###### Abstract

The gauge-Higgs grand unification in the Randall-Sundrum warped space is presented. The 4D Higgs field is identified as the zero mode of the fifth dimensional component of the gauge potentials, or as the fluctuation mode of the Aharonov-Bohm phase along the fifth dimension. Fermions are introduced in the bulk in the spinor and vector representations of . is broken to by the orbifold boundary conditions, which is broken to by a brane scalar. Evaluating the effective potential , we show that the electroweak symmetry is dynamically broken to . The quark-lepton masses are generated by the Hosotani mechanism and brane interactions, with which the observed mass spectrum is reproduced. The proton decay is forbidden thanks to the new fermion number conservation. It is pointed out that there appear light exotic fermions. The Higgs boson mass is determined with the quark-lepton masses given, which, however, turns out smaller than the observed value.

23 June 2016 OU-HET 893

MISC-2016-06

Toward Realistic Gauge-Higgs Grand Unification

Atsushi Furui, Yutaka Hosotani and Naoki Yamatsu

Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto 603-8555, Japan

## 1 Introduction

Up to now almost all observational data at low energies are consistent with the standard model (SM) of electroweak interactions. Yet it is not clear whether the Higgs boson discovered in 2012 at LHC is precisely what is introduced in the SM. Detailed study of interactions of the Higgs boson is necessary to pin down its nature. From the theory viewpoint the Higgs boson sector of the SM lacks a principle which governs and regulates the Higgs interactions with itself and other fields, quite in contrast to the gauge sector in which the gauge principle of completely fixes gauge interactions among gauge fields, quarks and leptons. Further the mass of the Higgs scalar boson generally acquires large quantum corrections from much higher energy scales, which have to be canceled by fine-tuning bare masses in a theory. It is called the gauge-hierarchy problem.

Many proposals have been made to overcome these problems. Supersymmetric generalization of the SM is among them. There is an alternative scenario of the gauge-Higgs unification in which the 4D Higgs boson is identified with a part of the extra-dimensional component of gauge fields defined in higher dimensional spacetime.[3]-[6] The Higgs boson, which is massless at the tree level, acquires a finite mass at the quantum level, independent of a cutoff scale and regularization scheme. The gauge-Higgs electroweak (EW) unification in the five-dimensional Randall-Sundrum (RS) warped space has been formulated.[7]-[13] It gives almost the same phenomenology at low energies as the SM, provided that the Aharonov-Bohm (AB) phase in the fifth dimension is . In particular, cubic couplings of the Higgs boson with other fields, , , quarks and leptons, are approximately given by the SM couplings multiplied by .[9][14]-[17] The corrections to the decay rates , which take place through one-loop diagrams, turn out finite and small.[13, 18] Although infinitely many Kaluza-Klein (KK) excited states of and top quark contribute, there appears miraculous cancellation among their contributions. In the gauge-Higgs unification the production rate of the Higgs boson at LHC is approximately that in the SM times , and the branching fractions of various Higgs decay modes are nearly the same as in the SM. The cubic and quartic self-interactions of the Higgs boson show deviations from those in the SM, which should be checked in future LHC and ILC experiments. Further the gauge-Higgs unification predicts the bosons, namely the first KK modes of , and , around 6 to 8 TeV range with broad widths for to , which awaits confirmation at 14 TeV LHC in the near future.[19, 20]

With a viable model of gauge-Higgs EW unification at hand, it is natural and necessary to extend it to gauge-Higgs grand unification to incorporate strong interaction. Since the idea of grand unification was proposed [21], a lot of grand unified theories based on grand unified gauge groups in four and higher dimensional spaces have been discussed. (See, e.g., Refs. [22]-[53] for recent works and Refs. [54, 55] for review.) The mere fact of the charge quantization in the quark-lepton spectrum strongly indicates the grand unification. Such attempt to construct gauge-Higgs grand unification has been made recently. gauge-Higgs grand unification in the RS space with fermions in the spinor and vector representations of has been proposed.[56, 57, 58] The model carries over good features of the EW unification. In this paper we present detailed analysis of the gauge-Higgs grand unification. Particularly we present how to obtain the observed quark-lepton mass spectrum in the combination of the Hosotani mechanism and brane interactions on the Planck brane. It will be shown that the proton decay can be forbidden by the new fermion number conservation.

There have been many proposals of gauge-Higgs grand unification in the literature, but they are not completely satisfactory in the points of the realistic spectrum and the symmetry breaking structure.[59]-[64] In the current model symmetry is broken to by orbifold boundary conditions, which breaks down to by a brane scalar on the Planck brane. Finally is dynamically broken to by the Hosotani mechanism. The quark-lepton mass spectrum is reproduced. However, unwanted exotic fermions appear. Further elaboration of the scenario is necessary to achieve a completely realistic grand unification model. We note that there have been many advances in the gauge-Higgs unification both in electroweak theory and grand unification.[65]-[71] Mechanism for dynamically selecting orbifold boundary conditions has been explored.[72] The gauge symmetry breaking by the Hosotani mechanism has been examined not only in the continuum theory, but also on the lattice by nonperturbative simulations.[73, 74, 75]

The paper is organized as follows. In Section 2 the model is introduced. The symmetry breaking structure and fermion content are explained in detail. The proton stability is also shown. In Section 3 the mass spectrum of gauge fields is determined. In Section 4 the mass spectrum of fermion fields are determined. With these results the effective potential is evaluated in Section 5. Conclusion and discussions are given in Section 6.

## 2 Model

The gauge theory is defined in the Randall-Sundrum (RS) space whose metric is given by

(2.1) |

where , , , , , and for . The topological structure of the RS space is . In terms of the conformal coordinate () in the region

(2.2) |

The bulk region () is anti-de Sitter (AdS) spacetime with a cosmological constant , which is sandwiched by the Planck brane at () and the TeV brane at (). The KK mass scale is for .

### 2.1 Action and boundary conditions

The model consists of gauge fields , fermion multiplets in the spinor representation and in the vector representation , and a brane scalar field .[56] In each generation of quarks and leptons, one and two ’s are introduced. , in the spinor representation of , is defined on the Planck brane.

The bulk part of the action is given by

(2.3) | |||||

(2.4) | |||||

(2.5) | |||||

where , . stands for the generation index. , , represent bulk mass parameters. We employ the background field method, separating into the classical part and the quantum part ; . The gauge-fixing function and the associated ghost term are given, in the conformal coordinate, by

(2.6) | |||||

(2.7) | |||||

where , etc. We adopt the convention , , , and .

Generators of are summarized in Appendix A in the vectorial and spinorial representations. We adopt the normalization and , With this normalization the weak gauge coupling constant is given by . The orbifold boundary conditions for the gauge fields are given, in the -coordinate, by

(2.8) | |||||

(2.9) | |||||

where . The ghost fields, and , satisfy the same boundary conditions as . is given in the vectorial representation by

(2.10) |

and in the spinorial representation by

(2.11) | |||||

(2.12) | |||||

The symmetry is broken down to by at , and to by at . With these two combined, there remains symmetry, which is further broken to by on the Planck brane as described below. is dynamically broken to by the Hosotani mechanism.

Fermion fields obey the following boundary conditions;

(2.13) | |||||

(2.14) | |||||

(2.15) | |||||

Eigenstates of with correspond to right- and left-handed components in four dimensions. For and one might impose alternative boundary conditions given by

(2.16) | |||||

(2.17) | |||||

It turns out that the model with (2.15) is easier to analyze in reproducing the mass spectrum of quarks and leptons.

On the Planck brane (at ) the brane scalar field has an invariant action given by

(2.18) | |||||

(2.19) | |||||

develops VEV. Without loss of generality one can take

(2.20) |

On the Planck brane symmetry is spontaneously broken to by . With the orbifold boundary condition , symmetry is broken to the SM symmetry .

To see it more explicitly, we note that mass terms for gauge fields are generated from (2.19) in the form . Making use of (A.15) and (A.17) for , one obtains

(2.21) | |||||

(2.22) | |||||

(2.23) | |||||

(2.24) | |||||

(2.25) | |||||

(2.26) | |||||

In all, 21 components in acquire large brane masses by , which effectively alters the Neumann boundary condition at to the Dirichlet boundary condition for their low-lying modes () as will be seen in Section 3. It follows that the generators are given, up to normalization, by

(2.27) | |||||

(2.28) | |||||

(2.29) | |||||

(2.30) | |||||

(2.31) | |||||

(2.32) | |||||

(2.33) | |||||

(2.34) | |||||

(2.35) | |||||

(2.36) | |||||

12 components of the gauge fields in the class (iv) have no zero modes by the boundary conditions. This leaves symmetry.

It will be seen later that symmetry is dynamically broken to by the Hosotani mechanism. The AB phase associated with becomes nontrivial so that picks up an additional mass term. Consequently the surviving massless gauge boson, the photon, is given by

(2.37) | |||||

(2.38) | |||||

where are generators of . The orthogonal component and mix with each other for . More rigorous and detailed reasoning is given in Section 3 in the twisted gauge. The gauge boson, , is given by

(2.39) |

The gauge couplings become

(2.40) | |||||

(2.41) | |||||

(2.42) | |||||

In other words 4D gauge couplings and the Weinberg angle are given, at the grand unification scale, by

(2.43) |

The content is determined with the EM charge given by (2.38). In the spinorial representation

(2.44) | |||||

(2.45) | |||||

We tabulate the content of in Table 1. We note that zero modes appear only for particles with the same quantum numbers as quarks and leptons in the SM, but not for anything else.

name | ||||||||
---|---|---|---|---|---|---|---|---|

decomposes into an vector and a singlet. The former further decomposes into an vector () and an vector (). An vector () transforms as of . Under and ,

(2.46) | |||||

(2.47) | |||||

(2.48) | |||||

An vector () decomposes into two parts;

(2.49) |

With this notation the contents of and in (2.15) are summarized in Table 2. Notice that and have no components carrying the quantum numbers of quark. With the boundary conditions (2.15), only and have zero modes. If the boundary conditions (2.17) were adopted, then and would have zero modes.

name | |||||||
---|---|---|---|---|---|---|---|