Yuhei Goto, E-mail:

Yoshiharu Kawamura, E-mail:

^{3}

We study the origin of electroweak symmetry under the assumption that

The discovery of the Higgs boson [

In this paper we tackle these problems through extensions of gauge symmetries and space-time. Concepts such as simplicity and variety are also adopted on a case-by-case basis. The SM gauge symmetry can be extended to contain a left–right symmetry. A typical one is the gauge group

We give an outline of our model. Particle physics above some high-energy scale ^{1}^{2}

This paper is organized as follows. In the next section, we formulate a 5D Pati–Salam model. We examine the Coleman–Weinberg mechanism and the vacuum stability in a four-dimensional (4D) model with

The space-time is assumed to be factorized into a product of 4D Minkowski space-time

In the following, we formulate a Pati–Salam model on

Gauge quantum numbers of bosons in the 5D Pati–Salam model.

Bosons | |||
---|---|---|---|

The gauge bosons possess several components such that

We suppose that all scalar fields have no bulk masses.

From the requirement that the Lagrangian density should be invariant under the translation

We impose the following BCs on

Only

We impose the following BCs on

We impose the following BCs on

For scalar fields, the following BCs are imposed:

Then, zero modes appear from the lower component of

We list gauge quantum numbers and mass spectra of bosons after compactification in

Gauge quantum numbers of bosons after compactification in the 5D Pati–Salam model.

Bosons | Mass | |||||
---|---|---|---|---|---|---|

0 | ||||||

0 | ||||||

After the dimensional reduction, we obtain the Lagrangian density:

Note that fields from zero modes are massless at

Let us study the 4D model with the gauge group

Gauge quantum numbers of massless fields in the 4D 3211 model.

Particles | ||||||
---|---|---|---|---|---|---|

We study the running of gauge couplings. For the sake of completeness, we consider the case that the 4D 3211 model holds beyond

The values of

Gauge couplings and their coefficients of

— | |||||

— |

Hereafter, we consider the case that the gauge symmetries are partially unified under the Pati–Salam-type gauge group

Let us estimate the running of gauge couplings beyond ^{3}

Typical runnings of ^{4}^{5}

The running of gauge couplings. The aqua, green, red, purple, and blue lines stand for the evolution of

We study the breakdown of

For the sake of completeness, we write down the RGEs of the Higgs quartic coupling

We obtain an effective potential improved by the RGEs at the one-loop level:

The effective potential

The first derivatives of

From the stationary conditions

After the breakdown of

The

Using the stationary conditions, we obtain the following formulae for mass matrix elements:

Here we choose

After diagonalizing the mass matrix, the mass of the

The third term on the right-hand side of Eqs. (

From a numerical analysis, we obtain the negative squared mass because

The runnings of

The running of

We have studied the origin of electroweak symmetry under the assumption that

Our 3211 model has the excellent feature that

Our 3211 model has almost the same particle contents as those in the minimal

^{1} The orbifold breaking mechanism was originally proposed in superstring theory [

^{2} The Coleman–Weinberg mechanism was originally proposed by S. Coleman and E. Weinberg [

^{3} A mass difference with

^{4} The mass bound of an additional neutral gauge boson of

^{5} Note that the running of gauge couplings and the unification scale change drastically due to the contributions from Kaluza–Klein modes, including incomplete multiplets [

The authors acknowledge Yasunari Nishikawa for collaborations in the early stages of this work. The authors thank Prof. S. Iso for valuable discussions. This work was supported in part by scientific grants from Iwanami Fu-Jukai and the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” under Grant No. S1511006 (Y. G.) and from the Ministry of Education, Culture, Sports, Science and Technology under Grant No. 17K05413 (Y. K.).

Open Access funding: SCOAP