Explaining Dark Matter and Neutrino Mass in the light of TYPE-II Seesaw Model

# Explaining Dark Matter and Neutrino Mass in the light of TYPE-II Seesaw Model

Anirban Biswas Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India    Avirup Shaw Department of Theoretical Physics, Indian Association for the Cultivation of Science,
###### Abstract

With the motivation of simultaneously explaining dark matter and neutrino masses, mixing angles, we have invoked the Type-II seesaw model extended by an extra doublet . Moreover, we have imposed a parity on which remains unbroken as the vacuum expectation value of is zero.  Consequently, the lightest neutral component of becomes naturally stable and can be a viable dark matter candidate.  On the other hand, light Majorana masses for neutrinos have been generated following usual Type-II seesaw mechanism. Further in this framework, for the first time we have derived the full set of vacuum stability and unitarity conditions, which must be satisfied to obtain a stable vacuum as well as to preserve the unitarity of the model respectively. Thereafter, we have performed extensive phenomenological studies of both dark matter and neutrino sectors considering all possible theoretical and current experimental constraints. Finally, we have also discussed a qualitative collider signatures of dark matter and associated odd particles at the 13 TeV Large Hadron Collider.

## I Introduction

The observation of various satellite borne experiments namely WMAP Hinshaw:2012aka () and more recently Planck Ade:2015xua (), establish firmly the existence of dark matter in the Universe over the ordinary luminous matter. The results of these experiments are indicating that more than 80% matter content of our Universe has been made of an unknown non-luminous matter or dark matter. In terms of cosmological language, the amount of dark matter present at the current epoch is expressed as Ade:2015xua () where is known as the relic density of dark matter and is the present value of Hubble parameter normalised by 100. In spite of this precise measurement, the particle nature of dark matter still remains an enigma. The least we can say about a dark matter candidate is that it is electrically neutral and must have a lifetime greater than the present age of the Universe. Moreover, N-body simulation requires dark matter candidate to be non-relativistic (cold) at the time of its decoupling from the thermal plasma to explain small scale structures of the Universe Frenk:2012ph (). Unfortunately, none of the Standard Model (SM) particles can fulfil all these properties and hence there exist various beyond Standard Model (BSM) theories in the literature Jungman:1995df (); Bertone:2004pz (); Hooper:2007qk (); ArkaniHamed:2008qn (); Kusenko:2009up (); Feng:2010gw () containing either at least one or more dark matter candidates. Among the different kinds of dark matter candidates, Weakly Interacting Massive Particle (WIMP) Gondolo:1990dk (); Srednicki:1988ce () is the most favourite class and so far, neutralino Jungman:1995df () in the supersymmetric extension of the SM is the well studied WIMP candidate. There are also a plethora of well motivated non-supersymmetric BSM theories which have dealt with WIMP type dark matter candidate Silveira:1985rk (); Burgess:2000yq (); McDonald:1993ex (); Barbieri:2006dq (); LopezHonorez:2006gr (); Kim:2008pp (); Hambye:2008bq (). Since the interaction strength of a WIMP is around week scale hence various experimental groups Akerib:2016vxi (); Amole:2017dex (); Armengaud:2016cvl (); Agnese:2014aze () have been trying to detect it directly over the last two decades by measuring the recoil energies of detector nuclei scattered by WIMPs. However, no such event has been found and as a result, dark matter nucleon elastic scattering cross section is getting severely constrained. Currently most stringent bounds on dark matter spin independent scattering cross section have been reported by the XENON 1T collaboration Aprile:2017iyp ()111Recently, PandaX-II collaboration 1708.06917 () has published their results on the exclusion limits of WIMP-nucleon spin independent scattering cross section (). Although, their results are most stringent for a WIMP of mass larger than 100 GeV, are very similar with the upper limits of XENON 1T.. Future direct detection experiment like DARWIN Aalbers:2016jon () is expecting to detect or ruled out the WIMP hypothesis by exploring the entire experimentally accessible parameter space of a WIMP (just above the neutrino floor).

On the other hand, neutrinos remain massless in the SM as there is no right handed counterpart of each where is the generation index. However, the existence of a tiny nonzero mass difference between and has been first confirmed by the atmospheric neutrino data of Super-kamiokande collaboration Fukuda:1998mi () from neutrino oscillation. Thereafter, many experimental groups Ahmad:2002jz (); Araki:2004mb (); Abe:2011sj (); Ahn:2012nd () have precisely measured the mass squared differences and mixing angles among different generations of neutrinos. In spite of these wonderful experimental achievements, we still have not properly understood the exact method of neutrino mass generation. There exist various mechanisms for generating tiny neutrino masses at tree level (via seesaw mechanisms) Minkowski:1977sc (); Mohapatra:1979ia (); Magg:1980ut (); Ma:1998dx (); Foot:1988aq (); Ma:2002pf (); Mohapatra:1986bd () and beyond Zee:1985id (); Ma:2006km (); Gustafsson:2012vj () by adding extra bosonic or fermionic degrees of freedom in the particle spectrum of SM. Moreover, the exact flavour structure in the neutrino sector, which is responsible for generating such a mixing pattern, still remains unknown to us. Furthermore, there are other important issues which are yet to be resolved. For example the particle nature of neutrinos (i.e. Dirac or Majorana fermion), mass hierarchy (i.e. Normal or Inverted), determination of octant for the atmospheric mixing angle , CP violation in the leptonic sector (i.e. measurement of Dirac CP phase ) etc. More recently, T2K collaboration Abe:2017vif () has reported their analysis of neutrino and antineutrino oscillations where they have excluded the hypothesis of CP conservation in the leptonic sector (i.e. or ) at 90% C.L. Their preliminary result indicate a range for lies in between third and fourth quadrant. Other neutrino experiments like DUNE Acciarri:2015uup (), NOA Adamson:2017gxd () etc. will address some of these issues in near future.

In the present article we try to cure both of these lacunae of the SM by introducing a Higgs triplet and an extra Higgs doublet to the particle spectrum of SM. Furthermore, we impose a discrete symmetry in addition to the SM gauge symmetry. Under this symmetry the triplet field and the SM particles are even while the the extra doublet field is odd222Here the odd doublet is analogous to the one in Inert Doublet Model (IDM)  Barbieri:2006dq (); LopezHonorez:2006gr (); Ma:2006km ().. This kind of BSM scenario has been studied earlier in Chen:2014lla (). To the best of our knowledge in such set up first time we derive the vacuum stability and unitarity constraints and use these constraints in our phenomenological study. This set up can serve our two fold motivations. First of all, as we have demanded that the extra doublet is odd under symmetry, consequently the lightest particle of neutral component of this doublet can play the role of viable dark matter candidate in this scenario. Secondly, with the small vacuum expectation value (VEV) of Higgs triplet field, required to satisfy the electroweak precision test, we can explain small neutrino masses by the Type-II seesaw mechanism Magg:1980ut (); Cheng:1980qt (); Lazarides:1980nt (); Mohapatra:1980yp (); Dev:2013ff (); Dev:2013hka () without introducing heavy right handed neutrinos. In the present work, we have explored both the normal and inverted hierarchies of neutrino mass spectra. At this point, we would like to mention that all the possible current experimental constraints have been taken into account while we investigate the dark matter related issues as well as the generation of neutrino masses and their mixings.

Apart from providing a viable solution to dark matter problem and neutrino mass generation, this scenario contains several non-standard scalars which can be classified into two categories. In one class we have even scalars originate from the mixing between triplet fields and SM scalar doublet fields while the three different components of the extra scalar doublet can be represented as odd scalars. Therefore, one has the opportunity to explore these non-standard scalars at the current and future collider experiments. In literature one can find several articles where the search of even scalars have been explored in context of the Large Hadron Collider (LHC) Chun:2003ej (); Han:2007bk (); Perez:2008ha (); Han:2015hba (); Han:2015sca (); Mitra:2016wpr () as well as at the International Linear Collider (ILC) Shen:2015pih (); Cao:2016hvg (); Blunier:2016peh (). However, in this work instead of even scalars, we have performed collider search of dark matter and the associated odd scalars at the 13 TeV LHC. Among the different final states, we find an optimistic result for signal at the 13 TeV LHC with an integrated luminosity of 3000.

One should note that, relying on the value of triplet VEV, decay modes of different non-standard scalars show distinct behaviour. From the consideration of electroweak precision test the triplet VEV can not be larger than a few GeV Aoki:2012jj (); Patrignani:2016xqp (). However, it can vary from GeV to GeV. Within this range the non-standard Higgs bosons decay in several distinct channels. To be more specific, for , the doubly charged Higgs dominantly decays into two same-sign leptonic final state. The latest same-sign dilepton searches at the LHC have already put strong lower limit on doubly charged Higgs mass ( 770 - 800 GeV) ATLAS:2017iqw (). On the other hand for , only gauge boson final state or cascade decays of singly charged Higgs (if they are kinematically allowed) are possible Perez:2008ha (); Melfo:2011nx (); Aoki:2011pz (); Han:2015hba (); Han:2015sca (). The collider search becomes more involved in this region of triplet VEV due to more complicated decay patterns of the doubly charged Higgs. As a result, the lower bound on the mass of the doubly charged Higgs is very relaxed. Therefore, in this region one can find scenarios where the mass of doubly charged Higgs may goes down to about 100 GeV Melfo:2011nx (); Chabab:2016vqn (). In this article, for all practical purposes we have considered the triplet VEV greater than GeV. For example, for the generation of neutrino mass we set triplet VEV at GeV. Whereas, for the purpose of dark matter analysis we show our results for two different values of triplet VEV e.g., GeV and 3 GeV respectively. This is in stark contrast to the Ref. Chen:2014lla () where the triplet VEV has been considered less than GeV. Further, for collider study we have fixed the the value of triplet VEV at 3 GeV and hence the doubly charged Higgs decays into with 100% branching ratio.

We organise this article as follows. First we introduce the model with possible interactions and set our conventions in Sec. II. Within this section we have also evaluated the vacuum stability and unitarity conditions in detail. In Sec. III, we discuss the neutrino mass generation via Type-II seesaw mechanism and explain neutrino oscillation data for normal and inverted hierarchies at range. The viability of dark matter candidate proposed in this work has been extensively studied in Sec. IV, considering all possible bounds from direct and indirect experiments. In Sec. V, we show the prospects of collider signature of the dark matter candidate of the present model at 13 TeV LHC. Finally in Sec. VI we summarize our results.

## Ii Type-II Seesaw with Inert Doublet

In this section, we discuss the model briefly. In order to produce a viable dark matter candidate, we introduce a symmetry in the SM gauge symmetry . Moreover, to generate the neutrino masses and also having a stable dark matter candidate, we incorporate a scalar triplet with hypercharge two and a scalar doublet with hypercharge one in the SM fields. Further, we demand that the SM particles and the triplet are even under parity while the new doublet is odd under parity. The field cannot develop a VEV at the time of electroweak symmetry breaking as this will break the symmetry spontaneously, which will jeopardize the dark matter stability. With this newly added symmetry, we discuss different interaction terms involving SM fields, and . The total Lagrangian which incorporates all possible interactions can be written as:

 L=LYukawa+LKinetic−V(H,Δ,Φ), (1)

where the relevant kinetic and Yukawa interaction terms are respectively

 Lkinetic = (DμH)†(DμH)+Tr[(DμΔ)†(DμΔ)]+(DμΦ)†(DμΦ), (2) LYukawa = LSMYukawa−Yνij2LTiCiσ2ΔLj+h.c.. (3)

The first two terms of generate the masses of gauge bosons and by electroweak symmetry breaking mechanism (EWSB), however the third term does not contribute to gauge boson masses as does not possess any VEV. Here represents doublet of left handed leptons where being the generational index, represents Yukawa coupling and is the charge conjugation operator. Further, denotes the Yukawa interactions for all SM fermions. Later, we will discuss the second term of Yukawa interactions in detail in the neutrino section (Section III). There is no term which involves the coupling between and the SM fermions as is odd under parity while the SM fermions are even under symmetry. Representations for the doublets and are chosen as and respectively. The triplet field transforms as under the gauge group, so one can write , which gives a representation given in the following:

 Δ=(δ+/√2δ++δ0−δ+/√2). (4)

In the above . The neutral component of the triplet field can be expressed as where and are vacuum expectation values of the doublet and triplet respectively. The covariant derivative of the scalar field is given by,

 DμΔ=∂μΔ+ig22[σaWaμ,Δ]+ig1BμΔ(a=1,2,3). (5)

Here ’s are the Pauli matrices while and are coupling constants for the gauge groups and respectively.

Let us discuss the scalar potential given in the following Chen:2014lla ():

 V(H,Δ,Φ) = −m2H(H†H)+λ4(H†H)2+M2ΔTr(Δ†Δ)+(μHTiσ2Δ†H+h.c.) (6) +λ1(H†H)Tr(Δ†Δ)+λ2[Tr(Δ†Δ)]2+λ3Tr(Δ†Δ)2+λ4(H†ΔΔ†H) +m2Φ(Φ†Φ)+λΦ(Φ†Φ)2+λ5(H†H)(Φ†Φ)+λ6(H†ΦΦ†H) +λ7(Φ†Φ)Tr(Δ†Δ)+λ8(Φ†ΔΔ†Φ)+λ9[(Φ†H)2+h.c.] +(~μΦTiσ2Δ†Φ+h.c.).

Here, , and () are dimensionless coupling constants, while , , , and are mass parameters of the above potential. Whereas , and are the only terms which can generate CP phases, as the other terms of the potential are self-conjugate. However, two of them can be removed by redefining the fields , and . Furthermore, we assume that for the spontaneous breaking of above mentioned gauge group.

After EWSB we obtain a doubly charged scalar including a singly charged scalar, , a pair of neutral CP even Higgs (), a CP odd scalar () and as usual three massless Goldstone bosons (). Further, we also have three particles (, , and ) which are members of the inert doublet. The mass eigenvalues for the even physical scalar are given by Arhrib:2011uy ():

 M2H±± = √2μv2d−λ4v2dvt−2λ3v3t2vt, (7) M2H± = (v2d+2v2t)(2√2μ−λ4vt)4vt, (8) M2A0 = μ(v2d+4v2t)√2vt, (9) M2h0 = 12(A+C−√(A−C)2+4B2), (10) M2H0 = 12(A+C+√(A−C)2+4B2), (11)

with

 A = λ2v2d, (12) B = vd[−√2μ+(λ1+λ4)vt], (13) C = √2μv2d+4(λ2+λ3)v3t2vt, (14)

while the mass eigenvalues of odd scalars are:

 M2ϕ0 = m2Φ+12(λ5+λ6)v2d+12(λ7+λ8)v2t+λ9v2d−√2~μvt, (15) M2a0 = m2Φ+12(λ5+λ6)v2d+12(λ7+λ8)v2t−λ9v2d+√2~μvt, (16) M2ϕ± = m2Φ+12λ5v2d+12λ7v2t. (17)

The mixing between the SM doublet and the triplet scalar fields in the charged, CP even as well as CP odd scalar sectors are respectively given by:

 (G±H±) = (cosβ′sinβ′−sinβ′cosβ′)(h±δ±), (18) (h0H0) = (cosαsinα−sinαcosα)(η0ξ0), (19) (G0A0) = (cosβsinβ−sinβcosβ)(z1z2), (20)

and the respective mixing angles are given by:

 tanβ′ = √2vtvd, (21) tanβ = 2vtvd=√2tanβ′, (22) tan2α = 2BA−C, (23)

where the expressions of , and are already given in Eq. 11.

### ii.1 Different constraints

Before going to study the phenomenological aspects of neutrino and dark matter sectors, it is necessary to check various constraints from theoretical considerations like vacuum stability, unitarity of the scattering matrices and perturbativity. Further, the model parameters also need to satisfy the phenomenological constraints arising from electroweak precision test and Higgs signal strength. Therefore, to serve the purposes we need to choose a set of free parameters of this model. In practice, a convenient set of free parameters are given in the following, however some of them are not independent:

 {tanα,MH±±,MH±,MH0(=MA0),Mϕ0,Mϕ±,λΦ,λ5,λ6,λ7,λ8,λ9}. (24)

#### ii.1.1 Vacuum stability bounds:

This section has been dedicated to derive the necessary and sufficient conditions for the stability of the vacuum. These conditions come from requiring that the potential given in Eq. 6 be bounded from below when the scalar fields become large in any direction of the field space. The constraints ensuring boundedness from below (BFB) of the present potential have not been studied in the literature so far. It would thus be very relevant to derive these constraints in the present model. For large field values, the potential given in Eq. 6 is generically dominated by the quartic part of the potential. Hence, in this limit we can ignore any terms with dimensionful couplings, mass terms or soft terms. So the general potential given in Eq. 6 can be written as in the following way which contains only the quartic terms,

 V(4)(H,Δ,Φ) = λ4(H†H)2+λ1(H†H)Tr(Δ†Δ)+λ2[Tr(Δ†Δ)]2+λ3Tr(Δ†Δ)2 (25) +λ4(H†ΔΔ†H)+λΦ(Φ†Φ)2+λ5(H†H)(Φ†Φ)+λ6(H†ΦΦ†H) +λ7(Φ†Φ)Tr(Δ†Δ)+λ8(Φ†ΔΔ†Φ)+λ9[(Φ†H)2+h.c.].

To determine the BFB conditions we have used copositivity criteria as given in Ref. Kannike:2012pe (). For this purpose we need to express the scalar potential in a biquadratic form , where . If the matrix is copositive then we can demand that the potential is bounded from below. Let us write down the matrix in our case:

 A=⎛⎜ ⎜ ⎜⎝14λ12(λ1+ξλ4)12[λ5+ρ2(λ6−2|λ9|)]12(λ1+ξλ4)(λ2+ζλ3)12(λ7+ξ′λ8)12[λ5+ρ2(λ6−2|λ9|)]12(λ7+ξ′λ8)λΦ⎞⎟ ⎟ ⎟⎠. (26)

The parameters , , and appearing in the matrix elements are required to determine all the necessary and sufficient BFB conditions. The detail illustrations of the parameters can be found in Arhrib:2011uy () where two fields (one doublet and a triplet) have been considered. However, in our case we have three different fields (two doublets and a triplet). Using the prescription given in Ref. Arhrib:2011uy (), we have defined the parameters in the following way,

 ζ≡Tr(Δ†Δ)2/[Tr(Δ†Δ)]2, (27) ρ≡|H†Φ|/|H||Φ|, (28) ξ≡(H†ΔΔ†H)/(H†H\leavevmode\nobreak Tr(Δ†Δ)), (29) ξ′≡(Φ†ΔΔ†Φ)/(Φ†Φ\leavevmode\nobreak Tr(Δ†Δ)). (30)

The and limits of these parameters are given as [], [], [] and [] respectively Arhrib:2011uy (). To determine the all possible BFB conditions of the scalar potential, we consider both the limits of these parameters and respect the copositivity criteria. Finally, we can write down the following BFB conditions by demanding the symmetric matrix is copositive Kannike:2012pe ().

 λ≥0, (31) (λ2+ζλ3)≥0, (32) λΦ≥0, (33) (λ1+ξλ4)+√λ(λ2+ζλ3)≥0, (34) λ5+ρ2(λ6−2|λ9|)+√λλΦ≥0, (35) (λ7+ξ′λ8)+2√(λ2+ζλ3)λΦ≥0, (36)
 √λ(λ2+ζλ3)λΦ+(λ1+ξλ4)√λΦ+[λ5+ρ2(λ6−2|λ9|)]√λ2+ζλ3+(λ7+ξ′λ8)2√λ (37) +√{(λ1+ξλ4)+√λ(λ2+ζλ3)}{(λ7+ξ′λ8)+2√λΦ(λ2+ζλ3)}{[λ5+ρ2(λ6−2|λ9|)]+√λλΦ}≥0.

Substituting the lower and upper limits of the parameters, one can get the full set of vacuum stability conditions given in Appendix B (see Eq. B-3 to Eq. B-10h).

#### ii.1.2 Unitarity bounds:

In this section we discuss the unitarity constraints on the parameters of scalar potential by using the tree-level unitarity of various scattering processes. One can find the scalar-scalar scattering, gauge boson-gauge boson scattering and scalar-gauge boson scattering in the context of SM in Appelquist:1971yj (); Cornwall:1974km (); Lee:1977eg (). In the case of various extended Higgs sector scenario, the generalizations of such constraints can be found in literature Kanemura:1993hm (); Akeroyd:2000wc (); Aoki:2007ah (); Gogoladze:2008ak (). It has been a well known fact that in the high energy limit using equivalence theorem Lee:1977eg (); Cornwall:1973tb (); Chanowitz:1985hj () one can replace longitudinal gauge bosons by those of the corresponding Nambu-Goldstone bosons in scattering. Hence, following this prescription in the current model, our main focus is to consider only the Higgs-Goldstone interactions of the scalar potential given in Eq. 6. Furthermore, under this situation the -body scalar scattering processes are dominated by the quartic interactions only.

To determine the unitarity constraints, it has been a usual trend to calculate the -matrix amplitude in the basis of unrotated states, corresponding to the fields before electroweak symmetry breaking. Because, in this situation the quartic scalar vertices have a much simpler form with respect to the complicated functions of , , and involved in the physical basis333For the inert Higgs doublet, the physical basis are equivalent to the gauge basis as in this case the vacuum expectation value is zero. (, , , , , , , , and ). So in the unrotated basis (, , , , , , , , and ), we study full set of -body scalar scattering processes which lead to a -matrix. This matrix can be decomposed into 7 block submatrices with definite charge. For example, , and corresponding to neutral charged states, corresponding to the singly charged states, corresponding to the doubly charged states, corresponding to the triply charged states and finally corresponding to the unique quartic charged state. These submatrices are hermitian, so the eigenvalues will always be real-valued.

To this end, we would like to mention that in the following cases we will determine the eigenvalues of the above mentioned submatrices. However, there is a caveat. The structure of some of the submatrices are very challenging, so it is not possible to find out the analytic form of all the eigenvalues of those matrices. However, using numerical technique given in Adhikary:2013bma () we can derive the remaining eigenvalues. Eventually, we will have all the full set of eigenvalues by which we will put the unitarity constraints on the model parameters.

The first submatrix corresponds to the scatterings whose initial and final states are one of the following:

 {h+δ−,δ+h−,ϕ+δ−,δ+ϕ−,h+ϕ−,ϕ+h−,η0z2,ξ0z1,z1z2,η0ξ0,ϕ0η0,ϕ0z1,η0a0,a0z1,ϕ0ξ0,ϕ0z2, a0ξ0,a0z2},

Eigenvalues of are:

 {λ1,λ1,λ1+λ4,λ1+λ4,λ1+3λ42,λ1+3λ42,λ5+λ6,λ5+λ6,λ7,λ7,λ7+λ8, λ7+λ8,λ7+3λ82,λ7+3λ82,λ5+2λ6−6λ9,λ5−2λ9,λ5+2λ9,λ5+2λ6+6λ9}.

The second submatrix corresponds to the scatterings whose initial and final states are one of the following:

 {h+h−,δ+δ−,z1z1√2,z2z2√2,η0η0√2,ξ0ξ0√2,ϕ+ϕ−,ϕ0ϕ0√2,a0a0√2,δ++δ−−},

Eigenvalues of are:

Rest of the six eigenvalues have been obtained by numerically solving the cubic Eqs. A-1 and A-2 given in Appendix A.

The third submatrix corresponds to the scatterings whose initial and final states are one of the following:

 {η0z1,ξ0z2,ϕ0a0},

Eigenvalues of are:

 {2(λ2+λ3),14(λ−√64λ29+(λ−4λΦ)2+4λΦ),14(λ+√64λ29+(λ−4λΦ)2+4λΦ)}.

The fourth submatrix corresponds to the scatterings, where one charge channels occur for scattering between the 20 charged states:

 {η0h+,ξ0h+,z1h+,z2h+,hδ+,ξ0δ+,z1δ+,z2δ+,η0ϕ0,ξ0ϕ+,z1ϕ+z2ϕ+,h+ϕ0,h+a0,δ+ϕ0,δ+a0, ϕ+ϕ0,ϕ+a0,δ++δ−,δ++h−,δ++ϕ−},

Eigenvalues of are:

 {λ1,λ1,2λ2,2(λ2+λ3),λ1% −λ42,λ1+λ4,λ1+3λ42,λ5−λ6,λ5+λ6,λ7,λ7,λ7−λ82, λ7+λ8,λ7+3λ82,λ5−2λ9,λ5+2λ9,14(λ+√64λ29+(λ−4λΦ)2+4λΦ), 14(λ−√64λ29+(λ−4λΦ)2+4λΦ)}.

Remaining three eigenvalues have been obtained from the cubic Eq. A-2 (see Appendix A) using numerical technique.

The fifth submatrix corresponds to the scatterings, where double charge channels occur for scattering between the 12 charged states:

 {h+h+√2,δ+δ+√2,δ+h+,ϕ+ϕ+√2,ϕ+δ+,ϕ+h+,δ++ξ0,δ++z2,δ++z1,δ++η0,δ++ϕ0,δ++a0},

Eigenvalues of are:

 {λ1,2λ2,2λ2−λ3,2(λ2+λ3),λ1−λ42,λ1+λ4,λ5+λ6,λ7,λ7−λ82,λ7+λ8, 14(λ+√64λ29+(λ−4λΦ)2+4λΦ),14(λ−√64λ29+(λ−4λΦ)2+4λΦ)}.

The sixth submatrix corresponds to the scatterings, where triple charge channels occur for scattering between the 3 charged states:

 {δ++h+,δ++δ+,δ++ϕ+},

Eigenvalues of are: {2(+),  +,  +}. Finally, there is unique quadruple charged state which leads to eigenvalue

 M7=2(λ2+λ3).

These eigenvalues, can be labelled as , then the -matrix unitarity constraint for elastic scattering demands Lee:1977eg (). Using this condition we generate the following relations. However, these conditions are not the full set of unitarity conditions as we have already mentioned that some of the eigenvalues of few submatrices are evaluated numerically. Hence, using the following conditions,

 |λ1|≤8π,|2λ2|≤8π,|λ1+λ4|≤8π,|2(λ2+λ3)|≤8π,∣∣∣λ1+3λ42∣∣∣≤8π,∣∣∣λ1−λ42∣∣∣≤8π, |2λ2−λ3|≤8π,|λ5+2λ6−6λ9|≤8π,|λ5+2λ6+6λ9|≤8π,|λ5+2λ9|≤8π,|λ5−2λ9|≤8π, |λ5−λ6|≤8π,|λ5+λ6|≤8π,|λ7|≤8π,|λ7+λ8|≤8π,∣∣∣λ7+3λ82∣∣∣≤8π,∣∣∣λ7−λ82∣∣∣≤8π, ∣∣∣14(λ+√64λ29+(λ−4λΦ)2+4λΦ)∣∣∣≤8π,∣∣∣14(λ−√64λ29+(λ−4λΦ)2+4λΦ)∣∣∣≤8π, (38)

and numerically evaluated six eigenvalues (whose absolute value should be ) we have imposed full set of unitarity constraints on the model parameters.

#### ii.1.3 Perturbativity:

If we demand that the model in the present work behaves as a perturbative quantum field theory at any energy scale, then we have to ensure the following conditions. For the scalar quartic coupling , the perturbativity criterion is,

 |λ|,|λΦ|,|λi|<4π. (39)

The corresponding constraints for the gauge and Yukawa interactions are,

 gi,yi<√4π, (40)

where, ’s and ’s are the gauge and Yukawa coupling constants respectively.

#### ii.1.4 Constraints from electroweak precision test:

Electroweak precision test (EWPT) can be considered as a very useful tool in constraining any BSM scenario. As the current scenario contains several non-standard scalars, hence they contribute to the electroweak precision observables, the parameters Lavoura:1993nq (); Barbieri:2006dq (); Chun:2012jw (); Aoki:2012jj (). The stringent bound comes from the -parameter which imposes strict limit on the mass splitting between the non-standard scalars. Therefore, we tune the relative mass splitting between the non-standard scalars in such a way for which the present scenario satisfy the constraints from EWPT Baak:2014ora (). Further, the electroweak precision data constraint the -parameter to be very close to its SM value of unity and from the latest data Patrignani:2016xqp () one gets an upper bound on GeV which we maintain in our analysis.

#### ii.1.5 Constraints from Higgs signal strength (μγγ=σ(pp→h)BSM×BR(h→γγ)BSMσ(pp→h)SM×BR(h→γγ)SM):

Moreover, apart from the above mentioned theoretical constraints, it is necessary to incorporate the constraints from LHC data in the model. As in the present model all the decay widths and cross sections are modified with respect to that of the SM predictions so in our analysis we have constrained the parameter space of this model by the present LHC Higgs data ATLAS:2016nke ().

## Iii Neutrino masses and mixings

In this section, we have tried to explain the origin of neutrino masses and their intergenerational mixing angles. In the present model, as we have one scalar triplet (Eq. 4), hence one can generate Majorana mass term for the SM neutrinos using Type-II seesaw mechanism Magg:1980ut (); Ma:1998dx (). The Yukawa interaction term which is responsible for the Majorana masses of SM neutrinos is given by

 LYukawa⊃−Yνij2LTiCiσ2ΔLj+h.c. (41)

where is the Yukawa coupling and are generational indices of the SM leptons. When the scalar triplet acquires a VEV , Majorana masses for the SM neutrinos are generated at tree level, which is

 Mνij=Yνij√2vt. (42)

Since this a Majorana type mass term for the SM neutrinos, must be a symmetric matrix. Therefore, for three generations of the SM neutrinos the Majorana mass matrix has the following form

 Mν=vt√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝y1\leavevmode\nobreak \leavevmode\nobreak y2\leavevmode\nobreak \leavevmode\nobreak y3y2\leavevmode\nobreak \leavevmode\nobreak y4\leavevmode\nobreak \leavevmode\nobreak y5y3\leavevmode\nobreak \leavevmode\nobreak y5\leavevmode\nobreak \leavevmode\nobreak y6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (43)

where, for notational simplicity we have redefined the Yukawa couplings as , , , , and . Now, our goal is to diagonalise the above mass matrix and find the mass eigenvalues and mixing angles. To diagonalise a complex symmetric matrix (all six independent elements of can be in general complex) we need a unitary matrix so that is a diagonal matrix (). This is however not the eigenvalue equation, which has usually been solved for the case of matrix diagonalisation. Therefore, instead of diagonalising a complex symmetric matrix , one can easily construct a hermitian matrix using , such that is a diagonal matrix with real non-negative entities at the diagonal positions. The unitary matrix is the usual PMNS matrix which has the following form

 UPMNS=UCKM⎛⎜ ⎜⎝1\leavevmode\nobreak 0\leavevmode\nobreak 00\leavevmode\nobreak expiα2\leavevmode\nobreak 00\leavevmode\nobreak 0\leavevmode\nobreak expiβ2⎞⎟ ⎟⎠, (44)

where is the usual CKM matrix containing three mixing angles , , and one phase , called the Dirac CP phase 444Because, any nonzero value of can generate CP violating effects in vacuum neutrino oscillations if , i.e. , (.) in vacuum oscillation when and Akhmedov:1999uz (). while , are known as the Majorana phases. If SM neutrinos are Dirac fermions then .

We have diagonalised the hermitian matrix by the unitary matrix and find the mass square differences and mixing angles between different generations of SM neutrinos. Dirac phase can be found by using a quantity known as Jarlskog Invariant () Jarlskog:1985ht (), which is related to the elements of matrix as,

 JCP=Im(h12h23h31)Δm221Δm232Δm231, (45)

where numerator represents the imaginary part of the product while in the denominator . One the other hand can also be written in terms of mixing angles and Dirac CP phases, i.e.

 JC