Advances in High Energy Physics

Volume 2019, Article ID 5431067, 11 pages

https://doi.org/10.1155/2019/5431067

## A General Method for Transforming Nonphysical Configurations in BPS States

^{1}Unidade Acadêmica de Física, Universidade de Federal de Campina Grande, 58109-970 Campina Grande, PB, Brazil^{2}São Paulo State University (UNESP), Campus de Guaratinguetá-DFQ, Avenida Dr. Ariberto Pereira da Cunha 333, 12516-410 Guaratinguetá, SP, Brazil^{3}Instituto Tecnológico de Aeronáutica, DCTA, 12228-900 São José dos Campos, SP, Brazil^{4}Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Correspondence should be addressed to R. A. C. Correa; moc.liamg@23140sif

Received 11 October 2018; Revised 12 January 2019; Accepted 18 February 2019; Published 5 March 2019

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2019 J. R. L. Santos et al. 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

In this work, we apply the so-called BPS method in order to obtain topological defects for a complex scalar field Lagrangian introduced by Trullinger and Subbaswamy. The BPS approach led us to compute new analytical solutions for this model. In our investigation, we found analytical configurations which satisfy the BPS first-order differential equations but do not obey the equations of motion of the model. Such defects were named nonphysical ones. In order to recover the physical meaning of these defects, we proposed a procedure which can transform them into BPS states of new scalar field models. The new models here founded were applied in the context of hybrid cosmological scenarios, where we derived cosmological parameters compatible with the observed Universe. Such a methodology opens a new window to connect different two scalar fields systems and can be implemented in several distinct applications such as Bloch Branes, Lorentz and Symmetry Breaking Scenarios, Q-Balls, Oscillons, Cosmological Contexts, and Condensed Matter Systems.

#### 1. Introduction

Topological defects are present in several scenarios of physics, covering areas like braneworld models, quintessence cosmological approaches, condensed matter, among others [1–8]. As examples of the applicability of defects solutions, we refer to studies involving defects in massive integrable field theories in 1+1 dimensions [9], in systems where the Lorentz symmetry is violated [10, 11], in 2D materials [12], and in Yang monopoles [13].

A well-established method to determine defect-like solutions is the so-called BPS method, proposed by Bogomol’ny, Prasad, and Sommerfeld [14, 15]. Such a method is based on the assumption that the fields obey first-order differential equations (BPS differential equations), in order to minimize their energy density. Therefore, the solutions corresponding to these first-order differential equations (here also called BPS solutions) must satisfy the equations of motion of given systems. The main advantage of the BPS method is that one needs to deal with first-order equations instead of second or higher order differential equations. Such a method has been mainly applied in the context of Lagrangian densities composed by real scalar fields, by complex scalar fields, and by gauge field theories, as one can see in [16–20]. Generalizations of the BPS approach can be found in the literature as in [21], where the authors show how the energy of the defects can saturate to a bound energy if different sets of BPS differential equations were obeyed. In this paper, we are going to explore the application of the BPS method in a complex field model proposed by Trullinger and Subbaswamy [22].

At the end of the seventies, Trullinger and Subbaswamy found topological solutions (or defects) which satisfy the equations of motion related to the following Lagrangian density with , where , , and are all positive constants. In their work, Trullinger and Subbaswamy analyzed the case , since this model has some interesting physical motivations in studies involving anisotropic ferromagnets [22], where the defects are analogous to magnetic domain walls. The mentioned domain walls were applied in the context of condensed matter physics and in statistical mechanics.

In this paper we are going to investigate the model introduced in [22] from the point of view of the BPS approach, moreover, we will show carefully what are the conditions to have BPS solutions which satisfy the equations of motion and the first-order differential equations for such a model. Despite the success of this methodology, we are going to unveil new sets of solutions for the first-order differential equations which do not satisfy the equations of motion coming from (1). These solutions also have energy different from the BPS sector of the Lagrangian above, and we named them as nonphysical ones. Our main objective in this paper is to transform such nonphysical solutions into new BPS states for other effective models, recovering their physical meaning. This approach also shows a new type of connection between scalar fields models which have not been observed in the literature, as far as we know.

In order to show the potential of our methodology, we apply the derived models in the context of hybrid cosmological scenarios. Since the seminal work of Kinney in this subject [5], several approaches to deal with cosmological models driven by more than one scalar field have been proposed [6, 23, 24]. The desire for hybrid models increased in the last few years after the work of Ellis* et al.* [25], where the authors unveiled that models composed by several scalar fields are compatible with the scalar index and with the tensor-to-scalar ratio parameters found by PLANCK collaboration [26, 27].

The ideas behind this investigation are divided into the following sections: Section 2 presents the main calculations of [22], starting by rewriting the complex scalar field Lagrangian in terms of a two real scalar fields model and then finding its second-order equations of motion. In Section 3 we determine the solutions obtained by Trullinger and Subbaswamy via the BPS approach; we also point what conditions the BPS solutions would have in order to satisfy the equations of motion. Furthermore, we determine the nonphysical solutions related to the first-order differential equations for this model. In Section 4 we show the methodology responsible to relate nonphysical solutions with new sets of two-field models. Sections 5 and 6 are dedicated to the application, and to the interpretation of the derived models in the context of cosmology. Our final remarks and perspectives are shown in Section 7.

#### 2. Generalities

In this section, we briefly review some generalities proposed by Trullinger and Subbaswamy [22] for the treatment of the Lagrangian density presented in (1). Let us start the procedures by rewriting the field as

Then, by substituting the above relation into (1), we obtain the following pair of Euler-Lagrange coupled equations:

Let us work with the following redefinitions: and , where , and . At this point, we emphasize that the new variable is known in the literature as travelling one, which is very useful in the analytical treatment of nonlinear systems. The previous procedures yield to

Now, the scalar fields and the variable can be rescaled as follows: resulting inwhere .

Looking at the previous results, it is natural to think that (8) and (9) could be derived from a two-field Lagrangian density with the form where the scalar potential can be written as

Note that, the potential (11) has four symmetric degenerated minima , which are localized in , , , and . Such minima are known as the vacua of the topological configurations, and the solutions connecting different topological configurations were named as and solutions [22]. The solution connects the topological sectors where while the solution is related to the vacua

In the literature about defects, one-dimensional solutions which connect two distinct vacua are called kinks; besides one-dimensional solutions related to only one vacuum are named lumps. In two scalar field models, these one-dimensional defects are combined to construct an orbit in the field space. Therefore, the solutions have orbits formed by the combinations of a kink with a lump defect, while the orbits of solutions are constructed by combinations of two kink-like solutions.

In [22] the authors determined two analytical solutions for the case , by directly integrating their equations of motion. However, it is possible to use the so-called BPS method to generalize such solutions [14, 15]. As we know, since this approach allows obtaining a first-order differential equation from the total energy, such insight becomes a powerful tool to solve nonlinear problems analytically.

#### 3. BPS Treatment

An advantage of the BPS method is that it simplifies considerably the integration process of equations of motion, and it also yields to new sets of analytical solutions which satisfy the BPS first-order differential equations. In this section, we will show that most part of the BPS solutions from [22] are not going to obey its equations of motion. Therefore, we will need to find models which are satisfied by these new sets of analytical solutions. In order to determine such models, we are going to use the methodology to construct scalar fields systems presented in [28].

From now on, we will be dealing with the problem of obtaining a BPS bound for the model under investigation. The first step to implement the BPS method to this context consists in rewrite (11) asMoreover, by defining the superpotential we are able to rewrite our potential as whereTherefore, if the analytical case studied by Trullinger and Subbaswamy in [22] is recovered naturally.

Furthermore, the total energy for the fields configurations is such that then repeating the BPS procedure we find So, if the first-order differential equations are obeyed, we have the following effective energy: which can be rewritten aswhere the BPS energy is simply

Thus, we can see that only if . In order to find general configurations, let us compute the possible analytical solutions of the first-order differential equations (21) and (22). One path to integrate such equations consists in rewrite them as Now, using the new variable , the above equation takes the form Solving the previous differential equation, we conclude, after straightforward manipulations, that the relation between and is where is an arbitrary integration constant. So, we directly see that the case (or ) means and by integrating (21) and (29) we determine the following solutions:

The above solutions with a general value of are new sets of configurations for the model proposed by Trullinger and Subbaswamy in [22], and they are graphically represented in Figures 1 and 2. In Figure 1 one can see two types of solutions: one called critical, where both graphics are kinks (left panel of Figure 1), and one called subcritical where is a kink while is a lump (right panel of Figure 1). Besides in Figure 2 we depicted the transition between the subcritical and the critical cases, where we have a double-kink defect for and a plateau-like lump for . Moreover, the solutions are divergent if . Defects like these were found for other two scalar fields models as one can see in [16, 17].