# Theory and Phenomenology of Two Higgs Doublet Type-II Seesaw Model at the LHC Run-2

###### Abstract

We study the most popular scalar extension of the Standard Model, namely the Two Higgs doublet model, extended by a complex triplet scalar (2HDMcT). Such considering model with a very small vacuum expectation value, provides a solution to the massive neutrinos through the so-called type II seesaw mechanism. We show that the 2HDMcT enlarged parameter space allow for a rich and interesting phenomenology compatible with current experimental constraints. In this paper the 2HDMcT is subject to a detailed scrutiny. Indeed, a complete set of tree level unitarity constraints on the coupling parameters of the potential is determined, and the exact tree-level boundedness from below constraints on these couplings are generated for all directions. We then perform an extensive parameter scan in the 2HDMcT parameter space, delimited by the above derived theoretical constraints as well as by experimental limits. We find that an important triplet admixtures are still compatible with the Higgs data and investigate which observables will allow to restrict the triplet nature most effectively in the next runs of the LHC. Finally, we emphasize new production and decay channels and their phenomenological relevance and treatment at the LHC.

###### Keywords:

: Lepton Number Violation, Seesaw model, 2HDM, LHC.## 1 Introduction

After the discovery of a Standard-Model-like Higgs boson at the Large Hadron Collider (LHC) in 2012 Aad et al. (2012); Chatrchyan et al. (2012), the Standard Model (SM) of particle physics has been established as the most successful theory describing the elementary particles and their interactions. Despite its success, the SM features several conceptual issues and practical limitations that have motivated the particle physics community to focus on theoretical developments and experimental studies of physics beyond it. In particular, the SM cannot provide any explanation for the observed non-zero neutrino masses and mixings Patrignani:2016xqp (). Indeed, the SM Higgs field is responsible for the generation of the masses of all known fundamental particles, but it cannot accommodate the tiny observed neutrino masses. Within a renormalisable theory where new heavy fields are introduced, neutrino masses can be generated through the dimension-five Weinberg operator Weinberg:1979sa (), this is the so called seesaw mechanism. Different realisations of such a mechanism can be classified into three types: type I Minkowski:1977sc (); GellMann:1980vs (); Yanagida:1979as (); Mohapatra:1979ia (); Schechter:1980gr () in which only right-handed neutrinos coupling to the Higgs field, type II Mohapatra:1980yp (); Lazarides:1980nt (); Wetterich:1981bx (); Schechter:1981cv () where a new scalar field in the adjoint representation of and type III Foot:1988aq () which involve two extra fermionic fields. In the above seesaw mechanisms heavy fields are supplemented to the SM in such a way the desired neutrino properties are reproduced after the breaking of the electroweak symmetry.

In the type II seesaw model, also dubbed Higgs triplet models (HTM), arhrib et al. (2011), the SM Lagrangian is augmented by a scalar triplet field with hypercharge . In HTM, neutrinos may obtain masses via the vacuum expectation value (vev) of a neutral Higgs boson in an isospin triplet representation. The smallness of neutrino masses is guaranteed by the smallness of the triplet vev assumed to be less than 1 GeV and the non-conservation of lepton number which is explicitly broken in the HTM scalar potential by a trilinear coupling . The latter is protected by symmetry and is naturally small which ensures small neutrino masses. The model predicts several scalar particles, including doubly charged Higgs boson and singly charged Higgs boson . In addition the HTM spectrum contains a CP-odd neutral scalar and two CP even neutral scalars, and . The rest degrees of freedom are absorbed by the vector bosons. In a large region of the HTM parameter space, the lightest CP-even scalar, , has essentially the same couplings to the fermions and vector bosons as the Higgs boson of the SM.

By the new discovery of 125 scalar boson at the LHC, the phenomenology of two-Higgs-doublet mod (2HDM) has been further investigated broadly in the literature and due to the similarity in mass generation mechanism between type-II seesaw and Higgs mechanism, we extend HTM and focus on the two Higgs double model extension to the type-II seesaw model, displaying phenomenological characteristics notably different from the scalar sector emerging from the HTM. In the present work, we study several Higgs processes giving rise to the production times branching ratios of heavy Higgs bosons and focus on. Unlike most of the earlier studies, we consider a framework where both type I and type II seesaw mechanisms are implemented and contribute to neutrino mass generation. We consider a high integrated luminosity of LHC collisions at a centre-of-mass energy of 13 TeV.

The content of the paper is laid out as follows. In Sec. 2, we derive some crucial features of the 2HDMcT, with a focus on the particle content and the scalar potential of the model, followed by discussions on the minimization conditions of the scalar potential and the scalar mass spectra. Besides, in this work we investigate the impacts of two new terms introduced in the model scalar potential. Sec. 3, is devoted to the study of the theoretical constraints on the scalar potential parameters from tree-level vacuum stability and perturbative unitarity of the scalar sector. In Sec. 4, we further impose the LHC constraints associated with the 125 GeV Higgs boson and its signal strength to delimit the parameter space. Finally, we present some phenomenological aspects in Sec.5, benchmark points in Sec.6 and summarise our findings in Sec. 7.

## 2 General considerations of 2HDM with triplet

### 2.1 The Higgs sector

In a model with two Higgs doublets , the Two Higgs Doublet Type-II Seesaw Model (2HDMcT) contains and additional triplet Higgs field with hypercharge and lepton number ,

(2.1) |

the most general renormalizable and gauge invariant Lagrangian of the 2HDMcT scalar sector is given by

(2.2) |

where the scalar potential , symmetric under a group , reads as Chen et al. (2014)

(2.3) |

where :

(2.4) | |||||

(2.5) | |||||

(2.6) | |||||

In the above, , i=1,2,3 and are mass squared parameters, , i=1,…,5 are dimensionless couplings not related to the triplet, , i=8,9 are dimensionless couplings related to the triplet field, while , i=1,2,3 with , i=6,…,9, are dimensionless couplings that mixes all three Higgs fields. In Eq 2.6, denotes the trace over matrices, where for convenience we have used the traceless matrix representation for the triplet. Also, the potential defined in Eq 2.6 exhausts all possible gauge invariant renormalizable operators. For instance, a terms of the form and Chen et al. (2014), which would be legitimate to add if contained a singlet component, can actually be projected on the and operators appearing in Eq 2.6 thanks to the identity which is valid because is a traceless matrix.

Subsequently, we will assume that all these parameters are real valued. Indeed, apart from the terms, all the other operators in are self-conjugate so that, by hermicity of the potential, only the real parts of the ’s and the mass parameters are relevant. As for , the only parameters that can pick up a would be CP-phases, these phases are unphysical and can always be absorbed in a redefinition of the fields and . One thus concludes that the 2HDMcT Lagrangian is CP conserving. The electroweak symmetry is spontaneously broken when the neutral components of the Higgs fields acquire vacuum expectation values , and . Thus we can shift the Higgs fields in the following way,

(2.7) |

finding minimization conditions, or tree-level tadpole equations, given by

(2.8) |

where it is safe to take ( i=1,2,3) in which it leads to

(2.9) | |||||

(2.10) | |||||

(2.11) |

with , and If the terms associated to are omitted in Eq 2.11, then we can derive a new expression for as a function of the triplet scalar mass,

(2.12) |

Furthermore, for sufficiently large compared to , we see that the above formula reduces to,
, which is referred as type II seesaw
mechanism^{1}^{1}1In the absence of , and the
becomes negative leading to a spontaneous violation of
lepton number. The resulting Higgs spectrum contains a massless triplet
scalar, called Majoron . This model was
excluded at the CERN Large Electron Positron Collider (LEP) .

The 2HDMcT model has altogether degrees of freedom: 21 parameters originating from the scalar potential, Eq 2.3 and tree vacuum expectation values of the Higgs doublets and triplet fields. However, by using the three minimisation conditions, i.e. the gauge boson mass and the correct electroweak scale of them (, , and ) can be eliminated.

### 2.2 Higgs masses and mixing angles

In what follows, we will use Eqs. 2.9, 2.10 and 2.11 to trade the mass parameters , and for the rest of parameters given in the potential. Thus, the squared mass matrix is given by

(2.13) |

by denoting the corresponding VEV’s

(2.14) |

Eq.2.3 can be recast in a block diagonal form of one doubly degenerate eigenvalue and three matrices denoted in the following by , and . The bilinear part of the Higgs potential is then given by

(2.15) | |||||

at tree-level. The elements of these mass matrices are explicitly presented below.

#### Mass of the doubly charged field

The double eigenvalue , corresponding to the doubly charged eigenstate , can simply be determined by collecting all the coefficients of in the scalar potential. It is given by reads,

(2.16) |

#### Mass of the simply charged field

The mass-squared matrix for the simply charged field in the () basis reads as:

(2.17) |

where , and the diagonal terms are given by,

(2.18) |

Among the three eigenvalues of this matrix, one is zero and corresponds to the charged Goldstone bosons , while the two others correspond to the singly charged Higgs bosons denoted by and given by

(2.19) |

where , , , stand for the , , , respectively, while , , and

(2.20) | |||||

The above symmetric squared matrix is diagonalized via as:

(2.21) |

where the rotation matrix is described by three mixing angles , and , and the corresponding expressions for the elements are give in Appendix.A as a function of the parameters inputs of our model.

#### Mass of the neutral pseudo-scalar field

As to the mass-squared matrix for the neutral pseudoscalar field in the basis (), it is expressed as:

(2.22) |

the diagonal terms are given by,

(2.23) |

here, has three eigenvalues, one is zero corresponding to the neutral Goldstone boson while the two others are the physical states and , by setting

(2.24) | |||||

theirs masses read

(2.25) |

while the diagonalization of such matrix is done in this case by the introduction of an unitary matrix described by three mixing angles , and whose expressions can be found in Appendix. A, as

(2.26) |

#### Mass of the neutral scalar field

In the basis () the neutral scalar mass matrix reads :

(2.27) |

Its diagonal terms are,

(2.28) |

while the off-diagonal terms are given by,

(2.29) |

The mass matrix can be diagonalised by an orthogonal matrix which we parametrise as

(2.30) |

where the mixing angles , and can be chosen in the range

(2.31) |

the rotation between the two basis () and () diagonalises the mass matrix as,

(2.32) |

and leads to three mass eigenstates are ordered by ascending mass as

(2.33) |

One choice of input parameters implemented in 2HDMcT consistes to use the following hybrid paramerization

(2.34) |

in which .

In Appendix.B, we discuss the second choice of input parameters in the physical basis 2HDMcT. Using the Eqs. 2.32 and 2.16, one can easily express the reset of Lagrangian parameters in terms of those given by. 2.34. These are given by

(2.35) |

where and

(2.36) |

the remaining 10 parameters consists of the 6 charged and sectors mixing angles given respectively by (i=1,2,3)

(2.37) |

and (j=1,2,3)

(2.38) |

and 4 Higgs bosons masses, two of them correspond to charged states masses, while the two others are matched to states as discussed previously.

### 2.3 Yukawa and gauge bosons textures

contains all the Yukawa sector of the SM plus one extra Yukawa term that leads after spontaneous symmetry breaking to (Majorana) mass terms for the neutrinos, without requiring right-handed neutrino states,

(2.39) |

where denotes doublets of left-handed leptons, denotes neutrino Yukawa couplings, the charge conjugation operator. The symmetry is imposed in order to avoid tree-level FCNCs. Furthermore, and in terms of the various which appear in the expressions of matrix elements, we liste in table-1, all the (i=1,2,3) Yukawa couplings for both type I and type II in the model.

type-I | ||||||
---|---|---|---|---|---|---|

type-II |