The Higgs boson mass constraint and the CP even-CP odd Higgs boson mixing in an MSSM extension

Tarek Ibrahim^{1}^{2}^{3}

University of Science and Technology, Zewail City of Science and Technology,

6th of October City, Giza 12588, Egypt^{4}

Department of Physics, Northeastern University,
Boston, MA 02115-5000, USA

Department of Physics, Faculty of Science, Alexandria University, Alexandria, Egypt

Abstract

One loop contributions to the CP even-CP odd Higgs boson mixings arising from contributions
due to exchange of a vectorlike multiplet are computed under the Higgs boson mass constraint.
The vectorlike multiplet consists of
a fourth generation of quarks and a mirror generation. This sector brings in new CP phases
which can be large consistent with EDM constraints. In this work we compute the
contributions from the exchange of quarks and mirror quarks
, and their scalar partners, the squarks and the mirror squarks. The effect of their
contributions to the Higgs boson masses and mixings are computed and analyzed.
The possibility of measuring the effects of mixing of CP even and CP odd Higgs in experiment
is discussed. It is shown that the branching ratios of the Higgs bosons into fermion pairs are sensitive to new
physics and specifically to CP phases.

Keywords: Neutral Higgs Spectrum, Higgs mixing, vector multiplet

PACS numbers: 12.60.-i, 14.60.Fg

## 1 Introduction

One of the important phenomenon in MSSM is the observation that the CP even-CP odd Higgs bosons can mix in the presence of an explicit CP violation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Such mixings give rise to effects which are observable at colliders. All of the early analyses, however, were done in the era before the experimental observation of the light Higgs boson at 125 GeV by ATLAS [14] and by CMS [15]. It turns out that the Higgs boson mass constraint is rather stringent and severely limits the parameter space of supersymmetry models. In this work we consider the effects of including a vectorlike multiplet in an MSSM extension. In this case the loop correction to the Higgs boson arises from two contributions: one from the MSSM sector and the other from the vectorlike multiplet. It is shown that such an inclusion leads to significant enhancement of the CP even-CP odd mixing. The explicit CP violation in the Higgs sector can be in conformity with the current limits on the EDM of quarks and leptons due to either mass suppression [16, 17] in the sfermion sector or via the cancellation mechanism [18, 19, 20, 18, 21, 22]. The neutral Higgs boson mixing is of great import since the observation of such a mixing would be a direct indication of the existence of a new source of CP violation beyond what is observed in the Kaon and the B-meson system (for a review see [23]).

The outline of the rest of the paper is as follows: In section 2 we describe the model and define notation. Inclusion of the vectorlike generation allowing for mixings between the vectorlike and the regular generations increases the dimensionality of the quark mass matrices from three to five and increases the dimensionality of the squark mass squared matrices from six to ten. In section 3 the effect of the vectorlike generation on the induced CP violation in the Higgs sector as a consequence of CP violation in the matter sector including the vectorlike matter is discussed. In section 4 a detailed computation of the corrections to the Higgs boson mass matrices is given. A numerical analysis of the mixing of the CP even-CP odd sector is discussed in section5. A discussion of the constraints arising from the EDM of the quarks is also given in this section. Conclusions are given in section 6. Further details of the squark mass squared matrices including the vectorlike squarks are given in the Appendix.

## 2 The Model and Notation

Here we briefly describe the model and further details are given in the appendix. The model we consider is an extension of MSSM with an additional vectorlike multiplet. Like MSSM the vectorlike extension is free of anomalies and vectorlike multiplets appear in a variety of settings which include grand unified models, string and D brane models [24, 25, 26, 27]. Several analyses have recently appeared which utilize vectorlike multiplets [28, 29, 30, 31, 32, 33, 34, 35, 36]

Here we focus on the quark sector where the vectorlike multiplet consists of a fourth generation of quarks and their mirror quarks. Thus the quark sector of the extended MSSM model is given by where

(1) |

(2) |

The numbers in the braces show the properties under where the first two entries label the representations for and and the last one gives the value of the hypercharge normalized so that . We allow the mixing of the vectorlike generation with the first three generations. Specifically we will focus on the mixings of the mirrors in the vectorlike generation with the first three generations. Here we display some relevant features. In the up quark sector we choose a basis as follows

(3) |

and we write the mass term so that

(4) |

The superpotential (as shown in the appendix) of the theory leads to the up-quark mass matrix which is given by

(5) |

This mass matrix is not hermitian and a bi-unitary transformation is needed to diagonalize it. Thus one has

(6) |

Under the bi-unitary transformations the basis vectors transform so that

(7) |

A similar analysis can be carried out for the down quarks. Here we choose the basis set as

(8) |

In this basis the down quark mass terms are given by

(9) |

where using the interactions of has the following form

(10) |

In general can be complex and we define their phases so that

(11) |

The squark sector of the model contains a variety of terms including F -type, D-type, soft as well as mixings terms involving squarks and mirror squarks. The details of these contributions to squark mass square matrices are discussed in the appendix.

## 3 Computation of correction to the Higgs boson mass

In MSSM the Higgs sector at the one loop level is described by the scalar potential

In our analysis we use the renormalization group improved effective potential where

(12) |

where and and are the soft SUSY breaking parameters, and is the one loop correction to the effective potential and is given by

(13) |

where where the sum runs over all particles with spin and counts the degrees of freedom of the particle i, and Q is the running scale. In the evaluation of one should include the contributions of all of the fields that enter in MSSM. This includes the Standard Model fields and their superpartners, the sfermions, the higgsinos and the gauginos. The one loop corrections to the effective potential make significant contributions to the minimization conditions.

It is well known that the presence of CP violating effect in the one loop effective potential induce CP violating phase in the Higgs VEV through the minimization of the effective potential. One can parametrize this effect by the CP phase where

(14) |

(15) |

The non-vanishing of the phase can be seen by looking at the minimization of the effective potential. For the present case with the inclusion of CP violating effects the variations with respect to the fields give the following

(16) |

(17) |

(18) |

where the subscript 0 means that the quantities are evaluated at the point .

The masses to be included in the analysis are the masses of three MSSM quark and their squark partners along with the masses of the generations in the vectorlike sector of the theory. In this case the phase is determined by

(19) |

where

(20) |

To construct the mass squared matrix of the Higgs scalars we need to compute the quantities

(21) |

where (a=1-4) are defined by

(22) |

and as already specified the subscript 0 means that we set after the evaluation of the mass matrix. The tree and loop contributions to are given by

(23) |

where are the contributions at the tree level and are the loop contributions where

(24) |

where e=2.718. Computation of the Higgs mass matrix in the basis of Eq.(22) gives

(25) |

where . In the above the explicit Q dependence has been absorbed in which is given by

(26) |

The first term in the second brace on the right hand side of the above equation is the tree term, while the rest ten terms are coming from the three generations of MSSM (six terms) and four terms from the vectorlike multiplet. One may reduce the matrix of the Higgs matrix by introducing a new basis where

(27) |

In this basis the field decouples from the other three fields as a Goldstone field with a zero mass eigen value. The Higgs mass matrix of the remaining three fields are given by

(28) |

## 4 Computation of Corrections to the Higgs boson mass squared matrix

We consider the exchange contribution from the quarks/mirror quarks and from the squarks/mirror squarks in the susy standard model enriched with the vectorlike generation.

(29) |

Note that in the supersymmetric limit, quark masses would be equal to the squark masses and the loop corrections vanish.

Using the above loop corrections we can calculate the corrections to the different Higgs mass elements as

(30) |

where

(31) |

For the up quarks/squarks we have the contributions

(32) |

where for , for , for , for and

(33) |

For the mirror contribution is given by