An inverse seesaw model with global $U(1)_H$ symmetry

We propose an inverse seesaw model based on hidden global symmetry $U(1)_H$ in which we realize tiny neutrino masses with rather natural manner taking into account relevant experimental bounds. The small Majorana mass for inverse seesaw mechanism is induced via small vacuum expectation value of a triplet scalar field whose Yukawa interactions with standard model fermions are controlled by $U(1)_H$. We discuss the phenomenology of the exotic particles present in the model including the Goldstone boson coming from breaking of the global symmetry, and explore testability at the Large Hadron Collider experiments.


I. INTRODUCTION
Inverse seesaw mechanism [1,2] is one of the promising candidates to induce neutrino masses and their mixing that are typically understood by an additional symmetry beyond the standard model (SM). The mechanism requires both chiralities of neutral fermions (N L/R ) that couple to the SM fermions with different manner, and their mass structures are also different. That is why an additional symmetry is needed [3][4][5][6]. Furthermore, it demands hierarchies among mass parameters associated with neutral fermions including active neutrinos where especially Majorana mass of N L should be tiny. Indeed, realizing such hierarchies in a natural way is rather challenging from model building perspective [4,5].
In this paper, we propose an inverse seesaw model under a hidden global U(1) H symmetry [7] 1 in which we try to realize natural hierarchies among neutral fermion mass matrix, taking advantage of experimental constraints of electroweak precision test to our model.
In our model, we introduce SU(2) L triplet scalar and exotic lepton doublets with non-zero U(1) H charges and scalar singlets. The Yukawa interaction among the triplet and exotic lepton doublet induces small Majorana mass term required for inverse seesaw mechanism due to small vacuum expectation value (VEV) of the triplet while Yukawa interaction between the SM lepton doublet and the scalar triplet is forbidden by the U(1) H symmetry. In addition, as a result of the global symmetry that accommodates a physical Goldstone boson (GB) [13,14], our model can be well-tested at future colliders since the GB couples to the SM fermions through their kinetic terms. 2 This letter is organized as follows. In Sec. II, we introduce our model and formulate the scalar sector, charged-lepton sector, and neutral fermion sector, and briefly discuss electroweak precision test. In the scalar sector, we show hierarchies among VEVs that connect to our desired hierarchies among neutral fermions. Using electroweak precision data, we show the constraint on the VEV of an isospin triplet scalar at tree level, and this bound is directly related to the hierarchies of neutral fermion mass matrices. Then we formulate the charged-lepton mass matrix that encapsulates with the mixing between SM charged-leptons and newly introduced heavier ones. From this we show that their mass 1 There are several studies utilizing hidden gauged symmetries to explain neutrino masses [8][9][10][11][12]. 2 Even in the case of a gauged symmetry, GB is induced one introduces two or more two bosons that breaks the additional U (1) symmetry; see, e.g., ref. [9]. hierarchies could be naturally realized, while maintaining the experimental bound on the new heavier leptons. Note that these hierarchies are also related to our neutral fermion mass matrix. In the last part of this section, we discuss the neutral fermion mass matrix and show how to realize these hierarchies. Also we discuss the bounds on non-unitarity effect originated from the inverse seesaw model. In Sec. III we discuss the phenomenology of our model. We mainly consider the decays of the exotic charged leptons and they can leave their footprints in the multi-lepton signatures of the collider experiments, like LHC.
Finally we summarize our results and conclude.

II. MODEL
In this section we formulate our model in which we introduce a global U(1) H symmetry. The fermionic sector is augmented by three families of vector-like fermions L ′ L/R ≡ [N L/R , E L/R ] T with charge (2, −1/2) under the SU(2) L × U(1) Y gauge symmetry, while under the global U(1) H right-and left-handed ones are assigned charges 2ℓ and ℓ respectively. As for the scalar sector, we add an isospin triplet scalar ∆ with hypercharge 1, and two singlet scalars ϕ 1,2 with zero hypercharge, while (2ℓ, ℓ, 2ℓ) are respectively assigned to (∆, ϕ 1 , ϕ 2 ) under the global U(1) H symmetry. The SM-like Higgs field is denoted by H.

The vacuum expectation values (VEVs
respectively. The field content and the respective quantum number assignments are shown in Table I. Note that the quark sector remains the same as that of SM. The renormalizable Yukawa Lagrangian under these symmetries is then given by where the indices a, b(= 1 − 3) represent the number of families,∆ ≡ iσ 2 ∆ † , and y ℓ is assumed to be diagonal matrix without loss of generality. After spontaneous symmetry breaking, one finds the charged-lepton mass matrix m ℓ = y ℓ v/ √ 2.

Scalar potential and VEVs:
First of all, we define each scalar field as follows: The scalar potential in our model is given by, where V trivial indicates trivial quartic terms containing scalar triplet. For simplicity we assume all the couplings to be real. The quartic coupling λ 0 plays a role in reducing the scale of VEV of ∆ to O(1) GeV. Applying the minimization condition ∂V/∂v = ∂V/∂v ′ 1,2 = ∂V/∂v ∆ = 0, we obtain the VEVs approximately as where we assume v ′ 2 ∼ v ∆ ≪ {v ′ 1 , v} and λ Hϕ 1 to be small. The above hierarchy of VEVs are motivated from the mass hierarchy in neutral fermions as we discuss below. The VEV hierarchy can be consistently obtained by choosing appropriate parameters; for example, we then take µ ϕ to be O(1) GeV or less in the following discussion.
Since we assume small λ Hϕ 1 coupling, the mass matrix for CP-even scalars from the SM singlets ϕ 1,2 is derived approximately Note that mixing between ϕ R 1 and ϕ R 2 would be small since off-diagonal elements are suppressed by v ′ 2 and µ ϕ unless values of diagonal elements are not too close. Thus we approximate ϕ R 1 and ϕ R 2 as mass eigenstates and corresponding mass eigenvalues are In CP-odd scalar sector from SM singlet ϕ 1,2 , we have one massless Goldstone boson associated with breaking of global U(1) H symmetry. Applying our assumption of v ′ 2 ≪ v ′ 1 , we can approximately identify ϕ I 1 as Goldstone boson while ϕ I 2 is massive scalar boson Thus the ϕ R 2 and ϕ I 2 have approximately degenerate masses.
The SM Higgs and scalar triplet sector are mostly same as the usual scalar triplet model since we assume the mixing among SM singlet scalar to be sufficiently small. The mass scale of scalar bosons in the triplet is given by M ∆ .
ρ parameter: The VEV of ∆ is restricted by the ρ-parameter at tree level that is given by [15] ρ where the experimental value is given by ρ = 1.0004 +0.0003 −0.0004 at 2σ confidence level [16]. On the other hand, we have v SM = v 2 + 2v 2 ∆ ≈246 GeV. Therefore the upper bound on v ∆ is of the order O(1) GeV 3 .

Charged-lepton sector:
The charged-lepton mass matrix consists of the component of the SM mass matrix and heavier one, after the spontaneous symmetry breaking. We define the mass matrix M to be Furthermore, we assume the mass matrices m D and M to be diagonal for simplicity.
Then, for each generation indicated by "a", we can write the charged-lepton fermion mass matrix as The mass matrix is diagonalized by the transformation (e a L(R) , E a L(R) ) → V † La(Ra) (e a L(R) , E a L(R) ). Thus we can obtain diagonalization matrices V La and V Ra which where we have assumed m ℓa , m Da ≪ M a . Then the mass eigenvalues for e a and E a are simply given by m ℓa and M a . Also their mixing angles θ Ra and θ La are very small and satisfy θ Ra ≪ θ La , since the lower bound on the mass of the heavier leptons is about 100 GeV [16] that is suggested by the current experimental data at LHC and LEP. This is because hierarchies in mass parameter m ℓa , m Da ≪ M a is comparatively natural.

Neutrino sector:
After the spontaneous symmetry breaking, neutral fermion mass matrix in the basis of (ν c L , N R , N c L ) T is given by where m D , M are diagonal while µ ≡ g L v ∆ / √ 2 is symmetric 3 × 3 mass matrix. Then the active neutrino mass matrix can be given as  [24], the upper bounds are given by [25]: Since F is assumed to be diagonal, the stringent constraint originates from 2-2 component of |F F † |. It suggests that |F | ∼ m D /M 10 −2 that is always safe in our model thanks to the small VEV v ′ 2 .

III. PHENOMENOLOGY
In this section, we discuss the phenomenology of the model focusing on the decay and production of exotic particles. One specific property of our model is the existence of physical Goldstone boson as a result of global U(1) H symmetry breaking. Here we first derive interactions associated with the Goldstone boson which is denoted by α G identified as Then ϕ 1 and ϕ 2 interact with α G as follows The covariant derivative of ∆ is rewritten including Goldstone boson as, where g is SU(2) L gauge coupling, T ± = (σ 1 ±iσ 2 )/2, θ W is Weinberg angle, andQ is electric charge operator acting on each component of the multiplet. Then we can obtain interactions of α G and ∆ from kinetic term Tr[(D µ ∆) † (D µ ∆)]. The triplet Higgs decay modes with α G are suppressed by v ∆ /v ′ 1 factor so that components in ∆ decay via gauge or scalar potential interactions. Also exotic lepton L ′a interaction with α G is derived from kinetic term as In passing we briefly comment about the constraints from cosmology on our physical GB. Actually the existence of our GB does not cause serious problem in cosmology since it does not couple to SM particles directly except for Higgs boson whose coupling is well controlled by the parameters in the potential which we assume to be negligibly small. Thus Higgs portal cosmological constraint can be avoided when the GB decouples from thermal bath at the scale larger than muon mass scale [13].

A. Decay of exotic particles
In this subsection, we discuss the decays of exotic particles in the model. At first, we write the Lagrangian relevant to decay of exotic charged lepton E a as follows where we have omitted subdominant terms. The partial decay widths are computed as where we have used m Da = y Daa v ′ 2 / √ 2. Thus E a± dominantly decay into ℓ a± ϕ R 2 (I 2 ) since the other modes are suppressed by small VEV v ′ 2 . Then we also estimate partial decay widths of ϕ R 2 (ϕ R I ) as, Thus ϕ R 2 dominantly decays into α G α G while ϕ I 2 dominantly decays into ℓ a+ ℓ a− . Therefore ℓ a− α G α G with branching ratio (BR) of 0.5 for both modes.

B. Collider physics
In this subsection we discuss collider signatures of our model. The exotic charged scalar bosons from Higgs triplet can be produced by electroweak production and they dominantly decay into gauge bosons as the triplet components have degenerate mass. This phenomenology is the same as scalar triplet model with relatively large triplet VEV case (v ∆ ∼ 1 GeV) where charged scalars in the triplet {δ ± , δ ±± } decays into SM gauge bosons. Phenomenology of such case can be found, for example, in refs. [26][27][28][29][30].
Then we focus on exotic charged lepton production at the LHC. The exotic charged lepton pair can be produced by pp → Z/γ → E a+ E a− via gauge interaction depicted in Eq. (16).
Furthermore single production process can be induced through mixing between E a and SM charged leptons. We obtain relevant interaction by where mixing effect in neutral current is canceled. Thus E a can be singly produced as pp → W −(+) → E a−νa (E a+ ν a ) at the LHC. For these processes cross sections are estimated by using CalcHEP [31] with the CTEQ6 parton distribution functions (PDFs) [32]. We show pair and single production cross sections as a function of exotic lepton mass in left and right panels in Fig. 1, where we fixed m Da /M a = 0.01 in our calculation and only the lightest exotic charged lepton E ± is considered. Then we find that single production cross section is much smaller than pair production one due to the suppression by mixing factor m Da /M a .
The pair production cross section is larger than ∼ 1 fb when exotic lepton mass is M a 700 GeV for √ s = 14 TeV. Since the exotic charged lepton decays into multi-lepton final state, the cross section can be constrained by multi-lepton search at the LHC where inclusive search indicates cross section producing more than three electron/muon is required to be σ · BR 1 fb at LHC 8 TeV [33]. We thus require our exotic charged lepton mass to be M a 500 GeV.
The signal from E + E − pair production is multi-lepton process, given by Here, signal events are generated using MADGRAPH5 [34] by implementing relevant interactions. In Fig. 2, we show event distribution for invariant mass for e + e + e − assuming E ± and ϕ I 2 dominantly decay into mode including electron and m ϕ I 2 = 100 GeV with integrated luminosity 300 fb −1 at the LHC 14 TeV 5 . We see the peak slightly larger than exotic lepton mass. The signal is clean and number of background events can be suppressed requiring multiple charged lepton final state. Thus the signal can be tested well at the future LHC experiments if exotic lepton mass is less than 1 TeV. More detailed simulation including detector effects is beyond the scope of this paper and it is left for future work. Note also that the decay chain of E ± including GB is less clear but searches for the signal with GB can be a good test of our model. 5 Invariant mass for e − e − e + provides similar distribution. Majorana mass term via triplet VEV due to appropriate U(1) H charge assignment. We thus realize neutral fermion mass matrix which has the structure similar to inverse seesaw mechanism. Furthermore small Majorana mass in inverse seesaw scenario can be naturally obtained from small triplet VEV as in the type-II seesaw model.
We have also discussed the phenomenology of the model focusing on decay and production processes of the exotic particles. The decay widths of exotic charged leptons E ± are estimated taking into account interactions with Goldstone boson from U(1) H symmetry breaking. We have found E ± will provide multi-lepton final states from decay chain. Then E ± production cross section has been estimated which is induced by electroweak interactions. Testable number of multi-lepton events could be obtained if mass of E ± is less than O(1) TeV scale.