Advances in High Energy Physics

Volume 2018, Article ID 4281939, 20 pages

https://doi.org/10.1155/2018/4281939

## Self-Dual Effective Compact and True Compacton Configurations in Generalized Abelian Higgs Models

^{1}Departamento de Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, 65080-805 São Luís, MA, Brazil^{2}Coordenação do Curso de Licenciatura em Ciências Naturais, Universidade Federal do Maranhão, Campus III, 65700-000 Bacabal, MA, Brazil^{3}IMDEA Nanociencia, Calle de Faraday, 9, Cantoblanco, 28049 Madrid, Spain^{4}IFISUR, Departamento de Física (UNS-CONICET), Avenida Alem 1253, Bahía Blanca, Buenos Aires, Argentina

Correspondence should be addressed to Rodolfo Casana; moc.liamg@anasac.oflodor

Received 14 December 2017; Revised 9 March 2018; Accepted 15 March 2018; Published 24 April 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Rodolfo Casana 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

We have studied the existence of self-dual effective compact and true compacton configurations in Abelian Higgs models with generalized dynamics. We have named an effective compact solution the one whose profile behavior is very similar to the one of a compacton structure but still preserves a tail in its asymptotic decay. In particular, we have investigated the electrically neutral configurations of the Maxwell-Higgs and Born-Infeld-Higgs models and the electrically charged ones of the Chern-Simons-Higgs and Maxwell-Chern-Simons-Higgs models. The generalization of the kinetic terms is performed by means of dielectric functions in gauge and Higgs sectors. The implementation of the BPS formalism without the need to use a specific* Ansatz* has led us to the explicit determination for the dielectric function associated with the Higgs sector to be proportional to , . Consequently, the followed procedure allows us to determine explicitly new families of self-dual potential for every model. We have also observed that, for sufficiently large values of , every model supports effective compact vortices. The true compacton solutions arising for are analytical. Therefore, these new self-dual structures enhance the space of BPS solutions of the Abelian Higgs models and they probably will imply interesting applications in physics and mathematics.

#### 1. Introduction

Topological defects produced by field theories including generalized kinematic terms have been an issue of great interest in the latest years. Usually these models include higher derivatives dynamic terms, but sometimes the generalizations are caused by the introduction of some generalized parameter or function in the kinetic terms. These modified theories named as -theories arose initially as effective cosmological models for inflationary evolution [1]. Later the -theories were permeating other issues of field theory and cosmology such as dark matter [2], strong gravitational waves [3], the tachyon matter problem [4], and ghost condensates [5]. It is worth emphasizing the possibility that these theories can arise naturally within the context of string theory. Several studies concerning the topological structure of the -theories have shown that they also engender topological solitons [6–22]; in general, they can present some new characteristics when compared those of the usual ones [23].

Compactons were defined as solitons with finite wavelength in the pioneer work [1] and so far they have been the subject of several studies, since models containing topological defects have been used to represent particles and cosmological objects such as cosmic strings [24]. A particular arrangement of particles can be represented by a group of compactons and in this case we will not have the problem of the superposition of particles (or defects) due to the fact that compactons do not carry a “tail” in its asymptotic decay. We also point out that compact vortices and skyrmions are intrinsically connected with recent advances in the miniaturization of magnetic materials at the nanometric scale for spintronic applications [25–27]. Compact topological defects have gained greater attention as effective low-energy models for QCD concerning skyrmions, where nonperturbative results at the classic level have been reached [28]. Compact solutions were also successfully employed in the description of boson stars [29] and in baby Skyrme models [30, 31].

At the classical level there is a widely employed mechanism to achieve field equations, namely, the Bogomol’nyi-Prasad-Sommerfield (BPS) formalism [32, 33]. The BPS method consists of building a set of first-order differential equations which solve as well the second-order Euler-Lagrange equations. One interesting aspect of this mechanism is that all the equations are built up for static field configurations. As a consequence, the first-order equations of motion coming out from the BPS formalism describe field configurations minimizing the total system energy. The static characteristic of the fields in the BPS limit has been applied to investigate topological defects in several frameworks. For example, in the context of planar gauge theories, vortices structures arise from the BPS equations, specially, magnetic vortices which were found in Maxwell-Higgs electrodynamics [24]. Also we can mention the Chern-Simons-Higgs electrodynamics [34, 35] and the Maxwell-Chern-Simons-Higgs model [36] both describing electrically charged magnetic vortices. Other interesting frameworks involving first-order BPS solutions are the nonlinear sigma models (NL*σ*M) [37] in the presence of a gauge field. These theories have been widely applied in the study of field theory and condensed matter physics [38, 39]. We can mention, in this sense, the topological defects in a nonlinear sigma model with the Maxwell term, as it is shown in [40–45]. Concerning the Chern-Simons term, topological and nontopological defects were analyzed in [46, 47]. Also, the gauged sigma model with both Maxwell and the Chern-Simons terms was studied in [48, 49].

The existence of vortex solutions with compact-like profiles in -generalized Abelian Maxwell-Higgs model was studied in [50, 51] and in -generalized Born-Infeld model [52]; however, only in [51] were the first-order vortices with compact-like profiles found. Therefore, the aim of this manuscript is to study the topological vortices engendered by the self-dual configurations obtained from the generalization of the following Abelian Higgs models: Maxwell-Higgs (MH), Born-Infeld-Higgs (BIH), Chern-Simons-Higgs (CSH), and Maxwell-Chern-Simons-Higgs (MCSH). Firstly, we have performed a consistent implementation of the Bogomol’nyi method for every model and obtained the respective generalized self-dual or BPS equations. The development of the BPS formalism has allowed fixing the form of the function , composing the generalized term , which only can be proportional to for (a similar result was obtained in [53]). Secondly, we use the usual vortex* Ansatz* to obtain the self-dual equations describing axially symmetric configurations. We have observed that independently of the model an effective compact behavior of the vortices arises for a sufficiently large value of the parameter and only in the limit are the true compacton structures achieved. Finally, we give our remarks and conclusions.

#### 2. The Maxwell-Higgs Case

The Maxwell-Higgs model is a classical field theory where the gauge field dynamics are controlled by the Maxwell term and the matter field is represented by the complex scalar Higgs field. The model presents vortex solutions when it is endowed with a fourth-order self-interacting potential promoting a spontaneous symmetry breaking. Although the model seems very simple it presents characteristics very similar to the phenomenological Ginzburg-Landau model for superconductivity [54] or superfluidity in . The applications of the Maxwell-Higgs model extend from the condensed matter to inflationary cosmology [55] or as an effective field theory for cosmic strings [24].

The generalized Maxwell-Higgs model [11] is described by the following Lagrangian density:The nonstandard dynamics are introduced by two nonnegative functions and depending of the Higgs field. The Greek index runs from to . The vector is the electromagnetic field, the Maxwell strength tensor is , and defines the covariant derivative of the Higgs field ;The function is a self-interacting scalar potential.

From (1), the gauge field equation readswhere is the conserved current density, that is, , and is the usual current density

Throughout the remainder of the section, we are interested in time-independent soliton solutions that ensure the finiteness of the action engendered by (1). Then, from (3), we read the static Gauss lawand the respective Ampère law

It is clear from the Gauss law that stands for the electric charge density, so that the total electric charge of the configurations is which is shown to be null () by integration of the Gauss law under suitable boundary conditions for the fields at infinity; that is, , , and a well behaved function. Therefore, the field configurations will be electrically neutral, like it happens in the usual Maxwell-Higgs model.

The fact that the configurations are electrically neutral is compatible with the gauge condition, , which satisfies identically the Gauss law (5). With the choice , the static and electrically neutral configurations are described by the Ampère law (6) and the reduced equation for the Higgs field

To implement the BPS formalism, we first establish the energy for the static field configuration in the gauge , so it readsTo proceed, we need the fundamental identitywhere . With it, the energy (9) is written as being

We observe that the term precludes the implementation of the BPS procedure, that is, expressing the integrand as a sum of squared terms plus a total derivative plus a term proportional to the magnetic field. This inconvenience already was observed in [56]; such a problem was circumvented by analyzing only axially symmetric solutions in polar coordinates.

The key question about the functional form of allowing a well-defined implementation of the BPS formalism was solved in [53]. In the following we reproduce some details on looking for the function . The starting point is the following expression:By considering to be an explicit function of , after some algebraic manipulations, the last term becomes expressed aswhich, after being substituted in (12), allows obtainingAt this point, we establish the function to satisfy the following condition:whose solutions provide the explicit functional form of to beguaranteeing that the vacuum expectation value of the Higgs field is .

With the key condition (15), (14) allows expressing the term in the following way:

By putting the expression (17) in the energy (11), it becomes We now manipulate the two first terms in such a form that the energy can be written asWith the objective being that the integrand has a term proportional to the magnetic field, we impose that the factor multiplying it must be a constant; that is, Consequently, the self-dual potential becomeswhere we have defined the potential given byWe can note that, for , it becomes the self-dual potential of the Maxwell-Higgs model.

Hence, the energy (77) reads as Now by imposing appropriate boundary conditions, the contribution to the total energy of the total derivative is null and the energy has a lower bound proportional to the magnitude of the magnetic flux,where for positive flux we choose the upper signal and for negative flux we choose the lower signal.

The lower bound is saturated by fields satisfying the first-order Bogomol’nyi or self-dual equations [32, 33]The function must be a function providing a finite magnetic field such that is sufficiently rapid to provide a finite total magnetic flux.

In the BPS limit the energy (9) provides the energy density of the self-dual configurationsit will be finite and positive-definite for . We here also require the function yielding a finite BPS energy density such that is sufficiently rapid to provide a finite total energy.

##### 2.1. Maxwell-Higgs Effective Compact Vortices for Finite

In the following, with loss of generality, we have chosen (for this one and for all other models analyzed throughout the manuscript) with the aim of studying the influence of the generalized dynamic in Higgs sector in the formation of effective compact vortices and true compactons. Such a generalization provided by the function defined in (16) and its effects in the formation of effective compact vortices apparently remains unexplored in the literature.

Thus, we seek axially symmetric solutions according to the usual vortex* Ansatz* [24]with standing for the winding number of the vortex solutions.

The profiles and are regular functions describing solutions possessing finite energy and obeying the boundary conditions,Under* Ansatz* (27), the magnetic field reads asThe correspondent quantized magnetic flux is given byas expected.

The BPS equations (25) are written asThe upper (lower) signal corresponds to the vortex (antivortex) solution with winding number ().

The self-dual energy density (26) is expressed byIt will be finite and positive-definite for .

The total energy of the self-dual solutions is given by the lower bound (24),it is proportional to the winding number of the vortex solution, as expected.

The behavior of and near the boundaries can be easily determined by solving the self-dual equations (31) around the boundary conditions (28). Then, for , the profiles behave aswhere the constant is computed numerically.

On the other hand, when , they behave as the constant is determined numerically and , the self-dual mass, is given byreminding us that is the mass scale of the usual Maxwell-Higgs model. The influence of the generalization in the mass scale explains the changes in the vortex-core size for large values of observed in Figures 1 and 2.