# Number of fermion generations from a novel grand unified model

## Abstract

Electroweak interactions based on the gauge group \(\mathrm{SU}(3)_\mathrm{L}\times \mathrm{U}(1)_\mathrm{X}\), coupled to the QCD gauge group \(\mathrm{{SU}}(3)_\mathrm{c}\), can predict the number of generations to be multiples of three. We first try to unify these models within \(\mathrm{{SU}}(N)\) groups, using antisymmetric tensor representations only. After examining why these attempts fail, we continue to search for an \(\mathrm{{SU}}(N)\) GUT that can explain the number of fermion generations. We show that such a model can be found for \(N=9\), with fermions in antisymmetric rank-1 and rank-3 representations only, and we examine the constraints on various masses in the model coming from the requirement of unification.

## Keywords

Gauge Group Sterile Neutrino Higgs Doublet Chiral Multiplet Electroweak Scale## 1 Introduction

In the standard model, the number of fermion generations appears as an arbitrary parameter, meaning that a mathematically consistent theory can be built up using any number of fermion generations. The same is true for many extensions of the standard model, including grand unification models based on the gauge groups SU(5) and SO(10). An interesting question related to the extension of the standard model is whether the number of fermion generations can in any way be explained through the internal consistency of the model. In the literature, there is some discussion of grand unification models based on large orthogonal groups like SO(18), where one spinor multiplet contains all known fermion fields of all generations, and much more [1]. It was shown that [2, 3], with a suitable symmetry breaking scheme, only three generations can remain light, whereas the others obtain masses at much above the electroweak scale.

In a quite different line of development, it was shown that if one extends the electroweak group to \(\mathrm{{SU}}(3) \times \mathrm{U}(1)\) and tries to accommodate the standard fermions into multiplets of this gauge group which must include some new fermions, cancelation of gauge anomalies can restrict the number of generations and one obtains consistent models with the number of generations equal to three or any multiple of it [4, 5, 6, 7, 8].

These models are described briefly in Sect. 2. Then, in Sects. 3 and 4, we try to see whether these models can be embedded into a simple SU(N) group. We conclude that if one uses only completely antisymmetric tensor representations, such an embedding cannot be found. Then, in Sect. 5, we start looking for general conditions that will specify the number of fermion generations for arbitrary SU(N) groups. In Sect. 6, we analyze one simple model, based on the group SU(9), which gives three generations. The renormalization group analysis of various scales in these models is performed in Sect. 7. We end with some concluding remarks in Sect. 8.

## 2 The 3-3-1 models

The 3-3-1 models are based on the gauge group \(\mathrm{{SU}}(3)_\mathrm{c} \times \mathrm{SU}(3)_\mathrm{L} \times \mathrm{U}(1)_\mathrm{X}\). The first factor in the gauge group is the group of QCD, whereas the other two factors pertain to electroweak interactions. There are two versions of such models, and we discuss them one by one.

Note that there are two kinds of generation indices: \(a\) goes from 1 to 3, whereas \(i\) takes only the values 2 and 3. An antifermion has been denoted by a hat. We emphasize that all representations given above pertain to the left-chiral components only. The representation of a right-chiral fermion would be the complex conjugate of that of the left-chiral antifermion, and vice versa. Note that there are extra quark fields, i.e., fields which are triplets of \(\mathrm{{SU}}(3)_\mathrm{c}\), but there is no neutrino field that is sterile under the standard model. Different generations of fermions are not copies of one another. Gauge anomalies cancel between the three generations [5, 6], but not within a single generation. Thus, the consistency of the model requires three generations of fermions. Of course, this pattern of three generations can be repeated, so one would obtain a number of generations that is a multiple of three.

## 3 Seeking embeddings into SU(6)

Since the group \(\mathrm{{SU}}(3)_\mathrm{c} \times \mathrm{SU}(3)_\mathrm{L} \times \mathrm{U}(1)_\mathrm{X}\) is of rank 5, the smallest unitary group that contains it as a subgroup is SU(6). Therefore, in this section, we analyze whether the models discussed in Sect. 2 can be embedded into an SU(6) grand unified model.

For the SVS model, the same kind of analysis can be performed keeping an eye towards the antilepton in Eq. (2a). In order to produce a singlet of both SU(3) factors, one needs the Kronecker product of equal numbers of \(6\) and \(6^*\). However, such products will give \(X=0\).

## 4 Generalities of SU(N) models with \(N>6\)

We now consider models based on the gauge groups SU(N), with \(N>6\). Henceforth we will use completely antisymmetric tensor representations only. Such a representation of rank \(n\) will be denoted by \({[\![} \, n \, {]\!]}\), as was done in Eq. (16). The fundamental representation will thus be denoted by \({[\![} \, 1 \, {]\!]}\) in this notation, whereas its complex conjugate will be \({[\![} \, N-1 \, {]\!]}\). We will show that, with antisymmetric representations only, neither the PPF nor the SVS model can be embedded into an SU(N) grand unified group.

The crucial aspect of both the PPF and the SVS model is that, among the left-chiral fields, the quarks, i.e., color triplets, appear in triplet or antitriplet of \(\mathrm{{SU}}(3)_\mathrm{L}\), whereas the antiquarks, i.e., color antitriplets, are all singlets of \(\mathrm{{SU}}(3)_\mathrm{L}\). In particular, then, there is no multiplet that transforms like \(\left[ 3^*,3,\star \right] \) or \(\left[ 3^*,3^*,\star \right] \). On the other hand, among the quark fields, some should be in the \(\big [3,3,\star \big ]\) and some in the \(\left[ 3,3^*,\star \right] \) representation, thereby ensuring anomaly cancelation among different generations. These features are not available in the decomposition of any \({[\![} \, m \, {]\!]}\) representation of an SU(N) group, as we show now.

## 5 Other embeddings into SU(N)

Number of different SM multiplets occurring in completely antisymmetric representations of SU(N)

SM multiplet | Representation | ||
---|---|---|---|

\({[\![} \, 1 \, {]\!]}\) | \({[\![} \, 2 \, {]\!]}\) | \({[\![} \, 3 \, {]\!]}\) | |

\(\left\{ 1,2,-\frac{1}{2}\right\} \) | \(-\)1 | \(-(N-5)\) | \(-{N-5 \atopwithdelims ()2}\) |

\(\left\{ 3,2,\frac{1}{6}\right\} \) | 0 | 1 | \(N-6\) |

\(\left\{ 3^*,1,-\frac{2}{3}\right\} \) | 0 | 1 | \(N-6\) |

\(\left\{ 3^*,1,\frac{1}{3}\right\} \) | \(-1\) | \(-(N-5)\) | \(-{N-5 \atopwithdelims ()2}\) |

\(\left\{ 1,1,1\right\} \) | 0 | 1 | \(N-6\) |

It should be noted that it is just as easy to obtain solutions of Eq. (22) with \(n_1, n_2\), and \(n_3\) all non-zero. One such model with an SU(8) gauge group was the subject matter of Ref. [10], where the authors took \(n_1=-9, n_2=1, n_3=1\) and obtained three generations of fermions. From our analysis, it seems that they could have obtained any other number of generations by adjusting the number of copies of various representations. For example, \(n_1=-13, n_2=2, n_3=1\) would give four generations. However, the merit of the \(n_2=0\) solutions is that the number of generations cannot be arbitrary: it must be a multiple of three.

## 6 Anatomy of an SU(9) model

## 7 Gauge coupling unification and the intermediate scales

Beta coefficients for the extra vector-like fermions

\(Q\) | \(U\) | \(D\) | \(L\) | \(E\) | |
---|---|---|---|---|---|

\(b_1\) | \(2/9\) | \(16/9\) | \(4/9\) | \(2/3\) | \(4/3\) |

\(b_2\) | \(2\) | \(0\) | \(0\) | \(2/3\) | \(0\) |

\(b_3\) | \(4/3\) | \(2/3\) | \(2/3\) | \(0\) | \(0\) |

Beta coefficients for the extra vector-like fermions

\(Q\) | \(U\) | \(D\) | \(L\) | \(E\) | |
---|---|---|---|---|---|

\(B_{12}\) | \(-28/15\) | \(16/15\) | \(4/15\) | \(-4/15\) | \(4/5\) |

\(B_{23}\) | \(2/3\) | \(-2/3\) | \(-2/3\) | \(2/3\) | \(0\) |

## 8 Conclusions

We have also shown that our model may be consistent with unification requirements. In this part of the analysis, we have assumed a direct breaking of SU(9) into the SM gauge group. A more detailed analysis, including possibilities of intermediate symmetry breaking scales, will be taken up in a future work.

## Notes

### Acknowledgments

The work of D.E.C. was supported by Associação do Instituto Superior Técnico para a Investigação e Desenvolvimento (IST-ID) and also by FCT through the projects PEst-OE/FIS/UI0777/2011 (CFTP), CERN/FP/123580/2011 and PTDC/FIS-NUC/0548/2012.

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