# Unification of Gauge Couplings and the Higgs Mass in Vector-Like Particle Theories Extended into NMSSM

###### Abstract

The minimal supersymmetry standard model (MSSM) extended with vector-like theories have long been discussed. If we extend the vector-like MSSM theory into NMSSM and let the vector-like particles couple with the singlet (S), we find out a natural way to generate the vector-like particle masses near 1TeV through the breaking of the group. Compared with MSSM+vectorlike models, vector-like models extended into NMSSM contain more yukawa couplings and can help us adjust the renormalization group (RG) trajectories of the gauge couplings in order to unify the intersections. They can also help press down the gauge RG- functions for a model, in order for the RG-trajectories of the gauge couplings to unify before the Landau-pole. We also discuss the higgs mass contributed from the vector-like sectors in this case.

###### pacs:

12.60.Jv, 14.60.Hi, 14.65.Jk, 14.80.Da## I Introduction

Minimal supersymmetry standard model (MSSM) is a way to extend the standard model (SM) Primer . Within this framework, every particle is paired up with a super-partner with a different spin. One of the features of this model is that it can also automatically unify the the gauge-coupling constants in the energy scale GeV SUSY_GUT , as is required by grand unification theory (GUT). Another way to extend the SM is to add extra copies of the multiplets. In order to construct an anomaly-free theory, the simplest way is just to add extra SM generations 4th1 ; 4th2 ; 4th3 ; 4th4 . Compared with the theories extended with chiral 4th generation, theories extended with vector-like (VL) generation is easier to survive among the experimental limits due to their particular vector-like mass parameters.

MSSM extended with VL generations have long been discussed VL ; Martin:2009bg ; Martin:2010dc ; Liu:2009cc ; Liu_With_Lu ; Chang:2013eia ; TopSeeSaw . In order not to disturb the gauge coupling unification scale, only copies of , or multiplets are the candidates to be added into the theory. Up to one-loop level, one multiplet modifies the renormalization group (RG) functions of the gauge coupling constants three times the same as one multiplet. If we would like the VL particles to be of the mass TeV, and require the gauge couplings to meet with each other before they knock into Landau Pole, we can only choose copies of , where , or one GMSB_GUT .

The situation of 4 copies of theory, or theory is subtle. One-loop calculation unifies the gauge coupling constants with a value of 2-5, which approaches the Landau pole so much. Two-loop corrections usually contribute a positive value to the gauge RG- functions, thus directly accelerate the gauge RG-trajectories to blow up before they meet.

In MSSM, effective higgs mass receives extra loop contributions from the top/stop yukawa coupling Primer . Higher order of stop mass raises the higgs mass while aggravates the tension of fine-tuning, and VL theories supply another source of the higgs mass. In this case, VL particles should directly couple with the SM-like higgs, and thus affect the Higgs phenomenology Elisabetta .

As is well-known, MSSM suffers from the problem, and adding VL generations cannot solve this problem at all. The most economic way to solve this problem is to extend MSSM into NMSSM by adding a singlet (S) NMSSMInt . The vacuum expectation value (vev) of the S naturally generates roughly , and the super-potential term can also add up to the higgs quartic couplings, thus raise up the higgs mass.

If we extend the VL-MSSM with a singlet S, just like the NMSSM, we may take the advantages of both these theories. Similar consequences have been discussed in NMSSMVLAncester1 ; NMSSMVLAncester2 ; NMSSMVLAncester3 . Models with a scalar singlet and VL fermions without supersymmetry is also studied in HeHongjianS . However, in our model, the VL-particles couple with S, so the VL-mass terms naturally come from the vev of the singlet Higgs rather than being inputed “by hand”. By setting appropriate value to the yukawa constants near GUT-scale, the mass-spectrum of the VL-fermions can be partly predicted. Gauge coupling trajectories also receive extra contributions from the yukawa coupling constants, thus the unification can be improved by adjusting the values of the yukawa couplings in and models. In model, Extra yukawa coupling constants also contribute a non-ignorable minus value in two-loop gauge RG- functions, thus the gauge-coupling RG trajectories might meet before the perturbative theories lose effect.

## Ii General Model

If we extend the ordinary NMSSM theory, we should assign the VL super-fields , , , , , with charges in order for them to be coupled with the S. These charges will also keep VL fermions massless before breaks. Appropriate assignment will also forbid the VL particles to mix with the SM , fields, which is highly limited by experiments. The quantum numbers assigned to the VL particles are listed in Table 1. However, only model involves all the fields listed in Table 1. In our discussion about model and model, only part of the fields are needed. We should also note that all MSSM quarks and leptons are ignored except that the effects of the top sectors are considered in our discussions due to their large yukawa coupling constant.

R-Parity | Description | |||||

Vector-like quark doublet. | ||||||

Vector-like anti-quark doublet. | ||||||

Vector-like right-handed up-type quark. | ||||||

Vector-like right-handed up-type anti-quark. | ||||||

Vector-like right-handed down-type quark. | ||||||

Vector-like right-handed down-type anti-quark. | ||||||

Vector-like lepton doublet | ||||||

Vector-like anti-lepton doublet. | ||||||

Vector-like right-handed electron | ||||||

Vector-like right-handed anti-electron. | ||||||

Up-type higgs doublet. | ||||||

Down-type higgs doublet. | ||||||

NMSSM Singlino higgs. | ||||||

SM 3rd generation quark doublet. | ||||||

SM right-handed top. |

Here we are going to take a short description about the basic NMSSM. The superpotential is NMSSMInt

(1) |

and together we show the supersymmetry soft-breaking terms

(2) |

The convention of the vacuum expectation values of the higgs fields is

(3) | |||||

so the MSSM-like superpotential term is generated.

Since the VL fermions receive mass terms from , their masses are actually in the same quantity as in most cases. In NMSSM, we usually concern about the and , and then . If , , which is usually applied for successful electro-weak symmetry breaking, we can derive that . Collider bounds on VL quarks have been reviewed in Earlier , and the CMS Collaboration recently published their lower bound of VL top-like quark mass to a value of 687-782 GeV CMS_Limit , so in our discussions below, we assume all of our VL fermions lie in the mass scale 1TeV, which is near the bound, although it is very easy to accumulate the VL mass towards TeV by lowering or raising . We also set 1TeV as the turning point in our RG-trajectory calculations.

### ii.1 Model

The model is the simplest model. It only contains vector-like down-type right-handed quark and anti-quark, and , together with vector-like leptonic doublet and . The VL particles can only couple with the , as the superfield showed below,

(4) |

In literature, right-handed neutrino might be introduced so that vertices are discussed, however, here we ignore them.

The supersymmetry breaking soft terms should be added,

(5) |

Notice that we have assumed that , , or , share the same soft-mass term only for simplicity. We can observe from (4, 5) that only singlet higgs sectors are involved here. However, as to be discussed below, this still contribute to the SM-like higgs mass.

### ii.2 Model

The model contains vector-like quark doublets , , vector-like up-type quark singlets and , and vector-like electron singlet and . The yukawa coupling structure is richer than theory due to the appearance of higgs doublets.

(6) |

The corresponding soft-terms are listed below,

(7) | |||||

### ii.3 Model

The model it not only a combination of and , new terms also rise up.

(8) |

The corresponding soft-terms are

(9) | |||||

During our discussions of the model, we would like to set and for simplicity.

## Iii Model

For the simplest model, the extra vector-like particles couple with the S, thus contribute to the Higgs mass. It is much easier to calculate this contribution than the circumstances in , or , because we only need to diagonalize mass(-squared) matrices here. There are two down-type squarks, and the corresponding mass-squared matrix is

(10) |

The mass matrix of the two slepton doublets is

(11) |

Gauge D-terms are ignored, and so is done in our remaining sections. Notice that for each slepton doublet, the masses of charged slepton and the neutral slepton are degenerate in this model. To diagonalize (10) and (11) we acquire

(12) | |||||

where are the two down-type squarks and , indicate the two charged sleptons and the two neutral sleptons separately.

The masses of the down-type vector-like quark and the charged (neutral) lepton are

(13) |

Thus, we can take (12) and (13) into Colemann-Weinberg potential under or scheme

(14) |

where Q is the renormalization scale, and , indicate the degrees of freedom of the particles. and take the value of 6 for colored fermionic or complex scalar particles, and 2 for colorless ones. Notice that the sum over fermions means to sum over all weyl-spinors, so each Dirac particle contributes an extra factor of 2 there.

If we assume that the SM-like Higgs mass-eigenstates to be in alignment with the vacuum expectation value (vev), that is to say, , where is the mixing angle of the Higgs mass-eigenstates, the SM-like higgs mass should be added with a term

(15) | |||||

It seems strange that the SM-like Higgs mass listed in (15) is Q-dependent, which is invisible in MSSM theories. In MSSM, the tree-level quartic coupling among Higgs fields only comes from the gauge D-terms, so loop contributions irrelevant to the gauge terms should not be renormalized in order not to break the gauge invariance. However, in the case of NMSSM, the appearance of also contribute to the Higgs quartic coupling, and receives the quantum correction from the field-strength renormalization constant of S. We can then define

(12,13) through (14) also contribute to the CP-even singlet Higgs mass. Expand the consequence up to and ,

The two leading terms are similar to (15), which result from the defined in the previous text, while the terms, especially the reflect the fact that the mass of the singlet Higgs also receive the corrections from the mass hierarchy of the corresponding vector-like fermions and sfermions, which is Q-independent.

To see the possible mass-spectrum of the vector-like fermions, we look into the RG trajectories of the coupling constants. We can learn from (29) that gauge terms contribute negative values to all yukawa RG- functions, while the yukawa terms always contribute positive ones. is a SM gauge-singlet, so the lack of minus terms decides the quasi-fixing point of to be actually 0. and are not SM gauge-singlets, however, we coupled a lot of things on and if and are too large, also tend to be small. At the GUT point, if we set , which is near the perturbative limit of , and apply with different values, and then we run the RG-trajectories down, we can see the relationship between these coupling constants near 1TeV through Figure 1.

From the two-loop functions of gauge couplings listed in (29), we can learn that the yukawa coupling constants play crucial roles if ever they’re large enough. It is ever-known that the unification of gauge coupling constants is not that good even in the circumstance of supersymmetry, although it has been improved a lot when compared with the case of SM. If we want to adjust the yukawa couplings in order to drive the gauge couplings into unifying in MSSM or NMSSM, there does not left much room in the parameter space because we do not have many notable yukawa coupling constants to be adjusted, and the top yukawa affects on all , , , making it difficult to converge the intersection points. In our case, strongly influences the trajectory of however slightly modifies due to and ’s relatively small super-charge , and only effects on and , so we can move the intersection point separately by adjusting and . After several attempts, we can reach a boundary condition

(17) |

and if we run down into Q=1TeV, the gauge coupling constants are accurately in accordance with the low-energy data , , . See Figure 2 for the trajectories.

## Iv Model

Without the help of extra ”vector-like neutrino” N, the model can only contribute to the SM-like higgs mass through S. However, the model contains direct vertices and . Unlike the case, the model contains four charged squarks and thus a matrix needs to be diagonalized, so it a simple analytical solution does not exist.

The mass-squared matrix of up-type squarks is shown below,

(18) | |||||

where we can observe that unlike the consequence of MSSM+()VL model (e.g. in Martin:2010dc ), lots of off-diagonal terms automatically appear, so the diagonalizing process becomes much more difficult. The mass-squared matrix of the two down-type squarks is

(19) |

The mass matrix of two VL fermionic up-type quark is

(20) |

while there is only one VL down-type quark

(21) |

Direct calculation diagonalizing (18) is so lengthy and troublesome, so we expand the result in series of , , and , and set , . According to experience, the coupling constants of leptons is usually smaller than quarks because leptons do not have colors, thus their quasi-fixing points are smaller. The leptons also do not receive accumulations, so, for simplicity, we ignore all leptonic contributions here. If we would like a relatively large , say, , the SM-like lightest higgs will mainly be and thus can also be ignored. Let’s define

(22) |

and expand the final result according to , , , . Similar to the process in (15), we acquire

(23) | |||||

where , are the estimated masses of fermionic and bosonic up-type quarks. We cut the series up to , , and . However, disappears in the final result, telling us that the difference between and does not exert a large effect on SM-like higgs mass.

Similar to (III), the singlet Higgs also receives one-loop quantum corrections. However, in spite of the similar , terms, the Q-independent terms are so complicated, that we don’t show them in this paper.

Now we are going to unify the gauge couplings. It is much more difficult to converge the intersection points in this circumstance than model, because the gauge couplings run into a larger value, , which is much less sensitive to the adjusting of the large yukawa couplings. If we set

(24) |

after running down to GeV, we get , , . See Fig 3 for trajectories. There is a little deviation from the (28) in Appendix A.

## V Model

If we ignore all the yukawa terms and put all the extra particles beyond SM at TeV, and calculate the gauge RG- functions up to 2-loop level, the trajectory actually blows up before can meet . See Fig 4.

Then we can add up yukawa couplings to modify the coupling constants’ trajectories. Now that runs the fastest, we add up , , to press the - function. However, the RG-equations are not stable enough if we run from TeV upwards to GUT-scale. If we apply the relatively large yukawa coupling constants to “press” the gauge RG- functions, it is easy for the yukawa coupling constants to blow up before the GUT is reached. However, it is much better to run from GUT-scale downwards to TeV, and by adjusting the yukawa coupling constants, we can acquire the correct values near TeV.

There is another severe problem that