Advances in High Energy Physics

Volume 2016, Article ID 3650632, 10 pages

http://dx.doi.org/10.1155/2016/3650632

## Matter Localization on Brane-Worlds Generated by Deformed Defects

^{1}Departamento de Física, Universidade Federal de São Carlos, P.O. Box 676, 13565-905 São Carlos, SP, Brazil^{2}Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC), 09210-580 Santo André, SP, Brazil

Received 14 March 2016; Revised 21 April 2016; Accepted 16 June 2016

Academic Editor: Barun Majumder

Copyright © 2016 Alex E. Bernardini and Roldão da Rocha. 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

Localization and mass spectrum of bosonic and fermionic matter fields of some novel families of asymmetric thick brane configurations generated by deformed defects are investigated. The localization profiles of spin 0, spin 1/2, and spin 1 bulk fields are identified for novel matter field potentials supported by thick branes with internal structures. The condition for localization is constrained by the brane thickness of each model such that thickest branes strongly induce matter localization. The bulk mass terms for both fermion and boson fields are included in the global action as to produce some imprints on mass-independent potentials of the Kaluza-Klein modes associated with the corresponding Schrödinger equations. In particular, for spin 1/2 fermions, a complete analytical profile of localization is obtained for the four classes of superpotentials here discussed. Regarding the localization of fermion fields, our overall conclusion indicates that thick branes produce a* left-right asymmetric chiral* localization of spin 1/2 particles.

#### 1. Introduction

The brane-world model is a prominent paradigm that has been addressed to solve several questions in physics. Within this framework, brane-worlds are required to render a consistent physics of our Universe, at least up to certain sensible limits [1]. In the brane-world scenario all kinds of matter fields should be localized on the brane. In the RS brane-world model [2], the brane is generated by a scalar field coupled to gravity [3, 4], in a particular scenario which may be interpreted as the thin brane limit of thick brane scenarios. Generically, a prominent test that thick brane-world models must pass, to be physically consistent, regards their stability, with respect to tensor, vector, and scalar fluctuations of the background fields that generate the field configurations, namely, the thick brane itself. At least the zero modes of Standard Model matter fields were shown to be localized on several brane-world models [5–10], suggesting that such kind of models is physically viable in high energy physics. Several alternative scenarios, including Gauss-Bonnet terms, gravity, tachyonic potentials, cyclic defects, and Bloch branes, have been further studied [11–15], and analogous scenarios in an expanding Universe have been approached [16, 17]. The curvature nature of the brane-world, namely, to be a de Sitter, Minkowski, or anti-de Sitter one, is in general obtained* a posteriori*, by solving the Einstein field equations. In fact, the bulk and the brane cosmological constants depend upon the brane and the bulk gravitational field content, governed by curvature, and must obey the intrinsic fine-tuning, in the Randall-Sundrum-like models limit.

The analytical study of stability can be uncontrollably intricate, due to the involved structure of the scalar field coupled to gravity. To circumvent the complicated and not analytical approaches, linearized formulations have been commonly worked out. In this context, supported by the stability of deformed defect generated brane-world models, scalar, vector, and tensor perturbations are investigated throughout this work.

Localization aspects of various matter fields with spin 0, spin 1/2, and spin 1 on analytical thick brane-world models are indeed a main concern in deriving brane-world models, since they must describe our physical world. The localization of the spin 1/2 fermions deserves a special attention, since there is no scalar field to couple with in this model, in contrast to thick branes generated by deforming defect mechanisms [18]. Otherwise, Kalb-Ramond fields, although already investigated [19], will not be the main aim here. The spin 1/2 issue has been previously studied in some other contexts [20], including further coupling of more scalar fields in the action [21] and asymmetric brane-worlds generated by a plenty of scalar field potentials [8, 22–25]. In particular, asymmetric Bloch branes in the context of the hierarchy problem have been addressed in [13].

Our aim is to investigate the localization of bulk matter and gauge fields on the brane, in the context where the mass-independent potentials of the corresponding Schrödinger-like equations, regarding the quantum mechanical analogue problem, can be suitably acquired from a warped metric. In particular, for a bulk mass proportional to the fermion mass term enclosed by the global action, the possibility of trapping spin 1/2 fermions on asymmetric branes is discussed and quantified.

To accomplish this aim, this paper is organized as follows. In Section 2, a brief review of brane-world scenarios supported by an effective action driven by a (dark sector) scalar field is presented. Warp factors and the corresponding internal brane structure are described for four different analytical models. In Section 3, the* left-right*-chiral asymmetric aspects of matter localization for spin 1/2 fermion fields on thick branes are investigated. Extensions to scalar boson and vector boson fields are obtained in Sections 4 and 5, respectively. Final conclusions are drawn in Section 6.

#### 2. Brane-World Preliminaries and Some Analytical Models

Let one start considering a space-time warped into . The most general metric compatible with a brane-world spatially flat cosmological background has the form given by where denotes the warp factor, and the signature is employed, with . stands for the components of the metric tensor (). One can identify as the infinite extra dimension coordinate (which runs from to ) and notice that the normal to surfaces of constant is orthogonal to the brane, into the bulk (brane tension terms have been suppressed/absorbed by the metric (c.f. Equations (24) and (25) from [10] for real scalar field Lagrangians in the context of thick brane solutions)).

The brane-world scenario examined here is set up by an effective action, driven by a (dark sector) scalar field, , coupled to gravity, given bywhere is the scalar curvature, is the Ricci tensor, and denotes the gravitational coupling constant, hereon set to be equal to unity, where is the Newton constant. The Einstein equations read where denotes the energy-momentum tensor corresponding to the matter Lagrangian, regarding the matter field . After solving the Einstein field equations, the bulk cosmological constant turns out, in general, to be positive or negative, thus realising a de Sitter or anti-de Sitter brane-world, respectively, generated by curvature. It realises and emulates the interplay involving the and cosmological constants. Some further possibilities are devised, for example, in [14, 26]; however it is worth mentioning that an additional scalar field can be still added in the action, whose isotropisation will precisely define the nature of the brane-world. This latter case is however beyond the scope of our analysis. Obviously, whatever the possibility to be considered, the thin brane limit must obey the fine-tuning relation [27] , among the effective and cosmological constants and the brane tension as well.

Considering the real scalar field action, (2), one can compute the stress-energy tensorwhich, supposing that both the scalar field and the warp factor dynamics depend only upon the extra coordinate, , leads to an explicit dependence of the energy density in terms of the field, , and of its first derivative, , as

With the same constraints on about the dependence on , the equations of motion currently known from [3, 4], which arise from the above action, arethrough a variational principle relative to the scalar field, , andthrough a variational principle relative to the metric, or equivalently to , manipulated to result intoafter an integration over .

For the scalar field potential written in terms of a* superpotential*, , asthe above equations are mapped into first-order equations [3, 4] asfor which the solutions can be found straightforwardly through immediate integrations [3] (see also [10] and references therein). The energy density follows from (9) as

The analysis of localization aspects of brane-world scenarios will be constrained by some known examples, I, II, III, and IV, for which the warp factor, , and the energy density, , can be analytically computed. Model I is supported by a sine-Gordon-like superpotential given bywhich reproduces the results from [4]. Model II corresponds to a deformed theory with the superpotential given byModels III and IV are deformed topological solutions from [28] supported by superpotentials likewhere the parameter fixes the thickness of the brane described by the warp factor, . Besides exhibiting analytically manipulable profiles, the above superpotentials have already been discussed in the context of thick brane localization [4, 5, 10]. Models I and II are, respectively, motivated by sine-Gordon and theories, and models III and IV are obtained (also analytically) from deformed versions of the model [12]. In particular, models III and IV can also be mapped onto tachyonic Lagrangian versions of scalar field brane models [6, 10, 28].

From the above superpotentials, the respective solutions for are set aswhere one has suppressed any additional (irrelevant) constant of integration for convenience, and one has just considered the positive solutions (in (16) there could be explicit constant of integration that amounts to letting , corresponding to the position of the brane in the extra dimension, for which one has set ).

The obtained expressions for the warp factor as resulting from (11) are, respectively, given bywhere integration constants are introduced as to set a normalization criterion for which .

The solutions for and are depicted in Figure 1. The corresponding localized energy densities computed through (12) are, respectively, given by