Advances in High Energy Physics

Volume 2017 (2017), Article ID 3926286, 10 pages

https://doi.org/10.1155/2017/3926286

## Localization of Massive and Massless Fermions on Two-Field Branes

^{1}Young Researchers and Elite Club, Islamic Azad University, Babol Branch, Babol, Iran^{2}Department of Physics, Faculty of Basic Science, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran

Correspondence should be addressed to Ali Tofighi

Received 19 September 2016; Revised 21 December 2016; Accepted 4 January 2017; Published 26 March 2017

Academic Editor: Sally Seidel

Copyright © 2017 Ali Farokhtabar and Ali Tofighi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study fermion localization and resonances on a special type of brane-world model supporting brane splitting. In such models one can construct multiwall branes which cause considerable simplification in the field equations. We use a polynomial superpotential to construct this brane. The suitable Yukawa coupling between the background scalar field and the localized fermion is determined. The massive fermion resonance spectrum is obtained. The number of resonances is increased for higher values of Yukawa coupling.

#### 1. Introduction

Brane-world scenarios have attracted considerable attention in the literature during the last two decades because these models can address some important issues in theoretical physics problems such as hierarchy [1–3] and cosmological constant problem [4, 5]. The branes in Randall and Sundrum (RS) models are fixed in some points along extra dimension and have a function form [1, 6]. This brane-world model is very ideal and its formation has no dynamical mechanism. But for realistic models thickness of brane should be considered. By now several thick brane construction mechanisms have been developed such as thick branes generated from pure gravity [7–11], fermion self-interaction branes [12, 13], and thick brane scenarios with the gravity coupled to five-dimensional scalar fields [14–21]. In the last scenario, the scalar field configuration is usually a kink. It is found that a single kink becomes unstable when it moves in a discrete lattice with a large velocity while multikink solutions remain stable [22]. This phenomenon is associated with interaction between kink and radiation, and the resonances were observed experimentally [23, 24]. Furthermore, in cosmology we encounter models in which our universe is the result of continuous collision of branes and nucleation and therefore splitting of branes is a fundamental scenario in these models [25–28].

Therefore, the universal aspects of thick brane splitting in warped bulk are important. Such branes are constructed from a complex scalar field potential [29] or from a real scalar field potential [30]. These branes can be constructed from deformation of scalar field potential as well [31–33].

Recently, Dutra and coworkers proposed a new model of thick brane in which multibrane scenario arises from scalar field models generating usual kink solutions [34]. It suggests a special type of brane splitting. In this method superpotential function and warp factor will decompose in a special form and field equations will be simplified significantly. In this work we deal with this thick brane model which arises from polynomial superpotential.

The localization of spin fermions on the brane is very interesting and important. Usually, in five dimensions fermion does not have a normalizable zero mode without scalar-fermion coupling [35–43]. In five dimensions, with a Yukawa scalar-fermion coupling, there may exist a massless bound state and a continuous gapless spectrum of massive Kaluza-Klein (KK) states [39], while, in some of other brane models, there exist some discrete KK states and a continuous gapless mass spectrum [44].

This paper is organized as follows. In the next section, we present the brane model that is constructed in [34]. In Section 3 we investigate localization of the zero mode of the fermion field on the brane which is derived from a polynomial potential. In Section 4 we study localization of massive fermionic modes. Finally, in the last section we present our conclusions.

#### 2. The Model

We consider the following action in which two scalar fields are coupled to gravity in 5 dimensions:where , . The coordinates on the brane are represented by () while the coordinate in the bulk is shown by . The line element is written aswhere is the usual Minkowski metric with and is called the warp factor. For this brane-world scenario, the equations of motion are obtained asand the potential can be written in terms of a superpotential asTherefore, equations of motion can be reduced to the following first-order equations:ButThe first-order equations in (5) are converted tofor polynomial superpotential and are given bywhere is an integration constant, representing the center of the kink. For solitonic solutions we consider and is defined as the average value of coordinates of center of the kinksIn order for this model to support brane splitting mechanism, we consider two symmetric kink solutions. Therefore, and . So and we can write (9) as

In Figure 1 we show the warp factor for different values of . For we have a single brane that warp factor has a sharp peak. For a plateau is formed in the interior of the brane where the energy density vanishes. This is attributed to the presence of a new phase inside the brane. The plateau region in the brane grows when increases.