gauge symmetry. As for the boson sector, we add quartet scalar field H 4 with 3/2 charge under the U (1) Y gauge symmetry, quintet scalar field H 5 with 2 charge under the U (1) Y gauge symmetry, where SM-like Higgs field is defined by H 2 . Here we denote each of vacuum expectation value(VEV) to be H i ≡ v i / √ 2 (i = 2, 4, 5) that is arisen after the electroweak spontaneously symmetry breaking. All the field contents and their assignments are summarized in Table I, where the quark sector is exactly same as the one of the SM. The renormalizable Yukawa Lagrangian under these symmetries is given by where SU (2) L index is omitted assuming it is contracted to be gauge invariant, and upper indices (a, b) = 1-3 are the number of families, and y ℓ , M , and M Σ is assumed to be diagonal matrix with real parameters. Each of the mass matrix is defined by m ℓ = y ℓ v/ √ 2 and m D = y D v 5 / √ 2.
Scalar potential and VEVs: The scalar potential in our where V trivial indicates other trivial 4-point terms and SU (2) L indices are implicitly contracted in the second line to be gauge invariant. Applying condition ∂V/∂v i = 0, we obtain the VEVs as Thus v 4 and v 5 can be naturally O(1) GeV scale if M 4 and M 5 are TeV scale. ρ parameter: The VEVs of H 4 and H 5 are restricted by the ρ-parameter at tree level that is given by where the experimental value is given by ρ = 1.0004 +0.0003 −0.0004 at 2σ confidence level [17]. While v SM = v 2 2 + 7v 2 4 + 10v 2 5 ≃ v 2 ≈246 GeV, therefore v 4 and v 5 are restricted via the constraint of ρ parameter. Here, we take these VEVs to be v 2 ≈245.9 GeV, v 4 ≈1.67 GeV, and v 5 ≈1.72 GeV, which are typical scale of VEVs. Exotic particles : The scalars and fermions with large SU (2) L multiplet provide exotic charged particles. Here we write components of multiplets as The masses of components in H 4 and H 5 are respectively given by ∼ M 4 and ∼ M 5 since v 4,5 ≪ M 4,5 . The charged components in ψ L(R) have Dirac mass M and neutral component is discussed with neutrino sector below. The septet fermion mass is M Σ and charged components have Dirac mass term constructed by pairs of positive-negative charged components in the multiplet. Charged particles in the same multiplet have degenerate mass at tree level which will be shifted at loop level [18].
Neutrino sector: After the spontaneously symmetry breaking, neutral fermion mass matrix in basis of (ν L , ψ 0c R , ψ 0 L ) T is given by where µ L/R is given by v 2 4 f L/R M −1 Σ f T L/R on the analogical manner of seesaw mechanism, as shown in Fig. 1. Then the active neutrino mass matrix can approximately be found as where µ L/R < m D ≪ M is naturally expected due to the constraint of ρ parameter and seesaw-like mechanism of µ R/L [31]. We thus obtain correlation among size of neutrino mass and other mass parameters such that Note that M Σ and M cannot be much larger than TeV scale, since v 4 and v 5 are GeV scale requiring perturbative limit for Yukawa coupling constants. The neutrino mass matrix is diagonalized by unitary matrix U MN S ; Here we apply a convenient method to reproduce neutrino oscillation data as follows [19]: 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 [20], which can uniquely be decomposed to be M −1 µ * R (M T ) −1 = L T N I N L N , since it is symmetric matrix. Note here that all the components of m D should not exceed O(1) GeV, once perturbative limit of y D is taken to be 1. Non-unitarity: Constraint of non-unitarity should always be taken into account in case of larger neutral mass matrix whose components are greater than three by three, since experimental neutrino oscillation results suggest nearly unitary. In case of the inverse seesaw, when non-unitarity matrix U ′ MN S is defined, one can typically parametrize the following form: where F ≡ M −1 m * D is a hermitian matrix, and U ′

MN S
represents the deviation from the unitarity. Considering several experimental bounds [25], one finds the following constraints [26]: Once we conservatively take F ≈ 10 −5 , we find µ R ≈1-10 GeV to satisfy the typical neutrino mass scale, which could be easy task.
where N f = 3 is the number of ψ and Σ R , µ is a reference energy, b SM g2 = −19/6, and we assume to be m in (= m Z ) < m th , being m th threshold masses of exotic fermions and bosons. The resulting flow of g 2 (µ) is then given by the Fig. 2. This figure shows that g 2 is relevant up to the mass scale µ = O(10) TeV in case of m th =500 GeV, while g 2 is relevant up to the mass scale µ = O(100) TeV in case of m th =5000 GeV. Thus our theory does not spoil, as far as we work on at around the scale of TeV.

III. COLLIDER PHYSICS
Here let us discuss collider physics of our model. We have rich phenomenology at collider experiments since there are many exotic charged particles from large SU (2) multiplet scalars and fermions. As the most specific signature we focus on the production of triply charged lepton Σ ±±± and its decay at the LHC. The gauge interactions associated with triply charged lepton are obtained asΣ where c W = cos θ W with Weinberg angle θ W and e is the electromagnetic coupling: covariant derivative for septet can be referred to ref. [16]. Then we estimate cross sections for triply charged lepton production processes using CalcHEP [28] by use of the CTEQ6 parton distribution functions (PDFs) [29], implementing relevant interactions. In Fig. 3, we show production cross section for triply charged lepton as a function of its mass; pair production pp → Σ +++ Σ −−− and associate productions pp → Σ +++(++) Σ −−(−−−) at the LHC 13 TeV. The cross section for pair production is the largest one and larger than 1 fb for 1 TeV mass due to large charge. The triply charged lepton can decay via Yukawa cou- is the electric charge of φ(ψ) with Q φ + Q ψ = 3, and we assume exotic scalars are lighter than exotic fermions in our discussion. In addition, ψ Q ψ decays as ψ Q ψ → φ There are several decay modes for exotic charged leptons due to combination of charges in final states which have similar size of branching ratio (BR). Here we discuss the representative decay chain: where φ ++ 4 and φ 0 5 decay into W + W + and ZZ via gauge interaction [14]: Note that BRs for φ ±± 4 → W ± W ± and φ 0 5 → ZZ are dominant when v 4 ∼ v 5 ∼ 1 GeV. When W + decays into leptons and Z decays into jets we obtain signal of three same sign charged leptons with jets and missing transverse energy, which provides products of BRs; BR(W + → ℓ + ν) 2 BR(Z → qq) 2 ∼ 0.02 with ℓ = µ, e. Thus we can obtain ∼ 6(60) signal events for integrated luminosity of 300(3000) fb −1 when products of Σ ±±± production cross section and BR(φ ±± 4 ψ ± )BR(ψ ± → φ 0 5 ℓ ± ) is ∼ 1 fb. This size of cross section can be obtained for M Σ ∼ 1 TeV. Since the SM background is very small for three same sign charged lepton signal we expect sizable discovery significance even if number of signal events is less than 10.

IV. SUMMARY AND CONCLUSIONS
We have constructed an inverse seesaw model with large SU (2) L multiplet fields in which we have formu-lated the neutrino mass matrix to reproduce current neutrino oscillation data, satisfying ρ parameter and nonunitarity bound. We have also checked the relevant energy scale of our theory via RGE of SU (2) L gauge coupling g 2 that gives the most stringent constraint. Then we have analyzed collider physics focusing on triply charged lepton production at the LHC as a representative process of our model and show a possibility of detection. We have found specific signal of the triply charged lepton as three same sign charged leptons with jets and missing transverse momentum. The number of events of the signal can be detectable level with integrated luminosity 300(3000) fb −1 when triply charged lepton mass is around 1 TeV. More detailed analysis will be given elsewhere.